TRANSIENT DYNAMICS OF NONLINEAR OSCILLATORS WITH APPLICATIONS
TO CENTRIFUGAL PENDULUM VIBRATION ABSORBERS
By
Ryan James Monroe

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mechanical Engineering
2011

ABSTRACT
TRANSIENT DYNAMICS OF NONLINEAR OSCILLATORS WITH
APPLICATIONS TO CENTRIFUGAL PENDULUM VIBRATION
ABSORBERS
By
Ryan James Monroe

We consider the transient behavior of centrifugal pendulum vibration absorbers (CPVAs),
speciﬁcally, the overshoot problem encountered when these absorbers are suddenly activated.
CPVAs are passive devices used to address torsional vibrations in rotating systems, for
example, helicopter rotors and crankshafts of internal combustion engines. They consist
of pendulums mounted on a rotor, driven by system rotation, and tuned in such a manner
that in steady-state operation they counteract engine-order ﬂuctuating torques acting on the
rotor, thereby smoothing vibrations. The primary feature of these devices is that they are
order tuned, that is, tuned to a given multiple of the rotation rate, as opposed to the more
common frequency tuned absorbers. Recently these absorbers have been proposed to expand
the operating envelope for cylinder deactivation in variable displacement engines, in order to
improve fuel economy. In these applications, the system encounters conditions in which the
absorbers are suddenly activated and undergo a beating-type transient motion, resulting in
overshoot of the absorber amplitude before it reaches steady-state. This overshoot depends
on a number of parameters, including the diﬀerence between the absorber’s natural frequency
and the frequency of excitation, the ratio of absorber inertia to rotor inertia, the system
damping, and system nonlinearities. An approximate analytical model is developed, using
perturbation methods, that predicts the overshoot in terms of these parameters, and the
model results are veriﬁed by simulations of the equations of motion and by experiments

using a fully instrumented spin rig. The predictive results are found to provide a useful
bound on the overshoot, and will be of use when designing absorber systems so that they
do not exceed rattle space constraints during startup. It is found that absorbers with near
tautochronic paths behave much like linear absorbers, and when lightly damped and start
from small initial conditions, they have an overshoot close to 100%. For absorbers with
softening paths, such as the commonly used circular path absorbers, the overshoot can reach
up to 173%, depending on system and input parameters.

To my wife, Anna, and my family

iv

ACKNOWLEDGMENT

I would like to thank my wife, Anna, for her support during my time as a graduate
student; her encouragement during the last ﬁve years has been instrumental in my success.
In addition, I would like to thank my family for their constant support and encouragement.
I would like to acknowledge my advisor, Steve Shaw, whom I simply cannot thank enough.
I am truly grateful to have been given the opportunity to work under his guidance on such
a fascinating topic. It has been a pleasure to work for him at Michigan State University,
where I have learned and grown so much over the last three years as a doctoral student.
Many thanks to Alan Haddow, Pete Schmitz, and Nate Verhanovitz for their assistance
with the experimental rig, and Bruce Geist, Brendan Vidmar, and Brian Feeny for the many
fruitful discussions. Thanks go to Ranjan Mukherjee and Hassan Khalil for being a part of my
committee. I am grateful for Neil Wright and Tom Pence for their mentorship during my ﬁrst
two years as a masters student. I would also like to acknowledge my fellow lab mates (both
past and present) of the vibrations lab, including Nick Miller, Venkat Ramakrishnan, Thomas
Theisen, Scott Strachan, Carl Coppola, and Rickey Caldwell. Thanks also to Cliﬀord E. Weil
for his L EX ﬁles that met the Graduate School formatting requirements. Many thanks to
AT
the wonderful Mechanical Engineering staﬀ, which includes: Suzanne Kroll, Aida Montalvo,
Mary Pease, and Lindsay Niesen. The work associated with this dissertation was supported
by grants from the National Science Foundation (CMMI-0700307) and Chrysler Group, LLC.
Additional support came from the Graduate School Dissertation Completion Fellowship.

v

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

viii

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1 Introduction
1.1 Motivation . . . . . . . . . .
1.2 Basic Operation of a CPVA
1.3 Background . . . . . . . . .
1.4 Summary of Results . . . .

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2 Accounting for Roller Inertia in the Biﬁlar CPVA
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.2 Dynamic Model . . . . . . . . . . . . . . . . . . . .
2.2.1 Formulation of the Absorber Path . . . . . .
2.3 The Tautochronic Path with Rollers . . . . . . . . .
2.4 Perturbation Analysis . . . . . . . . . . . . . . . .
2.4.1 Application of the Method of Averaging . .
2.4.2 Steady-State Response . . . . . . . . . . . .
2.4.3 Example . . . . . . . . . . . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . .

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CPVA System,
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System
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3 Transient Response of Forced Nonlinear Oscillators
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Model, Averaged Equations, and Scaling . . . . .
3.2.1 The Steady-State Response . . . . . . . . . .
3.2.2 The Averaged Equation Phase Portraits . . .
3.3 Percent Overshoot for the Undamped System . . . .
3.3.1 Transient Response Amplitudes . . . . . . . .
3.3.2 Basin Separation Trajectory . . . . . . . . . .
3.3.3 Numerical Examples . . . . . . . . . . . . . .
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . .
4 Transient Dynamics of a Single
4.1 Introduction . . . . . . . . . .
4.2 System Modeling . . . . . . .
4.2.1 Equations of Motion .
4.2.2 The Absorber Path . .

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4.3

4.4
4.5

4.2.3 Scaling and Asymptotic Analysis . . . .
Analytical Predictions of Transient Amplitudes
4.3.1 The Pseudo Energy Method . . . . . . .
4.3.2 Ramping the Excitation Torque . . . . .
Results and Discussion . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . .

5 Transient Dynamics of a Single CPVA System,
5.1 Introduction . . . . . . . . . . . . . . . . . . . .
5.2 The Experimental Apparatus . . . . . . . . . .
5.2.1 System Components and Parameters . .
5.2.2 Transient Testing Capabilities . . . . . .
5.3 Parameter Identiﬁcation . . . . . . . . . . . . .
5.4 Results & Discussion . . . . . . . . . . . . . . .
5.5 Conclusions . . . . . . . . . . . . . . . . . . . .

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6 Conclusions and Directions for Future Work

127

A Order 2 terms

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B Pendulum Path Details

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C Averaging Integrals

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D Steady-State Amplitudes

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E Transient Response Symmetry

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F Undamped Transient Response Amplitudes

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G Amplitude Correction for Small Damping

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H The Damped Transient Amplitude Response Polynomial

152

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

vii

LIST OF TABLES

2.1

Relationship between the non-dimensional roller and physical system parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

4.1

Parameter scalings for general and near-tautochronic paths. . . . . . . . . .

81

5.1

The main system parameters for the spin rig needed for comparison with
analytical predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2

System parameters for the spin rig and circular path absorber; ﬁnal values
chosen that give the best match between experiments and theory. . . . . . . 117

5.3

System parameters for the spin rig and tautochronic path absorber; ﬁnal
values chosen that give the best match between experiments and theory. . . 117

viii

LIST OF FIGURES

1.1

2.1

2.2

2.3

2.4

(a) Depiction of the basic CPVA model, consisting of a pendulum (absorber)
attached to a rotor, which is driven by a mean torque and and ﬂuctuating
torque of order n. (b) The basic elements of a biﬁlar absorber shown in a
deﬂected (oﬀ-center) position (the device shown was designed and fabricated
by Chrysler for experimentation in the MSU lab); elements A are the rollers
used for the biﬁlar suspension, B is one of the machined surfaces that dictates
the path followed by the absorber, C is a ﬂange that is attached to the rotor,
and D is the absorber mass; note that the machined surfaces on the ﬂange
and absorber are identical but inverted. (c) Helicopter rotor with 4 biﬁlar
absorbers attached (photo by S. Shaw, aboard the USS Nimitz, 1991). For
interpretation of the references to color in this and all other ﬁgures, the reader
is referred to the electronic version of this dissertation. . . . . . . . . . . . .

3

Picture taken by Steve Shaw of a helicopter rotor with four biﬁlar absorbers
attached (taken aboard the USS Nimitz). . . . . . . . . . . . . . . . . . . . .

20

Depiction of a biﬁlar roller-suspended pendulum (left) attached to a rotor,
and the paths of the pendulum COM and rollers’ COM (right). . . . . . . .

24

Inﬂuence of roller inertia on the absorber COM path, for = 0.10 and =
−0.25. (a) Five sample paths: a circle, a cycloid, and three epicycloids, for
n = 1.5. (b) Vertex curvature vs. n for three values of roller mass. . . . . . .
˜
˜

34

A table depicting the nine roller mass cases used in the simulations shown in
Figs. 2.5 and 2.6. The zero roller mass (δ = 0) case (Case x.1) is the same for
all three Cases I, II, and III. Whereas, the δ = 1/4 (Case x.2) and δ = 1/2
(Case x.3) roller mass Cases will have diﬀerent linear and nonlinear tuning
values as shown. Therefore, seven of the nine total Cases have unique pairs
of linear and nonlinear tuning values which is why only seven response curves
are shown in Fig. 2.5 with linetypes indicated in this table. . . . . . . . . . .

43

ix

2.5

2.6

2.7

2.8

3.1

3.2

3.3

Steady-state absorber response amplitude z vs. applied torque amplitude
¯
Γ for the three Cases described in the text. The linetypes for the Case I
responses are indicated in the ﬁrst row of Fig. 2.4. The linetypes for the Case
II responses are indicated in the second row of Fig. 2.4. The linetypes for the
Case III responses are indicated in the third row of Fig. 2.4. Only one response
line exists for Cases (x.1) because for δ = 0 the linear and nonlinear tuning
are equal for all three Cases. The ﬁnal three lines of the legend correspond
to results from simulations of equations (2.3) and (2.4). Only simulations of
Case I is depicted since all are very close to one another. Parameter values:
= 0.10, µ = 0.30, = −0.25, n = 1.5. . . . . . . . . . . . . . . . . . . . . .

45

Order n harmonic amplitude of the rotor response, |w |n , vs. applied torque
amplitude Γ for Case I. All absorber designs shown in the legend of Fig. 2.5
result in indistinguishable rotor response amplitudes, but diﬀerent torque
ranges. Only a simulation of the full nonlinear equations using δ = 0 (Cases
x.1) is depicted since the other roller mass values are indistinguishable. Parameter values: = 0.10, µ = 0.30, = −0.25, n = 1.5. . . . . . . . . . . . .

46

Steady-state absorber response normalized by the applied torque amplitude
Γ vs. the rotor angle θ for δ = 1/2 from Case III (i.e. Case III.3). The results
demonstrate the hardening nature of the system. Parameter values: = 0.10,
µ = 0.30, = −0.25, n = 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Steady-state rotor acceleration w vs. rotor angle θ for δ = 1/2 from Case
III (i.e. Case III.3). Note that the rotor acceleration is composed of multiple
harmonics and the relative contribution of these harmonics depends on the
torque amplitude. Parameter values: = 0.10, µ = 0.30, = −0.25, n = 1.5.

49

Steady-state response of the scaled averaged system as a function of χ. (a)
Steady-state amplitude; (b) Steady-state phase for D = 0.30, relevant for
σ > 0; (c) Steady-state phase for D = −0.30, relevant for σ < 0. . . . . . . .

57

Sample phase portraits and transient time traces for three values of χ, with
D = 0.30. (a) Phase portrait for χ = −1 < χ1 ; (b) Phase portrait for
χ = 0.10 ∈ (χ1 , χ2 ); (c) Phase portrait for χ = 1 > χ2 ; (d) Transient
time response for zero initial conditions IC#1 for χ = −1; (e) Transient time
response for χ = 0.10 and initial conditions IC#2; (f) Transient time response
for χ = 0.10 and initial conditions IC#3. Note that the initial amplitudes p0
are the same for IC#2 and IC#3, but the diﬀerent initial phases Φ0 result in
diﬀerent steady state outcomes. . . . . . . . . . . . . . . . . . . . . . . . . .

67

Phase plane plotted using level curves of equation (3.12) by varying c1 , for
χ = 0.066 and D = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

x

3.4

3.5

3.6

4.1

Stable and unstable manifolds of B for three values of χ. (a) χ = 0.02;
(b) χ = 0.10; (c) χ = 0.20. Figure 3.4 (a) and (b) show lines of constant
amplitude pcr , which indicate the maximum amplitude of initial conditions
p0 such that the response will approach the lower branch for all values of
the initial phase Φ0 . The topology of the homoclinic orbit about A changes
from Fig. 3.4 (a),(b) to that of Fig. 3.4 (c) for χ > 4/27. It is seen that
for χ ∈ (4/27, 8/27), small amplitude initial conditions will result in a large
amplitude steady state C, for all initial phases. . . . . . . . . . . . . . . . .

69

Numerical simulations of equation (3.1) for = 0.03 and ω = 2, and three sets
of system parameter values that yield χ = 0.094, with a predicted overshoot
of 115.8%. (a) F = 1/2, σ = 2, ξ = 2, simulated percent overshoot = 116.6%;
(b) F = 0.125, σ = −1, ξ = −4, simulated percent overshoot = 115.3%; (c)
F = 0.177, σ = 1, ξ = 1, simulated percent overshoot = 116.1%. . . . . . . .

70

(a) The analytical results for percent overshoot versus χ for zero initial conditions. The percent overshoot is computed relative to either the upper branch
C, shown by the solid gray curves, or the lower branch equilibria A, shown by
the solid black curve, depending on χ, as indicated in the inset. At χ = 4/27,
zero initial conditions results in a trajectory on the stable manifold of B.
When χ < 0 or χ > 4/27, zero initial conditions results in transient trajectories about equilibria C, as shown in Fig. 3.4 (c). Note that the phase at
which equation (F.2) is evaluated to obtain pmax for the percent overshoot is
diﬀerent for χ < 0 and χ > 4/27, as described in Section 3.3.1. (b) Comparison of the analytical solution and simulations of both the Duﬃng oscillator
(equation (3.1)) and the averaged equations (equations (3.6) and (3.7)) for
the case of zero initial conditions and diﬀerent values of . The parameter
values used in the simulations include: F = 1/2, σ = 1.75, ω = 2. The
nonlinearity ξ is varied from -4 to 7 to sweep the χ domain shown, and it is
noted that the percent overshoot is asymptotic to 58.74% as χ → ±∞ (i.e.,
as one approaches resonance, σ → 0). . . . . . . . . . . . . . . . . . . . . . .

71

Depiction of the CPVA system model showing three possible paths for the absorber COM. The outermost (circular) and innermost (cycloidal) paths bound
the two-parameter family under consideration; the middle path corresponds
to the tautochrone. The paths are shown enlarged, and not to scale relative
to the rotor, to exaggerate the diﬀerences in the paths. . . . . . . . . . . . .

75

xi

4.2

4.3

A damped zero initial condition transient trajectory, IC #1, obtained from
simulations of the general path averaged equations, indicating the peak and
steady-state amplitudes; (a) shown in the phase space, and (b) as the absorber
response versus θ. System parameter values: Dc = 0.30 and χc = 0.10. Also
shown in (a) is the trajectory from the unstable manifold of the saddle that
tends toward the desired steady state. . . . . . . . . . . . . . . . . . . . . . .

89

Simulations of the full EOM (equations (4.1) and (4.2)) for = 0.03 and
n = 1.5, and three sets of system parameter values that yield χc = 0.112,
with a predicted overshoot of 122%. (a) Γc = 1.171, n = 1.52 (σc = −4.263),
˜
λ = 0 (ξc = −4.22), simulated percent overshoot = 119%; (b) Γc = 0.799,
n = 1.51 (σc = −3.253), λ = 0.1 (ξc = −4.03), simulated percent overshoot
˜
= 121%; (c) Γc = 0.477, n = 1.5 (σc = −2.25), λ = 0.2 (ξc = −3.732),
˜
simulated percent overshoot = 124%. Additional parameters used to simulate
the full EOM include: Γ0 = Γ/2, α = 0, and β = 1. . . . . . . . . . . . . . .

91

4.4

The percent overshoot of an absorber system subject to zero initial conditions,
computed using the pseudo-energy method for Di = 0, and using the averaged
equations for Di = 0; for (a) general path and (b) near-tautochronic absorbers. 93

4.5

The rotor and absorber response to a step input of sinusoidal torque. (a) Rotor
and (b) absorber response to input +Γ sin(nθ); (c) rotor and (d) absorber
response to input −Γ sin(nθ). . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Comparison of the general path pseudo-energy method with simulations of
the full EOM for µ = 0, n = 1.5, = 0.03, Γ0 = Γ/2, α = 0, and β = 1.
Solid line is the pseudo-energy prediction. Simulation data: Sweep of χc
by varying λ from 0.80 to 0 with Γ = 0.005, and σc = −3.25 (˜ = 1.51).
n
Sweep of χc by varying Γ from 0.001 to 0.006 with ξc = −4.09 (λ = 0), and
σc = −3.25 (˜ = 1.51). Sweep of χc by varying n from 1.58 to 1.501 with
n
˜
Γ = 0.003 (note that in this case varying n results in sweeping ξc from -5 to
˜
-4 for a ﬁxed λ = 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

4.6

xii

4.7

Comparison of the general path pseudo-energy method with simulations of
the full EOM for µ = 0, n = 1.5, Γ0 = Γ/2, α = 0, and β = 1. Solid line is
the pseudo-energy prediction. (a) = 0.07 with simulation data: Sweep of
χc by varying λ from 0.80 to 0 with Γ = 0.012, and σc = −2.68 (˜ = 1.51).
n
Sweep of χc by varying Γ from 0.003 to 0.013 with ξc = −4.09 (λ = 0),
and σc = −2.68 (˜ = 1.51). Sweep of χc by varying n from 1.58 to 1.501
n
˜
with Γ = 0.01 (note that in this case varying n results in varying ξc from
˜
-5 to -4 for a ﬁxed λ = 0) (b) = 0.10 with simulation data: Sweep of χc
by varying λ from 0.80 to 0 with Γ = 0.019, and σc = −2.55 (˜ = 1.51).
n
Sweep of χc by varying Γ from 0.005 to 0.021 with ξc = −4.09 (λ = 0), and
σc = −2.55 (˜ = 1.51). Sweep of χc by varying n from 1.58 to 1.501 with
n
˜
Γ = 0.016 (note that in this case varying n results in sweeping ξc from -5 to
˜
-4 for a ﬁxed λ = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.8

Comparison of the general path pseudo-energy method with simulations of the
full EOM for Dc = 0.05 (ζ = 0.001) and Dc = 0.10 (ζ = 0.002). Simulation
data: Sweep of χc by varying λ from 0 to 0.80 with Γ = 0.013. Sweep of χc
by varying Γ from 0.003 to 0.013 with ξc = −4.09 (λ = 0). Other parameter
values used in simulations are: n = 1.51, n = 1.5, = 0.07, Γ0 = Γ/2, α = 0,
˜
and β = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.9

Comparison of the near-tautochronic path pseudo-energy method with simulations of the full EOM for λ = λe , µ = 0, n = 1.5, = 0.07, Γ0 = Γ/2,
α = 0, and β = 1. (a) Simulation data: Sweep of χt by varying Γ from
0.009 to 0.033 and σt = −2.68 (˜ = 1.51). (b) Simulation data: Sweep of
n
χt by varying n from 1.67 to 1.50 with Γ = 0.028. . . . . . . . . . . . . . . 102
˜

5.1

The experimental apparatus, with components: (A) computer for system control and data processing, (B) motor controller, (C) servo motor, (D) rotor
encoder, (E) absorber with encoder. . . . . . . . . . . . . . . . . . . . . . . . 107

5.2

Absorber pendulums: (a) near-tautochronic absorber (dark gray), suspended
by two steel bands that wrap around cheeks, generating the desired path,
shown with an encoder attached; (b) circular path absorber (inverted “T”),
suspended by a pin and needle bearing, as indicated. . . . . . . . . . . . . . 110

5.3

An experimentally simulated cylinder deactivation event showing the torque
input and the resulting transient absorber response. (a) Input torque T =
T0 + Tn sin(nθ) applied to rotor by the servo motor; (b) absorber transient
amplitude response to the input torque; (c) zoom-in of the input torque at
the transition point; (d) zoom-in of the absorber response at the transition
point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
xiii

5.4

Comparison of theory and experiments for the circular path absorber. Percent
overshoot vs. n for (a) Γc = 0.223, (b) Γc = 0.372, and (c) Γc = 0.50. Percent
overshoot vs. Γc for (d) n = 1.27, (e) n = 1.28, and (f) n = 1.30. . . . . . . . 118

5.5

Comparison of theory and experiments for the near-tautochronic path absorber. Percent overshoot vs. n for (a) Γt = 0.11, (b) Γt = 0.184, and (c)
Γt = 0.21. Percent overshoot vs. Γt for (d) n = 1.41, (e) n = 1.43, and (f)
n = 1.44. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.6

Comparisons of theory and experiments for the (a) circular path absorber and
the (b) near-tautochronic path absorber; percent overshoot vs. χc . . . . . . . 123

5.7

Bounding the experimental data using simulations of the damped general
path averaged equations for the (a) circular path absorber and (b) the neartautochronic path absorber; percent overshoot vs. χc . . . . . . . . . . . . . . 126

6.1

Non-synchronous transient motion to a steady-state near-synchronous response
using two circular path absorbers at torque order n = 1.292: (a) experiment
and (b) simulation. Parameter values used for simulation: n = 1.309 (both
˜
absorbers), ζ1 = 0.002 (response shown in red), ζ2 = 0.0035 (response shown
in black), T0 = 1.069 Nm, Tn = 2.919 Nm, Ω = 36.652 rad/s, J = 0.0636
kg·m2 , m = 0.241 kg (both absorbers), cc = 0.162 m (both absorbers, see
Tab. 5.2), ρ0c = 0.041 m (both absorbers, see Tab. 5.2), and a ramp of half
a torque cycle in of the ﬂuctuating torque. . . . . . . . . . . . . . . . . . . . 133

6.2

Non-synchronous transient motion to a non-synchronous steady-state at torque
order n = 1.302: (a) experiment and (b) simulation. Parameter values used
for simulation: n = 1.309 (both absorbers), ζ1 = 0.0013 (response shown in
˜
red), ζ2 = 0.0042 (response shown in black), T0 = 0.937 Nm, Tn = 1.40
Nm, Ω = 36.652 rad/s, J = 0.0636 kg·m2 , m = 0.241 kg (both absorbers),
cc = 0.162 m (both absorbers, see Tab. 5.2), ρ0c = 0.041 m (both absorbers,
see Tab. 5.2), and a ramp of half a torque cycle in of the ﬂuctuating torque. 134

6.3

Experimental comparison of the transient overshoot of a circular path absorber
when forced at n = 1.26 using two types of ﬂuctuating torque inputs that have
the same magnitude but a diﬀerent ramp duration. (a) Response to a near
step input of the torque, resulting in an overshoot of 94%. (b) Response to a
ramp input of the torque with a duration of 10 torque cycles, resulting in an
overshoot of 15%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
xiv

G.1 Depicting the percent overshoot for the damped system subject to zero initial
conditions. The damping levels include D = 0, D = 0.05, and D = 0.10
as labeled in the ﬁgure. The solid curves are obtained by numerically solving
equation (G.6) which is the constant of motion for small damping. The circles
are obtained by simulating the average equations (equations (3.6) and (3.7))
for the two levels of damping. The damping increases the critical system
parameter χ as shown in Fig. 3.6 and this increases the operating regime for
a transient response about the lower branch equilibria (A). The constant of
motion provides a very good approximation of the damped transient response
of the system up to a damping level of about D = 0.10. . . . . . . . . . . . . 151

xv

Chapter 1
Introduction
Centrifugal Pendulum Vibration Absorbers (CPVAs) are passive devices used to address
torsional vibrations in rotating systems, for example, helicopter rotors [77] and internal
combustion engine crankshafts [55]. As shown in Fig. 1.1 (a), a CPVA consists of a pendulum mounted on a rotor, driven by system rotation, and tuned in such a manner that
in steady-state operation the system dynamics lead to a torque from the pendulum that
counteracts engine-order ﬂuctuating torques acting on the rotor, thereby smoothing rotor
torsional vibrations. Multiple CPVAs are placed around a rotor for balancing, and each is
typically constructed using a biﬁlar (two point) suspension, as shown in Fig. 1.1 (b). These
absorber designs consists of the following essential components: an absorber mass (D) and
two rollers (A), which suspend the absorber from a ﬂange (C) that is ﬁxed to the rotor.
The rollers ride on surfaces (B) on the absorber and ﬂange, which are identical but inverted,
and are machined so that the absorber follows a user-designed path. This path, along with
the absorber mass and placement (distance from the rotor center), are the essential design
ingredients for an absorber system. For example, Fig. 1.1 (c) shows an implementation of
1

four absorbers mounted on a helicopter rotor, for which the absorbers follow a cycloidal
path [38]. In this chapter we describe some current motivations for using CPVAs in automotive engines, provide a basic description of how they work, and include a discussion of
previous work in the area.

1.1

Motivation

As pressures to improve fuel economy and reduce emissions continue to increase [1], cylinder
deactivation, which shuts down inessential cylinders during low-power operating conditions,
remains a viable technology. The fuel savings for these engines is typically in the 5-10% range,
accruing from reductions in pumping losses [22]. However, there is a current limitation on
the range of engine speeds over which one can deactivate cylinders, and this is limited on
the low-rpm end primarily by vibration issues. Therefore, if absorbers can be designed
such that they reduce torsional vibration (in the cylinder deactivated mode), then one can
expand the engine speed range over which the engine can deactivate cylinders, resulting in
additional fuel savings. Absorber design considerations for steady-state operation are already
well understood, and have proven to work well, even down to idle speeds for a prototype
V8/V4 engine [55]. However, transient absorber motions, speciﬁcally, those induced by
abrupt transitions from full cylinder (non-resonant excitation) to reduced cylinder (resonant
excitation) are not well understood, and the issues associated with these transient responses,
for example, bumper impacts and non-synchronous responses, are directly relevant to the
acceptance and implementation of CPVAs in these engines. Furthermore, a literature search
on analytical treatments of transient dynamics in nonlinear systems turned up only a handful
of publications [2, 15, 20, 32, 39, 42, 64], and none have considered the percent overshoot
2

(a)

C
D
B
A
(c)
(b)
Figure 1.1: (a) Depiction of the basic CPVA model, consisting of a pendulum (absorber)
attached to a rotor, which is driven by a mean torque and and ﬂuctuating torque of order
n. (b) The basic elements of a biﬁlar absorber shown in a deﬂected (oﬀ-center) position (the
device shown was designed and fabricated by Chrysler for experimentation in the MSU lab);
elements A are the rollers used for the biﬁlar suspension, B is one of the machined surfaces
that dictates the path followed by the absorber, C is a ﬂange that is attached to the rotor,
and D is the absorber mass; note that the machined surfaces on the ﬂange and absorber are
identical but inverted. (c) Helicopter rotor with 4 biﬁlar absorbers attached (photo by S.
Shaw, aboard the USS Nimitz, 1991). For interpretation of the references to color in this
and all other ﬁgures, the reader is referred to the electronic version of this dissertation.

3

of a nonlinear oscillator suddenly subjected to harmonic excitation, which is precisely the
information needed for the current investigation.
Successful reduction of torsional vibrations using CPVAs has other applications for improving fuel economy. Several automotive manufacturers are working to develop high-torque,
fuel-eﬃcient, four-cylinder engines, which typically operate with high compression ratios and
at low speeds, which inherently lead to large torsional oscillations. These engines can be
more aggressively tuned for fuel economy, for example, by running at lower engine speeds,
if torsional vibration issues are addressed. Also, in cars with automatic transmissions, the
torque converter is the source of signiﬁcant losses. When the torque converter is locked there
is an essentially rigid link between the engine and wheels, which minimizes these losses. The
ability to initiate torque converter lockup is dictated by the level of torsional vibrations between the engine and downstream powertrain components, and thus improved eﬃciency of
the powertrain can be achieved if one can reduce engine torsional vibrations. Another feature
of future engines that is relevant to transient absorber response is that of start-stop technology, which stops the engine entirely for zero load conditions, for example, when stopped
at traﬃc lights. These engine technologies provide transient environments under which the
absorbers must be able to survive without unwanted dynamics.
Thus, the primary goal of this work is to develop a fundamental understanding of the
transient dynamics in nonlinear systems, so that it can be applied to the design of absorber
systems for automotive applications. Thus, a systematic analytical and experimental study
of transient absorber response is carried out for a parameterized family of absorber paths
that includes those of practical interest. This is accomplished by building on the current
knowledge of absorber responses, including work on linear and nonlinear absorber models [58,
4

59, 69], as well as taking advantage of the existing experimental facility used in previous
studies [26, 52, 53, 70].
The main technical results of this dissertation are organized around chapters on various topics, as follows: in Chapter 2 we improve the absorber system model by accounting
for the roller dynamics and investigating the eﬀects of the rollers on system response and
performance; in Chapter 3 we derive an analytical method for the computing the percent
overshoot of a nonlinear Duﬃng oscillator subject to a suddenly applied harmonic force with
general initial conditions, and testing the results against simulations; in Chapters 4 & 5 we
apply the analytical results to CPVA models, including theoretical predictions compared
with simulations (Chapter 4), and experimental veriﬁcation (Chapter 5) for circular path
and near tautochronic path absorbers.

1.2

Basic Operation of a CPVA

In an internal combustion engine, engine-order ﬂuctuating torques are a result of the net
torque applied to the crankshaft as each cylinder ﬁres in succession as the crank rotates. In
many situations, this applied torque is dominated by a mean component T0 and the ﬁrst
harmonic of order n, Tn sin nθ, where θ ≈ Ωt and Ω is the mean rotor speed and n is the
order of excitation. 1 For example, for a four-stroke engine with K cylinders, n = K/2. The
order of excitation n has the same connotation as an angular frequency, except that n is
dimensionless. The mean torque T0 maintains the mean rotor speed Ω, and the ﬂuctuating
excitation torque produces torsional oscillations about this mean speed. If an absorber can
1 Higher harmonic torques, at orders 2n, etc., exist, but their amplitudes are generally
negligible in comparison to the ﬁrst and are often ignored in practice.
5

be tuned correctly, it will respond to this excitation and produce a counter-torque, thereby
reducing torsional oscillations of the rotor. The primary feature of CPVAs is that they
are order tuned, as opposed to the more common frequency tuned absorbers, so that the
absorber’s natural frequency (ωa = nΩ) stays tuned relative to the excitation frequency
˜
(˜ ≈ n) at all engine speeds (Ω). Therefore, the absorber is eﬀective at reducing torsional
n
vibrations across an entire range of rotor speeds. In contrast, a frequency tuned device can
be tuned only to a single rotor speed.
In order for a CPVA to be eﬀective at reducing the torsional vibration of the rotor, the
absorbers tuning order should be near to, or slightly greater than, that of the excitation
order, that is, n ≈ n with n ≥ n; this is referred to as over-tuning [31]. This tuning
˜
˜
results in steady-state operation in which the absorber response is out of phase relative to
the excitation torque, which reduces torsional vibrations of the rotor. If one under-tunes
an absorber, it is closer to the absorber/rotor system resonance and its amplitude grows
rapidly as the torque is increased, leading more quickly to nonlinear behavior. A possible
disastrous nonlinear behavior is the jump bifurcation that results in large amplitude steadystate absorber response that is in phase relative to the excitation, resulting in ampliﬁcation
of rotor torsional vibrations [5,56]. Another important nonlinear eﬀect is symmetry-breaking
bifurcations in systems composed of multiple absorbers [4,5,11,68]. All of these bifurcations
are possible due to the system nonlinearities that arise from both the absorber path, as well
as the kinematic coupling to the rotor.
The absorber natural frequency of oscillation is determined by its free vibration response
when the rotor spins at a constant speed Ω; this frequency is proportional to Ω and generally
depends on amplitude through nonlinear eﬀects determined by the absorber path. This
6

frequency consists of a linear part nΩ, determined by the curvature of the path at its vertex,
˜
and a nonlinear part dictated by a parameter λ that describes how the path curvature
varies as a function of travel along the path. The restoring force acting on the absorber
is the component of the centrifugal load tangential to the absorber path, and the path
curvature determines the nature of this force. For instance, a circular path, which has
constant curvature, results in an oscillator containing a nonlinear softening restoring force
(that is, the eﬀective stiﬀness decreases as a function of amplitude), resulting in a softening
nonlinear dynamic response. As one might expect, there exists a path which results in a
linear restoring force, resulting in a simple harmonic response for all amplitudes; this path
is referred to as the tautochronic path, and is known to be a particular epicycloid. Section
2.2.1 provides more details about the path parameters and their geometric interpretation.
Paths also exist which exhibit nonlinear hardening restoring forces, such as the cycloidal
path described in the patent by Madden [38]. Denman [17] has introduced a very convenient
framework for describing paths that include circles, cycloids, and epicycloids, using a two
parameter (˜ and λ) path, as described in Section 2.2.1.
n

The path nonlinearities described above directly aﬀect the forced response of the rotor/absorber system. However, in the forced vibration case, the excitation acts on the rotor,
and rotor/absorber inertial coupling comes into play, which results in additional system nonlinearities. For circular path absorbers, the path nonlinearity is dominant over a large range
of absorber motions. However, for the tautochronic path, which has a neutral path in terms
of nonlinearity, the system is slightly softening as a result of kinematic coupling to the rotor.
However, when the rotor is driven, the eﬀect of the applied torque is reduced at large amplitudes by kinematic eﬀects, with the net eﬀect being an overall slight hardening behavior
7

for forced vibration [49]. In addition, for all absorbers, the response is also aﬀected by the
dynamics of the rollers, which introduce additional kinematic path nonlinearities through
their inertia. Lastly, the situation can become even more complicated when multiple absorbers are attached to a rotor [4, 5, 11, 68]. For such systems the absorbers can exhibit
non-synchronous steady-state responses, even when the absorbers are identical [4, 5, 11, 68].
Therefore, it is important to understand how system parameters and nonlinearities aﬀect the
dynamic response, so that these systems can be designed to operate eﬀectively in a passive
manner.

1.3

Background

CPVAs have been in use for about 100 years, and studied scientiﬁcally for about 80 years. The
contributions, in chronological order, go from initial experimentation and implementation
in industrial applications, to linear modeling of the steady-state response, and ﬁnally to
the more recent contributions about the nonlinear steady-state response, both analytically
and experimentally; this history is brieﬂy summarized below. Up to the present, the focus
has always been on the steady-state response, which is applicable in aerospace applications.
However, the renewed interest in automotive applications has raised the issue of system
transient response, and that is the main thrust of the study described herein.
Our knowledge of absorber system dynamics has come along way since the ﬁrst known
conceptual design, which dates back to 1911 [31]. In 1911, the ﬁrst frequency tuned vibration
absorber was proposed in a U.S. patent by Frahm [23]. In the early 1930s the basic principles
of operation of order-tuned vibration absorbers were ﬁrst disclosed in patents [62, 63]. The
general idea behind the order-tuned absorbers was that of mounting a mass to a rotating
8

system such that it is free to oscillate and react in the desired manner to reduce ﬂuctuations
in the rotor speed, that is, torsional vibrations. Taylor was the ﬁrst to realize the importance
of the order-tuned device and introduced it into industry [73]. Constructions of order-tuned
devices were ﬁrst carried out by R. Chilton of the Wright Aeronautical Corporation, who
used the biﬁlar design, and E.A. Ryder and G.L. Williams of the Pratt and Whitney Aircraft
Company, who used a roller design [27]. The biﬁlar design (see Fig. 1.1 (c)) is still the typical
implementation used in applications today, including light aircraft engines and helicopter
rotors.

The mathematical and dynamics modeling of absorber systems began with investigating the absorber system’s linear dynamic response, in order to determine how to tune an
absorber to be eﬀective at counteracting an order n excitation. The absorber COM paths
initially used were circular, with curvature selected to meet the small amplitude (linear)
tuning condition [41]. Den Hartog [28] was the ﬁrst to clearly describe the detuning caused
by nonlinear eﬀects that arise at modest amplitudes for these absorbers. In regards to linear
theory, Den Hartog [27] investigated how viscous damping and detuning an absorber (the relative tuning between the absorber tuning order and the excitation order) aﬀect the resulting
linear steady-state torsional vibration of the rotor. Den Hartog was also the ﬁrst to account
for the nonlinearity in the pendulum free vibration response, showing that the softening
restoring force of the circular path resulted in a detuning of the pendulum, which grew with
increasing amplitude (i.e. the absorber tuning order decreased with increasing amplitude of
oscillation). He suggested overtuning the absorber to account for this eﬀect. Newland [56]
(a Den Hartog Ph.D. student), investigated the forced vibration response of circular path
absorbers and showed that the softening restoring force can result in the devastating non9

linear jump bifurcation described above. To avoid this jump, Newland [56], quantiﬁed the
required level of over-tuning for circular path absorbers. He provided guidelines for robust
tuning to avoid the jump bifurcation, while maintaining absorber eﬀectiveness in terms of
reducing torsional vibrations. These types of results, which were originally carried out using
harmonic balance methods [56], were recently extended using perturbation calculations to
consider a more general family of paths, and to determine the dynamic stability of single
and multi-absorber systems [5, 68].
Due to the potentially disastrous non-linear jump response associated with the circular
path, another phase of research has involved the study of non-circular paths that can avoid
the jump. These studies also consider the eﬀects of the path on absorber eﬀectiveness at
reducing order n vibrations, while avoiding ampliﬁcation of higher harmonic vibrations, i.e.,
2n, 3n.... It turns out that for an over-tuned absorber, the jump bifurcation is only possible for softening paths, and this is why non-circular paths that are either tautochronic or
hardening are advantageous [5, 68]. These paths, when over-tuned, have only one possible
steady-state amplitude response (for a system with a single absorber), which is the desired
response that is out of phase with the excitation. Madden [38] was the ﬁrst to suggest a
non-circular path that was capable of overcoming the jump response, speciﬁcally by using
a cycloid, which is the solution of the classical brachistochrone problem, and the less wellknown tautochrone problem.2 The cycloidal path suggested by Madden produces a slightly
hardening response. The brachristochrone/tautochrone epicycloidal path for a centrifugal
2 The brachistochrone is the solution to the problem posed in June 1696 by Johann
Bernouilli: “Given two points A and B in a vertical plane, ﬁnd the curve that a point
M, moving on a path AMB must follow such that, starting from A, reaches B in the shortest
time under its own gravity.” It turns out that the cycloid actually solves both the minimum
time (brachristochrone) and the equal time independent of starting amplitudes (tautochrone)
problem for a gravitational ﬁeld.
10

ﬁeld was known to Huygens, but brought to use in absorbers by Denman [16]. Denman’s
work simpliﬁed the study of absorber paths, since it provides a general framework in which
the equations of motion can be developed for a two-parameter family of paths. The parameter n sets the linear tuning, while the parameter λ allows one to vary paths from circular
˜
(strongly softening), to tautochronic (neutral), to cycloidal (hardening) [17]. This formulation provided a means by which perturbation studies can be performed to systematically
investigate the eﬀects that path parameters have on the system response. A study by Lee &
Shaw [33] showed that, although the epicycloidal path results in a favorable simple harmonic
absorber response (for constant rotor speed), the torque that the absorber puts back on
the rotor is not a pure harmonic of order n, due to the nonlinear kinematic coupling of the
absorber to the rotor. This implies that the tautochronic epicycloid, while avoiding detuning, is not able to exactly counteract a purely harmonic torque, as is desired. A solution
to this problem was a subharmonic absorber system, which consists of a pair of identical
absorbers riding on epicycloidal paths tuned to one half the order of the excitation torque.
These generated, via a subharmonic response, a torque that is exactly a pure harmonic over
a wide range of amplitudes [34, 35]. Further analyses by Chao & Shaw [12, 13] incorporated
path imperfections in the analysis and determined design guidelines for systems of identical,
multiple, subharmonic absorber pairs.

The subharmonic absorber system is actually one of a number of non-synchronous steadystate responses that can result from symmetry-breaking bifurcations inherent in systems
with multiple identical absorbers. Except for the subharmonic absorber, non-synchronous
responses are not desired in practice, since the absorbers do not function in an optimal
manner with minimal amplitudes. This observation has motivated several investigations
11

into determining system parameter conditions that result in non-synchronous responses,
so that absorbers can be designed to operate synchronously in the steady-state. In fact,
multiple absorber systems are typically required in applications, in order to balance the
rotating system, by placing sets of absorbers symmetrically about the rotor center. Chao &
Shaw [11] and Shaw & Geist [68] showed that for tautochronic and nearly-tautochronic path
absorbers, a critical excitation torque magnitude exists which, if exceeded, results in loss
of stability of the synchronous response, resulting in nonsynchronous absorber responses.
This bifurcation can be avoided by suﬃcient levels of absorber damping and/or intentional
absorber mistuning, both of which degrade absorber performance in terms of reducing rotor
torsional vibrations. In a related study, Chao & Shaw [14] found that the post-bifurcation
dynamics are dominated by a stable steady-state solution branch in which one absorber
undergoes a much larger amplitude of motion relative to the others, essentially a form of
nonlinear localized response. A similar study by Alsuwaiyan and Shaw showed that for
hardening and softening paths (i.e., paths that are not nearly tautochronic), the bifurcation
to a non-synchronous response can be avoided for softening paths if the absorber is designed
with an over-tuning level that is greater than or equal to a value set by the viscous damping
coeﬃcient and the excitation order n [4, 5]. It was also found that for suﬃciently over-tuned
neutral (tautochronic) and hardening paths, the non-synchronous response is not possible,
nor is the jump, and these properties make epicycloidal and cycloidal paths quite appealing.
To extend these results to nearly tautochronic paths, a recent study by Shaw & Geist [68]
describes analytical results for the stability of the synchronous absorber response against
both jump and symmetry-breaking bifurcations, in terms of the two path parameters. The
mathematical tools needed to investigate the stability of the synchronous response in multiple

12

absorber systems include circulant matrices, which are thoroughly covered in [57].

To validate the steady-state response analytical results, a number of experiments were
performed using both single and multiple absorber systems. These were carried out using an
experimental rig described in detail in [26] and Chapter 5 of this dissertation. This apparatus
allows one to input a speciﬁed excitation torque signal (in terms of order and amplitude),
and measure the resulting response of the rotor and up to four attached absorbers. The
torque signal is speciﬁed in LabVIEW and then sent to a servo motor which excites the
rotor and the attached absorbers. The responses of the rotor and absorbers are recorded
in LabVIEW for post-processing the data. The steady-state experiments performed to date
have included excitation order sweeps (at constant excitation amplitude) and excitation
amplitude sweeps (at constant excitation order). These results show excellent agreement
with theoretical predictions for circular path pendulum-type absorbers [26, 52–54], as well
as for tautochronic path absorbers [65, 70]. As expected, non-synchronous motions were
observed in the multiple absorber experiments for cases when the absorbers tuning order
was near the excitation order (i.e. n ≈ n) [52–54]. However, the non-synchronous motions
˜
observed in the experiments involving four absorbers did not agree in form with the previous
analytical predictions. In particular, Nester et al. [52–54] found that for low torque levels,
a subset of the four absorbers were essentially inactive (almost zero amplitude), and that
as the excitation amplitude was increased, more absorbers became active, and eventually
synchronous with the active set of absorbers, one at a time, until all four were active. These
results deserve further investigation, and some suggestions based on recent observations
during the course of this work are given in Chapter 6. More recently, Vidmar [75] performed
an analytical and experimental study accounting for both Coulomb and viscous damping
13

using the same circular path pendulum-type absorber as described in Nester [53, 54]. The
Coulomb friction was adjusted by tightening the screw on the bearing about which the
absorber swings, and it was found that the Coulomb friction resulted in a sticking (zero
rotational velocity relative to the rotor) of the absorber up to a certain level of torque. As
expected, this resulted in no reduction in torsional vibration, until the absorbers broke free
and began to move [75].

As discussed in Section 1.1, these absorbers have been proposed to expand the operating
envelope for cylinder deactivation in variable displacement engines, which oﬀers improved
fuel economy by reductions in pumping losses. In these engines, cylinders are deactivated
depending on the loading conditions. This is accomplished by closing the intake and exhaust
valves and stopping fuel transmission to the deactivated cylinders. Currently, these engines
are required to operate in full cylinder mode in the low-RPM range due to vibration issues
associated with the reduced cylinder mode at low speeds. Therefore, vibration absorbers
tuned to the reduced cylinder mode excitation order have the potential to allow such engines
to run at lower speeds, and higher loads, in reduced cylinder mode. In fact, if idle can be
reached within allowable vibration limits, signiﬁcant improvements in mileage will be realized
for the standard driving cycle that is used to establish oﬃcial vehicle mileage. It was shown
in an experimental study on a prototype Ford V8/V4 engine that these absorbers allowed one
to operate at much lower engine speeds, thereby expanding the operating range for reduced
cylinder operation, and improving fuel savings [55]; this is the only known experimental study
of absorbers on a variable displacement automotive engine. Another experimental study by
Albright et al. [3] using a roller type absorber, showed the absorber to work well when the
excitation exists at integer orders of crankshaft rotation, without the added ampliﬁcation of
14

a torsional resonance which occurred in their high power V8 drag motors.

While known to be eﬀective for steady-state operation, the transient behavior of absorbers
is a looming question when considering them for automotive implementations. A transient
event that is especially important in engines that employ cylinder deactivation is that of
the deactivation event, in which the ﬂuctuating torques on the crankshaft change quite
suddenly in terms of the order and amplitude. In general, the engine order is reduced
(typically by a factor of 1/2, when half the cylinders are deactivated) and the amplitude is
increased by a factor of about 2-3 (due to fewer harmonic cancellations). If the absorbers
are tuned for the reduced-cylinder mode of operation, this abrupt change corresponds to
a transition from non-resonant to near-resonant excitation for the absorber response, the
eﬀects of which are ampliﬁed by the increase in excitation level. This leads to a beating-type
transient motion, resulting in overshoot of the absorber response before it reaches steady
state conditions [58, 69]. An experimental and theoretical transient investigation by Palmer
for softening paths showed that simulations of the averaged equations provided a satisfactory
prediction of the experimental results for circular path absorbers [59]. Palmer suggested that
the averaged equations provide an eﬃcient way to determine the bounds on the overshoot,
the beating frequencies, and the decay rates. A subsequent study investigated the transient
response of exactly tautochronic paths, and scaled the amplitudes so that a closed form
solution of the envelope of the transient response could be obtained [53, 54]. This linear
analysis provided explicit formulae for many important quantities, including peak amplitude,
steady-state amplitude, settling time, and beat period. This thesis extends and ampliﬁes
these previous studies.
15

1.4

Summary of Results

The main contributions of this work include the following: We have provided a model of the
absorber that includes the dynamics of the support rollers used in biﬁlar suspensions, and
derived, for the ﬁrst time, a tautochronic path for this case. In terms of problem formulation,
we combined the two types of path scalings introduced by Shaw et al. [5,11,14,26,52,53,68,70]
into a single framework from which an approximate analytical method for the transient
overshoot of a CPVA system can be obtained. In addition, a unique rescaling of the averaged
equations was introduced that combines the order detuning, the amplitude of excitation,
and the system nonlinearity, into a single parameter, χ. The analytical predictions are
based on the method of averaging, and for an undamped system the prediction of overshoot
involves solving the roots of a pseudo-energy equation. When the dominant nonlinearity
arises from the path, the undamped overshoot result is obtained in closed form, as the roots
of a quartic polynomial that depend solely on χ. This result follows from an analysis of
the transient response of a standard Duﬃng oscillator with suddenly applied near-resonant
harmonic excitation, a treatment that is, to the author’s knowledge, original. For paths that
are nearly tautochronic, an additional parameter γ is required, and the overshoot roots must
be found numerically. The addition of damping introduces a dissipation parameter D for all
paths, and the predictions require simulation of the averaged equations. These analytical
predictions are compared against simulations of the original equations of motion, simulations
of the averaged equations, and experiments performed using a circular path absorber and a
near-tautochronic path absorber. The main results are presented in Fig. 5.6 (a) & (b), which
show that the analytical overshoot predictions provide a useful upper bound to the data,
thus making it a convenient tool for conservative absorber designs. As expected, the full
16

equations of motion do the best at predicting the overshoot in general cases, but simulations
of the averaged equations are also quite accurate. Lastly, as shown in Fig. 5.7 (a) & (b),
more aggressive absorber designs can be obtained if a bound on the absorber damping is
known.

17

Chapter 2
Accounting for Roller Inertia in the
Biﬁlar CPVA System Model

2.1

Introduction

A biﬁlar suspended CPVA employs a pair of cylindrical rollers which suspend the absorber
mass as shown in Fig. 2.1. This arrangement results in the absorber mass translating, but
not rotating, relative to the rotor. Other implementations do exist such as a roller type
absorber system [30], but the biﬁlar suspension has been most common in practice. An
important aspect of these absorbers is their tuning, which is determined by the path along
which their center of mass moves. This tuning, which for these absorbers is order tuning
rather than frequency tuning, consists of linear and nonlinear components. The linear part
sets the small amplitude vibration characteristics, and has long been understood [28, 31].
The linear tuning order is set to be close to the order of excitation, so that the absorber
responds to the excitation in such a manner so as to reduce torsional vibrations of the rotor.
18

The softening nonlinear aspects of the commonly-used circular path absorbers were known
to cause problems, since their frequency of free oscillation decreases as a function of vibration
amplitude, thereby resulting in detuning [28,56]. Absorber paths that overcome this problem
have been proposed and studied, and include cycloids, epicycloids, and generalizations of
these [5,17,38]. Much of this work simply ignored the inertial eﬀects of the rollers. However,
Denman [17] included the roller inertia in a study of absorbers with quite general paths,
and derived a means of including the eﬀects of roller inertia for the linear tuning. His
work included the nonlinear dynamics of the rollers in the equations of motion, but does
not provide a systematic investigation of how to design nonlinear aspects of the dynamics
to account for rollers. That work also included a thorough description of the zero roller
inertia tautochronic absorber path, that is, the path that renders the absorber frequency
as constant for all physically realizable amplitudes of response, i.e., a path such that the
absorber is neither softening nor hardening out to large amplitudes.
Although the roller masses are small in comparison to the absorber mass, it is important
to know how the overall system dynamics are aﬀected by the inertia of the rollers. This
is obvious for the linear tuning, since a slight oﬀ-tuning dramatically aﬀects the system
response in terms of absorber eﬀectiveness and operating range [5, 68]. The purpose of the
present work is to systematically quantify the eﬀects of rollers on the linear and nonlinear
system response, so that one can design absorber systems that fully account for the dynamics
of the rollers.
We begin with a derivation and description of the equations of motion in a nondimensional
form that is convenient for analysis; this, and many subsequent steps, follow previous work
on CPVAs, cf. [11, 68]. The absorber path and the attendant linear and nonlinear tuning
19

Figure 2.1: Picture taken by Steve Shaw of a helicopter rotor with four biﬁlar absorbers
attached (taken aboard the USS Nimitz).

parameters are then considered, and the tautochronic path that accounts for rollers is derived.
This is followed by a perturbation analysis of the system dynamics, and an example is
presented that demonstrates the inﬂuence of rollers on the system response. It is seen that
so long as the rollers are accounted for in the linear tuning, and the nonlinear tuning is close
to tautochronic, then the overall system response is largely insensitive to small variations in
the nonlinear tuning, which is inﬂuenced by the inertia of the rollers. In fact, the rollers are
found to provide a slight hardening eﬀect on the absorber response, which actually shifts
the system away from potential jumps in the absorber response. This chapter closes with a
brief discussion of the results.
20

2.2

Dynamic Model

In this section the dynamic model for a CPVA suspended in a biﬁlar conﬁguration by a
pair of rollers is constructed. The model is introduced by deﬁning parameters for a single
absorber with general inertial properties and a pair of rollers that are assumed to roll without
slipping.1 After these basic features and some nondimensional parameters are introduced,
the equations of motion for a system composed of a rotor with N CPVAs riding on rollers are
derived by a Lagrangian approach. Here we simply outline the steps of the derivation, since
the details are available in previous work [5,17]. The equations of motion are then presented
for a single absorber which are non-dimenionalized and re-scaled for the purpose of doing
analysis. This includes a perturbation calculation which is used to determine the eﬀects that
the rollers have on the system response; these steps also follow previous work [11, 68], but
here include the eﬀects of rollers.
The biﬁlar construction of CPVAs using rollers consists of four essential components: a
rotor, an absorber mass, and two rollers, as shown in Fig. 2.2. The rotor is subjected to
a ﬂuctuating torque, resulting in torsional vibrations of the rotor. The absorber mass is
a generalized centrifugal pendulum and will be referred to herein simply as the pendulum.
The pendulum is suspended from the rotor by the rollers, which roll along two cutouts in
the rotor, as shown in Fig. 2.2. The rollers also roll along identical, inverted cutouts on the
pendulum. This arrangement kinematically forces the pendulum center of mass (COM) to
follow a path that is dictated by the shape of the cutouts. In many applications, for ease of
manufacturing, these cutouts are simply circles. In general, the dynamic model for the biﬁlar
construction is described by the motion of the COM of N identical pendulums, each of mass
1 We have examined the motions of such rollers in an experimental rig and found that
they indeed roll without slipping in all situations observed.
21

mP and moment of inertia IP about the pendulum’s COM. These are attached to a rotor
which has a moment of inertia Ic about the center of rotation, point 0, as shown in Fig. 2.2.
As previously mentioned, it is assumed that the two rollers, each of mass mR , radius A, and
moment of inertia iR about their COM, roll without slip along the cutouts. Note that in
this arrangement the arclength distance S displaced by the COM of the pendulum along its
path is twice that moved by the COM of each of the rollers along their respective paths [17].
The positions of both the pendulum and the rollers’ COM relative to the rotating coordinate
system at point O can be conveniently expressed in terms of a function RP (S), which is the
distance from the center of the rotor O to the pendulum COM at a position corresponding
to S. Here all absorber paths are taken to be identical and symmetric about a vertex at
S = 0 such that RP (−S) = RP (S). Here we denote R(0) = c, which is the largest value of
RP (S). Under these assumptions the dynamic system consists of N + 1 degrees of freedom,
corresponding to the positions Si of the pendulums and the rotor orientation θ. We present
a derivation for this system and then specialize to the case N = 1 which is suﬃcient for the
present study.

The system kinetic energy is given by
N
Tt = Tc +

(TP + TR ),
i
i
i=1

(2.1)

where Tc accounts for the rotational kinetic energy of the rotor, TP is the total kinetic
i
energy of the ith pendulum, and TR is the total kinetic energy of the the ith pair of rollers.
i
For the pendulum and rollers we compute the kinetic energy using the translation of the
22

COM and rotation about the COM. The total kinetic energy can then be written as
N
˙
T = Tc (θ) +
i=1
=

1
J+
2

N

˙ ˙
˙ ˙
TR (θ, Si , Si ) + TP (θ, Si , Si )
i
i

2
mP RP (Si ) + 2mR H 2 +

i=1

1 2
+ (c/2 − D)2 + (c/2 − D)YP (Si ) + RP (Si ) +2iR
4
N
+
i=1
N
+
i=1

(2.2)

˙
θ2

∂X (S )
1
1
˙ ˙
(mP + mR )G(Si ) + mR (2D − c) P i − iR /A Si θ
2
2
∂Si
1
1
1
˙2
mP + mR + (iR /A2 ) Si ,
2
2
2

where J = Ic + N IP is the total inertia of the rotor and the pendulums about their respective COM, A is the roller radius, and D and H are, respectively, the vertical and
horizontal distances from the rotor center O to the roller COM when the pendulum is at
its vertex; see Fig. 2.2. The terms XP (S) and YP (S) locate the pendulum’s COM posi2
2
2
tion relative to the rotor-ﬁxed X − Y coordinate system, and RP (S) = XP (S) + YP (S)
1
2
2
(see [17] or Appendix B for further details). Lastly, G(S) = RP (S) − 4 (∂RP (S)/∂S)2 =
RP

1 − (∂RP (S)/∂S)2 , is RP times the cosine of the angle between the local tangent to

the absorber path at S and the normal to the radial line passing through O and the point at
S on the path [68]. Therefore, G(0) = R(0) = c, and G(S) = 0 corresponds to the absorber
moving directly towards the rotor center, which gives the maximum achievable value of S,
denoted by Smax , for many absorber paths. This limit will be investigated further during the perturbation calculations, for which the absorber path and the maximum possible

23

absorber amplitude will be important.

H
mR

90

90

S

90

90

90

RP (S)

mP

90

mR

1
2S

D

Figure 2.2: Depiction of a biﬁlar roller-suspended pendulum (left) attached to a rotor, and
the paths of the pendulum COM and rollers’ COM (right).

Using the kinetic energy in equation (2.2), and generalized forces from dissipation and
applied loads, one can derive the equations of motion for the system using the Euler-Lagrange
method. These derivations, and subsequent non-dimensionalization and rescaling, with the
rollers absent, are described in previous work [11,14]. An important step in the development
24

of the equations of motion is that the rotor angle θ is used to replace time as the independent
variable, which converts nonlinear θ-dependent torques into external excitation terms. The
equations of motion are presented by ﬁrst introducing a number of variables and parameters,
after which we present the absorber and rotor equations of motion. These equations are,
of course, coupled in the following way: the absorber response is driven by the torsional
acceleration of the rotor, and the rotor dynamics are aﬀected by torques produced by the
absorbers as they move along their paths. The goal is to design the absorbers such that
these torques counteract an applied ﬂuctuating torque that acts on the rotor, that is, the
torque that produces the unwanted torsional vibrations.
We ﬁrst introduce s, g, xP , yP , rP , and a as non-dimensional versions of their corresponding upper case versions given in equation (2.2), in which physical system lengths are
normalized by the distance c, is deﬁned as the absorber to rotor inertia ratio mP c2 /J, and
many parameters that are known to be small are scaled by

for convenience. The nondi-

˙
mensional rotor speed is denoted by ν, which is the rotor speed θ normalized by the mean
rotor speed Ω; thus torsional vibrations are represented by ﬂuctuations in ν and the desired
constant rotor speed operating state is represented by ν = 1. The eﬀects of damping and
applied torques are included by their generalized forces, and are speciﬁed below. Using these
deﬁnitions, and following the steps of previous derivations [11, 14], the equation of motion
for a single absorber (N = 1) is found to be

(1 + δ + ξ)ν 2 s + (1 + δ + ξ)s +
+ (1 + δ)g(s) − 2 ξa − 2 δ
− ν2

∂xP (s)
ν ν+
∂s

2
∂yP (s)
1 ∂rP (s)
(1 + δ) + 2 δ
= − µs ν,
2 ∂s
∂s
25

(2.3)

where primes indicate derivatives with respect to θ, µ is an eﬀective viscous damping coefﬁcient along the absorber path, and δ, ξ, and

are the non-dimensional terms introduced

by the rollers. Table 2.1 provides deﬁnitions of these nondimensional roller parameters in
terms of physical variables, from which one can see that

is related to the placement of the

roller relative to the absorber (see Fig. 2.2), while δ and ξ are related to the roller mass
and moment of inertia, respectively. Note that these parameters are related to one another
in a manner that depends on the details of the roller’s inertial properties and placement on
the rotor. These three coeﬃcients capture the eﬀects of the rollers, which is the primary
subject of this work.2
The rotor equation of motion is similarly derived, and is found to be relatively complicated. It is therefore given here in expanded form, which to ﬁrst order in

νν

=

Γ sin(nθ) + Γ0 − µ0 ν −

is given by,

2
νν rP (s) +

2
2 ∂rP (s) s + ν 2 ∂g(s) s 2 + ν 2 g(s)s
+ νν g(s)s + ν
∂s
∂s

(2.4)
+O( 2 ),

¨
where νν is the rotor angular acceleration (the normalized value of θ with θ as the in¨
dependent variable, speciﬁcally, νν = θ/Ω2 [11]), Γ sin(nθ) is the order n component of
the ﬂuctuating torque (the source of the rotor torsional vibrations), Γ0 is the mean torque
acting on the rotor, µ0 ν is the equivalent viscous rotor bearing resistance torque, and the
remaining terms are the torque on the rotor resulting from the motion of the absorber (as
can be seen since they are proportional to the nondimensional absorber inertia, ). Details
2 Note that many designs require that > −0.5, which according to Fig. 2.2 means that
the roller COM is closer to the rotor center then the pendulum COM. The situation of
< −0.5, which is the conﬁguration shown in Fig. 2.2, is avoided because it can cause
out-of-plane instabilities in the absorber motion unless the absorber motion is restricted.
26

of this derivation and the relationships between physical system parameters and the nondimensional quantities (other than those associated with the rollers) can be found in [11,14],
and the full equation is given in Appendix A. It is worth noting that the eﬀects of the
rollers do not appear explicitly at this order, since they are order compared to the absorber
inertia, and therefore appear at order 2 in νν . However, the rollers have a signiﬁcant aﬀect
on the absorber response s, since the system is acting near resonance and the rollers aﬀect
the linear tuning. Finally, it should be noted that the assumptions that the nondimensional
Table 2.1: Relationship between the non-dimensional roller and physical system parameters.
δ
ξ
1 (m /m )
1 i /(m A2 )
1 − D/c
P
2 R P
2 R
2

damping parameters, torques, and absorber to rotor and roller to absorber inertia ratios, are
small, and thus scaled by , are consistent with parameter ranges found in most applications.

2.2.1

Formulation of the Absorber Path

In order to completely specify the equations of motion, the path of the absorber COM must
be prescribed. In practical implementations this path is dictated by the shape of the cutouts
shown in Fig. 2.2, which are designed to achieve desired absorber performance. After an
absorber path is speciﬁed, equations (2.3) and (2.4) can then be solved (by numerical or
perturbation methods) for the response of the absorber and rotor. As one might expect, the
shape of the path plays an important role in the system response and the eﬀectiveness of the
absorber in attenuating the rotor torsional vibrations, and several studies have considered
these eﬀects [5, 68]. As in previous work, we specify these paths using the two-parameter
family of curves introduced by Denman [17]. This class of paths is uniquely determined
27

by linear and nonlinear tuning parameters, as follows: we denote n as the linear absorber
˜
tuning order and λ as the nonlinear tuning parameter. The linearized absorber frequency
of oscillation for constant rotor speed Ω is given by nΩ in the dimensional case, or simply
˜
by the order n in the nondimensional case. The nonlinear tuning parameter controls the
˜
frequency-amplitude dependence of the absorber’s vibration response, that is, its hardening
or softening eﬀects. Both of these path parameters can be introduced into the equations of
motion through the pendulum’s position rP (s), as is now described. It should be noted that
in the following n is the tuning order and that n0 is the tuning order for the special case of
˜
˜
zero roller inertia, and it has a special geometric interpretation, described subsequently.
In order to specify the equations of motion in terms of the parameters n (or n0 ) and λ, we
˜
˜
need to formulate the pendulum position function rP (s) as a function of these parameters.
Denman [17] provides a detailed development of this formulation, the highlights of which are
presented here and in Appendix B. This path formulation is valid for a family of epicycloidal
paths described by nonlinear tuning values in the range 0 ≤ λ < 1, with limiting cases of
circles (λ = 0) and cycloids (λ = 1) [17, 24].
We ﬁrst consider the linear tuning problem. The linear tuning order without rollers n0
˜
is well known [17] and is dictated by the ratio of the distance from the eﬀective pendulum
pivot point to the rotor center, c − ρ0 , to the eﬀective pendulum length, ρ0 , speciﬁcally,

n0 =
˜

c − ρ0
=
ρ0

c
− 1.
ρ0

(2.5)

We will use n0 throughout the rest of this chapter as a convenience instead of using its full
˜
expression which is related to the path curvature ρ0 as well as the distance c as shown in
equation (5); however, it is important to note that only when considering the zero roller
28

inertia case does n0 correspond to the linear tuning parameter n.
˜
˜
In order to consider the entire path, including nonlinear eﬀects, one utilizes the nondimensional version of the general path expressions provided in Appendix B, speciﬁcally,

2
rP (s) = 1 +

2
2
λ2
2 + 2 λ − λe0
s
λ2 − 1
(λ2 − 1)2

∂x
1 − λ2 − P
e0
∂s

∂y
(1 − λ2 )2 − s2 λ2 − sλ2 P (2.6)
,
e0
∂s

where
∂yP
1
= sin
sin−1 λ(˜ 2 + 1)s
n0
∂s
λ

,

∂xP
1
= cos
sin−1 λ(˜ 2 + 1)s
n0
∂s
λ

,

˜0
λe0 = n0 / n2 + 1.
˜

(2.7)

Note that λe0 is a special zero-roller nonlinear tuning parameter described below. The reader
is reminded that n0 is a tuning parameter only for zero roller inertia, that is, for = 0.
˜
It is important to account for the rollers when selecting the tuning parameters n and λ,
˜
because improper path selection can lead to poor performance, and even instabilities in the
desired response [5]. To avoid these undesirable eﬀects, one typically takes n to be slightly
˜
larger than n (i.e., the absorber is slightly over-tuned linearly), and the nonlinear tuning
parameter is selected so that the path is neutral, λ = λe , or slightly stiﬀening, that is,
λe < λ ≤ 1, where λe is deﬁned below [5, 56, 68].

2.3

The Tautochronic Path with Rollers

It is known that if roller inertia is neglected the pendulum can be modeled as an equivalent
point mass, with its moment of inertia about its COM included with that of the rotor, since
29

in a biﬁlar suspension the absorber rotates identically with the rotor [17]. In that case there
is a special absorber path for which the absorber behaves like a linear oscillator, even for
large amplitude motions. This tautochronic, or equal time, path is known to be a particular
epicycloid [17], and it renders the absorber to have the same frequency, or, more precisely,
the same order, out to large amplitudes. The absorber amplitude in this case is limited
by the cusp on the epicycloidal path, which occurs at s = smax , at which g(smax ) = 0.
A previously unanswered question is whether or not such a tautochronic path exists when
roller inertia is accounted for in the model. In this section we show that such a path indeed
exists, and we derive its form.
To consider this problem, we take the absorber equation of motion (2.3) with zero damping (µ = 0) and a constant rotor speed (ν = 1, ν = 0), yielding

(1 + δ + ξ)s −

2
∂yP (s)
1 ∂rP (s)
(1 + δ) − 2 δ
= 0,
2 ∂s
∂s

(2.8)

2
where ∂rP /∂s is given by
2
∂rP

λ2 − λ2 ∂yP
λ2
=2
.
s + 2 e0
∂s
∂s
λ2 − 1
λ2 − 1

(2.9)

The goal here is to show that there exists a nonlinear tuning parameter λ = λe that renders
this as a linear oscillator with a ﬁxed order n for all amplitudes. It will be shown that this
˜
special path is described by an epicycloid in the two parameter family developed above.
Before turning to the general case, we ﬁrst consider the roller-free tautochrone, for which
δ = ξ = 0 in equation (2.8). In this case it is not diﬃcult to show that the choice of λ = λe0 =
2
n0 / n2 + 1 renders equation (2.9) as ∂rP /∂s = −2˜ 2 s, resulting in the linear oscillator
˜
˜0
n0
30

s + n2 s = 0 for equation (2.8). The attendant path function is rP 0 (s) =
˜0

1 − n2 s 2 ,
˜0

corresponding to the roller-free tautochronic epicycloid of order n0 .
˜
For the development that includes roller inertia it is convenient to express the equation
2
of motion (2.8) in a Taylor series in s. Using the given expressions for ∂yP /∂s and ∂rP /∂s,
expanding in s, and recognizing a pattern in the series, the equation of motion can be written
as

s + n2 s +
˜

χ
1+ δ+ ξ

∞

(˜ 2 + 1)s 2k+1 k
n0

k=1

(2k + 1)!

(2i + 1)2 λ2 − 1

= 0,

(2.10)

i=1

where χ = (1+ δ)(λ2 −λ2 )−2 δ (λ2 −1). It is immediately seen that a (linear) tautochronic
e0
absorber of order n is recovered for χ = 0.3 This condition results in the following simple
˜
expression for the tautochronic value of λ,

λe =

n2 (1 + α)
˜
,
1 + n2 (1 + α)
˜

(2.11)

where α = ξ/(1 + δ) contains the eﬀects from the rollers. Selection of this value for λ yields
the linear oscillator s + n2 s = 0, as desired. Note that although we have retained the
˜
(typically small) parameter

in this analysis, the results are exact and do not depend on

small roller inertia, or any other small parameters. The (small) correction in the nonlinear

3 One can also obtain the tautochrone condition by substituting equation (2.9) into equation (2.8) and realizing that tautochronic motion implies that terms proportional to ∂yP /∂s
must vanish. The present approach, however, is cleaner in terms of keeping conditions on
the tuning parameters separate.
31

tuning parameter due to roller inertia can be approximated by expanding in , which yields,

λ2 = λ2 −
e
e0

2 δ
1 + n2
˜0

+O( 2 ),

(2.12)

showing that λe may be larger or smaller than λe0 , depending on the sign of , since , δ > 0.

The linear tuning order n falls out directly from the coeﬃcient of s in the expansion and
˜
can be expressed in terms of n0 , which dictates the vertex curvature of the path (and the
˜
roller-free linear tuning) and the roller parameters, as follows,
n2 (1 + δ(1 − 2 )) − 2 δ
˜
n2 = 0
˜
.
1+ δ+ ξ

(2.13)

This expression, which is a nondimensional form of the linear tuning equation derived by
Denman [17], directly shows the inﬂuence of the rollers on the linear tuning parameter. To
achieve good absorber performance, n and n0 will diﬀer by at most a few percent, but this
˜
˜
diﬀerence is important since the system operates near resonance.

For absorber path design one chooses a value of n that gives the desired system response
˜
from the coupled rotor/absorber equations of motion, and the corresponding path is determined by the following steps. First, solve equation (2.13) for n0 in terms of n and the roller
˜
˜
parameters, yielding a special value denoted as n0e , which ﬁxes the path vertex curvature
˜
by equation (2.5). The absorber path is then generated by utilizing n0 = n0e and λ = λe in
˜
˜
equations (B.4) and (B.5) in Appendix B, and scaling up (xP (s), yP (s)) and s by the factor
c to obtain (XP (S), YP (S)), and taking −Smax < S < Smax . From this one can generate
the cutouts for the rollers shown in Fig. 2.2, which are given by 1 (XP (S), YP (S)) and placed
2
32

at coordinates given by D and H as shown in Fig. 2.2.4 We now turn to examples.
In the following comparisons of paths we assume that the distance c is ﬁxed, which is
typically the case, since a designer wants to maximize the absorber moment arm within the
allowable space. In designing a vibration absorber for a given application one must satisfy
constraints on absorber mass, absorber amplitude, and the absorber placement, all of which
are limited in some manner by c. Within these constraints one chooses an absorber path
that will give good performance and will avoid instabilities throughout the torque range of
interest. Typically, the required path is linearly overtuned and nonlinearly tuned between
an epicycloid and a cycloid, that is, is neutral or hardening (i.e. n ≥ n and λe ≤ λ ≤ 1) [68].
˜
Shown in Fig. 2.3 (a) are cycloidal, circular, and three epicycloidal paths, all with the
same linear tuning n. The epicycloidal paths are tautochronic paths for three values of the
˜
roller inertia, corresponding to δ = 0, 1/4, and 1/2. The cycloidal and circular paths have
δ = 0 and are shown to convey the boundaries of typical paths between the circular and
cycloidal paths. To generate the tautochronic paths, which are seen to be nearly identical,
we ﬁx the linear tuning n, and choose δ and ξ, and then solve equation (2.11) for λe . We also
˜
use the assumption of a solid roller which implies that δ = 2ξ for the tautochronic paths.
Note that while the three tautochronic paths appear to be nearly identical, they have slightly
diﬀerent vertex curvatures, as required to have the same n for diﬀerent roller inertias.
˜
Figure 2.3 (b) depicts how the path vertex curvature depends on n for three values of the
˜
roller inertia, with tuning near 1.5. Note that in order to achieve a speciﬁc linear tuning the
path curvature increases as the roller inertia increases, assuming all other parameters are
4 Note that achieving a speciﬁc geometry that provides the desired value of tuning is a
complicated iterative process since the shape of the cutouts, their placement, the location of
the absorber COM, the roller paths, and the tuning parameters are all interdependent. This
process can be automated in a computer program [24].
33

ﬁxed. Note that the roller inertia also aﬀects the value of the nonlinear tuning coeﬃcient
λe , as seen from equation (2.12), and that it may increase or decrease, depending on the
sign of .

(a)
x

(b)
Figure 2.3: Inﬂuence of roller inertia on the absorber COM path, for = 0.10 and = −0.25.
(a) Five sample paths: a circle, a cycloid, and three epicycloids, for n = 1.5. (b) Vertex
˜
curvature vs. n for three values of roller mass.
˜

34

2.4

Perturbation Analysis

Here we consider the dynamic response of the system, paying special attention to the eﬀects
of the rollers. We take a perturbation approach similar to previous work [5, 11, 70], here
incorporating the eﬀects of the relatively small rollers by inclusion of the terms involving δ,
ξ, and . The perturbation analysis will make use of a base system that has the tautochronic
path without rollers, represented by parameters n0 and λe0 . It is noted that in light of the
˜
existence of a tautochronic path with rollers, a perturbation could be performed about that
path instead of the roller-free tautochronic path. This was investigated and it was found that
the formulation is signiﬁcantly more complicated than the perturbation approach presented
here. One minor complication with the present approach is that one has to transform results
obtained for the values of n0 and λe0 into values of n and λe , using equations (2.13) and
˜
˜
(2.11), in order to obtain the desired path.

To obtain the equations in a form suitable for perturbation calculations, further expansions on equations (2.3) and (2.4) are carried out such that we obtain an oscillator equation
that governs the absorber dynamics, with the rotor coupling and roller corrections appearing
at order , along with the damping. This is possible due to the small inertia ratio, which
represents the rotor-absorber coupling, as described in [11, 68, 70]. To begin, we note from
equation (2.4) that the torques acting on the rotor are all of order , which results in the
rotor speed being nearly unity. It is then reasonable to assume the form ν = 1 + w + O( 2 ),
where w captures the torsional ﬂuctuations of the rotor. If we use this assumption, and the
fact that steady rotor oscillations require a balance of mean torque and rotor bearing resistance, stated as Γ0 − µ0 ν = O( ), we can use equation (2.4) to express the rotor acceleration
35

νν = w + O( 2 ), where

w = Γ sin(nθ) −

2
∂rP (s)
∂s

s −

∂g(s) 2
s − g(s)s .
∂s

We next expand the absorber equation of motion (2.3) about
expansion ν = 1 + w + O( 2 ), and keep only order
2
1 ∂rP (s)
=
s −
2 ∂s

(2.14)

= 0 and substitute the

terms, one obtains

2
∂yP (s) ξ ∂rP (s)
2δ
−
− µs − (s + g(s))w
∂s
2 ∂s

+O( 2 ).

(2.15)

Equation (2.14) can then be substituted into equation (2.15), which uncouples the absorber
equation of motion from the rotor equation of motion. Note that this uncoupling process
results in an oscillator equation that governs the absorber dynamics, and it includes the
excitation acting on the absorber through the rotor acceleration, which is expressed to leading
order in terms of the applied torque and the absorber dynamics. To get the desired form of the
absorber equation, we introduce a nonlinear detuning parameter κ, substitute λ = λe0 + κ
2
into the absorber path derivatives (∂yP /∂s and ∂rP /∂s) in equation (2.15), and expand
about

= 0. This results in an oscillator that is weakly nonlinear, with weak forcing, and

weak damping, expressed in the form

s + n2 s = f (s, s , θ) + O( 2 ),
˜0

36

(2.16)

where

∂yP
|
+ ξ˜ 2 +
n0
∂s λe0
∂y
−2κ˜ 0 1 + n2 (1 + n2 )s − P |λ
n
˜0
˜0
+
∂s e0
∂g (s) 2
−[s + g0 (s)] Γ sin(nθ) + 2˜ 2 s s − 0 s + g0 (s)˜ 2 s ,
n0
n0
∂s

f (s, s , θ) = −µs + 2δ

(2.17)

and (∂yP /∂s)|λ is the path derivative (∂yP /∂s) evaluated at λ = λe0 . Also, the term
e0
g0 (s) =

˜0
1 − n2 (1 + n2 )s2 is the leading order part of g(s), that is, its
˜0

= 0 version,

or, equivalently, its roller-free tautochronic version. In addition, to obtain equation (2.17),
−˜ 2 s is substituted for s in equation (2.14), which is valid to leading order in , as can be
n0
seen from equation (2.16). Note that the substitution of λ = λe0 + κ gives the desired form
for equation (2.16), amenable to the method of averaging. It introduces a parameter κ in
the order

terms which controls the variation of the nonlinear tuning parameter λ from the

roller-free case. Speciﬁcally, when accounting for rollers in the averaged equations one must
select κ = (λ − λe0 )/ for the desired value of λ and λe0 (that is, n0 ). This, along with n,
˜
˜
allows one to investigate a two-parameter family of absorber paths near the perfectly tuned
tautochronic path, with one parameter each for linear and nonlinear tuning. See [68] for a
similar formulation of multi-absorber systems without rollers.

Finally, it is important to note that these equations of motion, while weakly nonlinear,
are not restricted to small amplitudes, that is, no amplitude expansions have been carried
out. This is a consequence of the tautochronic nature of the unperturbed absorber path,
and it results in equations that are valid for all physically realizable amplitudes, that is, up
to limits imposed by motion-limiting snubbers, or at most, the cusps on the epicycloidal
37

absorber path.

2.4.1

Application of the Method of Averaging

To obtain approximate solutions to the equations of motion, we apply the method of averaging. The absorber response is expressed in terms of polar coordinates of the form

s(θ) = Z(θ) cos nθ + ψ(θ)
s (θ) = −nZ(θ) sin nθ + ψ(θ) ,

(2.18)

where Z(θ) and ψ(θ) are the absorber amplitude and phase, which will, under conditions
of practical operation, be slowly varying functions of the rotor angle θ . This coordinate
transformation imposes a constraint on Z (θ) and ψ (θ), as follows

Z (θ) cos nθ + ψ(θ) −ψ (θ)Z(θ) sin nθ + ψ(θ) = 0.

(2.19)

Substitution of the transformation in equation (2.18) into equation (2.16) results in an equation with only ﬁrst order derivatives of Z(θ) and ψ(θ). This result and equation (2.19) are
linear in Z (θ) and ψ (θ) and are easily solved for (Z (θ), ψ (θ)). The resulting equations
each have terms of order

and a term with the factor (n2 − n2 ). This term results from
˜0

the fact that the solution to equation (2.16) with

= 0 is simply the linear solution with

frequency (order) n0 , whereas the coordinate transformation assumes a frequency (order) of
˜
n. The presence of the << 1 terms results in slowly varying amplitudes and phases, except
for the fast phase drift due to the diﬀerence between n and n0 . Since absorbers are eﬀective
˜
only when tuned near the excitation order, the diﬀerence between n and n0 is small, and so
˜
38

we deﬁne an excitation order detuning parameter ∆ via

∆ = n2 − n0 2 .
˜

(2.20)

With this assumption, the equations for Z (θ) and ψ (θ) become slowly varying, that is,
all terms are O( ), and can thus be averaged in θ over one period, 2π/n. This results in
autonomous equations for the average values of Z and ψ, which are denoted as z and φ.
The averaging in this problem is non-trivial due to the nature of terms like ∂yP /∂s and
g0 (s). Exact analytical expressions for the averages of these terms is not available, and so
some terms will be left in the form of integrals that are computed numerically. Approximate
solutions to these integrals can be computed using amplitude expansions in z, although we
do not pursue that here.
The results for the averaged equations for z and φ are given by

z =
φ =

−µzn
+ Γ cos(φ)F1 (z) +O( 2 )
n
2
zn

−2M (z) κ˜ 0
n

1 + n2 + δ
˜0

−Γ sin(φ)F2 (z)+

(2.21)
(2.22)

z
2˜ 0 (1 + n2 )3/2 κ − ∆ − n2 (ξ − 1) +
n
˜0
˜0
2
z3
˜0
+O( 2 ),
˜
˜0
n0
+ n2 n2 (3 + n2 ) − 3˜ 2 (1 + n2 )
8 0
+

where F1 , F2 , and M are integrals related to the path function, and are given in Appendix
C5 . It is important to note that the only restriction on the amplitude of these equations is

5 It is noted that the integrals F (z) and F (z) can be expressed as elliptic functions and
1
2
this is done in Chapter 4.
39

that imposed by the cusps on the path, speciﬁcally, |s| < smax , where g(smax ) = 0.

2.4.2

Steady-State Response

Now that the equations of motion have been averaged and reduced to two ﬁrst-order autonomous diﬀerential equations describing the slow evolution of the amplitude and phase,
one can investigate the steady-state response of the system in terms of amplitude, phase, and
stability. To do this we set z = 0 and φ = 0, denote the steady-state values as (¯, φ), solve
z ¯
¯
¯
¯
¯
equations (2.21) and (2.22) for cos(φ) and sin(φ), and use cos2 (φ) + sin2 (φ) = 1, resulting in
an equation relating the applied torque amplitude Γ to the steady-state absorber amplitude
z and the system parameters, as follows,
¯

Γ2 =

µ¯n
z
2F1 (¯)
z

2
+

1
z
¯
2˜ 0 (1 + n2 )3/2 κ − ∆ +
n
˜0
2 (¯) 2
F2 z

− n2 (ξ − 1) −2M (¯) κ˜ 0
˜0
z
n

1 + n2 + δ
˜0

z3
¯
+ n2 n2 (3 + n2 ) − 3˜ 2 (1 + n2 )
˜
˜0
n0
˜0
8 0

(2.23)
+

2
.

For plotting purposes it is convenient to choose a steady-state amplitude z and compute the
¯
applied torque (which involves evaluating the integrals in F1 , F2 , and M as a function of
z ). Once the amplitude and applied torque have been computed, the absorber steady-state
¯
¯
phase φ can be obtained using equations (2.21) and (2.22) set to zero.
It is also of interest to know the resulting torsional vibration amplitude of the rotor. A
good measure of this is the rotor angular acceleration νν = w + O( 2 ), which is conveniently captured by w . Therefore, one can obtain an approximation of the rotor vibration
amplitude from w using equation (2.14). Of particular interest is the amplitude of the order
40

n harmonic of this acceleration, since that is the order which the absorber is designed to
reduce. This component can be computed via the magnitude of the corresponding complex
Fourier coeﬃcient,
|w |n =

2π/n
n
e−inθ w (θ)dθ | .
|
π
0

(2.24)

The reader should note that the w (θ) used here is the reduced form of equation (2.14),
2
obtained by substitution of λ = λe0 + κ into ∂rP /∂s and g(s), which results in the following
expression for w (θ), after keeping only leading order terms in ,

∂g (s) 2
w (θ) = Γ sin(nθ) + 2˜ 2 ss − 0 s + n2 g0 (s)s,
n0
˜0
∂s

(2.25)

and where, as mentioned above, −˜ 2 s is substituted for s . Also, note that the eﬀects of
n0
the rollers are not explicit here, since they appear at higher order, but will aﬀect w (θ) via
their eﬀects on the absorber response. The steady-state response conditions for s and s are
substituted here, and the attendant non-dimensional steady-state rotor acceleration is given
to leading order by νν = w . Note that the absorber and rollers add rotational inertia,
and therefore their presence reduces torsional vibrations by a ﬂywheel eﬀect, even if they
don’t move along their paths, and this must be accounted for in order to assess the dynamic
eﬀectiveness of the absorbers [68]. We therefore compare the value of |w |n experienced
with the absorber free to move, as compared to its value with the absorber locked at its
vertex (s = s = 0). According to equation (2.25) the reference condition is given simply by
|w |n,ref = Γ and this is denoted as the “reference line” in the following section.

In order to assess the maximum torque range over which the absorbers can operate, it will
be useful to know the maximum allowable amplitude of the absorber motion. As discussed
41

above, this can be determined by solving g(smax ) = 0. For purposes of doing steady-state
analysis within this amplitude level, we will use the approximate cusp amplitude expression
derived by Denman [17], which assumes a roller-free order n0 tautochronic path. This
˜
approximation for the maximum absorber amplitude is given by

smax ≈ smax,0 =

1
n0
˜

1 + n2
˜0

,

(2.26)

which is exact in the roller-free case. Here it is assumed that the steady state absorber
amplitude must satisfy z < smax,0 . In the examples to follow we will be using a linear
¯
tuning of n0 ≈ 1.5, for which the cusp amplitude is approximately smax,0 = 0.37. This will
˜
be a suﬃcient approximation for demonstrating parameter trends in the allowable torque
range. In all cases we will compute steady-state responses for torque ranges up to the point
where z = 0.36. For further details on the path cusp points for general paths, see [17, 68].
¯
With expressions describing the steady-state response of the absorber and rotor in hand,
we now turn to a speciﬁc example of practical interest, which will emphasize how the rollers
aﬀect the steady-state system response.

2.4.3

Example

In this example we use the results derived above to investigate how the roller inertia, as
determined by the parameters δ and ξ (see Tab. 2.1), aﬀects the response of the system.
Our focus here will be on a single absorber that is designed to be exactly tuned with respect
to an excitation order of 1.5, that is, we take n = n = 1.5. As shown in Fig. 2.4, we will
˜
consider a 3 × 3 matrix of cases with three values of roller inertia and three paths, all of
which are close to, or exactly, tautochronic.
42

We make some simplifying assumptions in order to ﬁx some parameters. First, to simplify
the number of roller parameters we can relate ξ to δ by assuming a solid uniform roller, as is
typically used in practice. This assumption implies that iR = mR A2 /2, which corresponds
to δ = 2ξ. In addition, we ﬁx the absorber geometry throughout all comparisons by taking
= −0.25.

Figure 2.4: A table depicting the nine roller mass cases used in the simulations shown in
Figs. 2.5 and 2.6. The zero roller mass (δ = 0) case (Case x.1) is the same for all three Cases
I, II, and III. Whereas, the δ = 1/4 (Case x.2) and δ = 1/2 (Case x.3) roller mass Cases
will have diﬀerent linear and nonlinear tuning values as shown. Therefore, seven of the nine
total Cases have unique pairs of linear and nonlinear tuning values which is why only seven
response curves are shown in Fig. 2.5 with linetypes indicated in this table.

This investigation considers three values for the roller masses, which includes a range of
practical interest. All of the details for the nine roller mass cases are depicted in Fig. 2.4.
The ﬁrst value is zero roller mass (δ = 0), corresponding to linear tuning n = n0 and a
˜
˜
tautochronic path with λ = λe = λe0 ; these are labeled as Case x.1 in Fig. 2.4, where x
takes on I, II, or III. We also consider δ = 1/4 (Case x.2) and δ = 1/2 (Case x.3), for which
43

the linear tuning is n (> n0 ) and the tautochronic path has λe (> λe0 ). Each of the three
˜
˜
values of roller mass will be used for the following three cases: In Case I we ﬁx n0 = 1.5
˜
which then also ﬁxes λe0 ; this is the case in which the roller inertia is ignored, and thus the
zero roller mass path will be exactly tautochronic. In this case, we will be able to quantify
the eﬀects of ignoring roller mass in both the linear and nonlinear tuning. The reader should
realize that in Case I, the absorber COM paths for δ = 0, 1/4, and 1/2 are exactly the same
because the initial radius of curvature of the path ρ0 is ﬁxed for a ﬁxed n0 (see equation
˜
(2.5)) and λe0 is independent of roller mass (recall λe0 = n2 /(˜ 2 + 1)).
˜ 0 n0
In Case II, we ﬁx n = 1.5 and use the λ = λe0 corresponding to the roller-free linear
˜
tuning. This corresponds to the case in which the roller inertia is accounted for in the linear
tuning, but ignored for nonlinear tuning eﬀects. This was done in Denman [17], and was
standard practice, since it was not known until the present work that a tautochronic path
existed that accounts for the inertia of the rollers. As noted above, the initial radius of
curvature depends on the roller mass for ﬁxed values of n, c, and . So, in this situation the
˜
radius of curvature depends on the value of δ, both initially (ρ0 ) and continually up to the
cusps (λe0 ).
For Case III we ﬁx n = 1.5 and use the correct tautochronic path for rollers, for which
˜
λ = λe . Similar to Case II, the absorber paths have diﬀerent radii of curvature. In fact, the
initial radius of curvatures will be the same for Cases II and III since the linear tuning is
computed in the same way. However, Case III will have diﬀerent non-linear tuning, that is,
λe instead of λe0 .
Figure 2.5 shows the steady state absorber response amplitude z versus the applied torque
¯
amplitude Γ for the three values of roller inertia for each of the three cases described above
44

Figure 2.5: Steady-state absorber response amplitude z vs. applied torque amplitude Γ for
¯
the three Cases described in the text. The linetypes for the Case I responses are indicated
in the ﬁrst row of Fig. 2.4. The linetypes for the Case II responses are indicated in the
second row of Fig. 2.4. The linetypes for the Case III responses are indicated in the third
row of Fig. 2.4. Only one response line exists for Cases (x.1) because for δ = 0 the linear and
nonlinear tuning are equal for all three Cases. The ﬁnal three lines of the legend correspond
to results from simulations of equations (2.3) and (2.4). Only simulations of Case I is depicted
since all are very close to one another. Parameter values: = 0.10, µ = 0.30, = −0.25,
n = 1.5.

and in Fig. 2.4. Note that there are seven response curves, since for zero roller inertia,
n = n0 and λe = λe0 , and thus the response is the same for Cases (x.1) for x = I, II,
˜
˜
III (see the ﬁrst column (δ = 0) in Fig. 2.4). The line types for the seven response curves
are indicated in Fig. 2.4. Additionally, note that in plotting the steady-state amplitude
response in all Cases, except the simulated ones as shown in Fig. 2.5, the non-linear tuning
parameter λ is controlled using the parameter κ (as deﬁned in Section 2.4). Case I, in which
the rollers are ignored, is described by the ﬁrst row of Fig. 2.4, and these three response
curves in Fig. 2.5 show the largest deviation of the three Cases. The deviation present in
Case I shows a (desirable) hardening response as the roller mass is increased. Therefore, if
45

an absorber is designed with a tautochronic path that ignores the rollers, they will actually
gain some torque range before the cusps are reached, albeit at a slight reduction in absorber
eﬀectiveness (as shown in Fig. 2.6 and described below). For Cases II and III all plots are
virtually identical to the roller-free curve from Case I (i.e. Case I.1), which implies that
for the roller inertia values used here, if one accounts for the rollers in the linear tuning n,
˜
the nonlinear tautochronic roller correction is essentially inconsequential. The main point of
these results is that one need only account for roller inertia in the linear tuning.
Simulations of the non-linear equations of motion (using equations (2.3) and (2.4)) are
also shown in Fig. 2.5 to illustrate the accuracy of the averaged solutions. Note that only
Case I is simulated because the results are virtually indistinguishable in terms of amplitude,
only the torque range diﬀers.

Figure 2.6: Order n harmonic amplitude of the rotor response, |w |n , vs. applied torque
amplitude Γ for Case I. All absorber designs shown in the legend of Fig. 2.5 result in indistinguishable rotor response amplitudes, but diﬀerent torque ranges. Only a simulation of
the full nonlinear equations using δ = 0 (Cases x.1) is depicted since the other roller mass
values are indistinguishable. Parameter values: = 0.10, µ = 0.30, = −0.25, n = 1.5.
46

Figure 2.6 shows the amplitude of the order n harmonic of the rotor, |w |n , as it depends
on the level of applied torque Γ, along with the reference line. Here we show only the three
curves from Case I, since all these curves fall very close to the roller-free curve of Case I,
although they have diﬀerent end points, set by the cusp limits. This shows the eﬀectiveness
of the absorber at attenuating torsional vibration levels at this order. It is important to note
that if one ignores the eﬀects of the rollers, the resulting response predictions are conservative in terms of the torque range, since the rollers provide an increased hardening (that is,
increased eﬀective stiﬀness) for the absorber response. In terms of the rotor response, if the
absorber is slightly over-tuned using n0 with respect to the excitation order n, adding roller
˜
mass results in the absorber being slightly less eﬀective at attenuating torsional vibrations,
as is seen in Fig. 2.6. This is because the tuning order will increase, which moves the absorber
further away from the resonant excitation of order n. Lastly, simulation of the non-linear
equations of motion using equations (2.3) and (2.4) are shown in Fig. 2.6 to illustrate the
accuracy of the averaged equations. Note that only the δ = 0 case is presented here because
results from the other cases are virtually indistinguishable.

To compliment Figs. 2.5 and 2.6, simulations of the non-linear equations of motion are
displayed in Figs. 2.7 and 2.8. Note that in these plots the independent variable is the rotor
angle θ, playing the role of time. Shown in Fig. 2.7 is the steady-state absorber response,
normalized by the amplitude of the order n applied torque Γ. Figure 2.7 shows that the
normalized absorber amplitude speak /Γ decreases as the applied torque Γ increases, which
is a result of the non-linear hardening behavior of the absorber response, as shown in Fig. 2.5.
Shown in Fig. 2.8 is the normalized steady-state rotor acceleration, w /Γ, over the same θ
interval, for the applied torque levels given in Fig. 2.7, plus the case Γ = 0.40. Note that
47

0.5
Γ = 0.20
Γ = 0.70
Γ = 0.90

s
Γ

0

−0.5
3000

3001

3002

3003

3004

3005

3006

3007

3008

θ
Figure 2.7: Steady-state absorber response normalized by the applied torque amplitude Γ
vs. the rotor angle θ for δ = 1/2 from Case III (i.e. Case III.3). The results demonstrate the
hardening nature of the system. Parameter values: = 0.10, µ = 0.30, = −0.25, n = 1.5.
the rotor acceleration contains a few signiﬁcant harmonics, in fact, three, and their relative
contributions change as Γ increases. Recall that if the absorbers were locked there is only
an order n harmonic in the steady-state rotor acceleration, and therefore, the additional
harmonics are a result of the absorber dynamics and their nonlinear interactions with the
rotor. The higher harmonics are prominent since the order n harmonic is largely reduced by
the absorbers.

2.5

Conclusions

There are two main contributions of this investigation. First, a tautochronic absorber path
that accounts for roller inertia has been found. Second, a perturbation analysis of the
48

Γ
Γ
Γ
Γ

0.8
0.6
0.4

w′
Γ

= 0.20
= 0.40
= 0.70
= 0.90

0.2
0
−0.2
−0.4
−0.6
−0.8
3000

3001

3002

3003

3004

3005

3006

3007

3008

θ
Figure 2.8: Steady-state rotor acceleration w vs. rotor angle θ for δ = 1/2 from Case III
(i.e. Case III.3). Note that the rotor acceleration is composed of multiple harmonics and
the relative contribution of these harmonics depends on the torque amplitude. Parameter
values: = 0.10, µ = 0.30, = −0.25, n = 1.5.
equations of motion shows that the response of a system with a nearly tautochronic absorber
that includes roller inertia in the linear tuning parameter n is largely insensitive to the choice
˜
of the nonlinear tuning parameter λ. In fact, if the rollers are ignored altogether in the
absorber design, the analysis shows that the system can handle a greater torque range with
virtually no change in the vibration levels of the rotor. This is mostly due to the fact that
the rollers will aﬀect the linear tuning which results in a roller-free absorber design that will
be oﬀ its ideal tuning value. To a lesser degree, it is also because the rollers will provide a
slight hardening eﬀect due to their inﬂuence on the nonlinear tuning. However, it appears
that for the sizes of rollers typically in use, one needs to include their eﬀects only in the
linear tuning in order to achieve satisfactory performance.

49

Chapter 3
Transient Response of Forced
Nonlinear Oscillators

3.1

Introduction

Most studies of the response of nonlinear vibratory systems subjected to harmonic excitation
focus on steady state response. Typical investigations consider periodic responses, stability
and bifurcations of these, quasiperiodic responses, and chaotic responses. In this chapter we
consider the transient response of the Duﬃng equation subjected to harmonic excitation,
focusing on overshoot that occurs when the excitation is suddenly switched on. The Duﬃng
oscillator is an archetypical nonlinear system that arises in many engineering problems [5,
7, 20, 21, 29, 51, 61, 67]. There exists a large body of work on the steady state response and
its stability for this classical oscillator; see, for example, [51]. However, in certain design
scenarios it is useful to know the short-term transient dynamics of the response [40]. For
instance, one may have design constraints on the peak amplitude of a system response,
50

and the transient response amplitude may violate this constraint, even when the steady
state amplitude does not. Such an application motivated the present study, speciﬁcally, the
response of a vibration absorber that starts in an inactive state and is suddenly brought into
conditions for it to be active, leading to beating type transients. Another consideration is
that a transient can sometimes result in the system ending up in an undesirable steady state,
in situations where multiple stable steady state responses exist. For nonlinear systems one
can investigate such transient responses through numerical simulations, and in some cases
this is the only feasible approach. However, for simple systems, analytical results can be
derived and provide predictive results that are useful in design. Here we derive such results
for the Duﬃng oscillator.

The approach for investigating the transient response of a linear oscillator is straightforward, since one can express the exact solution of the diﬀerential equation governing
the system response in terms of the system and excitation parameters and initial conditions [28, 69, 74]. Another approach is to ﬁnd a constant of motion for the system that
relates the position, velocity, system parameters, and initial conditions, from which the peak
response can be determined [10, 36, 37]. Both approaches provide exact analytical results
for the transient response. However, the constant of motion approach typically does not
apply for nonautonomous systems. As for nonlinear systems, the direct solution approach
can be used only in a few special cases [8], whereas the latter approach can be applied to
a number of nonlinear systems [9, 18, 19, 60]. In addition, the constant of motion approach
is well suited for problems such as determination of the overshoot, in which the velocity is
known (zero in this case) at some desired system position (peak amplitude in this case).
A limitation with the constant of motion approach is that it is not directly applicable to
51

systems with time-dependent excitation. However, the method of averaging provides a mean
of removing this time dependence for a certain class of systems, so that a constant of motion
for the averaged system can be obtained [71]. Only a few treatments of transient response in
non-linear systems are known to the authors, see [15,20,25,32,39,42,64], and we know of no
published analytical results for the overshoot of the Duﬃng oscillator, as considered here.
The primary focus here is an analytical treatment of the transient overshoot problem of
the Duﬃng oscillator that allows for predictions. The main result is a closed form expression
for the overshoot for the case of zero damping when the system starts with zero initial
conditions. It is shown that this undamped result provides a useful, conservative bound for
system overshoot, one that is quite accurate for systems with light damping. The analysis is
aided by a convenient scaling of the position and time such that the three system parameters
(force magnitude, frequency detuning, and nonlinearity) are combined into a single parameter
χ governing the response of the scaled system. The results are compared against simulations
of the original non-linear oscillator and the averaged equations to demonstrate their accuracy.

3.2

The Model, Averaged Equations, and Scaling

The system of interest here is a single degree of freedom oscillator with small damping,
nonlinearity, and harmonic excitation, driven near resonance, expressed as

2
x + ω0 x =

F sin(ωt + φ) − ξx3 − µx
52

,

(3.1)

where is a small parameter. In order to apply the method of averaging, we use the standard
change of coordinates,

x(t) = a(t) sin (ωt + ψ(t)) ,

(3.2)

x (t) = a(t)ω cos (ωt + ψ(t)) ,

which will have slowly varying amplitude and phase when

<< 1 and the forcing is near

resonance, ω ≈ ωn . Constraining the response of equation (3.1) to be in the form (3.2),
2
deﬁning the frequency detuning as σ = ω 2 − ω0 , and averaging over one period of the
excitation, results in the following standard averaged equations

r =−
rΦ =

2ω
2ω

F sin(Φ) + µrω ,

(3.3)

3 3
ξr − F cos(Φ) − σr ,
4

(3.4)

where r is the averaged amplitude and Φ = ϕ − φ is the relative phase between the averaged
oscillator phase ϕ and the phase of the forcing. In addition, since the forcing phase φ is
constant we have replaced ϕ with Φ in equation (3.4) for convenience in the subsequent
analysis. The steady state response of equations (3.3) and (3.4) has been studied extensively.

We reduce the number of system parameters by scaling the dependent variable r and the
independent variable t as follows

t=

4ω
σ

τ,

r=2

F
σ

p,

(3.5)

where it is assumed that σ = 0. Utilizing this scaling in equations (3.3) and (3.4) results in
53

the rescaled averaged equations,

dp
= − sin(Φ) − Dp
dτ
dΦ
p
= 4χp3 − cos(Φ) − 2p,
dτ

(3.6)
(3.7)

where
3 F2
χ= ξ
2
σ3

and D = 2

µω
.
σ

(3.8)

Note that this scaling introduces a complication of the results exactly at resonance, where
σ = 0. However, it will be shown that this is not a limitation for the overshoot problem
since the results can be computed in the limit σ → 0. Also, it is important to note that
when σ < 0 both position and time of the scaled system become negative, since

> 0 and

ω > 0. In this case, the system is solved in reverse time, which may lead one to think the
system amplitude response will grow exponentially. However, the sign of the scaled damping
coeﬃcient depends on σ as well, such that solving reverse time will result in exponential
decay, as expected. In addition, the results presented subsequently are simpliﬁed by making
use of the inherent symmetry in the hardening vs. softening Duﬃng responses, in such a
manner that negative amplitude responses can be ignored. However, it will be shown that
this symmetry between softening vs. hardening Duﬃng oscillators does not carry over to
the phase response, and therefore we will need to consider the individual cases for the phase
response.

54

3.2.1

The Steady-State Response

The steady state response of the scaled system is obtained by letting dp/dτ = 0 and dϕ/dτ =
¯
0, which results in the following equations for the steady state amplitude p and phase Φ,
¯

(Dp)2 + (4χ¯3 − 2¯)2 = 1,
¯
p
p
¯
tan Φ =

−D
.
4χ¯2 − 2
p

(3.9)
(3.10)

It is seen that this scaling has reduced the system parameter dependence of the steady state
response from four (ξ, F , σ, µ) to two (χ, D). In general, one can solve equation (3.9)
analytically to obtain six roots governing the steady state amplitude p as a function of D
¯
and χ. However, for plotting the steady state response we use only the three positive roots,
since the other three are symmetric with respect to p = 0 and don’t provide any additional
¯
information about the steady state amplitude. These roots are given explicitly in Appendix
D. More speciﬁcally, the three ignored roots are negative, which for σ < 0 lead to a positive r
¯
(see equation (3.5)) that match the positive roots for p when σ > 0 case. It is also important
¯
to note that there is a symmetry in the steady state frequency responses of two Duﬃng
oscillators wherein one is hardening (ξ1 ) and the other is softening (ξ2 ), such that ξ1 = −ξ2 ,
with all other parameters ﬁxed. As seen by the deﬁnition of χ, the frequency response of
these two systems are symmetric about σ = 0, provided the other parameters are the same.
When considering the steady state phase response, this symmetry does not exist, that is,
¯
¯
Φ1 = Φ2 for σ1 = −σ2 , as can be seen from equation (3.10), where the steady state phase
depends on the sign of σ through the damping coeﬃcient D. Therefore, in the scaled system,
for a given value of the damping coeﬃcient D, one distinct steady state amplitude exists as
55

χ is swept, an example of which is shown in Fig. 3.1 (a). However, the steady state phase can
take one of two forms, depending on the sign of D, which is governed by σ; these are shown
in Fig. 3.1 (b) and (c). As expected, Fig. 3.1 (a), (b), & (c) exhibit the familiar bistable
region over which three steady state responses exist for a single value of χ, two of which
are stable. We refer to these in terms of “the lower branch” (labeled as A), “the middle
branch” (labeled as B), and “the upper branch” (labeled as C), designated by their relative
amplitudes. The bistable region is given by χ1 < χ < χ2 , which depends on the damping
coeﬃcient D. Explicit formulas for χ1 and χ2 are given in Appendix D. For the case of zero
damping (D = 0), the results simplify to χ1 = 0 and χ2 = 8/27.

3.2.2

The Averaged Equation Phase Portraits

The (p, Φ) phase portraits of the averaged equations are very useful for analyzing the transient trajectories of the Duﬃng equation, and the remainder of the chapter will make use
of this information. In fact, for many of the results we will focus on the undamped case,
D = 0, since this case yields analytical solutions that provide a good approximation, and a
useful bound, for lightly damped systems.
Figure 3.2 (a)-(c) show phase portraits of a lightly damped system (D = 0.30) for three
values of χ, displaying the three topologically distinct cases.1 Attendant sample transient
time responses are shown in Fig. 3.2 (d)-(f) for the initial conditions (p0 , Φ0 ) labeled as
IC#1, IC#2, and IC#3. Figure 3.2 (a) shows a sample phase portrait for χ < χ1 . For this
1 Note that we have not shown the case when D = 0 and χ ∈ (0, χ ), which is the case
1
of the lower branch A existing alone. However, as seen from Fig. 3.1 (a)-(c), this case will
be qualitatively similar to the case of χ < 0, which is shown in Fig. 3.2 (a), except that the
equilibrium point C is replaced with equilibrium A. Also note that this case will not exist
for D = 0.
56

(a)

(b)

(c)
Figure 3.1: Steady-state response of the scaled averaged system as a function of χ. (a)
Steady-state amplitude; (b) Steady-state phase for D = 0.30, relevant for σ > 0; (c) Steadystate phase for D = −0.30, relevant for σ < 0.

57

range of χ only the upper branch (C) exists. Figure 3.2 (d) depicts a sample transient time
response for this case with initial condition IC#1, speciﬁcally, with zero amplitude, p0 = 0.
Since p(τ ) describes the amplitude envelope of the oscillatory response with frequency near
ω, the oscillation in p demonstrates the beating nature of the x response. Note that p and x
both have a peak amplitude of ppeak , and that p decays to the steady state amplitude pC
¯
of branch (C). The more interesting phase portrait for the bistable case is shown in Fig. 3.2
(b), for a value of χ ∈ (χ1 , χ2 ). As depicted in the time responses shown in Fig. 3.2 (e)
& (f), depending on the initial conditions (p0 , Φ0 ), generic transient trajectories approach
either (A) or (C). The sample initial conditions shown have equal initial amplitudes and
diﬀerent initial phases, as indicated by IC#2 and IC#3 in Fig. 3.2. As usual, the basins of
attraction for these two stable outcomes are separated by the stable manifold of the saddle
point (B). Lastly, Fig. 3.2 (c) shows a sample phase portrait for χ > χ2 , for which C is the
only possible steady state.
We now turn to an analysis that oﬀers predictive results for the peak of the transient
response.

3.3

Percent Overshoot for the Undamped System

In this section an analytical result for the percent overshoot for the undamped Duﬃng
oscillator is derived. In general, the percent overshoot of a system is deﬁned as

% Overshoot =

pmax − pss
×100,
pss
58

(3.11)

where pmax is the peak amplitude during a transient response and pss is the resulting steady
state amplitude (given by one of the roots p). We begin by deriving an energy-like constant
¯
of motion for the undamped (D = 0) scaled averaged equations. This is a fourth order
polynomial in p whose roots dictate the important features of the transient response. These
roots are a function of the initial conditions (p0 , Φ0 ), the parameter χ, and the relative
phase of the response Φ. It will be shown that the peak amplitude of the response occurs
at a relative phase of either Φ = 0 or Φ = π, so that the roots can be expressed in terms
of the initial conditions and χ. One feature of interest will be the case where the transient
response approaches the saddle point B for χ ∈ (χ1 , χ2 ), since this will represent the basin
boundary in the bistable case.

3.3.1

Transient Response Amplitudes

The averaged equations (3.6) and (3.7) with D = 0 is conservative and has a constant of
motion that can be determined by combining equations (3.6) and (3.7) into a diﬀerential
equation for p(Φ). Integrating this exact equation leads to the following integral of motion,

−χp4 + p2 + p cos(Φ) = c1 ,

(3.12)

where c1 is an arbitrary constant determined by the initial conditions (p0 , Φ0 ) and χ. Equation (3.12) allows one to construct the phase portrait for the conservative averaged system in
the usual manner: Given initial conditions (p0 , Φ0 ) one can obtain c1 and then solve equation
(3.12) for phase plane trajectories in the form p(Φ) (or, simpler in this case Φ(p), since it has
only two branches). An example phase portrait is shown in Fig. 3.3 for χ = 0.066 ∈ (χ1 , χ2 ),
which is an undamped version of the phase portrait shown in Fig. 3.2 (b).
59

In general, equation (3.12) will have four roots for p that can be obtained exactly and will
generate the level curves shown in Fig. 3.3; these roots are presented in Appendix D. In order
to generate the level curves shown in Fig. 3.3, as well as the level curves when χ is outside
of the bistable region, one needs only three of the four roots. Furthermore, it turns out that
only two of these roots involve trajectories that include the peak amplitude about the steady
state equilibria, and therefore the other two roots are ignored for the present study. The two
roots governing the transient trajectories of interest are given in equations (F.1) and (F.2)
in Appendix F. Given a value of χ in the bistable region, and initial conditions (p0 , Φ0 ),
the root p1 (given in equation (F.1)) will generate a transient trajectory as a function of Φ
about the lower branch steady state equilibria A. On the other hand, for any value of χ and
initial conditions (p0 , Φ0 ), the root p2 (given in equation (F.2)) will generate a transient
trajectory as a function of Φ about the upper branch steady state equilibria C.
These results provide an analytical expression for the percent overshoot of the undamped
Duﬃng oscillator. For χ in the bistable region, the peak amplitude of transient trajectories
approaching branch A (branch C, respectively) can be determined by evaluating the root
p1 (p2 , respectively) at Φ = π (Φ = 0, respectively). Outside the bistable region, the
transient peak amplitude about the upper branch C can be determined by evaluating root
p2 at Φ = π when χ < χ1 = 0, and at Φ = 0 for χ > χ2 = 8/27.2 Note that for a negative
initial amplitude condition p0 < 0, corresponding to σ < 0 according to equation (3.5), the
determination of the peak amplitude pmax is slightly modiﬁed, the details for which are
given in Appendix E. To determine the percent overshoot, the steady state amplitudes for

2 Although we have not shown the undamped portraits when χ is outside the bistable
region, the interested reader can refer to the damped phase portraits shown in Fig. 3.2 (a)
& (c) to clarify where Φ is evaluated.
60

A and C are also required. These can be determined as a function of χ by solving equation
(3.9) for p, using the analytical expressions given in Appendix D (equations (D.1)-(D.2))
¯
evaluated at D = 0. From the peak and steady state amplitudes, the percent overshoot
can be computed in an explicit form. This result is rather cumbersome, due to the roots
involved, and is therefore not given explicitly, but it is straightforward to compute.

3.3.2

Basin Separation Trajectory

It is also of use to determine the location of the saddle point B (see equation (D.3)), since
this, along with the attendant transient trajectories, determines basin boundaries for steadystate ﬁxed points A and C. This information is useful if it is desired to have a system reach
a particular steady state when operating in the bistable region.
To begin, we ﬁrst derive the initial conditions that will result in a transient trajectory
on the stable or unstable manifolds of B when D = 0. This is done by evaluating the initial
conditions for equation (3.12) at the steady state amplitude p0 = pB and phase Φ0 = π,
resulting in the following condition on equation (F.3),

c1 (χ) = c1 (χ, pB , π)
=

(3.13)

9
1
4χ(3ν)2/3 + 1921/6 δν 1/3 ν 1/3 − 32/3 +
2
288χ3
+ 3χ2 16 − 18(3ν)1/3 + 9(3ν)2/3

61

,

where
√
ν = δ 3 − 9χ2 ,
δ = χ3/2

(3.14)

27χ − 8,

for χ ∈ (0, 8/27). One can now use c1 in the expressions for p1 and p2 to generate stable
and unstable manifolds, as shown in Fig. 3.4 (a)-(c). The trajectory of particular interest is
the stable manifold, which is obtained from c1 in the expression for p1 (equation (F.1)).
In general, these undamped basin boundaries provide a good approximation for the basin
boundaries of the lightly damped system near the saddle. These results can be used as
follows: Suppose that the steady state at A is desired for all initial phases, then the maximum
initial amplitude allowed is given by pcr = p1 (χ, 0, c1 ), as indicated by the horizontal lines
in Fig. 3.4 (a) & (b) for two diﬀerent values of χ. Note that for χ ∈ (4/27, 8/27), no such
pcr exists, since at χ = 4/27 the homoclinic orbit to B changes character such that all small
amplitude initial conditions lie in the basin of C, an example of which is shown in Fig. 3.4
(c).

3.3.3

Numerical Examples

In this section two examples are shown to illustrate the main overshoot results, and the
beneﬁts of introducing the parameter χ. Although these examples consider only the case of
zero initial amplitude (p0 = 0, Φ0 ∈ [0, 2π)), corresponding to c1 = 0, results for any initial
conditions can be obtained by varying c1 . The ﬁrst example compares the percent overshoot
for systems with three diﬀerent sets of original system parameters (F, ξ, σ) that all yield the
same value of χ, and thus the same percent overshoot. The second example shows how the
62

percent overshoot varies as a function of χ in the bistable region. Both examples show the
percent overshoot relative to the lower branch A, since this is the resulting steady state for
p0 = 0 when χ < 4/27. However, in the second example, for χ > 4/27 and χ < 0, the zero
amplitude initial conditions result in a percent overshoot relative to C.
For the case of zero initial conditions (c1 = 0) and zero damping (D = 0), the percent
overshoot is a function of χ only. Figure 3.5 (a)-(c) show simulations of the original equation
(3.1) for three diﬀerent sets of system parameter values that each give χ = 0.094. The
overshoot is calculated using the explicit expressions for the peak and steady state amplitudes
for response A in equation (3.11); these expressions are equation (D.1), from Appendix D for
the steady state amplitude, and equation (F.1) evaluated at Φ = π for the peak amplitude,
from Appendix F. The result for χ = 0.094 is a predicted percent overshoot of 115.8%. This
prediction is compared with the results from direct simulations of equation (3.1), which are
indicated in the ﬁgure caption, showing the accuracy of the results. Note that the diﬀerences
in system parameters result in diﬀerences in the amplitude scales and the transient beat
frequencies, but the percent overshoot is essentially identical in all three cases.
We now consider how the percent overshoot resulting from zero amplitude initial conditions depends on χ. The analytical results are shown in Fig. 3.6 (a), where the overshoot
relative to branch C is shown by the gray curves and those relative to branch A are shown
by the black curve. Note that the percent overshoot for a linear undamped system is 100%,
and there is transition at χ = 0 from percent overshoot values less than 100% to those larger
than 100%. Due to the nature of χ, this can be interpreted in various ways. For example,
the overshoot for a hardening system (ξ > 0) driven below resonance (σ < 0) will be less
than 100% and it can go only to A, while that for a softening system (ξ < 0) driven below
63

resonance (σ < 0) will be greater than 100% in its transition to A. Note that the maximum
possible overshoot is 173%, for systems, obtained just below χ = 4/27. The transition at
χ = 4/27 corresponds to the case when the zero amplitude initial trajectory is on the stable
manifold of B. For χ > 4/27, zero amplitude initial conditions result in transient trajectories
headed towards C, and in this case the percent overshoot is less than 100%, since the diﬀerence between the peak and steady state amplitude is smaller for upper branch trajectories.
Lastly, note that as χ → ±∞ the percent overshoot is asymptotic to the limiting value of
58.74%, which corresponds to the percent overshoot at resonance, σ = 0, for branch (C).
Figure 3.6 (b) shows a comparison of the analytical results (those shown in Fig. 3.6 (a))
and simulations of both the original Duﬃng equation (3.1) and the averaged equations (3.6)
and (3.7), for the case of zero amplitude initial conditions. These results demonstrate the
accuracy of the analytical method, and how it degrades as one increases the value of .

3.4

Conclusions

We have derived an approximate analytical method for predicting the overshoot for the
transient response of a weakly nonlinear Duﬃng oscillator that is harmonically forced near
resonance. The method makes use of the system’s averaged equations, which yield an integral
of motion in the undamped case, by which the system parameters and initial conditions can
be related to the amplitude and phase of the system response. The resulting equation for
the amplitude is a fourth order polynomial, which can be solved as function of the response
phase, system parameters, and initial conditions. This gives analytical expressions with
parameter dependence for the amplitude response, with useful information about the peak
amplitude and, for bistable systems, the ultimate steady state response achieved. A unique
64

scaling reduces the problem to two parameters, χ, that combines the force magnitude F , the
frequency detuning σ, and the nonlinearity ξ, and D, which involves damping. Thus, the
undamped results, which provide a useful bound for the overshoot in lightly damped systems,
depend on a single parameter. Of course, the presence of damping will alter these results. A
correction for small D has been computed and is given in Appendices G & H. However, the
payoﬀ is small in comparison with the extensive nature of these calculations and therefore it
seems to be more eﬃcient to simply obtain the damped overshoot by simulating the damped
average equations.

The results presented here are being used to predict overshoot for centrifugal pendulum
vibration absorbers, which, in some applications, are activated when suddenly subjected to
near resonant excitation [45–47,69]. These absorbers must be designed to not exceed speciﬁed
amplitudes, even during startup, and so transient predictions such as those derived here are
of great practical value. For the case of tautochronic absorbers [11, 14, 68, 70], the system
nonlinearities are much more complicated than the cubic term in the Duﬃng equation, yet
we have shown that their averaged equations are integrable in the undamped case, so that
similar predictive results can be achieved for that system [45–47]. Furthermore, this percent
overshoot approach has also been applied to the case of parametric resonance and it has
been found to produce simpler formulas for the overshoot than those given here. In fact, for
the case of near zero initial conditions (i.e. p0 ≈ 0) it is remarkably found that the percent
√
overshoot is simply 100( 2 − 1), and is therefore independent of the system parameters
contained. These results include experimental evidence of the constant overshoot for near
zero initial conditions, as well as an outline for computing the overshoot for non-zero initial
conditions [44]. These extensions suggest that the approach may be quite general for weakly
65

nonlinear systems of a more general form, and quite possibly for classes of conservative
strongly nonlinear systems with periodic excitation.

66

4
3
2
1
0
0

1

2

(a)

3

4

5

6

(b)

ppeak

0.6
2
1.5

C

p

0.4

1

0.2

0.5
0
0

2

4

6

0
0

IC#1

5

(c)

τ

10

15

(d)

3

3
C
IC#2

2

p

p

2

IC#3

1

0
0

A

5

τ
(e)

10

15

1

0
0

5

τ

10

15

(f)

Figure 3.2: Sample phase portraits and transient time traces for three values of χ, with
D = 0.30. (a) Phase portrait for χ = −1 < χ1 ; (b) Phase portrait for χ = 0.10 ∈ (χ1 , χ2 );
(c) Phase portrait for χ = 1 > χ2 ; (d) Transient time response for zero initial conditions
IC#1 for χ = −1; (e) Transient time response for χ = 0.10 and initial conditions IC#2;
(f) Transient time response for χ = 0.10 and initial conditions IC#3. Note that the initial
amplitudes p0 are the same for IC#2 and IC#3, but the diﬀerent initial phases Φ0 result in
diﬀerent steady state outcomes.
67

Figure 3.3: Phase plane plotted using level curves of equation (3.12) by varying c1 , for
χ = 0.066 and D = 0.

68

(a)

(b)

(c)
Figure 3.4: Stable and unstable manifolds of B for three values of χ. (a) χ = 0.02; (b)
χ = 0.10; (c) χ = 0.20. Figure 3.4 (a) and (b) show lines of constant amplitude pcr , which
indicate the maximum amplitude of initial conditions p0 such that the response will approach
the lower branch for all values of the initial phase Φ0 . The topology of the homoclinic orbit
about A changes from Fig. 3.4 (a),(b) to that of Fig. 3.4 (c) for χ > 4/27. It is seen that for
χ ∈ (4/27, 8/27), small amplitude initial conditions will result in a large amplitude steady
state C, for all initial phases.

69

(a)

(b)

(c)
Figure 3.5: Numerical simulations of equation (3.1) for = 0.03 and ω = 2, and three sets
of system parameter values that yield χ = 0.094, with a predicted overshoot of 115.8%. (a)
F = 1/2, σ = 2, ξ = 2, simulated percent overshoot = 116.6%; (b) F = 0.125, σ = −1,
ξ = −4, simulated percent overshoot = 115.3%; (c) F = 0.177, σ = 1, ξ = 1, simulated
percent overshoot = 116.1%.
70

% Overshoot

160
140

% O.S. (C)
% O.S. (A)

4
χ = 27

120
100
80

Bistable
Region

60
40

−0.2

0

χ

0.2

0.4

(a)

% Overshoot

160
140
120

Analytical % O.S.
Average Eqns.

100

Duﬃng ( = 0.03)

80

Duﬃng ( = 0.10)
Duﬃng ( = 0.20)

60
40

−0.2

0

χ

0.2

0.4

(b)
Figure 3.6: (a) The analytical results for percent overshoot versus χ for zero initial conditions.
The percent overshoot is computed relative to either the upper branch C, shown by the solid
gray curves, or the lower branch equilibria A, shown by the solid black curve, depending
on χ, as indicated in the inset. At χ = 4/27, zero initial conditions results in a trajectory
on the stable manifold of B. When χ < 0 or χ > 4/27, zero initial conditions results in
transient trajectories about equilibria C, as shown in Fig. 3.4 (c). Note that the phase at
which equation (F.2) is evaluated to obtain pmax for the percent overshoot is diﬀerent for
χ < 0 and χ > 4/27, as described in Section 3.3.1. (b) Comparison of the analytical solution
and simulations of both the Duﬃng oscillator (equation (3.1)) and the averaged equations
(equations (3.6) and (3.7)) for the case of zero initial conditions and diﬀerent values of .
The parameter values used in the simulations include: F = 1/2, σ = 1.75, ω = 2. The
nonlinearity ξ is varied from -4 to 7 to sweep the χ domain shown, and it is noted that the
percent overshoot is asymptotic to 58.74% as χ → ±∞ (i.e., as one approaches resonance,
σ → 0).
71

Chapter 4
Transient Dynamics of a Single CPVA
System, Theory

4.1

Introduction

In the present chapter we focus on the transient response characteristics of a single absorber
ﬁtted to a rotor that is subjected to engine-order torsional excitation. Absorbers with a wide
range of paths, including both circular and tautochronic (neither hardening nor softening
out to large amplitudes [11,68,70]), the most common types used in practice, are considered.
The results obtained are also applicable to the synchronous response of systems with multiple
identical absorbers. We derive an approximate analytical method for predicting the percent
overshoot of an absorber during startup, such as occurs during cylinder deactivation, which
leads to a nonlinear beating type of absorber response in the transition to steady-state. The
model is quite general and provides a relatively simple method for predicting the overshoot
when the absorber has zero damping and starts with zero initial conditions. The results
72

are found to accurately predict the transient overshoot observed in simulations of the full
nonlinear equations of motion (EOM) when damping is small (as is the case in practice).
Furthermore, we show in Chapter 5 that these overshoot results provide a useful upper bound
for transient absorber response, which is useful for design purposes.

The results are obtained by ﬁrst deriving the full nonlinear EOM for a single absorber
system and then scaling the equations in two ways that allow for the application of perturbation methods. The method of averaging yields equations that govern the slowly varying
amplitude and phase of the absorber response envelope. These equations have been used in
previous studies of steady-state response, and are used here to predict the transient absorber
amplitude response for a range of practical absorber paths. The prediction is based on properties of the conservative slow-ﬂow equations for zero absorber damping. Using the scaling
introduced in Chapter 3, these equations are rescaled to combine the excitation amplitude,
the diﬀerence in absorber and excitation frequencies, and the absorber path nonlinearity,
into a single system parameter that dictates the response. For absorber paths not neartautochronic, e.g., circles and cycloids, the problem reduces to the transient response of a
simple Duﬃng oscillator, which was the subject of Chapter 3. The approximate results obtained are shown to provide accurate predictions for the full EOM for a wide range of system
parameters and operating conditions. The results for undamped absorbers are expressed in
terms of either one or two nondimensional parameters, depending on the path type, while
the results for systems with damping require the simulation of the averaged equations, and
involve one more parameter.
73

4.2

System Modeling

The dynamic model for the CPVA system is based on the schematic shown in Fig. 4.1,
which is suﬃciently general to describe the two experimental absorber systems described in
Chapter 5. The pendulum absorber is modeled as a point-mass m located at the pendulum
center of mass (COM), which moves along a prescribed path on a rotor of inertia J that
is subjected to a harmonic, engine-order torque excitation. This system has two degrees of
freedom, described by the absorber arc-length S and the rotor/crank angle θ. Note that
absorber arc-length is used, as opposed to angular position, since it is more accommodating
to non-circular paths; see Fig. 4.1. A small scaling parameter, deﬁned to be the ratio of
absorber to rotor rotational inertia,

= mc2 /J, is used throughout this work in order

to scale small nondimensional parameters, which include damping, applied torques, and
absorber inertia; these assumptions are entirely consistent with parameters encountered in
practical applications.
We begin by deriving the non-dimensional EOM for the CPVA system model shown in
Fig. 4.1. Using a two-parameter formulation for the path along which the absorber rides relative to the rotor [17], the EOM can accommodate a family of paths such as those shown in
Fig. 4.1. The equations are scaled using in order to capture the leading order eﬀects of nonlinearity, damping, applied torque, and absorber-to-rotor coupling. Two diﬀerent parameter
scalings are used, depending on the absorber path type, since for near-tautochronic absorbers
the path nonlinearity is essentially zero. The averaged equations for these two types of systems, with zero damping, are then used to determine a constant of motion governing the
transient response, from which one can predict the percent overshoot. The physical assumptions used in obtaining both the full non-linear EOM, as well as the subsequent reduced
74

Figure 4.1: Depiction of the CPVA system model showing three possible paths for the
absorber COM. The outermost (circular) and innermost (cycloidal) paths bound the twoparameter family under consideration; the middle path corresponds to the tautochrone. The
paths are shown enlarged, and not to scale relative to the rotor, to exaggerate the diﬀerences
in the paths.

set of approximate equations include: (1) a rigid rotor; (2) small viscous damping in the
absorber, µ << 1 and rotor µ0 << 1; (3) single order harmonic torque, Γ sin(nθ); (4) the
2
˙
rotor spins at a nearly constant speed, θ ≈ Ω; (5) small inertia ratio, = mc << 1; (6)
J
applied torque is small (compared to the system kinetic energy), Γ << 1, Γ0 << 1; and (7)
the independent variable employed is the rotor angle θ rather than time t. For convenience,
these assumptions will be referred to by their respective numbers in the model development.
75

4.2.1

Equations of Motion

Only a brief summary of the derivation of the EOM for the system is presented here, since
this has been previously described in the literature, and these equations are similar to those
˙
derived in Chapter 21 . The absorber coordinate S and rotor speed θ are non-dimensionalized
using, respectively, the distance c from rotor center to absorber path vertex, s = S/c, and the
˙
mean rotor speed Ω, ν = θ/Ω. Also, the independent variable is switched from time t to θ,
in order convert the ﬂuctuating torque Tn sin(nθ) from a nonlinear term into a type of nonautonomous term, which facilitates the perturbation analysis. The resulting non-dimensional
EOM for the absorber (s) and the rotor (ν) are,

βνs + g(s) + α + βs

νν +

2
1 drP (s)
ν = − βµs ,
ν −
2 ds

g(s) + α (νν s + ν 2 s ) +
+

dg(s) 2 2
ν s +
ds

(4.1)

(4.2)

2
drP (s) 2
2
ν s + νν rP (s) + µ0 ν − Γ0 − Γ sin(nθ) = 0,
ds

where rP (s) is the radial distance from the center of the rotor to the absorber’s COM, µ is
the coeﬃcient of absorber viscous damping, µ0 is the viscous damping coeﬃcient in the rotor
bearing, Γ0 is the magnitude of the mean torque applied to the rotor, Γ is the magnitude
of the ﬂuctuating torque applied to the rotor, n is the excitation order, and the function
g(s) =

2
2
rP (s) − (drP /ds)2 /4 is a path dependent function, speciﬁcally, it is the cosine of

the angle between the local tangent to the path and a line normal to a radial line passing
1 The rollers are neglected in this derivation and pendulum-type absorbers are accommodated in addition to the biﬁlar conﬁguration.
76

through the rotor center and the path vertex. Coeﬃcients β and α are additional rotational
inertia terms related to the absorber geometry; these arise because the circular path absorbers
can be implemented as compound pendulums with a single point pivot suspension, in which
case their rotational inertia contributes to the absorber inertia. More common absorber
implementations use a biﬁlar (two point) suspension which constrains the absorber body
such that it only translates relative to the rotor. In the biﬁlar case, the point mass model
suﬃces, with the absorber rotational inertia bundled in with that of the rotor, J. The biﬁlar
suspension is able to easily accommodate both circular or non-circular absorber paths. For
example, a circular path absorber with a biﬁlar suspension has α = 0 and β = 1, whereas a
circular path absorber with a single point suspension has α = r2 /(cρ0 ) and β = 1 + (r/ρ0 )2 ,
where r is the radius of gyration of the pendulum about its center of mass [53], c is the
distance from the absorber path vertex to the center of rotation of the rotor (see Fig. 4.1) ,
and ρ0 is the initial radius of curvature of the absorber path at its vertex (see Fig. 4.1).

Clearly, the absorber and rotor responses are coupled. A perturbation study of the
system response is made convenient by uncoupling these EOM to leading order in

using

some approximations. The uncoupling is accomplished by using assumptions (4)-(6), as
follows. Brieﬂy, the applied ﬂuctuating torque Γ induces small speed ﬂuctuations about the
constant non-dimensional rotor speed, described by ν = 1 + w(θ). After substituting this
into equations (4.1) and (4.2) and expanding in , the rotor acceleration w , expressed in
expanded form, can be substituted into the absorber equation, which results in an oscillatortype equation governing the response of the absorber [49, 50]. This equation, which includes
the nonlinear eﬀects arising from the speciﬁed absorber path rP (s), as well as the eﬀects of
77

the rotor acceleration, is given by,
2
1 drP (s)
s −
= F (s, s , s , θ) + O( 2 ).
2β ds

(4.3)

where

F (s, s , s , θ) = −µs − (g(s) + α)/β + s

g(s) + α s +

Γ sin(nθ)−
2
dg(s) 2 drP (s)
s +
s
ds
ds

.

Note that the mean torque Γ0 and the rotor bearing resistance µ0 ν fall to higher order when
one assumes that they balance to order , i.e., µ0 ν − Γ0 = O( ). The next section describes
the absorber COM path formulation and how it aﬀects the absorber response.

4.2.2

The Absorber Path

The path followed by the absorber COM determines the form of its eﬀective restoring force,
speciﬁcally, by the component of the centrifugal load that is tangent to the absorber path.
2
In the absorber oscillator equation (4.3), this is described by the term drP (s)/ds. Here we
employ a two parameter family of paths, introduced by Denman [17], ranging from circles to
cycloids, that are described by the linear tuning order n and a nonlinear tuning parameter
˜
λ. Speciﬁcally, the local radius of curvature on the absorber path, normalized by c, is given
by ρ =

ρ2 − λ2 s2 where ρ0 = 1/(1 + n2 ) is the value at the vertex, which sets the linear
˜
0

tuning. This family of paths consists of epicycloids for λ ∈ (0, 1), with limiting cases of a
circle for λ = 0 and a cycloid for λ = 1 [17]. In this formulation the absorber restoring force
78

can be written, in an expanded form, as
2
drP (s)
1
= −2β n2 s + (λ2 − λ2 ) (1 + β n2 )3 s3 + O(s5 ) .
˜
˜
e
ds
3

(4.4)

As seen in equation (4.4), n (with β) sets the small amplitude, linear, free vibration natural
˜
frequency of the absorber. This linear tuning is a function of the absorber geometry, as
described above and shown in Fig. 4.1. The parameter λ sets the large amplitude, nonlinear,
free vibration response, which can be softening (λ < λe ), hardening (λ > λe ), or neutral
(λ = λe ), where λe = β n/ β n2 + 1 is a special value that results in a purely linear restoring
˜
˜
force, even at large amplitudes, thus realizing the tautochronic path [49, 50].
The middle absorber path shown in Fig. 4.1 is tautochronic, while the outermost and
innermost paths shown in Fig. 4.1 are circular (λ = 0) and cycloidal (λ = 1), respectively. It
is important to note that a CPVA system with a tautochronic path is nonlinear, due to the
inertial coupling to the rotor, described by terms of order

in equation (4.3). As described

below, for analysis by perturbations, the absorber amplitudes for paths that are nearly
tautochronic must be scaled diﬀerently than those for other paths in this family, in order
to introduce the dominant nonlinearity at leading order. Therefore, these scalings aren’t
speciﬁc for perturbation studies of an exact circular or an exact tautochronic path absorber,
but rather, they provide a good approximation of the response for all paths in the prescribed
family, speciﬁcally, for all epicycloids with λ ∈ (0, 1). The tautochronic path perturbation
results are valid for all feasible absorber amplitudes and capture the essential dynamics for
paths that are nearly tautochronic, that is, with λ ≈ λe ; here the dominant nonlinear eﬀects
arise from kinematic coupling with the rotor. The general path perturbation results capture
the essential dynamics for all paths that are not nearly tautochronic, although their scaling
79

limits the results to moderate amplitudes; for these paths the dominant nonlinear eﬀects
arise from the absorber path. These scalings are referred to herein, respectively, as neartautochronic path scaling and general path scaling. The approximate equations obtained for
these cases have a similar structure, as described below. For zero absorber damping, the
general path problem formulation can be expressed in terms of a single parameter, χ. The
near-tautochronic path problem can be expressed in terms of χ and an additional parameter,
γ, where the near-tautochronic results approach the general path results in the limit γ → 0.

4.2.3

Scaling and Asymptotic Analysis

Here we describe the scalings used for general path and near-tautochronic path absorbers.
The equations are scaled in such a way to extract to leading order the eﬀects of damping,
forcing, detuning from resonance, and nonlinearities. The method of averaging is employed
to obtain a set of ﬁrst-order diﬀerential equations governing the slowly varying amplitude
and phase of the absorber response. The derivations are only summarized here, since these
averaged equations and their steady-state responses have been investigated elsewhere for
both general paths [5, 26, 52, 53] and near-tautochronic paths [11, 14, 68, 70]. Here we focus
on an additional scaling of the previously studied averaged equations which further reduces
the number of system parameters, thus simplifying the transient analysis, and providing a
framework that describes the averaged equations for all paths in a uniﬁed manner.
To extract the desired eﬀects at order , one substitutes equation (4.4) into equation (4.3)
and employs the scalings shown in Tab. 4.1 for the two types of paths, where the subscript c is
used for terms in the general path scaling (c stands for circular, since that is the most common
of these paths) and the subscript t is used for terms in the near-tautochronic path scaling.
80

The essential diﬀerence in these scalings is that the absorber amplitude must be scaled for
the general path in order to capture the cubic path nonlinearity to order . In contrast, for
near-tautochronic paths the absorber response is nearly linear out to large amplitudes, and
thus the amplitude does not need to be scaled. Here we introduce a parameter κ to allow
the path to deviate slightly from exact tautochronicity. In all cases the absorber is tuned
such that it addresses the order n excitation, and thus we introduce the (small) detuning
parameters σi (i = c, t), which describe the diﬀerence between n and n, as employed in the
˜
averaging process. Explicit expressions for the σi are found below in equations (4.13) and
(4.15). As shown in Tab. 4.1, the general path case requires an adjustment to the scaling of
the torque so it will be retained at order . Finally, for the case of a compound pendulum
circular path absorber, the torque scaling also involves absorber geometry and inertia terms
β and α.
Table 4.1: Parameter scalings for general and near-tautochronic paths.
Type of Path
General Path
Near Tautochronic
√
Amplitude
s=
rc
s = rt
Path
λ
λ = λe + κ
√
Torque
Γ=
Γc β/(1 + α)
Γ = Γt
Damping
µ
µ

Expanding the absorber equations in

and retaining order

terms yields an absorber

oscillator equation with weak damping, forcing, and nonlinearity, in the following universal
form
ri + n2 ri = fi (ri , ri , θ) + O( 2 ),
˜

(4.5)

where subscript i = c, t indicates the path type, and the expressions for fi (ri , ri , θ) are given
by
2
3 − Γ sin(nθ) − µr + (1 + α) r ,
fc (rc , rc , θ) = −ξc rc
c
c
c
β
81

(4.6)

for the general path, where ξc is the eﬀective path nonlinearity coeﬃcient given explicitly
below in equation (4.14). For the near-tautochronic path scaling,

ft (rt , rt , θ)

κ
3
= −µrt − n(1 + n2 )5/2 rt +
˜
˜
3
n
−[rt + g0 (rt )] Γt sin(nθ) + 2˜ 2 rt rt −

where g0 (rt ) =

(4.7)
∂g0 (rt ) 2
r − g0 (rt )rt
∂rt t

1 − n2 (1 + n2 )rt is the leading order part of g(s), that is, its
˜
˜ 2

,

= 0

version2 . The perturbation function fi (ri , ri , θ) and its subsequent averaged form are well
known for both the general path and the near-tautochronic path systems [5, 11, 14, 26, 52, 53,
68, 70]. Here we make a small adjustment to the previous approach, before averaging, which
involves extracting the linear detuning term ri from the right hand side of equations (4.6)
and (4.7) and rewriting equation (4.8) as

ri + n2 ri = ˆfi (ri , ri , θ) + O(ˆ2 ),
ˆ

(4.8)

√
where n = n/ 1 − Λ , ˆ = /(1 − Λ ), Λ = (1 + α)2 /β, and we employ averaging on
ˆ
˜
this form of the oscillator. For averaging, one makes the assumption of a nearly harmonic
response at the forcing frequency via the coordinate transformation [ri , ri ] = [zi (θi ) sin(nθi +
ψi (θi )), zi (θi )n cos(nθi + ψi (θi ))], and averages over one period of the forcing, 2π/n. This
results in two ﬁrst order diﬀerential equations governing the slowly varying amplitude ai
and phase φi , where (ai , φi ) are the averages of (zi , ψi ) over one period. For brevity, these

2 The near tautochronic perturbation function f (r , r , θ) is the same as that obtained in
t t t
equation (2.17) of Chapter 2, except, the rollers are neglected here and the path derivative
∂yP /∂s is expanded in amplitude to extract up to the cubic Duﬃng nonlinearity term
3
(1/3)κ˜ (1 + n2 )5/2 rt .
n
˜
82

averaged equations are not presented explicitly here, but a scaled version of them is shown
next, and the interested reader can ﬁnd intermediate results in the papers cited above.

One can signiﬁcantly simplify the analysis of the averaged equations by scaling the absorber amplitude ai and independent variable (the rotor angle) θi by the system parameters
in the following way [48],

θi =

4n
ˆσi

ai = −2

Θi ,

Γi
σi

pi ,

(4.9)

for σi = 0. This scaling results in the following averaged equations that are valid for both
classes of paths,

dpi
i
= − sin(φi )F1 (pi ) − Di pi
dΘi

(4.10)

dφi
i
= 4χi p3 − cos(φi )F2 (pi ) − 2pi ,
i
dΘi

(4.11)

Γ2
3
i ,
χi = ξi
3
2
σi

pi

(4.12)

where
Di = 2

nµ
,
σi

and σi and ξi are the detuning and the eﬀective path nonlinearity of the respective oscillators,
respectively, which are given by

σc =
ξc =

n2 − n2
˜

(1 + α)2 2
−
n ,
˜
β

1
1 + β n2 2 λ2 + (λ2 − 1)β n2 ,
˜
˜
6β
83

(4.13)
(4.14)

for the general path, and

σt =

n2 − n2
˜

− n2 ,
˜

(4.15)

ξt = n2 n2 (1 + n2 /3) − n2 (1 + n2 ) +
˜
˜
˜
˜

(4.16)

κ
+ n 1 + n2 1 + n2 (2 + n2 ) ,
˜
˜
˜
˜
3
i
i
for the near-tautochronic path. The functions F1 and F2 are a primary diﬀerence in the equac
c
tions for the two types of paths. For general paths, F1 = F2 = 1 and one recovers a scaled
version of the well-known averaged Duﬃng equation for the case of weak nonlinearity, damping, and forcing near resonance which is investigated in Chapter 3. For near-tautochronic
t
t
paths F1 (pt ) and F2 (pt ) are based on elliptic functions and are given by
t
F1 =

4
3π

t
F2 =

4
3π

1

1
+ 1 EE (γp2 ) + 1 −
EK (γp2 ) ,
t
t
2
γpt
1
1
2−
− 1 EK (γp2 ) ,
EE (γp2 ) +
t
t
2
γpt
γp2
t
γp2
t

(4.17)

(4.18)

where
4˜ 2 (1 + n2 )Γ2
n
˜
t
γ=
2
σt

(4.19)

is the additional parameter introduced by the near-tautochronic scaling, and in the limit
t t
γ → 0, F1 , F2 → 1. The functions EK (γp2 ) and EE (γp2 ) are the complete elliptic integrals
t
t
of the ﬁrst and second kind, which depend on the absorber amplitude pt and the parameter
γ. These terms arise since the rotor acceleration has a diminished eﬀect on how it drives the
absorber at large amplitudes along the path, thus providing an amplitude-dependent eﬀective
84

forcing. In fact, parameter γ describes an eﬀective hardening of the system response, in the
sense that the excitation has a reduced eﬀect at large amplitudes, as described in more detail
below. For the general path case, the path nonlinearity dominates these eﬀects, which are
scaled out by the amplitude expansion.

The scaling given in equation (4.9) reduces the parameter dependence of the averaged
equations from four parameters (ξi , Γi , σi , µ) to two parameters (χc , Dc ) for the general
path and to three parameters (χt , Dt , γ) for the near-tautochronic path. For the undamped
system Di = 0, the general path averaged equations depend on a single parameter χc ,
making it particularly attractive for analysis.

In order to compute the percent overshoot of the absorber transient response, the steadystate response is needed. The steady-state response amplitude pi can be computed from
¯
equations (4.10) and (4.11) by setting dpi /dτi = dφi /dτi = 0 and using sin2 (φi )+cos2 (φi ) =
1, which results in the following implicit expression for pi ,
¯
4χi p3 − 2¯i 2
¯i
p
Di pi 2
¯
+
= 1.
i
i
F1 (pi )
F2 (pi )

(4.20)

For the general path, equation (4.20) is a cubic in p2 and can be solved explicitly [48]. For
¯c
t
the near-tautochronic paths, pt is numerically obtained, due to the elliptic functions F1 and
¯
t
F2 .
85

4.3

Analytical Predictions of Transient Amplitudes

The percent overshoot of a system is deﬁned as

P.O. =

pmax − p
¯
×100,
p
¯

(4.21)

where pmax is the peak amplitude achieved during the transient response and p is the steady¯
state amplitude. For this system the transient is a beating type of motion, composed roughly
of harmonic components at orders n and n, with the latter shifted by amplitude-dependent
˜
nonlinear eﬀects. For a linear system this overshoot can be computed exactly [28, 69]. For
the undamped nonlinear system the overshoot can be computed by using a constant of
motion for the averaged equations, providing a useful bound for systems with weak damping.
With this constant of motion, and the previous steady-state amplitude results, the percent
overshoot approximation can be constructed for both path types. This approach will be
referred to as the pseudo-energy method. In the present study we focus on the case of
zero initial conditions, since it is a close approximation to the absorber response in variable
displacement automotive engines when cylinders deactivate. However, the results easily
generalize to arbitrary initial conditions. In this section, we show results for the zero initial
condition overshoot problem using the pseudo-energy method. In the following section we
investigate the accuracy of the method by comparing it with simulations of the full EOM.
We also provide a comment on how the form of the input torque during cylinder deactivation
transition can eﬀect absorber overshoot.
86

4.3.1

The Pseudo Energy Method

For zero absorber damping (Di = 0), one combines equations (4.10) and (4.11) to obtain an
exact ﬁrst order diﬀerential equation,
i
−pi sin(φi )F1 (pi ) − Di p2
dpi
i .
=
3 − cos(φ )F i (p ) − 2p
dφi
4χi pi
i
i 2 i

(4.22)

which is integrated, resulting in the following constant of motion,

−χi p4 + p2 + Gi (pi ) cos(φi ) = c1 ,
i
i

(4.23)

t
where Gc (pc ) = pc , Gt (pt ) = pt F1 (pt ), and c1 is the constant of integration. Note that
for the general path this is a quartic (cubic) equation for c1 = 0 (c1 = 0), and for nearly
tautochronic paths the equation involves elliptic functions. Equation (4.23) allows one to
simply construct the phase portrait for the undamped averaged system in the usual way. A
thorough study of transient responses and overshoot using equation (4.23) for the general
path case, that is, for Duﬃng’s equation, is given in Chapter 3. Here we describe only those
results needed to compute the percent overshoot subject to zero initial conditions, that is,
for c1 = 0.
Only those absorber responses that reduce torsional vibrations are of interest, and for
the undamped system these satisfy φi = π, such that the absorber response is exactly
out of phase with the excitation. To obtain this response, and also avoid undesirable nonsynchronous responses [68], one over-tunes the absorbers by making n > n (σi < 0) with
˜
|˜ −n| << 1, which results in torsional vibration reduction. For hardening absorber systems,
n
ξi > 0 (χi < 0), the over-tuned system will have only a single order n steady-state response,
87

which is described by a stable equilibrium of the averaged equations [48]. On the other
hand, for softening absorber systems (ξi < 0), for 0 < χi < χi the over-tuned system is bistable and the lower amplitude steady-state equilibrium is the one that results in the desired
response with φi = π. At χi , the initial conditions (pi0 , φi0 ) result in a trajectory that lies
on the stable manifold of the saddle point, and for χi > χi the system transitions to the
upper branch steady-state response which is in phase with the excitation, φi = 0, resulting
in vibration ampliﬁcation. The jump parameter value, χi , is further described below.

A sample response with zero initial conditions that leads to the desired response using the
general path scaling is shown in the phase space (pc , φc ) in Fig. 4.2 (a), and its equivalent
s(θ) response is shown in Fig. 4.2 (b); this is for a damped system with parameters χc = 0.10,
ξc < 0, and Dc = 0.30. Note that p(Θ), obtained from simulating the averaged equations,
accurately captures the envelope of the beating behavior of s(θ). The transient response
of a system with a hardening path is identical to that in Fig. 4.2 (a) & (b), except that
the trajectory labeled as coming from the saddle point would come in from inﬁnity, since
the saddle doesn’t exist for χi < 0. For undamped systems, the beating response continues
indeﬁnitely (it is motion on a torus), and the averaged equation response will follow a closed
curve around the target steady-state equilibrium. For small damping, the peak amplitude
should be well approximated by the undamped value, unless the beat period is long compared
to the envelope settling time. Using the pseudo-energy method, a closed form result for
the percent overshoot of general path absorbers is possible because both the steady-state
amplitude (equation (4.20)) and the peak transient amplitude (equation (4.23)) can be solved
exactly. In fact, because of the parameter scaling employed, this result depends on the single
parameter χc [48]. For near-tautochronic paths, a closed form expression is not available,
88

due to the presence of elliptic functions, but the percent overshoot can be computed in a
straightforward manner using numerical root-ﬁnding techniques.

(a)

(b)
Figure 4.2: A damped zero initial condition transient trajectory, IC #1, obtained from
simulations of the general path averaged equations, indicating the peak and steady-state
amplitudes; (a) shown in the phase space, and (b) as the absorber response versus θ. System
parameter values: Dc = 0.30 and χc = 0.10. Also shown in (a) is the trajectory from the
unstable manifold of the saddle that tends toward the desired steady state.

To demonstrate the utility of the general path scaling, shown in Fig. 4.3 (a)-(c) is the
transient absorber response obtained from simulations of the full EOM for three undamped
(µ = 0) systems using an inertia ratio of

= 0.03, excitation order n = 1.5 and three
89

diﬀerent sets of system parameters n, λ, and Γ, as indicated in the caption. These ﬁve
˜
system parameters specify the averaged system parameters σc , ξc , and Γc . For all three
systems χc = 0.112, for which the pseudo-energy method gives an overshoot prediction of
122%, and all three simulated system overshoots are very near the predicted value. Note
that the diﬀerences in system parameters result in diﬀerences in the amplitude scales and
the transient beat frequencies, but the percent overshoot is essentially identical in all three
cases. Lastly, we note that the two noncircular path cases (λ = 0.1 and 0.2) are not typical
for applications, but serve the purpose of demonstrating the accuracy of the prediction in
its most sensitive regime, which is the softening range of χc > 0, as shown in Fig. 4.4 (a).

The percent overshoot for zero initial conditions computed using the pseudo-energy
method for D = 0 and the averaged equations for D = 0 for both general path and neartautochronic path absorbers are shown in Fig. 4.4 (a) & (b), respectively. These ﬁgures
depict how the parameters χi , γ, and Di aﬀect the overshoot for the two types of paths. For
an over-tuned absorber (σi < 0), the overshoot for absorbers with hardening paths occurs
for χi < 0 and with softening paths it occurs for χi > 0. Note that the general path scaling
captures the zero initial condition overshoot of an undamped linear system (ξc = 0 and
χc = 0), which is 100%, and the absorber path in this case is the tautochrone, λ = λe . On
the other hand, due to the diﬀerent parameter deﬁnitions, for the near-tautochronic scaling,
λ = λe corresponds to κ = 0 and makes ξt small and negative, that is, nonzero, in a manner that depends on n and n (see equation (4.16)). And, although ξt is softening for the
˜
tautochronic path, the parameter γ results in a hardening eﬀect, as shown in Fig. 4.4 (b),
and thus these eﬀects compete in a manner that depends on the system parameters Γt and
σt (see equation (4.19)). One interpretation of Fig. 4.4 (a) & (b) is, that for a ﬁxed tuning
90

(a)

(b)

(c)
Figure 4.3: Simulations of the full EOM (equations (4.1) and (4.2)) for = 0.03 and n = 1.5,
and three sets of system parameter values that yield χc = 0.112, with a predicted overshoot
of 122%. (a) Γc = 1.171, n = 1.52 (σc = −4.263), λ = 0 (ξc = −4.22), simulated percent
˜
overshoot = 119%; (b) Γc = 0.799, n = 1.51 (σc = −3.253), λ = 0.1 (ξc = −4.03), simulated
˜
percent overshoot = 121%; (c) Γc = 0.477, n = 1.5 (σc = −2.25), λ = 0.2 (ξc = −3.732),
˜
simulated percent overshoot = 124%. Additional parameters used to simulate the full EOM
include: Γ0 = Γ/2, α = 0, and β = 1.

91

σi , and a ﬁxed excitation amplitude Γi , increasing (decreasing) the nonlinearity from the
linear system (ξi = 0) will result in a decrease (increase) in the overshoot. The softening
paths will eventually reach the jump point at χ = χ , which is the last overshoot point
shown on all seven curves for χi > 0 in Fig. 4.4. Therefore, for zero initial conditions, one
can see how the damping and the scaling will aﬀect the location of the jump point. The
presence of damping reduces the absorber overshoot and extends the parameter range over
which the transient response from zero initial conditions results in the desired response. In
addition, for the near-tautochronic scaling, the parameter γ has a similar eﬀect of reducing
the system overshoot as the jump point is approached. However, this eﬀect diminishes as
the system moves in the opposite (hardening) direction, and eventually the γ = 0 overshoot
will converge to the γ = 0 overshoot curve as χt → −∞3 .

4.3.2

Ramping the Excitation Torque

At this point an important issue regarding the form of the excitation torque will be addressed.
It is observed in simulations of the full EOM that the sign of the torque immediately after
it is switched on (i.e., +Γ sin(nθ) or −Γ sin(nθ)) has a signiﬁcant eﬀect on the level of
absorber overshoot. While the steady-state amplitude of the absorber does not depend on
this sign, the peak amplitude of the ﬁrst transient pulse in the beat response does depend
on the manner in which the torque is initially applied. This is because a torque input
that is initially positive (negative) will cause a rapid, but temporary, increase (decrease)
in the rotor speed, and this will temporarily increase (decrease) the centrifugal restoring
3 We note that the near-tautochronic averaged equations can be evaluated in the limit
γ → 0, resulting in the same averaged equations obtained for the general path scaling,
except for the diﬀerences in deﬁnitions of the system parameters χt and χc .
92

% Overshoot

160
140

Pseudo Energy (Dc =0)
Avg. Eqns. Sims. (Dc =0.066)
Avg. Eqns. Sims. (Dc =0.132)

120
100
80
−0.2

−0.1

0

χc

0.1

0.2

0.1

0.2

(a)

% Overshoot

160
140
120

Pseudo Energy
(Dt =0, γ=0)
Pseudo Energy
(Dt =0, γ=1/4)
Pseudo Energy
(Dt =0, γ=1/2)
Avg. Eqns. Sims.
(Dt =0.066, γ=1/2)

100
80
−0.2

−0.1

0

χt

(b)
Figure 4.4: The percent overshoot of an absorber system subject to zero initial conditions,
computed using the pseudo-energy method for Di = 0, and using the averaged equations for
Di = 0; for (a) general path and (b) near-tautochronic absorbers.

93

force acting on the absorber mass, which results in a decrease (increase) in the absorber
overshoot. This situation is depicted in Fig. 4.5 (a)-(d), which shows the simulated rotor
and absorber response for sinusoidal inputs started with opposite signs, and the resulting
diﬀerence in rotor transient and absorber overshoot.

This eﬀect is not captured by our

(a)

(b)

(c)

(d)

Figure 4.5: The rotor and absorber response to a step input of sinusoidal torque. (a) Rotor
and (b) absorber response to input +Γ sin(nθ); (c) rotor and (d) absorber response to input
−Γ sin(nθ).

model, since the scaling employed assumes a constant mean rotor speed (ν = 1) in equation
(4.3), an assumption used to uncouple the absorber and rotor responses to leading order in .
Therefore, the perturbation analysis cannot capture this transient change in the mean rotor
speed. This diﬀerence in the overshoot is not observed in experiments [47], due to the fact
94

that the torque input is not an exact step input of a sinusoid, but, due to the electronics, has
a short transition period. Furthermore, this short transition in the torque input is even more
likely in automotive applications, since cylinder deactivation is not instantaneous. Therefore,
we minimize this dependence on the torque sign by quickly ramping up the sinusoidal torque
from zero to its full value. The duration of the ramp θr used in simulations is the minimum
crank angle duration that removes the torque sign dependence, and this is found to be onehalf a forcing cycle, θr = π/n. This ramp time is small compared to the time at which
the absorber peak amplitude is reached. Using this ramp, the diﬀerence in the absorber
overshoot is essentially eliminated; for example, for the case shown in Fig. 4.5 (b) & (d), the
ramped input results in overshoot values of 118.30% and 118.56% for negative and positive
torque, respectively.

4.4

Results and Discussion

We now compare results from the analytically predicted overshoot with those obtained from
simulations of the full EOM, equations (4.1) and (4.2). The analytical predictions for zero
absorber damping (µ = 0, Di = 0) are obtained using the pseudo-energy method and
those with non-zero absorber damping are obtained by simulating the averaged equations
(equations (4.10) and (4.11)). It is shown that the general path scaling provides an accurate
estimate of the overshoot for absorber systems throughout the entire two-parameter family
of paths when peak amplitudes are kept at moderate levels. In fact, it is noted that as the
nonlinear absorber path parameter λ increases from a circle towards a cycloid, the absorber
cusp amplitude smax decreases [17]. Therefore, for paths near tautochronic, the general path
pseudo-energy is found to provide the same accuracy as the near-tautochronic path pseudo95

energy when considering moderate absorber amplitudes riding on paths with λe ≤ λ ≤ 1.
However, the accuracy of the general path breaks down as the excitation and tuning order
increase, so that the absorber is pushed out near the cusp of the epicycloid, and the neartautochronic scaling is able to predict the overshoot for these cases. In all comparisons we
ﬁx the excitation order n = 1.5, which corresponds to the transient response from cylinder
deactivation to 3 cylinders, from, say 6 cylinders, assuming a four-stroke engine. Lastly, in
all simulations of the full EOM we set the mean torque Γ0 = Γ/2.4
Shown in Fig. 4.6 is a comparison for the general path pseudo-energy theory and simulations of the full EOM for the case of zero damping and a system inertia ratio of

= 0.03.

The three sets of simulations shown in Fig. 4.6 are obtained by individually sweeping the
three system parameters contained in χc , namely ξc , Γc , and σc . In the full EOM, the sweep
of ξc is performed by sweeping the nonlinear tuning parameter λ, the sweep of Γc is done
by sweeping the non-dimensional torque Γ, and the sweep of σc is performed by sweeping
the linear tuning order n. The exact values used in the three sweeps can be extracted from
˜
the information given in the caption of Fig. 4.6. The results show that the pseudo-energy
method gives a very accurate prediction of the system response, except for the sweep of σc ,
which shows some deviation in the simulation points closest to χ = 0 and χ = χ = 4/27.
For the simulations near χ = 0, the deviation is due to the fact the absorber is far from
resonance (as far as n = 1.58 which results in σc = −10.463), where it is not surprising that
˜
−3
the averaging method breaks down. Furthermore, recall that χc scales like σc , making
it relatively sensitive to changes in the detuning. For simulations near χ , the deviation is
due to the large absorber amplitudes that occur when the system is excited very close to
4 We note that the results shown here break down when the mean torque is Γ > 2Γ,
0
which is expected since both torques are scaled in the same manner.
96

resonance, as close as n = 1.501 at the simulation point closest to χ .5 This is in contrast
˜
to the Γ and λ sweeps, which were performed at a linear tuning of n = 1.51, resulting in
˜
moderate peak absorber amplitudes in all cases.

Dc = 0, Pseudo-energy

160

% Overshoot

180

Full EOM Sims., Sweep λ

140

Full EOM Sims., Sweep Γ
Full EOM Sims., Sweep n
˜

120
100
80
60
40
0

0.05

χc

0.1

0.15

Figure 4.6: Comparison of the general path pseudo-energy method with simulations of the
full EOM for µ = 0, n = 1.5, = 0.03, Γ0 = Γ/2, α = 0, and β = 1. Solid line is the
pseudo-energy prediction. Simulation data: Sweep of χc by varying λ from 0.80 to 0 with
Γ = 0.005, and σc = −3.25 (˜ = 1.51). Sweep of χc by varying Γ from 0.001 to 0.006
n
with ξc = −4.09 (λ = 0), and σc = −3.25 (˜ = 1.51). Sweep of χc by varying n from 1.58
n
˜
to 1.501 with Γ = 0.003 (note that in this case varying n results in sweeping ξc from -5 to
˜
-4 for a ﬁxed λ = 0).

Shown in Fig. 4.7 (a)-(b) is a comparison between the general path pseudo energy method
and simulations of the full EOM for the case of increasing the system inertia ratio . Note
that increasing has the same eﬀect as increasing the absolute magnitudes of the three main
system parameters, as seen from the caption of Fig. 4.7. The results of Fig. 4.7 indicate that
5 Recall that for general path absorbers, the absorber amplitude is scaled by 1/2 , and
the resulting perturbation equations break down for large amplitudes.
97

the full EOM overshoot results begin to deviate from the theory in a manner similar to that
seen for the detuning sweep shown in Fig. 4.6. As is standard in perturbation equations,
as one increases the small scaling parameter, the accuracy of the approximation diminishes.
In addition, increasing

will increase the torque magnitude Γ, and the magnitude of the

softening nonlinearity ξc which results in larger transient amplitudes, which violates the
moderate amplitude assumption used for these paths. This eﬀect becomes more prominent
near χ . Near χc = 0, only the detuning sweep deviates signiﬁcantly; this is expected,
since the absorber is tuned far from resonance (˜ = 1.58). On the other hand, there is
n
less deviation for the overshoot near χc = 0 for the full EOM sweeps of λ and Γ, because
the nonlinearity ξc and torque amplitude Γc tend to zero as χc → 0. Overall, the theory
provides a valuable prediction for the absorber overshoot up to ≈ 0.10, and the predictions
are generally conservative, except near the jump point.
Shown in Fig. 4.8 are results for lightly damped systems, using the general path averaged
equations for the theory and comparing with simulations of the full EOM. Two sets of
data are shown, for which the value of the damping ratio ζ is doubled from ζ = 0.001 to
ζ = 0.0026 . Note that only system parameters λ and Γ are swept, since the scaled damping
coeﬃcient Di , which is held constant for these parameter sweeps, depends on n and n (see
˜
equation (4.12)). The damping dependence on the excitation and tuning parameters is an
interesting consequence of the scaling, and for a linear system χc = 0, the damped overshoot
is remarkably only a function of the damping coeﬃcient Dc . Additionally, as one approaches
resonance, σc → 0, Dc → ∞, which recovers the zero overshoot result for a damped linear
system excited exactly at resonance. As seen in Fig. 4.8, the averaged equations provide an

6 The viscous damping coeﬃcient µ is related to the damping ratio ζ by µ = 2ζ n.
˜
98

DC = 0, Pseudo-energy

160

% Overshoot

180

Full EOM Sims., Sweep λ

140

Full EOM Sims., Sweep Γ
Full EOM Sims., Sweep n
˜

120
100
80
60
40
0

0.05

χc

0.1

0.15

(a)
DC = 0, Pseudo-energy

160

% Overshoot

180

Full EOM Sims., Sweep λ

140

Full EOM Sims., Sweep Γ
Full EOM Sims., Sweep n
˜

120
100
80
60
40
0

0.05

χc

0.1

0.15

(b)
Figure 4.7: Comparison of the general path pseudo-energy method with simulations of the
full EOM for µ = 0, n = 1.5, Γ0 = Γ/2, α = 0, and β = 1. Solid line is the pseudo-energy
prediction. (a) = 0.07 with simulation data: Sweep of χc by varying λ from 0.80 to 0
with Γ = 0.012, and σc = −2.68 (˜ = 1.51). Sweep of χc by varying Γ from 0.003 to
n
0.013 with ξc = −4.09 (λ = 0), and σc = −2.68 (˜ = 1.51). Sweep of χc by varying n
n
˜
from 1.58 to 1.501 with Γ = 0.01 (note that in this case varying n results in varying ξc from
˜
-5 to -4 for a ﬁxed λ = 0) (b) = 0.10 with simulation data: Sweep of χc by varying λ
from 0.80 to 0 with Γ = 0.019, and σc = −2.55 (˜ = 1.51). Sweep of χc by varying Γ
n
from 0.005 to 0.021 with ξc = −4.09 (λ = 0), and σc = −2.55 (˜ = 1.51). Sweep of χc
n
by varying n from 1.58 to 1.501 with Γ = 0.016 (note that in this case varying n results in
˜
˜
sweeping ξc from -5 to -4 for a ﬁxed λ = 0)

99

accurate prediction of the damped system overshoot, with the same limitations discussed
previously, plus the additional requirement of small damping, required since the damping
coeﬃcient µ is scaled by

170

for all paths.

DC = 0.05, Avg. Eqns

% Overshoot

160

DC = 0.10, Avg. Eqns

150

Full EOM Sim, Sweep λ
ζ = 0.001

140

Full EOM Sim, Sweep Γ
ζ = 0.001

130

Full EOM Sim, Sweep λ
ζ = 0.002

120
110

Full EOM Sim, Sweep Γ
ζ = 0.002

100
90
80
0

0.05

0.1

χc

0.15

0.2

Figure 4.8: Comparison of the general path pseudo-energy method with simulations of the
full EOM for Dc = 0.05 (ζ = 0.001) and Dc = 0.10 (ζ = 0.002). Simulation data: Sweep of
χc by varying λ from 0 to 0.80 with Γ = 0.013. Sweep of χc by varying Γ from 0.003 to
0.013 with ξc = −4.09 (λ = 0). Other parameter values used in simulations are: n = 1.51,
˜
n = 1.5, = 0.07, Γ0 = Γ/2, α = 0, and β = 1.

Shown in Fig. 4.9 (a) & (b) is a comparison of the near-tautochronic pseudo-energy results
and simulations of the full EOM for λ = λe and

= 0.07. Two pseudo-energy curves are

generated, obtained by sweeping n and Γ, since both χt and γ depend on these two system
˜
parameters. As seen in Fig. 4.9 (a) & (b), the near-tautochronic pseudo-energy method
100

provides a conservative estimate of the absorber overshoot. The results show the tautochronic
absorber to be almost linear, that is, with 100% overshoot nearly all the way up to the cusp
amplitudes. This near-linear behavior is a result of the fact that the path nonlinearity is zero
(by deﬁnition of the tautochrone), and the competition between the small softening value of
ξt for λ = λe and the mild hardening eﬀect of γ (which diminishes the eﬀect of the torque on
the absorber), resulting in an almost net neutral eﬀect. Therefore, the general path pseudoenergy method, which predicts 100% overshoot for the tautochronic path absorber, is as good
as the near-tautochronic pseudo-energy for a range of n values, in this case near 1.5. As the
˜
tuning order n and excitation order n increase, the magnitude of the softening nonlinearity
˜
ξt increases, with negligible change in the hardening parameter γ (for a slightly over-tuned
absorber, n > n) which eventually results in the general path failing to capture the amount
˜
of softening observed in the overshoot. In fact, this is observed at n = 1.51 when λ is
˜
very close but just below tautochronic λe , in this case the softening observed in simulations
of the full EOM overshoot is only accurately captured using the near-tautochronic scaling.
However, it seems that for practical non-circular paths (i.e. λe ≤ λ ≤ 1), the general path
scaling provides a good approximation of overshoot for slightly over-tuned systems up to an
excitation order of n = 3.

4.5

Conclusions

The results of this chapter provide a simple way to estimate the overshoot of centrifugal
vibration absorbers when the rotor is suddenly subjected to near-resonant torsional excitation. A simple, compact formulation for the percent overshoot of the absorber amplitude
is derived from the full equations of motion by a series of rescalings and approximations,
101

% Overshoot

160

Dt = 0, Pseudo Energy (Sweep Γ)
Full EOM Sims. (Sweep Γ)

140
120
100
80
0

0.05

0.1

χt

0.15

0.2

(a)

% Overshoot

160

Dt = 0, Pseudo Energy (Sweep n)
˜
Full EOM Sims. (Sweep n)
˜

140
120
100
80
0

0.05

0.1

χt

0.15

0.2

(b)
Figure 4.9: Comparison of the near-tautochronic path pseudo-energy method with simulations of the full EOM for λ = λe , µ = 0, n = 1.5, = 0.07, Γ0 = Γ/2, α = 0, and β = 1. (a)
Simulation data: Sweep of χt by varying Γ from 0.009 to 0.033 and σt = −2.68 (˜ = 1.51).
n
(b) Simulation data: Sweep of χt by varying n from 1.67 to 1.50 with Γ = 0.028.
˜
including the method of averaging. For the case when the dominant nonlinearity arises from
the absorber path, and damping is negligible, the formulation boils down to a convenient
102

closed form expression (speciﬁcally, roots of a cubic or quartic equation, depending on initial
conditions) that depends on a single parameter. In fact, this case reduces to the transient
response of a simple Duﬃng oscillator, considered in Chapter 3. For absorber systems where
the dominant nonlinearity arises from kinematic coupling to the rotor, the overshoot estimate for undamped systems depends on two parameters, and requires ﬁnding the root of an
equation that involves elliptic functions. For all path types, more accurate estimates that account for damping are available via simulation of the averaged equations. Simulations of the
full nonlinear equations of motion demonstrate the accuracy and utility of the approximate
results over a wide range of system parameters. From a design point of view, the undamped
system results are particularly convenient, since they are easy to compute and provide a
conservative estimate of the overshoot. These results will be of use for a variety of applications, including use in automotive engines that utilize cylinder deactivation. Experimental
veriﬁcation of these results are provided in Chapter 5.

103

Chapter 5
Transient Dynamics of a Single CPVA
System, Experiment

5.1

Introduction

The results of an experimental investigation of the transient dynamics of centrifugal pendulum vibration absorbers (CPVAs) are presented, and compared with the theoretical and
simulation results described in Chapter 4. Here we describe the experimental apparatus,
the methods used for determining system parameters, and the results of systematic experiments performed using the apparatus. The experiments conducted consist of responses due
to the instantaneous activation of near-resonant torsional excitation of the rotor, so that the
absorber response transitions from an essentially quiescent state to a periodic steady-state.
This mimics the conditions encountered during cylinder deactivation in variable displacement automotive engines ﬁtted with pendulum absorbers that are tuned for the torsional
excitation encountered in the reduced cylinder mode of operation. This input leads to a
104

beating-type transient motion, resulting in an overshoot of the absorber amplitude response
before it reaches steady state. The purpose of this chapter is to experimentally measure this
overshoot in order to verify the analytical predictions in Chapter 4.
We consider the response of a system consisting of a rotor and a single absorber. The
fully instrumented apparatus can provide prescribed torque inputs and measure the resulting responses of the absorber and rotor. The overshoot experiments are performed using
absorbers with two diﬀerent path types, circular and near-tautochronic. Both paths have a
softening nonlinearity when coupled to the rotor, one quite strong (circular path) and one
close to neutral (near-tautochronic). These diﬀerences will be useful for validating the model
equations of motion (EOM) and verifying the analytical overshoot predictions.
The apparatus absorbers used in this study have been used previously for studies of
steady state responses, which considered single and multi-absorber systems, and were used
to validate perturbation analytical results for general path absorbers [26, 53, 76] and neartautochronic path absorbers [70]. Previous studies considering the transient dynamics of
pendulum vibration absorbers include simulations for circular path absorbers [59] and analytical closed form results for damped linear absorbers [69], which closely approximate the
results for near-tautochronic absorbers. In this chapter we describe the ﬁrst known systematic experimental investigation of transient absorber response, and compare it with analytical
predictions.
The chapter begins with a description of the experimental test apparatus, whose main
components are explained in detail, along with a description of how the rig is used to perform
cylinder deactivation experiments. The processes for identifying the required experimental
system parameters, speciﬁcally, the absorber tuning order n, the absorber damping µ, the
˜
105

nonlinearity in the absorber path λ, and the system inertia ratio , are explained. With
these parameters in hand, comparisons between analysis, simulations, and experiments can
be carried out for a range of loading conditions. It is shown that the EOM provide a very
accurate prediction of the experimentally observed overshoot, that the damped averaged
equations also provide accurate, and mostly conservative, predictions, and that the pseudoenergy predictions provide an upper bound for the overshoot. The beneﬁts of the pseudoenergy results are that they are simple to obtain, are expressed in terms of minimal system
parameters, notably, χc , and are conservative, making them very convenient for accounting
for transient responses in absorber designs. One conclusion of this study is that the general
path model is suitable for most absorber paths, and the near-tautochronic results are required
only in special situations. In fact, for the experiments described here, the general path results
suﬃce for both absorbers.

5.2

The Experimental Apparatus

We begin by providing a description of the components of the experimental spin rig shown
in Fig. 5.1, and explain how the rig is used to perform the transient experiments. This basic
rig was described and characterized previously in works considering steady state absorber
responses [26, 52, 53, 70]. For the present studies the rig was upgraded with a new servo motor (C) and controller (B) and has undergone signiﬁcant software updates for motor control,
response readouts, and data processing. In addition, ﬂywheel weights were fabricated to
allow the inertia ratio of the system

to be adjusted [76]. Lastly, to study the gravitational

inﬂuence on the absorber dynamics, the rig has been modiﬁed so that it can be run horizontally, i.e., rotated by 90o from the orientation shown in Fig. 5.1. Gravity will have very little
106

Figure 5.1: The experimental apparatus, with components: (A) computer for system control
and data processing, (B) motor controller, (C) servo motor, (D) rotor encoder, (E) absorber
with encoder.
inﬂuence on the absorbers response at speeds when cylinder deactivation is likely to occur.
Therefore, all experiments described here are carried out with the test rig in the orientation
shown in Fig. 5.1. This eliminates the eﬀects of gravity and allows the experiments to be
run at low speeds, speciﬁcally, Ω ≈ 350 RPM.

5.2.1

System Components and Parameters

The experimental apparatus uses an electric motor and control system capable of applying
a user-speciﬁed torque to the rotor, including transient inputs, while maintaining a nearconstant rotation speed of the rotor. The rotor and absorbers are ﬁtted with encoders for
measuring their responses, and a current feedback measurement from the motor is utilized
to determine the instantaneous torque acting on the rotor. A primary advantage of this
apparatus is that it allows one to systematically vary the torque order n, and thus the
107

detuning between the input order and the absorber order, a parameter not easily changed
in engines and other rotors, since these orders are typically integers and ﬁxed by hardware.
The apparatus consists of ﬁve main components, labeled (A)-(E) in Fig. 5.1. Component
(A) is a computer running a LabVIEW program that generates the user speciﬁed harmonic
torque signal and sends it to the control box (B), which then sends the signal to the motor
(C), commanding it to perform the desired torque. The computer is also used for data
collection and processing. Note that the applied torque in an automotive engine depends on
the rotor angle θ (e.g., Tn sin(nθ)), rather than time, and therefore the harmonic torque signal
generated in LabVIEW uses feedback from encoder (D), which instantaneously measures the
rotor orientation. This encoder produces 1000 pulses per revolution and informs LabVIEW
how to increment the torque during rotor revolution. The torque signal contains a mean
component T0 , used to maintain a desired rotor speed Ω, plus a user-speciﬁed varying part,
which can be harmonic (Tn sin(nθ)), step, etc. The mean rotor speed is held close to constant
by making slight incremental adjustments to T0 , determined by an onboard slow-time PID
controller in LabVIEW. The mean torque T0 and ﬂuctuating torque Tf are related to their
non-dimensional counterparts Γ0,f by T0,f /(JΩ2 ) = Γ0,f , where J is the rotor inertia and
= mc2 /J is the (small) ratio of absorber rotational inertia to rotor inertia, expressed in
terms of the absorber mass m and the distance from the rotor center to the absorber zero
position c, as shown in Fig. 4.1.
The vibration absorbers are placed at position (E) in Fig. 5.1. Here we consider two
diﬀerent absorber types, one with a circular path and the other with a nearly tautochronic
path. This allows us to experimentally investigate the eﬀects of the path nonlinearity on the
absorber overshoot, and make comparisons with model predictions [46]. The absorber shown
108

in Fig. 5.2 (a) is a near-tautochronic path absorber with a biﬁlar suspension, as considered
in [70]. The tautochronic path is accomplished by suspending the absorber mass (the dark
gray block in Fig. 5.2 (a)) from two tempered blue spring steel bands that wrap around
aluminum cheeks that are machined with the required evolutes, so that the center of mass
follows the desired involute. In addition, to achieve the biﬁlar type suspension, there are
bearings at each point where the steel bands connect to the absorber, and these constrain the
absorber such that it only translates relative to the rotor, and does not induce any bending
in the bands. The absorber shown in Fig. 5.2 (b) is a circular path pendulum type absorber
suspended at a single pivot point, as considered in [53]. In contrast to the near-tautochronic
absorber, the circular path absorber is a compound pendulum absorber suspended by a
pin and needle bearing. Note that this absorber will translate and rotate relative to the
rotor, which aﬀects its eﬀective inertia, and thus its linear tuning order n, as explained
˜
below. A rotary encoder is attached to the absorbers (E) (only shown on the tautochronic
absorber in Fig. 5.2 (a)), which provides an angular measurement of the absorber position.
Note that although the path followed by the near-tautochronic absorber is not circular, the
angular measurement taken by the encoder using a pin on the absorber and an arm with a
slot on the encoder, varies almost linearly with the arc-length [43]. Therefore, the encoder
angular measurement is used to determine the position for both types of absorbers, with a
measurement resolution of about 0.25o .

5.2.2

Transient Testing Capabilities

The rig is capable of performing experiments which emulate cylinder deactivation events in
variable displacement automotive engines. Typically, half of the cylinders are shut down
109

(a)

(b)
Figure 5.2: Absorber pendulums: (a) near-tautochronic absorber (dark gray), suspended
by two steel bands that wrap around cheeks, generating the desired path, shown with an
encoder attached; (b) circular path absorber (inverted “T”), suspended by a pin and needle
bearing, as indicated.

during deactivation, which results in a torque order transition from 2n to n where, for a four
stroke N cylinder engine, n = N/4. This is accompanied by an increase in the amplitude
110

of the dominant harmonic torque Tn by a factor of about 2-3 over that experienced in
full cylinder operation. This increase in amplitude results from the fact that the remaining
active cylinders must generate more torque per cylinder in order to maintain the mean torque
through the transition [66]. The experimental rig can perform these transitions by specifying
step changes in torque amplitude and order, according to the scenario described. The rig is
very ﬂexible and can be programmed to investigate a wide variety of transient torque inputs.
Since the absorber response in full cylinder mode is nonresonant, and thus nearly zero, zero
initial conditions are used for all experimental runs. The step input is achieved by suddenly
applying and maintaining a ﬂuctuating torque Tn sin(nθ), superimposed on the mean torque
T0 . A sample of such an experiment using the circular path absorber is displayed in Fig. 5.3
(a)-(d), where the torque input measured by the motor feedback is shown in Fig. 5.3 (a),
and the absorber response is shown in Fig. 5.3 (b). To see the rapid oscillations and initial
transitions in these signals, close ups of Fig. 5.3 (a) & (b) are shown in Fig. 5.3 (c) &
(d), respectively. In this simulated cylinder deactivation event (depicted in Fig. 5.3 (a)),
the absorbers abruptly experience a near-resonant excitation, resulting in the beating-type
transient response shown in Fig. 5.3 (b), composed of dominant harmonic components at
order n and approximately order n, the latter shifted by nonlinear eﬀects. The absorbers
˜
eventually settle down into the order n steady-state, on a time scale set by the frequency
nΩ and damping.

5.3

Parameter Identiﬁcation

In order to compare the experimental and analytical results, the experimental system parameter values need to be determined for both types of absorbers. The torque order n and
111

Absorber Amplitude [Deg.]

5
4

T [Nm]

3

Fluctuating Torque, Tn

2

Mean Torque, T0

1
0
−1
−2
0

500

1000

θ

1500

2000

2500

40

Peak Amp.
S.S.

20
0
−20
−40
0

500

4

T [Nm]

3
2
1
0
−1
100

110

120

θ

130

140

150

Absorber Amplitude [Deg.]

5

90

θ

1500

2000

2500

(b)

(a)

−2

1000

40
20
0
−20
−40

90

100

110

120

θ

130

140

150

(d)

(c)

Figure 5.3: An experimentally simulated cylinder deactivation event showing the torque
input and the resulting transient absorber response. (a) Input torque T = T0 + Tn sin(nθ)
applied to rotor by the servo motor; (b) absorber transient amplitude response to the input
torque; (c) zoom-in of the input torque at the transition point; (d) zoom-in of the absorber
response at the transition point.

amplitude Tn , as well as the mean rotor speed Ω, can be varied by the user. The parameters
taken to be ﬁxed for each absorber type include the absorber equivalent viscous damping
coeﬃcient µ, the absorber tuning order n, and the absorber/rotor inertia ratio . Only a
˜
brief summary of how each system parameter is estimated is given, since these methods are
abstracted from previous works [26,52,53,70]. An error analysis in regards to the uncertainty
in the estimated system parameters is performed following the NIST Type A method [72].
112

Within the computed uncertainty bounds, the system parameters are chosen based on best
agreement with transient simulations of the full EOM. The system parameters used to make
analytical predictions are given in Tab. 5.1, and all of the estimated system parameters
with their respective uncertainties for the circular path and the tautochronic path absorbers
are given in Tabs. 5.2 & 5.3, respectively. In the following section, absorber overshoot results from transient experiments that mimic cylinder deactivation will be shown, along with
comparisons with analytical predictions.
Table 5.1: The main system parameters for the spin rig needed for comparison with analytical
predictions.
Type of Path General Path Near Tautochronic
Damping
µ = 0.0395
µ = 0.0675
Tuning Order
n = 1.312
˜
n = 1.449
˜
Inertia Ratio
= 0.0864
= 0.1074
Path Curvature
λ=0
κ = −1.518 (λ = 0.66)

The absorber linear tuning order n is obtained through a combination of physical length
˜
measurements and experimental tests. For the near-tautochronic path absorber, the linear
tuning order is deﬁned as
n=
˜

c − ρ0
,
ρ0

(5.1)

and for the circular path pendulum, the linear tuning order is deﬁned as

n=
˜

c − ρ0
,
ρ0 β

(5.2)

where c is the distance from the center of rotation O of the spin rig to that of the absorbers’
COM and ρ0 is the radius of curvature of the absorber path at its vertex, as depicted in
Fig. 4.1. The circular path absorber tuning order n contains the additional term β that
˜
accounts for the inertial eﬀects associated with a compound pendulum; speciﬁcally, the
113

rotational inertia scales the eﬀective pendulum length ρ0 , as described in equation (5.2).
Recall that parameters β and α were introduced in Chapter 4 to account for the compound
pendulum absorber implementation, and these are related to the absorber geometry and
inertia as follows
β =1+

r 2
,
ρ0

α=

r2
,
cρ0

(5.3)

where r is the radius of gyration of the circular path absorber about its COM; see Nester [53]
for details. Note that these arise from the single point suspension of the absorber, and β = 1
and α = 0 for all absorbers with biﬁlar suspensions, regardless of the path.
Following the method given in [53,70], the tuning order n was determined experimentally
˜
by performing a sweep in the excitation order n, i.e., a frequency sweep. According to linear
¨
theory, when the rotor is excited at n = n, the rotor vibration amplitude |θ| is zero for zero
˜
damping, or a minimum for small damping. Therefore, the order n is experimentally swept
and the corresponding rotor vibrations at each order are measured from the encoder on the
˙
rotor shaft. Speciﬁcally, the rotor encoder gives an instantaneous speed signal θ(t) whose
˙
harmonic magnitudes in steady-state |θ|j (j = n, 2n, . . .) are computed using an FFT, which
¨
are multiplied by jΩ to get the desired acceleration harmonic magnitudes, |θ|j . Using this
method, with j = n, the circular path and near-tautochronic absorber tuning orders are
found to be n = 1.312 and n = 1.449, respectively. The radius of gyration for the circular
˜
˜
path absorber is found to be r = 0.0347m, as computed using the physically measured
parameters ρ0 , c (see Tab. 5.2), along with the measured value for n. This results in α = 0.181
˜
and β = 1.714. Due to the biﬁlar suspension of the near-tautochronic path absorber, one
can can simply use the measured geometry, speciﬁcally, c and ρ0 , given in Tab. 5.3, which
yields values that precisely match the tuning obtained by order sweep measurements.
114

The experimental order sweep must be performed at a low level of ﬂuctuating torque level
Tn in order to keep the absorber amplitudes in the linear response range. On the other hand,
the tautochronic path nonlinearity is only very lightly softening, from the kinematic coupling
to the rotor, and the low torque amplitude requirement is not required for this absorber.
However, the near-tautochronic path absorber used here exhibited some nonlinear frequencyamplitude dependence that was observed from an experimental order sweep performed over
a range of torque levels, Tn . From the measured results, the nonlinear path curvature for
the near-tautochronic absorber is found to be slightly softening (λ = 0.66), which results
in a value of κ = −1.518 that describes the nonlinear detuning for near-tautochronic path
absorbers [46, 68]1 .
The absorber dissipation is modeled as an equivalent viscous damping and the attendant coeﬃcient is measured experimentally. Although some Coulomb damping has been
observed and quantiﬁed for the circular path absorber [76], for amplitudes greater than 10o
the absorber decay is captured very accurately with a viscous model. To obtain the damping
coeﬃcients, the log decrement method is applied to 13 consecutive peaks of a ringdown in
amplitude, which is obtained by switching oﬀ the oscillating torque during a steady-state
oscillation, resulting in a free decay of the absorber/rotor system. Note that in fact the
rotor and absorber are coupled, and the rotor and absorber have independent damping characteristics; however, the experimentally measured values provide an eﬀective damping that
captures the combined eﬀect that these dissipation sources have on the absorber, and therefore best reﬂects the damping coeﬃcient needed for the absorber oscillator model considered
in Chapter 4. The log decrement restricts the amplitudes used in calculating the damping
1 The experimental results presented in [70] were limited in amplitude such that they
match very closely those for an exact tautochronic path.
115

ratio to the linear range, that is, below 25o . The damping ratio ζ calculated from the log
decrement is related to the absorber damping coeﬃcient by,

µ = 2ζ n,
˜

(5.4)

where µ is the non-dimensional damping coeﬃcient used in Chapter 4. As shown in Tabs. 5.1
& 5.2, the circular path absorber damping ratio is found to be 0.13% of critical, which yields
a damping coeﬃcient of µ = 0.0395. As shown in Tabs. 5.1 & 5.3, the tautochronic path
absorber damping ratio is found to be 0.25% of critical, which yields a damping coeﬃcient
of µ = 0.0675.

The system inertia ratio is deﬁned as

=

mc2
,
J

(5.5)

where m is the absorber mass, and J is the inertia of the rotor. The mass m and the distance
c are obtained from direct measurements and the rotor inertia J is obtained through experiment. The rotor inertia is experimentally obtained by locking down the absorbers and applying increasing levels of ﬂuctuating torque Tn sin(nθ), measuring the resulting steady-state
¨
rotor vibrations, θ, at each torque level, and then computing the magnitudes of harmonics
¨
|θ|n as previously described. Using a linear input-output relationship, the locked rotor in¨
ertia Jlocked can be obtained from a line of best ﬁt through the data points (Tn ,|θ|n ). The
locked rotor inertia Jlocked is related to the rotor inertia via J = Jlocked − mc2 , where J
contains the inertia of the rotor system with no absorbers plus the inertia of the absorbers
about their respective COM; see Shaw et al. [70] for more details.
116

Table 5.2: System parameters for the spin rig and circular path absorber; ﬁnal values chosen
that give the best match between experiments and theory.
Parameter
Range
Value Chosen
c (m)
0.160 ± 0.002
0.162
ρ0 (m)
0.040 ± 0.001
0.041
ζ
0.002 ± 0.0007
0.0013
m (kg)
0.241 ± 0.001
0.241
2 ) 0.0735 ± 0.0003
J (kg·m
0.0732
Table 5.3: System parameters for the spin rig and tautochronic path absorber; ﬁnal values
chosen that give the best match between experiments and theory.
Parameter
Range
Value Chosen
c (m)
0.1783 ± 0.001
0.1785
ρ0 (m)
0.0576 ± 0.0001
0.0576
ζ
0.003 ± 0.0006
0.0025
m (kg)
0.173 ± 0.001
0.173
2 ) 0.0514 ± 0.0002
J (kg·m
0.0512

5.4

Results & Discussion

The full predictive capabilities of the modeling and analysis are demonstrated by comparing
simulations of the full EOM, simulations of the averaged equations, the zero damping pseudoenergy predictions, and the experimental results for both types of absorbers. The full EOM
simulations use a short ramp input of the ﬂuctuating torque, with a build-up duration of
one half of a torque cycle θr = π/n, as described in Chapter 4. All of the experimental
results shown here are performed at a mean rotor speed of Ω = 350 RPM. Percent overshoot
results obtained by sweeping the individual system parameters n and Γi are ﬁrst shown,
followed by a summary of all results in which the parameters n, Γ and ξ are collapsed into
χ, demonstrating the utility of the scaling.
Shown in Figs. 5.4 & 5.5 (a)-(f) are the percent overshoot results obtained from experiments using the circular path (Fig. 5.4) and near-tautochronic path (Fig. 5.5) absorbers,
compared with simulations of the averaged equations and the full EOM. Figures 5.4 & 5.5
117

140

Gen. Path Avg. Sims.
Experiment
Full NL EOM Sims.

120

% Overshoot

% Overshoot

140

120

Γc = 0.223

100

100

80
60

1.26

1.28

1.3

n

Γc = 0.372

80
60

1.32

1.26

1.28

(a)
140

Γc = 0.50

120

120

100

Gen. Path Avg. Sims.
Experiment
Full NL EOM Sims.

n = 1.27

1.26

1.28

1.3

n

80
60
0

1.32

0.2

120

0.6

n = 1.30

120
100

100
80
60
0

0.4

140

n = 1.28

% Overshoot

140

Γc
(d)

(c)

% Overshoot

1.32

100

80
60

1.3

(b)

% Overshoot

% Overshoot

140

n

0.2

Γc

0.4

80
60
0

0.6

0.2

Γc

0.4

0.6

(f)

(e)

Figure 5.4: Comparison of theory and experiments for the circular path absorber. Percent
overshoot vs. n for (a) Γc = 0.223, (b) Γc = 0.372, and (c) Γc = 0.50. Percent overshoot vs.
Γc for (d) n = 1.27, (e) n = 1.28, and (f) n = 1.30.

118

120

140

Near Taut. Avg. Sims.
Gen. Path Avg. Sims.
Experiment
Full NL EOM Sims.

% Overshoot

% Overshoot

140

120

Γt = 0.11

100

100

80
60

1.42

1.44

Γt = 0.184

1.46

1.48

n

80
60

1.5

1.42

1.44

(a)

Γt = 0.21

120

120
100

100

80
60

1.42

1.44

1.46

n

1.48

60
0

1.5

n = 1.41

80

0.1

n = 1.43

% Overshoot

% Overshoot

0.2

Γt

0.3

0.4

Γt

0.3

0.4

(d)
140

n = 1.44

120

120

100

100
80
60
0

1.5

Near Taut. Avg. Sims.
Experiment
Full NL EOM Sims.

(c)
140

1.48

(b)
140

% Overshoot

% Overshoot

140

1.46

n

0.1

0.2

Γt

0.3

80
60
0

0.4

(e)

0.1

0.2

(f)

Figure 5.5: Comparison of theory and experiments for the near-tautochronic path absorber.
Percent overshoot vs. n for (a) Γt = 0.11, (b) Γt = 0.184, and (c) Γt = 0.21. Percent
overshoot vs. Γt for (d) n = 1.41, (e) n = 1.43, and (f) n = 1.44.

119

(a)-(c) show the eﬀect that sweeping the torque order n has on the system overshoot for
a ﬁxed torque amplitude Γi . As the torque order n approaches the resonant tuning orders
n = 1.312 (circular path) and n = 1.449 (tautochronic path), the overshoot increases, as
˜
˜
expected for lightly damped systems. Figures 5.4 & 5.5 (d)-(f) show the eﬀect that sweeping
the torque amplitude Γi has on the system overshoot for a ﬁxed torque order n. This shows
that increasing the torque amplitude increases the percent overshoot for a ﬁxed amount of
detuning, as expected for a softening system. In comparing Figs. 5.4 & 5.5, it is seen that
the observed overshoot levels are signiﬁcantly less for the near-tautochronic absorber as compared to the circular path absorber, speciﬁcally, a maximum of 100% compared to 124%.
This is primarily a result of the relative levels of softening between the two paths. A smaller
secondary eﬀect in the reduction of the overshoot for the near-tautochronic paths is due to
the eﬀective hardening parameter, γ, that limits the eﬀect of the torque at moderate and
large amplitudes. For paths that are near-tautochronic, and for hardening paths, the combination of the kinematic softening and the hardening parameter γ, result in a system that
does not experience a jump in the system response (unless the absorbers are undertuned,
which is undesirable and never used in practice) [68].

Order n sweeps are shown in Fig. 5.5 (a)-(c) for the near-tautochronic absorber, along
with the pseudo-energy results for both the general path and the near-tautochronic path
models, showing that the near-tautochronic path theory is more conservative than the general
path theory. Due to the fact that the torques Γc and Γt are scaled diﬀerently, three additional
plots would be needed to compare both pseudo-energy methods for the torque sweep of the
near-tautochronic absorber; these are not shown because they demonstrate the same trend
as that shown in Fig. 5.5 (a)-(c). Therefore, the near-tautochronic absorber with nonlinear
120

tuning λ = 0.66 is in the region where either model can provide accurate estimates of
the overshoot. However, as one moves closer to a circular path, by reducing λ, the neartautochronic theory fails to capture the amount of softening in the path and will underpredict the amount of overshoot. Similarly, moving the path towards the tautochrone, that
is, increasing λ towards λe , the near-tautochronic theory performs the best because the
general path theory is unable to capture the amount of system softening, and therefore
under-predicts the overshoot2 . As discussed in Chapter 4, for nonlinear tuning between
a tautochrone and a cycloid, λe ≤ λ ≤ 1, both theories provide similar accuracy in the
overshoot estimate for excitation orders up to n = 3, beyond which the softening is not
accurately captured with the general path theory.
As a whole, the simulations of the full EOM in Figs. 5.4 & 5.5 perform the best at
predicting the experimental system overshoot. However, the averaged equations oﬀer the
convenience of having fewer parameters and being much faster to obtain, and they are
generally conservative at estimating the experimentally observed overshoot, which is useful
for design. Lastly, the small amount of scatter in the experimental overshoot seen in Figs. 5.4
& 5.5, which is not observed in the simulations of the full EOM, is due to a combination of
the torque sign (+Γi or −Γi ) at transition, as well as the transition time (ramp duration),
both of which vary among the experimental results, but are not accounted for in the model.
The scatter is not observed in the simulations of the full EOM, due to the consistency in
the torque sign (+Γi for all simulations) and the ﬁxed ramp duration of θr = π/n. This
shows the sensitivity of the absorber overshoot to the torque proﬁle at transition, making
2 Recall that a tautochronic path system with λ is slightly softening (ξ < 0) in the
e
t
near-tautochronic theory and is linear (ξc = 0) in the general path theory. For the neartautochronic theory, the level of softening increases as the excitation order n and tuning
order n increase.
˜
121

it an important design consideration that is most easily treated through simulation. The
worst case torque proﬁle is an abrupt transition with a torque acting in the direction that
instantaneously slows down the rotor speed, thereby reducing the absorber restoring force
and resulting in more overshoot. For non-zero initial conditions, this scenario can be made
even worse, depending on the dynamic state of the absorber at the time of the transition.
Shown in Fig. 5.6 (a) & (b) is the percent overshoot obtained from the general path
pseudo-energy solution, simulations of the averaged equations (with damping), simulations
of the full EOM, and experimental results, for the circular path absorber (Fig. 5.6 (a))
and the near-tautochronic path absorber (Fig. 5.6 (b)), which are collapsed onto the single
system parameter χc . As expected, the circular path absorbers jump shortly beyond the
point χ , where the jump oﬀset is due to damping. In fact, the circular path absorber jumps
at very small torque levels when the excitation order reaches the pendulum tuning order
n ≈ n = 1.312. In contrast, the near-tautochronic absorbers do not jump, but instead reach
˜
their physical limits. In fact, they run well beyond their tuning order n = 1.449, before they
˜
began to limit out at n = 1.48. As noted previously, it is observed that the full EOM perform
the best at predicting the experimentally measured overshoot, while the averaged equations
provide a mostly conservative estimate. In addition, not a single experimentally measured
overshoot value was observed to be above the (undamped) pseudo-energy curve, making
this curve a useful upper bound for the overshoot that one would expect during a cylinder
deactivation event. An attractive feature of this upper bound is that it is the solution of a
cubic equation, which has a closed form, and it depends on the single parameter χc .
Lastly, the introduction of the damping parameter Di oﬀers a more accurate bound on the
overshoot. However, recall that the scaled damping coeﬃcients Di depend on the absorber
122

% Overshoot

180

Dc = 0, Pseudo-energy

160
140

χ = 4/27

Circ. Path Experiment
Full NL EOM Sims.
Gen. Path Avg. Sims.

120
100
80
60
0

0.05

0.1

0.15

χc

0.2

(a)

180

% Overshoot

Dc = 0, Pseudo-energy
160

Circ. Path Experiment

140

Full NL EOM Sims.
Gen. Path Avg. Sims.

120
100
80
60
0

0.02

0.04

0.06

χc

0.08

0.1

0.12

(b)
Figure 5.6: Comparisons of theory and experiments for the (a) circular path absorber and
the (b) near-tautochronic path absorber; percent overshoot vs. χc .

123

damping coeﬃcients µ, the excitation order n, the tuning order n, and the inertia ratio ,
˜
so that one cannot show a single overshoot curve versus χ that represents a ﬁxed level of
damping. Fig. 5.7 shows three curves for various levels of damping, along with the same
experimental data presented in Fig. 5.6. The three curves are the zero damping general path
pseudo-energy result, Dc = 0, along with the curves generated by simulating the damped
general path averaged equations. The damping coeﬃcient Di in the damped simulations is
computed by using the values of the damping ratio that bound the range of uncertainty (given
in Tabs. 5.2 & 5.3) for each absorber, as well as the smallest (largest) excitation order n used
in the experiments which will minimize (maximize) Di . For the circular path absorber, the
damping ratios used are ζ = 0.0013 and ζ = 0.0025, with excitation orders of n = 1.261 and
n = 1.312, which yield Dc = 0.036 and Dc = 0.152, respectively. For the near-tautochronic
path absorber, the damping ratios used are ζ = 0.0024 and ζ = 0.0036, with excitation
orders of n = 1.413 and n = 1.485, which yield Dc = 0.062 and Dc = 0.193, respectively. It
is seen that the curves that account for damping bound most of the experimentally observed
overshoot data, thus providing a more precise prediction. Such results can be used to develop
more aggressive absorber designs, provided estimates of the absorber damping are known.

5.5

Conclusions

The results presented here demonstrate the validity of the analytical predictions in Chapter
4, and provide the tools required for estimating, with conﬁdence, absorber overshoot for
a wide range of system parameters. While the experimental results presented focused on
zero initial conditions, the analysis is quite general and is able to predict overshoot levels for
arbitrary initial conditions. In this experimental study, two types of absorbers were used, one
124

with a circular path (λ = 0) and the other with a near-tautochronic path (λ = 0.66). These
two paths span about 2/3 of the parameter range for the Denman family of epicycloidal
paths. Both paths are nonlinearly softening, which are the most sensitive to changes in
the excitation parameters that lead to transient response; this makes them well suited for
testing the theoretical predictions. The experimental results show that numerical solutions
of the full equations of motion perform the best at capturing the observed overshoot, while
the general path and near-tautochronic path results obtained by simulating the damped
averaged equations were mostly conservative, but still quite accurate. In fact, the general
path results also worked well at predicting the response of the near-tautochronic absorbers,
but the reverse was not true, as expected. The pseudo-energy method provides a quick and
convenient method for determining a useful upper bound for absorber overshoot, which for
the general path is available in closed form and depends on a single system parameter χc .
For the near-tautochronic paths, the pseudo-energy method depends on two parameters,
and requires numerical solution for the root of a single equation involving elliptic functions.
However, the overshoot for most such paths, with small damping, is essentially that of a
linear system, that is, about 100%. It is worth noting that it appears that the general path
results can be used to estimate absorber overshoot for most practical absorber designs [46].
However, the full near-tautochronic results are required for excitation orders greater than
n = 3 that will operate at large amplitudes. Lastly, if estimates of the absorber damping are
known, an accurate bound on the overshoot can be obtained by simulations of the averaged
equations, and this allows for more aggressive absorber designs when compared to using the
conservative upper bound obtained from the pseudo-energy method.

125

180

% Overshoot

160

Dc = 0, Pseudo-energy
Circ. Path Experiment
Dc = 0.036, Avg. Sims.

140

Dc = 0.152, Avg. Sims.

120
100
80
60
0

0.05

0.1

χc

0.15

0.2

(a)

180

% Overshoot

160

Dc = 0, Pseudo-energy
Circ. Path Experiment
Dc = 0.062, Avg. Sims.

140

Dc = 0.193, Avg. Sims.

120
100
80
60
0

0.02

0.04

0.06

χc

0.08

0.1

0.12

(b)
Figure 5.7: Bounding the experimental data using simulations of the damped general path
averaged equations for the (a) circular path absorber and (b) the near-tautochronic path
absorber; percent overshoot vs. χc .
126

Chapter 6
Conclusions and Directions for Future
Work
The ﬁrst part of this study considering improving the absorber system model by accounting
for the rollers which suspend the pendulum mass in biﬁlar absorber conﬁgurations. This
analysis led to the discovery of a tautochronic absorber path that accounts for roller inertia,
a previously unknown result. It was shown that when the rollers are accounted for in the
linear tuning n, the tautochronic path with rollers results in a forced absorber response that
˜
is slightly more hardening than the roller-free absorber path. In addition, a perturbation
analysis extracted the leading order roller inﬂuences on the absorber kinematics, in order
to determine the accuracy of the previous roller-free assumptions on the absorber’s tuning
parameters n and λ. The perturbation results show that for practical roller sizes, the rollers
˜
have a very small inﬂuence on the nonlinear tuning. Therefore, for the sizes of rollers
typically in use, one needs to include their eﬀects only in the linear tuning in order to
achieve satisfactory performance.
127

The roller investigation employed a near tautochronic path scaling [11, 14, 68, 70] on the
full equations of motion in order to perform the perturbation analysis, and for these paths
the dominant nonlinearity arises from kinematic coupling to the rotor. The other type of
scaling employed in this work is the general path scaling [5,26,52, 53], which applies in cases
where the dominant nonlinearity arises from the absorber path. The resulting approximate
models obtained from these two scalings have a very similar form and can be analyzed in a
similar manner. The general path results involve polynomial nonlinearities arising from the
path, whereas the near tautochronic path results involve elliptic function nonlinearities that
arise from the absorber/rotor coupling.

At the heart of the model for the coupled absorber/rotor system is a harmonically forced
(near resonance) nonlinear oscillator, and the analysis of the transient response starts with
an investigation of the harmonically forced Duﬃng oscillator with cubic stiﬀness nonlinearity. The major contributions of this study include an approximate method to compute
the nonlinear percent overshoot, based on a unique scaling of the position and independent
variable that signiﬁcantly simpliﬁes the oscillators parameter dependence. Speciﬁcally, the
excitation amplitude, the system nonlinearity, and the order detuning can be combined into
a single parameter, χ. This parameter, and a damping parameter D, completely dictate
the response. The overshoot results derived here are quite fundamental and, to the author’s
knowledge, original. The nonlinear percent overshoot method makes use of the system’s
averaged equations, which yield an integral of motion in the undamped (D = 0) case [71],
by which the amplitude and phase of the system response are related to χ and the initial
conditions. This integral of motion is quartic in amplitude for non-zero initial conditions and
cubic in amplitude for zero initial conditions, and the roots can be found in closed form. The
128

roots give phase space trajectories that are a function of the response phase, χ, and initial
conditions. These transient trajectories contain the peak amplitudes and, for bistable systems, the ultimate steady state response achieved, both of which are important to absorber
design.

The scaling employed here reduces the averaged system to two parameters, χ, described
above, and D, which involves the damping. Thus, the undamped results, which provide
a useful bound for the overshoot in lightly damped systems, depend only on χ. With the
presence of damping, the averaged equations can no longer be integrated exactly, however,
an approximate integral of motion for small damping is possible, and this computation is
summarized in Appendices G & H. The payoﬀ, in terms of analytical approximation and
its accuracy, is small in comparison with the extensive nature of these calculations, and
thus it appears to be more eﬃcient to simply obtain the damped overshoot by simulating
the averaged equations with damping. The undamped results provide quick estimates of the
overshoot for CPVAs, which must be designed to not exceed speciﬁed amplitudes, even during
abrupt startups, making them of great practical value to absorber design. Additionally, for
the case of tautochronic absorbers [11, 14, 68, 70], the system nonlinearities are much more
complicated than the cubic term in the Duﬃng equation, yet the averaged equations are
integrable in the undamped case, so that similar predictive results can be achieved for that
system. Furthermore, this percent overshoot approach has also been applied to the case of
parametric resonance and it has been found to produce simpler formulas for the overshoot
than those given here. In fact, for the case of near zero initial conditions (i.e. p0 ≈ 0)
√
it is remarkably found that the percent overshoot is simply 100( 2 − 1), and is therefore
independent of the system parameters contained. Completion of these results, which include
129

experimental evidence of the constant overshoot for near zero initial conditions, and an
outline for computing the overshoot for non-zero initial conditions [46], are still in progress.
These extensions suggest that the pseudo-energy approach may be quite general for weakly
nonlinear systems of a general form, and also possibly for classes of conservative, strongly
nonlinear systems with periodic excitation.
Another contribution of this work is that, for the ﬁrst time, the general path [5,26,52,53]
and near tautochronic path [11, 14, 68, 70] perturbation results are formulated together in a
single, compact framework. This is useful for all investigations, and is used here to solve
the nonlinear overshoot problem. For the general path case, with negligible damping, the
formulation boils down to previously mentioned closed form expression (speciﬁcally, roots
of a cubic or quartic equation, depending on initial conditions) that depends on a single
parameter. For the near tautochronic path case, the overshoot estimate for undamped
systems depends on two parameters, and requires ﬁnding the root of an equation that involves
elliptic functions. For all path types, more accurate estimates that account for damping are
available via simulation of the averaged equations.
An experimental study of the transient response of pendulum vibration absorbers during
a simulated cylinder deactivation event demonstrated the validity of the analytical approximations. While these results focused on zero initial conditions, the analysis is quite general
and is able to predict overshoot levels for absorbers with all paths used in practice. In the
experiments, two types of absorbers are used, including a circular path absorber (λ = 0) and
a near tautochronic path absorber (λ = 0.66); these paths span about 2/3 of the total range
in the Denman family. Both paths are in the softening region of the family, which is most
sensitive to changes in the excitation parameters and the torque proﬁle at startup, which
130

makes them well suited for testing the theoretical predictions. The experimental results from
both absorbers show that the equations of motion perform best at capturing the observed
overshoot, while the averaged equations were mostly conservative, but still quite accurate.
In fact, the general path averaged equations worked well at predicting the response of the
near tautochronic absorbers, except at large amplitudes, but the reverse was not true, as
expected. The pseudo-energy method provides a quick and simple method for determining
a useful upper bound for absorber overshoot. This result is available in closed form for the
general path and depends on a single system parameter χc , and it appears that the general
path results can be used to estimate absorber overshoot for most practical absorber designs.
Lastly, if estimates of the absorber damping are known, an accurate bound on the overshoot
can be obtained by simulation of the averaged equations, and this allows for slightly more
aggressive absorber designs, when compared to using the upper bound obtained from the
pseudo-energy method.

A major issue yet to be explored is the possibility of non-synchronous transient dynamics
in multi-absorber systems. A preliminary experimental investigation of transient dynamics
using two circular path absorbers was carried out during the course of the present work.
An interesting behavior, observed in both experiments and simulations of absorbers with
slightly diﬀerent levels of damping, is a transient response wherein the two absorbers start
at zero, then exchange energy in a non-synchronous manner as they begin to settle down,
but ultimately end up in a near-synchronous response, with the diﬀerence in steady-states
attributed to the diﬀerent damping values. Such a response, from experiments and simulations, is shown in Fig. 6.1 (a) & (b). This occurred at a torque order n = 1.292, where the
absorber motion was initially synchronous, until a critical torque amplitude was reached. As
131

the torque amplitude was incremented further, the non-synchronous transient motion shown
in Fig. 6.1 (a) & (b) was observed, until a level beyond which the absorbers eventually
jumped to the upper branch steady-state, as expected. The simulation shown in Fig. 6.1
(b), using the experimentally measured system parameters, captures this bifurcation very
accurately, which is believed to be a result of the diﬀerence in the values of the equivalent
viscous damping coeﬃcients µi between the two absorbers, since the absorbers have nearly
identical tuning orders n = 1.309. Also, as the diﬀerence in damping values was decreased
˜
in simulations, this bifurcation disappeared.

In addition, when the torque order was in-

creased to n = 1.302, the steady-state behavior observed by Nester was reproduced [52–54].
Speciﬁcally, at near resonance excitation n ≈ n, one absorber was in an inactive state (near
˜
zero steady-state amplitude), while the other absorber was active. The transient response
leading to this must, of course, be non-synchronous. As shown in Fig. 6.2 (a), after the
ﬁrst beat cycle, the transient amplitude response shows the absorber with larger damping
(black) decays to a near zero state, while the other grows to nearly twice the predicted
synchronous steady-state amplitude. A simulation using the experimentally measured parameters is shown in Fig. 6.2 (b), where it is seen that diﬀerences in viscous damping cannot
fully capture this behavior. However, the addition of Coulomb damping may capture this
behavior since Vidmar [75] observed that Coulomb damping plays a more prominent role at
smaller amplitudes, i.e, less than 10o swing angle, in these circular path absorbers. As the
torque amplitude was incremented further at order n = 1.302, a non-synchronous motion
similar to that observed in Fig. 6.1 (a) & (b) was observed, until the absorbers eventually
jumped to the upper branch. To date, this bifurcation has not been captured analytically,
but this investigation is underway. This problem is more easily tackled by performing a

132

Absorber Amplitude (Deg.)

40

20

0

−20

−40
0

500

1000

1500

2000

2500

θ
(a)

40

20

0

- 20
- 40

0

500

1000

1500

2000

2500

(b)
Figure 6.1: Non-synchronous transient motion to a steady-state near-synchronous response
using two circular path absorbers at torque order n = 1.292: (a) experiment and (b) simulation. Parameter values used for simulation: n = 1.309 (both absorbers), ζ1 = 0.002
˜
(response shown in red), ζ2 = 0.0035 (response shown in black), T0 = 1.069 Nm, Tn = 2.919
Nm, Ω = 36.652 rad/s, J = 0.0636 kg·m2 , m = 0.241 kg (both absorbers), cc = 0.162 m
(both absorbers, see Tab. 5.2), ρ0c = 0.041 m (both absorbers, see Tab. 5.2), and a ramp
of half a torque cycle in of the ﬂuctuating torque.

133

Absorber Amplitude (Deg.)

20

10

0

−10

−20
0

500

1000

1500

2000

1500

2000

θ
(a)

20

10

0

- 10
- 20

0

500

1000
(b)

Figure 6.2: Non-synchronous transient motion to a non-synchronous steady-state at torque
order n = 1.302: (a) experiment and (b) simulation. Parameter values used for simulation:
n = 1.309 (both absorbers), ζ1 = 0.0013 (response shown in red), ζ2 = 0.0042 (response
˜
shown in black), T0 = 0.937 Nm, Tn = 1.40 Nm, Ω = 36.652 rad/s, J = 0.0636 kg·m2 ,
m = 0.241 kg (both absorbers), cc = 0.162 m (both absorbers, see Tab. 5.2), ρ0c = 0.041
m (both absorbers, see Tab. 5.2), and a ramp of half a torque cycle in of the ﬂuctuating
torque.

134

steady-state stability analysis along similar lines to that performed in [6, 68], before a transient stability analysis is considered. Following the stability analysis using circulant matrices
in [68], one could investigate the transient stability of such a response. Of course, one would
have to assume identical absorber systems with zero damping, but such results may provide
useful bounds for lightly damped systems.
Other topics for future consideration include: more gradual transitions, transient responses of systems with absorbers tuned to multiple orders, and a more detailed investigation of the transition during cylinder deactivation, using more realistic engine torque data.
Along these lines is the design of torque proﬁles that minimize overshoot, such as the ramp
used in Chapter 4. An example of an experimental run demonstrating this idea is shown
in Fig. 6.3, which results in signiﬁcantly less overshoot when compared to the step input.
A similar approach that may be easier to physically implement involves discretely jumping
the excitation order between oﬀ-resonant and resonant during a transition between these
two cases, by turning on and oﬀ cylinders, thus allowing the absorber amplitude to gradually build up. Analysis of these behaviors will generally require simulations, but the eﬀort
involved in a detailed parameter study can be minimized by making use of the rescaling
developed in this work, and the averaged equations, which are much faster to simulate.

135

Absorber Amplitude [Deg.]

30

P.O. = 94%

20
10
0
−10
−20
−30
0

10

20

30

time (s)
Absorber Amplitude [Deg.]

(a)

30

P.O. = 15%

20
10
0
−10
−20
−30
0

10

20

30

time (s)
(b)
Figure 6.3: Experimental comparison of the transient overshoot of a circular path absorber
when forced at n = 1.26 using two types of ﬂuctuating torque inputs that have the same
magnitude but a diﬀerent ramp duration. (a) Response to a near step input of the torque,
resulting in an overshoot of 94%. (b) Response to a ramp input of the torque with a duration
of 10 torque cycles, resulting in an overshoot of 15%.

136

APPENDICES

137

Appendix A
Order 2 terms
The order 2 terms for the rotor equation of motion are given by

1
1
− 2 −µνg(s)s + 4ξa2 νν − νν s − ν 2 s
2a
2a

+

1
1 ∂xP (s)
1 2
+
+ 4δ νν h2 + 2 + rP (s) + yp (s) + g(s)s − s
4
4
2
∂s
2
2 1 ∂rP (s) s + 1 ∂g(s) s 2 + s ∂yP (s) + 1 g(s)s +
+ν
4 ∂s
4 ∂s
∂s
4
2 (s)
∂x
1 ∂xP (s)
1
−
,
s − s2 P
2
∂s
2
∂s2
which completes the rotor equation of motion (2.4), since there are no higher order terms.

138

Appendix B

Pendulum Path Details

Here we follow the path formulation of Denman [17], in which the local radius of path
curvature ρ is described by ρ2 = ρ2 − λ2 S 2 . Denman derives the position of the pendulum
0
2
in terms of rotating Cartesian coordinates XP and YP , which can be used to ﬁnd RP =
2
2
XP + YP . The expressions for XP and YP are

XP =

ρ0
1 − λ2

1
λS
sin−1
λ
ρ0

sin

−

λS
cos sin−1
ρ0

+

1
λS
λ2 S
cos
sin−1
ρ0
λ
ρ0

(B.1)
,

and

YP = −c −

ρ0
1 − λ2

cos
+

1
λS
sin−1
λ
ρ0

λS
cos sin−1
ρ0

λ2 S
1
λS
sin
sin−1
ρ0
λ
ρ0
139

−1 .

+

(B.2)

Using trigonometric relations one can obtain the following expression,

λS
cos sin−1
ρ0

=

λS 2
.
ρ0

1−

(B.3)

If one substitutes equation (B.3) into equations (B.1) and (B.2), normalizes the path expressions and the arc length S by c, and makes use of equation (2.5) to replace ρ0 with n0 , the
˜
following non-dimensional path expressions are obtained,

xP =

1
1
sin
sin−1 λ(1 + n2 )s
˜0
2
λ
1−λ

1

2

1 + n2
˜0

−λ2 s2 +

1
sin−1 λ(1 + n2 )s
˜0
λ

− sλ2 cos

(B.4)

,

and

yP = −1+
−

(B.5)

1
1
cos
sin−1 λ(1 + n2 )s
˜0
2
λ
1−λ

1
sin−1 λ(1 + n2 )s
˜0
+ sλ2 sin
λ

140

−

1
1 + n2
˜0
1
1 + n2
˜0

.

2

−λ2 s2 +

Appendix C
Averaging Integrals
The integrals that result from the averaging process, and are not available in convenient
closed form, are deﬁned by

F1 (z) =

2π
1
sin2 (x) 1 − n2 (1 + n2 )z 2 cos2 (x) dx,
˜0
˜0
2π 0

(C.1)

F2 (z) =

2π
1
˜0
˜0
cos2 (x) 1 − n2 (1 + n2 )z 2 cos2 (x) dx,
2π 0

(C.2)

and
M (z) =

2π
1
1
cos(x) sin
sin−1 z˜ 0
n
2π 0
λe0

141

1 + n2 cos(x)
˜0

dx.

(C.3)

Appendix D

Steady-State Amplitudes

The steady state amplitudes for branch A, branch C, and saddle branch B are given by
three of the six possible roots to equation (3.9) when χ = 0. The three roots are as follows,

1
pA (χ, D) = √
3χ

√
√
1
1
1 + (1 + i 3)(3D2 − 4)λ−1/3 + (1 − i 3)λ1/3 ,
8
8

1
pC (χ, D) = √
3χ

1
1
1 + (4 − 3D2 )λ−1/3 + λ1/3 ,
4
4
√
√
1
1
1 − (1 − i 3)(3D2 − 4)λ−1/3 − (1 + i 3)λ1/3 ,
8
8

1
pB (χ, D) = √
3χ

(D.1)
(D.2)
(D.3)

where
√
λ =2(27χ − 4 − 9D2 ) + 3 3 D6 + 8D2 (D2 + 2 − 9χ) + 4χ(27χ − 8).

(D.4)

The root pC is valid for all values of χ = 0. The roots pA and pB are valid in the bistable
region χ ∈ (χ1 , χ2 ). This range is determined by the conditions on χ for these roots to be
142

real, speciﬁcally,

χ1 =

1
54

8 + 18D2 −

4 − 3D2

3

χ2 =

1
54

8 + 18D2 +

4 − 3D2

3

,

(D.5)

.

(D.6)

For the undamped system results, roots are simply evaluated at D = 0. The linear (ξ =
0, χ = 0) steady state forced response steady is recovered from equation (3.9), and is given
by p =
¯

1
.
4+D2

143

Appendix E
Transient Response Symmetry
Similar to the steady state amplitude response, the transient amplitude response also possesses a symmetry that allows one to use the derived results to capture the case when p < 0,
which arises when σ < 0 in the rescaled system. To see this, substitute p = −k into equation
(3.12) to obtain
−χk 4 + k 2 − k cos(Φ) = c1 ,

(E.1)

which, upon replacing Φ with Φ + π, one obtains

−χk 4 + k 2 + k cos(Φ) = c1 ,

(E.2)

which is the same as equation (3.12). Therefore the transient trajectories for p < 0 are the
same as those for p > 0 with a phase shift of Φ + π. The percent overshoot results for p0 < 0
are obtained by computing c1 using initial conditions (|p0 |, Φ0 + π). The resulting c1 is
then used in the roots given in equations (F.1) and (F.2) to ﬁnd the peak amplitude pmax
for the overshoot calculation.

144

Appendix F

Undamped Transient Response
Amplitudes

Here we give the full analytical results needed for the percent overshoot calculation given in
equation (3.11). The percent overshoot calculation uses the peak transient amplitude and
the steady state amplitude for a given χ and initial conditions (p0 , Φ0 ). The two roots of the
transient equation (equation (3.12)) that contain the peak amplitudes about steady states
A and C, as explained in the text, are given by,
√
√
1
6 3χ cos(Φ) 1/2
p1 (χ, Φ, c1 ) = √
2+κ− 4−κ+ √
,
2 3χ
2+κ
√
√
1
6 3χ cos(Φ) 1/2
p2 (χ, Φ, c1 ) = √
2+κ+ 4−κ+ √
,
2 3χ
2+κ

where κ is
κ=β

2 1/3
α 1/3
+
,
α
2
145

(F.1)

(F.2)

and β and α are deﬁned as follows,

β =1 + 12c1 χ,
α =γ +

γ 2 − 4β 3 ,

γ =9χ 3 cos2 (Φ) + 8c1 −2.

The constant c1 is determined by the initial conditions (p0 , Φ0 ) and χ, speciﬁcally,

c1 (χ, p0 , Φ0 ) = −χp4 + p2 + p0 cos(Φ0 ).
0
0

(F.3)

The roots p1 and p2 give transient trajectories as a function of the relative phase Φ, and for
D = 0 the peak occurs at Φ = 0 or Φ = π, depending on the system parameters χ and the
initial conditions (p0 , Φ0 ). With the transient amplitude roots given here and the steady
state amplitude roots given in Appendix D, one can formulate the percent overshoot about
equilibria A and C by following the procedure given in Section 3.3.1.

146

Appendix G
Amplitude Correction for Small
Damping
It is now of interest to investigate a correction to the percent overshoot calculation for
the case of small damping. As mentioned earlier, no exact integral of motion is known for
equations (3.6) and (3.7) when D = 0, however, an approximate solution for the case of small
damping D << 1 will be derived. The approach used here follows that of Denman’s [18],
who expands the constant of motion in a power series using the damping coeﬃcient D and
derives n partial diﬀerential equations to be solved at each order Dn . This expansion will
only be solved up to ﬁrst order in D, where the D0 order solution will recover the result
derived in equation (3.12) and the D1 solution is summarized in Appendix H. The resulting
constant of motion which is corrected for small damping is then compared with simulations
of the average equations (equations (3.6) and (3.7)) to show its accuracy. This correction
only provides an analytical result for the constant of motion, analytical results for the peak
amplitude pmax of this constant of motion are intractable and these amplitudes are obtained
147

numerically and then used in the percent overshoot calculation.

Following the work of Denman’s [18], one can determine a corrected constant of motion
for the damped system by expanding the constant of motion Ψ(p, Φ, χ, D) in a power series

Ψ(p, Φ, χ, D) = Ψ0 (p, Φ, χ) + DΨ1 (p, Φ, χ) + D2 Ψ2 (p, Φ, χ) + .....,

(G.1)

where Ψ0 is the constant of motion when D = 0 as found in equation (3.12) and Ψ1 , Ψ2
are the ﬁrst and second order corrections for the damped system, respectively. As will be
shown, the exact solution for the ﬁrst order damping correction Ψ1 will be quite complicated
rendering the second order or higher corrections not practical even if an exact solution exists.
Therefore, for the remainder of the derivations the terms of order D2 and above will be
dropped from the series. To formulate the governing partial diﬀerential equation (PDE), one
takes the total derivative of equation (G.1) w.r.t ϕ which results in

∂Ψ0 ∂Ψ0 ∂p
dΨ
=
+
+D
dϕ
∂ϕ
∂p ∂ϕ

∂Ψ1 ∂Ψ1 ∂p
+
∂ϕ
∂p ∂ϕ

+ .... = 0,

(G.2)

which is equal to zero since Ψ = constant and dΨ = 0. The term ∂p/∂ϕ was obtained earlier
by dividing equation (3.6) by equation (3.7) resulting in
dp
p sin(Φ) + Dp2
=
,
dϕ
cos(Φ) + 2p − 4χp3

(G.3)

and this equation is no longer an exact ﬁrst order diﬀerential equation when D = 0. The
governing PDE’s for the constants of motion Ψ0 and Ψ1 are obtained after plugging equation
148

(G.3) into equation (G.2) and equating powers of D

∂Ψ0
∂Ψ0
cos(Φ) + 2p − 4χp3 +
p sin(Φ) = 0,
∂ϕ
∂p

(G.4)

∂Ψ1
∂Ψ1
∂Ψ0 2
cos(Φ) + 2p − 4χp3 +
p sin(Φ) +
p = 0.
∂ϕ
∂p
∂p

(G.5)

As expected, the solution to equation (G.4) is the constant of motion for the undamped
system found in equation (3.12). Using the result for Ψ0 in equation (G.5), the ﬁrst order
correction Ψ1 can be obtained in closed form with the aid of Mathematica and the result
is summarized in Appendix H. Using the results for Ψ0 and Ψ1 , the corrected constant of
motion to ﬁrst order can be constructed from equation (G.1)

Ψ = −χp4 + p2 + p cos(Φ) + DΨ1 (p, Φ, χ) = c2 ,

(G.6)

where c2 is the constant of integration determined by the initial conditions (p0 , Φ0 ), system
parameters χ, and the damping coeﬃcient D. Note that there are two separate Ψ1 functions
that satisfy equation (G.5), however, the function given in Appendix H contains the desired
peak amplitude pmax which is corrected for small damping.
Shown in Fig. G.1 is the percent overshoot for zero initial conditions (c2 ≈ 0) which
includes the analytical zero damping (D = 0) result from Fig. 3.6 as well as two additional
curves for two levels of damping indicated in the Figure. Note that equation (H.1) cannot
be evaluated at p = 0 or Φ = nπ for all whole numbers n but this isn’t a problem because
as seen in Fig. 3.2 (a), (b), & (c), the damping slightly shifts the phase at which equation
(G.6) is evaluated to get the peak amplitude so that it doesn’t occur at Φ = 0 or π as in
the undamped case. The solid lines shown in Fig. G.1 are obtained by numerically solving
149

equation (G.6) for pmax after setting c2 = 0 and evaluating Φ at its steady state value1
using equation (3.10). The percent overshoot is then calculated by substituting pmax and
the corresponding damped steady state amplitude obtained from either equation (D.1) or
(D.2) depending on χ. The circles shown in Fig. G.1 are simulations of the average equations
(equations (3.6) and (3.7)) for the two damping levels indicated in the Figure and undamped
simulation is not shown since it was shown in Fig. 3.6. As may be expected, the damping
increases the critical system parameter condition χ

= 4/27 shown in Fig. 3.6 and this

increases the system parameter regime for operation about the lower branch equilibria (A).
The corrected constant of motion for small damping shows a good approximation up to
about D = 0.10. Therefore, for small damping, equation (G.6) provides an alternative to
simulations of the average equations when analyzing the transient response of a system.
Lastly, it is noted the damping coeﬃcient D of the scaled system (p, Φ) is related to the
damping coeﬃcient µ of the original system (r, Φ) through equation (3.8) and this can be
related to the damping ratio ζ which will depend on the excitation frequency ω and the
natural frequency ω0 .

1 Note that this value of Φ satisﬁes p = 0 (see equation (3.6)) which corresponds to the
peak amplitude of a transient trajectory.
150

160

% Overshoot

140

Average Equations
Damping Correction

D=0
D = 0.05
D = 0.10

120
100
80
60
40
−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

χ
Figure G.1: Depicting the percent overshoot for the damped system subject to zero initial
conditions. The damping levels include D = 0, D = 0.05, and D = 0.10 as labeled in the
ﬁgure. The solid curves are obtained by numerically solving equation (G.6) which is the
constant of motion for small damping. The circles are obtained by simulating the average
equations (equations (3.6) and (3.7)) for the two levels of damping. The damping increases
the critical system parameter χ as shown in Fig. 3.6 and this increases the operating regime
for a transient response about the lower branch equilibria (A). The constant of motion
provides a very good approximation of the damped transient response of the system up to a
damping level of about D = 0.10.

151

Appendix H

The Damped Transient Amplitude
Response Polynomial

This appendix gives the exact analytical results needed for the percent overshoot calculation
given in equation (3.11) when small damping (D << 1) is present. The damped percent
overshoot calculation uses the peak transient amplitude and the steady state amplitude for
a given set of system parameters χ, initial conditions (p0 , Φ0 ), and the damping coeﬃcient
D. The peak amplitudes are obtained by numerically solving equation (G.6) where Ψ1 is
the ﬁrst order correction for the damped system. There exists two solutions for Ψ1 that
satisfy the PDE in equation (G.5), but the solution containing the desired peak amplitude
152

information is given below

Ψ1 (p, Φ, χ) =

−3(G1 − G3 )(G1 − G4 )(G4 − G2 )(G2 − p2 )2 H2 H4 χ+

(H.1)

− (G1 − G4 )H1 H4 (G2 − p2 )2 3G2 χ − 3G1 G4 χ + G2 (3G1 χ + 3G4 χ − 2) − 2p2 +
2
+ 2χp4 − 2p cos(Φ) +
+ (G1 − G2 ) (G1 − G4 )H3 H4 (3G1 χ + 3G2 χ + 3G3 χ + 3G4 χ − 2)(G2 − p2 )2 +
− (G2 − G4 ) 3G4 χp4 − 3χp6 + 3G1 χ(G3 − p2 )(G4 − p2 ) + 3G3 χp2 (p2 − G4 )

/

2(G1 − G2 )(G2 − G4 )p sin(Φ) ,

where H1 , H2 , and H3 are elliptic integrals of the ﬁrst, second, and third kind, respectively;
which are deﬁned below
θ
H1 = F (θ, m) =
=F

0
(G2 − G4 )(G1 − p2 )
(G1 − G4 )(G2 − p2 )

sin−1
θ

H2 = E(θ, m) =

(1 − m sin(q)2 )−1/2 dq =
,

(G2 − G3 )(G1 − G4 )
(G1 − G3 )(G2 − G4 )

,

(1 − m sin(q)2 )1/2 dq =

0

(G2 − G4 )(G1 − p2 )
(G − G3 )(G1 − G4 )
, 2
,
2)
(G1 − G3 )(G2 − G4 )
(G1 − G4 )(G2 − p
θ
H3 = Π(n, θ, m) =
(1 − n sin(q)2 )−1 (1 − m sin(q)2 )−1/2 dq =
0
(G − G3 )(G1 − G4 )
G1 − G4
(G2 − G4 )(G1 − p2 )
, 2
=Π
, sin−1
2)
G2 − G4
(G1 − G3 )(G2 − G4 )
(G1 − G4 )(G2 − p
=E

sin−1

153

,

and H4 is deﬁned as follows

H4 =

(G1 − G2 )(G3 − p2 )
(G1 − G3 )(G2 − p2 )

(G1 − G2 )(G2 − G4 )(G1 − p2 )(G4 − p2 )
.
(G1 − G4 )2 (G2 − p2 )2

To compute H1 through H4 and ultimately equation (H.1), the functions G1 through G4
are required which are deﬁned below

1
G1 = −
2χ
1
G2 =
2χ
1
G3 = −
2χ
1
G4 =
2χ

2
1
J2 (1 − χ) − 1 +2 −
δ
3
2
1
J2 (1 − χ) − 1 +2 −
δ
3
2
1
J2 (χ − 1) + 1 +2 −
δ
3
1
2
J2 (χ − 1) + 1 +2 −
δ
3

J1
+ 4J2 +
21/3
J1
+ 4J2 +
21/3

J3
J1

J3
J1

J
J1
+ 4J2 + 3
J1
21/3
J1
+ 4J2 +
21/3

J3
J1

−δ+1 ,
−δ+1 ,

+δ+1 ,

+δ+1 ,

where
δ=

1+

1
3

J1
− 2J2 +
21/3

154

J3
.
J1

(H.2)

Finally, to compute the functions G1 through G4 , the functions J1 through J3 are required
which are deﬁned below

J1 = 2 − 18χ + 27χ2 − 24χp2 + 72χ2 p2 + 120χ2 p4 − 72χ3 p4 − 320χ3 p6 + 480χ4 p8 +
− 384χ5 p10 + 128χ6 p12 +
− 24χp 1 + 24χ2 p4 − 32χ3 p6 + 16χ4 p8 − χ(3 + 8p2 ) cos(Φ)+
√
+ 96χ2 p2 (1 − 2χp2 )2 cos(Φ)2 − 128χ3 p3 cos(Φ)3 + 3 3 χ3 896χ3 p8 − 768χ4 p10 +
+ 256χ5 p12 − 4(1 + 4p2 ) − 16χ2 p4 (9 + 32p2 ) + 9χ(3 + 16p2 + 16p4 )+
− 16p 1 + 64χ2 p4 − 96χ3 p6 + 48χ4 p8 − χ(9 + 16p2 ) cos(Φ)+
+ 128χp2 (1 − 6χp2 + 6χ2 p4 ) cos(Φ)2 − 256χ2 p3 cos(Φ)3
J2 = 1 + 2χp p − χp3 + cos(Φ)

,

J3 =21/3 1 + 24χ2 p4 − 32χ3 p6 + 16χ4 p8 − 2χ(3 + 4z 2 )+
− 8χp(1 − 2χp2 )2 cos(Φ) + 16χ2 p2 cos(Φ)2 .

155

1/2 1/3
,

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