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Chaotic behavior of bouncing systems

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Chaotic behavior of bouncing systems
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Lee, Chi-Wook, 1957-
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vii, 101 leaves : ill. ; 29 cm.

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Chaos ( jstor )
Damping ( jstor )
Datasets ( jstor )
Degrees of freedom ( jstor )
Locomotion ( jstor )
Natural frequencies ( jstor )
Phase plane ( jstor )
Spectroscopy ( jstor )
Standard deviation ( jstor )
Stiffness ( jstor )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1991.
Bibliography:
Includes bibliographical references (leaves 97-100).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chi-Wook Lee.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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CHAOTIC BEHAVIOR OF BOUNCING SYSTEMS


BY


CHI-WOOK LEE
















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1991






























To My Mother, Won-Ae Ro, who passed away in 1981














ACKNOWLEDGEMENTS


The author wishes to thank his committee chairman,

Dr.Joseph Duffy, for his invaluable guidance, support, and

encouragement throughout this work. The author is truly

indebted to his committee cochairman, Dr. Ali Seireg, for his

precious guidance and support.

Special thanks should be extended to the members of his

graduate committee, Dr. Carl Crane, Dr. Gary Matthew, and Dr.

Keith Doty, for the support and the comments each provided in

preparing this dissertation. Many thanks go to fellow

students in CIMAR (Center for Intelligent Machines and

Robotics) for their friendship.

Finally, the author sincerely thanks his father, Nam-Soo

Lee, and his wife and son, Woo-Sun and Nicholas, for their

patience and support.


iii















TABLE OF CONTENTS


ACKNOWLEDGEMENT .... .. ........................ ... ...... ..

ABSTRACT ....................... .............. ........ .

CHAPTERS

1. INTRODUCTION ...........................................

1.1 Problem Statement and Previous Work ...............

1.2 Chaotic Systems ....................................

1.3 The Goal and Organization of the Work ...........

2. SIMPLIFIED BOUNCING SYSTEMS ..........................

2.1 Background .........................................

2.2 One Degree of Freedom Spring-Mass System ..........

2.3 Phase Planes of One Degree of Freedom Bouncing
Systems ........................... ................ ..

2.4 Two Degree of Freedom Spring-Mass Systems .........

2.5 Phase Planes of Two Degree of Freedom Bouncing
Systems ............................................

3. CHAOS IN TWO DEGREE OF FREEDOM BOUNCING SYSTEMS .......


iii

vi


3.1 Background .........................................

3.2 Sum of Standard Deviations and Area in Phase
Plane Plot for the Specified Region ...............

3.3 Spectrum Analyses for the Chaotic Bouncing
Systems with h = 0.2m .............................
3.3.1 Case of 570N/m as Foot Stiffness (K2) ........
3.3.2 Case of 3670N/m as Foot Stiffness (K2) .......

3.4 Spectrum Analyses for the Chaotic Bouncing
Systems with h = 0.5m .............................
3.4.1 Case of 445N/m as Foot Stiffness (K2) ........
3.4.2 Case of 1860N/m as Foot Stiffness (K2) .......










3.5 Spectrum Analyses for the Chaotic Bouncing
Systems with h = 1.0m .............. .......... ..... 60
3.5.1 Case of 2010N/m as Foot Stiffness (K2) ....... 60
3.5.2 Case of 2745N/m as Foot Stiffness (K2) ....... 66

3.6 Summary ............................................ .72

4. ELIMINATION OF CHAOS IN TWO DEGREE OF FREEDOM
SYSTEMS .................................................. 81

4.1 Background .................. ....... ......... .... ... 81

4.2 Spring Selections ................................... 81

4.3 Use of Damping Elements ............................ 84

5. CONCLUSIONS AND RECOMMENDATIONS ........................ 90

APPENDICES

A. EFFECTS OF TIME STEP SIZE .......................... 92

B. PERIODS FROM PHASE PLANE ............................ 94

REFERENCES .................. ............... .. ............. 97

BIOGRAPHICAL SKETCH .................................... 101















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


CHAOTIC BEHAVIOR OF BOUNCING SYSTEMS

By

CHI-WOOK LEE

August, 1991




Chairman: Dr. Joseph Duffy
Cochairman: Dr. Ali Seireg
Major Department: Mechanical Engineering


This study investigates the behavior of simple bouncing

systems, namely simple oscillators which are released from a

certain height. In particular, a nonlinearity exists in the

discontinuity of two different dynamic system modes, which

are the flight mode and the ground mode, although the

behavior of such a system is linear in each mode. Such

oscillators provide models for mechanical systems such as

legged systems for hopping, in which components make

intermittent contact.

The phase plane technique and the power spectrum

analysis, which provide simple yet powerful tools of the

dynamic analysis for linear and nonlinear systems, permit not

only the determination of parameters for the stability of

bilinear bouncing systems, but also the investigation of the










chaos that may occur. It is necessary to examine what causes

the chaotic behavior of the simple bouncing systems in order

to design a practical leg system.

From the analyses of the linear single degree of freedom

and the linear two degree of freedom oscillatory systems,

some of the chaotic responses at the critical frequencies for

the bilinear two degree of freedom bouncing systems can be

predicted.


vii















CHAPTER 1
INTRODUCTION


1.1 Problem Statement and Previous Work


The motivation for this study came originally from the

area of legged locomotion. Legged locomotion can be attained

by a motion that combines a vertical bouncing of the body

with a fore-aft swinging of the leg. Legs are the elements

that exert forces on the body to propel the body forward for

transport, and to keep the body in an upright posture.

Spring and damping elements in the leg systems can reduce

maximum loading and simplify control.

In this research, leg systems for hopping are

considered because of the interest in emulating human or

animal locomotion as a biped or a quadruped. Central to the

design of a legged machine is the mechanical design of the

leg itself. Legged systems should be able to generate

bouncing motions of the body and swinging motions of the leg

for transport. In particular, repeated bouncing motions as a

part of legged locomotions are investigated to design a leg.

Chaotic behaviors of the simplified bouncing system are found

to exist depending on the system parameters and initial

conditions.










There has been much research on the subject of legged

locomotion. Those may be divided into two major fields:

(dynamic) modelling and control of a legged locomotion, and

construction of experimental legged vehicles.

The present major obstacle for development of highly

mobile and practical legged vehicles stems from the lack of a

solid mathematical description of legged locomotion. As a

result, a deficiency exists in mastering the associated

control problems. Lagrangian dynamics and modern control

theory have been applied to the study of legged locomotion,

especially biped [1-12]. Although these techniques have been

successful in providing an understanding of the unstable

dynamics associated with bipedal postural stance and optimal

trajectories for bipedal systems as coupled rigid body

systems, the models which were used for these studies are far

from realistic for human or animal locomotion systems. For

any given motion, certain degrees of freedom are dominant

while others can be neglected. Hence, to study the dynamics

of legged locomotion systems, it becomes imperative to select

mechanical models having few degrees of freedom to keep the

equations of motion to a manageable level, yet having enough

degrees of freedom to represent the motions involved.

Compound inverted pendulums with no more than three

degrees of freedom have been employed to study the dynamics

of postural stance as mechanical models of a biped. Hemami

et al. [3-9] examined periodic motion generation, trajectory

stabilization, and trajectory transition controls for










inverted pendulums. Since controls for practical bipeds

should be capable of producing stable locomotion gaits, and

of providing transitions from one periodic motion to another,

the results for the inverted pendulums may be useful for

further development of robot locomotion controls.

McMahon and Mochon [11, 12] described the swing phase of

human gait as a ballistic motion of a pure pendulum. They

assume that the muscles act only to establish an initial

configuration and velocity of links at the beginning of the

swing phase. The swinging leg and the rest of the body then

moves through the remainder of the swing phase entirely under

the action of gravity. The computed range of times of swing

for the model was found to be very close to experimental

results.

However, neither the inverted pendulum nor the pure

pendulum alone can be used to model complete cycles of

walking or running. While one leg supports the body like an

inverted pendulum, another leg is in the swing phase at the

same time in bipedal locomotion. Thus, both the inverted

pendulum and pure pendulum should be combined to achieve the

bipedal locomotion cycles.

Seireg and Townsend [1] presented a decomposition scheme

which separates the dynamics and the kinematics, and a

numerical algorithm which provides time optimal control

functions for the nonlinear dynamic problems of systems of

coupled rigid bodies with application to a bipedal locomotion

system. The time-optimal control functions and trajectories










may be used for the synthesis of closed-loop controls for the

system [2]. Although the methods demonstrated the existence

of a solution to given systems of coupled rigid bodies, the

uniqueness cannot be guaranteed.

A historical review of research on legged machines can

be found in Raibert [13] and Todd [14]. However, some of the

outstanding works will be described here briefly. In the

late 1960's and the 1970's some work on exoskeletons was done

by Grundman and Seireg [15]. A series of hydraulic

exoskeletons for paraplegics was built, which were programmed

for standing up, sitting down, and stair climbing. A fully

computer controlled quadruped walking machine was built by

Frank and McGhee at the University of Southern California

[14, 16]. Each leg possessed two degrees freedom, and the

joint coordination was performed by a computer. In the

1970's, Ohio State University (OSU) started a series of

investigations on legged locomotion. The OSU Hexapod was

built by McGhee and his associates [14]. This machine was

fully controlled by a PDP 11/70 computer via an umbilical

cord and was powered externally through a cable. Each leg

had three degrees of freedom and was made of two links

connected by joints. The group at OSU is currently

developing a much larger hexapod (Adaptive Suspension

Vehicle). The Adaptive Suspension Vehicle is intended to

operate on rough terrain with a high degree of autonomy [17].

Other experimental walking machines include TITAN III, PV II,

and ODEX I [14]. Even though these research efforts for






5



statically stable multi-legged vehicles have generated good

results, few of them contribute to dynamically stable legged

locomotion.

Miura and Shimoyama built the BIPER series [18]. The

BIPER3 has knee joints but no ankle joint assuming a point

contact between the foot and the ground. A continuous

walking gait is required to prevent the BIPER3 from falling.

The BIPER4 has a shape similar to human legs. Both are

designed to walk following a preplanned trajectory.

Frusho and Masubuchi developed a hierarchical control

structure [19]. In the lower level control, a local feedback

at each joint is used, while in the upper level the reference

inputs to the local feedback are corrected by using a reduced

order model.

Zheng and Shen proposed a control scheme, using position

and force sensors [20], which enables a biped to walk from a

level surface to a slope. For a statically stable biped

locomotion, the projection of the center of mass remains

within the supporting area by moving the main body of the

robot back and forth. The experimental biped robot, which

has nine links and eight joints without knee joints, was

built to verify the proposed control scheme.

For legged locomotions, some means for balance must be

provided, since the body's center of mass is above the

ground. The results of biped research have been limited to

only slow walking gaits so that the dynamic effects of the

system can be neglected and the static balance for the system










can be maintained. However, because of the need to maintain

stability at all times, statically balanced locomotion is

limited in speed and maneuverability. Recently, dynamic

balance has been used to overcome these limitations. Dynamic

balance requires attention not only to position and forces,

but also to temporal aspects of limb control. For example,

if a biped does not put its foot down in the right place at

the right time, it falls down.

The first machines that balance actively were

automatically controlled inverted pendulums. Cannon and

Higdon [21] used a cart, on which one or more simple inverted

pendulums were mounted, as the physical model. The pendulums

were to be balanced by a controller which moves the cart back

and forth. Using analysis based on bang-bang control logic,

the regions of controllability were expressed as explicit

functions of the physical parameters of the system. This

study of balance for inverted pendulums was an important

precursor to later work on locomotion. The inverted pendulum

model for walking became the primary tool for studying

balance in legged systems as discussed before. In 1983, a

hopping machine was built by Raibert [13] at Carnegie-Mellon

University. This hopping machine has only one leg, and must

hop continuously to maintain balance. The leg has three

degrees of freedom. The vertical motion is provided by a

pneumatic cylinder which is mounted on the body frame via a

gimbaled joint. Two hydraulic actuators control the lateral

motion of the leg. This important piece of research has










provided considerable insight into the balancing problems of

walking machines, and has been extended to a four legged

bouncing machine. A variety of control procedures were used

for a steady state motion which repeats itself after each

hop. In fact, most research on legged locomotion

concentrated mainly on controls for stability.

Numerous studies of legged locomotion deal with regular

periodic running or walking. However, in order to design a

leg system for a practical legged vehicle, the specific

nonlinear dynamics should be analyzed, since nonlinearities

could produce nonperiodic or chaotic motions [22, 23].

Nonlinearity exists between the flight mode and the ground

mode of the system when bouncing motion is considered as a

part of legged locomotions.


1.2 Chaotic Systems


Recently, new phenomena have been observed in all areas

of nonlinear dynamics. Chaotic oscillations, which seem to

be random, are motions from completely deterministic systems.

Such motions had been known in fluid mechanics, but they have

been explored in low-order mechanical and electrical systems

and even in simple one degree of freedom systems. A certain

level of mathematical understanding has to be acquired in

order to study chaotic phenomena.

The study of chaotic vibrations is important to

engineering vibrations for several reasons. First, in










mechanical systems, a chaotic system makes life prediction or

fatigue analysis difficult because the precise history of the

system is not known. Second, the recognition that simple

nonlinearities can lead to chaos raises the question of

predictability in classical physics. For nonlinear systems

with chaotic dynamics, the time history is sensitive to

initial conditions and definite knowledge of the future may

not be possible even when the motion is periodic.

Thompson et al. [32] defined chaos in the negative as

recurrent behavior that is not an equilibrium, a cycle or

even a quasi-periodic motion. Chaotic motion has some random

aspects. The randomness arises from sensitive dependence on

initial conditions, resulting for example in broadband noise

in the power spectra of observable time histories. This

seems remarkable because the dynamic systems need no

stochastic input to achieve this. Even more surprising is

that chaotic motions can be observed in quite simple dynamic

systems [32]. It is necessary to distinguish between random

and chaotic motions. The former one is for problems where

the input forces are totally unknown or only some statistical

measures of the parameters are given. The term chaotic is

for those deterministic problems for which there are no

random or unpredictable inputs or parameters. In fact,

Thompson et al. [32] replaced the negative definition of

chaos by a more positive one: chaos is recurrent motion with

some random aspect in nonlinear dynamic systems. Exponential










divergence from adjacent starts while remaining in a bounded

region of phase space is a signature of chaotic motion [32].

Chaotic vibrations may occur when nonlinearity exists.

Examples of nonlinearities in mechanical systems include

nonlinear elastic or spring elements, nonlinear damping, and

backlash. To identify nonperiodic or chaotic motions, the

following tests can be performed:

a) Identify nonlinear elements in the system.

b) Check for sources of random input in the system.

c) Observe time history.

d) Look at phase plane trajectories.

e) Examine Fourier spectrum of signal.

When the motion is periodic, the phase plane trajectory

is a closed curve. For example, the phase plane trajectory

of a linear spring-mass system exhibits an ellipse. However,

a nonlinear system may show an orbit that crosses itself but

is still closed. This can represent a subharmonic

oscillation. Systems for which the force does not depend

explicitly on time are called autonomous. For autonomous

nonlinear systems without harmonic inputs, periodic motions

are referred to as limit cycles and are closed in the phase

plane. Chaotic motions exhibit different behavior. Their

phase plane trajectories are never closed or repeated. Thus,

the trajectory will tend to fill up a section of the phase

space. This is a strong indication of chaos.

One of the methods for detecting chaotic vibrations is

the presence of a broad spectrum of frequencies. This










characteristic of chaos is very important for the low

dimensional system. Often, if there is an initial dominant

frequency, a precursor to chaos is the appearance of

subharmonics in the frequency spectrum. One must be

cautioned against concluding that multiharmonic outputs imply

chaotic vibrations since the system in question might have

many hidden degrees of freedom.

For chaotic vibrations, one or more of the system

parameters must be varied to see if the system has steady or

periodic behavior for some range of the parameter. In this

way, it can be decided if the system is in fact chaotic and

if there are no hidden inputs or sources of truly random

noise. In changing a parameter, a pattern of periodic

responses are searched for. One characteristic clue to

chaotic motion is the existence of subharmonic periodic

vibrations.

Although chaotic phenomena have been observed in many

physical systems, chaotic systems which are closely related

to bouncing motions will be listed here.

Impact-type problems have emerged as an obvious class of

mechanical examples of chaos. Lichtenberg and Lieberman [24]

used a model for the motion of a particle between two walls,

where one wall is stationary and the other is oscillatory.

Numerical studies of this model reveal that stochastic-type

solutions exist so that most of the phase space is filled up.

This model is similar to a bilinear oscillator whose mass

slides freely on a shaft with viscous damping until it hits










stiff springs on either side [25, 26]. Another impact-type

mathematical model is a bouncing ball on a table which is

investigated by Holmes [27]. Experiments on the chaotic

bouncing ball have been performed by Tufillaro and Albano

[28].

A practical realization of impact-induced chaotic

vibrations is the impact print hammer experiment by Tung and

Shaw [29]. In this printing device, a hammer head is

accelerated by magnetic force and the kinetic energy is

absorbed in pushing ink from a ribbon onto paper. When the

print hammer is excited by a periodic voltage, it will

respond periodically as long as the frequency is low.

However, as the frequency is increased, the hammer has little

time to damp or settle out and the impact history becomes

chaotic. Thus, chaotic vibrations restrict the speed at

which the printer can work.

Compliant offshore structures and articulated mooring

towers which have been modelled by a bilinear oscillator have

been studied by Thompson et al. [30-32]. A bilinear

oscillator with different stiffness for positive and negative

deflections arises frequently in off-shore marine technology

due to the slackening of mooring lines. When one of the

stiffness becomes infinite, the system becomes an impact

oscillator. Harmonic, subharmonics, and chaotic motions were

found to exist for this model.

Numerical simulations have been carried out to study

chaotic phenomena. The time history of chaotic systems is










sensitive to slight changes in initial conditions and system

parameters. This sensitive dependence of chaotic systems

also raises questions about the accuracy of predicting time

response by numerical simulations. Toungue [33] and Koh et

al. [34, 35] discussed the effects of time step size on the

numerical solutions of chaos.

In view of the great variety of behavior observed for

nonlinear systems, it would be impossible at this time to

make sweeping generalizations about routes to chaos. The

question of which types of response may be found, and under

what conditions they are stable, especially in relation to

the choice of initial conditions, is extremely difficult, and

it cannot be answered with full rigor.


1.3 The Goal and Organization of the Work


The goal of this research is to understand chaotic

bouncing motions in order to design a practical leg system

for bouncing. Since bouncing systems may exhibit chaos, it

is desirable to predict system behavior with selected system

parameters.

In the following chapter, the phase planes of various

simplified bouncing systems will be displayed. Chaotic

bouncing motions are found to exist for two degree of freedom

bouncing systems depending on system parameters and initial

conditions. In Chapter 3, spectrum analyses for chaotic

bouncing systems will be described in detail. A separation






13



method is used to obtain dominant frequencies for each

dynamic mode of the system. Frequency relationships are

found after examining the results of spectrum analyses. They

can be used to predict the system behavior with selected

system parameters and given initial conditions. Chapter 4

explains elimination methods for chaotic bouncing motions.















CHAPTER 2
SIMPLIFIED BOUNCING SYSTEMS


2.1 Background


Running is a motion that combines a vertical bouncing

motion with a swing motion of the leg. Legged systems for

running should be able to generate a lift and a thrust by the

leg simultaneously. In his pioneering work, Raibert has

examined theoretical and experimental models of one-legged

and multi-legged hopping robots [13]. One of the major

functions of his hopping machines is the bouncing motion by

the telescopic pneumatic leg system. Since point contact

between the tip of the leg and the ground was assumed, i.e.,

without consideration of a foot, the one-legged hopping robot

is a simple one degree of spring-mass bouncing system [22,

23]. The mass represents the body while the spring itself is

a springy leg. However, for a smooth motion, a foot with an

ankle joint would be required. A legged system with a foot

(without an ankle) for the bouncing motions can be modelled

as a two degree of freedom spring-mass system. One mass is

for the body, and another mass is for the foot. A leg is

modelled as a spring between the two masses, and the

compliance of the foot is modelled by a second spring. It is

assumed that there is no energy dissipation for the









simplified models.


2.2 One Degree of Freedom Spring-Mass System







M




K





r / / / / / /1/ / //

Figure 2.1 One Degree of Freedom Bouncing System

Consider the one degree of freedom spring-mass system

which is dropped from a certain height (h) and bounces back

(Figure 2.1). This represents the simplest model for the

bouncing motion of a legged system. The mass (M) and the

spring (K) denote the body and the leg, respectively. When

the system hits the ground, the potential energy of the

system is converted to the elastic strain energy in the

compressed spring. This stored energy will be used for the

rebound when the spring loses contact with the ground.

This model has two different modes. One is a linear

oscillatory mode (Ground Mode) when the spring maintains

contact with the ground, and the other is a free fall mode in

a gravitational field (Flight Mode) when the spring is not in










contact with the ground. A nonlinearity occurs due to the

discontinuity of two different dynamic system modes, although

the behavior of such a system is linear in each mode. The

dynamic equations of the system for each mode are:

(Ground Mode) Mx + Kx = -Mg (2.1)

(Flight Mode) Mx = -Mg (2.2)

where M is the mass, K is the stiffness, and g is the

gravity.

The displacement x is measured from the position of the

body M at the instant when the spring first reaches the

surface. The position x is negative if the body is below

this reference point. Thus, if x is greater than zero, the

system is in the flight mode while the system is in the

ground mode with x less than or equal to zero.


2.3 Phase Planes of One Degree of Freedom Bouncing Systems


The phase planes of one degree of freedom spring-mass

bouncing systems are all obtained for a fixed body mass of M

(= 10 Kg). Since the displacement x is measured from the

position of the body M at the instant when the spring first

touches the ground, x = 0 and v = 2gh at t = 0 are used as

initial conditions for the bouncing motions.

Six different cases were selected to simulate bouncing

motions of one degree of freedom bouncing systems by Runge-

Kutta method with 0.001 second as a time step.

Cases of Dropping Height (h) = 0.2 m










When the dropping height is 0.2 m, i.e., the system is

raised 0.2 m and is dropped from that height, K = 575 N/m

(Figure 2.2.a) and K = 1900 N/m (Figure 2.2.b) are used as

examples of one degree of freedom systems.

Cases of Dropping Height (h) = 0.5 m

With h = 0.5 m, K = 1860 N/m (Figure 2.3.a) and K = 3080

N/m (Figure 2.3.b) are applied as system parameters.

Cases of Dropping Height (h) = 1.0 m

With h = 1.0 m, K = 430 N/m (Figure 2.4.a) and K = 2010

N/m (Figure 2.4.b) are assigned to examine bouncing motions.

For all cases of single degree freedom systems, stable

bouncing motions were demonstrated with phase plane plots.

In the upper half of the phase plane which is the flight

mode, displacement has a parabolic relationship with

velocity. For the ground mode, displacement and velocity

display the shape of an ellipse. If the velocity divided by

the natural frequency of the system in the ground mode is

used for the horizontal axis of the phase plane, then the

phase plane trajectory for the ground mode is a circle

instead of an ellipse.
















0.2



0.1



X 0.0



-0.1



-0.2


-2 -1 0 1 2
v


-2 -1 0 1 2


(a) (b)

Figure 2.2 One Degree of Freedom Bouncing
with h = 0.2 m, M = 10 Kg
(a) K = 575 N/m, (b) K = 1990 N/m




0.5-

0.4-

0.3-

0.2-
x
0.1-

0.0-

-0.1

-0.2r

-2 0 2 -2 0
V v

(a) (b)

Figure 2.3 One Degree of Freedom Bouncing
with h = 0.5 m and M = 10 Kg
(a) K = 1860 N/m, (b) K = 3080 N/m


0.2

0.1

0.0

-0.1
X
-0.2

-0.3

-0.4


0.0



-0.2












1.0



0.5



S0.0



-0.5


-4 -2 0 2 4
v


1.0

0.8

0.6

0.4

0.2

0.0

-0.2


-4 -2 0 2 4
v


Figure 2.4 One Degree of
with h = 1.0m
(a) K = 430 N/m, (b)


Freedom Bouncing
and M = 10 Kg
K = 2010 N/m











2.4 Two Degree of Freedom Spring-Mass Systems






Mi


Ki


M2


K2



\///////////\

Figure 2.5 Two Degree of Freedom Bouncing System

For Raibert's hopping machines, a point contact between

the end of the leg and the surface with enough frictional

forces between them to prevent slipping, was assumed for

simplicity. However, for a smooth motion as seen in animal

locomotion, a foot is also required. Another spring-mass

system as a foot is added to the one degree of freedom

spring-mass system. This two degree of freedom spring-mass

bouncing system also has two different modes, the flight mode

and the ground mode. For the ground mode, it is a simple

linear oscillatory system with two masses and two springs,

and the motion of the system for the flight mode is a

combination of a free fall in a gravitational field and

oscillations between the two masses.









The equations of motion are:

(Ground Mode) Mixi + Kixi Kx,2 = -M1g (2.3)

M2x2 Kjx, + (K, + K2)x2 = -M2g (2.4)

(Flight Mode) Mjxi + Kixi Kx2, = -M1g (2.5)

M2x2 K1xi + Kx,2 = -M2g (2.6)

where M1 and M2 are the body and the foot mass, KL and K2

are the leg and the foot stiffness, and g is gravity.

The displacements (xl and x2) are measured from the

positions of the bodies (MI and M2) at the instant when the

foot spring (K2) contacts the floor for the first time. If

the system is initially raised to a certain height (h) and

dropped from that height, it is assumed that there are no

interactions between the two bodies until the foot spring

touches the surface at least once, i.e., the starting free

fall motion of the system is a rigid body motion so that

initial conditions for the two different bodies are the same.

The condition for the ground mode is that x2 is less than or

equal to zero. As soon as the foot spring (K2) hits the

surface, the system begins the ground mode since the

displacement of the foot (M2) becomes zero and negative

thereafter. The system is in the flight mode when the

displacement of the foot (x2) is greater than zero. In other

words, it is the ground contact reference point when x2 is

equal to zero. The conditions for the ground mode and the

flight mode are independent of the body displacement (xi).










2.5 Phase Planes of Two Degree of Freedom Bouncing Systems


For two degree of freedom spring-mass bouncing systems,

the body mass (MI) = 10Kg, the foot mass (M2) = 1Kg, and the

leg stiffness (Kl) = 1000N/m were used as the fixed system

parameters. The fixed parameters for the body and the foot

were selected in order to emulate animal mass ratio of body

and foot. Since the system is initially raised to a certain

height (h) and dropped from that height, xl = x2 = 0.0 and vl

= v2 = 2gh at t = 0 are used as initial conditions for the

bouncing motions where xl and vj, x2 and v2 are displacements

and velocities of the body and the foot, respectively.

Six different cases were selected to simulate bouncing

motions of two degree of freedom systems by Runge-Kutta

method with 0.001 second as a time step'.

Cases of Dropping Height (h) = 0.2m

When the dropping height is 0.2m, i.e., the system is

initially raised to the height of 0.2m and is released from

that height, K2 = 575N/m (Figure 2.6) and K2 = 1900N/m (Figure

2.7) are used as examples of two degree of freedom bouncing

system parameters.

Cases of Dropping Height (h) = 0.5m

With h = 0.5m, K2 = 1860N/m (Figure 2.8) and K2 = 3080N/m

(Figure 2.9) are applied as system parameters in order see

how systems behave with changes of initial conditions and


1 see Appendix A for the effects on time step size










system parameters.

Cases of Dropping Height (h) = 1.0m

With h = 1.0m, K2 = 430N/m (Figure 2.10) and K2 = 2010N/m

(Figure 2.11) are assigned to examine the bouncing motions of

the two degree of freedom systems with different initial

conditions and system parameters.

The phase planes in Figure 2.7.a, Figure 2.9.a, and

Figure 2.10.a for the body (Mi) have repeated bouncing

motions which are similar to those of the single degree of

freedom bouncing systems. However, according to Figure

2.6.a, Figure 2.8.a, and Figure 2.11.a, chaotic motions can

be seen in the phase planes for the body. The dynamic

responses of the body and the foot seem to have strong

relationships. When the shape of the phase plane for the

foot (M2) is simple as in Figure 2.7.b, Figure 2.9.b, and

Figure 2.10.b, the body has repeated bouncing motions which

are desirable aspects for bouncing systems. On the other

hand, the trajectories of the phase plane for the body have

deviations with obvious chaotic motions of the foot (Figure

2.6.b, Figure 2.8.b, and Figure 2.11.b). By introducing

another mass-spring as a foot and a foot spring to the single

degree of freedom system, a two degree of freedom bouncing

system may have chaotic behavior depending upon the initial

conditions and the system parameters.
















0.2



0.0



-0.2


-0.4


-0.6


-2 -1 0 1 2


0.2-



0.0



-0.2-


-04.4




-6 -4 -2 0 2 4 6


Figure 2.6 Phase Planes of Two Degree of Freedom System with
h = 0.2m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, and K2 = 575N/m
(a) Body Phase Plane (M1), (b) Foot Phase Plane (M2)


-0.1

-0.2

-0.3

-0.4


0.2



0.1



0.0



-0.1


-2 -1 0 1 2
V1


-2 -1 0 1 2


Figure 2.7 Phase Planes of Two Degree of Freedom System with
h = 0.2m, K1 = 1000N/m, M1 = 10Kg, M2 = IKg, and K2 = 1900N/m
(a) Body Phase Plane (MI), (b) Foot Phase Plane (M2)

















0.4


0.2


0.0


-0.2


-0.4


-0.6


-2 0 2
vi


0.4-



0.2-


0.0 --4


-0.2



-8 -4 0 4 8


(a) (b)

Figure 2.8 Phase Planes of Two Degree of Freedom System with
h = 0.5m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, and K2 = 1860N/m
(a) Body Phase Plane (MI), (b) Foot Phase Plane (M2)


0


x 0


-0.2


0.1

0.0

-0.1


-2 0 2
vi


-2 0 2
V2


Figure 2.9 Phase Planes of Two Degree of Freedom System with
h = 0.5m, KI = 1000N/m, MI = 10Kg, M2 = 1Kg, and K2 = 3080N/m
(a) Body Phase Plane (MI), (b) Foot Phase Plane (M2)















1.0



0.5



0.0



-0.5


-4 -2 0 2 4
vi


-4 -2 0 2 4
V2


Figure 2.10 Phase Planes of Two Degree of Freedom System with
h = 1.0m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, and K2 = 430N/m
(a) Body Phase Plane (M) (b) Foot Phase Plane (M2)


1.0

0.8

0.6

0.4

H 0.2

0.0

-0.2

-0.4

-0.6


-4 -2 0 2 4


1.0

0.8-

0.6 -

0.4-

0.2-

0.0

-0.2

-0.4- |
-10 -5 0


Figure 2.11 Phase Planes of Two Degree of Freedom System with
h = 1.0m, K1 = 1000N/m, M1 = 10Kg, M2 = IKg, and K2 = 2010N/m
(a) Body Phase Plane (Mi), (b) Foot Phase Plane (M2)


1.0


-0.5



-1.0


5 10















CHAPTER 3
CHAOS IN TWO DEGREE OF FREEDOM BOUNCING SYSTEMS


3.1 Background


The chaotic behavior of two degree of freedom bouncing

systems has been observed with the fixed system parameters

for the body mass (MI = 10Kg), the leg stiffness (K1 =

1000N/m), and the foot mass (M2 = lKg), i.e., the foot

stiffness (K2) is the only system variable. These fixed

system parameters will be used throughout the investigation

of the chaotic system behavior.

The two degree of freedom bouncing system, which is

linear in each mode, can be considered as three different

systems depending on the dynamic modes of the system. The

three different systems can be identified as the two degree

of freedom linear oscillatory systems (Figure 3.1.a for the

normal ground mode and Figure 3.1.b for the normal flight

mode) along with the one degree of freedom linear oscillatory

system (Figure 3.1.c for the rigid body ground mode). The

first system for the ground mode (see Figure 3.1.a) is

labelled system a, the second system for the flight mode (see

Figure 3.1.b) is labelled system b, and the third system for

the ground mode (see Figure 3.1.c) is labelled system c.

When the bouncing system is represented by system c, it is




























(a) (b)









Equivalent





K2






(C)


Figure 3.1 Three Different Linear Oscillatory Systems
(a) System a for the normal ground mode
(b) System b for the normal flight mode
(c) System c for the rigid body ground mode
(Mequivalent = M1 + M2)


Mi








M2


l\ 7/7 7// 7 77////\









assumed that there is no relative motion between the body

mass and the foot (as in rigid body motion).

The natural frequencies (oal and Wa2) of system a are


K 2M+KM2+K M _+ (KMI+KMy+KiM)M2 --4MK Ki
Wal,a2 = (3.1)

The natural frequencies (obl and (b2) of system b are

S(Mi+M,)K,
Wbl,b2 = 0 MM2 (3.2)
MAM,


Since 0bl is equal to zero (i.e., the system is in rigid

body mode) (b will be used as ob2 for the natural frequency

of system b.

The natural frequency of system c (0c)is


w, = (3.3)
Equivalent


where M1 is the body mass, M2 is the foot mass, Kl is the leg

stiffness, K2 is the foot stiffness, Mequivalent is equivalent

mass for the rigid body ground mode (Mequivalent = M1 + M2).

It should be noted that the natural frequency of system c

(oc) is a convenience and does not correspond to any natural

frequencies of the two degree of freedom system.

The objective of this chapter is to find the

relationships between the dominant frequencies of the flight

and the ground modes and the natural frequencies of the

specified linear oscillatory systems. A Fast Fourier

Transform (FFT) was used to obtain the dominating frequencies









of the flight and ground modes of the bouncing system.


3.2 Sum of Standard Deviations and Area in Phase Plane Plot
for the Specified Region


First, a statistical approach is used to select the foot

stiffness which makes bouncing motions chaotic. Standard

deviations for the data points of the vertical axis (xl) with

respect to each data point of a horizontal axis (vi) in the

body phase plane plot are obtained. These standard

deviations are added together for the designated range of the

horizontal axis. The range is set so that the data for the

ground and the flight modes can be separated easily. For the

cases of h = 0.5m and h = 1.0m, the velocity (vl) range

between -2.0 and 2.0 is used. The horizontal axis range

between -1.0 and 1.0 is used with h = 0.2m. The procedure to

get the sum of the standard deviations is repeated for the

minimum foot stiffness (K2) = 100N/m and the maximum foot

stiffness (K2) = 4500N/m with an increment of 5N/m and with h

= 0.2m and 1.0m. However, the value of 4000N/m is used for

the maximum foot stiffness (K2) with h = 0.5m. The results

are shown in Figure 3.2.a, Figure 3.3.a, and Figure 3.4.a.

When the body phase plane trajectories are repeated, the sum

of standard deviations is small. On the contrary, when the

system with chosen foot stiffness has chaotic bouncing

motions, then the sum of standard deviations is increased.

Second, the area of the body phase plane with the same

velocity range is calculated to check whether it has









different criteria from those of the sum of the standard

deviations. The area of the phase plane of the body for the

given horizontal axis range versus foot stiffness is plotted

in Figure 3.2.b, Figure 3.3.b, and Figure 3.4.b. These are

almost identical to the results of the sum of standard

deviations with respect to the foot stiffness.

The foot stiffness which makes the sum of standard

deviations or area high is selected to obtain the dominant

frequencies of the flight and ground modes.

In the following sections, spectrum analyses with

selected system parameters will be discussed. For each case,

the data for the regular two degree of freedom bouncing

system will be initially used. The complete set of data will

be separated into two sets of data as the flight mode and the

ground mode to obtain the critical frequencies for each

system mode. As a matter of fact, there are discontinuities

between the end of the previous flight mode and the beginning

of the the following flight mode, since there is a ground

mode in the middle of two successive flight modes. However,

it will be assumed that the end of the previous flight mode

is followed by the beginning of the subsequent flight mode

without any discontinuity. This assumption is also applied

to the data of the ground mode. Then, the data sets with and

without separations for the regular one degree of freedom

bouncing system will be utilized. In this case, the sum of

the body mass and the foot mass for the two degree of freedom

system is the system mass (Mequivalent).









K2 = 3670 N/mI


(IK2 = 570 N/mr


1000 2000 3000 4000
Foot Stiffness
(a)


K, = 570 N/mr


20,;


1000 2000 3000 4000
Foot Stiffness
(b)
Figure 3.2 Sum of Standard Deviations and Area
(Mi = 10Kg, M2 = lKg, Ki = 1000N/m, and h = 0.2m)
(a) Sum of Standard Deviations
(b) Area


80x10


K2 = 3670 N/m



LAI































1000 2000 3000
Foot Stiffness

(a)


4000


4000


1000 2000 3000
Foot Stiffness

(b)


Figure 3.3 Sum of Standard Deviations and Area
(M1 = 10Kg, M2 = 1Kg, K1 = 1000N/m, and h = 0.5m)
(a) Sum of Standard Deviations
(b) Area


0.20



0.15



0.10



0.05



0.00










80 K2 = 2745 N/m

0
.-
60

K2 = 2010 N/m

0 40



o 20



0 I. .. .. 1 1 I '

1000 2000 3000 4000
Foot Stiffness

(a)



iK = 2745 N/m
0.5


0.4

K2 =2010 N/m
S$0.3-


0.2


0.1-
0.0- EA


1000 2000 3000 4000
Foot Stiffness

(b)

Figure 3.4 Sum of Standard Deviations and Area
(M1 = 10Kg, M2 = lKg, K1 = 1000N/m, and h = 1.0m)
(a) Sum of Standard Deviations
(b) Area










The period diagrams will also be used to study the

chaotic system behavior. Since the period diagram is the

intervals of the flight mode and the ground mode, the

frequencies of each mode can be obtained easily for the one

degree of freedom system. However, for the chaotic bouncing

motions, sudden mode changes with a very short period can be

observed with these period diagrams.


3.3 Spectrum Analyses for the Chaotic Bouncing Systems

with h = 0.2m

570N/m and 3670N/m for the foot stiffness (K2) are

selected as system parameters for the spectrum analyses of

the chaotic bouncing systems with h = 0.2m. According to

Figure 3.2, they correspond to the high values of the sum of

standard deviations and area in the phase planes. The

natural frequencies with the chosen system parameters for

system a, system b, and system c are listed in Table 3.1.


Table 3.1 Natural Frequencies (Hz) for system a, system b,
and system c

K2 0al 0a2 Wb COC
570 0.9397 6.4357 5.2786 1.1457
3670 1.4076 10.9016 5.2786 2.9071


3.3.1 Case of 570N/m as Foot Stiffness (Kzl


Two Degree of Freedom System

The sets of separated data for the flight mode and the

ground mode are displayed as wave forms in Figure 3.5.
















0.20

0.15

0.10

0.05

0.00

-0.05

-0.10

-0.15


0 500 1000 1500
Number of Data Points


-0.4



-0.6


0 500 1000 1500
Number of Data Points


2000


2000


(b)


Figure 3.5 Wave Forms of the Separated Data Set
(M1 = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 570N/m, and h = 0.2m)
(a) Flight Mode, (b) Ground Mode










The results of spectrum analyses are shown in Figure

3.6. The interesting frequencies are summarized in Table

3.2. For the flight mode, the most dominant frequency is

2.8320Hz while it is 1.1719Hz for the ground mode. Sudden

changes of modes in the period diagram (Figure 3.7) can also

be observed.


0.8


0..6


0-.4


0..2


0.0


5 10 15


f [Hz]
(a)


4.0


3.0


42.
-2.0


0 5 10 15
f [Hz]
(b)

Figure 3.6 Spectrum Analysis for Two Degree of Freedom System
(Mi = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 570N/m, and h = 0.2m)
(a) Flight Mode, (b) Ground Mode











Table 3.2 Dominant Frequencies (Hz)


Flight Mode
i Odfi
1 0
2 1.9351
3 2.3438
4 2.8320*
5 3.3203
6 5.9570
7 6.1523
8 6.4453
9 6.8359
10 8.7891
11 0.0920
12 9.2773
13 10.0586


(* the most dominant frequency)


Flight.
Mode







Ground
Mode


0 2 4 6 8
time [sec]


10 12 14


Figure 3.7 Period Diagram


Ground Mode
i Odgi_
1 0
2 1.1719*
3 2.4414
4 3.6133










One Degree of Freedom System


The total data set without separation and the separated

data sets for the ground mode and the flight mode are used to

obtain the dominant frequencies. First, the result of FFT

for the data without separation (total data) is shown in

Figure 3.8 and the dominant frequencies are listed in Table

3.3


2.0


1.5


1.0


0.5


0.0


f [Hz]

Figure 3.8 Spectrum Analysis with the Total Data


Table 3.3 Dominant Frequencies (Hz)

i Osti
1 0
2 0.9766*
3 1.9351
(* the most dominant frequency)

The most dominant frequency for the total data set for

the one degree of freedom system is 0.9766Hz. The results of

the spectrum analyses with the separated data sets are

displayed in Figure 3.9 and Table 3.4. The most dominant


2


1





- 3






40



1.5



1.0 1
0 0 ... ... .. .- .- ...-


IX:
-0.5 2


S 3 4
0.0 A
0 5 10 15
f [Hz]

(a)

4.0


3.0
3.0 --------..........................................................

1

2.0
422 .0 ---- -----.- .- .-.- ... .. ........- ........ -- ---------... -.--

2
1 .0 ..........

3
S 4
0.0
0 5 10 15
f [Hz]

(b)

Figure 3.9 Spectrum Analysis of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode


Table 3.4 Dominant Frequencies (Hz)


(* the most dominant frequency)


Flight Mode
i 3sfi
1 0
2 2.4414*
3 4.9805
4 7.4219


Ground Mode
i Osgi
1 0
2 1.6602*
3 3.3203
4 4.9805










frequency is 2.4414Hz for the flight mode and 1.6602Hz for

the ground mode.

The frequencies of the flight mode and the ground mode

can be obtained either directly from the period diagram

(Figure 3.10) or analytically from the phase planes (Appendix

B). The period of the ground mode is 0.603 second and that

of the flight mode is 0.404 second. Thus, the corresponding

frequencies of the ground mode and the flight mode become

1.6584 Hz and 2.4752 Hz, respectively (Table 3.5). Those

frequencies are very close to the results of the spectrum

analyses.



Flight -
Mode







Ground
Mode

1 I I I I I I I
0 2 4 6 8 10 12 14
time [sec]

Figure 3.10 Period Diagram of One Degree of Freedom System


Table 3.5 Frequencies from Period Diagram


Mode Frequency (Hz)

Ground 1.6584

Flight 2.4752






42





3.3.2 Case of 3670N/m as Foot Stiffness (K21


Two Degree of Freedom System


In Figure 3.11, the sets of separated data for the

flight mode and the ground mode are displayed as wave forms.


0.2


0.1


0.0


-0.1


-0.2


-0.3


0 500 1000 1500
Number of Data Points

(a)


0.0


-0.1


-0.2


-0.3


-0.4


0 500 1000 1500
Number of Data Points

(b)


2000


2000


Figure 3.11 Wave Forms of the Separated Data Set
(Ml = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 3670N/m,and h = 0.2m)
(a) Flight Mode, (b) Ground Mode










The results of spectrum analyses are displayed in Figure

3.12 and summarized in Table 3.6. It is interesting to see

that the most dominant frequency is 2.2461Hz for both the

flight mode and the ground mode. Abrupt changes of modes are

illustrated in the period diagram (Figure 3.13).


0.8


0.6


(M.
x


0.4 3--3 .. 4


0 2 .... .- .. .......


0.0 ~
0 5 10
f [Hz]

(a)


2.5

2.0
2 0 .... -- .


1.5.--------------

1.0
1 0 ..... 0 ---------- ... ..... .------. -.-... --




2
0.5 3
0~~- -- -- -------- 3 .... ... ..
4 5
0. 0
0 5 10 15
f [Hz]
(b)

Figure 3.12 Spectrum Analysis for Two Degree of Freedom
System
(Mi = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 3670N/m,and h = 0.2m)
(a) Flight Mode, (b) Ground Mode









Table 3.6 Dominant Frequencies (Hz)


Flight Mode
i Odf i
1 0
2 0.5859
3 2.2461*
4 2.4414
5 4.2969
6 4.6875
7 4.8828


(* the most dominant frequency)


Flight-
Mode







Ground
Mode


I I I I I
2 4 6 8 10
time [sec]

Figure 3.13 Period Diagram


I I
12 14


One Degree of Freedom System


The data sets without and with separation for each

system mode are utilized to determine the dominant

frequencies by means of spectrum analysis. At first, Figure

3.14 presents the result of the spectrum analysis of the

total data set. The dominant frequencies are itemized in

in Table 3.7.


Ground Mode

i Odgi
1 0
2 2.2461*
3 2.4414
4 2.6367
5 4.4922


N











1.0

0.8 ---...-1..- 2..---- -....

0 6 .. .. .. .........
1
X 0 .4 .. .. ......... .... -

3
0 2 ... .. .--

4
0.0
0 5 10 15
f [Hz]

Figure 3.14 Spectrum Analysis of the Total Data Set


Table 3.7 Dominant Frequencies (Hz)


i Osti

1 0

2 1.6602*

3 3.3203

4 4.9805
(* the most dominant frequency)


The most dominant frequency of the total data for the

one degree of freedom system is 1.6602Hz. The results of the

spectrum analyses for the separated data sets are shown in

Figure 3.15 and Table 3.8. The dominant frequencies are

2.4414Hz and 4.9805Hz for the flight mode and the ground

mode, respectively.

The frequencies of the flight mode and the ground mode

can be attained directly from the period diagram (Figure









1.5


1.
1 0 ... .......... .. .. ... ...........


0.5


0.0




1.0

0.8

- 0.6
4-4
S0.4

0.2


f [Hz]
(a)


0 5 10 15
f [Hz]
(b)


Figure 3.15 Spectrum Analysis of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode


Table 3.8 Dominant Frequencies (Hz)


(* the most dominant frequency)


. ... .


13 4
-h A_ a


Flight Mode
i Qsfi
1 0
2 2.4414*
3 4.9805
4 1 7.4219


Ground Mode
i osgi
1 0
2 4.9805*
3 9.9609









3.16). The period of the ground mode is 0.201 second and

that of the flight mode is 0.404 second. Thus, the

corresponding frequencies are 4.9751Hz and 2.4752Hz for the

ground mode and the flight mode, respectively (Table 3.9).

These two frequencies are very close to the most dominant

frequencies for each system mode obtained by the spectrum

analyses.


Flight
Mode








Ground
Mode


Figure 3.16 Period Diagram


1
6
time

of One


I I
8 10
[sec]

Degree


of Freedom System


Table 3.9 Frequencies from the Period Diagram


3.4 Spectrum Analyses for the Chaotic Bouncing

Systems with h = 0.5m

445N/m and 1860N/m for the foot stiffness (K2) are


Mode Frequency (Hz)

Ground 4.9751

Flight 2.4752










selected as system parameters for the spectrum analyses of

the chaotic bouncing systems with h = 0.5m. According to

Figure 3.3, they correspond to the high values of the sum of

standard deviations and area in the phase planes. The

natural frequencies with the chosen system parameters for

system a, system b, and system c are listed in Table 3.10.


Table 3.10 Natural Frequencies (Hz) for system a, system b,
and system c

K2 Wal 0a2 Ob Wc
445 0.8624 6.1961 5.2786 1.0123
1860 1.2755 8.5645 5.2786 2.0696



3.4.1 Case of 445N/m as Foot Stiffness (Kz)


Two Degree of Freedom System

Figure 3.17 exhibits the sets of separated data for the

flight mode and the ground mode as wave forms. The results

of spectrum analyses are shown in Figure 3.18 and Table 3.11.

For the flight mode, the most dominant frequency is 1.8555Hz

while it is 1.2695Hz for the ground mode.

Abrupt mode changes in the period diagram (Figure 3.19),

i.e., the very short intervals for each mode, can also be

recognized.

One Degree of Freedom System

The complete data set without separation and the

separated data sets for the ground mode and the flight mode

are adopted to acquire the dominant frequencies. The result
















0.5

0.4

0.3

0.2


0.1

0.0

-0.1













0.2-


0.0-

-0.2-


x -0.4-

-0.6-


-0.8-

-1.0-


0 500 1000 1500
Number of Data Points

(a)


0 500 1000 1500
Number of Data Points

(b)


2000


2000


Figure 3.17 Wave Forms of the Separated Data Set
(MI = 10Kg, M2 = IKg, K1 = 100N/m, K2 = 445N/m, and h = 0.5m)
(a) Flight Mode, (b) Ground Mode









3.0

2.5

2.0

S1.5

S1.0

0.5

0.0
0




6.0

5.0

4.0

S3.0

- 2.0

1.0

0.0


5 10 15
f [HZ]


(b)

Figure 3.18 Spectrum Analysis of Two Degree of Freedom System
(MI = 10Kg, M2 = IKg, K1 = 1000N/m, K2 = 445N/m, and h = 0.5m)
(a) Flight Mode, (b) Ground Mode

Table 3.11 Dominant Frequencies (Hz)


Flight Mode
i Qdfi
1 0
2 1.5625
3 1.8555*
4 3.1250
5 3.7109
6 5.5664


(* the most dominant frequency)


5 10
f [Hz]


Ground Mode
i Odgi
1 0
2 1.2695*
3 2.4414
4 3.7109










Flight-
Mode








Ground
Mode


I I I I I I
0 2 4 6 8 10
time [sec]

Figure 3.19 Period Diagram


I I
12 14


of the spectrum analysis with the complete data set in in

Figure 3.20 and the dominant frequencies are listed in Table

3.12.


4.0


3.0


- 2.0
41.


1.0


0.0'
0


5 10


f [Hz]

Figure 3.20 Spectrum Analysis with the Complete Data Set

Table 3.12 Dominant Frequencies (Hz)


i sti
1 0
2 0.7813*
3 1.5625
(* the most dominant frequency)


2







1
3
A





0










The most dominant frequency with the complete data set

(without separation) for the one degree of freedom system is

0.7813Hz. The results of the spectrum analyses with the

separated data sets are shown in Figure 3.21. The most

dominant frequency is 1.5625Hz for both the flight mode and

the ground mode (Table 3.13).


Table 3.13 Dominant Frequencies (Hz)

Flight Mode Ground Mode


i fsfi
1 0
2 1.5625*
3 3.1250
4 4.6875
5 6.2500


(* the most dominant frequency)

The period of the flight mode is 0.639 second directly

from the period diagram (Figure 3.22). The corresponding

frequency to this period is 1.5649Hz (Table 3.14) which is

approximately the same as the dominant frequency for the

flight mode obtained by the spectrum analysis. It is also

true for the ground mode since both periods are exactly same

with selected system parameters.


3.4.2 Case of 1860N/m as Foot Stiffness (K21


Two Degree of Freedom System

The separated data sets for the flight mode and the

ground mode are shown as wave forms in Figure 3.23. The


i Qsgi
1 0
2 1.5625*
3 3.1250
4 4.6875















S2
X


5.0

4.0

- 3.0

- 2.0

1.0

0.0
0


0 5 10
f [Hz]


1



2



3
4


f [Hz]


(b)

Figure 3.21 Spectrum Analyses of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode)



Table 3.14 Frequencies from Period Diagram


Mode Frequency (Hz)

Ground 1.5649

Flight 1.5649










Flight-
Mode







Ground
Mode

I I I I I I I
0 2 4 6 8 10 12 14
time [sec]

Figure 3.22 Period Diagram


results of the spectrum analyses with these separated data

sets are displayed in Figure 3.24. For the flight mode, the

most dominant frequency is 1.5625Hz while it is 2.2461Hz for

the ground mode according to Table 3.15. Sudden changes of

system modes in the period diagram (Figure 3.25) can also be

observed.

One Degree of Freedom System

The data sets with and without separation for the flight

mode and the ground mode are used to find the dominant

frequencies by the spectrum analysis. The results of the

spectrum analysis for the data set without separation are

shown in Figure 3.26 and Table 3.16. The most dominant

frequency in this case is 1.0742Hz. The spectrum analysis

results for the data sets with separation for each system

mode are shown in Figure 3.27 and Table 3.17.

















0.4-



0.2

x
0.0



-0.2



0 500 1000 1500 2000
Number of Data Points

(a)






0.1

0.0

-0.1

H-0.2-

-0.3-

-0.4-

-0.5-


0 500 1000 1500 2000
Number of Data Points

(b)


Figure 3.23 Wave forms of the Separated Data Set
(Mi = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 1860N/m,and h = 0.5m)
(a) Flight Mode, (b) Ground Mode









3.0

2.5

2.0

1.5

1.0

0.5

0.0


5 10 15
f [Hz]


4.0


3.0 1


2.0 ------------------


2
1 0 2. .. .. .. ..... .

3 4
0.0
0 5 10 15
f [Hz]
(b)
Figure 3.24 Spectrum Analyses of Two Degree of Freedom System
(Ml = 10Kg, M2 = IKg, K1 = 1000N/m, K2 = 1860N/m,and h = 0.5m)
(a) Flight Mode, (b) Ground Mode


Table 3.15 Dominalt )


Flight Mode
i Odfi
1 0
2 1.5625*
3 3.2227
4 3.6133
5 4.7852
6 6.4453
7 6.8359


(* the most dominant frequency)


Ground Mode
i Odgi
1 0
2 2.2461*
3 4.4922
4 6.6406
























I I I I I I I
0 2 4 6 8 10 12 14
time [sec]

Figure 3.25 Period Diagram


5 10
f [Hz]


Figure 3.26 Spectrum Analysis of
Separation


the Data Set without


Table 3.16 Dominant Frequencies


i sti
1 0
2 1.0742*
3 2.1484
4 3.3203
(* the most dominant frequency)


(Hz)


Flight
Mode








Ground
Mode


0.0L
0









4.0


3.0


2.0


1.0


n n


f [Hz]
(a)


2.0


1.5


1.0


0.5


0.0 -
0


f [Hz]
(b)

Figure 3.27 Spectrum Analysis of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode


Table 3.17 Dominant Frequencies (Hz)


(* the most dominant frequency)


1



2

3
4:
-


*-. --- -



2



3 4
A


Flight Mode
i Osfi
1 0
2 1.5625*
3 3.1250
4 4.6875


Ground Mode
i Isgi
1 0
2 3.6133*
3 7.2266
4 10.8398


V










The dominant frequencies are 1.5625Hz and 3.6133Hz for

the flight mode and the ground mode, respectively (Table

3.17).

The period of the ground mode is 0.277 second and that

of the flight mode is 0.639 second as taken directly from the

period diagram (Figure 3.28). Therefore, the corresponding

frequency for the ground mode is 3.6101Hz while it is

1.5625Hz for the flight mode (Table 3.18).


Flight_
Mode








Ground
Mode


I I
0 2


Figure 3.28


I I I
6 8 10
time [sec]

Period Diagram


Table 3.18 Frequencies from the Period Diagram


Mode Frequency (Hz)
Ground 3.6101
Flight 1.5625


I I
12 14


I I I











3.5 Spectrum Analyses for the Chaotic Bouncing

Systems with h = 1.0m

2010N/m and 2745N/m for the foot stiffness (K2) are

selected as system parameters for the spectrum analyses of

the chaotic bouncing systems with h = 1.0m. According to

Figure 3.4, they correspond the high values of the sum of

standard deviations and area in the phase planes. The

natural frequencies with the chosen system parameters for

system a, system b, and system c are listed in Table 3.19.

Table 3.19 Natural Frequencies (Hz) for system a, system b,
and system c

K2 Wal a2 0b Wc
2010 1.2933 8.7809 5.2786 2.1514
2745 1.3577 9.7751 5.2786 2.5142



3.5.1 Case of 2010N/m as Foot Stiffness (K21


Two Degree of Freedom System

Figure 3.29 exhibits the separated data sets for the

flight mode and the ground mode as wave forms.

With these data sets, the spectrum analyses are carried

on to obtain the dominant frequencies. The results are shown

in Figure 3.30 and they are summarized in Table 3.20. The

most dominant frequency for the flight mode is 1.1719Hz while

it is 2.5391Hz for the ground mode. Rapid changes between

the ground mode and the flight mode exist in the period

diagram (Figure 3.31).





























0.0

-0.2

-0.4


0.2


0.0


-- -0.2


-0.4


-0.6


0 500 1000 1500
Number of Data Points


0 500 1000 1500
Number of Data Points


2000


2000


(b)


Figure 3.29 Wave Forms of the Separated Data Set
(Ml = 10Kg, M2 = IKg, K1 = 1000N/m, K2 = 2010N/m,and h = 1.0m)
(a) Flight Mode, (b) Ground Mode









5.0


4.0 1


3.01


2.0

1.0

0.0
0


2


... 3--... -.. ..... -..... -...............
4 5

5
f [Hz]
(a)


f [Hz]
(b)


Figure 3.30 Spectrum Analysis of Two Degree of Freedom System
(M1 = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 2010N/m,and h = 1.0m)
(a) Flight Mode, (b) Ground Mode


Table 3.20 Dominant FreQuencies (Hz)


(* the most dominant frequency)


Flight Mode
i _dfi
1 0
2 1.1719*
3 2.3438
4 3.5156


Ground Mode
i Odgi
1 0
2 2.2461
3 2.5391*
4 4.7852
5 5.0781
6 5.4688











Flight
Mode







Ground
Mode

I I I I I I I I
0 2 4 6 8 10 12 14
time [sec]

Figure 3.31 Period Diagram



One Degree of Freedom System

The data set without separation and the data sets with

separation for the flight mode and the ground mode are

employed to get the dominant frequencies by the spectrum

analysis. The result of spectrum analysis for the data set

without separation is displayed in Figure 3.32.

The most dominant frequency of the data set without

separation for the one degree of freedom system is 0.8789Hz

(Table 3.21).

Table 3.21 Dominant Frequencies (Hz)

i 3sti
1 0
2 0.8789*
3 1.7578
4 2.5391
5 3.4180
(* the most dominant frequency)






64


5.0
4.0 -------..----..--..-...................---------

3 0 -- .-------.. ..... .. -- ----- -. -



2 .. 0 .. .--. -.. --... -- -. -.-.-.-.. -... -.. .
4.0
1

3.0 2


-2.0

1.0 .3------------
4 5
0.0
0 5 10 15
f [Hz]

Figure 3.32 Spectrum Analysis of the Data Set Without
Separation


The results of the spectrum analyses for the data sets

with separation are shown in Figure 3.33. The most dominant

frequencies are 1.0742Hz and 3.9063Hz for the flight mode and

the ground mode, respectively (Table 3.22).

The period of the ground mode is 0.256 second and that

of the flight mode is 0.903 second as taken directly from the

period diagram (Figure 3.34). Therefore, the corresponding

frequency for the ground mode is 3.9063Hz while it is

1.1074Hz for the flight mode (Table 3.23).


Table 3.22 Dominant Frequencies (Hz)

Flight Mode Ground Mode


(* the most dominant frequency)


i 3sfi
1 0
2 1.0742*
3 2.2461
4 3.3203


i Qsgi
1 0
2 3.9063*
3 7.8125
4 11.7188









8.0


6.0


-I

4.0


2.0


0.0


f [Hz]
(a)


0 5 10 15
f [Hz]
(b)

Figure 3.33 Spectrum Analysis of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode



Table 3.23 Frequencies From Period Diagram


Mode Frequency (Hz)
Ground 3.9063
Flight 1.1074


1



2

3
- -













Flight
Mode







Ground
Mode

0 2 4 6 8 10 12 14
time [sec]

Figure 3.34 Period Diagram




3.5.2 Case of 2745N/m as Foot Stiffness (K2


Two Degree of Freedom System

The separated data sets for the flight mode and the

ground mode are displayed in Figure 3.35 as wave forms.

The results of the spectrum analyses with these data

sets are displayed in Figure 3.36. The most dominant

frequency for the flight mode is 1.1719Hz while it is

3.6133Hz for the ground mode (Table 3.24).

There are sudden changes between the flight mode and the

ground mode in the period diagram (Figure 3.37).
























0.2

0.0

-0.2

-0.4














0.2


0.0*


-0.2-


-0.4-


-0.6.


0 500 1000 1500
Number of Data Points


0 500 1000 1500


2000


2000


Number of Data Points

(b)


Figure 3.35 Wave Forms of the Separated Data Set
(M1 = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 2745N/m,and h = 1.0m)
(a) Flight Mode, (b) Ground Mode
























f [Hz]
(a)


2.0

1.5

1.0

0.5

0.0
C


f [Hz]
(b)

Figure 3.36 Spectrum Analysis of Two Degree of Freedom System
(M1 = 10Kg, M2 = 1Kg, K1 = 1000N/m, K2 = 2745N/m,and h = 1.0m)
(a) Flight Mode, (b) Ground Mode

Table 3.24 Dominant Frequencies (Hz)


(* the most dominant frequency)


1



3


2 5.2.
4 5
"4
L' A'J


Flight Mode
i Odfi
1 0
2 1.1719*
3 2.2461
4 3.4180
5 4.4922
6 5.6641


Ground Mode
i Odgi
1 0
2 2.6397
3 3.6133*

5 7.1289


"^~ u? -^-`I^"`


-










Flight
Mode








Ground -
Mode

0 2 4 6 8 10 12 14

time [sec]

Figure 3.37 Period Diagram

One Degree of Freedom System

In order to get the dominant frequencies for the

selected one degree of freedom system, spectrum analyses are

carried out for the data set without separation and the data

set with the separation. Figure 3.38 displays the result of

the spectrum analysis of the data set without separation.

The most dominant frequency in this case is 0.8789Hz (Table

3.25).


Table 3.25 Dominant Frequencies


i sti
1 0
2 0.8789*
3 1.7578
4 2.7344
5 3.6133
(* the most dominant frequency)














- 3.
X 2.
} 2.1


0 5 10 15
f [Hz]

Figure 3.38 Spectrum Analysis of the Data Set Without

Separation


The results of the spectrum analyses for the data sets

with separation are shown in Figure 3.39. The most dominant

frequencies are 1.0742Hz and 4.5898Hz for the flight mode and

the ground mode, respectively (Table 3.26).

The period of the ground mode is 0.216 second and that

of the flight mode is 0.903 second as taken directly from the

period diagram (Figure 3.40). Thus, the corresponding

frequency for the ground mode is 4.6296Hz while it is

1.1074Hz for the flight mode (Table 3.27).



Table 3.26 Dominant Frequencies (Hz)

Flight Mode Ground Mode


(* the most dominant frequency)


i J sfi
1 0
2 1.0742*
3 2.2461
4 1 3.3203


i Qsgi
1 0
2 4.5898*
3 9.2773









8. 01


6.0


t 4.0
2x0

2.0


0.


(


f [Hz]
(a)


2.5

2.0


1.5

1.0


0.5


0.0


5 10 15
f [Hz]
(b)


Figure 3.39 Spectrum Analysis of One Degree of Freedom System
(a) Flight Mode, (b) Ground Mode


Table 3.27 Frequencies From Period Diagram


Mode Frequency (Hz)
Ground 4.6296
Flight 1.1074


1




2-
3 4
KA-












Flight
Mode







Ground
Mode

I I I I I I I I
0 2 4 6 8 10 12 14

time [sec]

Figure 3.40 Period Diagram



3.6 Summary


Before proceeding with the summary, it is necessary to

note that the ground mode of a one degree of freedom system

begins as soon as the spring touches the surface. However,

the natural frequency of system c is obtained from the

equation of motion based on the equilibrium position of the

system. There exists a difference between the ground contact

reference and the equilibrium position of the system. This

is a static displacement (6st). Thus, the system has to move

the static displacement to reach the equilibrium position

after the beginning of the ground mode. At the end of the

ground mode, the system moves up the static displacement

beyond the equilibrium position before reaching the ground

contact reference. It is assumed that the velocity of the

system during that period is V2gh. The static displacement









can be expressed as


bet Mg
K


(3.4)


Therefore, the time required for the system to travel

the static displacement at the beginning and the end of the

ground mode is


28 t
tt gh


(3.5)


The ground mode frequency (og) obtained from the

spectrum analysis must be corrected to include the time

required for the system to travel the static displacement.

The corrected frequency of the ground mode with consideration

of the equilibrium point can be obtained as


9 = 1 1
-g t6t
(A) S


(3.6)


This corrected frequency (0g) can be used to get

relationships with dominating frequencies of each mode and

the natural frequency of system c.

The results of spectrum analyses for two degree of

freedom and one degree of freedom bouncing systems with

different system parameters and initial conditions were

obtained. Those results are now summarized to establish

relationships between the dominant frequencies and the

natural mode frequencies.










Case of 570N/m as Foot Stiffness (h = 0.2m)

At first, the ground mode frequency of one degree of

freedom system (Osg2 = 1.6602 Hz from Table 3.4) is

considered. The static displacement is 0.1893m by Equation

(3.4), and the traveling time is 0.1911 second by Equation
(3.5). Thus, the ground mode frequency (0g) with

consideration of the equilibrium position is 2.4317 Hz

obtained from Equation (3.6), which is approximately twice

the natural frequency of system c (ic = 1.1457 Hz from Table

3.1), i.e., Og 20c. The natural frequency of system c is

very close to the most dominant ground mode frequency of the

two degree of freedom system (adg2 = 1.1719 Hz from Table

3.2).

The dominant frequencies for the one degree of freedom
system without separation are 0.9766 Hz (Ost2) and 1.9351 Hz

(Ost3) (Table 3.3). If the most dominant ground mode

frequency of the two degree of freedom system is subtracted

from 20st3, then the result is 2.6983 Hz and it is close to

the most dominant flight mode frequency of the two degree of

freedom system (Odf4 = 2.8320 Hz from Table 3.2). In short,

for the flight mode, the frequency relationship can be

expressed as f4 : 22)st3 Odg2.


Case of 3670N/m as Foot Stiffness (h = 0.2m)

For the two degree of freedom system, the most dominant
ground mode frequency (Wdg2) and the most dominant flight

mode frequency (adf3) are identically 2.2461 Hz (Table 3.6).









For the one degree of freedom system, the most dominant

flight mode frequency is 2.4414 Hz (Osf2) and the most

dominant ground mode frequency is 4.9805 Hz (Osg2) from Table

3.8. The most dominant ground mode frequency (tsg2) is

approximately twice the most dominant flight mode frequency

(Osf2), i.e., Qsg2 s 2sgf2.

The dominant frequencies for the one degree of freedom
system without separation are 1.6602 Hz (Ost2), 3.3203 Hz

(Ost3), and 4.9805 Hz (Wst4) (Table 3.7). If the most

dominant ground mode frequency of the two degree of freedom
system is subtracted from 2flt3, then the result is 4.3945 Hz

and is close to two times of the most dominant flight mode
frequency of the two degree of freedom system (cdf3 = 2.2461

Hz from Table 3.6). In short, for the flight mode, the
frequency relationship can be expressed as 2fdf3 a 2Qst3 -

Sdg2

Case of 445N/m as Foot Stiffness (h = 0.5m)

For the two degree of freedom system, the most dominant
flight mode frequency (daf3) is 1.8555 Hz and the most

dominant ground mode frequency (adg2) is 1.2695 Hz (Table

3.11). For the one degree of freedom system, the most

dominant flight mode frequency (Osf2) and the most dominant

ground mode frequency (Wsg2) are identically 1.5625 Hz from

Table 3.13.

The dominant frequencies for the one degree of freedom
system without separation are 0.7813 Hz (Ast2) and 1.5625 Hz









(Ist3) (Table 3.12). If the most dominant ground mode

frequency of the two degree of freedom system is subtracted
from 20st3, then the result is 1.8555 Hz and is the same as

the most dominant flight mode frequency of the two degree of
freedom system (Odf3). In short, for the flight mode, the

frequency relationship can be expressed as Qdf3 = 2tst3

fdg2. Also the average of the most dominant flight mode

frequency and the most dominant ground mode frequency is

1.5625 Hz.


Case of 1860N/m as Foot Stiffness (h = 0.5m)

The most dominant ground mode frequency of the one
degree of freedom system (9sg2) is 3.6133 Hz from Table 3.17.

By using Equation (3.4), Equation (3.5), and Equation (3.6),
the corresponding ground mode frequency (0g) with

consideration of the equilibrium position is obtained as

4.1709 Hz which is approximately twice the natural frequency

of system c (0c = 2.0696 Hz from Table 3.10), i.e., Qg 9 20c.

The natural frequency of system c is very close to the most

dominant ground mode frequency of the two degree of freedom

system (fdg2 = 2.2461 Hz from Table 3.15). The dominant

frequencies for the one degree of freedom system without
separation are 1.0742 Hz (Ust2), 2.1484 Hz (Wst3), and 3.3203

Hz (Ost4) (Table 3.16). If Ostl is subtracted from Qst4, then

the result is the most dominant ground mode frequency of the
two degree of freedom system, i.e., Qdg2 = Lst4 Qst2 =

2.2461 Hz.









The most dominant flight frequency is 1.5625 Hz for both

one and two degree of freedom systems.


Case of 2010N/m as Foot Stiffness (h = 1.0m)

For the ground mode of the one degree of freedom system,
the most dominant frequency (Osg2) is 3.9063 Hz from Table

3.22. If the equilibrium position and the ground contact

reference at the beginning and the end of the ground mode is
considered, the most dominant frequency (Osg2) becomes 4.3141

Hz by using Equation (3.4), Equation (3.5), and Equation

(3.6). This corrected ground mode frequency (Og = 4.3141 Hz)

based on the equilibrium position is twice the natural
frequency of system c (,c = 2.1514 Hz from Table 3.10), i.e.,

Og m 20c. For the two degree of freedom system, the most

dominant ground mode frequency (Qdg3) is 2.5391 Hz and the

second most dominant ground mode frequency (Odg2) is 2.2461

Hz from Table 3.20. In this case, the natural frequency of

system c (Oc) is very close to the second most dominant

ground mode frequency (adg2) of the two degree of freedom

system, i.e., Odg2 Wc. The dominant frequencies for the

one degree of freedom system without separation are 0.8789 Hz

(Qst2), 1.7578 Hz ((st3), 2.5391 Hz (Ost4), and 3.4180 Hz

(Osts) from Table 3.21. The most dominant ground mode

frequency (Odg3) for the two degree of freedom system is

equal to Lst4.

For the flight mode of the two degree of freedom system,
the most dominant frequency is 1.1719 Hz (ndf2) from Table









3.20. The most dominant flight mode frequency is 1.0742 Hz

(nsf2) and the second most dominant flight mode frequency is
2.2461 Hz (Isf3) for the one degree of freedom system. Qdf2

is the same as the difference between Osf2 and Lsf3, i.e.,

Qdf2 = Ksf3 Osf2-

Case of 2745N/m as Foot Stiffness (h = 1.0m)

For the ground mode of the one degree of freedom system,
the most dominant frequency (Osg2) is 4.5898 Hz from Table

3.26. With consideration of the equilibrium position and the

ground contact reference at the beginning and the end of the
ground mode, the most dominant frequency (Osg2) becomes

4.9981 Hz by using Equation (3.4), Equation (3.5), and
Equation (3.6). This ground mode frequency (Qg) based on the

equilibrium position is twice the natural frequency of system
c (wc = 2.5142 Hz from Table 3.10), i.e., Og 20c. For the

two degree of freedom system, the most dominant ground mode
frequency (fdg3) is 3.6133 Hz and the second most dominant

ground mode frequency (Odg2) is 2.6397 Hz from Table 3.24.

In this case, the natural frequency of system c (Oc) is very

close to the second most dominant ground mode frequency

(Odg2) of the two degree of freedom system, i.e., Odg2 s Oc.

The dominant frequencies for the one degree of freedom system
without separation are 0.8789 Hz (Sst2), 1.7578 Hz (Ost3),

2.7344 Hz (Qst4), and 3.6133 Hz (Ost5) from Table 3.25. The

most dominant ground mode frequency (Odg3) for the two degree

of freedom system is equal to Ost5.









For the flight mode of the two degree of freedom system,

the most dominant frequency is 1.1719 Hz (Odf2) from Table

3.24. The most dominant flight mode frequency is 1.0742 Hz

(nsf2) and the second most dominant flight mode frequency is

2.2461 Hz (nsf3) for the one degree of freedom system. ldf2

is same as the difference between ,sf2 and (sf3, i.e., df2 =

fsf3 Osf2*

Two most common frequency relationships can be found

among chaotic systems. The first one is that the most (or

the second most) dominant ground mode frequency of a two

degree of freedom system is equal to the natural frequency of

system c, which is equal to half of the most dominant ground

mode frequency of a corresponding one degree of freedom

system. The second one is that the most dominant frequencies

are identical for both the ground mode and the flight mode

either for corresponding one degree of freedom systems or two

degree of freedom systems. Even though there might exist

different criteria of chaotic bouncing motions other than

these two common frequency relationships, one can use these

two frequency relationships to check whether or not a system

with selected parameters has chaotic bouncing motions.

In order to verify the frequency relationships, 2415N/m

is used as the foot stiffness with a 0.5m dropping height.

The most dominant ground mode frequency of one degree of

freedom system (Osg) is 4.1991 Hz and the corresponding

frequency (fg) with consideration of the equilibrium position

becomes 4.7708 Hz. The most dominant ground mode frequency









of the two degree of freedom system (Odg) is 2.4414 Hz while

the natural frequency of system c (Oc) is 2.3582 Hz. Since

Odg is approximately equal to Oc or 2 the first frequency

relationship holds in this case. According to Figure 3.3, in

fact, the system has chaotic bouncing motion since the sum of

standard deviations and the area are high.

The value of 1815N/m as foot stiffness is selected with

a 0.2m dropping height to check the established frequency

relationships. The most dominant flight mode frequencies of

the corresponding one degree of freedom system and the two

degree of freedom system are identically 2.4414 Hz. The most

dominant ground mode frequency of the one degree of freedom

system (0sg) is 3.3203 Hz and the corresponding frequency

(0g) with consideration of the equilibrium position becomes

4.1468 Hz. The most dominant ground mode frequency of the
two degree of freedom system (Odg) is 1.8555 Hz while the

natural frequency of system c (wc) is 2.0444 Hz. The system

does not fall in the first frequency relationship since adg

is not equal and/or close to oc. Neither can the second

frequency relationship be found for this system. In fact,

the system with chosen system parameters and initial

conditions does not have chaotic bouncing behavior since the

sum of standard deviations and area are small in Figure 3.2.

With these two more examples, it can be concluded that

the most significant system mode for chaos is not the flight

mode but the ground mode as in rigid body motion.















CHAPTER 4
ELIMINATION OF CHAOS IN TWO DEGREE OF FREEDOM SYSTEMS


4.1 Background


Previous chapters have presented the conditions for

chaotic bouncing motions depending on the initial conditions

and system parameters. It was assumed that there is no

energy dissipation throughout the investigation of the system

behavior.

Since the chaotic behavior is not desirable, elimination

methods for this objectionable characteristics of the system

should be provided. This can be done by varying system

parameters or by introducing damping elements to the system.

However, with damping elements, the loss of energy must be

compensated by external energy sources in order to make the

system keep bouncing.


4.2 Spring Selections


The sum of standard deviations and area in a phase plane

plot have been discussed in Chapter 3 (Figure 3.2, Figure

3.3, and Figure 3.4). The range is set so that the data for

the ground and the flight modes can be separated easily.

These two results are almost identical. If the system has

chaotic motions, the sum of standard deviations and the area











are increased. On the other hand, the sum of the standard

deviations or the area becomes small for the repeated

bouncing motions.

The range for the foot stiffness (K2) is a system

parameter which can easily be selected for the repeated

bouncing motion from Figure 3.2 when the dropping height (h)

is 0.2m. The foot stiffness, which makes the sum of standard

deviations or area small in Figure 3.2, can also be chosen

for obtaining repeated bouncing motion. For this case, the

range of foot stiffness is approximately between 1000N/m and

2000N/m. For example, if 1500N/m is selected as the foot

stiffness, there are repeated bouncing motions. Figure 4.1

illustrates that the chaotic behavior exhibited in Figure 2.6

can thus be avoided by an appropriate choice of system

parameters.

0.2- 0.2-

0.1-
0.1
0.0-


Fie0.1 0.0
X X
-0.2

-0.3- -0.1-

-0.4-
-0.2-

-2 -1 0 1 2 -2 -1 0 1 2
V1 V2

(a) (b)
Figure 4.1 Phase Planes with h = 0.2m, M1 = 10Kg, M2 = 1Kg,
Kl = 1000N/m, and K2 = 1500N/m
(a) Body Phase Plane, (b) Foot Phase Plane










The foot stiffness (K2), which makes the sum of standard

deviations or area small, can be obtained from Figure 3.3 for

the case of 0.5m dropping height and from Figure 3.4 for the

case of 1.0m dropping height. However, there are no

selective ranges for the system parameter in Figure 3.3 and

Figure 3.4. For example, if the foot stiffness is chosen

among the values of 195N/m, 295N/m, 550N/m, and 3080N/m, the

system would have repeated bouncing motions when dropping

height is 0.5m. For the case of 1.0m dropping height,

125N/m, 165N/m, 245N/m, 430N/m, and 3775N/m are the values of

the foot stiffness to be selected for repeated bouncing

motions. Figure 4.2 and Figure 4.3 display the repeated

bouncing motions for the cases of 0.5m and 1.0m dropping

heights with 550N/m and 3775N/m as selected foot stiffness,

respectively.


0.4- 0.4-

0.2-
0.2-
0.0
0.0
x -0.2 x

-0.4- -0.2-

-0.6- -0.4

-0.8-
-0.6

-2 0 2 -2 0 2
V1 V2

(a) (b)
Figure 4.2 Phase Planes with h = 0.5m, M1 = 10Kg, M2 = 1Kg,
K1 = 1000N/m, and K2 = 550N/m
(a) Body Phase Plane, (b) Foot Phase Plane










1.0- 1.0-

0.8-
0.8-
0.6-

0.4- 0.6-

0.2- -
x X 0.4
0.0-
0.2-
-0.2-
-0.4-- 0.0-

-0.6-
--0.2
-4 -2 0 2 4 -4 -2 0 2 4
vi V2

(a) (b)
Figure 4.3 Phase Planes with h = 1.0m, M1 = 10Kg, M2 = 1Kg,
KI = 1000N/m, and K2 = 3775N/m
(a) Body Phase Plane, (b) Foot Phase Plane


4.3 Use of Damping Elements


It has been illustrated that chaotic system behavior can

be eliminated by properly choosing system parameters under

the assumption of no energy dissipation. However, it may be

impossible to build a physical system without any energy

dissipation. One of the major energy dissipation sources is

friction which can be modelled as a damping element.

Although the actual description of the damping force is

difficult, ideal damping models often result in satisfactory

prediction of the response. Of these models, a viscous

damping force, proportional to velocity, leads to the

simplest mathematical treatment. Therefore, viscous damping

would be used as the energy dissipation source and the

application of it may provide a potential elimination method










for chaotic bouncing motions.

There are three different ways to add damping elements

to two degree of freedom systems: damping element to the

foot, damping element to the body, and damping elements to

both the body and the foot. However, only the first case

will be discussed, since an addition of a damping element to

the foot could contribute for the elimination of chaotic

behavior. If the viscous damping constant (C2) is too small,

addition of a damping element may not help to eliminate

chaotic bouncing motions. The larger damping constant the

system has, the more likely chaotic bouncing motions can be

removed. However, if the damping coefficient is too high,

the system may not bounce at all. If the damping constant is

bigger than 20N*sec/m with 570N/m of foot stiffness and 0.2m

of dropping height, the system never leaves the ground after

releasing from the given height without any external energy

source. When the system starts to have this behavior with a

damping element, the damping constant is called critical

system damping for bouncing. Critical system damping

coefficients for bouncing (Ccr) are shown in Table 4.1 for the

previously discussed cases. If the damping constant is

smaller than critical system damping coefficient for

bouncing, the system has both ground mode and flight mode

initially. However, the system eventually would have only

ground mode due to gradual energy loss, and finally the

system would stop without any motion. Therefore, in any

cases, the dissipated energy due to damping must be










compensated by external energy sources for continued bouncing

motions.

Table 4.1 Critical System Damping Coefficients for the
Previously Discussed Cases

Foot Stiffness Dropping Height Damping Constant
(K2) [N/m] (h) [m] (Ccr) [N'sec/m]
570 0.2 20
3670 0.2 420
445 0.5 28
1860 0.5 260
2010 1.0 470
2745 1.0 745


In order to see the effects of changes of damping

constants, the bouncing system with 1860N/m of foot stiffness

and 0.5m of dropping height, which is the most chaotic system

for 0.5m of dropping height, will be used as an example.

Phase planes without damping are shown in Figure 2.8. At

first, 5N*sec/m is used as a damping coefficient (Figure

4.1). For the body phase plane, most of high frequency

motions are removed. Although, with this damping, the motion

of the foot still displays a little bit of chaos. If the

damping coefficient is increased to 10N'sec/m (Figure 4.5)

and 25N'sec/m (Figure 4.6), chaotic motions of the foot can

be eliminated more. However, bouncing motions would also be

decreased and motions of the system will be stopped due to

energy dissipation. In fact, with 10N*sec/m of damping

constant, the system has 14 bouncing cycles while the system

has only 6 bouncing cycles with 25N*sec/m of damping










constant. If the damping constant is between 100N-sec/m

(Figure 4.7) and 260N*sec/m (Figure 4.8), then the system

would bounce only once after releasing from 0.5m of dropping

height. Thus, there is a tradeoff. With a high damping

constant, it is easy to eliminate chaotic bouncing motions.

However, it is necessary to have a powerful external energy

source, which can produce required power at once. On the

other hand, chaotic bouncing motions may still exist with a

very low damping constant.

Therefore, further research for an external energy

source should be combined with damping elements to build a

practical legged system.


S0.5
0.4- 0.5-
0.4-
0.2-
0.3-
0.0- 0.2-

0.1
-0.2
0.0-

-0.4- -0.1-

-0.2-
-0.6 --7
-2 0 2 -4 -2 0 2 4
V1 V2

(a) (b)

Figure 4.4 Phase Planes of Two Degree of Freedom System with
h = 0.5m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, K2 = 1860N/m, and
C2 = 5 N'sec/m
(a) Body Phase Plane (Ml), (b) Foot Phase Plane (M2)











0.4


0.2


0.0


-0.2


-0.4


-2 0 2


-4 -2 0 2 4


Figure 4.5 Phase Planes of Two Degree of Freedom System with
h = 0.5m, Ki = 1000N/m, M1 = 10Kg, M2 = 1Kg, K2 = 1860N/m, and

C2 = 10 N*sec/m
(a) Body Phase Plane (Mi), (b) Foot Phase Plane (M2)


-0.2


-0.4


0.0

-0.1

-0.2


-2 0 2
vi


-4 -2 0 2
V2


Figure 4.6 Phase Planes of Two Degree of Freedom System with
h = 0.5m, K1 = 1000N/m, M1 = 10Kg, M2 = iKg, K2 = 1860N/m, and

C2 = 25 N*sec/m
(a) Body Phase Plane (MI), (b) Foot Phase Plane (M2)


0.0

-0.1

-0.2



























-0.1


-2 0 2
vI


-4 -2 0 2
V2


Figure 4.7 Phase Planes of Two Degree of Freedom System with
h = 0.5m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, K2 = 1860N/m, and

C2 = 100 N*sec/m
(a) Body Phase Plane (M1), (b) Foot Phase Plane (M2)


0.0


-0.1


-0.2


-0.3


-0.4


0.00

-0.02

-0.04

-0.06

-0.08

-0.10

-0.12

-0.14


-3 -2 -1 0 1 2


-3 -2 -1 0
V2


Figure 4.8 Phase Planes of Two Degree of Freedom System with
h = 0.5m, K1 = 1000N/m, M1 = 10Kg, M2 = 1Kg, K2 = 1860N/m, and

C2 = 260 N*sec/m
(a) Body Phase Plane (MI), (b) Foot Phase Plane (M2)


0.1

0.0

-0.1

-0.2

-0.3

-0.4

-0.5














CHAPTER 5
CONCLUSIONS AND RECOMMENDATIONS



Chaotic behavior of simplified bouncing systems has been

observed depending on not only system parameters but also

initial conditions. Chaotic bouncing systems with different

parameters and initial conditions have been analyzed to find

some prediction schemes for chaos. A Fast Fourier Transform

has been used for the spectrum analysis. Data sets have been

separated in order to obtain dominating frequencies for the

ground and flight modes of the system. Frequency

relationships between dominating frequencies of each system

mode and natural frequencies of some linear oscillatory

systems have shown that the rigid body ground mode is the

most important for chaotic behavior of two degree of bouncing

systems. Methods for elimination of chaos have also been

discussed.

Although the research results demonstrate that frequency

relationships can be used to predict chaotic bouncing

motions, it does not provide, by any means, a general design

technique for nonlinear intermittent contact problems.

However, it is hoped that this work will be considered as a

viable method for the design and analysis for various

intermittent contact problems such as bilinear oscillatory

systems.





91



One immediate extension of this work is to use a

rotating unbalance as external power source. A

counterrotating eccentric weight exciter, along with damping

elements to remove chaos, may be used for maintaining steady

state bouncing motions by adjusting an eccentric mass with

eccentricity which is rotating with angular velocity, since

it provides mainly vertical excitation while horizontal

excitation is minimized.














APPENDIX A
EFFECTS OF TIME STEP SIZE



Numerical integration have been carried out to study

the bouncing systems. Small time steps must be used to get

the correct responses of two degree of freedom bouncing

systems, since there might be chaos which is very sensitive.

If the time step is not small enough, the numerical solutions

can lead to spurious existence of chaos. For example, 0.02

second is used in Figure A.1 as time step for the body phase

plane plot with Ki = 1000N/m, K2 = 1860N/m, Mi = 10Kg, M2 =

1Kg, and h = 0.5m. With the same system parameters, Figure

A.2 displays the body phase plane when the time step size is

0.001 second. Even though both phase planes do not have

repeated bouncing motions, it proves that if the time step is

not small enough, the system response by the numerical

integration may lead to false conclusion of chaos or

subharmonics.

It is confirmed that one millisecond (0.001 second) is

small enough to be used as time step throughout the

investigation of chaotic behavior of bouncing systems by

reducing it to ten micro-second(0.00001 second). In fact,

the time responses by numerical integration with one

millisecond and ten micro-second for chaotic bouncing systems

are identical.
















0.4


0.2



S0.0-


-0.2-



-0.4-


-0.6 -

-3 -2 -1 0 1 2 3
Vi

Figure A.1 Body Phase Plane of Two Degree of Freedom System
with 0.05 second as time step
(Ki=1000N/m, K2=1860N/m, Mi=1OKg, M2=lKg, and h=0.5m)


-0.2



-0.4


-0.6
-3 -2 -1 0 1 2 3
V1
Figure A.2 Body Phase Plane of Two Degree of Freedom System
with 0.001 second as time step
(K1=1000N/m, K2=1860N/m, Mi=1OKg, M2=lKg, and h=0.5m)




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