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This is to certify that the

thesis entitled

EXPERIMENTAL INVESTIGATION OF NONLINEAR
OSCILLATIONS OF A BASE EXCITED FLEXIBLE
CANTILEVER BEAM

presented by

Syed Masroor Hasan

has been accepted towards fulfillment
of the requirements for

M. S . degree in Mechanical Engineering

 

 

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EXPERINIENTAL INVESTIGATION OF NONLINEAR
OSCILLATIONS OF A BASE EXCITED
FLEXIBLE CAN TILEVER BEAM

By

Syed Masroor Hasan

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1989

ABSTRACT

Experimental Investigation of Nonlinear
Oscillations of a Base Excited

Flexible Cantilever Beam.
By

Syed Masroor Hasan

A study is made of the nonlinear response of a flexible cantilever beam
excited by a sinusoidal base motion. Experimental results are presented which
exhibit a variety of unusual and potentially dangerous phenomena such as chaos,
multi-mode interaction, multiple steady-states and a very high order subharmonic
response. Particular attention is focused on resonances which can occur at frequen-
cies well removed from the excitation frequency. In addition, the partial integro—
differential equation is obtained which governs the planar motion of the beam. It
takes into account the effects of large curvature and axial deformations and
includes up to order three nonlinearities. For a single mode response, the equation
can be reduced to Mathieu’s equation with a cubic nonlinearity. The stability of
the trivial solution of this equation is studied using a perturbation technique and
the results are compared to experimentally obtained data. In conclusion, the
pointwise dimension is estimated for the chaotic motion observed, and the results

suggest that the chaotic motion can be modelled by three orthogonal modes.

Dedicated to my father, Syed Ghiasul Haq.

iii

ACKNOWLEDGNIENTS

I would like to express my sincere gratitude to Dr. Alan Haddow, my major
professor, for his patient guidance and assistance during the period of research

and graduate study. It has enhanced the content of this thesis considerably.

The other members of my Master Committee also deserve my thanks. Drs.
Clark Radclifle and Joe Whitesell were all helpful, and their comments and obser-

vations concerning this work were very useful.

Special thanks to Bob Rose from the machine shop for his valuable aid in the

construction of the experimental set-up.

My family has been a source of strength, love, and moral support for me,
especially over the last two years. I only hope that I can put back as much as I
have drawn. To my father and mother, Ghias and Rabia, my brothers and sisters,
Masood, Shahida, Mahmood, Rukhsana, and Sarwat... thank you, I couldn’t have
done it without you. I would also like to thank my fiancee Lubna, and her family,

for their patience and understanding.

Finally a sincer thanks to my friends. I would especially like to thank Brian
D’Mello, and Mohammad Usman for their extra support and help. To all my

other friends, thanks for being there when I needed you.

1v

Table of Contents

List Of Tables ....................................................................................................... vi
List of Figures ...................................................................................................... vii
Chapter 1: Introduction ..................................................................................... 1
1.1: Scope of the Work ............................................................................. 1

1.2: Literature Review .............................................................................. 2
Chapter 2: The Governing Equation for Planar Motion ............................ 4
Chapter 3: Stability Analysis ........................................................................... 11
3.1: Introduction ........................................................................................ 11

3.2: Reduction to Mathieu’s Equation .................................................. 11

3.3: Stability Analysis .............................................................................. 12
Chapter 4: Experimental Results ..................................................................... 16
4.1: Experimental Set-up ......................................................................... 16

4.2: Linear Stability and Planar Steady-State Motion ..................... 16

4.3: Chaos .................................................................................................... 22
4.3.1: Transient Chaos ........................................................................ 27

4.3.2: Intermittent and Steady-State Chaos .................................. 28

4.4: Multi-Mode Interaction .................................................................... 35
4.4.1: Extremely Low Sub-harmonic Response .............................. 35

4.4.2: Modal Interaction and Beating Phenomenon ..................... 39

4.4.3: Combination Resonance .......................................................... 42
Chapter 5: Dimension Estimate ....................................................................... 45
5.1: Introduction ........................................................................................ 45

5.2: Acquisition of Data ........................................................................... 48

5.3: Estimation of Optimal Time Delay .............................................. 48

5.4: Reconstructed Coordinates and Dimension Estimate ............... 52

5.5: Conclusions ......................................................................................... 58
Chapter 6: Conclusions ...................................................................................... 59
Appendix A ........................................................................................................... 61
Appendix B ........................................................................................................... 62
List of References ................................................................................................. 63

List of Tables

Table 2.1: Values of r,- and K,- ......................................................................... 9
Table 2.2: Values of [3,, 1,, and x, ................................................................. 10

vi

List of Figures

Figure 2.1: Cantilever beam and coordinate system ................................................ 5
Figure 2.2: Angle of rotation of a difl'erential element of beam ............................ 5
Figure 3.1: Transition curves seperating stable and unstable regions ................. 15
Figure 4.1: Sketch of existing experimental set-up ................................................... 17
Figure 4.2: Block diagram of the experimental set-up ............................................ 18
Figure 4.3: Stability diagram for fourth mode .......................................................... 20
Figure 4.4: Comparision of theoretical Strutt diagram with experimental
data points for fourth mode .......................................................................................... 21
Figure 4.5a: Sketch of frequency reponse curve for nonlinear Mathieu’s
equation. ............................................................................................................................ 23
Figure 4.5b: Experimental frequency response curve. ............................................ 23
Figure 4.6: Time trace showing a steady-state parametric response .................... 24
Figure 4.6: FFT of time trace ....................................................................................... 25
Figure 4.7: Experimental frequency response curve. Level of force con-
stant but higher than Figure 4.5b ................................................................................ 26
Figure 4.8: Transient chaotic response for a forcing frequency of 87.0 Hz 29
Figure 4.9: Robust chaotic response for a forcing frequency of 87.0 Hz ............. 31
Figure 4.9: FFT of input and output .......................................................................... 32
Figure 4.10: Steady-state parametric response for a forcing frequency of
45.0 Hz ................................................................................................................................ 33
Figure 4.11: Chaotic response for a forcing frequency of 43 Hz ............................ 34
Figure 4.12: Time traces for initial time close to zero at 195 Hz. ....................... 36
Figure 4.13: Time traces at time t+30 sees at 195 Hz ............................................ 37
Figure 4.14: FFT of time trace at t+30 sees ............................................................. 38
Figure 4.15: Waterfall plot showing output close to an initial time zero
at 385.6 Hz ......................................................................................................................... 40
Figure 4.16: Waterfall plot showing output some t+10 sees later for a
forcing frequency of 385.6 Hz ........................................................................................ 41
Figure 4.17: Combination of responses at a forcing frequency of 85.5 Hz 43
Figure 4.18: Chaotic response for a forcing frequency of 85.5 Hz ........................ 44
‘ Figure 5.1: Flow chart for dimension estimate ......................................................... 49
Figure 5.2: The unnormalized autocorrelation function for various time
delays .................................................................................................................................. 51
Figure 5.3: Phase portrait of measured time signal SGS (t) from cantil-
ever beam using the reconstructed coordinates ......................................................... 53
Figure 5.4a: log P plotted against log r for m = 3,9 and 15 ................................ 56

vii

Figure 5.4b: The embedding dimension In plotted against the pointwise
dimension of the attractor ............................................................................................. 57

Figure A.l: An element of beam ................................................................................... 66

viii

CHAPTER 1
INTRODUCTION

1.1 Scope of the Work

Recently, there has been a growing interest in the dynamical behaviour of
flexible structures. Such structures are becoming more common as designers are
continualy striving to make eflicient use of newly available materials and make
lighter, less expensive products. There is often the need to reduce the weight to a
bare minimum, in order to reduce the inertial loading on the system. This is par-
ticularly desirable if the application involves high speed machinery e.g. robot
arms. Another area in which flexibility is a problem is that of space structures. In
a zero gravity environment a structure does not have to be designed to withstand

its own weight. The resulting designs are often very flexible.

The main aim of this work is to experimentally investigate the dynamical
behaviour of a flexible structure. A relatively simple model is chosen, viz. a cantil-
ever beam of dimensions 55.88 cm x 1.27 cm x 0.0508 cm and it is excited by a
sinusoidal base motion in the longitudanal direction. Chapter 4 catalogues the
various types of vibration phenomena observed. A number of these have received
attention in the past, such as main parametric resonances, multiple steady-states
and the jump phenomenon. However, less well documented behaviour was also
observed, including chaotic response, multi-mode interactions and very high order

sub-harmonic responses.

Before discussing the experimental observations, the governing equation of
motion of the system is derived in Chapter 2. It is valid for in-plane motions with
large curvatures and it includes up to third order nonlinear terms. In this thesis,
limited use is made of the equation of motion, but it will act as a sound starting

point for future studies in this area. The equation is also used to predict the

1

theoretical stability of the trivial solution. This is undertaken in Chapter 3 and

the results are compared with the experimental results in Chapter 4.

Chapter 5 is concerned with a more detailed study of the chaotic response. A
discussion is presented as to the use of dimension calculations and and such a cal-
culation is undertaken for the chaotic motion observed in the cantilever.

1.2 Literature Review

A review article by Sathyamoorthy [1982] surveys recent advances in the area
of nonlinear analysis of beams. Most of the works dealing with planar motion are
devoted to the study of axially restrained simply supported beam which take into
account the nonlinearities due to mid-plane stretching. See, for example Wojnow-
sky Kreiger [1950], Evensen [1968], Ray and Bert [1969], Mei [1973], Nayfeh et al.
[1974], and Bhashyam and Prathap [1980]. A few studies have dealt with the
oscillations of beams with no axial restraint in which the nonlinearities arising
from the efl'ects of large curvature and longitudinal inertia are also considered, see

Wagner [1965], Alturi [1973], and Luongo et al. [1986].

There have been a number of works related to the non-planar response of
beams. For example, Crespo da Silva and Glynn [1978a] investigated the
bending-bending-torsion of a beam. In part two of the paper, Crespo da Silva and
Glynn [1978b], they extended the work to include the effects of a transverse force.
Ho et al.[l975, 1976] studied a similar problem but did not consider the torsional
motion. Hyer [1980] investigated the non-planar response of a base excited cantil-
ever tusing the method of multiple scales, and presented numerical results for a
specific beam excited near its fundamental and its second natural frequencies.
Luongo et al. [1984a] considered the free non-planar motions of an inextensible
elastic beam, supported in an arbitary manner without any axial restraints. A
thorough mathematical investigation of Euler’s elastica has been completed by

Caflish and Maddocks [1984].

Tseng and Dugundji [1971] reported on what seems to be the first observa-
tion of chaotic motion in a physical structure. They completed a theoretical and
an experimental investigation of the bending response of a buckled beam sub-
jected to a harmonic excitation. Although it was not named as chaos a "snap-
through" response was observed both numerically and experimentally. The work
was later expanded to include torsional responses, Dugundji and Mukhopadhyay
[1973]. The text by Evan-Iwanowski [1976] reports on a number of experimental
results relating to the steady-state, or almost steady-state response of beams.
Evensen and Evan-Iwanowski [1966] investigated the stability of a column sub-
jected to axial excitation. Handoo and Sundararajan [1971] studied a similar prob-
lem. Haight and King [1969] investigated planar and the non-planar stability of a
rod due to an axial excitation. In a related work, Haight and King [1971] studied
a similar problem but with a lateral excitation. Takahski [1979] reported on the
stability of multi-mode, steady-state, planar response of a nonlinear beam. Dowell
et al. [1977] devised an experiment to investigate the dependence of the bending
and torsional natural frequencies on the static deflection of a hingless rotor blade
A detailed investigation by Bux and Roberts [1986] reported on the complex
modal interactions that can occur as a consequence of internal resonances. They
studied both theoretically and experimentally, the behaviour of a system of cou-
pled beams. Housner and Belvin [1986] studied the transient response of a slender

guyed boom.

Works reporting on the experimental existence of chaos in structures include
those by Moon and Holmes [1979], Moon [1980], Shaw [1985] and Burton & Kolo-
with [1988].

CHAPTER 2
THE GOVERNING EQUATION FOR PLANAR MOTION

The planar vibration of a base excited cantilever beam is studied. The equa-
tion of motion is derived using Hamilton’s Principle (see Goldstein [1980] pp 35-
37), and terms arising from large curvature and from the resulting axial
deflections are included. The equation of motion contains nonlinearities up to
order three. Galerkin’s method is used to reduce the partial difl'erential equation
of motion to a nonlinear ordinary difl'erential equation by making use of the eigen-

functions of the linearized system.

The following assumptions are made to simplify the problem,
i) the beam is inextensible,
ii) there is no warping or shear deformation,
iii) the effects of rotary inertia are neglected,
and
iv) the cross-sectional dimensions and the material properties of the

beam are constant along the length.

Consider a slender beam of length l and mass m per unit length as shown in
Figure 2.1. The base of the beam is located at the origin of the coordinate axes
and has a known harmonic displacement in the x-direction of bsin(fl,t), 6 being
the magnitude of the displacement and n, the excitation frequency. Let S be the
path coordinate measured along the undeformed arc center line. For the planar
motion, the deformed shape of the beam is defined by a displacement of «(54)

and v(S,t) in x and y direction respevtively, and by a rotation of the beam’s cross

section of 0(S ,t ).

Noting assumption (iii) and (iv), the total kinetic energy of the beam is given

by,

 

 

x..¢.ugi5
A — A,-
(5'0) (5+ use), v5.13 3
BSMQS’C fl? ‘3’“‘5
Figure 2.1: Cantilever beam and coordinate system
,//
/
i /
is ]
. d5
f
/"//

 

 

Figure 2.2: Angle of ratation of a differential
element of beam.

T(:)-{—;-m(.22(s,¢)+.32(s,¢))ds (2.1)

where a dot denotes partial diflerentiation with respect to the time t and m is the

mass per unit length.

Noting assumption (i), the total potential energy of the beam is the elastic

potential energy due to bending , and employing the usual notation, is given by,
t l
V (t) .. f -2- E! u? (5,!) as (2.2)
0

In the above equation, it is the curvature of the center line.

The Lagrangian function (see Meirovitch [1967] pp 44) is given by

L(t)=T(t)—V(t)
l
L(t)={[-§-m(d2+62)—-;-E1p2]d5 (2.3)

We now seek to express p(S,t) in terms of the displacement component v(S,t).
From assumption (ii) and Figure 2.2, the rotation 0(S,t) is related to the displace-

ment derivatives by

I

tan 0 = T13,— (2.4)

where the prime denotes the partial difl'erentiation with respect to S. The expres-

sion for the strain (see Appendix A) in terms of displacement components is
e(S,t)=[(1+u' )2+v'2]‘/2—l (2.5)
Expanding this using the Taylor series we get,
£(S,t) = u’ + -;—v'2 (2.6)
Noting assumption (i), equation (2.6) reduces to

u' =-%v'2 (2.7)

The curvature expression is given by,

#(5,t)= 9' (5.1) (2-8)
Using equation (2.4) and equation (2.7) the curvature can be written as (see
Appendix B)

143,1) = 0' (5,1). 0 ”(Hi-.1 '2) (2.9)

Substituting equation (2.9) into equation (2.3) yields
1
L(1)=f[-;-m(a’+ a?)——;-Er(v"(1+-;—v'2))2)ds (2.10)
0

The partial differential equations of motion can now be obtained by using the

Hamilton’s principle (see Goldstein [1980] pp 36), which requires that
‘2 l

6!{L(t)=0

‘2 t
i -2 -2 _ l n _1_ t2 2 -
6{{[2 m(u+v) 251” (1+2 0 )) [as o (2.11)
After taking variations with respect to e and v, performing integration by parts

of the terms in equation (2.11), and rewriting them as the coefficients of variations

611 and 511, which must vanish so that equation (2.11) holds, we arrive at:

‘21

II[(—mii)6u +(—mii—E1(v""+v""v'2+4v'v"u'”+v”3))6v]dS=0 (2.12)
1,0

In order to eliminate the function u(S,t) from the above equation, we integrate

equation (2.7) with respect to S to obtain
5 l
a (3,1) = 8 (0,1) - f? v '2 d3 (2.13)
0

At the base of the beam (S=0), the u displacement is given by :1 (0,1) = bsin(n,t),

hence

S
u (s, 1)= bsin( (—n,1) $.11; '2 as (2.14)
0
Therefore
1 S ..
" = -(-2—jv ”43) — b a,2 sin (n, 1) (2.15)
0

Substituting equations (2.14) and (2.15) into equation (2.12) we get

‘2!

fI[-mv '(—fv '2dS)+ mv "(f(% Iv ”115)

mbfl, 25in“), t)[(l—S)v ”— 1,]
(-mi5-EI(v "H +v""v "+4.; '1; "v M +11 '18))511) as -0 (2.16)

The stationary condition leads to the partial governing integro-diflerential equa-

tion which determines v(S,t). The equation is

mii+EI(v""+v"”v’2+4v'v"v"'+v"3)
1s” 1 1s
+ mu ’(—]v ”115) — mu "(fl—Iv ”115)
2o s 2o
= mbfll2sin(fl,t)[(l—S)v "—v '] (2.17)
Introducing the following non-dimensional terms,

~ 5
5'7

{=01}! (where 111=(%)1/2 )
m

.. Q, ~ b

we can non-dimensionalize equation (2.17). Dropping tilde for convenience, we get
8
'9' +0 III! + v ”H” 12+“) 'v "v H! +0 113+ v '(-l—Iv ”(5')
2 0

1 5 ..
v H ”(21” ”11.9)115] = 6025in(0t)[(1—S)v "-v '1 (2.19)
S 0

where dots and primes represent partial derivatives with respect to the new time I

and new arc length 5‘ respectively.

The Galerkin’s Method (see Meirovitch [1967] ) can be used to reduce equa-
tion (2.19) to a ordinary differential equation. The displacement v(S,t) is assumed

to be an eigenfunction of the linearized system, and a solution is sought of the
form
1) (5,!) = ¢j(S)q,-(t) (2.20)
where j = 1,2,3 ...........
The mode shape ¢,~(S) of the linearized system is given by (see, for example
Thompson [1981] pp 218-221),
43,-(5 ) :- cosh(r,- 5 )—cos( 1', S )-K,-(sinh( r,- S )—sin(r,- S) (2.21)

where

cosh(r,- S )+cos( r, S )
j - sinh(r,~ S )+sin( r, S)

 

(2.22)

The values of r, and K,- as tabulated by Young and Felgar [1949] are as follows:
Table 2.1

Values of K, and r,-

 

J' Ki '1'
1 0.7341 1.8751
1.0185 4.6941

0.9992 7.8548

AGO”

1.0000 10.99554

 

 

 

 

 

The substitution of equation (2.20) into equation (2.19) results in

qj¢j + qj¢jlllf + Qj3[¢j'”'¢j'2+4¢j'¢j H¢j Ill+¢j II

10

S l 5
+q1'(qi q:'+91'2)(¢i' Id’i I 2d$‘¢i” f(f¢1"245)d5)
0 0 0

= bflzsin(flt)[(l-S)¢, "—11, '] (2.23)

An application of the Galerkin Method in the usual manner leads to the ordinary

diflerential equation of the motion,

91' + (01+51'5923inmtll91 + 71' 913+>15l012412+95021 = 0 (2.24)

The constants which appear in equation (2.24) are defined by

l
a,=f¢,""(5)¢,(5)45= r," (2.2511)
0

1
7j={(¢j'” I¢jt 2¢i+4¢i' ¢jt 1¢j1 u ¢’,+¢jus¢j)d5 (2.255)

1 S l S

x1'31”,"(I¢j'2ds)’¢j” (£(I¢j'2)]¢jd5 (2.25c)
0 0 o

1
fi.= — 11(1—S)¢.-"- 21' 111,15 (2250
0

The constants 11,, 13,, 7,, and x, can be obtained by numerical integration of equa-

tions (2.25). For j = 1-4, these evaluate to:
Table 2.2

Values of 13,, 7,, and >.,

 

 

 

 

 

1' fl,- 1, X,-

l 1.57 40.44 4.60

2 8.65 13.4x103 145

3 25.00 26.41110‘ 103

4 50.21 1.75x10° 3.6511103

 

 

CHAPTER 3
STABILITY ANALYSIS

3.1 Introduction

The governing equation for planar motion obtained previously admits a
trivial solution (q,(t)=0). In this section we study the stability of this solution for a
main parametric resonance, i.e for a forcing frequency in the region of twice the
natural frequency. The boundaries between stable and unstable solutions are
obtained using Lindstedt-Poincare Method. The method is explained in detail in

Nayfeh & Mook [1979] (pp 54-56) and only those steps pertinent to this particular

problem will be included here.

3.2 Reduction to Mathieu’s Equation

Recalling equation (2.24), the general equation of motion is
if! +(01' +315923mm!))91'+’71'91'3+’11(¢112?ii +q,1},~’)=0

Since the stability of the trivial solution is sought we can drop the nonlinear

terms, which yields
2;, + (a, + )6,bflzsin(flt) )q, = O (3.1)
The above equation is now reduced to the form whose stability has been studied

by Nayfeh & Mook [1979] (pp 300-301). Let 11%: and substitute this into equa-

tion 3.1, noting that the dots are now derivatives with respect to new time 1., we

get
.. 40; . -
Qj+[F4-4flj0810(2t)]qi = 0 (3.2)
Let

11

12

4a,

 

(3.3a)

and
e,=2fl,b (3.33)
and dropping the over bar for convenience, the equation 3.2 becomes
Ej,-+(6+21,sin(21) )q,=o (3.4)

which is the standard form of Mathieu’s equation.

3.3 Stability Analysis

The stability analysis is carried out using Lindstedt-Poincare technique which
is valid for small e(ez0.035 in our case). Before application of the method, a
damping term should be introduced into equation (3.4) to account for small
energy losses in the physical system. The general form of the stability diagram is
not afl'ected by the value of the damping coeflicient, provided the damping is
small. The system we are considering is very lightly damped. Therefore, instead of
accurately modelling the damping, we will assume it to be viscous with a
coefficient, 1,11, of 0.001. Even if we choose ten times higher damping coefficient, it
results in altering the boundaries between stable and unstable regions, by less
than 1%. Adding this to equation (3.4) and dropping subscript j for convenience

results in
6+(6+215in(2t))q +2u11j=o (3.5)

Now we will follow the method as outlined by Nayfeh & Mook [1979] (pp 300-
301), to obtain boundaries between the stable and unstable regions. Using a first

order approximation, let

<I(t =¢)=qo(l)+691(t)

l3

5(6)=l+£61
Substituting these into equation (3.5) results in
(<11:+e<ii)+[(1+16,)+(2esin(21))](qo+eq,)+21p(q'o+¢q'l)=o

Equating like powers of e, we get

602

Mac (3611)

21+21= -[51+25in(2t)190 4141111 (13-56)
The solution of equation (3.6a) is
qo=aocos(t)+bosin(t)
Substituting this back into equation (3.6b) we get
21+91= -[51+2sin(2¢)llaoCOS(t)+bosin(t)l-2Fiolboc05(¢)-aosin(t)l
and expanding the terms in the above equation, we get
01+q1= —[(61a0+bo+2pbo)]cos(t)—[6lbo+ao—2uao]sin(t)+ Non Secular Producing Terms

Now eliminating secular producing terms, we get

6(ao+(l+2p)bo=0 (3.7a)

(1—2p)a,+3,1,=o (3.71)
For a, non-trivial solution to exist

a, 1+2u

6,2—(1—4112)=0

=> 51 ‘3 i l 1-4#2 11/2 (3-8)

14

The transition curves seperating stable regions from unstable region are
6 = l + c 61
=> 6 = 1 :1: [c2 — 4(£;1)"’]1/2 (3.9)
These are approximate transition curves seperating stable region from
unstable region in 6- 1 space as shown in Figure 3.1. They hold for any single

mode response (j=1,2,3...). In the next chapter we shall compare this theoretical

result to the experimentally obtained stability boundaries for the fourth mode.

15

STRUTI' DIAGRAM

   

  

 

 

0.0% STABLE REGION UNSTABLE REGION smaLE 12501011
I

0.01s
.

°'°° ‘ l a T 1 I ' I f I
0.97 0.93 0.99 1.00 1.01 1.02

Figure 3.1: Transition curves seperqting stable
and unstable regIons.

 

1.03

CHAPTER 4
EXPERJIVIENTAL RESULTS

4.1 Experimental Set-up:

A sketch of the existing experimental set-up is shown in Figure 4.1. The test
specimen consists of a flexible steel cantilevered beam attached to a B&K type
4808 electromagnetic shaker. The beam is forced through a sinusoidal base excita-
tion in the axial direction. The acceleration of this motion is proportional to the
magnitude of the forcing term, “3,502 I as it appears in equation (2.24). It is

measured by an accelerometer attached to the head of the shaker.

The response of the beam is measured by two strain gages attached near the
root of the cantilever. They are positioned on each side of the beam and can
detect both in-plane and out-of-plane motions. The signals from the accelerometer
and strain gages are analyzed and recorded on a HP 5423A structural dynamic

analyzer. A block diagram of the experimental set-up is shown in Figure 4.2.

The first five linear natural frequencies associated with a motion in the x-y
plane (i.e a transverse vibration in the flexible direction) were measured as n, =
1.0 Hz, 02 = 8.0 Hz, 03 = 22.5 Hz, 11, = 44.8 Hz, and 05 = 72 Hz. These com-
pare well with the theoretical, linear natural frequency values of 1.24 Hz, 7.75 Hz,
21.74 Hz, 42.63 Hz and 70.47 Hz respectively (see, for example, Hartog [1980] pp
153,432)

4.2 Linear Stability and Planar Steady State Motions

It is well documented that the dynamic stability of a beam excited by a
sinusoidal displacement in the axial direction is given by a Strutt diagram (see
Haight & King [1969]). In the displacement-frequency parameter space there are

wedge shaped boundaries which separate the stable and unstable regions. The

16

17

static equilibrium position

lateral vibration v(S,t)

<l-—--—l>

beam dimensions 55.88 cm x 1.27 cm
x 0.0508 cm

 

.4“ strain gages

accelerometer ii |
. bsin H’ t

 

 

IZ/// //

shaker

Figure 4.1: Sketch of existing experimental set—up

GAGES

\H

bsin {at

18

BEAM

 

 

 

 

  

 

4 AMPLIFIER

 

 

 

 

  

 

 

 

AMPLIFIER

 

SHAKER

 

 

 

__1__1

 

 

WAVETEK
SIGNAL
GENERATOR

 

 

 

Figure 4.2:

ACCELEROMETER
/ H

 

AMPLIFIER

 

 

.__L

 

CHANNEL
2.

 

 

 

L

CHANNEL

 

 

HP 5423A

Block diagram of the experimental set-up

 

19

nose of these regions occur at forcing frequencies that are equal to twice the linear
natural frequencies of the system (see Chapter 3). They also occur at multiples

and combinations of these natural frequencies.

Figure 4.3 presents experimental data which defines one of these regions for
the cantilever beam. It is associated with instability of the fourth in-plane mode
of the system which results from a forcing frequency in the region of 89.6 Hz.
Similar results can be obtained for other modes. The level of force corresponds to
a range in the base acceleration between zero and 40 111/32. The data points

correspond to a transition from a trivial response to a non-trivial response.

Figure 4.4 presents a comparision of the experimental data with the theoreti-
cal Strutt diagram, obtained using equation (3.9). The comparision can be made
by noting the following relationships for mode 4 from equations (3.3a) and (3.3b),

and recalling from equation (2.25a) that a, - 114‘,

4 rt
02

 

6-

(4:23‘6

where

“21099554 from table 2.1
13,-5021 from table 2.2
It should be noted that the experimental natural frequency is used in place of the
theoretical linear natural frequency, to transform the experimental values of the
forcing frequency to the variable 6. The experimental data points compare reason-

ably well with the theoritical predictions.

One should recall that the stability diagram presented in Figure 4.3 only
predicts what happens to the trivial solution. Linear theory, which is valid for the
trivial case, predicts that the trivial solution will grow exponentially in an

unstable region (see, for example Nayfeh & Mook [1979] pp 338). However, as the

Level of Force (mVrms)

1 400.0

20

 

1200.01

1000.0-1

800.0 -4

600.0 -1

400.0 -1

200.0 -1

0.0

STABLE . 0111511131.: . STABLE

0 Experimental Data

 

 

87

03.0 96.0
Forcing Frequency (Hz)
FIgure 4.3: Stability diagram for fourth made.

1.0 311.0 91.0

92..

 

0.06

21

 

0.05-

0.04-

0.03-

0.02 -4

0.014

 

0 Experimental Data Pahb

 

 

I T

f 1 . I ' I ' If 1
0.95 0.00 0.97 0.90 0.00 1.00 1.01

6

‘l

1 1 r
1.02 1.03

T

r
l .04

Figure 4.4: Comparision of theoretical Strutt diagram with

experimental data points for fourth mode.

1

 

lJ

22

response grows the linear theory is no longer valid and the efl'ects of non-linearities
cannot be ignored. If the response remains inplane and contains essentially one
mode, the motion is governed by the nonlinear Mathieu equation as derived in
Chapter 2, equation (2.24) (also see Evan-Iwanowski [1976]). This equation admits
nontrivial, steady-state solutions. Approximate solutions to this equation can be
found (see, for example, Nayfeh & Mook [1979] pp 169, 338-348) and it can be
shown that non-trivial, steady-state solutions exist over a range of forcing fre-
quency. The general form of the variation of the amplitude of the steady-state
solutions, as a function of the forcing frequency, is sketched in Figure 4.5a. The
experimental results presented in Figure 4.5b substantiate these theoretical predic—
tions. The data for Figure 4.5b was obtained by measuring the amplitude of the
response at various values of the forcing frequency. The base acceleration is kept
constant at 7.81 m/32 throughout. A typical time trace of the forcing term and
the associated response are shown in Figure 4.6 along with their corresponding
Fourier transforms. It can be noted that the frequency of the response is one half

of the frequency of the excitation. This corresponds to a main parametric reso-

nance.

The frequency response curve of Figure 4.5b shows the posible existence of
multiple steady state solutions. At a frequency, say of 89 Hz, two stable steady
state solutions can be obtained, one at A (trivial) and the other at B (non-trivial).

The response adopted depends on the initial conditions of the system.

4.3 Chaos

If the level of force is increased, corresponding, say, to a base acceleration of
31.44 111/32, then the amplitude response curve presented in Figure 4.7 results. As
might be expected, the response amplitudes are larger. However, an interesting

additional feature develops. At large amplitudes the response becomes chaotic.

23

 

 

 

 

 

 

 

 

 

a:
U3
2:
c:
D.-
ID
a:
a:
I}.
c:
a:
a.
:1
E2
2:
c:
E
FREQUENCY
lOOOJPT-
g C
1.1 800.0-1
‘E .
m a ”\i
:3
3'1 ‘ -\
g 4000- '\_
a \.\
u;
3 200.0--1 '
t: I \
a:
‘9 mg
g 0.0., ‘— _. 1 f r ?—: 2 .— ,
'-20043 .

 

 

 

35.0 35.0 372033.0' :90 9009110921? 93094.? 95.0
FORCING FREQUENCY (3:)

Figure 4.5: a) Sketch of frequency-response curve for
nonlinear Mathieu's equation.

b) Experimental frequency-response curve
for cantilever beam. Level of force constant.

TI AVG 1
1.-

24

Y: '1. -

1

A70 2. -

 

 

(Volts

 

 

TI AVE 2
1.-

ll

 

 

.1

d

(Volts)
b) am.“

at

 

 

 

Figure 4.6:

I I

parametric response.

a) input

b) output

1‘. I

Time trace showing a steady-state

25

 

 

 

 

 

 

 

 

 

 

 

Yet! ‘10-.- e
L m 1 III. 04 0h 1 m
I.
I
mV ‘
( rms)
a) III ‘
A
‘I I r 1 ‘l T I I I
ll I! 1..-
Ye‘l 0'01.-
l. a 2 U. 03 lb 1 ma
1.-
(vrms) '
.l
b) mu ‘ n
t. I T I 1 I j I I I

 

 

ll I! 1..-
a) FFT of input in Figure 4.6

b) FFT of output in Figure 4.6

26

 

 

 

 

Alm0.o//////’//s'l./W/l’ , l-
.;>20NE/

a /, '\ / //_:’;,/'/

> 300.0-..;,_ ,://,}\

a -. . e

v J ;\:/

.2 3000-1 \.

5 4000-)

m ‘\\\3

O cl

‘3 200.0- ]

E .

Z

E 0.0-1 7 ‘v‘ : #r i L—fl—

-200. 0+

 

 

33.0 33.0 37'.0 33.0 39.0 90.0 91.0 92.0 93.0 94.0 95.0
FORCING FREQUENCY (112)

Figure 4.7: Experimental frequency-response curve
for cantilever beam. Level of force
constant but higher than Figure 4.5b.

.q. I
“a“

 

27

The chaotic motions observed for the model are associated with the non-planar
response of the cantilever. The particular type of non-planar response observed
has, to the best of the author’s knowledge, not been reported in the literature. To
appreciate the physics of this response, the reader is referred to Figure 4.1 in
which the static equilibrium position (s.e.p) of the cantilever is depicted. In this
position the stiffness associated with a rotation about the x-axis (torsion) and the
stifl'nese associated with the bending in the x-z plane are very high compared to
the bending stiffness in the x-y plane. However, when the beam deflects in the x-y
plane, these stiflnesses reduce drastically. An oscillation out of the x—y plane can
now occur but can only be sustained while the beam remains away from the s.e.p..
When this out of plane oscillation occurs, the beam gets locked-over on one side of
the s.e.p.. It is this out-of-plane motion that is at the heart of the chaos which has
been observed. For example, if the phasing of the out of plane motion is just
correct, then, as this motion passes through the x-y plane, it is possible for the
beam to snap through to the other side of the s.e.p.. Even if this snap through
does not occur, the resulting motion is highly sensitive to small perturbations,

especially as the beam approaches the s.e.p..

The various kinds of chaotic response observed in and about the "zones of
chaos" (see Figure 4.7) are classified as:
i) Transient Chaos,
ii) Intermittent and Steady-State Chaos,
The experimental results for these responses are presented and discussed in the fol-
lowing sub-sections.

4.3.1 Transient Chaos

The appearence of chaotic motions for a short period of time, before settling
down to steady-state trivial or non-trivial value is termed transient chaos. Such

transient chaos appears in the system when the response is quite large, e.g point C

28

on Figure 4.7. It is then possible to observe transient chaos, if the system is per—
turbed from its steady-state. The transient chaotic response never remains choatic,

but always returns to either the trivial or the non-trivial, single mode steady-state

value.

Transient chaos in the beam was observed when the system was perturbed
from its steady- state, in-plane, mode 4 motion by gently tapping the model with
one’s hand. The system was being forced at 87 Hz and level of force was 31.44
m/a’. Time traces of the forcing term and the response along with their associated
Fourier transforms are shown in Figure 4.8. Note that the Fourier transform of
the response shows a broad band power spectra which is typical of a chaotic
motion. The transient chaotic motion subsided and the beam returned to single

mode 4, in-plane, steady-state motion after approximately 250 cycles of forcing

term.

As in all nonlinear phenomena, it is important to allow a long enough period
of time to pass before pronouncing that the final form of the response has been
reached. A sufiicient time to classify chaos as transient or steady state is a matter
of judgement of the investigator, but we considered 30,000 cycles of forcing fre-
quency as a sufficient time for pronouncing the chaos as steady state.

4.3.2 Intermittent and Steady-State Chaos

Intermittent chaos is a burst of chaotic motion occuring between periods of
regular motion (see Moon [1988] pp 59,181), whereas steady-state chaos refers to
the fact that the response remains chaotic for all time even when perturbed more

than an infinitesimal amount.

Refering to Figure 4.7, if the frequency is decreased enough, an out of plane
mode is excited and a chaotic response results. The physical system’s response was
observed to cycle back and forth between chaotic and mode 4 motions indicating

a transition to chaotic motion via the intermittency route, as reported by Burton

29

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a: c.5m «c accosaouu mcqoaou a nob uncommon ouuowzo acoumcmub "w.c ousmam

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BBdBv a; Id- .Ilidlu “mm _-d
p - I11 L b 14¢ l? L ad - p p L p p
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T I
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30

and Kolowith [1988]. The intermittent chaos was observed for a downward sweep
of frequency at a constant level of force. It appeared that the time spent in
chaotic bursts increased as compared to time spent in periodic motion as the level

of force was increased.

Steady-state chaos in this system occurs if the frequency is decreased enough.
The system may achieve steady-state chaos with or without going through the
intermittency route. However, if intermittent chaos is present, an increase in the
level of force results in steady state chaos. An example of such a steady state
chaotic response is shown in Figure 4.9. The cantilever was being forced at 87.1
Hz and the level of the force corresponds to 40 m/a’. The system was responding
in a single, mode 4 motion prior to the onset of the chaos. The chaotic response
shown is very complex in the sense that although only a few modes are present,
they come and go in a chaotic manner. The Fourier transform of the response
presented in Fig. 4.9 suggests that the 2nd, 3rd, and 4th, in-plane (x-y plane)
vibration modes are involved in the chaotic motion. A similar observation was

made by Burton and Kolowith [1988].

An example of the system attaining steady-state chaos without going through
the intermittent chaotic stage was observed while the system was responding in
the third transverse in—plane mode. The cantilever was forced at a frequency of 45
Hz. The time traces shown in Figure 4.10 show the cantilever’s response for the
third mode in plane transverse vibration. When the frequency was decreased to 43
Hz while keeping the level of force constant, the cantilever jumped into chaotic
motion without going through intermittent chaos. The time trace along with its
Fourier transform of this response are shown in Figure 4.11. Clearly,the response
has developed into a very complex form. Even when perturbed a little it remains

in this state for all time.

1’! AVG!
z—

31

Yn-Z— AYIQ-
lb 1 ND

 

(Volts)

a)

TIAVGZ
:—

. 1‘ . 11', . 1' ’!"

 

Y: 4.- Ah 4.-
82 0h 1 sumo

 

(Volts)

b)

 

 

 

ll

Figure 4.9: Robust

' I
a: ---

- chaotic response for a
forcing frequency of 87.0 Hz.

a) input b) output

32

 

a) .

 

 

 

 

 

t. I 1 I I ‘I I I I I

ll IE 3.-

Yetl £701... -

(mvrms)

b) m

 

 

 

 

 

ll HZ ‘-
a) FFT of input in Figure 4.9

b) FFT of output in Figure 4.9

33

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Id

 

 

 

 

 

 

 

d

 

'1.”

 

 

e.- ‘3 1n.- .

 

mV
rms

(b)

 

 

 

 

'Figure 4.11: Chaotic'reSponserfor a forcing
frequency‘Of 43 Hz.

a) FFT of output
b) output

35

4.4 Multi-Mode Interaction

Internal resonance, i.e when the natural frequencies of a system are commen-
surable, can greatly enhance the coupling between modes. Haddow et al. [1984]
showed experimentally and theoretically the efiect of internal resonance on a two
degreeof-freedom system. Barr [1980] described the possibility of a "cascading" of
energy through the modes of a multi—degree-of-freedom system as a result of inter-
nal resonances. It is believed that such a "cascading" is occuring in this experimen-
tal model when a high frequency input gives rise to a low frequency, high ampli-
tude response. Experimental results showing the existence of this phenomenon are
presented and discussed in the sub-sections to follow.

4.4.1 Extremely Low Subharmonic Response

Although Dugundji and Mukhopadhyay [1973] reported on the theoretical
possibility of exciting a low frequency mode in an axially excited thin beam, no
experimental results have been reported in the literature . Such a response was

observed in our physical system.

Figure 4.12 shows a time trace of the input and the output for an initial time
close to zero. This is for a forcing frequency of 190 Hz. Figure 4.13 again shows the
input and output some 30 seconds later. Note that the scales have been changed
in order to better show the form of the response. It should be stressed that the
forcing function remains the same in Figures 4.12 and 4.13. Over the 30 seconds, a
high order sub-harmonic with a period about 40 times higher than the forcing
term has developed (see Figure 4.14 which shows Fourier transforms of Figure
4.13). Moreover, there has been an extremely large increase in the amptitude of
the response. This is a consequence of the energy input being transformed from a
small amplitude at a higher frequency to a large amplitude at a very low fre

quency.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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37

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l.-

 

 

 

 

 

 

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ll C -I n
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Volts x
33L ' N
(b) n
h
-1 _ 1 1 ‘
II 931: 5... I

Figure 4-13=Traces at time +30 secs. (3) Time trace of the
input (the variation in the amplitude is a
consequence of the digitizing).

(b) Time trace of the
output.

38

Miami:

 

 

(a)

 

 

A A A
t. T I I g 1 '

ll ' n

 

‘-

 

 

 

 

t' ‘¢ 4uaen

Figure 4.]4: FFT of time trace t+30 secs

a) input
b) output

39

4.4.2 Modal Interaction and Beating Phenomenon

Another class of interesting response has been observed. It results from a
combination resonance which in turn excites a regular parametric response. Figure
4.15 shows the waterfall plot of the frequency content of this interaction for an
initial time close to zero. The waterfall plot gives the variation of the frequency
content of a time signal as a function of time. The system was forced at a fre
quency of 385.60 Hz, which is close to 12th linear transverse natural frequency.
In this case, the response frequency was the same as the forcing frequency. Such a

response is consistent with a secondary parametric instability.

The forcing frequency acts as a parametric excitation term for the combina-
tion resonance case of 0, M1, + 0,2, where {2,— 44.8 Hz and 0,,- 337.5 Hz are the 4th
and 12th linear transverse natural frequencies respectively. We believe that the
nonlinearity adjusts the frequencies of the 4th and 12th mode so that the resonant
frequency combination is satisfied. This results in exciting the 4th and 12th linear
mode of the system, and we see the model responding at 45 Hz and 337.5 Hz,
simultaneously, after approximately 350 cycles of forcing frequency. See Figure
4.15. After 2200 cycles of the forcing frequency, another regular main parametric
response is generated at 22.5 Hz. This occurs because the response at 45 Hz acts as
a parametric excitation on the third linear mode (i.e 0,— 22.5 Hz). Thus the
response is now a combination of several modes and the model is responding

simultaneously at frequencies of 22.5 Hz, 45 Hz, 337.5 Hz and 385.6 Hz.

Figure 4.16 shows the waterfall plot of the response discussed above some 10
seconds later (i.e some 3,850 cycles of the forcing frequency). The forcing fr3~
quency is the same. Note that the third and fourth modes have not attained
steady-state. Instead, there is a continual exchange of energy between these
modes. This is an example of a beating phenomenon. Another response is gen-

erated at 409 Hz which is not accounted for. This response appears after response

40

 

 

 

 

 

 

 
 
 
  
 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.03 [24 n
/ 51.. 5713“” '1
time ‘1 / /
(secs) , ‘1‘ ,-
EiS‘V' :3“ A
‘ 1 :3.
5%: _
se- 1
~1“ .i‘
“:1 .
=5. ' A
t ’—
=.‘ . :
—:1‘ :,
=5: 3',
=3 4— ,-l' —
.h , ,1: :
HE ‘ -
{C23 1‘ 2
g I n :
Vr-mo E: A;
START: B H: STOP: SUB Hz
Figure 4.15: Waterfall plot showing output close to an initial
time zero for a forcing frequency of 385.6 Hz.
£3 .1. 22.15 Hz

J14 : 45'0 HZ.

$7.12 -: 337-5 H7.
3355 Hz

.51.; v

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:11 A

:1.

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STOP:

1‘;
:: _iL__aeA:
fl&:::?:;~
53B H:

BU: 4.7743 H:

 

Vrmo
START: 81H:
Figure 4.16: Waterfall plot showing output some t+10 secs
later for a forcing frequency of 385.6 Hz.

523; 22.5 Hz

524: 4510 H2
12,2 = 3375 Hz
.12; - 2385-6112.
51 409 Hz.

-
—
’

42

at 22.5 Hz is generated and the system starts exhibiting beating phenomenon.

Both the waterfall plots presented in Figures 4.15 and 4.16 span an interval
of time corresponding to approximately 3500 cycles and 2400 cycles of forcing
term respectively. The form of the response shown in Figure 4.17 remained

unchanged even after some 30,000 cycles of the forcing term.

The response presented above is a consequence of internal resonance. It is
important to note that a high frequency, low amplitude input, eventually leads
to a large amplitude response which is at a very low frequency.

4.4.3 Combination Resonance

An interesting observation regarding a combination resonance was made
while the physical system was in a chaotic regime born from a fourth mode oscil-
lation. If the system was preturbed in a suitable manner, the chaos disappeared
and the solution becomes periodic. The response was observed to be a combination
of several modes. A time trace depicting this motion is presented in Figure 4.17
along with its Fourier transform. The system was being forced at a frequency of
85.5 Hz with a base excitation of 40 m/a’. Figure 4.18 shows the response before

the combination resonance was attained.

It was interesting to note that this combination could not be attained by
starting from the trivial solution. Presumably, the domain of attraction of the
combination resonance was very small and, perhaps coincidently, the chaotic
motion passed closed to it. However, its response was relatively robust in the
sense that once the combination resonance had been attained, it was insensitive to

perturbations.

TI AVGZ

2.-

43

Y: '1I. -
file 31 Oh i

“I 2.. ”
EPA“)

 

Volts ‘

(a)

4.1.-

 

 

 

ll 8

 

(b)

 

 

 

ll

FIGURE 4- 17

II HZ

 

Combination of responses at a forcing frequency

of 85.5 Hz. (a) Time trace of the output.

(b) FFT of (a).

44

11-25.! 11.5.”
n ave 2 3:. 29 m 1 3mm
2.-

 

Volts

 

4".“

 

 

 

II it 1a.! a

 

 

 

 

 

 

3.3 12 2...:

Figure 4.18: Chaotic response for a forcing
frequency of 85.5 Hz.

a) timetrace of output
b) FFT of output

CHAPTER 5
DINIENSION ESTI'NIATE

5.1 Introduction

For continous systems the number of modes necessary for a full dynamical
description becomes large, and in theory sometimes infinite. From a practical
standpoint one must truncate this model to just a few mode shapes, i.e those
modes which contain the most information about the ’true’ system dynamies.
Most often modal truncations are based upon retaining only the most energetic

mode shapes.

Since chaotic systems produce power spectra with many peaks, or even of
continous form, it is often unclear how many modes should be retained. The

Hausdorfl’ dimension D (see Burge et.al [1984] pp 146), is defined as

D - lim 11131?- (5.1)

e-oo
1n—
6

where N(e) is the smallest number of hyperecubes necessary to cover the attractor
(see Guckenheimer and Holmes [1986] pp 256-257). This can be regarded as the
average number of linearly independent coordinates necessary to describe a dissi-
pative dynamical system. Thus the Hausdorff dimension gives the modeller a tool

by which to approximate the number of active modes in a system.

However, the Hausdorff dimension has the following drawbacks:
i) The computation is very time consuming since it converges slowly when the
dimension of the phase space is greater than two (see Burge et al. [1984] pp
149)
ii) It is a geometric measure. It does not account for the frequency with which

the orbit might visit the covering cube (see Moon [1987] pp 214).

45

46-

T herfore, we must use an alternative way of estimating the dimension of a
dynamical system. Some of these which can be used are (see Moon [1987] pp 214):
i) Pointwise dimension as discussed by Farmer et al. [1983] and defined as

maize-5m

d =- lim lim ‘ (5.2)
a‘uoo r—o logr

 

where n, is the total number of data points , and it consists of counting the

number of data points N;;(r) within a hyper-sphere of radius r, centered at a

point 2'; on the attractor.

ii) Correlation dimension C(r) as described by Grassberger and Procaccia

[1983b] and estimated by

C(r)=1im—2'— 20(r-IzT- ,~|) (5.3)

noon, j-l "J {-1

where n, is the number of reference points, 2'; are the reference points (vectors),
£7 are the rest of the points on the attractor, 11, is the number of data points,
and 0 equals 1 when its argument is positive, and 0 when its argument is nega-

tive.

In this work the dimension of the chaotic motion of the cantilever beam is
computed using the procedure and computer programs as described by Klewicki et
al. [1988]. Pointwise dimension is used to estimate the dimension of the system,
since Holzfuss and Meyer-Kress [1985] showed it to be more accurate than the
correlation dimension estimate used by Grassberger and Procaccia [1983b]. Before

discussing this, the concept of coordinate reconstruction will be introduced.

Often, one cannot measure all of the variables (coordinates) necessary to
describe (embed) the phase trajectories of a given system. It is for this reason
that techniques of phase portrait reconstruction from a single measured variable

have been developed. The basic idea behind phase portrait reconstruction is that

47

any N-dimensional system can be described, at any instant, by the measurement
of N independent variables pertinent to that system. Implicit in the following
method is that, at least indirectly, any measured variable (assuming the resolution
of the probe is adequate) contains all of the information about the dynamies as
prescribed by the equations of motion. Thus the method involves constructing

independent coordinates from a single time series.

There are two known methods by which independent coordinates may be
constructed from a single time series. These methods were developed concurrently
in a heuristic manner by Packard et al. [1980], and formally by Takens [1981].
Takens proved that for a N-dimensional dynamical system, one may be assured to

embed its attractor with at most 2N+l reconstructed coordinates of the form;
f;(1)=[X(1),X(1-r), ...... ,X(t—n 1)]
or

d"‘[X]

f;(1)-[X('¢),X(1), ......... , am]

where X(t) is the experimentally measured time series, and r is a time delay. That
is the reconstructed coordinates may be created either by time delaying or
differentiating the measured variable. Due to constraints such as the order of the
system, and problems with higher order differentiation of the experimental data,
the time delay method is usually preferred. In this work, the time delay method

is used to compute the dimension estimate.

The computation of a dimension estimate using measurement from only one
location on the cantilever beam involves four basic steps, i.e
i) acquisition of data,
ii) estimation of the optimal time delay 1,

iii) phase coordinate reconstruction using the optimal time delay,

48

and

iv) computation of the dimesion estimate.

These four steps, as they pertain to the computer programs used, are out-
lined in the flow chart of Figure 5.1. For more details the reader is referred to

Klewicki et al. [1988].

5.2 Acquisition of Data

The data used for dimensional analysis was collected from a chaotic motion
arising from the cantilever’s third mode motion. The base excitation frequency

was 45.0 Hz, and a sampling rate of 1000.0 Hz was used. Record lengths of

40,000 samples were used.

5.3 Estimation of Optimal Time Delay

The process of choosing the time delay used in the phase coordinate recon-
struction procedure is subjective (see Klewicki et a1. [1988]). Figure 5.2 presents
the unnormalized autocorrelation of the data collected from the cantilever beam
for various time delays. The curve exhibits a sharp decrease in the first 25 ele-
ments (0.025 seconds), then there is a sharp increase for the next 25 elements
(intermediate points have been calculated, but not plotted, to check that this
trend is correct). The first zero-crossing is at approximately 150 elements. Con-
trary to the use of the first zero crossing as the criteria for the optimal time delay,
(see Klewicki et al. [1988]), the optimal time delay is chosen to be 0.025 seconds.
This choice of time delay corresponds to the fact that data has almost lost all
information about its past, as evident by the sharp decrease of the autocorrelation

function. Notice that the value of Optimal time delay chosen is very small.

 

[mmle .-

49

 

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Figure 5.1: Flow chart for the estimation of
dimension

 

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51

 

 

 

 

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.1
0.44
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0.0.1
.1
'0" 1 r r I F l T l I I r f r 1 F
0.0 100.0 200.0 300.0

TIME DELAY i.e Number of Data Points in 0 Column

Figure 5.2 The unnormalized autocorrelation
functlon for venous time delays.

 

3.0

 

52

5.4 Reconstructed Coordinates and Dimension Estimate

The following variables will be used in this section:
N = number of linearly independent coordinates necessary for a full dynamical
description of a system,
J = pointwise dimension,
M = integer part of the pointwise dimension,

m = number of reconstructed coordinates used.

To determine the minimum N we calculate a series of pointwise dimensions,
11,, using an increasing number of reconstructed coordinates, m-1,2,3 ...... As m
increases, the pointwise dimension, reaches an asymptote, say d 3M +14, where u<1.
Then the minimum number of linearly indepedent coordinates necessary for a full

dynamical description for this system is N-M+1 (see Moon [1987] pp 232).

As discussed in the Section 5.3, we have chosen a time delay of 0.025 seconds
to reconstruct the coordinates. Figures 5.3a and 5.3b show projections of the
reconstructed attractor. Notice that the attractor is spread out over the recon-

structed space, and that it is being stretched in one direction and folded in the

other direction.

Recalling the definition of the Pointwise Dimension from equation (5.2) we
can see that there will be some practical difficulties in its evaluation. Firstly, the
number of data points, 11,, will have to be finite, and secondly, the limit r—oO will

have some lower cut-off point. Now, consider equation (5.2) in the form:

J =- lim lim(D)

y-ooor-co
where

pgkdfl

103(r)

 

scs (t—TAU) (Volta)

53

 

NPOlNTS - 4000 SAMPLING FREQUENCY - 1000.0 HZ
TAU - 0.025 SECS FORClNG FREQUENCY - 45.0 HZ

  

 
 

 
  

- ._ n'.‘ :‘-
2,, 4

..-"
:7
.1" _ ‘

 

 

 

1 T l r l ' 1 ' 1 r
—3.0 —2.0 -1.0 0.0 1.0 2.0 3.0 4.0
Strain Gage Signal SGS (t) (Volts)

Figure 5.30: Phase portrait of measured time
signal SGS (t) from cantilever_ beam usnng the
reconstructed coordinates.

SGS (t—2TAU) (Volt?)

4.0-—

3.0 -—

 

—4.0

54

NPOINTS - 4000 SAMPUNG FREOUWW - 1000.0 HZ
TAU = 0.025 SECS FORCING FREQUENCY - 45.0 HZ

 

l F ' I ‘ l r i
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
Strain Gage Signol SGS (t) (Volts)

Figure 5.3b: Phase portrait of measured time
Signal SGS (t) from cantilever. beam usmg the
reconstructed coordinates.

 

4.0

55

and where

__ N41)

"4

 

P

The first difficulty can be alleviated by taking n, as large as possible (in our case
40,000 data points) and by calculating P at various reference points, 2‘0, over the
attractor. The various values of P are then averaged and a plot of log (P) vs log
(r) constructed. Examples of such a plot are presented in Figure 5.4a for m=3,9
and 15. Referring to this figure, we can clearly see there exists a range of r
(represented by solid line) over which the slope of the curve (i.e. d) is non zero.
This region is often called the scaling region. The upper bound of this scaling
region occurs as r approaches the size of the attractor, and the lower limit occurs
when r becomes so small that there are not enough data points within the hyper-
sphere to give a meaningful result. Also note that the scaling region shifts to
higher values of radii as the embedding dimension increases. This is due to the
fact that the attractor stretches as the phase space dimension increase and there-

fore the number of data points lying within a set radius decreases.

Figure 5.4b shows the estimated dimension for each embedding space in the
range of m = 1 through 15. It is evident that the dimension 4 reaches an asymp-
tote of approximately 4:6.63 when mzls. This is in agreement with Taken’s

theorem [1981] previously mentioned in Section 5.1.

Hence
d = 6.63
M = 6
N 2 M+l

Therefore N 2 7

Log P

1.0

56

 

0.0—(

-1.0—

_2.0 .—

l

—3.04

.4

4.04

d

-5.0-J

1]]
mars-moonaar-sousrP-soo //,/ /’
TAU - 0.025 secs FORGING FREQUENCY- 45.9 H2, /

 

 

-l .5

/ u/ugg. ‘
fl
//° / 1/.
/" /‘/
/'l //’
D O,
,/ /
/ /
I
/
I /
o/ 0/
/.°/
'/
. °/ ,/
/ . . O / I. eta-15
/ , em-O
// [I / Illa-3
fi—WTr TIFWITIIIIIrffIIIFfIIIIf
-1.0 -O.5 0.0 0.5 1.0
LOG(r)

Fi are 5.40: lo P lotted o oinst lo r for
g g p m=3,ggond 15g U

 

1.5

pointwise dimension d

7.0—r

6.0—

5.0—4

4.0-4

 

57

Figure 5.4- b The embedding .dimension m plotted

ogomst the pomtwise dimensuon of the attractor.

 

58

5.5 Conclusions

The minimum number of linearly independent coordinates required to com-
pletely describe the dynamics of the cantilever beam, while in this chaotic regime,
is 7. Hence it can be modelled mathematically by three orthogonal linear modes
which account for 6 of the 7 linearly independent coordinates. The dynamical sys-
tem equations contain a parametric excitation term which accounts for the 7th

coordinate, since time acts as an additional coordinate.

The motion of the beam is chaotic as the pointwise dimension is non-integer.

CHAPTER 6
CONCLUSIONS

The behaviour of a flexible cantilever beam excited by a sinusoidal base exci-
tation in the axial direction has been studied. The governing partial integro-
differential equation has been derived assuming the motion to be planar. The
equation includes upto third order nonlinear terms arising from large curvature

and the associated axial deflections.

The main thrust of the research was to explore, through physical experimen-
tation, the various types of resonant behaviour that the cantilever exhibits. There
were found to be two main classifications, viz.

1) Planar Motions,

and

2) Non Planar Motions.

Under classication 1 were observed multi-valued steady-state responses, jump
phenomenon, main parametric resonances, multi-mode interactions, almost
periodic solutions and finally, very high order sub-harmonic responses. Although a
number of these phenomena have received attention in the past, very little work
has been reported in the literature on the physical evidence of multi-mode interac-
tion and very high order sub-harmonic responses. These are particularly interest-
ing in that they result in modes being excited at frequencies very far removed
from the excitation frequency. The energy put into the system "cascades" down to
a lower frequency and results in very large response amplitudes even although the
input amplitude is extremely small. If not allowed for at the design stage, this

could be potentially a very dangerous situation.

It is recommended that this "cascading" behaviour be studied in more detail.

The partial integro-difi'erential equation obtained in Chapter 2 could be a starting

59

 

 

60

point for an analytical study of the phenomenon.

Under classification 2 were observed transient chaos, intermittent chaos, and
steady-state chaos. The dimension of the steady—state chaos was estimated using
Pointwise dimension, and it suggests that the physically observed chaotic response

can be modelled mathematically by three orthogonal modes.

 

 

 

APPENDICES

APPENDIX A

 

X- axis

|

I /

l

(was, 1321—. / /

J1:- /(S+ (44- (H u’)J5) 17+ V3,?)
to. §*

l ‘<

| ”(n-u 2r)

' //

l

 

goat's
Figure A.1: An element of cantilever beam

Consider an small undeformed element of beam of length «15 as shown in
above Figure. Referring to the above figure, the deformed length dc of beam ele-

ment is given by
d: = [(1 + 11 ')2+ 0 ”1V2 d5

Now the extensional strain is defined as

 

_ change in length
((5,!) length

=> 6(S,t)=([(l+u ’)2+v '2]‘/2dS-d3)/ 45

-> c(S,t)=[ (1+, ')2+v 1211/2 _ l

61

 

APPENDIX B

We know from equation (2.6) that

tan0=

 

l-l-n '

-19,

1+: ’

 

0=tan

Taking partial derivatives with respect to S, we get

0 Il(l+u I)_v I" H
(1+u')’+v"

 

0'-

Substituting 3 ’ from equation (2.8) into the above relation and dropping higher

order terms, we get

91:“ Il(l+.;_v 02)]

62

 

 

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