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"15838
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2004
5M>850§7

LIBRARY
Michigan State
University

This is to certify that the
thesis entitled

NON-LINEAR DYNAMICS OF A FRICTIONALLY EXClTED
CANTILEVER BEAM

presented by

Jeffrey Larkin Quinby

has been accepted towards fulfillment
of the requirements for the

MS. degree in Mechanical Engineering

 

 

 

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9/24/03

Date

MSU is an Affirmative Action/Equal Opportunity Institution

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
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DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 c:/C|RC/DateDue.p65—p. 15

——-—_—_

 

NON-LINEAR DYNAMICS OF A FRICTIONALLY
EXCITED CAN TILEVER BEAM

By

Jeffrey Larkin Quinby

A THESIS

Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2003

ABSTRACT

NON-LINEAR DYNAMICS OF A FRICTIONALLY EXCITED
CANTILEVER BEAM

BY

Jeffrey Larkin Quinby

The main motivation of this thesis is to study a phenomenon of extremely low
frequency responses to periodically excited structural oscillators, and to observe this and
other non-linear dynamic phenomena in an experiment. The tip of a cantilevered beam in
an experiment exhibited extreme low frequency (on the order of 1/20 to 1/100 of the
driving frequency) as well as chaotic and quasi-periodic responses under different forcing
frequencies. Performing a frequency sweep experiment, regions of dynamic activity
were identified. Then, to examine these regions more closely, fixed frequencies that
belonged to the regions were selected. Data gathered by a strain gages attached to the
cantilever beam were examined in the Poincaré section. This in turn allowed for graphic
representations of the dynamics occurring in the oscillating beam. The Poincaré section
data was further processed into torus-angle return maps and Poincaré section of Poincare
section data to help characterize the dynamic responses of the system. The FFT was also
taken using the original data as well as the Poincaré section data. These methods of

examining the gathered data confirmed the types of responses observed in the beam.

 

 

For Paul Texeira Jr. (1978-2001)

iii

ACKNOWLEDGMENTS

This work was conducted with the support of the Michigan State University
College of Engineering and the department of Mechanical Engineering as well as support
from the National Science Foundation grant No. CMS-0099603. This research was
conducted due to the support of Dr. Brian F eeny and the faculty of the System Dynamic
and Vibrations Laboratory at Michigan State University. I would also like to thank all of

my fellow colleagues for the help they have given me over the past three years.

iv

 

 

 

 

TABLE OF CONTENS

LIST OF TABLES

LIST OF FIGURES

1

Introduction
1.1 Motivation
1.2 Background on Low-Frequency Behavior
1.2.1 High-to-Low Frequency Energy Transfer
1.2.2 Ultra-subharmonic Responses
1.2.3 Frequency Demultiplication
1.2.4 Dither
1.2.5 Beating
1.2.6 Other Studies
1.3 Background on Frictionally Excited Systems
1.4 Overview of Thesis

Description of the Experiment
2.1 Background

2.2 Experimental Setup

2.3 System Characteristics

2.4 Friction Measurement

2.5 Method of Poincaré Sections

Frequency Sweeps

3.1 Frequency Data Sweeps

3.2 Subharrnonic Order in each Region
3.2.1 Region I
3.2.2 Region 11
3.2.3 Region Hybrid, III & IV
3.2.4 Region V
3.2.5 Region V1

3.3 Summery

Fixed Frequency
4.1 Fixed frequency Examination
4.1.1 Region I (8 & 9.8 Hz)
4.1.2 Region II (21.9, 22.1 & 22.3 Hz)
4.1.3 Region III & Hybrid (25.7 Hz)
4.1.4 Region VI (43.8, 44.4, 45.2 & 45.6 Hz)
4.1.5 Analysis of 37.9 Hz

vii

viii

\DOONOlON-b-hNN—‘H

10
ll
14
17
22

26
26
30
32
34
36
39
41
46

48
48
49
59
73
78
87

 

4.2 Torus Doubling 88
5 Conclusion 91
APPENDIX 96

BIBLIOGRAPHY 101

vi

4.1

4.2

4.3

4.4

 

LIST OF TABLES

FNN data of8 and 9.8 Hz
FNN data of21.9, 22.1, and 22.3 Hz
FNN data of 25.7 Hz

FNN data of 44.4, 45.2, and 45.6 Hz

vii

57

69

77

83

1.1

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

3.1

3.2

3.3

3.4

LIST OF FIGURES

A hand sketch of a figure taken from Van Der Pol & Van Der Mark
1927 Nature paper [13].

Experimental set up

Diagram of experimental setup

Rotated View of sections A-A and BB

F FT response to the unloaded cantilever

F FT response of the loaded cantilever

F FT response of the impulse test of vertical motion of the loaded beam
Kappagantu’s friction measurement setup

Kappagantu’s sample time-series of displacement, velocity, friction
force & effective normal load

Kappagantu’s friction-displacement relation
Kappagantu’s friction-velocity relation
Phase representation of the Poincaré plot

An example of a data and the sequence of strain gage volts taken form
the Poincaré section

An example of actual data from the Poincaré section

Frequency up sweep data showing the Poincare, the DC drift and the
Poincaré minus the DC drifi

Frequency down sweep data showing the Poincare, the DC drift and
the Poincare minus the DC drift

Frequency up sweep of region I and the subharmonic orders of the
response

Frequency down sweep of region I and the subharmonic orders

viii

12

12

13

16

16

19

20

21

21

22

23

24

28

29

33

33

3.5

3.6

3.7

3.8

3.9

3.10

3.11

3.12

3.13

3.14

3.15

3.16

3.17

3.18

3.19

3.20

4.1

4.2

4.3

4.4

4.5

Frequency up sweep of region II and the subharmonic orders
Frequency down sweep of region 11 and the subharmonic orders
Frequency up sweep of the Hybrid region and the subharmonic orders

Frequency up sweep of the Hybrid region sub 1 and the subharmonic
orders

Frequency down sweep of region III and the subharmonic orders

Frequency up sweep of the Hybrid region sub 2 and the subharmonic
orders

Frequency down sweep of region IV and the subharmonic orders
Frequency up sweep of region V and the subharmonic orders
Frequency down sweep of region V and the subharmonic orders
Frequency up sweep of region VI and the subharmonic order
Frequency down sweep of region VI and the subharmonic order
Frequency up sweep region VI sub 1 and the subharmonic order
Frequency up sweep region VI sub 2 and the subharmonic order
Frequency down sweep of region VI sub 1 and the subharmonic order
Frequency down sweep of region VI sub 2 and the subharmonic order
Apparent demultiplication of region VI sub 2

Poincaré section delay map of 8 Hz

Poincaré section delay map of 9.8 Hz

Delay map of the Poincare section of the Poincare section delay
map at 8 Hz

Torus angle model

Torus angle dynamics of 8 Hz

ix

35

35

37

37

38

38

39

40

40

41

42

42

43

43

44

46

51

51

52

53

54

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

 

Torus angle dynamics of 9.8 Hz

F FT and the Poincare FF T of 8 Hz

F FT and FFT of the Poincaré section of 9.8 Hz
Poincaré section delay map of 21.9 Hz

Poincare section of the Poincaré section of 21.9 Hz plotted
against its index

Sample of the Poincaré section data at 21.9 Hz plotted against its index

Poincaré section of the Poincaré section of 22.1 Hz plotted against its
index

Delay map of the Poincare section of 22.1 Hz of data segment 1

Delay map of the Poincare section of 22.1 Hz of data segment 2
Poincaré section of 22.3 Hz with a delay of 1

Torus angle dynamics of 22.1 Hz of sub region 2

Torus angle dynamics of 22.3 Hz

Sample of the Poincaré section data at 22.3 Hz plotted against its index
FFT and the Poincaré FFT of21.9 Hz

FFT of22.1 Hz segments 1 & 2

FFT and the Poincare F FT of 22.3 Hz

Poincare section of the Poincaré section for 25.7 Hz and its two sub
regions

Delay map of the Poincare section of 25.7 Hz
Torus angle dynamics at 25.7 Hz of sub 1
Torus angle dynamics at 25.7 Hz of sub 2
FFT of22.1Hz Sub 1 & Sub 2

Poincaré section of 44.4 Hz with a delay of 1

54

58

58

60

61

61

63

64

64

65

67

67

68

70

72

72

73

74

75

76

78

79

4.28

4.29

4.30

4.31

4.32

4.33

4.34

4.35

4.36

4.37

4.38

Al

A2

A3

A4

A5

A6

Poincaré section of 45.2 Hz with a delay of 5
Poincaré section of 45.6 Hz with a delay of 3
Torus angle dynamics of 44.4 Hz

Torus angle dynamics of 45.2 Hz

Torus angle dynamics of 45.6 Hz

F F T and the Poincare FFT of 44.4 Hz

F FT and the Poincaré FFT of 45.2 Hz

FFT and the Poincaré FFT of 45.6 Hz

The FFT and sample time trace of 37.9 Hz

Poincare section of the Poincaré section of 45.6 Hz plotted against its
Index

Example of torus doubling with increasing excitation frequency

Poincaré section of the Poincaré section of 8 Hz plotted
against its index

Poincaré section of the Poincaré section delay map of 9.8 Hz
Correlation dimension and derivative of 8 and 9.8 Hz

Delay Map of the Poincare section of the Poincaré section of 21.9 Hz
with Delay of 1

Correlation number and their derivatives vs. log(r) for 21.9, 22.1 and
22.3 Hz

Correlation number and their derivative vs. log(r) for 44.4, 45.2 and
45.6 Hz

xi

80

80

82

82

83

85

86

87

88

89

90

96

97

98

99

100

101

Chapter 1

Introduction

This thesis presents a study of a cantilevered beam undergoing frictional
excitation. Under different forcing frequencies the tip of the cantilevered beam appeared
to exhibit extreme subharmonic as well as chaotic and quasi-periodic responses. This
phenomenon was referred to in the doctoral thesis of Ramana V Kappagantu [1].
However, the subharmonic phenomenon was noted but was not of primary interest in his
thesis. In this work, we follow-up those initial observations with more detailed
observations of the responses of the same experiment. We look at very low-order

subharmonic responses, and characterize the dynamics of these and other responses.

1.1 Motivation

The main motivation of this thesis is to study a phenomenon of extremely low
frequency responses to periodically excited structural oscillators, and to observe this
phenomenon in an experiment. Subharmonics are defined as response frequencies that
are smaller than the primary forcing frequency. In some works subharmonics are a
periodic response that can be thought of in terms of a Fourier series with fundamental
frequency a). If Q is the forcing frequency, then co/Q defines the order of the
subharmonic response. Typical subharmonic orders are 1/2, 1/3, or 1/4. In this work, we
observe subharmonic responses with much lower orders, in addition to typical
subharmonic responses. Because we are interested in the denominator we will use the

inverse of the definition. In this thesis the subharmonic order will be defined as Q/co. It

should be clarified that what we are after is not a subharmonic-resonance periodic
response of frequency a), rather we are looking for a dominate frequency in which the

response is chaotic or quasi-periodic.

1.2 Background on Low-Frequency Behavior

The phenomena that are present in this thesis deal with low frequency behaviors.
In the following sections examples of different types of low frequency responses will be
given. These examples have been characterized into the following four groupings: high-
to-low frequency energy transfer, ultra sub harmonic, frequency demultiplication, and

dither.

1.2.1 High-to-Low Frequency Energy Transfer

The concept of an energy transfer from high-to-low frequency modes was the
focus of a lot of research in the mid 90’s. Some of the groundwork was laid in the
summer of 1988 at a conference at the Virginia Polytechnic Institute and State
University. Haddow & Hasan [2] gave a talk that dealt with the experimental results of a
base excited cantilever beam. They noticed that if the beam was excited at a relatively
high resonance it would react in the expected mode shape. But as time elapsed it
appeared that energy would ‘cascade’ from the higher modes down to the lower modes.
What ultimately came about was a combination of first-mode steady-state response at the

lowest modal frequency, plus the resonated mode.

This type of phenomena became of great interest to Ali Hasan Nayfeh who is an
accomplished scholar in the field nonlinear dynamics. In the early 90’s A. H. Nayfeh
was second author to his son S. A. Nayfeh in two journal papers that discussed an energy
transfer from hi gh- to low-frequency modes in a flexible structure [3,4]. The first paper,
which was published in 1993, gave an analytical study of a nonlinear two-degree-of-
freedom system with widely spaced modes being excited by a simple-harmonic. The
results from this analytical study, which largely used the method of averaging, gave
valuable insight when the jump was made to an experiment of a multi—degree-of-freedom
system in their second paper. The experiment that the father and son team investigated in
their second paper was a cantilever beam undergoing base excitation. Their experimental
analysis revealed how nonlinear coupling between linear modes provided an avenue for
the energy cascade.

In 1995 Nayfeh & Mook [5] wrote a similar journal paper, which concluded that a
low amplitude high frequency excitation can generate a large amplitude low frequency
response at a modal frequency. In the same year Nayfeh & Chin [6] discussed a
parametrically excited system with a comparable outcome in a paper for Nonlinear
Dynamics. Over the next couple of years Oh & Nayfeh [7, 8] worked on resonance and
modal interactions in composite cantilever plates. Once again the energy transfer between
high and low frequency modes had an effect on the dynamic system. In 1996 Anderson,
Nayfeh & Balachandran [9] published work on the coupling between high-frequency and
low—frequency modes. Their findings showed that interaction between the first mode and
the third and fourth modes of their cantilever beam resulted in a transfer of energy from a

high-frequency excitation to a low-frequency mode.

The signature for this phenomenon is an input at a frequency near a higher modal
frequency, and a steady-state response with a low-mode component at the corresponding
lower modal frequency. This is relevant because in our research the dynamic response of
our beam seems to following a similar trend, but with an added twist. A high forcing
frequency with low amplitude is producing a low frequency large amplitude response.

However, the low frequency may be lower then the lowest modal frequency.

1.2.2 Ultra-subharmonic Responses

An “ultra-subharmonic” response is defined as a subharmonic response of a
fractional order [10]. Nayfeh & Mook’s book Nonlinear Oscillations [10] suggests that
ultra-subharmonic resonances exist if the constant m is even in the following relation:

Q, = m0, and (00 z1/2(Q2 i: Q, ), where Q] & Q; are forcing frequencies and (no is the
natural frequency. In 1994 Qriouet & Mira [11] took the Duffing-Rayleigh equation and
subjected it to an external periodic driving force. According to the authors “this first
elementary study” produced complex fractional harmonics that forced them to limit their
numerical approach to low frequency periodic solutions. Four years later Yagasaki [12]
theoretically described ultra-subharmonic resonance motion using a third-order averaging

technique.

1.2.3 Frequency Demultiplication

In 1927 Van Der Pol & Van Der Mark [13] designed an electrical system that

when a sinusoidal forcing frequency is applied the output is has a response that is whole

submultiples of the applied sinusoid. The response frequency becomes a fraction of the
initial applied frequency, i.e. (0/2, (0/3, (0/4 up to (0/40. The system would only oscillate
with “discrete frequencies” or submultiple frequencies. In the experiment, as the
capacitance of the circuit was changed the response frequency remained constant until the
capacitance was large enough to cause a jump in the frequency, creating a bifirrcation
diagram, showing that the response followed steps of submultiple frequencies with
respect to the capacitance parameter. The demultiplying frequencies followed a minor

scale and “strongly reminds one of the tunes of a bagpipes.”

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0.0005 10 15 20 25 3o 35 0,0040 11F

Figure 1.1: A hand sketch of a figure taken from Van Der Pol & Van Der Mark 1927
Nature paper [13].

1.2.4 Dither

“Dither” is the application of a high-frequency excitation (much higher than the
system’s characteristic frequency) to a system with the hopes of affecting the low-
frequency behavior of the system. Various oscillatory systems with poor response
behavior can be restored by dither. Examples of this are quenching limit cycle behavior
[14], stabilizing an equilibrium (inverted pendulum) [15], and improving friction
behavior [16]. Dithering excitation makes a system compliant to averaging because its
frequency is much higher then the characteristic frequency of the system. This creates
two time scales, one slow scale coupled with the oscillator itself and one fast scale
coupled with the dither. F eeny & Moon [17] investigated this phenomena and its affect
on stick-slip chaos in 2000. By applying dither to a low-frequency forced stick-slip
friction oscillator they found that the chaos that was typically intrinsic to the system was
extinguished.

This alleviation of stick-slip chaos by using dither seems to be ironic because the
experimental research done for this thesis appeared to have the opposite effect. We will
see that “dither” was applied to our frictionally excited system, which in turn created a

low-frequency chaotic or quasi—periodic stick-slip response.

1.2.5 Beating

Linear and non-linear systems can exhibit beating if responses have two similar

frequency components, a)I and (02. The result is a low frequency amplitude variation at

the frequency cob = ((02 — a), )/ 2 . An FFT of the beating signal will show spikes at (01

and (02, not (0b. Non-linear effects can give rise to spikes that are some type of linear

combination of the two frequencies, like a), = 2602 — 60,.

1.2.6 Other Studies

Some other studies that can be classified as having low-frequency behavior are
presented here. Miranda & Thomsen [18] set up a simple model for predicting vibration
induced sliding of a mass, which was able to slide along a beam under going base
excitation. The results of the model retained features of a real system, which would allow
the model to be used in research in using vibration induced sliding for passive vibration
damping. One can also notice in the response graphs that the collar slides along the beam
at a very low frequency. Zhang & Tao [19] observed beats between subharmonic and
super subharmonic responses in an experiment that dealt with nonlinear vibrations in a
thin loudspeaker cone. The results of this experiment showed that the nonlinear
vibrations are associated with the bending resonance of the loudspeaker cone.

It is also worth mentioning squeak and squeal. These are characterized by high
frequency noises resulting form low frequency, or steady sliding motions. While quite
the opposite of subharmonic interest, it may involve large-order frequency ratios between
input and output. On the other side, brake grunt, growl, and groan may be characterized
by lower-frequency noises. If this results form steady sliding (as in braking) it may not

be obvious how to define the “order” of the response harmonics.

1.3 Background on Frictionally Excited Systems

Friction is a complex non-linear property that has been studied extensively. Dry
fiiction gives rise to stick-slip motion. As the term stick-slip suggests, at any give time
an object my be sticking to a base object while a short time later the same object may
now be sliding along the base object. This time independent change between static and
dynamic friction can result in chaotic motions that are extremely interesting to watch.
Stick-slip oscillators correspond to velocity-field discontinuities, and low dimensional
systems have a fundamental characteristic that allows their dynamic responses to be one-
dimensionally mapped [20, 21, 22].

There has been a huge global research effort on friction problems in dynamics.
Phenomena studied are friction-induced vibration, fiiction-induced noise (squeak &
squeal), such as with stick slip dynamics, control issues, dynamical fiiction modeling,
and friction damping.

R. A. Ibrahim [23] provided an extensive review on the mechanics of friction,
giving the science community a better understanding of this complex force involving
sliding surfaces. A system’s behavior, if it encountered friction, strongly relies on several
factors, including: materials and the geometry of sliding surfaces, surface roughness,
sliding velocity, temperature, and normal load. In 1994 B. Armstrong-Hélouvry, P.
Dupont & C. Canudas de Wit discussed how friction affects the controls community [24].
Using the results from tribology, lubrication and physics literature and combining it with
the findings in the controls community, a set of tools and models were created to
incorporate friction with control systems. B. Feeny et.al. gave a historical review of the

dynamics of systems with fiiction in 1998 [25] and S. Shaw presented a study on

dynamic response of a system with dry friction [21]. Kappagantu’s Ph.D. thesis [1] dealt
with the modal reduction of a system undergoing frictional excitation. The experiment for

our research is an offshoot of Kappagantu’s work.

1.4 Overview of Thesis

In Chapter 2 a description of the experiment will be given. It will include a
background on Ramana Kappagantu’s Ph.D. thesis; an in-depth look at the experimental
set up and system characteristics; a qualitative summary of Kappagantu’s friction
measurements; and a look at the methods that were used to analyze data gathered during
the experiment. In Chapter 3 and Chapter 4 the results of the analysis of the experiment
will be given. This will include a discussion on how the system dynamic responses were
characterized. Chapter 3 includes dynamics during frequency sweeps and identifies
frequency ranges of interesting dynamics are identified. Chapter 4 includes in-depth
studies of responses at fixed, selected parameter values. Chapter 5 contains the
conclusion, giving a summery of the experimental research conducted and possible

research that could be taken on in the future.

Chapter 2

Description of the Experiment

This chapter, as the title suggests, is a description of the experiment. It contains
an in-depth look at the physical set up, equipment, materials and contact geometry used
in the experiment, as well as the methods used to gather measurement data such as
nonlinear response data, frequency response data, modal frequencies and shaker contact

frequencies.

2.1 Background

This experiment exhibits a phenomenon that was referred to in the doctoral thesis
of Ramana V. Kappagantu. Kappagantu’s thesis [1], as his title suggests, dealt with “A
method of order reduction in modeling multi-degree—offreedom and continuous systems
with frictional excitation The tip of his cantilevered beam experiment with frictional
excitation appeared to behave in subharmonic as well as chaotic and quasi-periodic
response under different forcing frequencies. This phenomenon was noted but was not of
primary interest in Kappagantu’s thesis.

The research presented in this thesis took the same experimental setup of
Kappagantu and looked more closely at this interesting phenomenon. Poincare' methods
were used to get a better view of the dynamical system behavior. It was found that the
system did in fact have chaotic and quasi-periodic motion with very large order

subharmonic frequencies.

10

2.2 Experimental Setup

The shaker was a PM Vibration Exciter Type 4808 with a Power Amplifier Type
2712, both products by Brfiel & Kjaar. A Stanford Research Systems Model D8340
15MHz Synthesized Function Generator was used to produce the forcing function. These
experimental components were used to force the cantilever beam via friction excitation.
This accrues because a rubber nub at the tip of the cantilever beam rests on a contact
surface that is being shaken by the B&K shaker.

The cantilever beam was instrumented with strain gages. A 2100 System Stain
Gage Conditioner and Amplifier System, by Instruments Division Measurements Group
IN C., amplified the strain gage data. That data was then sent to the TEAC GX-l
Integrated Recorder, which was used as a data acquisition system. Matlab and Fortan
were used to post process the experimental data.

Figure 2.2 is a general diagram of the experimental setup and Figure 2.3 shows a
rotated view of the two sections. It is important to understand the coordinate system
XYZ in Figures 2.2 and 2.3. X is along the length of the cantilever beam, Y is along the
thickness of the beam and Z is along the width/height of the beam. According to this

coordinate system the crosswise oscillation is in the Y direction.

11

 

Figure 2.1: Experimental set up.

s—B

A—b O

T
I Week
A» —” «B
z
x J

Figure 2.2: Diagram of experimental setup.

 

 

 

 

 

 
    
   
 
 
    

      
         
   
 

B-B Section view

I l

A-A Section view

 

Asinwt
«———»

[o—

 

 
 

Friction Contact —

\ B&K Shaker
‘ 4808

2 H

Y_l

Beam—pl Z

._l

 

     
  

 

 

 

N

fiff/I/TIIIIIII I

 

 

 

Figure 2.3: Rotated view of sections A-A and B-B.

The shaker was fixed to the vibration table with clamps. This effectively
eliminated all normal and rotational vibration of the shaker. Kappagantu had the shaker
placed at a slight angle in the Y-Z plane, which made the contact friction between the
cantilever beam and the shaker vary depending on the location of the contact. In our
experiment this was achieved by rotating the fixed end of the cantilever in the Y-Z plane.
This rotation as it turns out this is very important to the behavior of the cantilever. The
tilt with the loading beam provided a variation in the normal load of about 0.035 N/cm,
where the tip position was estimated in a static displacement, and the load at the midpoint
of the surface was 0.63 N. Over the 7 cm span of motion on the surface, the load ranges
from 0.466 N to 0.655 N.

The cantilever beam used by Kappagantu was also used in this experiment.
Unlike Kappagantu, who was concerned with modal principles, our experiment aimed at
a qualitative description of the dynamics. We used only the strain gage pair closest to the
clamped end of the cantilever to produce time series data. Thus, as we are interested in

the qualitative nature of the behavior of the cantilever beam, we are using the strain gage

13

pair as an observer. The cantilever beam was constructed from a piece of mild steel with

dimensions of 0.4000 x 0.0128 x 0.00086 m3, elasticity ofE = 128x 109 N/mz, and a

density of p = 7488 kg/m3. Another piece of mild steel, with dimensions

0.064 x 0.0134 x 0.00056 m3, elasticity of E = 126 x109 N/mz, and a density

of ,o = 6777 kg/m3, was attached to the free end of the cantilever beam with its flexural

direction rotated 90° to act as a leaf spring. The leaf spring, referred to as a loading
beam, had a small rubber nub attached at its free end. The mass of the fixture that
connected the cantilever beam to the loading beam was 0.01226 kg. The total length of
the cantilever beam and the loading beam setup was 0.464 m. The bending axes of the
two beams were perpendicular, meaning the loading beam thickness was along the Z-axis
and the width was along the Y-axis while the cantilever thickness is along the Y-axis and
the width/height was along the Z-axis. This allowed for forces on the tip of the loading

beam to be transmitted to the cantilever.

2.3 System Characteristics

This experiment can be classified as a frictional stick-slip problem. As such, there
are two different dynamic states. In the first state the rubber tip sticks to the slider
surface, much like a pinned beam. In the second case, the rubber tip slips over the slider
surface, such that it deforms like a free beam. A stick-slip response can be seen as an
interchange between pinned beam and free beam dynamics. With these two different
dynamic cases it is beneficial is to see the Fast Fourier Transform (FFT) of the system

pinned and free to locate the model frequencies of the each state.

14

We examined the linear behavior of both the free and the pinned beam. Loading
the tip with a large normal force, such that sliding was not likely to occur, approximated
the linear pinned beam case. Figure 2.4 shows the F F T of the unloaded or free floating
cantilever and Figure 2.5 shows the FFT of the loaded case, where there was a high
normal load between surfaces. The first five modal frequencies for the unloaded case
were approximately 2.5, 19.5, 57.5, 112 and 180 Hz. The small peak that can be seen at
approximately 37 Hz is believed to be a distortion. The lowest frequencies of the loaded
modes were approximately 13.5, 43.5, 93 and 171.5 Hz. It was determined later that it
was also important to know the modal frequencies in the vertical (normal) direction
because the beam appeared to resonate in the normal direction at several different
frequencies. The normal modal frequencies were found by giving the pinned (loaded)
beam an impulse along the Z-axis at the fixture between the beam and the loading beam.
The output signal taken from the paired strain gages on the loading beam where then FFT

by the TEAC. The first normal modal frequency was found to be approximately 47 Hz.

15

 

 

MB 77 i r i 7

 

 

 

 

 

 

 

 

-20dB
was w ~ —:~~~ e—e ,_ ~~~
50Hz ‘ 100m 150m 200Hz '250Hz
Figure 2.4: FFT response to the unloaded cantilever.
OdB 7 7 7 7 7 7
—43dB
-80dB ~— ~~ 7~—7+i—-4— ~77 $777 77—77- 7—» 7~ 7% 7V ——7

 

 

 

50 Hz 100 Hz 150 Hz 200 Hz 250 Hz
Figure 2.5: FFT response of the loaded cantilever.

 

 

OdB

-20 dB

~80 dB

 

 

. .
,___ . _ ._ __ _ _ ._....7_7 _ _ - .___ ._.-__ ._.._ _ __..._________ n .—
I . I

 

 

mm. s f
« «WWW

 

50 Hz 100 Hz 150 Hz 200 Hz 250 Hz

Figure 2.6: FFT response of the impulse test of vertical motion of the loaded beam.

With the modal frequencies we are now interested in finding the forcing

frequencies in which the tip dynamics were present and showed appealing characteristics.

The signal generator was set to approximately 5H2 and slowly increased to

approximately 60Hz during which the tip of the beam was observed for interesting and

exciting dynamic behavior. In the end we focused on 11 different frequencies: 8, 9.8,

21.9, 22.1, 22.3, 25.7, 37.9, 43.8, 44.4, 45.2 and 45.6Hz. Each one of these frequencies

had similar but distinct tip movements that were fascinating to watch.

2.4 Friction Characteristics

Kappagantu, in his thesis, did an extensive study on the friction characteristics of

his cantilever beam and the surface that it rested on. Since our experiment is using the

17

 

same beam and a frictional surface made of the same material, the friction characteristics
should be similar to the results of Kappagantu. Since we expect the same qualitative
friction behavior, and our interest is in a qualitative description of the dynamics, we have
not quantified the current fiiction behavior; rather we summarize Kappagantu’s friction
measurements.

The Figure 2.7 (which came from Kappagantu’s thesis) shows the apparatus used
by Kappagantu to investigate fi'iction characteristic for his experiment. This setup used
equipment and materials that were very similar to his main experiment and allowed him
to gather information about the displacement, velocity, friction force and effective normal
load which the cantilever beam would experience during his experiment (Figure 2.8 taken
from Kappagantu’s thesis shows this). He was also able to see the relationship between
the fiction force and displacement of the beam (Figure 2.9 taken from Kappagantu’s
thesis) as well as the relationship between the friction force and the velocity of the beam
(Figure 2.10 taken from Kappagantu’s thesis).

The hysteresis in the force-velocity curve in Fig. 2.9 is consistent with a
compliant contact with negligible mass. Kappagantu measured the contact compliance
and crosschecked it with the stick-slip transition velocity. The contact stiffness was

estimated to be Ky: —20 kN/m and the coefficient of sliding friction was u=0.55 [1].

18

 

Asln(wt)

 

 

 

 

 

 

 

fl
\ j r—x
r \
Strain Gages v

 

Steel Bracket

 

    
 

 

 

 

 

I‘

B&K Shaker
4808

 

 

Pine-wood block

 

 

Figure 2.7: Kappagantu’s friction measurement setup.

19

 

 

 

disp in m
0 U!
E

 

 

 

 

 

 

 

 

 

 

 

_5 1 I 1 J i
1 1.2 1.4 1.5 1.8 2 2.2
0.2 . . , r .
w
E
.E 0 a
3
>
-02 I 1 l L 1
l 1.2 1.4 1.6 1.8 2 2.2
2 T I I I 1
2
E
.5 0 ‘
L3
"_2 1 1 1 1 1
Z 1 1.2 1.4 1.6 1.8 2 2.2
IE I 7 I T r I I I I I
131.865 '
a
:9 1.86
E1855
i) 1 1 1 1 1 1 1 1 1 1 1

 

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1
timeins

Figure 2.8: Kappagantu’s sampled time—series of displacement, velocity, friction force &
effective normal load.

20

friction force in N

friction force in N

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 I Y T I 1 fl 7 I
‘ 1- ( . . ‘ ‘ -1
0.5 1. . . . . .1
o b y 7 .4
-0.5 + q
‘ ’ ’ ‘ , - ~. ~’. ‘ !1‘. ‘
-1.5 A l 1 l l 1 1 1 l
-2.5 -2 .-1.5 71 -0.5 0 0.5 1 1.5 2 2.5
displacement in m x 10"
Figure 2.9: Kappagantu’s friction-displacement relation.
1.5 I 1 r Y Y
-18 1 l L 1 L
.15 -0.1 -0.05 o 0.05 0.1 0.15

velocity in m/sec

Figure 2.10: Kappagantu’s friction-velocity relation.

21

2.5 Method of Poincaré Sections

Jules Henri Poincare was a French mathematician who lived from the mid-
nineteenth to the early twentieth century. He is considered by many as the grandfather of
topology and qualitative dynamics. His work helped the development of deterministic
chaos [26]. With our experiment we used one of Poincaré’s techniques to help us view
our data.

Since the input fit) is periodic, if you make 0 equal to the phase of f(t), you are
effectively adding a periodic or cyclic axis to the state space. If the response states are If
and ii , then the Poincaré section is a cross section, 2, of the 3-D state space. Response
trajectories form points as they pierce 2. As a result, a discrete map represents the

continuous flow on 2. This is depicted in Figure 2.11.
More generally, there are n response states, and so the Poincare section 2 is n-
dimensional. In our work, we extract one component from the n-dimensional Poincare

section, and look at the sampled series of this component.

i
/ x1-
M

\
—r—’_'."—PI

¢ = phasero]

 

 

 

\Si

 

/.

Figure 2.11: Phase representation of the Poincaré plot.

22

 

Volts

 

 

 

 

l J
0 1 000 2000 3000 4000 5000 6000 7000
Sample Data Solid = Input Dashed = Output

 

Volts
O
l
J

 

 

l 1 1
0 2 4 6 8 10 12
Index

p
.—

 

-5

Figure 2.12: An example of a data and the sequence of strain gage volts taken form the
Poincaré section.

With the sinusoid from the signal generator used as the input signal and the strain
gage response-signal as the output signal, the strain gage signal was mapped at specific
moments with respect to the input signal. More specifically, every time the leading edge
of the signal generator input (which in all cases was a sine wave at some specified
frequency) passed through zero with a positive slope, the corresponding point in the
output signal was flagged and saved. This results in the sampling of one coordinate from
the Poincaré section. Figure 2.12 shows an example of this; however in this case both

signals are sine waves.

23

VoIts

Volts

Volts

I

p

.5
f

 

 

 

 

 

l
2000

3000

4000

I
5000
Sample Data Dashed = Input Solid=0utput

 

 

I

1

fl

 

 

Index

12

 

 

l

.—

 

 

6
Index

10

11

Figure 2.13: An example of actual data from the Poincaré section.

With our data the output signal is not going to be a perfect sine wave like the

example given. Rather it is the response signal from the strain gage closest to the base of

the cantilever beam.

Figure 2.13 shows a small sample of actual response data and the series of data of

a component of the Poincare section with respect to the forcing phase. This Poincaré data

allows us to investigate what type of dynamic motion the cantilever is undergoing. Even

from the small sample provided above it can be seen that the Poincare data allows a

pattern that might otherwise be over looked to be seen. The Poincare data is then mapped

as a delay map that allows the observer to see the path the Poincare data follows in a

reconstruction of the Poincare section. This was a starting point to see what type of

analyses should follow. In some cases it was easy to tell that the data behaved as if it was

24

moving through a torus or a torus doubling. In others however it was difficult to
determine the behavior. Visual inspection of the delay map allowed us to see which way

our investigation should head regardless of the case examined.

25

Chapter 3

Frequency Sweeps

This chapter contains the results from frequency sweeps. Up and down sweeps
were performed, which consisted of sweeping though frequencies from 8 to 50 Hz and 50
to 8 Hz receptively. The input and output data were collected via a strain gage. The
main question was whether there were frequencies where the cantilever beam behavior
was relevant to our study. The simple answer to this is a resounding yes. The frequency
sweeps identified frequency intervals of interesting dynamics, and helped estimate the

order of the response frequencies.

3.] Frequency Data Sweeps

A frequency sweep window of 8 to 50 Hz was chosen after the casual inspection
from 5 to 60 Hz mentioned in section 2.3. It was felt that this window of 42 Hz was
convenient because it incorporated all the regions that presented the phenomena we
wanted to study. For these sweeps the amplifier gain of the B&K power amplifier was
set to approximately 5.5 with the output impedance set to “high”. The function generator
amplitude was set to 4.00 peak-to-peak voltage. The function generator frequency was
increased (or decreased for the down sweep) at approximately 0.01 Hz every 0.31
seconds. The total time of the sweep was just under 22 minutes. Thus, at 8 Hz, there
were about 2 1/2 cycles of excitation for each frequency setting in the sweep. If the
response had, say, a subharmonic at order 5, then there would only be half of a response

cycle occurring before the sweep parameter changed. At the high-frequency end of the

26

sweep, say 50 Hz, there were about 15 input cycles per 0.01 Hz sweep parameter
increment. Still, if the response subharmonic had high order, insufficient response cycles
would take place for each sweep parameter value for the system to approach steady-state.
As such, the resulting sweep responses do not approach steady state at each parameter
setting. Rather, the sweeps really represent transient responses with a slowly varying
parameter. (The effect of the slowly varying sweep vs. steady state in a nonlinear system
has been examined by Raman, Bajaj, & Davies [27].) Nonetheless, we use these sweeps
to give hints at how the responses depend on frequency and what frequency values
generate interesting forced responses.

The sampling rate was 5000 Hz. Twenty-two minutes of data amounted to 6.6
million data. To reduce this, we retained the strain responses in the Poincaré section.
When the data was extracted from the Poincare section, the period between extracted
Poincare-sections data was also determined, thereby providing the driving frequency
associated with each data point. As a result, we obtained, for our sweep data, strains in
the Poincaré section, and driving frequencies.

A response point in the Poincare section does not have constant drift, amplitude
and phase information. It became clear that we could not tell from the frequency sweep
data if the large amplitudes being presented were real or just the beam being displaced
about a new equilibrium and having small oscillation about this new point. To reduce
this problem we created a low-pass filter program in Matlab to draw out the DC drift for

the original data sample. The low pass filter was set to 0.25 Hz. The filter was applied

 

using the transfer function, where a = 0.25 * 2 * 7r (rad/sec), b = 24a with

s2 +bs+a2

J = 0.707 , and s is the Laplace transform variable. The filter was applied to the data

27

first by integrating the signal as input to the filter, using Matlab’s lsim command. This
filtered signal was then reversed and applied to the same transfer function using the same
Matlab function. The reversal of the data was applied to nearly cancel the filter’s phase
effects. The double application of this transfer function amounts to a 4th order filter with
a cutoff frequency at a. Subtracting this DC drift from the original data effectively
allowed one to view the data as if it was oscillating about the same equilibrium the entire
time. Figures 3.1 and 3.2 show the up and down frequency sweep data respectively. The

nonzero DC components of .1: indicate an effective quadratic nonlinearity in the system.

Frequency Up Sweep Data

5 I l I I l 1 I I

 

Pomcare
X(N)
O
>
L

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

' 5 1o 15 20 25 30 35 4o 45 50 55
5 I l I I I I I I I
g (”N 1 l\
D 2 _ L, x __ /‘\ _
8 $2 0 (,2/ \\ "/1/ \T‘cwfl'v“ __,..\rJ/ o
_5 4 1 1 1 1 1 L 1 1
5 1o 15 20 25 30 35 4o 45 50 55
o 5 I I I I I I I I I
D
10 VI
.3 A EE N 1: "WM V
E a o ‘— WN H—fi “
2 >< 1 \,\~
(U
U
C
'5
0' _5 1 1 1 1 1 1 1 1 m
5 1o 15 20 25 30 35 4o 45 50 55

Frequency Hz

Figure 3.1: Frequency up sweep data showing the Poincare, the DC drift and the Poincare
minus the DC drift.

28

Frequency Down Sweep Data

 

Poincare
X( N)
O
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 10 55
5 T If If f I I I I I
c ’M
5 :2: 0 '- tr-A If \!\F“ A n \— -4
8 >< \\ M MM i/
5 1 1 1 a 1 1 1 1
5 10 15 20 25 30 35 40 45 50 55
U 5 I I I I *I I I I
O
01 VI
.3 A n in IV V
e >‘<’ ‘
3
c
'6
0' _5 1 1 1 1 1 1 1 1 1
5 10 15 20 25 30 35 40 45 50 55

Frequency Hz

Figure 3.2: Frequency down sweep data showing the Poincare, the DC drift and the
Poincaré minus the DC drift.

The response in the Poincare variable minus dc drift figures are not the actual
amplitude that is experienced doing the experiment, but rather Poincaré section samples
taken for the experimental data. The data shown is more analogous to the real part of the
complex oscillation. An approximate calibration for the dc drift is that one-volt positive
or negative is equal to about one centimeter on either side of the strain gage equilibrium.
Frequency intervals for which significant oscillations occurred are labeled as Regions 1-
VI in Figures 3.1 and 3.2. It should be remembered that the modal frequencies for the
unloaded case were approximately 2.5, 19.5, 57.5, 112, and 180 Hz, and that none of the
regions of dynamics drift occurred at these frequencies. The first two loaded model
frequencies were approximately 13.5 and 43.5 Hz, which occurred at the edges of regions

I and VI respectively. The nonnal-degree-of-freedom frequency, which is approximately

29

47 Hz, lines up with region VI and is mostly likely the reason for the response that
occurs. This normal vibration became noticeable in region hybrid and VI. In fact the
normal vibration was very large in the first subregion, but became somewhat subdued in
the second subregion. The comparison of these two regions will be discussed in more
detail in section 3.3. Labeled on both of the Poincaré-minus—DC graphs are regions of
oscillatory activity. These regions, though different, do appear to be comparable. Since
Figure 3.2, the down sweep, shows the most distinct regions, it was used to number the
regions. The “hybrid ” region of the Figure 3.1 does not fit into the mold set by Figure
3.2. Rather the “hybrid ” region appears to be a combination of regions [I] and IV from
the down sweep, where region [11 has slide up the frequency scale and region IV has
moved down the frequency scale. This could be the case since the initial conditions of
the beam will differ between the up and down sweeps making the corresponding

responses of the beam analogous but still unique.

3.2 Subharmonic Order in each Region

From the behavior of the beam we felt that it was important to look at the
“subharmonic order” of the regions of interest. We accomplished this by taking a region
and performing a short-time average FF T calculation on a small window of data. This
window of data started at the beginning of the region. Index by index we calculated the
average FFT for each window. The end product was a short-time average of the FFT
across each window, whence we obtain the average subharmonic at each point of the

region.

30

In this context, what we mean by “subharmonic order” is the ratio Q/a) , where a)

is the dominant frequency in a response that may not be periodic. This reference to a
“subharmonic response” may thus be different from the traditional reference to a periodic
response of a period which is a multiple of the excitation period. We use the terminology
“subharmonic order” since many of our response have a dominant component at a
frequency which is lower then the excitation frequency.

In each frequency region, a short-time window length q was chosen to be large

enough to capture the lowest frequency expected (from previous inspections of the data),

yet small enough to represent a narrow frequency interval. Given q , the frequency
window for each short-time FFT was from the sweep parameter window, Um , fmax ],
from which we estimate the average driving frequency as f d = (fmin + m )/ 2. Given q
and f d , the approximate time record for a short-time FFT was q / f d . This defined the
frequency resolution (minimum frequency) in the short-time F FT as f d / q . The
maximum frequency of the short-time FFT was based on the Nyquist criterion as f d / 2.
Hence, a given short-time FFT had a minimum detectable frequency of f d / q and a
maximum detectable frequency of f d / 2. The short-time FFT frequency axis has values
fd/q, Zfd/q, 3fd/q kfd/q,-~, fd/Z-

In the calculation of the subharmonic order, we found the frequency for which the

short-time FFT was a maximum, say k f d / q . Then we computed the subharmonic order
as f d / (k f d / q) = q / k , where k can have an integer value from 1 to q / 2 . The reciprocal

of the order would be k/ q . Comparing k =1 and k = 2 corresponds to a factor of two in

the order. Signals with aberrant low frequency components (for example due to

31

transients and changes in dynamic behavior across the window) lead to small k .
Because of this, if k = 1, we set the order to zero to omit that case and limit the scale of

the order axis in the plots. Thus, the maximum presentable subharmonic order is q / 2 .

3.2.1 Region I

In this case, q = 50 , leading to a maximum presentable subharmonic order of 25,

and a frequency window width for the short time FFTs of the up and down sweep of
approximately 0.15 and 0.153 Hz respectively. Region I is located approximately
between 8 and 13.5 Hz. Figure 3.3 shows the frequency up sweep Poincare-section strain
signal of the region along with its subharmonic order. Figure 3.4 displays the same as
Figure 3.3 but is of the down sweep.

Both up and down sweeps are rather consistent in indicating subharmonic orders
in ranging from of 3 to 7, depending on the input frequency. Isolated incidents of large
subharmonic orders are likely due to shifts in the sweep response, which would be
represented as large changes in the course of the short-time signal window, and hence
correspond to low frequency components. It is also possible that chaotic responses with

broad band FFTs are present, such that the lower frequencies in the FFTs dominate.

32

X(N)

Sub-Order

Up Sweep Region I

 

 

 

 

 

 

 

 

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10 ‘1
5 F— .fi - u
Hh—fit“ é .
0 1 : I: 1 A 1 : 1 t l: .1-
8 9 1o 1 1 1 2 1 3 1 4
Frequency Hz

Figure 3.3: Frequency up sweep of region I and the subharmonic orders of the response.

X(N)

Subhan'nonic Order

3.5

2.5

N

1.5

0.5

25

20

15

10

Down Sweep Region I

 

 

 

 

 

 

 

 

 

 

8 9 10 11 12 13

»—r—o—-——~—-—~ war » >w-———-——r—~——————«o—oor—o-Ha——-oor————-———&o—o1
]
.:.o-—-.-‘. 0....- I :- . . o e —-l

'.' w “.3... . '—fi ‘ :o h ‘. f— -
' 1: 1 1 : j :5 _ _1 =
8 10 11 12 13
Frequency Hz

Figure 3.4: Frequency down sweep of region I and the subharmonic orders.

33

3.2.2 Region II

In this case, q = 200 , leading to a maximum detectable subharmonic order of 100,

and an average short-time F FT frequency window for the up and down sweep of
approximately 0.29 and 0.3 Hz respectively. Region II is located approximately between
20.8 and 23.2 Hz. Figure 3.5 shows the Poincare' section of the up sweep of the region
along with its subharmonics and Figure 3.6 shows the Poincaré section of the down
sweep of the region along with its subharmonics. Viewing Figure 3.5, the subharmonics
jump around like so: 40, 28.5, 50, 67, 5, 40, 18, 33.5 and 40 Hz. To compare, in Figure
3.6 the subharmonic orders are as follows: 25, 28.5, 33.5, 40, 50, 28.5, 25 and 33.5Hz.
Here, the isolated “regular-order” subharmonic is not observed (as in the order 5 seen in
the upsweep). It is possible that the finite sweep rate glossed over it. Thus, in general the
values of the subharmonic orders do differ but a basic pattern stays the same between the
up and down sweeps, which implies that the up and down sweeps in fact share a similar
dynamic behavior in region 11 although the absolute response may very slightly. Again,
the incidents of zero-subharmonic orders in the plot arise in fact from the maximal
subharmonic order, reset to zero for the plot, disregarded as they are likely to indicate

low frequency transients during the sweep.

34

Up Sweep Region II

 

 

 

 

 

 

 

 

 

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21.4 21.6 21.8 22 22.2 22.4 22.6 22.8 23 23.2
Frequency Hz

Figure 3.5:

Frequency up sweep of region 11 and the subharmonic orders.

Down Sweep Region II

 

 

 

 

 

 

 

 

 

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20.8 21 21.2 21.4 21.6 21.8 22 22.2 22.4 22.5 228

Frequency Hz

Figure 3.6: Frequency down sweep of region 11 and the subharmonic orders.

35

3.2.3 Region Hybrid, III & IV

In this case, q = 200, leading to a maximum presentable subharmonic order of
100, and the average short-time FFT frequency windows for the hybrid, III, and IV
regions are approximately 0.24, 0.26, and 0.23 Hz respectively. The “Hybrid” region is
located approximately between 25 and 28.3 Hz. Region III and IV are located
approximately between 22.5 to 26.5 Hz and 27.2 to 28.8 Hz respectively. As mentioned
earlier the Hybrid region of the frequency up sweep appeared to be a combination of
regions 11] and IV of the frequency down sweep. By looking more closely at Figure 3.7,
which shows the Hybrid region, it is unmistakable that there are two regions present.
Figure 3.8 and Figure 3.10 separates these distinct regions of the Hybrid region into two
sub regions. For simplicity Figure 3.9, region 111, and Figure 3.11, region IV, have been
placed where they are because of the uncanny similarity to their previous Figures 3.8 and
3.10.

Region III is characterized by subharmonic orders in the range of about 10-13.
The hybrid region shows subharmonic orders that are consistently around 12-13 for the
first segment of the up sweep, then jump to over 20, thereafter decreasing markedly to ten

as the parameter increases. A similar trend will be seen again in another region.

36

Up Sweep Hybrid Region

 

 

 

 

 

 

 

 

 

 

 

1 1 r 1 1 —1 1
0.5 — _
2 _ _
52' 0
-0.5 ~ —
_1 I 1 I 1 I I
25 25.5 26 26.5 27 27.5 28 28.5
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O
E
O
.2 60 L _
E
a 40 ~ ~
5
_. ‘— A _
0 _ I I l l l L
25 25.5 26 26.5 27 27.5 28 28.5
Frequency Hz

Figure 3.7: Frequency up sweep of the Hybrid region and the subharmonic orders.

Up Sweep Hybrid Region Sub 1

 

 

 

 

 

 

 

 

 

 

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. ' o ' 0'... o .0 ‘.
A O >- 2: 3: o . ... . ::‘.1:.. ... 32:4.53... .' o”, 1. _
a at. . . - -~ ., =.:, :5. ~ .. ..
X '0 2 I. . . . 2‘0...’ 0 00 . .‘. . .° 0 ‘ :. 3." ‘51:: :4 ;\. —1
..W W We: -- : :-:.. 1 '
5 e I . , e
4W ‘~ ‘ ° :....‘2‘ . _
'06 l i i l I 1
25 25.2 25.4 25.5 25.8 25 25.2 25.4
20 I I r 1 I 1
g 15 ~ -
o ..
.9 ow
5 10 7 " _
E
a
a:
D
a, 5 7 _
0 1 _ 1 L 1 1 1
25 25.2 25.4 25.5 25.8 25 25.2 25.4

Frequency Hz

Figure 3.8: Frequency up sweep of the Hybrid region sub 1 and the subharmonic orders.

37

Subharmonic Order

Down Sweep Region III

 

 

 

 

I I
23.5 24.5 25

 

I
25.5

26.5

 

100

80

60

40

20

 

 

M

A . 1

 

L

 

 

 

24 24.5 25

Frequency Hz

23.5

1
25.5

26 26.5

Figure 3.9: Frequency down sweep of region III and the subharmonic orders.

X(N)

Subharmonic Order

Up Sweep Hybrid Region Sub 2

 

 

 

 

 

 

 

 

 

I I f I I I I
.0. Q . E f
0.5 - ’4. _
t 2 3 .012. O‘ . 2.933": of? ..." «.4 5;: 5:310:54: .J
01%: L£o$£°l¢ :%&M\ j}:o:'b.:'0 g” ...fsz'. ~ :;o:. ::’f':;:;:': ‘0}0 ‘1’..; 5"”. '9
mama“. .. c. .. ~= . .... “35:33pm? w
0 “ 111332;: 7‘ tax: '. .' .': .:. 1’1 ”-’°’~r"..=> -:'-.-.’.~.-- "’I-‘w ‘ ° “
° ’11 “2:3: 'r',."-.,'- .. H" , ': :‘p:°°:i°;$:?z"x.“:; '3“
17:71:23.3; .- 21"“ .x. 75:31:22“
'05 P— . 25213.9: .'0 :g, 00.0:00 1"? '4
_1 I; 1 1 1 1 1 1 1 1
26.2 26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 28 28.2
20 I I I I I I I I r
18 e e
16 7 I
14 *- °. 71
10 1 1 L 1 I ‘7- l 1 "
26.2 26.4 26.6 26.8 27 27.2 27.4 27.6 27.8 28 28.2

Frequency Hz

Figure 3.10: Frequency up sweep of the Hybrid region sub 2 and the subharmonic orders.

38

Down Sweep Region IV

 

 

 

 

 

 

 

 

1 I I I I I fl
0-5'7 O ; ;: .. :::o: ' . . 0'.‘. :0 .4
A 1.. an. ..e 2.. .. .'.~. .' «.13 . :.= '
E - 5.?!“ ~-..'. '4'- .:. -. x ' '.- . W"
X M .\ E:‘.:“ ,. 0 ....18.)‘.. ‘ . u ' ” 8 o . a. .z :10. ... .0 .0 8 o
o __ : 3': .0. .:0~.'0.’.'. ? '0 8. 8 .° '. o 00'. ”W?¢¢
0'9"O:... I. “"0 ..Jo‘ ‘.. 00“ ‘ WI. —1
to .oz;~:.xx'.._‘:c. J .0: g «z: ”7:???“ 4;". 3; ’0 a“:
o 1. g. .9 . w . O .0 . 0 '0 0 .
x. . ‘ .O 3:: .4 r . z.“
: 8
_0'5 1 1 1 1 1 1 1
27.2 27.4 27.5 27.8 28 28.2 28.4 28.6 28.8
20 T l I I T T j
g 15 ~ g
5 o
.2 *— ¢ -
E 10 r- W—-__-_ - —
a
5
a 5 ~ 1
0 1 I 1 l 1 ' —1
27.2 27.4 27.6 27.8 28 28.2 28.4 28.6 288

Frequency Hz

Figure 3.1]: Frequency down sweep of region IV and the subharmonic orders.

3.2.4 Region V

Region V is located approximately between 30.2 and 34 Hz. With q=200, the
subharmonics of this region feature a general harmonic around 2.2 Hz, though there are
short instants where the harmonics jumped to around 20 Hz, but this still did not create a
beam response that was interesting in our study. Figure 3.12 and Figure 3.13 in the
appendix to see the graphs of the frequency up and down sweeps of region V and their

sub-order harmonics.

39

Up Sweep Region V

 

 

 

 

 

 

 

 

 

 

 

1 I I I I
0.5 ’- *3
12:
X
0 " 4
_0.5 I l I l l
31 31.5 32 32.5 33 33.5 34
100 =° T T l 1 —: r =
3 80 F _
E
O
.2 50 “ 3
g 40 ~ 3
.C
D o —
(3 n - o o- o _
20 I" a - r . . . a o -J
0 __v —- 10 1 __
31 31.5 32 32.5 33 33.5 34

Frequency Hz

Figure 3.12: Frequency up sweep of region V and the subharmonic orders.

Down Sweep Region V

 

 

 

 

 

 

 

 

 

1 T f r I r T
' . ..I . I r
05 ” , , 0 . .. 9.0.}...r’: ‘8. z! ‘ _
A ‘2"... .‘O/ .‘ 3‘ 9" it!" my “I: . I II , ..
z wrung. ,, £*’,«,‘~ « t. ° -
z . .. .. J1 .. <0 drum . ~ . -
, ‘. . " \.- ‘r. W3; . o'go‘.’ .2
0 _ z 3‘“ ..k‘ 0’9:'o6 o.) ”-:mq’. 0 ‘ o A
'05 l l 1 L i 1
30 30.5 31 31.5 32 32.5 33 33.5 34
100 V“ I I r I * l
a; so» A
E
O - -. .
.9 60F n
g - -
m 40» -' - ° ‘
C
D
3 O . -
‘0 20- . ” " ~
0 1 ' - fl 5— - 1
30 30 5 31 31.5 32 32.5 33 33.5 34

Figure 3.13:

Frequency Hz

Frequency down sweep of region V and the subharmonic orders.

40

3.2.5 Region VI

In this case, q = 400, leading to a maximum presentable subharmonic order of
200, and short-time F FT frequency windows for the up and down sweeps were 0.28 and
0.284 Hz respectively. Region V1 is located approximately between 43 and 47.3 Hz.
Figure 3.14 shows the frequency up sweep Poincare section of region VI along with its
subharmonic order and Figure 3.15 shows the down sweep. Upon looking at the
frequency up and down sweep graphs of region VI it becomes clear that there are two
distinct regions, similar to the Hybrid region. Figure 3.16 to Figure 3.19 show region VI

subdivided into to separate regions.

Up Sweep Region Vi

 

 

 

 

 

 

 

 

 

 

g o a
X
-2 I
A L I J l l I
43.5 44 44.5 45 45.5 46 46.5 47
150 I T If I I I I
5
E
0 100 - — ~
.9
E —
Q 50 - —_ _
3 _
w —
O I l l I i l I
43.5 44 44.5 45 45.5 46 46.5 47

Frequency Hz

Figure 3.14: Frequency up sweep of region VI and the subharmonic order.

41

Down Sweep Region VI

 

 

 

 

 

 

 

 

 

 

_4 1 L 1 4 4 1 4 1
43 43.5 44 44.5 45 45.5 46 46.5 47
150 I I I I I I I I
(‘5 100 ~ . _ _
.2
g _
g 50 " __ . .1
3 —
(D
0 1 1 1 _.L 1 1 1 __ 1_
43 43.5 44 44.5 45 45.5 46 46.5 47
Frequency Hz

Figure 3.15: Frequency down sweep of region VI and the subharmonic order.

Up Weep Region VI Sub 1

 

 

 

 

 

 

 

 

 

 

2 I I I I l I I I
1.5 I" «W “
1 . Wmfi " _
2 0 5 4 5 .v 3‘, 335.35.655.51; .0”... .S
SE “;:.:::?.0.:.:. 0 :5... 0'0 .’*?$0
0 _ ....‘3; ° ".3130: cs. .. .f _
“3.0:: J...” ..trw’é'.’
-o.s - ' ”1:“: a
_1 1 1 1 1 1 1 1 1
43.4 43.6 43.8 44 44.2 44.4 44.6 44.8 45 45.2
9 I I I I I I I
Q" m
U
<5 8.5 ~ -
.2
8
E ...
G
s a - — .
(3 —
75 I— 1 1 1 I 1 1 1
43.4 43.6 43.8 44 44.2 44.4 44.6 44.8 45 45.2
Frequency Hz
Figure 3.16: Frequency up sweep region VI sub 1 and the subharmonic order.

42

 

Up Sweep Region VI Sub 2

 

 

4 , j I I i I
- I IIIétifi
2 -,3‘ I“? ”III; I. I
2 0» ..‘SAJIEIIIIIIIIH ”I‘M“ i
’7 Signiiséuti 1‘
-2- :21” I
-4 1 1 4 1 1

 

 

 

L I I I
45 45.2 45.4 45.6 45.8 46 46.2 46.4 46.6 46.8 47

 

 

100 I I T I I I h I T

I. 80 >— _ _.
*3

O o—

g 40 I- — ~I
5

3

‘0 20 ~ 1

I

 

 

 

0 1 I I 1 I J 4 I
45 45.2 45.4 45.6 45.8 46 46.2 46.4 46.6 46.8 47
Frequency Hz

Figure 3.17: Frequency up sweep region VI sub 2 and the subharmonic order.

Down Sweep Region VI Sub 1

 

 

 

 

 

 

 

 

 

 

2 I j I I 7 I I I I
1.5 '— 8...” {.:.ée
8:53:32; 03"‘fffip
1 - 53:15.2, ‘.-':...I:..’ .
E, 0.5 . #:"a; ° . 3
X . " zf'oa‘si-Iflrar .\ °
o- “32»:- .- ...--. -
’7’}?
-0.5 r I
_ 1 1 4 4 1 I 1 1 1
43 43.2 43.4 43.6 43.8 44 44.2 44.4 44.6 44.8 45
9 I I I T— I 7 I I I
3 8.5 —
5 ...
'2 — I
g 8 ’- —_ _1
a .—
S —
5 7 5 I —
7 I - l I I I I I J; I
43 43.2 43.4 43.6 43.8 44 44.2 44.4 44.6 44.8 45
Frequency Hz

Figure 3.18: Frequency down sweep of region VI sub 1 and the subharmonic order.

43

Down Sweep Region VI Sub 2

 

 

 

 

 

 

 

 

 

H";
IIIIIIIIIIIIUKI: fl
...: .. ‘1 ‘11?!
II{’{§?8'°iF°1° '
2 fig-r4554» “g f} ~
>'< :12?! 4;;5?‘ ..i
{5:31‘ dz
3 :2 A
-4 l 1 I I 4
45 455 46 46.5 47
150 r r r n l
1‘3
.2
8 ——
E —
cu ...—......
fi 50" —_ _1
3 _— .
U)
0 I I 1 4 = 1
45 45.5 46 46.5 47

Frequency Hz

Figure 3.19: Frequency down sweep of region VI sub 2 and the subharmonic order.

In the 43-45 Hz window, we see rather subharmonic orders ranging from about 7
to 9 (Figures 3.16 & 3.18). The 45 to 46.5 Hz window shows a gradual change in
subharmonic order, quantized in steps due to our order resolution defined by q / k .
Following driving frequency (in either the up or down sweep) downward from 46.5 Hz,
for which the subharmonic order is about 40, the subharmonic order increases, actually
according to a smooth curve to a remarkably high value (well over 100) before the
system bifurcates into the “order-10” subharmonic region.

This gradual change in subharmonic orders is predominately represented by

quasi—periodic dynamics, as the order is likely typically irrational. The “order-10”

44

subl

dIII;

rm

{/3
p-‘
V.)—

IICL

Ila

subharmonic region is also quantized, and typically represented by quasi-periodic
dynamics.

Our graph measures subharmonic orders up to 130, as a quantization of some
irrational order, during a test with a slow sweep rate. It is interesting to fathom the
behavior for true steady-state dynamics as a function of driving frequency.

If we plot the reciprocal of the subharmonic order, i.e. f / f M, , for regions VI

sub 2, we get plots, like Figure 3.20, which resemble the frequency demultiplication plots
of Van Der P01 and Van Der Mark [13]. It is not the same behavior. Our steps are due to
frequency resolution in the short-time FFT plots. The steps of Van Der Pol and Van Der
Mark were observed as discrete subharmonic responses, sustained for a parameter
interval, and the bifurcating to the next subharmonic, often with accompanying noise at

the bifurcation.

45

Demultiplication Region VI Sub 2

 

 

 

 

 

I I f l I I I I
E
O
D a
3 _—
g 0.01 — .
0005 I I I I I I I I I
45 45.2 45.4 45.6 45.8 46 46.2 46.4 46.6 46.8 47
I I I 7 I I I I I
5 0.025 " _ _.
s .
3o
0 2 002 >- 0”— —
5 8 “
C
3 g O_015 »— ——-—- —r
o f: __
3
5".) 0,01 .. __ q
0005 I I I %

 

 

 

I I I I I
45 45.2 45.4 45.6 45.8 46 46.2 46.4 46.6 46.8 47
Frequency Hz

Figure 3.20: Apparent demultiplication of region VI sub 2.

3.3 Summary

The similarity between the hybrid region and region V] are almost unmistakable.
By comparing Figure 3.7 with Figure 3.14 or 3.15 you can see a bifurcation between two
zones of the subharmonic responses. With the region VI having magnified subharmonic
orders compared to the hybrid region. Region VI involved vibrations in the normal
degree of freedom, and it occurs close to the normal modal frequency of 47 Hz. The
hybrid region also was affected by normal vibration and it occurred close to 47/2 Hz.

From this information we can interpret that the normal degree of freedom plays a
roll in the dynamics, in one case (VI) the 47 Hz normal degree of freedom is resonated

directly, and in the other case (hybrid) through a superharmonic resonance of order 2

46

(from comma, /a)d,,,,e, 5 2 ). Such a superharmonic resonance of order two requires a

quadratic non-linearity, which must already exist by virtue of DC drift figures.

47

Chapter 4

Fixed-Frequency Responses

In this chapter the frequencies noticed, during the sweeps, to cause the tip of the
cantilever beam to under go what appeared to be chaotic or quasi-periodic drift were
examined in more detail. As stated in Chapter 2, we focused on 11 different frequencies:
8, 9.8, 21.9, 22.1, 22.3, 25.7, 37.9, 43.8, 44.4, 45.2 and 45.6 Hz. We will group these
frequencies with their corresponding regions from Chapter 3. Region I holds 8 and 9.8
Hz; region 11 holds 21 .9, 22.1, and 22.3 Hz; region III/Hybrid holds 25.7 Hz; and region
VI holds 43.8, 44.4, 45.2, and 45.6 Hz. The only frequency that is not in a region is 37.9
Hz; this frequency was examined not because it caused the chaotic tip drift we wanted to
study, rather because its strain gage response signal during the sweep was almost a

perfect triangle wave which was a curious quirk.

4.1 Fixed Frequency Examination

Once the fixed frequencies of interest were chosen, a quick estimation of the
cantilever tip characteristic drift period was used to estimate the length of the time trace
data needed to effectively examine the response behavior of the cantilever beam at the
given frequency. In general the higher driving frequencies needed a longer time trace
than the lower driving frequencies because they generated lower frequency responses.
This time trace data was then post-processed by Matlab to gather data in the Poincare
sections, through which most of the dynamical analyses were performed. Using the

Poincaré section condensed the amount of data to be analyzed. A time trace with

48

millions of data points was reduced to only tens of thousands of Poincaré section data
points.

Data from the Poincare section was viewed in a delay map to provide a
reconstruction of the true Poincaré section. The pseudo Poincaré section enabled us to
glimpse how the cantilever beam was behaving. This allowed us to come up with other
types of analyses, such as taking the Poincare section of the Poincaré section, or a
looking at the torus angle of possible tori, to gather more information about this dynamic
phenomenon.

To get a sense of the dimension of the dynamics, we looked at the false nearest
neighbors (FNNs) [28] and the fractal (correlation) dimension [29]. The correlation
dimension is a measure of the dimension of the attractor (or attracting set), while the
FNNs give information on the dimension of the phase space needed to describe the
attractor.

We also looked at F F T’s of the response signals to understand the frequency
content. In performing the FFT’s we applied a Hanning window to the data to reduce
leakage distortions. We performed the F FT’s using the first 10,000 data points of the
collected data of each frequency. We omitted data to reduce the sampling rate and
thereby fix the Nyquist frequency, and hence the scale of the FF T plot, to a reasonable
value (two to five times the driving frequency). In some cases we displayed the FFT of

the Poincaré section data to magnify the subharmonic contributions.
4.1.1 Region I (8 & 9.8 Hz)

8 and 9.8 Hz were the forcing frequencies selected from region I. The cantilever

beam was then forced at the given frequency for an extended amount of time to allow it

49

to settle into a steady state after which the collection of data was started. Data was
collected for 15 minutes at 8 Hz and for 20 minutes at 9.8 Hz. This time trace data was
then imported into Matlab and the Poincaré section was taken. Plotting the Poincaré
section in a delay map helped in visualization of the data. Figure 4.1 and Figure 4.2 show
the delay map with a delay of one index of the Poincaré section samples of 8 and 9.8 Hz
respectively. Three iterates of the 8 Hz data travel clockwise in the Poincaré section,
moving from spot to spot, occasionally getting trapped in the area of (-0.2, -0.2) for
several iterates. The iterates of the 9.8 Hz data also circle clockwise. While the distance
from the center of the data (2, 2) increases and decreases, circular motions in the Poincare
sections suggest toroidal dynamics, although in these cases the torus would be fat.
Thinking this way, we took the Poincaré section of the Poincaré section and the torus
angle to look for dynamical structure. The torus angle was taken about the approximate

center of the torus of the delay map of the original Poincaré section.

50

X(N+1)

X(N*1)

Poincare of 8 Hz with a Delay of 1

 

 

 

 

 

 

 

 

 

 

0.2 I I I I I
0 _ ..
-o.2 ~ -
-o.4 ~ —
-06 >- —
—o.a . —
.1 I. _
I l I l I l
.1 -o a - -o.4 -o 2 o 0.2
X(N)
Figure 4.1: Poincare section delay map of 8 Hz.
Poincare of 9.8 Hz with 8 Delay of
35 l7 I f I I
3 ’ -I
2.5 — .
2 - -
1.5 » ~
1 , _
0.5 I l I I I
0.5 1 1 5 2 2.5 3 3.5
X(N)

Figure 4.2: Poincare section delay map of 9.8 Hz.

51

Figure 4.3 shows the Poincaré section of the Poincare section for 8 Hz. The
Matlab program that performed the Poincaré section of the Poincaré section flagged
Poincare section data that crossed the diagonal from positive to negative, the positive
being the bottom right half and thus the negative being the top left half. By viewing the
delay map of the 2"d Poincaré section in Figure 4.3 we see that, for the most part, the
Poincare data wraps around the torus, fluctuating at (-0.7, -0.70, but there is also another
mini torus the Poincaré periodically visits at (-0.3, -.03). However, Figure 4.3 does not
reveal any structure in these toruidal fluctuations, suggesting higher dimensional
dynamics between the two components of the torus. Refer to Figure A.1 in the Appendix
to see the Poincare section of the Poincaré section verses its index to see when the mini

torus is visited with respect to the larger torus.

8 Hz Poincare of Poincare
'0.1 I I I I I j I I

 

o . . ’0‘.
o ' o 0 ° .0 0..
' o . {Jo’s-5‘ '1" o
0 3 . ' ' .. . o . o .0 ... . ...
- . r r ’ ' . o 0' 0.- e P
o - ° . ° :. s. :° °'.':"° ‘ o
' o o ' . . . . 00
0 . . o . 0 O 0 .
_0 4 ... .0 O ' oo o: 0 _,
. . o 0 . 0 o
0 ° . 0' .

. O
o o
. .0 o
-05~ -
- 0 o
O o
o

X(N+1)

-O.6 F

~0.7 F

 

-0.8 F

-o.9 » ' ~

 

 

 

' -1 -o.9 -0.8 -o.7 -0.6 -o.5 -o.4 -o.3 -o,2 -o.1
X(N)

Figure 4.3: Delay map of the Poincaré section of the Poincare section delay map at 8 Hz.

52

The Poincaré section of the Poincaré section for 9.8 Hz is unlike the 8 Hz
case because this time the Matlab program flagged the Poincare section data that crossed
the diagonal from negative to positive. This was chosen after looking at Figure 4.2 and it
was felt that it was a clearer crossing than if we were to go from positive to negative.
Figure A2 in the Appendix shows the outcome, which is that the Poincaré data crosses

the diagonal in one continuous area.

 

 

 

Figure 4.4: Torus angle model.

53

Theta(N+1)

Theta(N+1)

Toms Angle Return Map of 8hz

 

 

   

 

 

 

  
   

 

4 I I T I I f
3 _
2 ~ Winding # : -1.3376
1 ~ . . , .
0’0"". .';. " '
0 . ~.
g \
.. 0 (a.
0 .,, r .
. co . . '. v .
"a ....
o o : ‘ ...” r
-2 a . . , “3“.
0‘ . "o‘ . ...: ' 00.
-3 — f
_4 L I I I 1 1 I
-4 -3 -2 -1 0 1 2 3
Theta(N)
Figure 4.5: Torus angle dynamics of 8 Hz.
Torus Angle Return Map of 9.8112
4 T I T I I f
3 t
z
2 I
1 ~ Winding #: 4.445
01
-1 _
-2 a
-3 F
_4 I I I I I L I
-4 -3 -2 -1 0 1 2 3
Theta(N)

Figure 4.6: Torus angle dynamics of 9.8 Hz.

54

 

 

Figure 4.4 shows a simple schematic model of how the torus angle return map
was created from the Poincaré section delay map. Matlab was used to calculate the angle
0 in each Poincare section with respect to a given center. Figure 4.5 and Figure 4.6 show
the delay maps of the torus angles, 0, for 8 Hz and 9.8 Hz respectively. At 8 Hz the
center coordinates x-bar and y-bar were at -0.5 on Figure 4.1. The x-bar and y-bar, for
9.8 Hz, were 2.1 on Figure 4.2. Figure 4.5 shows that Poincaré section data at an angle of
approximately -0.3 rad will map to an angle of approximately -0.2 rad. These data then
map to around 0 equal to -O.5 rad. Data at 0 approximately equal to -0.5 rad mostly map
to 0 equal to 1-2 rad, but occasionally map to values of 0 approximately equal to 0-1 rad.
In this region the data is almost trapped, most often mapping back to the same region of
0. This is the “mini torus.” Occaisionally they map back to the previous region, but data
in the positive end of the 0-1 rad region get cleanly mapped to 0 values of +1 .5-2.5 rad.
These data go to 0 approximately equal to it, and wrap to 0 equal to negative 7:, where the
process starts over again.

Hence the 0 map indicates rather deterministic travel around the main turns, but
dynamical interaction which is not a simple l-D map at the region of the “mini torus.”

Figure 4.6 shows, for the 9.8 Hz case, a strong band resembling a circle map,
clouded by additional dynamics off the circle map. It suggests that some higher
dimensional dynamics occurs with some strong tendency torward circle map, dynamics,
yet it does not “settle in” on the circle map. This makes sense, as a thin torus is not
apparent in Figure 4.2. The underlying circle map has a near tangency with the indentity

diagonal, suggesting intermittency in the dynamics [30].

55

The figures also give us information on the winding number, which has units of
radians per cycle. The winding number for 8 Hz is -l .3376 rads/cyc (or -0.2129 cyc/cyc)
and for 9.8 Hz it is -1.445 rads/cyc (or -0.23 cyc/cyc).

After viewing the torus angle figures we see that although there is a definite
pattern to the Poincare section, it appears that the dynamics are only loosely based on a
torus. Figure 4.5 show how the Poincaré section jumps from one stage to the next as well
as how it might get hung up between two stages. Figure 4.6 appears to fit the
classification of a torus little better, but a more of special type of torus known as a fat
torus.

The Poincare sections and torus angle maps show that the dynamics are of higher
dimension than 3. It is important to see exactly what dimensions we are dealing with. To
do this, a F ortran and a Matlab program were created to find the FNNs (False Nearest
Neighbors) [28] and the correlation dimension [29] respectively. Table 4.1 shows the
number of FNNs as well as the percent of F FNs to total number of Poincaré section data
in each corresponding reconstruction dimension. It is difficult to pinpoint the embedding
dimension of the deterministic dynamics, as experimental noise contributes to the FNN
counts. If we were to (arbitrarily) prescribe a 1% criterion, it would lead to embedding
dimensions of 5 for the 8 Hz data, and 6 for the 9.8 Hz case. The correlation dimension
program allows user to view log C(r) vs. log (r) (where C(r) is the correlation sum as a
function of distance r), choose a scaling region, and find the slope in the scaling region.
This slope is an approximation of the fractal (correlation) dimension. The correlation
program also allows the user to view an approximation of the derivative of the log C(r)

vs. log (r). This works as a nice check to see if the fractal dimension is right. Refer to

56

Figure A.3 in the Appendix to see the correlation data for 8 and 9.8 Hz. It should be said
that the correlation dimension slope, dc, is r dependent, and suggesting that the fractal
scaling varies with the size scale. The correlation dimension might be represented by the
maximum dc, or alternatively by the slope of a in the linear region of the log C (r) vs.
log(r) plot. The FNN and dc should agree, i.e. the embedding dimension of the FNN, d5,

should have the relationship of d C < d E S 2dC + 1 . At any rate the correlation dimension

is 3.3275 at 8 Hz and 3.7394 at 9.8 Hz.

 

 

8 Hz 9.8 Hz
Dimension F FN Percent FFN Percent
First 7086 98.43% 12144 99.44%
Second 3499 48.60% 8022 65.69%
Third 802 11.14% 2680 21.95%
Fourth 151 2.10% 635 5.20%
Fifth 44 0.61% 197 1.61%
Sixth 16 0.22% 62 0.51%
Seventh 4 0.06% 1 1 0.09%
Eighth 0 0.00% 0 0.00%
Ninth 0 0.00% 0 0.00%

 

 

 

 

 

 

 

Table 4.1: FNN data of 8 and 9.8 Hz.

Figure 4.7 and Figure 4.8 show the F FTs as well as the FFTs of the Poincare
sections of 8 Hz and 9.8 respectively. These figures both show the driving frequency as
the dominant spike, but the first spikes on both of the FFT graphs occur at 1/4 the driving
frequency. This type of subharmonic resonance suggests that the beam’s behavior is
affected by period doubling [31] at both 8 and 9.8 Hz, although clear period doublings
are not apparent in the frequency sweeps of Chapter 3. This may be due to the speed of

the sweeps, as discussed in Chapter 3.

57

d8

d8

d8

d8

FFT of 8 Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

100 T I I I I I I T I
0 ~ -
-100 «
-2ooI ' ‘ ‘ ' " "
_300 L n I I 1 1 I 1 1
0 5 10 15 20 25 30 35 40 45 50
Poincare FFT of 8 Hz
0 z I fl I I I I
-50
-100
‘. ‘I ' “ ,.. , .V I" : . .. ‘ I
-150 -“ . "21 ' - ' . V . 1 . . ‘r .,' q
I I ' , 1. I ] 1, , I 1. 1
I I ' ' - I 1‘
-200 I
_250 I 1 L I 7 1 1 I
0 0.5 1 1.5 2 2.5 3 3.5 4
Hz
Figure 4.7: FFT and the Poincaré FFT of 8 Hz.
FFT of 9.8 Hz
0 T I I I I I F I
-50
-100 .
-150
-200
-250 I I 1 1 I 1 I L I
0 5 10 15 20 25 30 35 40 45 50
Poincare FFT of 9.8 Hz
50 I I I I l l I if I
0 F _I
-50 F -
-100 . . .1 , . ”‘ 1 . . 1. _ ,u
-150" ,I' I ' 1b} 3 4| I ‘ ‘,' I“ . '1? I! -
~200 ~ 4
-250 1 1 1 1 I 1 1 I I
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Hz

Figure 4.8: FFT and FFT of the Poincaré section of 9.8 Hz.

58

After viewing the F FT figures and understanding the winding number given from
the torus angle dynamic figures, we can see a direct correlation between the subharmonic
spike and the winding number. It is also noticeable that the driving frequency divided by
the subharmonic spike frequency yields values that are comparable to the subharmonic
orders found from the frequency sweep data of Chapter 3.

For the case of 8 Hz, the winding number in cycles per cycles multiplied by the
driving frequency produces a subharmonic frequency value of 1.703 Hz (ignoring the
minus sign). This in not close to the subharmonic spike frequency of 2 Hz but this may
be because of the mini torus winds faster than the larger torus, and does not contribute to
the changes in 8, causing this error. Indeed, with the coupling between the “mini torus”
and the large torus, using one torus angle may not be representative of the dynamics.

In the case of 9.8 Hz the subharmonic frequency value produced is 2.2534 Hz.
This is value is closer to the subharmonic spike frequency. Dividing the driving
frequency by the subharmonic frequency will yield the subharmonic order; these values
of 4.6976 for 8 Hz and 4.349 for 9.8 Hz correspond closely with the subharmonic orders

of region I of Chapter 3.

4.1.2 Region II (21.9, 22.1 & 22.3 Hz)

The forcing frequencies selected from region 11 were 21.9, 22.1, and 22.3 Hz.
The cantilever beam was then forced at the given frequency for an extended amount of
time to allow it to settle into a steady state, after which the collection of data was started.
Data was collected for 15 minutes at 21.9 Hz and for 20 minutes at 22.1 and 22.3 Hz.

This time trace data was then imported into Matlab and the Poincaré section was taken.

59

By plotting the Poincaré section in a delay map we were able to observe what path the
Poincaré data follows in higher dimension. Figures 4.9 and Figure 4.10 shows the
Poincaré section and the Poincaré section of the Poincaré section of 21 .9 Hz respectively.
In Figure 4.9, the map has an interchange between dynamics around the circular cloud,

and the dynamics in the diagonal dark band.

 

Poincare of 21.9 Hz with 1: Delay of 1

 

 

 

 

I T
-1.5 — —
.2 - I
‘+-
E
X
.25 — _.
.3 - _
.3; l l I 1
3'35 -3 -2.5 -2 -1.5 -1

X(N)

Figure 4.9: Poincaré section delay map of 21 .9 Hz.

60

21 .9112 Poincare of Poincare

'1 I I I I I

 

-1.51- 1

-2 _. . o 0 .. . .0 . . . . . o .. —<
. O . . C
.. . . . ‘ . ' o .. ,' o '05:. : :: .. ~. .8. . :. " a. .0 .
.... . e O ..: .:4. 0:02.: ‘0? 3". .... ... .. .0 . .0. .. ..g ... 00:,
«'2. *7. 13162. ' 3‘: -' .: . .: z :1: 2 ~. I}, z :1.» kn. 5? °.- :5; (m
\° ‘ 0 fi . o ‘ 0 ' or ' e 0 Co a ' “'0
': 0‘ :O «I "r '3 3",0‘ 0 a“. ‘5'};5305 ’3 ’ 15 ‘Oor".g;r'~) } ‘1' G. .o ~
2 2 5 -' .9236; I.) ‘a- ,i '3’3’34‘3 "if! 3%"! 43"" 3,"! '97'3§&.3§3a .9 d
. ...‘ . . O. O . . 0.0 . o O . ...:

i I s. . . 4’ ' {.§ ”“0”“.ga ~05 .fi” *0 A‘. A%%3 “Q:~).’°. {$.Lzap? 0‘.
I‘:¢~;‘."....:‘ “\ '9 .. -,’,; 43:53 «53:: s"; {633.5 ..-’ 534-? my :‘fi' ’5 .. - 132?:
h”:.’.”.:' .é‘qrz .34 00 g ’3'... . .0 . '9 ’. . . z I . .0 . 0‘ o. .

‘ ’0 O..\O!. fl. : {... \. 0 ... o .' o . lb. ’ o .0 o o . ‘0 . . ~ .
.0 0 ° .: o . .° .00 ' ° . .0 g .. ' co '3 0.... ' . .
O t o. \ .3 o. . a... . . '0... ' : .. o . . . . 5 ’. . o .. O . . . ..0 O .
.3 . fi;’:; 0 ‘ . .... ‘ 9. ... :’ ..O‘ I . . 0.. o ....f .. 0 0 0.. . .“ ...~ "1
' . o. co 0 .o .\e .' ‘ .. o . . e ’ ..." .0 f I...’ ' z. .' . ....
Lo. . Q .0 . .... ' .. ’0 o $‘0~0‘ .:.Iz“ . ‘9 .0 o: ... ... ...o'c‘o....~o ...“, . “’*oo~o.:. ’0: .0... ... ‘5 ’0’ s. ...:0..:‘”
". .fi :0... {{‘z‘ ‘ a... .9... .0 '0..e’:‘?::. ‘. ' . "".‘ ~ ' i. ‘ ’ O 0 o.

.500 .
e ‘.

 

 

 

l l I
0 500 1000 1 500 2000 2500 3000
Index

Figure 4.10: Poincare section of the Poincaré section of 21 .9 Hz plotted against its index.

Sample of the Poincare Section Data at 21.9 Hz

‘1 I I I I I I I I I

 

4.5- a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_3.5 1 I 1 1 1 m 1 I I
0 50 1 00 150 200 250 300 350 400 450 500

Index

Figure 4.11: Sample of the Poincaré section data at 21.9 Hz plotted against its index.

61

The delay map of the Poincare section of the Poincare section of 21.9 Hz did not
give any insight into how Figure 4.9 was behaving; refer to the Appendix Figure A.4 to
see the delay map of the cross-section of the Poincaré section of 21 .9. Figure 4.10 on the
other hand tells us that the Poincaré section of Figure 4.9 represents an attractor, rather
than a case were the dynamics of the cantilever beam suddenly change to give us two
different responses.

The next frequency to be presented is a case were during the data collection there
is a change in the responses and thus two different responses are present. The Matlab
program that preformed the cross-section of the Poincare section of 21.9 Hz flagged
Poincaré section data that crossed the diagonal from positive to negative, the positive
being the bottom right half and thus the negative being the top left half. Figure 4.11
allows one to see how long the dynamic behavior tends to spend in the slow part (“flat

torus”) versus how long it spends in the oscillatory part (“round torus”).

62

22.1hz Poincare of Poincare

'1 I I I I I

 

 

   
 

 

 

-1.2 ~ -I
1.4 - _
—1.6 e _.
—1.8 - a
. a o o .. o e . . o .
A we: $4“ '=' " ~ f‘“ "s" ‘3 ' ‘1' ”'.'°- . ' "1'; "
z -2 K? ' ‘. ‘ R if 7': ;:3. A 0 ,v‘ ‘ '.‘R I f 'k' ' /:.;'..- "g :: V. f _1
sz 0' "$3 3"" . a "6*; ‘ . I . , 5“,»): ti“, (fay 9 u I
1 :«“.’. . ' . 1&- . ... 'k a... o . {4"4‘1?’ Wh'
.‘Ifl. . 2’}’\;!$ 0' . ..r‘ ‘I‘ ‘H. ‘33,: .$’(y{; 8 ‘0 fi' 0. o ‘ y {i g. ?
'aa‘Sti r W4 km r * a. a? .-.- . .. a. 2' : ~ I
2.2 '1’“ ‘ :93?” ”f ‘3. .I. a.
...,ae‘wi, ...-e a? .. . .
1’ o e ..‘ o e ...." :0 :' ... . .. e o
-2.4 '"g’“‘;t-’{!’-}$":r\v‘° . . - ‘2- . . -
-2.6 a
2.8 _
-3 1 I 1 I I
0 1 000 2000 3000 4000 5000 6000

Index

Figure 4.12: Poincaré section of the Poincaré section of 22.1 Hz plotted against its index.

Figure 4.12 shows the Poincaré section of the Poincaré section at a frequency of
22.1 Hz plotted against its index. This figure clearly shows that there are two different
segments of the response. It is likely that the slight changes in parameter, such as
friction, due to, e. g., temperature and wear, led to a change in the nonlinear system
response. As such we have taken the liberty of breaking the response data in to two
different sub regions. Below, Figure 4.13 and Figure 4.14 show the Poincaré section
delay maps of each region. The figures have the same scale so the reader will be able to
see how the data segments would have overlapped and important information would have
been missed. Figure 4.14 appears to be a better-defined torus which allows us to use the

torus angle program in Matlab to see if in fact it is a torus.

63

Poincare 01221 Hz with a Delay of 1 Sub 1

i I I I

 

1,

X(N+1)
1L:
N
I

 

 

 

 

4|; 4 A; l J
-3 ~28 -26 ~24 -22 -2 -1.3 -1.6 -1.4
X(N)

Figure 4.13: Delay map of the Poincaré section of 22.1 Hz of data segment 1.

 

Poincare 0122.1 Hz with a Delay of 1 Stb 2
I I I

 

 

 

_3 I 1 1 l 1 a 1
-3 -28 -26 -24 ~22 -2 -1.8 ~1.6 -1.4
X(N)

 

Figure 4.14: Delay map ofthe Poincaré section of 22.1 Hz of data segment 2.

64

Figure 4.15 shows the Poincaré section of 22.3 Hz. The dynamics are similar to
those of the 21.9 Hz case, in that the map of the data alternates between going around the
toroidal loop, and iterating on the diagonal dark band. After viewing this figure we
decided that we would be unable to take the Poincare section of the Poincare section of
the 22.3 Hz data. The problem was that the majority of the data appeared to be along the
diagonal axis we wanted to take the cross section along. But as one can see there is still a

clear hole in the torus, which allows us to define a center point to use the torus angle

 

 

 

 

 

program in Matlab.
Poincare of 22.3 Hz with a Delay of 1
“0.5 I I I I
-1 .— -
-1.5 — ~
'3‘
E
x
-2 [ I
-2.5 e ~
-3 l l l
-3 2 5 2 -1.5 1 0 5

X(N)

Figure 4.15: Poincare section of 22.3 Hz with a delay of 1.

Figure 4.16 and Figure 4.17 show the delay map of the torus angle return maps of
the Poincare section of 22.1 Hz data segment 2 and 22.3 Hz respectively. These are

more clearly defined, compared to the torus-angle figures in section 2.1.1. The graphs

65

inform us that the Poincare sections are in fact toruses, and the dynamics are governed by
circle maps. The winding numbers are -1.3367 rads/cyc (-O.2127 cyc/cyc) and -O.24981
rads/cyc (-0.0398 cyc/cyc). The first circle map is monotonic, suggesting quasi-periodic
dynamics with some higher dimensional fluctuation. The second circle map has a nearly
zero-slope portion, and may therefore be on the brink of chaos by torus wrinkling. The
dynamics are similar to those of he 21.9 Hz case, in that the map of the data alternates
between going around the toroidal loop and interacting on the diagonal dark band.
Considering a diagonal line along the identity (separating the top left from the bottom
right) in the plot there are near tangencies at —2 and —2 suggesting intermittency [30].
The existence of some higher dimensional dynamics (evident in the fatness of the plot)
might combine with intermittency to generate chaos. Figure 4.18 shows a sample of the
Poincaré section data versus its index, which suggests how long the dynamic behaviors

tend to spend in each part of the torus.

66

Theta(N+1)

Iheta(N+1)

Torus Angle Return Map of 22.1hz of Sub Region 2

 

 

   
   

 

4 . , , , I 1
3 c
2 -
1 _
0 ~ “funding it : -1.3367
-1 _
-2 _
-3 F .
-4 A I 1 1 I I
-4 -3 -2 -1 0 1 2
Theta(N)

Figure 4.16: Torus angle dynamics of 22.1 Hz of sub region 2.

.1

-2

-3

.-

 

Torus Angle Return Map of 22.3hz

 

 

I I I I I

VWnding # : 024981

 

 

Theta(N)

Figure 4.17: Torus angle dynamics of 22.3 Hz.

67

 

Sample of the Poincare Section Data at 22.3 Hz

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

'0.5 I I I I fl T I I I
-1 _ a
-1.5 ~ I‘\ J
12‘, .
><
-2 .. —i
-2.5 ~ ~
_3 I 1 I I 1 1 1 I I
0 50 1 00 1 50 200 250 300 350 400 450 500
Index

Figure 4.18: Sample of the Poincare section data at 22.3 Hz plotted against its index.

For of 21 .9, 22.1, and 22.3 Hz, the FNN and correlation dimension programs were
run to gather information about the dimension of the dynamics at these frequencies. Table
4.2 contains the FNN information. Refer to the Appendix, Figure A5, to examine the
correlation dimension information which yields a correlation dimension of 2.4439 at 29.1

Hz, 3.3973 at 22.1 Hz, and 3.1805 at 22.3 Hz.

68

 

 

 

 

 

 

 

 

 

21.9 Hz 22.1 Hz 22.3 Hz
Dimension FFN Percent FFN Percent FNN Percent
First 19640 99.43% 26458 99.67% 2661 1 99.21%
Second 1 1526 58.35% 18125 68.28% 15293 57.02%
Third 2244 11.36% 4919 18.53% 3399 12.67%
Fourth 41 1 2.08% 750 2.83% 689 2.57%
Fifth 83 0.42% 99 0.37% 191 0.71%
Sixth 27 0.14% 18 0.07% 58 0.22%
Seventh 8 0.04% 3 0.01% 15 0.06%
Eighth 0 0.00% 0 0.00% 0 0.00%
Ninth 0 0.00% 0 0.00% 0 0.00%

 

 

Table 4.2: FNN data of21.9, 22.1, and 22.3 Hz.

Figure 4.19, Figure 4.20, and Figure 4.21 show the F FT and the Poincare-section
FFT of 21 .9, 22.1, and 22.3 Hz respectively. Figure 4.19 highlights some very interesting
frequencies. There are two pairs of frequency peaks on either side of the driving
frequency of 21.9 Hz. These paired frequencies have distances of 0.4 and 4.27 Hz from
the driving frequency. These distances match up with the first two peaks in the FF T,
which have frequencies of 0.4 and 4.27 Hz respectively. Taking these frequencies and
dividing them by the driving frequency produces (probably) irrational numbers. If
0.4/21.9 and 4.27/21.9 were truly representing rational ratios, then we would know we
have some sort of period doubling going on, which is similar to what was taking place in
the pervious section with 8 and 9.8 Hz. But here the fraction is (seemingly) irrational,
which means these frequencies arise on their own, and are independent of the driving
frequency. This means there are three frequencies that are affecting the dynamic
behavior of the cantilever beam. The slightly elevated spectrum near the frequency peaks
suggests a narrow-band chaos. As such were classified this as chaos with 3-frequency

quasi-periodicity [30].

69

The order of the subharmonic is 21.9/0.4 as well as 21.9/4.27 giving subharmonic

orders of 54.75 and 5.129 respectively, which are similar to the subharmonic orders of

 

region II in chapter 3.
FFT of 21.912
0 I r I I I I I I I
-50 a 0'4 m 21.3 a: 22.3 a:

-100 '

dB

-150

 

-200

 

 

 

_250 I I I I I I 1 1 I
0

Poincare FFT of 21 .9hz
1 00 I I I l I I I I I I

 

dB

 

-100 g

-200

 

 

 

.—
>-
I—
I'"
I—
)—
..
b

-300 1 L
0

 

Figure 4.19: FFT and the Poincaré FFT of21.9 Hz.

In Figure 4.20 there are several frequency peaks but only three of these peaks
have a convincing affect on the driving frequency. It seems likely that the behavior that
was noticed with 21.9 Hz is also appearing here. Figure 4.20 has a pair of frequencies
that accompany the driving frequency on either side. These frequencies have distances
from the driving frequency of 5.05 Hz in the first sub region and 4.83 in the second sub
region. Once again the first peak on each FFT has a frequency that matches this
difference. And similarly this frequency divided by the driving frequency yields an

irrational number. The small peak to the right of 22.1 Hz occurs at a distance of 22.1

70

minus 2(10.5). The F FT is slightly raised. Adding the 10.5 Hz peak, we conclude that
the dynamics is either very narrow-band chaotic or quasi-periodic with 3-frequencies, and
that this structure occurs in both the sub intervals 1 and 2. Therefore we concluded that
the behavior of the cantilever beam at 22.1 Hz can be classified as qusai-periodic or
narrow band chaos with 3-frequency quasi-periodicity.

Highlighted in Figure 4.21 for the 27.3 Hz case are the frequencies that are paired
to the driving frequency and their governing response frequency of 0.68 Hz. This
subharmonic order is approximately 33. There is a broadband of the spectrum elevated
above the noise level, suggesting chaos or stochasticity. Again the response frequency
divided by the driving frequency yields (probably) an irrational number and we can
conclude that the behavior of the cantilever beam at 22.3 Hz can be classified as chaos
with 2-frequency quasi-periodicity and intermittency.

In summary, in Region II we see chaotic and quasi-periodic dynamics associated
with either 3-frequency quasi-periodicity or 2-frequency quasi-periodicity with torus

wrinkling and intermittency.

71

d3

d8

d8

dB

-100

-150 ;.

-200

-250
0

-100

-150

-200

-250

~100
-150
-200
-250

-300

FFT 0122.1 Hz Sub 1

 

 

5.05 Hz

I I I I

17.05 Hz
/
27.15 Hz
/

l l l l

 

 

15

20 25 30 35

FFT 01221 Hz Sub 2

45 50

 

I

 

I I l

17.27 Hz
/ 25.93 Hz

 

 

Figure 4.20: FFT of 22.1 Hz segments 1 & 2.

10

15

20 25 30 35

FFT of 223112

40

45 50

 

0.68 Hz

 

I I I l

21.62 Hz

1 l 1 I

20 25 30 35

Poincare FFT 0122.311:

45 50

 

 

I I I I

Hz

 

Figure 4.21: FFT and the Poincaré FFT of 22.3 Hz.

72

4.1.3 Region III & Hybrid (25.7 Hz)

Figure 4.22 displays the six graphs that show Poincare section of the Poincaré
section, in both delay and index from, at a frequency of 25.7 Hz and its sub divided

regions which were picked form the obvious change in the response signal.

Delay of The Poincare of the Poincare

'2 ’ o‘tw'". .:

Poincare of the Poincare

 

 

X(N+1)
1‘5
0'!
\
X(N)

:. .
.h’ 0’

 

 

 

 

 

 

-2 4.5 o 500 1000 1500 2600

0- f f Sub1

 

 

f

X(N+1)
lb
or
‘
I
X(N)
:1:

' .°: 9 0° “63:396va '4

 

 

 

 

 

-35 . . I _4 . . 1 1 .
~35 -3 -2.5 -2 -1.5 0 200 400 600 800 1000 1200

Sub 2 Sub 2

 

 

-1.5

 

-2»

-2.5r m.

-3I

X(N+1)
X(N)

 

 

 

 

 

 

-3.5 . . . . . . .
-3.5 -3 -2.5 -2 -15 o 200 400 600 800

X(N) Index

Figure 4.22: Poincare section of the Poincaré section for 25.7 Hz and its two sub regions.

These graphs give valuable insight in to the behavior the cantilever beam. We
can clearly see that there are two distinct responses going on in the first sub region. If we
were to plot the Poincare section in a delay map of just this region it would appear to be
just a thick torus, that is presented in Figure 4.23, but after analyzing data from 45.6 Hz

we can conclude that what is going on at 25.7 Hz is a fat torus shrinking to a thinner torus

73

llE

as the dynamics changes. Section 4.2 will focus on this phenomena, is also showing the

delay map of the 25.7 Hz Poincaré section.

Poincare Section of 25.7 Hz with a Delay of 1
'1.‘ l 1’ l I I l I

 

-2.2 .

.24 _

X(N+1)

 

 

 

 

Figure 4.23: Delay map ofthe Poincaré section of 25.7 Hz.

The 0 dynamics was taking with respect to each sub segment, which both have a
common center, the x-bar is -2.3 and the y-bar is -2.3 on the Poincare section delay map.
The winding number was calculated to be -0.52297 rads/cyc (—0.0832 cyc/cyc) for the
first segment and —0.55395 rads/cyc (-0.0882 cyc/cyc) for the second segment. Figure
4.24 and Figure 4.25 present the delayed 9 dynamics plot and we can conform that the
plot is in fact a torus governed by circle map dynamics. There is no strong indication of
torus winkling (negative slope regions on the circle map), and hence the dynamics look
quasi-periodic. However, the segment 1 plot has a “kink” in it, so maybe there is a very

narrow banded chaos involved, maybe related to the fatness of torus. Once again if you

74

 

 

consider the diagonal line in the plots there are near tangencies at -2, -2 and 1,1. The
near tangency of the torus angle return map to the diagonal indicated that intermittency

plays a role in the quasi-periodic dynamics.

Torus Angle Retum Map of 257112 Sub 1

4 I I I I T I I

 

      

Winding It : ~052297

Theta(N+1)
O

 

 

y—
I—

l I
-4 -3 -2 -1 0 1 2 3 4
Theta(N)

is
_
_
_

 

Figure 4.24: Torus angle dynamics at 25.7 Hz of sub 1.

75

 

Torus Angle Relum Map 01 25.7hz Sub 2

4 I I I I T I I

 

2 F \Mnding it : -0.55395 ~

Theta(N+1)
O

 

 

 

I L l
-4 -3 -2 -1 0
Theta(N)

...

I
A
...
I—
.—

 

.5
N
U
&

Figure 4.25: Torus angle dynamics at 25.7 Hz of sub 2.

To get an idea of the dimensionality at 25.7 Hz, the FNN and correlation
dimension programs where run. Table 4.3 contains the FNN information. Refer to the
Appendix, Figure A6, to examine the correlation dimension information, which yields a

correlation dimension of 2.4579.

 

 

 

 

 

1 25.7 Hz
Dimension FFN Percent
First 23056 99.59%
Second 1 1572 49.98%
Third 1673 7.23%
Fourth 89 0.38%
Fifth 6 0.03%
Sixth 0 0.00%
Seventh 0 0.00%
Eighth 0 0.00%
Ninth 0 0.00%

 

 

 

Table 2.3: FNN data of 25.7 Hz.

76

 

Figure 4.26 highlights the F FT frequencies that are paired to the driving
frequency and the subharmonic response frequencies of 2.08 Hz in the first data segment
and 2.37 Hz in the second data segment. In the second region the subharmonic frequency
minus the driving frequency does not match up with the spike highlighted in the figure
but is only off by 0.01 so this maybe because of frequency resolution effects and thus is
treated as if it was at 23.33 Hz. Again the frequency divided by the driving frequency
yields an irrational number and we can conclude that the behavior of the cantilever beam
at 25.7 Hz can be classified as dominantly 2-frequency quasi-periodicity. Using the
winding numbers given in the torus angle figures for both segments to estimate the
subharmonic frequencies of each segment yields values of 2.139 Hz for the first segment
and 2.266 Hz for the second segment. From the FFT it appears that these frequencies are
the torus frequencies. These subharmonic frequencies yield a similar subharmonic order
for region 111 of Chapter 3. Looking at the peak that is to the right of the driving
frequency in Figure 4.27 it is interesting to note that the frequency is close to half the
fundamental normal frequency of 47 Hz. Therefore the normal degree-of-freedom

resonance that was noticed could have come about through this relationship.

77

 

 

 

 

 

 

 

 

 

FFT 0125.7 Hz Sub 1
100 I I IT I I I I I
0 2.03 Hz a
23.62\Hz I?“ m
9,! -100 —
I .
II
-200
-300 I 1 L I 1 41 I L I
0 5 1 0 15 20 25 30 35 40 45 50 ’
FFT of 25.7 Hz Sub 2

100 6 I I I T I T I I

0 ~ 2.37 Hz ' a
/ 133“ “I 28.07 Hz
\
g -100 ~ / — .!
-200
-300 I 1 1 g L 1 1 I I
0 5 10 15 20 25 30 35 40 45 50
hz

Figure 4.26: FFT of 22.1 Hz Sub l & Sub 2.

4.1.4 Region VI (43.8, 44.4, 45.2 & 45.6)

43.8, 44.4, 45.2 and 45.6 Hz were the forcing frequencies selected from region
VI. The cantilever beam was then forced at the given frequency for an extended amount
of time to allow it to settle into a steady state, after which the collection of data was
started. Data was collected for 15 minutes at 45.2 Hz, for 30 minutes at 45.6 Hz, and for
40 minutes at 43.8 and 44.4 Hz. This time trace data was then imported into Matlab and
the Poincare section was taken. Unfortunately the data that was taken at 43.8 Hz was
corrupted and was rendered useless. But the other frequencies of region VI revealed
some of the most interesting dynamics yet.

Figure 4.27 shows the delay map of the Poincare section at 44.4 Hz, which

displays a very nicely defined torus. The Poincare section of the Poincare section was

78

 

not performed on this data because no new information could be gathered. Figure 4.28
shows the delay map of the Poincaré section at 45.2 Hz, which displays an unusually
shaped torus. A delay of 5 was used to see this torus in its “optimal” form. Again the
Poincaré section of the Poincare section was not used because no new information could
be obtained. Figure 4.29 shows the delay map of the Poincaré section at 45.6 Hz, which
displays what appears to be some sort of doubled torus. A delay of 3 was use to see this
torus in its optimal form. For this case the Poincare section of the Poincare section could
come in handy to see the torus size variation in this response data. Refer to Section 4.2

for more on this phenomenon.

Poincare of 44.4 Hz with 3 Delay of 1
1.5 l I I I

 

0.5 F

X(N+1)

-0.5 ~

 

 

 

 

X(N)

Figure 4.27: Poincaré section of 44.4 Hz with a delay of 1.

79

5.5

4.5

X(N+5)

X(N+3)

-1

-2

-3

Poincare of 45.2 Hz with a Delay of 5

 

 

I I T I I r I I I

 

 

 

X(N)

Figure 4.28: Poincaré section of 45.2 Hz with a delay of S.

Poincare of 45.6 Hz with a Delay of 3

5.5

 

 

I I I I I I I I

 

 

 

X(N)

Figure 4.29: Poincaré section of 45.6 Hz with a delay of 3.

80

 

The torus angle return plots of 44.4, 45.2, and 45.6 Hz are Figure 4.30, Figure
4.31, and Figure 4.32 respectively, and give some insight into the structure of the tori.
The winding numbers were -O.76796 rads/cyc (-O.1222 cyc/cyc) at 44.4 Hz, -0.10303
rads/cyc (-0.0164 cyc/cyc) at 45.2 Hz, and —0.090993 rads/cyc (-0.0145 cyc/cyc) at 45.6
Hz. No wrinkling is evident in the circle map at 44.4 Hz and hence the dynamics are
dominantly quasi-periodic. The circle maps for 45.2 Hz and 45.6 Hz lie close to the
diagonal, corresponding to the small winding numbers. If there is any wrinkling, as

might be suspected from Figure 4.28, it is only on a very small scale in 6. Hence, the

I'

dynamics are dominantly quasi-periodic. Intermittency may play a role in the dynamics
that take place at 44.4 Hz, but it is unlikely that intermittency is involved with the
dynamics that take place at 45.2 and 45.6 Hz because in this case the torus return maps
place the dynamic response along the diagonal for the entire time.

The doubled torus at 45.6 Hz is clear in the double circle map on 9. A single-
valued circle map could be obtained by redefining the angle to cover two loops (47:
radians) around the torus. A question may arise as to whether the data from 45.2 and
45.6 Hz surround a torus doubling bifurcation. This is addressed later. If torus doubling
has occurred, it is accompanied by the doubling of the circle map in a higher dimensional

manifold.

 

81

Theta(N+1)

Theta(N+1)

Torus Angle Relum Map of “AM

 

-3.-

 

-4

I I I I

      

Winding It : -0.76796

1 L l l l l

 

 

-4

Theta(N)

Figure 4.30: Torus angle dynamics of 44.4 Hz.

Torus Angle Relum Map of 45.2hz
I I I

 

 

Winding # : -0.10303

 

l 1 .1 l

    

 

 

-4
-4

-3 -2 -1 0 1 2
Theta(N)

Figure 4.31: Torus angle dynamics of45.2 Hz.

82

 

 

 

 

Theta(N+8)
O

 

 

Winding It :

Torus Angle Return Map of 45.6hz

-0.090993

I

l

    

,—

.—

 

 

0
Theta(N)

Figure 4.32: Torus angle dynamics of 45.6 Hz.

To get an idea of the dimensions of dynamics at 44.4, 45.2, and 45.6 Hz, the FNN
and correlation dimension programs where run. Table 4.3 contains the FNN information.

Refer to the Appendix, Figure A.6, to examine the correlation dimension information.

The correlation dimensions are 35714 at 44.4 Hz, 2.8328 at 45.2 Hz, and 2.9595 at 45.6

 

 

 

 

 

 

 

 

 

Hz.
44.4 Hz 45.2 Hz 45.6 Hz
Dimension FFN Percent FFN Percent FNN Percent
First 107159 99.54% 40139 98.58% 109221 99.57%
Second 68768 63.88% 16755 41 .15% 45828 41 .78%
Third 14677 13.63% 2725 6.69% 6974 6.36%
Fourth 1848 1 .72% 402 0.99% 977 0.89%
Fifth 180 0.17% 85 0.21% 213 0.19%
Sixth 9 0.01% 23 0.06% 48 0.04%
Seventh 0 0.00% 6 0.01% 22 0.02%
Eighth 0 0.00% 0 0.00% 0 0.00%
Ninth 0 0.00% 0 0.00% 0 0.00%

 

Table 4.4: FNN data of 44.4, 45.2, and 45.6 Hz.

83

 

The FNNs yield information about how many delay cords are needed to describe
the data, not the bifurcation. A torus doubling bifurcation requires at least a four-

dimensional phase space.

 

Figure 4.33 to Figure 4.35 give the FFT and the F F T of the Poincaré section at
44.4, 45.2, and 45.6 respectively. The FFT of 44.4 Hz displays a similar reponse to the
frequencies in region 11. In this case the subharmonic frequency is 5.32 Hz, which

complements the driving frequency at 39.08 and 49.72 Hz. This informs us that once

 

again we are dealing with 2-frequency quasi-periodicity. We were unable to find a
correlation between the other spikes, which can be seen in the FFT, and the driving
frequency. But these are of relatively low order, and are likely to be associated with the

torus being slightly fat. By multiplying the winding number in cycles per cycles by the

 

driving frequency of 44.4 Hz the result is 5.423 H2 is close to the subharmonic frequency
spike at 5.32 Hz and thus the yields that the torus frequency. (If examined closely one
will notice a peak right up against the Y-axis, at a frequency of 0.4 Hz. Notice that there
are similar peaks on either side of the driving frequency peak. If these were judged as
significant, we could classify the response behavior to be 3-frequency quasi-periodic).

The subharmonic order of 8.3 matches well with Chapter 3.

84

FFT 0144.412

0 I I I I I I I I 1

 

-50 49.72 Hz
-100 \ p

4501‘ I i 'i‘l:

dB

-200' I ' -“.' "‘W.l
-250

-300 # 4 L 1 1 1 1 n

J_
0 1 0 20 30 40 50 60 70 80 90 1 00

Poincare FFT 0144.411:
0 r I ' I

 

-100

dB

~200

 

-300

 

 

400 1 1 L g
0 5

Figure 4.33: FFT and the Poincaré FFT of 44.4 Hz.

The F F T of 45.2 Hz held some very interesting subharmonics. The many spikes

 

that can be see Figure 4.34 possess a pattern that that follows: 0.44, 0.9, 1.34, 1.8, 2.24,
2.68, 3.14, 3.58, 4.04, 4.48, 4.92,. . ., etc. It may be easier to see the pattern like this: 0, 1,
0, 1, O, O, 1, O, 1, O, O, 1, ..., etc. where 0 represents an addition of 0.44 Hz to the
pervious frequency and 1 represents adding 0.46 Hz to the previous frequency. This
pattern continues though out the entire FFT; 0, 1, O, 1, 0 back to O, 1, O, 1, O and again,
and again, and again. The combination of these frequencies eventually adds up to the
driving frequency of 45.2 Hz. This interesting pattern is probably due to the frequency
resolution, which is mentioned in Chapter 3. So, this behavior appears to be a case where
there are two predominant frequencies with the subharmonic occurring on the order of
about 100. This order matches with region V1 in Chapter 3. The dynamics corresponds

to 2-frequency quasi-periodicity.

85

 

FFT of 45.2hz

 

 

.100 1

dB

-150 7 g- i ll ; i , ..., . . 1 1' I I
-200 ' ,:i ' “Ill"
-2so~ ‘ ‘

 

 

 

_3000 1 1 1 1 1 1 1 1 1 I

Poincare F FT of 45.2hz
1 00 I I I r

 

dB

-100

 

-2oo "

 

-300
0

 

 

Figure 4.34: FFT and the Poincaré FFT of 45.2 Hz.

The FFT of 45.6 Hz, displayed in Figure 4.35, has a similar behavior to the 45.2
Hz case, in that the subharmonic is extremely dominant. In this case, the subharmonic
frequency of 0.6 Hz is associated with the main torus, and the winding number. The 0.3
Hz is then a subharmonic resonance of the 0.6 Hz component, associated with torus
doubling. This subharmonic is at an order of 152 with respect to the excitation! The
dynamics are classified as 2-frequency quasi-periodic with torus doubling. The winding
number of —0.0145 cyc/cyc matches well with the subharmonic of order 76, which

corresponds to the main (undoubled) torus.

 

86

FFT of 45.6hz

 

-50 .

-100

dB

-150 -

-200

 

-250

 

_300 4 1 1 1 1 1 1 1 1

1 0 20 30 40 50 60 70 80 90 100

 

-100

dB

-200

 

-300

 

 

 

-400 1 1 1 1
0

Figure 4.35: FFT and the Poincare' FFT of 45.6 Hz.

4.1.5 Analysis of 37.9 Hz

The behavior of the cantilever beam when it was forced at 37.9 Hz was unlike
anything that has been presented thus far. In fact the beam did not really “do” anything;
the cantilever was stationary and it appeared that nothing was going on. What caught our
attention was that during the frequency sweeps it was noticed that the response strain
gage signal became what appeared to be a triangle wave accompanied by an audible ring
coming from the beam. Figure 4.36 presents the FFT and a sample of the time trace
response signal. The first peak is the driving frequency; this peak is followed by a peak at
double the driving frequnecy and then a peak at triple the drive frequency and so forth
and so on; even the 10th and 11th harmonics are significant. This would then be a

superharmonic resonance. The response of the cantilever beam in Matlab does not really

87

 

appear do be a triangle wave. This may be do to the way the GxView and TEAC
interpretation of the response data, or the response could be a bit different than during the

frequency sweeps.

 

 

 

 

 

 

 

 

 

 

FFT of 37.9 Hz
100 I I I I I I I I I
0 ~ ..
g -100 «
-200—' “ .. .- - 1. . ' 1' . .
_300 L 1 1 1 1 1 1 1 1
0 so 100 150 200 250 300 350 400 450 500
Sample of the Time Trace of 37.9 Hz
5.8 I I I I I I I I I
5.7 ~ 4
g 5.6 l 1 . .1 ‘ 4
X 1 1
l 1
5.5
_: i I
5.4 ~ —
I 1 l l l I l A 1
o 200 400 600 800 1000 1200 1400 1600 1800 2000
Index
Figure 4.36: The FFT and sample time trace of 37.9 Hz.
0
4.2 Torus Doubling

Figure 4.37 shows the Poincaré section of the Poincare section plotted against its index of
45.6 Hz, and present what appears to be four torus cycles. Indeed the top two infrequent
toroidal crossings are due to a warping of the torus in higher dimensions, and are not
indicative of data circumnavigating the torus. But viewing the torus doubling in Figure
4.26 we can see that there is definitely a double torus involved in the behavior of the

cantilever beam’s response.

88

45.6hz Poincare of Poincare

1 I I I I I T T I

 

0F '1
s ", ~- ~
1~ z
1,93. ..
‘0. O
2 -21:&”::' .5: g‘ .4
SE 2, 151.9. ~°, , 8c. 41"
tarsus :‘m «a.» at: ...... ’1 h
. s O 9 ” g 0" ‘0.
. :- :‘so’hflfit, My,“ , , {334 $3,ng
- ’ 89¢$~xu<¢nzé gruff, "
o '0. . 0‘ {.:.f 0‘.” I f o
.0 °
3 . 9 : —1

 

 

 

1 l l I 1 I 1 l
0 200 400 600 800 1000 1200 1400 1600 1800
Index

 

Figure 4.37: Poincaré section of the Poincaré section of 45.6 Hz plotted against its index.

This torus warping phenomenon has appeared in two of our cases, at 25.7 and
45.6 Hz. The case at 25.7 Hz was a little weird because the torus was not really visible,
but after viewing the Poincaré section of the Poincaré section in both the delay maps and
against its index (Figure 4.22), what at first glimpse looks like torus doubling turned out
to really be torus warping. In Figure 4.38 we present samples of short data records at
various frequencies in Poincare section delay maps. This data was taken in the given
order: starting at 46 Hz we swept down slowly to 45.84 Hz where the torus doubling
phenomena began. Notice that there are two cases at 45.84 Hz. In the first one case there
is no indication of a torus doubling, while a little will later data was taken at that same
frequency and the torus doubling had become active and as the frequency was increased
again, the doubled torus disappeared. Figure 4.38 is interesting to see because it gives a

visual idea of how torus doubling becomes active as well as how it becomes dormant.

89

 

 

 

   

     

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

         

 

 

 

 

 

 

 

 

 

 

4 . . . . 4 .
a) 45 Hz N=3 ; , b) 45.88 Hz N=3
a 2 > / 2 »
+
z
\X’ 0 P M L- 0 I I
-2 1 1 1 k -2 ‘
-1 0 1 2 3 4 -2 0 2 4
4 . . 4 . - . -
c) 45.84 Hz N=3 d) 45.84 Hz N=3
a 21 2'
i
>7 01 °’
-2 i - 1 '2 1 . 1 . 1 1
-2 0 2 4 -2 -1 0 1 2 3
4 . 4 . - . .
e) 45.85 Hz N=3 1) 45.857 Hz N=3
8 2 2'
...
z
>"<’ 0 0'
-2 t -2 4
-2 0 2 4 -2 -1 0 1 2 3
4 ~ . - - X(N)
9) 45.859 Hz N=3
8 2 J
+
2
S? 0'
_2 1 1 1 1
-2 -1 0 1 2 3
X(N)

Figure 4.38: Example of torus doubling with increasing excitation frequency.

 

9O

 

Chapter 5

Conclusion

This thesis has given a better understanding into the dynamic behavior of a
cantilever beam undergoing friction forced excitation at its tip. From the frequency
sweep stand point we were able to see regions in which active dynamics were taking
place. Dynamics in these regions were then investigated more closely by examining the
cantilever beam behavior at specified frequencies.

In region 1, the two frequencies chosen were 8 and 9.8 Hz. These frequencies
induced the beam to behave with the desired dynamics. Upon examination it was found
that the subharmonic frequencies of 8 and 9.8 Hz approximately matched the
subharmonic order of region I. It was also determined that the cantilever beam was
probably behaving due to the effects of period doubling.

In region 11, three frequencies were chosen to inspect the characteristics of the
cantilever beam. These frequencies were 21 .9, 22.1, and 22.3 Hz. Each one of these
frequencies presented its unique behavior. In the case of 21.9 Hz, the inspection of the
strain gage data yielded a system which behaved something like coupled tori, with one of
the tori fat and the other skinny. As it turned out, at a driving frequency of 21.9 Hz the
system had two independent subharmonic frequencies, which meant there were three
frequencies that affected the dynamic behavior of the cantilever beam. This behavior was
classified as chaos arising from 3-frequency quasi-periodicity. Also the subharmonic
order approximately matched the subharmonic order of region 11 in chapter 3. The lowest

frequency was 54.75 times smaller than the driving frequency.

91

 

In the case of 22.1 Hz, the examination indicated that there were two different
dynamic time segments. How this change in the dynamics took place can only be
speculated, but it most likely came about because of small changes in the parameters. By
splitting up the dynamic segment we were able to see the different types of dynamic
response that arose. The main difference between the two segments was that the second

segment produced a better-defined torus; the first segment appeared to be a cloud like I

torus with one edge relatively well defined and the other scattered. The subharmonic

 

frequency of the system excited at 22.1 Hz in both segments yielded the same basic
response, which was classified as quasi-periodic or narrow band chaos with 3-frequency
quasi-periodicity. Once again the subharmonic frequency gave way to a subharmonic
order the resembled that of region [I in chapter 3.

In the final case in region 11, the driving frequency of 22.3 Hz produced a
dynamic response that appeared to be on the brink of chaos by torus wrinkling. This was
interpreted from the fact that the torus angle return map, Figure 4.16, had a nearly zero-
slope region. The FFT in Figure 4.19 implies that the dynamic behavior had the same

basic type of response as the 22.1 Hz case and thus was classified as chaos with 2-

 

frequency quasi-periodicity. Plus, the subharmonic frequency yielded a subharmonic
order of about 33, which was equivalent to that of region 11 in chapter 3.

Region III and the Hybrid region shared the next frequency of interest of 25.7 Hz.
Once again the strain gage data was divided into two segments after it was noticed that
the response changed significantly. In the first segment it appeared that the dynamic
response was due to a fat torus shrinking into a thinner torus. The subharmonic

frequencies for both segments allowed us to classify the systems behavior as dominantly

92

 

 

2-frequency quasi-periodicity. These subharmonic frequencies gave way to a
subharmonic order (around 12 for the first segment of data) that resembled those in
region 111 of chapter 3.

In region VI, four frequencies were selected to investigate the dynamic properties
of the cantilever beam they were: 43.8, 44.4, 45.2, and 45.6 Hz. The data taken at 43.8
Hz was corrupted and was rendered useless. But data from the other frequencies data
revealed some of the most interesting results. In the case of 44.4 Hz, the dynamics were
determined to be dominantly quasi-periodic.

In the case of 45.2 Hz, the small winding number and proximity to the diagonal in
the torus angle return map, suggested that there might be some torus wrinkling, but it was
only on a very small scale in the torus angle, and thus the dynamics were considered to be
dominantly quasi-periodic. The subharmonic order in this case was on the order of about
100, which resembled the subharmonic order seen in the sweeps of region VI sub 2.

In the case of 45.6 Hz, there was what would appear to be a perfect example of
torus doubling. By examining the F FT of the strain gage response and the winding
number a connection between the two became apparent. The main torus was associated
the winding number with a subharmonic order of 75, and the other subharmonic
frequency (order of 150) was associated with the torus doubling.

The normal fundamental frequency became a factor in the response of region VI.
The pattern that the response created was similar to that of the hybrid region but involved
a larger response order. It was deduced that the hybrid region response was because of
the affect of superharmonic resonance of order 2. The fundamental normal frequency

was a 47 Hz, the hybrid region lies close to 47/2 Hz.

93

The last frequency that was examined was not part of any regions that were
defined in chapter 3. The frequency was 37.9 Hz and it was interesting because the
response signal appeared to be a triangle wave and an audible ring came from the beam.
The F F T yield information that showed that the superharmonic resonances are an
important part of the behavior of the beam.

The extremely low-frequency responses (observed with subharmonic orders of 33,
54, 100, and even 150) were note worthy. They tended to occur in the range of 0.4 to 0.7
Hz. These frequencies are well below the 2.5 Hz fundamental frequency of the beam.
As such, this beam phenomenon differs from the high-to-low frequency modal energy
cascade documented in the literature. In two parameter regimes, the extreme
subharmonic order was seen to vary continuously, typically incommensurately with the
excitation frequency, yielding quasi-periodic behavior on tori. This differs from
frequency demultiplication, observed by Van der P01 and Van der Mark [13], which
occurs at fixed, pure subharmonics, and then bifurcates. The low frequencies are not the
result of beating as the FFTs indicate the low frequency spikes to be independent ones.

Although is project was not directly tied to any specific engineering problem, it
does have valuable information about the response behaviors of a system that is
undergoing non—linear friction forced excitation. Two of the fundamental frequencies
(2.5 Hz unload and 13.5 Hz load) did not even have an effect the cantilever beam. And
cases where linear modeling would say that nothing should happen; chaos and or quasi-
periodic responses became apparent. This reaffirms the idea that non-linearities can not
be taken lightly and are extremely important if one is to fully understand and appreciate

the dynamics of a given system.

94

Some future work can be done to mathematically model the dynamic response of
this system. I would suggest finding the equations of motion for the cantilever beam
using Newtonian and Lagrangian methods; it might also be helpful to use finite element
method as well. The normal degree of freedom was seen to be active in some of the
responses. It might be necessary to include contact compliance in the friction law, as it
was observed in earlier friction measurements [1]. The experiment has a displacement-
dependent normal load, due to the tilt in the beam fixture, influencing the friction. As the
FNN studies indicated four to six delay coordinates needed in the reconstructed Poincare
sections, a low order model may need two transverse (bending) beam modes, plus the
normal degree of freedom. Using the reduced equations of motion, a simulation that
mimics the parameters in the experiment can be run, and the simulation response data can

be compared with the experimental data.

95

Appendix

This appendix contains figures that deal with the fixed frequency examination. These

figures are of interest, but were not placed in Chapter 4 to allow for an easier read.

8 Hz Poincare of Poincare

 

 

 

 

 

 

T I r r 1 I f 1 r
-0.2F “
’.o’. z ' ’ .0 .0 U .. o _‘
-03 a 5.0”. l ' . 0 '0 :0 o z z 1 ..
i t; ' : °. 2 - ' .:. l I.
.. . : . “..
-o.4 - - —
o .
-o.5 i _
’2; .
X 0 o . .’ '0 ° .9 o
‘0 6 ’- 8 6. t . 2 ’0‘. ‘0. .0. ‘3. . o. o. ‘...o. ., -<
fl ”0 ‘ : .a..l.'?}. ‘o.;?. 83;". J . :. :. o. o t .0 . ' .: :0 ‘€ ' ‘0‘ . .
'5 ~° ,.’ 3° "- 43.2%“ '3‘ 13% 3:. ’ V9841 NEW-'1‘ a ' "53?"; - r
.07.— o‘ J §,'o’“ot ‘5’. n .' ' ° 3; 0’8 "8:: 0’ ‘ 3 .0): .0 “.:.-o _
. o O o ' . .0 C o 0‘ o. ..&‘ i '0 ,‘ o o 5‘ ‘ . '
1' ~. .11 1;-,..;.~ o. an: .11: ~ a: 1- I." ~: 4 or)"
o. '0‘ ; ‘ r ”0.. . .0 .. o k.‘ .oy“..l :t a“ . 5" 0.. .. 1 o . ...":3
o o 0.0 . .0. o o ‘ o . . ‘0‘. .:. . ‘00.... “.0. O .. O...’ ’ j
-08 H . O ’. b . o... 0 g . 0 ~ ‘ a .
419*- _
'1 1 ° 1 l l l l l l l
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Index

 

Figure A. 1: Poincare section of the Poincaré section of 8 Hz plotted against its index.

 

96

9.8 Hz Poincare of Poincare

 

3 I I
O
a ' o o.
. 9.. .0 O. .‘u 9.... .
o a. . . . 'k . ...o. A. ...’”o. :0: o
‘ oo 9" j. 'a .200. .0 0‘

‘0
.5”!

 

2.5 '- .o o. . 3 .0. ... d. I .9330. ?.‘. g f. . ‘ . 0
. o o " . o ’ l. o . e w. ’ ’ ' 5.. ~. .. .0:
' z, 00 - ?§‘ ‘90,... ' '

X(N+1)
.I
.3
t
.6.
&. ‘fiéj. a.
Y ‘b
'1‘
.1“
.32.
..
é:

. : 0' '0 O. .‘ '33:} t . o o
o. g y - ’or‘a O o - . .... ° 0
. ?. ... . ... . . ‘ 00‘ O. .... .":.’ '... v . 0‘. ..

 

 

 

o
o
2 o . 'H' 0". ’ § O U ' "fl’...’ .0 .. 0.0 .0 _1
b ...“. ‘D .0. "°‘b: "" 7. e o
...“ 3.0 a. {.9 ..O.‘ ..‘ool.. .
$0.. ' ' .. .:- . o 0‘... .
. . “i”. 0‘ ' . o .
t 0 ‘ . .. ... ... 0 O
00 . "' .0 o O
O Q o o
o . .
O : . '0
. 9
o
a
l l

 

X(N)

Figure A.2: Poincaré section of the Poincare section delay map of 9.8 Hz.

97

 

log(Cr)

log(Cr)

8hz Correlation Dimension

 

 

 

 

 

 

 

 

0
00000&
-1 . O
00
o
-2 I f
o
0
,0
-3» f Slope: 3.3275
-4 1 .
-2 -1.5 -1 -0.5
9.8hz Correlation Dimension
0 . . .
1 IO
0"
-1 O0
0
-2 1 Slope: 3.7394 (3‘2?)
gf
X?)
'3 8'
-‘O
ooon::-:-:0«:OO
-4 1 1
-2 -1.5 -1 -0.5
'09“)

Derivative of Correlation

 

 

 

 

 

 

 

 

4 - -
o 0 0
o o
A 3 i O I
t; O O
8’ o
E 2 1 O O .
i6,
3 o
“O 1 Q +
o
0 O O O 1 1
-2 -1.5 -1
Derivative of Correlation
4 ’ o O O o i
A O O
t: 3.
.8 o
8
§ 2 t O 1
5 0
1 1
o
o 003000 4 .
-1.5 -1 -0.5
mm

Figure A.3: Correlation dimension and derivative of 8 and 9.8 Hz.

98

 

 

 

 

 

21 .9hz Poincare of Poincare

 

 

-1.8 T I I fir I I o I
a
t‘ °
1"
a ’ ...
-2 - . . —<
o
-2.2 F 4
o
a o
.:.
-2.4 F ~
0 o o
. “0‘ O
A 050 e O . o 0.. o ‘
1- "." 0 I O’ I ' 'l V.-
* 26 ' ' -. :-‘ . ' 1 .. ’
g - I '— 0'... I... 0 .0. ~ ’08:.gw ' '. CI
x ° 0” 9 I o. g . ‘ 8 0"
$4314., . 3-0 2; ~ ~13 ‘11
a 0.. ‘t" o . . o . o
2 8 I (.‘h' . ‘I I"? I . 1:. I I. .
- O o —1
‘.. 8. "O O
O I O. s . :0 O . ‘
0‘ I. 8' ' 0 o
... o " .... o o '
’8 :0 ’0‘. ' . 0 fl
0 ”A. . O. . o. .
-3 P . 0“, I 3. .... . o .0. q
‘.o:.. . .g .0. '3. o . o
'0'. ...... .0. “.:;.¢ . ° "0 o
..v.o,(°oo " 0°: . ' a '
0’; 0‘. .‘dc‘z‘fil...ozk 0 o: .. . .‘ .0 O o
-32 1— 00 "r‘ 00. ' o. 0". fl ’ o .k .0 o . _I
”I... O. ‘0' . e 0 0 o. . . .0
. ‘ . 1° 0 . o '0 ....3 .
o . o o O

 

 

 

3.4 -3.2 -3 -2.8 -2.6 -2.4 -2.2 -2 -1.8
X(N)

Figure A.4: Delay Map of the Poincaré section of the Poincare section of 21 .9 Hz with
Delay of 1.

99

 

log(Cr)

21 .9hz Correlation Dimension

 

 

 

 

 

 

 

 

 

 

 

 

c-
000
-1 1 000
”>00
fiery
-31 9“ Slope:
-4 000000. 14439 .
-2 -1.5 -1 -0.5 0
22.1hz Correlation Dimension _
0 r z r ' -000
a OIJ
'1 i OOU
-2. 00
-3» 9" Slo e: 3.3973 .
009 p
:4 OOCCODO
-2 -1.5 -1 -0.5 0
0 22.3hz Correlation Dimension
. . -0000
00
-1 . Oo 1
-0004
-2: .fy
-3» dog Slope: 3.1805
4(fip 1 1 .
-2 -1.5 -1 -0.5 0
mm

21 .9hz Derivative of Correlation

 

 

 

 

 

 

 

 

 

 

 

 

3
€2.5i (DOOOC) 4
8 00 00
fi 2’ OOQU
E 0
§ 1.5 o O
6 1 ' O 1
o
0.5 1 1
-2 -1.5 -1 -0.5
4 22.1hz Derivative of Correlation
A o O O
b 3 . O O O o
g 0 O o
B 2* 0. 1
8. -
3 1 1 U
U o
OIZILO 1 1
-2 -1.5 -1 -0.5
22.32 Derivative of Correlation
3.5 . .
o O o
S 3 ' O O 1
8’ O n
225‘ O 'OO 1
Q 2* 000
o 0
é’ 1.51 O 01
1 O 1 1
-2 -1.5 -1
mm

Figure A.5: Correlation number and their derivatives vs. log(r) for 21 .9, 22.1 and 22.3

Hz.

100

 

 

 

44.4hz Correlation Dimension Derivative of Correlation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 4 o -
o o O
‘1 ' O 100 1 S 3 l O O i
e 2 0000 8’ O 0o
o ' O 5 O 0
V 00 E 2 0 O . ‘
_8_’ -3» .10 S O
‘4’ adj Slope: 3.5714 é’ 1 O O
_5<:>COCOQO 1 g 1 0 9 o o ‘ . 1
-2.5 -2 -1.5 -1 -0.5 0 -2.5 -2 -1.5 —1
0 45.2hz Correlation Dimension Derivative of Correlation
_ 00 A O O
'1 ’ 0001.300" ‘ b 3 O O 1
C -2 1 . 000 g, 0 Q n
o -00 U 2 ~ 0 *
3 '3’ o3" §1 O 00 000
4130000 Slope: 2.8328 5 O
_5 1 1 1 1 1 0 O 1 a 1 1
-2 -1.5 -1 -0.5 0 -2 -1.5 -1 -0.5
0 45.6hz Correlation Dimension Derivative of Correlation
1 00 1 o J
'1 r 00000 ‘ E 3 O O O
A (,OOC) 8 O O O
5 -2. £00» 22 O 00
3‘3? @909 l %1 O OOOOJ
-4 . 1 0'3 Slope: 2.9595 J fi 0
5 CO?” . . . . 0 O . . 1 .
-2 -1.5 -1 -0.5 0 0.5 -2 -1.5 -1 -0.5
log(r) log(r)

Figure A.6: Correlation number and their derivative vs. log(r) for 44.4, 45.2 and 45.6 Hz.

101

 

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