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CHIGANS TA

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00909 6417

This is to certify that the

thesis entitled

CHAOTIC AND PERIODIC DYNAMICS OF A
SLIDER CRANK MECHANISM WITH SLIDER
CLEARANCE

presented by

FARAMARZ FARAHANCHI

has been accepted towards fulfillment
of the requirements for

—M~$.___degree in MECHANICAL ENGINEERING

MW

Major professor

Date 11-11-1991

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

CHAOTIC AND PERIODIC DYNAMICS OF A
SLIDER CRANK MECHANISM WITH
SLIDER CLEARANCE

By

Faramarz Farahanchi

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1991

ABSTRACT

CHAOTIC AND PERIODIC DYNAMICS OF A SLIDER CRANK

MECHANISM WITH SLIDER CLEARANCE

BY

Faramarz Farahanchi

Bearing clearance is one of the most important factors influencing the dynamic
performance and durability of mechanisms. In this thesis, the problem of a planar
slider crank mechanism with clearance at the sliding bearing is investigated. This
problem is relevant to the phenomenon known as piston slap to automotive engineers.
In this study the influence of the clearance gap size, bearing friction, crank speed,
and impact parameters on the response of the system are investigated. Three types
of response are observed: chaotic, transient chaos, and periodic. It is shown that
chaotic motion is prevalent over a range of parameters which corresponds to high
crank speeds and / or low values of the bearing friction with relatively ideal impacts.
Periodic response is generally observed at low crank speeds and also at low values
of the coefficient of restitution. Poincare maps and statistical profiles of the impact
locations and severity are used to characterize the motion and to obtain information
regarding possible patterns of wear due to repeated impacts. As expected, chaotic
motions lead to quite uniform distributions of impacts while periodic motions lead to
highly localized impact locations. Hence, chaotic motions may be beneficial, as they

provide a more desirable pattern of impacts.

TO MY PARENTS

iii

ACKNOWLEDGEMENTS

I would like to this opportunity to express my sincere gratitude to the following
people. First of all, I would like to thank my thesis advisor Dr. Steven W. Shaw. His
endless patience, guidance, help, and suggestions during my research and graduate
work is greatly appreciated. It has been a great privilege to work with him.

My especial thanks goes to Dr. Alan G. Haddow for being in my graduate commit-
tee and agreeing to be my supervising advisor. I would also like to thank Dr. Alejan-
dro Diaz and Dr. Philip Fitzsimons for agreeing to serve as my committee members.

I would also like to thank National Science Foundations for their support of this
investagation.

Finally, I would like to thank my parents and my grandparents for their support

and encouragements.

iv

Contents

List of Figures
1 Introduction

2 Mathematical Model
2.1 Basic Assumptions ............................

2.2 Derivation of the Equation of Motion for Free Flight Motion .....

2.3 Nondimensionalization ..........................
2.4 Linearization in 1,0 .............................
2.5 Impact Conditions ............................
2.6 Sliding Conditions ............................

3 Methods of Analysis

3.1 Poincare Section .............................

3.2 Impact Distribution Profile ........................

3.3 Impact Spectra ..............................
4 Results

vii

10

12

13

16

29

29

32

34

35

4.1 Introduction ................................ 35

4.2 General Observation of Chaotic Motion ................. 37
4.3 Non-sliding Dynamics .......................... 41
4.4 Dynamics Involving Sliding ....................... 49
5 Discussions and Future Direction for Research 60
A Derivation of 1pm,, and 1pm,” 64
B Simulation Routine 68
Bibliography 71

vi

List of Figures

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

3.1

4.1

4.2

4.3

4.4

4.5

Schematic view of the model ................ ' ....... 6
The Constraints for 1p .......................... 15
The model through the virtual displacement of the roller ....... 19
Constraint force on the roller at A = 0.01 ................ 23
Constraint force on the roller at A = 0.1 ................ 24
Constraint force on the roller at A = 0.25 ................ 25
Constraint force on the roller at A = 0.267, p = 0.02 .......... 25
Blow up of the constraint force on the roller .............. 26
Constraint force on the roller at A = 0.35 ................ 27
Regions of Sliding on the Supporting Structure. ............ 28
Three Dimensional phase space, with the restriction .......... 31
Time Trace ................................ 38
Phase Portrait ............................... 38
Poincare Map, A = 0.05 and p = 0.01 .................. 39
First Magnification of Poincare Map .................. 40
Second Magnification of Poincare Map ................. 40

vii

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

4.16

4.17

4.18

4.19

4.20

4.21

4.22

4.23

4.24

4.25

4.26

4.27

4.28

Impact Distribution Pattern, A = 0.025 and p = 0.010 ......... 42

Impact Distribution Pattern, A = 0.025 and p = 0.015 ......... 42
Impact Distribution Pattern, A = 0.025 and p = 0.020 ......... 42
Impact Distribution Pattern, A = 0.05 and p = 0.010 ......... 43
Impact Distribution Pattern, A = 0.05 and p = 0.015 ......... 43
Impact Distribution Pattern, A = 0.05 and p = 0.020 ......... 43
Impact Distribution Pattern, A = 0.075 and p = 0.010 ......... 44
Impact Distribution Pattern, A = 0.075 and p = 0.015 ......... 44
Impact Distribution Pattern, A = 0.075 and p = 0.020 ......... 44
Poincare Map, A = 0.05 and p = 0.015 ................. 45
Impact Spectra, Upper Wall A = 0.05 and p = 0.015 .......... 45
Impact Spectra, Lower Wall A = 0.05 and p = 0.015 .......... 45
Poincare Map, A = 0.025 and p = 0.01 ................. 46
Impact Spectra, Upper Wall A = 0.025 and p = 0.01 .......... 46
Impact Spectra, Lower Wall A = 0.025 and p = 0.01 .......... 46
Poincare Map, A = 0.075 and p = 0.02 ................. 47
Impact Spectra, Upper Wall A = 0.075 and p = 0.02 .......... 47
Impact Spectra, Lower Wall A = 0.075 and p = 0.02 .......... 47
Impact Distribution Pattern, A = 0.15 and p = 0.010 ......... 50
Impact Distribution Pattern, A = 0.15 and p = 0.015 ......... 50
Impact Distribution Pattern, A = 0.15 and p = 0.020 ......... 50
Impact Distribution Pattern, A = 0.20 and p = 0.010 ......... 51

Impact Distribution Pattern, A = 0.20 and p = 0.015 ......... 51

viii

4.29

4.30

4.31

4.32

4.33

4.34

4.35

4.36

4.37

4.38

4.39

4.40

4.41

4.42

4.43

4.44

A.l

3.1

Impact Distribution Pattern, A = 0.20 and p = 0.020 ......... 51

Impact Distribution Pattern, A = 0.25 and p = 0.010 ......... 52
Impact Distribution Pattern, A = 0.25 and p = 0.015 ....... . 52
Impact Distribution Pattern, A = 0.25 and p = 0.020 ......... 52
Poincare Map, A = 0.2 and p = 0.015 .................. 53
Poincare Map, A = 0.25 and p = 0.015 ................. 53
Time Trace, e = 0.9, A = 0.25, and p = 0.015 .............. 54
Poincare Map, A = 0.25 and p = 0.01 .................. 54
Poincare Map, A = 0.25 and p = 0.02 .................. 55
Poincare Map, 6 = 0.7, A = 0.05, and p = 0.02 ............ 56
Poincare Map, 6 = 0.5, A = 0.05, and p = 0.02 ............ 56
Impact Dist. Pattern, e = 0.7, A = 0.025, and p = 0.020 ....... 57
Impact Dist. Pattern, e = 0.5, A = 0.025, and p = 0.020 ....... 57
Time Trace, e = 0.9, A = 0.05, and p = 0.02 .............. 58
Time Trace, e = 0.7, A = 0.05, and p = 0.02 .............. 59
Time Trace, e = 0.5, A = 0.05, and p = 0.02 .............. 59
The maximum angle ........................... 64
Flow Chart: Simulation Routine ..................... 70

ix

Chapter 1

Introduction

Of the many factors which influence the dynamic performance and durability of mech-
anisms, one of the most important is bearing clearance. Such clearances often lead to
high localized stresses and thus to increased fatigue, noise, and wear. In this thesis
we choose a simple model problem which is of interest in automotive applications in
order to show that the dynamic response of such systems, which has been known to be
quite complicated, can, in fact, be chaotic. We also describe the influence of various
system parameters on the response and provide means of displaying data which may
be helpful in understanding patterns of fretting and wear.

The problem under investigation is a slider crank mechanism with clearance be-
tween the slider and its supporting structure. This problem is relevant to the phe-
nomenon known as piston slap to automotive engineers. Due to the clearance, re-
peated impacts occur between the slider and its supporting structure, resulting in
fatigue, dynamical stress, noise, and wear. In order to control these effects, one must

understand the underlying dynamics.

2

In recent years, the effects of clearance at the joints of mechanical systems has
been studied by many researchers. Most closely related to the present work is that
of Wilson and Fawcett [11] who considered the case of a slider crank mechanism with
clearance at the sliding bearing. They consider all possible impact configurations of
the slider using a two degree of freedom model and present simulation results for that
model. Other researchers have considered clearance at revolute joints in mechanisms
using a variety of methods. The research done in this area includes studies of the
response of rigid link systems with clearances at joints [15, 16, 6, 17], as well as those
with joint clearance and flexible links [18, 19]. Some investigations have considered
systems in which only a single joint has clearance [15, 16, 6, 17], while others consider
clearances in several of the joints of the mechanism [20, 21].

In these investigations, several different approaches are proposed. Grant and
Fawcett [7] suggested methods for predicting contact loss and a method of preventing
loss of contact. Dubowsky and Norris [8] provide an analytical and experimental study
for prediction of impact and they defined the Impact Prediction Number (IPN) which
correctly predicts the trends observed in their study. Dubowsky and Gardener [18]
considered multi-link flexible mechanisms with multiple clearance connections. The
methodology employed in that study was a perturbation coordinate approach and it
was shown that simpler models yield useful insights into behavior of the more complex
systems. Mansour and Townsend [9] took a momentum-exchange approach in their
study of impacts at a joint. They studied the local and path-spectra of the impacts
between the socket and the connecting pin and from these the dominant impacts at

the joints were identified and investigated.

3

In the work presented here, the problem of the planar slider crank mechanism with
clearance between a cylindrical slider and its supporting structure is investigated. The
equations of motion for the system are derived by using Lagrange’s method along with
a simple impact rule, and these are investigated by numerical solutions and Poincare
maps. The response of the system appears to be generally chaotic over a wide range
of parameters although periodic motions become more common as dissipation effects
are increased. Having determined the response of the system, techniques developed
in recent years, such as Poincare maps, are applied for further investigation of the
response.

Other studies along similar lines include that of Karagiannis [13] who studied
the problem of gear backlash. He has developed an understanding of the periodic
and chaotic behavior of gear backlash by applying similar methods. Pheiffer and
Kunert [14] have also applied these methods to the problem of rattling in gears due
to backlash. In addition, Shaw [1] has studied the response of an oscillator with
constraints subjected to harmonic excitation by using similar methods.

This thesis is arranged as follows. The underlying assumptions and the equations
of motion for the system under study are provided in Chapter 2. Chapter 3 describes
the methods of analysis. Chapter 4 contains the main results and describes the
influence of various important system parameters on features of the response by using
simulations. A discussion and directions for future research are provided in Chapter 5.
Some detailed derivations and a summary of the simulation routine are provided in

the appendices.

Chapter 2

Mathematical Model

In this chapter we describe the model employed and derive the equations which gov-
erns its dynamics. The basic assumptions are provided first. Next, the differential
equation which describes the free flight dynamics, that is, the motion between im-
pacts, is derived. It is nondimensionalized and then linearized based on a small clear-
ance assumption. The rules which govern the impact dynamics are then presented.
Also, a complete discussion regarding the possible regions of potential sliding and
the attendant constraint forces is included. This provides a complete set of dynamic

equations which are studied in subsequent chapters.

2.1 Basic Assumptions

In order to provide some insight into the effects of clearances on the motion of a con-
necting rod / piston assembly in a slider crank mechanism, we introduce a simplified

model, shown in Figure 2.1, which employs the following assumptions:

All motions occur in a fixed plane.

All components are rigid.

The bearings joining the connecting rod to the crank and the piston have no

clearance.

The bearing which connects the connecting rod to the crank provides a viscous
type frictional moment which is proportional to the relative rotational rate

between the crank and connecting rod.

The crank speed is constant.

The nominal mechanism, that is, the one without clearance, is an on-line slider

crank.

The clearances for the piston are symmetrically placed about the nominal piston

path and have a fixed magnitude along the cylinder wall.

Gravitational effects are ignored. This is valid when the mechanism lies in a
horizontal plane or when the crank speed is sufficiently large that inertial forces

dominate the response.

The piston is a cylindrical roller which is attached to the connecting rod at its
geometric center. This simple geometry allows for the isolation of the effects of
A the clearance from complications associated with the piston geometry. It also
reduces the mechanism to a single degree of freedom system, since the piston

orientation does not affect the dynamics.

6

o Impacts between the piston and the walls are instantaneous and modeled by a

simple impact rule which employs a constant coefficient of restitution.

2.2 Derivation of the Equation of Motion for Free

Flight Motion

For deriving the equation of motion, the Lagrangian approach is used. To apply
the Lagrangian method, the kinetic energy and potential energy of each component
is required. Based on the assumptions made in the previous section, there is no
potential energy in any component of the system. In order to obtain the kinetic
energy of each component, the center of mass velocity and angular velocity of each

component is required.

Y

 

Figure 2.1: Schematic view of the model

For the generalized coordinate which describes the configuration of the system we
choose 2/2 , the angle which measures the deviation of the connecting rod centerline
from its nominal (that is, zero clearance) orientation; see Figure 2.1. This angle is

chosen since it will remain small when the clearance is small.

7

From Figure 2.1 the displacement of the center of mass of the uniform connecting
rod can be expressed in terms of Kb , the angle measuring the deviation of the
connecting rod form its nominal position, (,6 , the angle between the nominal position
of the connecting rod and the horizontal, and 0, the crank angle measured as indicated

in Figure 2.1, as follows:

2:, = rc059+ élcosw—ib)

y, = r sin9 — élsinM—zp)

Note that if) and 0 are related; this will be subsequently exploited.
By taking the time derivative of the above displacements, the horizontal and

vertical velocities of the connecting rod center of mass can be expressed in terms of

96, «p, and 0 as follows

i, = —r0.sin0-%I(¢.3—¢l) 3in(¢—¢)

g, = r0 cosH—%I($—¢3) WSW-'1’)

By using the above velocities and the angular velocity of the connecting rod, ¢+¢ ,

the kinetic energy of the connecting rod is obtained

1:: %m(x'3+ :23) + g J. «is + a)” (2.1)

 

 

01‘

T, = émfl—résinO—élfi—Jt) sin(¢—¢)]2

+[ro' cow—gnaw) cos(¢—¢)P}+§J. (4w)? (2.2)

where m is the mass of the connecting rod and J, is the moment of inertia of the
connecting rod about its mass center.

From Figure 2.1 the displacement of the center of slider in terms of ¢, 1p, and 0

is determined to be

:c, = r c030 + Icos(¢-¢)

y, = rsin0 —Isin(¢—z/2)

By taking time derivatives of these, the horizontal and vertical velocities of the

slider are obtained

is = —résin0—l(ql—zp) sin(¢—z/))
37: = récOSO—Htp—zp) cos(¢—¢)

Note that in the absence of clearance the vertical position and velocity of the
slider piston must be zero. This can be verified by using the geometry of the nominal
mechanism. Due to the rotational symmetry of the slider mass, its angular orientation

is inconsequential to the response of the system and thus only its translation is of

9

interest here. The relevant kinetic energy of the slider is

01'

T. = g M (at: + :23) (2.3)

T, =: %M{[—r03in0—l(<,iS-1/})si11(<15--¢)l2

+ M coso —I(¢-¢) case—«pm (2.4)

where M is the mass of the slider piston.

Having obtained expressions for the kinetic energies of the components, the total

kinetic energy of the system is then expressed as

Ttotal —
Tidal _

fI‘total _

T. + T.

émctz + :23) + éJAJ» + 213)” + $4403 + 93)

gm { [st sin0 -— $1 (4.5-113) sin(¢—¢)12

+[ra cosO— guts—:13) cow-11)) PM? (43%)”
+ éMfl—ré sin0 — 1(45 — t) sin (43 - #012

+ W cow - 103 — ti) cos (¢ — 31)) 1’} (2.5)

The frictional moment at the crank-connecting rod connection is assumed to be

proportional to the relative angular velocity of the components connected there. The

10

resulting dissipative moment on the connecting rod can be expressed as
M2 = -02(-é - ¢ + Kb.) (2-6)
For determining the equation of motion, Lagrange’s method is then applied:

_(____) _ _ = Q' (2.7)

where

q = 1b
Q, = -62("é-¢.5+t/3)

L = Ttotal

which results in the following equation of the motion for the system

%m{—flzrl [sinflt cos(¢ — 1p) — cos Qt sin ((15 — 1.0)] - :- 12 ( (I; — '2; l}
— M {921:1 [sinflt cos(¢—1/J)+ cosflt sin(¢—¢) + 12( 45—1/3)”

+ J. (513+ d3) = —c2 (—9 - 43+ «13) (2-8)

2.3 Nondimensionalization

It is convenient to present the equation of motion in dimensionless form. By nondi-

mensionalizing, the number of parameters associated with the equation of motion is

11

reduced. In order to nondimensionalize the equation of motion, both sides of the

equation are first divided by M12512 as follows:

figs—23992,»: [sin nt cos (¢ — xi) + cos nt sin (¢ — 1101—; I” ( 55- 113)}

- W132); {flzrl [sinflt cos (¢ — 11)) + COS Qt sin (‘A - ¢) + 12(9- I; M}
Jr ,. ,, _c2 . . .
+ W (<15 + 1P) = M1202 (-9 - ¢ + 4)) (2'9)

 

 

 

 

T
6 = 7
_ L’i
0‘ “ M
Jr
3 = M12
A M129
A = 1—{231n21'
T = at

M
£2

(Y

the dimensionless equation of motion is obtained by straightforward rescaling. It is

given by

12

—%a{£[sinr cos(¢—¢) + cosr sin(¢—¢)1+§(¢"—¢")}
— {élsinr cos(¢—z/2)+ sins-01+ (¢”—¢")}
+,B (¢"+1,0") = —A (—1 — ¢'+z0') (2.10)

2.4 Linearization in 1,0

Note that, from Figure 2.1, (0 can be written in terms of 0 based on the following

constraint equation

I sin¢= r sin0 (2.11)

or in dimensionless form

sin¢= é sinr (2.12)

since T=Qt=0.
Since 10 will be restricted to small angles for all reasonable clearances, the following

approximations can be used for 10

c0310 = 1+0(z,02)

sim0 = wows

By using the above assumptions, retaining only those terms linear in t0 and

derivatives of 1,0 , and utilizing equation 2.12, the equation of motion can be expressed

13

in linearized form in terms of 10. The dimensionless, linearized equation of motion is

given by

where

a a" + b w + chw = f(r) (2.13)
40 + 1 + 3
A

(1+é-a)§A cosr—(l +31?!) {2 sinz'r

 

1 ((3—0 sinr 1 .

(Ea-i-l-fl) A3 +(EA+-2-a£A) smr
l 1 . A6 cosr

+(§E2 +162 a)s1n2r+A+—-A———

Note that in the equation of motion (i.e., equation 2.13), the only dimensionless

system parameter that depends on the crank frequency fl is the dissipation param-

eter , A . This implies that a study which varies A is equivalent to one which varies

the crank operating speed. Also note that the equation of motion has both external

and parametric excitation which arise from the gross motion of the connecting rod.

The solution of this equation of motion describes the motion of the connecting rod

during flights between the barriers.

2.5 Impact Conditions

The magnitude of the clearance is described by the difference between the slider

radius (in the plane of motion), r“, and the distance from the nominal piston motion

14

centerline to the cylinder wall, (1 . Thus the piston is free to move a distance d — 7'"
up or down from its nominal position.

Based on this gap the motion of the connecting rod is restricted to lie in a region
10min S 10 5 10m” . From the geometry of the model, for I 10 |<< 1 the maximum

and minimum values that 10 can achieve can be shown to be given by

 

 

 

fpmaz = P (2'14)
\/1 — £2 sin2 7'
2l’min = “P (2.15)
\/l — {2 sin2 1'
where
_ d — r“
” ' 1
is the dimensionless gap size. Note that 10",“. = -—z/)m,-,, is valid only for linearized

case (i.e. under the assumption of small gap size). A detailed derivation of 10m“,
and 10min are provided in Appendix A.

Figure 2.2 depicts the time dependent region in which 10 can lie without impact
occurring. Note that impact occurs whenever 10 = 10",” or 1,0 = 10",," is satisfied
during a given motion. A simple impact rule is assumed in which the relative velocity
after impact is proportional to the relative velocity before impact, with the constant
of proportionality equal to the negative of the coefficient of restitution [24]. This is
done by using the vertical component of the absolute velocity of the slider center,

3}. . This is the natural velocity to use since it is normal to the impact surface and

15

 

 

 

 

 

 

 

 

 

 

p=0m
l I l I W l
0.01 —' 1pm“: _
Ibms'n _"
0.005 - -
20 o
—0.005 - -
—0.01
l l l l l I
0.0 1.0 2.0 3.0 4.0 5.0 6.0
0 - Crank Angle

Figure 2.2: The Constraints for 20

since it gives the relative velocity between the slider and the stationary supporting

structure. The impact rule is given by

y‘aftcr = _e ysbefore (2.16)

Using the expression for g, from section 2.2, nondimensionalizing, linearizing in 1,0 ,

and solving for 10“,," leads to the following expression for the velocity after impact

in terms of 10 :

£2 sinr cosr

A2 III) I — eibbcfore (2'17)

0.}... = (1 + e)[

 

where

Coefficient of restitution

8

 

16
and 1/1 = 10m: 01' '11 = tbmtn-

Note that 10“,," is found by assuming simple impacts between the roller and
the wall of the cylinder. In simulations the exact location and time of the impact
is determined in the simulations by applying Newton’s root solving method to the
conditions 10 = 20",” or 10 = 10",," (see Appendix B for details).

We now have a complete set of dynamic equations. Equation 2.13 gives the free
flight and equations 2.14, 2.15, and 2.17 determine the impact conditions. It should
be noted that since an infinite number of impacts can occur in finite time, sliding
motions may occur in which 20 = 10",“ or 10 = 10".," for finite time durations (see

Wilson and Fawcett [11]); this is considered in the following section.

2.6 Sliding Conditions

In the problem under investigation there are occasions in which the roller may slide
along the supporting structure. This occurs when several impacts occur close together
in such a manner that the relative velocity goes to zero via an infinite number of im-
pacts which occur in finite time. This is analogous to the simple problem of dropping
an inelastic ball on a rigid surface and letting it bounce; it similarly comes to rest
in a finite time after an infinite number of impacts ( Greenwood [25], pp 160-161
). In the mechanism under consideration, this typically occurs when the friction at
the joint connecting the crank and the connecting rod is large, or if the coefficient of
restitution is small, or some combination of these conditions.

The analysis of sliding motions is carried out as follows. We begin by computing

17

the normal forces between the roller and the supporting structure under the assump-
tion that the roller moves along paths «,0 = 1,0,“; or 10 = 10”,,“ . These are the forces
which would be required to maintain contact and will represent the actual contact
forces in these cases in which the force is compressive. A tensile force is non-physical
and occurs in regions where sliding will not occur. For a given set of parameters
( A, p ) as the crank advances these forces vary and change direction at certain crank
angles. At those angles where the force changes from compression to tension, the
roller, if sliding, will be released into free motion between the constraints. Crank
angles at which the force changes from tension to compression correspond to the
beginning of crank angle intervals in which sliding can occur on the corresponding

constraint. This analysis indicates that there exists four types of crank angle regions:

0 those in which sliding can occur only along the upper constraint,

0 those in which sliding can occur only along the lower constraint,

0 those where sliding can not occur along either constraint.

0 those where sliding can occur along either constraint.

Depending on p and A , the crank angle may be broken into distinct intervals
of these types. It should be remembered that free flight can occur in any region, and
that sliding will occur in the allowed region only under certain conditions.

The constraint force is derived using the principle of virtual work and Lagrange’s
method. The procedure is to first obtain the equation of motion for an uncon-

strained connecting rod with and applied vertical force on the roller. The constraints

18

10 = 10",“ and 1,0 = 10",,-,, are then substituted into the equation of motion and
the resulting applied forces , i.e. those required to maintain contact on the lower and
upper constraints, respectively, are solved for directly.

Lagrange’s equation for this case is given by

— —- — — = ' 2.18
where
Q = The generalized constraint force due to the supporting structure,
Q' = Generalized forces due to dissipation,

T = Kinetic Energy of the system.

The principle of virtual work is applied for deriving the generalized constrained
force Q . The virtual work done by an applied force is given by the applied force

times an arbitrary virtual displacement

Note that in this case, the coordinate y is a function of the generalized coordinate
and time ( i.e. y = f (z0,t)). Thus, virtual displacements of y can be expressed in

terms of the corresponding virtual displacement of 1.0 by differentiating y with

 

Up]

and

19

respect to 10 while holding time fixed:

This results in the corresponding virtual work

__ 33’
6W _ F (9—20- 6¢ (2.21)

'<
\

   

Sn"-..-

  

Figure 2.3: The model through the virtual displacement of the roller

Figure 2.3 presents a schematic view of the the model. Note that in this figure the
uPper supporting structure is replaced by a force F,, acting on the roller. From the
geometry of the model, as shown in Figure 2.3, the following relationship between y

a-Ild 1,0 is obtained

3] = Isin¢ — I sin(¢ — 20) (2.22)

20

which results in the following derivative

3% = 1cos(¢ — (b) (2.23)

Substituting the above in to the virtual work equation results in
(SW = —F,, lcos(¢ — 20) 610 (2.24)

which can be written as

6W = Q 51/) (2.25)

where Q is the generalized force associated with the generalized coordinate 10 and
is given by

Q = —F., I cos(¢ — 10) (2.26)

The following expression, equation 2.5, for the kinetic energy was derived in section

2.2

11..., = gMurn sinm +1 sin(¢—¢).(J>—¢)]”
+[rfl coth—%I(ti>—z,0) cos(¢—1,0)]2}
+-;-m{[rfl sinflt 4%: sin(¢—z0)(<0-10)]2

+ I’” ”8‘" “i" cosmos—43m + $2054.12)”

By using the above equation, the equation of motion which takes into consideration

 

 

 

IO

0U

21

the constraint generalized force is found by applying equation 2.19 with the following:

d 8T 8T

$577)) _ .07). = % m{—92rl [sinflt cos(d> — (,0) + COS Qt sin (45 — 10)]

—-;-I2(<0—10)}- M {Q2r1[sinflt cos(¢—(0)
+ coth sin(¢-t0) + 12(55—10)]}+ J,($+1,Z)
Qu = —F,,Icos(¢—10)

Q' = —c1,0+cfl+cql

The above is non-dimensionalized by dividing both sides of the above equation by
M I 02 , as in section 2.2 . The dimensionless form of the equation of motion which

includes the generalized force is then given by:

—%a£[sin 7 cos ((0 — 10) + COS 7' sin (<0 - 10)] - i495” - ‘1’")
—£ [sin 1’ cos ((0 - 10) + cos 1 sin ((0 - 10)] — (43” — t0”)
— 5 (¢" — ‘0’”) = -fucos(¢ " ‘0’) + A + ,\¢’ - All), (2-27)

where the dimensionless parameters are described in section 2.2.

The new dimensionless parameter fu corresponds to the dimensionless generalized
force from the upper surface acting on the roller. A similar analysis can be carried out
for the lower supporting structure. The dimensionless generalized constraint forces

on the roller from the upper and lower structures are given by

Fu/l

f,” = M102 (2.28)

 

22

where f; is the force of the lower supporting structure on the roller. It is obtained
by using the same approach as above with F} taken in the opposite direction of
Fu for consistency in sign: a positive force implies compression between the relevant
constraint and the roller.

The forces of the constraints on the roller at 10",“ and 10m” are obtained by

simply solving the equation of motion for f“), , yielding:

f“); = SSS—(Z—i—tp-fi—éaflsinrcos ((0 — 10) + cos T sin (<0 -— 10)]

-€ [sinT COS (45 - *0) + COS T sin (<0 - 0)] - (¢” - t0”)

‘ i002” - 112”) + 3 (¢” + 112”) — A - M’ + W} (229)

with 1,0 = 10m” for the ”u” and 10 2 10min for the ”I” subscript. Note that
the ”— ” corresponds to the force of the upper supporting structure on the roller
(i.e. f“) and ”+ ” corresponds to the force of the lower supporting structure on the
roller (i.e. f; ). Based on this sign convention, whenever the force is positive, the
roller may be sliding and whenever it is negative the roller cannot be sliding on the
corresponding supporting structure. Thus, in order to have the roller sliding on the
supporting structure, 10 must be at its maximum / minimum, and fu / f; must be
positive, respectively.

The parameters which influence the possibility of sliding are the gap size and the
bearing friction (or, equivalently, the crank speed). The gap size affects the geometry
While the bearing friction (crank speed) influences the moment on the connecting rod

Which in turn influences the constraint force. The coefficient of restitution cannot

23

affect the constraint force, and hence does not play a role in determining the ( A, p )
parameter regions or crank angle intervals in which sliding may occur. However, it
does influence how frequently sliding actually does occur in these regions since it has
a direct effect on the time required for settle-out of the impacts into a sliding motion.
A set of results are presented here that demonstrates the effect of these parameters
on constraint force. These results are based on the parameter set given in section 4.1.

For p = 0, and A = 0 the constraint forces possess the following symmetries:
(i) fu = -f; and (ii) fu(9 + 7r) = —fu(0). For p = 0,A 54 0 symmetry (2') holds
but (if) is broken. For p 79 0, A = 0 symmetry (2) is broken while (ii) holds. As

is demonstrated below, these facts have some interesting consequences.

p = 0.01, A = 0.01
0.4

0.3

0.2

0.1

Force 0
—0.1
—0.2
—0.3

_0.4 l 1 I J | J

0.0 1.0 2.0 3.0 4.0 5.0 6.0
0 - Crank Angle

 

 

 

 

 

 

Figure 2.4: Constraint force on the roller at A = 0.01

Figure 2.4 presents the forces of the upper and lower supporting structures on the
roller for p = 0.01 and A = 0.01. Note that, there are two crank angles at which the

constraint forces change direction. From approximately 0 to 1r in crank angle the

24

roller may slide along the upper surface while from approximately 7r to 27r it may

slide along the lower surface.

0.5

0.4

0.3

0.2

0.1

Force 0
—0.1
—0.2
—0.3 - e
—0.4 — A

__05 1 1 1 1 1 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0
0 - Crank Angle

 

 

 

 

 

 

Figure 2.5: Constraint force on the roller at A = 0.1

Figure 2.5 shows the case with p = 0.01 and A = 0.1. As the dissipation
parameter increases, the force tending to hold the roller against the upper supporting
structure increases, in which case the roller can be sliding along the upper supporting
structure for a longer period. Figure 2.6 demonstrates the effect of increasing the
dissipation parameter to 0.25. Comparison between Figures 2.4, 2.5, and 2.6 suggests
that as A increases, as expected, the force of the supporting structure on the roller
increases and the sliding region on the upper supporting structure is extended.

Although in the figures shown to this point it appears that f,, = — f1 , this is not
the case in general. It is true only for p = 0 . The quantity (fu + f,) deviates
from zero as p is increased from zero. This lack of symmetry allows for a situation

in which the roller may slide along the upper constraint over the entire crank cycle

25

p 2: 0.01, A = 0.25
0.8 I I I r w 1

0.6

0.4

0.2

Force 0
—0.2
—0.4 ~
—0.6 - -

_08 1 1 1 1 1 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0
0 - Crank Angle

 

 

 

 

 

 

Figure 2.6: Constraint force on the roller at A = 0.25

p = 0.020, A = 0.267

 

 

 

 

 

 

0.005 I l I

0.004 -

0.003 -

0.002 -

0.001 - /—\

Force 0
—0.001 - —
—0.002 - d
—0.003 ~ ' -
-0.004 I- d
_0.005 1 l I 1 I l L
4.85 4.9 4.95 5 5.05 5.1 5.15 5.2 5.25

0 - Crank Angle

Figure 2.7: Constraint force on the roller at A = 0.267, p = 0.02

26

and may also slide along the lower constraint over a small crank interval. Such a case

is shown in Figure 2.7 for p = 0.02 and A = 0.267

p = 0.01, A = 0.1

0.05 T I T I f
0.04 - f r
0.03 r "
0.02 " ‘
0.01 '- "
Force 0
—0.01 - -
—0.02 e _
-0.03 - -*
—0.04 - -

_005 1 1 1 1 1

3 3.5 4 4.5 5 5.5 6
9 - Crank Angle

 

 

 

 

p

 

Figure 2.8: Blow up of the constraint force on the roller

The lack of symmetry also permits the existence of small crank angle intervals
over which sliding may occur on either the upper or the lower structure. Figure 2.7
clearly shows such a interval in the range 4.96 < 0 < 5.14 . A similar situation occurs
near the leftmost crossing point (i.e., the continuation of the crossing at 0 = 7r for
p = A = 0) of f“ and f; for all values of p and A for which crossings occur. Similarly,
near the second crossing point (the continuation of the one originating at 0 = 21r
for p = A = 0), there exists an interval over which sliding cannot occur along either
constraint. Figure 2.8 shows a blow up of the case from Figure 2.5 (A = 0.1, p = 0.01)
which clearly shows these regions. These intervals are typically small and for small
p do not have a significant effect on the dynamics of the system.

Figure 2.9 represents the results of having very large friction at the joint (or low

 

 

 

 

511

Ml.

27

p = 0.01, A = 0.35
0.8 I 1 T I l l

0.6

0.4

0.2

Force 0
—-0.2
-—0.4
—0.6

—08 1 1 1 4 1 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0
9 - Crank Angle

 

 

 

 

 

 

Figure 2.9: Constraint force on the roller at A = 0.35

speed), A = 0.35. Here the roller may slide along the upper supporting structure
throughout the entire crank cycle. Note that in this situation, if started in a sliding
mode of motion, the roller will remain sliding along the upper surface and will never
release.

Figure 2.10 presents an overview of the effect of the gap size and the dissipation
coefficient on the sliding of the roller on the upper and lower supporting structures.
Region I corresponds to the parameter ranges where there can be sliding on both
parts of the supporting structure during one cycle, for example, as shown in Fig-
ures 2.4, 2.5, and 2.6. Region II represents the parameter range in which the roller
can be sliding on the upper supporting structure during the entire cycle and there is
some interval in each cycle for which the roller can also slide on the lower supporting
structure, for example, as shown in Figure 2.7. Finally, region III corresponds to the

range of parameters for which the roller can slide on the upper surface over the entire

28

Dissipation Coefficient vs. Gap Size

0.08 I I l
0.07
0.06
0.05
P 0.04
0.03
0.02
0.01

 

I
1

Region Region Region
I II III -

I

I
l

I
l

I
l

I
l

I
l

 

 

 

0.26 0.265 0.37 0.275 0.28

Figure 2.10: Regions of Sliding on the Supporting Structure.

crank cycle and nowhere can it slide in contact with the lower surface, for example,

Figure 2.9.

Chapter 3

Methods of Analysis

 

In this chapter, three methods for presenting the results obtained from the equations
of motion, equation 2.13 and equation 2.17, are described. First, the concept of a
Poincare map is introduced as a geometrical means of presenting simulation data.
Next, a statistical analysis which presents the average number of impacts and the
associated impact velocities along locations of the supporting structure is described.
This method of presentation provides information regarding possible patterns of wear
in the cylinder. Finally, impact spectra are defined that provide information regarding
the number of impacts and the associated impact velocities as a function of the crank

angle. This is essentially a polar representation of the statistical analysis.

3.1 Poincare Section

An important conceptual tool for understanding the behavior of a time periodic sys-

tem is the Poincare map. Typically this map is defined to sample the system’s

29

 

 

displa

perio<

ten '1

in set

tion

152

wh

tht.

30

displacement and velocity, or, more generally, its dynamics state variables, once per
period of the excitation. Using such a map, a continuous-time non-autonomous sys-
tem is reduced to a discrete-time system with one less dimension.

The linearized and dimensionless equation of motion, equation 2.13, was obtained

in section 2.2, and is restated here:

a 10” + b 10’ + C(TW = f(T)

where C(T), and f (T) are all periodic in T with period 21r.
Note that this equation is valid for 10",,“ S 10 5 10",“. Also, based on the assump-
tions given section 2.1, whenever 1,0 = 10",” or 10 = 10min the simple impact rule [24]

is applied (i.e. equation 2.17) which results in a jump in velocity given by:

 

[52 sin 1 cos 7'

Illaftcr = (1 + e) A2 71") l - czi’before

where e is a coefficient of restitution with a value between zero and unity. Note that
the above equations completely determine the dynamics of the system.

To define the Poincare section for this system, we first write the equation of motion
in the form of a first order system of equations with (1,0, 10, r(mod21r)) = (2:1, 0:2, 0)

as follows:

331’ = $2 (3.1)

$2, —-b-$2 — Ema + E (3.2)
a a a

 

 

 

 

 

 

(the:

$90.

liq,

Ia};

31
0’ = 1 (3.3)

where ( )’ corresponds to the dimensionless time derivative (as described in
section 2.3 ).

The three variables, (2:1 , x2, 6), are required for specifying the state of the system.
In this three-dimensional phase space, solutions are restricted to

10min < 2:1 3 10m”. The time variable is taken to be the crank angle so that it

takes on values between 0 and Zr , i.e. 0 S 0 S 27r .

 

 

 

 

 

Figure 3.1: Three Dimensional phase space, with the restriction

Figure 3.1 illustrates the three dimensional extended phase space (10 ,10 ,r)
and its restrictions for the present system. It can be seen that a Poincare section is

conveniently defined in this case using the surface of 10",“ or 10,,,,-n in the three

32

dimensional space. The Poincare section for this system is thus defined as

Z = {($11 $216) I 131 = Il’maan $2 > 0} (3.4)

The Poincare’ map P is a rule that takes points in 2 back into 2 under the

action of the equation of motion. It can be represented as

P i Z —" Z a 01', (3214120914) = P0323300 (35)

where the points (2:2,, 9,) are in Z . Note that points in 2 correspond to the velocity
of the roller as it is just coming in to contact with the constraint at 10 = :21 2 10m“,
at the ith impact, that is 2:2,- : 10,- , and the crank angle at which the impact occurs,
0; .

The map P simply relates points between successive impacts in an implicit func-
tional form. The map can not be written in closed form for this system and will
be generated by directly sampling the points as defined during simulations. Simi-
lar maps have been used in the investigations of other impacting systems and are
known to possess interesting singular behavior (see Shaw [3], Shaw and Holmes [4],

Karagiannis [13], Whiston [22], and Pheiffer and Kunert [14] ).

3.2 Impact Distribution Profile

The following approach is taken for performing the statistical analysis. We define the

domain of the possible dimensionless impact locations on the supporting structure as

33

D = [1 — C, 1 + C], and the coordinate a: as the location of an impact measured such
that :1: = 1 corresponds to the mid point of D . Thus a: E D. Note that a: = 1 —6
corresponds to the crank angles 0 = (2n + 1)1r (for n =0, :l:1, :i:2, . . .) and a: = 1 +6
corresponds to the crank angles 0 = 2n1r ( for n = 0, i1, i2, . . .).

The domain D is subdivided into j equal subintervals of width A2: = Z]; , desig-
nated as Asa-,1 = 1,2, - . - , j. In each Ax,- , there will be N,- impacts during 12 crank
rotations. The total number of impacts during this time is given by N = £{=1N;.

We designate the impact velocities over this period according to their impact
location by labeling them as 10k(A$,'), k = 1,2,---,N,-. Based on this, we can
compute an average impact velocity for each interval Ax,- as

< 10(Axt) >= 2&1 110M“) i= 1,2, - - - ,J' (35)

 

Bar graphs of < 1,0(A:r,-) > versus A3,- for i = 1,2,~ - ~ , j provide a measure
of the number of impacts and their severity, given by the impact velocity, versus
the dimensionless location of the impact on the supporting structure. It is worth
noting that this distribution is the product of the average impact velocity in each
Am.- and the fraction of impacts which occur in each Ax,- , that is %. If 10 is
a stationary process, then as N —-> 00 these graphs will converge on to a limiting
distribution. These plots provide useful information regarding possible locations of
wear, heat generation, noise generation, and fatigue due to repetitive impacts. They
will be referred to as impact profiles or impact distribution profiles. In these diagrams

average impact velocities on the upper supporting structure are shown with positive

34

sign while those on the lower supporting structure are shown with a negative sign. In

this manner a single diagram can capture the entire impact distribution.

3.3 Impact Spectra

Impact spectra provide information relating the number and severity of impacts to the
crank angle. Note that impact spectra are a polar representation of the data defined
in the previous section. Here we define the domain as D = [0, 2w) . We subdivide
the domain into j equal subinterval of width A0, labeled as A0;,z' = 1, 2, - - - , j . Let
N, be the number of impacts which occur in A0.- in n revolutions of the crank and
let N = 2le N.- be the total number of impacts. Here the impact velocities are
labeled according to the crank angle at impact as 43,.(A0.), k = 1, 2, - ~ - , N,- . Based

on this, we compute an average velocity for each interval A0,- as

 

N- '
- .1 A0,
< 10(A05) >= 2"" I)“ ) (3.7)
N
This polar representation of the impact velocity provides useful information re-
lating the impacts and the corresponding crank angles. Note that separate spectra

figures must be produced for impacts on the upper and lower supporting structure.

Chapter 4

Results

4.1 Introduction

In this chapter we present results from simulations. Results are presented in various
forms: simple time traces (10 vs. t), phase planes (1]) vs. 10), Poincare maps, impact
distribution profiles, and impact spectra. We emphasize the use of impact distribution
profiles since these are the most relevant to the practical issues of wear.

Some general observations for a typical chaotic response are given first, and then
these are followed by sets of simulations which describe some general behaviors which
are observed as the gap size, p, the dissipation parameter, A, and the coefficient of
restitution, e, are varied. These results are grouped into three sets. The first is
for parameter ranges where sliding is not observed during the steady state response,
this range is for small values of A (A << 1) and e values slightly less than unity
((1 — 6) << 1). The second set of simulations is also for 6 near unity, but A values
are large enough so that sliding occurs during the steady state motion. The final set

35

36

of simulations is carried for fixed p and A, with e decreasing. This set indicates the
general trend observed as the impacts become more dissipative.

In the simulations three general types of responses are observed. The simplest
type of response is periodic, in which impact patterns are repeated and the motion is
regular. The second type of motion is ultimately periodic, but experiences “transient
chaos” . This occurs when chaos is observed over a significant period well beyond what
would be typically considered as transient, and the motion then transits into a periodic
pattern, in which it remains thereafter. The last type of response is chaos which is
sustained over at least 1000 crank cycles. Each of these three types are observed
in sliding and non-sliding cases, although periodic motions are more prevalent when
sliding occurs.

Typical values for a slider-crank in an automotive four-cylinder gasoline engine,
taken from a particular Ford Motor Company engine [12], are used in this study.

They are

m = 0.00399 lb. sec.2 in.‘1
M = 0.00210 lb. sec.2 in.‘1
r = 1.6535 in.
I = 5.4570 in.

I, = 0.0295 lb. sec.2 in.

37

which result in the following dimensionless parameter values

5 = 0.3030
0 = 1.9000
fl = 0.4719

Note that throughout this investigation, these values are kept constant and the
study is performed based on varying the values of the dissipation parameter, A , the

gap size, p , and the coefficient of restitution, e.

4.2 General Observation of Chaotic Motion

Figure 4.1 represents a time trace of the steady state behavior of the system (i.e.
10 vs. 1' ) for two crank cycles, for parameter values A = 0.05,p = 0.01,and e = 0.9,
obtained after 25 cycles during which transients have decayed. Note that, while
certain patterns are repeated, no strict periodicity is observed in this figure. In fact,
the system has a chaotic response to the periodic excitation provided by the crank.
Also, note that from this figure it is explicitly seen that 10 is constrained by 10,“,
and 10mg... Figure 4.2 shows the phase portrait (1,0 vs. 10) of the system for the same
two cycles of the crankshaft with the same parameters. A Poincare plot of the
system showing points corresponding to the absolute velocity before impact at the
top side of the cylinder and the crank angle at impact is presented in Figure 4.3. This

plot, taken over 3000 crank revolutions, clearly demonstrates the chaotic behavior of

38

p = 0.01, A = 0.05, Steady State Behavior for 2 cycles

 

0.015

 

 

 

 

 

 

 

 

I I l I I n
0.01 ‘
0.005 - -
0

-—0.005 - A

-0.01 -——__...————-— .
—0.015 ‘ ' 4 1 ' ‘

0 2 4 6 8 10 12

1' - dimensionless time

Figure 4.1: Time Trace

p = 0.01, A = 0.05, Steady State Behavior for 2 cycles

I I

 

0.06

0.04 -

0.02 ~

 

0

 

 

 

—0.02

—0.04 ~

 

W

 

 

 

—0.06 ' ‘
—0.015 —0.01 —0.005 3) 0.005 0.01 0.015

 

Figure 4.2: Phase Portrait

39

p OOI.A‘005

 

012
011
0‘0
9091

0004
007
,1 0064
005
0004
003

 

001
001

 

 

on 3'0 ‘0 "‘ '9

Crank Anqlc 0

Figure 4.3: Poincare Map, A = 0.05 and p = 0.01

the system. During an initial transient period, the points from the map appear to be
randomly placed. However, as more data points are taken the highly ordered pattern
shown is formed. Figure 4.4 and Figure 4.5 contain two consecutive magnifications
of the Poincare plot (Figure 4.3) which demonstrate the fractal nature of this chaotic
attractor (Moon [26]).

An interesting feature of this strange attractor is that it is composed of several
”lobes”, some of which are distinct but others of which conjugate near zero velocity.
These lobes are correlated to the physical motion in the following way: A series of
impacts occur which has points in these lobes moving sequentially from the upper left
to the lower right. This corresponds to a simple sequence of impacts which occur as
the crank advances and for which the impact velocity being reduced at each impact.
Eventually the moment acting on the connecting rod reverses sign, and it is released
from this sequence for some time. As the time trace of Figure 4.1 shows, there is some
pattern to the motion and although is not periodic, certain features are repeated each

crank cycle. This results in the observed structure of the Poincare map.

40

 

 

 

 

 

p B 0.01 .A 3 0.05
0.03.
007'-
.d'-
006- ° .. ’
0.05‘ ”a
v, . IF":W
0.04-' .- 4a; -' ‘ ' '
0‘ .. p
..c o ”a. . .. ..
0.03- #~ -.' '00 -. .. a. o. ’J
.o. .‘fi.. . . a.
I“. 9 '-"""'
0.02‘ an.“
0014 ., vvesrv .,
0.9 1.1 1.4 1,5 1.9

Crank Angle 0

Figure 4.4: First Magnification of Poincare Map

 

 

 

 

p I 0.01 I ' 0.05
0.050
0
0,045-1 . . \.0I
. no
...?..: ..q
0046-] w .‘ "
g ’. ' ’0 s 2'.
o I C '
‘I’ ~- v. I '3'
'. P. I o.
0.044 0' .H *3" '
.I I, ' o a“.
O ’ O. I
4 O 0. d. I
o .3 . I. T ... F... d
0.042“ ’1‘? .. .:.o aeg'oh .- .
t O o .. ... . .
s a". “:5":
0.0404 ' ' -, ........ ,-.-'. ...... fl
1.2 1.3 1.4 1.5 1.6 1.7

Crank Angle 9

Figure 4.5: Second Magnification of Poincare Map

41

4.3 Non-sliding Dynamics

For the simulations presented in this section, we keep the coefficient of restitution,
e, fixed at a value of 0.9 and carry out simulations for the 3 x 3 parameter matrix
with A = 0.025, 0.05, 0.075 and p = 0.01, 0.015, 0.2. The results are depicted in
the form of impact distribution patterns and are presented in Figures 4.6 - 4.14. It is
noted that all three types of response occur. The cases (A, p) = (0.025, 0.01), (0.025,
0.015), and (0.075, 0.01) exhibit transient chaos (i.e., Figures 4.6, 4.7, and 4.12),
cases (A,p) = (0.025, 0.02), (0.05, 0.01), (0.05, 0.02), (0.075, 0.015), and (0.075,
0.02) (i.e., Figures 4.8, 4.9, 4.11, 4.13, and 4.14) exhibit sustained chaos, and case
(A, p) = (0.05, 0.015) (i.e., Figure 4.10) is periodic without a chaotic transient. The
impact distribution patterns reflect these facts in the following way: chaos results
in a widely distributed set of impacts, periodicity is reflected by clean spikes, and
transient chaos is dominated by spikes but has superimposed some more uniformly
distributed impacts whose level depends on the duration of the chaos in relation to the
duration of the total data sample. As the sample length is increased, this background
will decrease in amplitude.

We show Poincare maps and impact spectra for three cases that represent the
three types of motions: (A,p) = (0.05, 0.015) for periodic motion, see Figures 4.18,
4.19, and 4.20; (A,p) = (0.025, 0.01) for transient chaos, see Figures 4.15, 4.16, and
4.17; and (A, p) = (0.075,0.02) for for sustained chaos, see Figures 4.21, 4.22, and 4.23.
A comparison of the Poincare maps, impact spectra, and impact distribution patterns

for these cases indicates the sufficiency of the impact distribution for characterizing

/\
e.
V

I l I
coasts-wowumcsoo

42

 

 

 

 

 

_ IXIO-IJ I 1 J1 IAl I _‘
: ' If I H
b 1X10'1'3 1 1 1 1 1 .0
0.7 0.8 0.9 1 1.1 1.2 1.3

1: - Dimensionless Distance

Figure 4.6: Impact Distribution Pattern, A = 0.025 and p = 0.010

 

 

 

 

 

 

g)- X1043 1 1 T I l-

4— _
(1),) (2)- lII ll IiI '-
_2_111 1111mm
_4_ .1

:g ~4X10'13 1 1 1 1 1—

0.7 0.8 0.9 1 1.1 1.2 1.3

:1: - Dimensionless Distance

Figure 4.7: Impact Distribution Pattern, A = 0.025 and p = 0.015

 

 

 

 

_1.5 I: 1X10-13 1 1 1 1 1
0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

Figure 4.8: Impact Distribution Pattern, A = 0.025 and p = 0.020

43

 

 

/\
a.
llV

9999 9999
washrooms-0:00

 

 

1.0-13 1 1 1 1 1

 

0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

Figure 4.9: Impact Distribution Pattern, A = 0.05 and p = 0.010

 

30 I +3 1 l I I

20 _ xTO

. 10»

<10> 0 I I ’ I 1’ l
—10
—20
_30 1xln‘I3 1 1 1 1 1

0.7 0.8 0.9 1 1.1 1.2 1.3

:1: - Dimensionless Distance

 

 

I
l

i
l

 

 

 

Figure 4.10: Impact Distribution Pattern, A = 0.05 and p = 0.015

 

 

 

 

‘1 1x 1073 1 1 1 1 1 T

0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

 

Figure 4.11: Impact Distribution Pattern, A = 0.05 and p = 0.020

44

 

 

13W 1 I I 1 l
< 10 > 3 b L I I I - I I I
-5 _ ] ] I
—10 9
_15 9419-13 1 1 1 1 1

 

 

 

0.7 0.8 0.9 1 1.1 1.2 1.3

2: - Dimensionless Distance

Figure 4.12: Impact Distribution Pattern, A = 0.075 and p = 0.010

 

 

I
l

 

 

_ 1X10?3 1 1 1 1 1 _
0.7 0.8 0.9 l 1.1 1.2 1.3

a: - Dimensionless Distance

 

Figure 4.13: Impact Distribution Pattern, A = 0.075 and p = 0.015

 

IX 10*? I I I I I

IIII

    

 

llIl LLlll

 

TIIT

1x IIL'Ta 1 J 1 1 1

0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

 

Figure 4.14: Impact Distribution Pattern, A = 0.075 and p = 0.020

,4;
C)!

the response of the system.

0 - an A - 0.025
0.1:

0.111
0.101
0.0!]
o OM
0071

w ’ 0004
I]
0.001 . .0 A
0031 °' . '
0.02] - “h
and _

ox . A!

 

 

 

 

0.0 1.0 2,0 1 0 d 0 5:0 5:0
Crank Angle 0

Figure 4.15: Poincare Map, A = 0.025 and p = 0.01

 

 

 

 

 

4.. -
2L- —
<10>0 9
-2- _
_4)- _

 

Figure 4.16: Impact Spectra. Upper Wall A = 0.025 and p = 0.01

 

 

4t. —1
- 2“ .91
<10>0 \

-2- /\ -

-4). _

 

 

 

 

Figure 4.17: Impact Spectra. Lower Wall A = 0.025 and p = 0.01

46

p-OJNLMIOM
'.

 

0.11]
0W1
00M
Nit
0.07] ‘
1,! com
0051 o
00¢]
00“
0.021
00H
0:: . . . . .
00 10 20 30 ‘0 SO 60
Crank Angie 0

 

 

 

Figure 4.18: Poincare Map, A = 0.05 and p = 0.015

 

 

<10)

II I II II
1111§1

 

 

 

 

Figure 4.19: Impact Spectra, Upper Wall A = 0.05 and p = 0.015

 

 

 

 

6

4

- 3
< >

¢_2

-4

-6

 

 

Figure 4.20: Impact Spectra. Lower Wall A = 0.05 and p = 0.015

47

D-OQAOOUN

 

0.1:
0.11]

0101
001M
QMI
0.071 t. ‘
VI 0001 (I, '
COSI 0': c'!.'
1
3:11! If f;.:’\\ ‘l
* a...»

 

 

 

0.021 “4"“ ”s ..\ ‘
0.011 fishy... :0
0.: . " '
00 0 20 so so so so
Cram: Anal. 0

Figure 4.21: Poincare Map, A = 0.075 and p = 0.02

 

/\

a.

IV
999 999

bis-bowed:
I I

 

 

 

 

Figure 4.22: Impact Spectra, Upper Wall A = 0.075 and p = 0.02

 

I

I

 

 

 

llll
III

 

Figure 4.23: Impact Spectra, Lower Wall A = 0.075 and p = 0.02

48

The following observation can be made based on the impact distribution patterns
shown in Figures 4.6 - 4.14. Chaotic motions lead to a quite uniform spreading of
the impacts over a range of locations while periodic motions lead to highly localized
impact locations. Also, as a general rule, the average impact velocities increase as
the gap size is increased. This is expected since the free motion has a longer time in
which to build up momentum before impact.

Also note that, as expected, the impact velocities are greater on the upper con-
straint than on the lower. Also, this difference increases as A is increased. This is due
to the increase of the moment acting on the connecting rod from the bearing, and
in particular from the A (0 + (0) term which is biased by the rotation direction. In
all cases, there are regions in which very small impact velocities occur (this is most
directly seen by considering the Poincare maps). These small impact velocities will
result, as A increased or e is decreased, in motions in which sliding occurs during
some part of the cycle.

Another point worth noting is that there is no apparent correspondence relating
the parameter variation to specific trends in the types of motion observed (at least in
this parameter range). Chaos and periodicity occur for (A, p) values without rhyme
or reason. In fact, for the periodic motions observed, very slight changes in the
parameters rendered the motion chaotic. It is therefore impossible to draw substantive
conclusions regarding non-sliding motions, other than the general observations given
above. In addition, a priori prediction of the nature of the response at a given set
of parameters is impossible. One must simply run the simulation to determine the

steady state motion.

 

49
4.4 Dynamics Involving Sliding

In the simulation routine, the appearance of an extremely low relative velocity at
impact is assumed to initiate sliding. At such a point the simulation program simply
advances the crank angle to the release value and starts the free flight conditions at
1(’(9) = ¢max(0) (or 10(9) = 10m;n(0), respectively) and 10(0) = 10,,,,,,,(0) (or 10(0) =
10m,n(0), respectively) with 9 set at the release value. Details of these algorithms are
given in Appendix B.

In these simulations 6 is again fixed at 0.9 and a 3 x 3 matrix of (A, p) values
is considered with A = 0.15, 0.20, 0.25 and p = 0.010, 0.015, 0.020 . Note that the
A values are larger here, corresponding to values where sliding occurs during some
part of the motion. Again the results are presented in the form of impact distribution
diagrams given in Figures 4.24 - 4.32. Periodic motions are much more prevalent as
A is increased in this parameter range. This is quite simply due to the fact that in
general dissipation tends to discourage chaos.

As in the case of non-sliding motions, an increase in the gap size results in larger
impact velocities and increases in A tend to increase the difference between the impact
velocity magnitudes on the upper supporting structure and the lower supporting
structure. Increases in A also tend to suppress chaotic dynamics. This can be seen in

the impact distribution patterns provided in this section (Figures 4.24 - 4.32).

50

 

 

2__IX]0-13 l 1 1 I I__1

1~ -
<¢> 0 __-....1111lll I I LI_II

-1- _

—2-1)(lfl‘f2 1 1 1 1 IF

 

 

 

 

0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

Figure 4.24: Impact Distribution Pattern, A = 0.15 and p = 0.010

 

2 r leo-I3 I I I I I _

 
  

 

 
 

_

-2 I 1X1110'I2 1 1 1 1 1 _
0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

 

 

 

Figure 4.25: Impact Distribution Pattern, A = 0.15 and p = 0.015

 

 

 

 

_2 _ 1xlfl‘I3 1 1 1 1 1 -

0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

 

Figure 4.26: Impact Distribution Pattern, A = 0.15 and p = 0.020

51

 

 

 

 

 

2_'X10'*3 1 1 1 1 1_1
1..
<15 0 Ms“... W
-1-
—2-1xlflj3 J
0.7 0.8 0.9 1.11.21.3

:r- Dimensionless Distance

Figure 4.27: Impact Distribution Pattern, A = 0.20 and p = 0.010

 

 

 

 

2 IxIU'Iy I I I I I _

1 - -1

< 10 > 0 W
_1 - _

“2 ” 1x1013 . t . 1 t ‘

 

0.7 0.8 0.9 1 1.1 1.2 1.3

x - Dimensionless Distance

Figure 4.28: Impact Distribution Pattern, A = 0.20 and p _= 0.015

 

2 _fix 1013 1 1 1 1 1 q

_

 

 

 

—2 ’- 1xlfl13 1 1 1 1 1

0.7 0.8 0.9 1 1.1 1.2 1.3

1: - Dimensionless Distance

 

Figure 4.29: Impact Distribution Pattern, A = 0.20 and p = 0.020

A
$.

| V

Figure 4.30: Impact Distribution Pattern, A = 0.25 and p = 0.010

/\
$.

I V
ooo ooo

obboicia'o

Figure 4.31: Impact Distribution Pattern, A = 0.25 and p = 0.015

/\
$.

Figure 4.32: Impact Distribution Pattern, A = 0.25 and p = 0.020

999 999
outsmowsxo

IV
999 999

52

 

 

 

 

 

__ 1x101" T 1 1 1 r d
- ”ll" -1
- -I
I. 'XIQ'Is 1 1 1 1 1 _
0.7 0.8 0.9 1 1.1 1.2 1.3

:1: - Dimensionless Distance

 

 

 

 

 

_ IXIUWG I 1 I 1 1 _
P 0(1913 1 1 1 1 1 '-
0.7 0.8 0.9 I 1.1 1.2 1.3

1: - Dimensionless Distance

 

 

 

 

 

0 W 1 1 1 1 1 _

2 .. _.

0

2 .. _

4 1- -

'6 '- 1x1933 1 1 1 1 1 '-
0.7 0.8 0.9 1 1.1 1.2 1.3

a: - Dimensionless Distance

53

Figures 4.33 and 4.34 present the two Poincare maps for p = 0.015, with A =
0.2 and 0.25. At A = 0.2 the motion is chaotic. During this chaotic motion, sliding
occurs intermediately, but not infrequently. As A is increased , at approximately
A = 0.22 only transient chaos is observed, and for A = 0.23 and above, simple periodic
motion occurs. The case of A = 0.25 indicates that these periodic motions involve

cycles of low velocity impacts concluding in sliding, and then release. A time trace

p 9 0.015) - 0.8
0.1: ‘

 

 

 

 

 

tic 2:0 3:0 4:0 530 do
Crank Anglo 0

Figure 4.33: Poincare Map, A = 0.2 and p = 0.015

p - 0.010. A- 0.28

 

a t 0.01.

 

 

0.01"“\
0.0 ' 170 2:0 3:0 3'0 . - 5:0 370
Crank Anqu 0

Figure 4.34: Poincare Map, A = 0.25 and p = 0.015

 

for the case of (A, p) = (0.25,0.015) is shown in figure 4.35. Note that this is shown
for three crank cycles and the response is periodic.
Note that in the case of relatively large values of the dissipation coefficient A ,

unlike the case of lower values of A , the response of the system does not change

54

0.02 I I H I I I I I I
0.019 ~ -
0.018 I -
0.017 - —
0.016 - -

2 0015 WWW

0.014 r 9
0.013 - -
0.012 ~ -

0.011 - -

0.01 l l l l I l l l l
0 2 4 6 8 10 12 14 16 18

T - dimensionless time

 

 

 

 

Figure 4.35: Time Trace, e = 0.9, A = 0.25, and p = 0.015

from periodic to chaotic with small variations of p. In the previous section it is seen
that with the slightest change of the gap size at a fixed A, the response changes
from periodic to chaotic, while in the case of high A , if the response periodic,
small variations of the gap size do not affect the periodicity. This can be seen in

Figures 4.36, 4.34, and 4.37.

p 9 0.01 . A. 0.20

 

 

 

 

0.0 1.0 2 o .1 0 0'0 5'0 5'0
Crank Angle 9

Figure 4.36: Poincare Map, A = 0.25 and p = 0.01

Note that at the highest dissipation coefficient used in this study, A = 0.25,

55

p-OOD.A-025

 

0.111
0110‘
0.001
0.00
0.071

0.051
0.04-I
0.03:
0 ()2I

0m]”vm\\

0m: . .. , . , "a
00 10 20 10 so so 60

Crank Angle 0

 

 

 

Figure 4.37: Poincare Map, A = 0.25 and p = 0.02

there are no impacts on the lower supporting structure. Comparison of these impact
distribution patterns with the ones presented in the previous section ‘suggest that as A
increases the locations of the impacts on the lower supporting structure become more
limited and the impacts on the lower supporting structure become more sever. Note
that the impact locations shift toward the end of the supporting structure (i.e. head
of the cylinder) and for A above 0.25 there is no impact on the lower supporting
structure. These observations are consistent with the constraint forces shown in
chapter 2. There it was observed that as A increases the region over which the
net moment on the connecting rod which leads to impacts on the lower structure
becomes smaller and moves towards the piston position corresponding to 0 = 0. For
A beyond a critical value, depending on p, over no range of crank angle intervals
does the net moment push the connecting rod toward the lower structure. In such
a case impacts will not generally occur on the lower structure during steady state
operation.

To this point we have not varied the coefficient of restitution 13. Reduction in

the value of e has a predictable outcome: motions increase their sliding duration and

 

56
become periodic. This is true over all ranges of A and p considered in this work. As an
example we take (A,p) = (005,002) (a case considered with e = 0.9 in section 4.3)
and reduce c. This case is one in which the bearing dissipation is small (or the crank
speed is high) and the gap is quite large, conditions ripe for chaos. Figures 4.21, 4.38,
and 4.39 present the Poincare maps for 6 values of 0.9, 0.7, and 0.5. It can be seen
that as the coefficient of restitution decreases, the response changes from chaotic to
transient chaos and eventually periodic motion appears. Note that at values of 6 near
unity no sliding occurs and as 0 decreases sliding occurs more often. Figures 4.8, 4.40,
and 4.41 provide the impact distribution pattern for those values of the coefficient of

restitution.

a 9 0.7 .A- 0.05 . 01-002
012
OJH
0101 .°
009 ,
... l
0.071 -
, , a... K
0.054
0.04- :1-
0.031

aozI """ -
0.01] "K

 

 

 

 

ox . FOE—g

0.0 1.0 in .10 lo 570 370‘
Crank Angle 0

Figure 4.38: Poincare Map, 6 = 0.7, A = 0.05, and p = 0.02

01-05. 0‘002.A 005

 

1111;

 

 

 

Figure 4.39: Poincare Map, e = 0.5, A = 0.05, and p = 0.02

 

57

 

 

 

 

 

10 1x10a3 1 ’7 1 1 1
5 r 1
< 10 > 0 9 A
-5 _ 1
_10 1x1013 1 1 A 1 1
0.7 0.8 0.9 1 1.1 1.2 1.3

:1: - Dimensionless Distance

Figure 4.40: Impact Dist. Pattern, e = 0.7, A = 0.025, and p = 0.020

15

 

 

 

 

 

IX1043 I I I I I
10 -
. s— I
< 1,0 > 0 ’ I
:ig _ 1X10?2 1 1 1 1 1
0.7 0.8 0.9 l 1.1 1.2 1.3

:1: - Dimensionless Distance

Figure 4.41: Impact Dist. Pattern, e = 0.5, A = 0.025, and p = 0.020

58

Figures 4.42, 4.43, and 4.44 show results in the form of time traces over three
crank revolutions for these three cases. As expected, the motion develops substantial
sliding and becomes periodic. Note that in the limit 6 —+ 0 sliding will nearly always
occur since the rebound velocity is zero. While transient chaos may occur even in the

case of e = 0 , periodic motions will dominate the response (see Shaw/ Holmes [5]).

 

0.025 n

1 r 1 1 f 1
0.02
0.015 " I
0.005 ‘

1
l

l

.O

O

H
I

 

S
O
I\ H

 

 

 

 

0 2 4 6 8 10 12 14 16 18

T - dimensionless time

Figure 4.42: Time Trace, 12 = 0.9, A = 0.05, and p = 0.02

59

 

 

 

 

 

 

0 2 4 6 8 10 12 14 16 18

1' - dimensionless time

Figure 4.43: Time Trace, 12 = 0.7, A = 0.05, and p = 0.02

 

0.025 1 1 1 F 1 1 1 1 1
0.02 ""
0.015 ‘ "‘
0.01 ' ‘
0.005 "' "

 

 

—0.005
-—0.01
—0.015
—0.02

_0.025 1 I I l I l l I L
0 2 4 6 8 10 12 14 16 18

T - dimensionless time

 

 

 

 

Figure 4.44: Time Trace, e = 0.5, A = 0.05, and p = 0.02

Chapter 5

Discussions and Future Direction

for Research

This study is a first step in a series of studies which should be carried out in order
to more completely understand the dynamics of piston-slap. The model developed
and investigated herein was designed so as to facilitate the analysis and simulations in
order that some general observations could be made regarding the influence of various
system parameters on the response of a slider-crank mechanism with slider clearance.
Of particular interest was the possibility and characterization of chaotic motions in
this system.

It has been shown that chaotic motions are prevalent over a range of parameters
which correspond to high crank speeds and/or low bearing friction with relatively
ideal impacts (that is, coefficient of restitution near to unity). In applications, one
tends to design so as to minimize bearing friction in order to maximize efficiency and

reduce energy consumption. This may have an additional benefit if it encourages

60

61

chaos (for the reasons given below). However, in typical automotive applications, the
effective coefficient of restitution is on the order of e = 0.4 (Fawcett and Wilson [11])
which indicates that sliding motions will most likely dominate the response of pistons
in automotive engines. The data also indicates that impact velocities are more severe
for larger gap sizes. This is expected and is due to the fact that the connecting
rod/piston assembly has a longer time duration over which to build up momentum
between impacts.

One must be very cautious in using information obtained from simulations of the
type presented here due to their demonstrated sensitivity on parameter changes in
the model and. due to the limitations of the model itself. Due to this, it may be
practically impossible to construct a model which is capable of predicting the nature
of the system response over a range of speeds with any confidence. Chaos, transient
chaos, and periodic motions are all possible and, in fact, due to the very possibility
of transient chaos, it is possible to observe either chaos or periodicity for a given set
of parameters depending entirely on the initial conditions and on the time scale over
which the response is observed.

The results presented here indicate that chaotic motion may have a beneficial effect
with regards to wear in mechanisms with clearances. This is simply due to the fact
that the distribution of impacts is more uniform when the system responds in a chaotic
manner than when the motion is periodic. The repeated impact patterns associated
with periodic motions can lead to more highly localized wear. One must be careful in
drawing too firm conclusions from this observation, however, since periodic motions

with significant sliding may induce less wear than motions with a greater number of

62

impacts, independent of how well distributed they are. In addition, consideration of
noise generation and fatigue may offset any potential benefits in reducing wear, since
increases in sliding duration are typically correlated with fewer impacts per cycle.
These are topics which lie outside the scope of this thesis, but should be considered
in future work on this subject.

The statistical means of data presentation used in this thesis are, for two reasons,
of more practical use than the Poincare maps which are typically used to present
chaotic dynamics. First, by showing normalized average impact velocities after the
completion of a long-term simulation, in place of individual velocities at each impact,
one obtains an integral average measure which cannot be obtained from the pattern
of the Poincare maps. Second, and more significant, is the fact that the impact
distribution diagrams and impact spectra can be used for more complex models,
including those with more than one degree of freedom, by indicating actual impact
locations and impact velocities regardless of the particular configuration of the system
at impact. In contrast, the Poincare map for even a two degree of freedom model (such
as the one described below) contains all information about the state of the system at
impact; it is four dimensional, and therefore difficult to represent in graphical form.

The following are some possible directions for improving the model for future

studies:

0 Consideration of the piston geometry. A first step in this direction would be to
make the piston rectangular and include its dynamics, resulting in a two degree

of freedom model.

 

63

0 Consideration of an end-load acting on the piston. This would represent gas-

pressure effects in engine applications.

0 Non-symmetric placement of the connecting rod-piston bearing joint. This tech-

nique is currently used to suppress piston-slap ( [23]).
0 Inclusion of friction in all bearing joints which influence the dynamics.

By implementing the above improvements, the resulting two degree of freedom

model will provide a more accurate prediction of the motion of a piston in a cylinder.

Appendix A

Appendix A

Derivation of gbmax and ¢min

The maximum and minimum values of d) that are possible due to the existence of

the clearance are found by directly using the geometry of the model. Figure A.1

   

Y i
EA
: '- Ibmaz
.......... .. -----.p.
0’ :0 D X

Figure A.1: The maximum angle

contains a schematic view of the model showing the connecting rod in its maximum
possible position (i.e. 1,!) = 1pm”). In this figure, 0’ corresponds to the center of crank
rotation, line if corresponds to the line along which the slider center will travel when

the roller moves along the upper slider boundary, and the X -axis corresponds to the

64

65

nominal path of center of the roller. Also, A is the clearance that exists between
the upper supporting structure and the upper edge of the roller (the size of A is
exaggerated in the Figure A.1 for clarity). Note that 1,0sz depends on the crank angle
9, or equivalently, is time dependent. Note also that as the gap size A decreases to
zero, the model approaches a system without clearance. The maximum possible angle

is found by applying some simple trigonometric identities. Note that in this figure

E is equal to KC- which is the length of the connecting rod (i.e. AD =AC = 1).

Considering A AOD we obtain

mzl sin¢

The height of A ABC is

IKE-=1 Sin¢—A

and also the [ACE is known to be

ZACB = ¢ — 112",“

Using the following identity

all 3|

Sin(¢ _ 11157103) =

 

66

with the known values yields

Isind—A

1 (A.1)

SID(¢ - Ipmax) =
The rules of summation of angles results in
I sin ()5 — A

sin d) cos 1pm” — cos ¢ sin 2pm,, = I

For small gap sizes, and thus small angles 11) , a Taylor expansion in terms of 1,!) is

employed. Using

cosz/J = 1+0(¢2)

simb = MOO/)3)

and considering only linear terms in 112, the following is obtained:

 

sin¢ —cos¢¢....= ’ 8m", ’ A
Solving for 11)me gives
A
= — A2

and recalling that

 

Icos¢= fp—rzsinzflt

 

67

 

results in
A
max: = A.3
¢ x/l2—r2sin2flt ( )
The dimensionless form is obtained by dividing through by 1 resulting in
1pm“: = p (A4)

 

\/ 1 — 62 sin2 7'
where p is the dimensionless gap size given by

_ A
P " 1
where A = d — r“ in the notation of section 2.5.

By applying a similar approach the minimum possible angle for the connecting

rod can be found. It is given by

 

min = -p A5
1,0 \/1—£2 sinz'r ( )

Note that am and dam-n , the maximum and minimum angles that the connect-
ing rod can deviate from its nominal position due to the existence of the clearance,
depend on the crank angle orientation as well as the system’s fixed geometrical prop-

erties.

‘ I
ry-«..:_

 

Appendix B

Appendix B

Simulation Routine

The routine used for the simulations takes into consideration non-sliding motion and
sliding motion. As a result, it is capable of handling a wide range of parameter values.

In this routine, the equation of motion (equation 2.13) is written in terms of two
first order differential equations as given in section 3.1 (equations 3.2, 3.3). The
two differential equations which describe the motion of the connecting rod angle
are numerically solved by using a forth order Runge-Kutta method with a nominal
step size of 0.005. Starting from some initial conditions inside the constraints and
stepping through time, eventually a violation of the constraints on the roller occurs
(i.e. 2:1 > zbma, or 2:1 < I/ngn). By using Newton’s root solving method, the crank
angle at which x1 = 1pm,,- ( or $1 = 2pm..) is found. Then, the impact rule is applied
and the post impact velocity is determined by equation 2.17. Once the post impact
velocity and the location of the impact have been determined, the solver continues
to solve for :01 and 922, using these as initial locations for the free flight equations

of motion. A period of 30 crank cycles is allowed for the system to achieve “steady

68

 

69

state“ and then the required data is collected over a a number of crank cycles (typically
1500 - 3000). Occasionally Newton’s method does not converge, for example when the
impact velocity is very small. This difficulty occurs most frequently for larger values
of A , in which case the following procedure takes place. In such cases a reduced step
size of 0.0001 is used for the integrator from the pre-impact point until the clearance
condition is violated. If Newton’s method fails again, the step size is reduced further.

Whenever the relative post-impact velocity is less than some very small value,
WJ — Kbmaxl 01‘ Id) — 15min] < e (e = 0.005 is used) , the roller is assumed to slide from
that point until it is released at the crank angle at which the constraint force becomes
zero. At this location the system will once again begin free flight with initial conditions
of 2:1 = 112",” and x2 = dim” for the upper supporting structure and x1 = 2pm... and
x2 = 35",,” for the lower supporting structure. Figure B.1 provides a flow chart for

the simulation routine for the dynamics of the system under investigation.

 

70

Suing to mm
and LC.

 

 

 

 

 

 

F—u—p +6

 

RX 411! and»
ms SOLVE

 

 

 

No > Yum:

Y<Ynin

 

 

ya

 

Nm‘: Method
‘—> ms Roarsovan

M’é

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1’
ya
lambda
swam..-
ya min; the roller In
the rule-u beads:
No 1
Set than!“ LC.

 

'I' (t) ‘0'” (t) .‘I’nu (1:)
‘I” (1) ‘P'nu (t) S's-u“- (t)
1 I Rel-o in.

 

 

 

 

 

 

 

 

 

 

 

Figure B.1: Flow Chart: Simulation Routine

Bibliography

Bibliography

[1] Shaw, Steven W., ”The Dynamics of a Harmonically Excited System Having
Rigid Amplitude Constraints Part 1: Sub-harmonic Motions and Global Bifuri-
cations,” Journal of Applied Mechanics ,Vol. 52, pp. 453-458, 1985.

[2] Shaw, Steven W., ”The Dynamics of a Harmonically Excited System Having
Rigid Amplitude Constraints Part 2: Chaotic Motions and Global Bifurcation,”
Journal of Applied Mechanics ,Vol. 52, pp. 459-464, 1985.

[3] Shaw, Steven W., ”Forced Vibration of a Beam with One-Sided Amplitude Con-
straint: Theory and Experiment,” Journal of Sound and Vibration ,Vol 99, pp.
199-212, 1983.

[4] Shaw, Steven W., and Holmes, P. J ., ”A Periodically Forced Piecewise Linear
Oscillator,” Journal of Sound and Vibration ,Vol 90, pp. 129-155, 1983.

[5] Shaw, Steven W., and Holmes, P. J., ”A Periodically Forced Impact Oscillator
with Large Dissipation,” Journal of Applied Mechanics, Vol. 105, pp. 849-857,
1983.

[6] Miedema,B. ,and Mansour, W. M., ”Mechanical Joints With Clearance: a Three-
Mode Modal,” Journal of Engineering for Industry, pp.1319-1323, 1976.

[7] Granat, S. J. , and Fawcett, ”Effects of clearance at the Coupler-Rocker Bearing
of a 4-Bar Linkage,” Mechanism and Machine Theory, Vol. 14, pp. 99-110, 1979.

[8] Dubowsky, S. , Norris, M., Aloni, E. ,and Tamir A. , ”An Analytical and Exper-
imental Study of the Prediction of Impacts in Planar Mechanical Systems With

Clearances,” Journal of Mechanisms Transmissions and Automation Design, pp.
1-8, 1984.

[9] Mansour, W. M., and Townsend, M.A., ”Impact Spectra and Intensities for High-
Speed Mechanisms,” Journal of Engineering for Industry, pp. 347-358, 1975.

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