Bicycle Dynamics

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Using Bicyclesto Teach System Dynamics

Richard E. Klein

ABSTRACT: This paper reports on an in-

novative approach, based on open-ended de-sign questions related to bicycles, for the

teaching of dynamic systems concepts in an

undergraduate mechanical engineering en-

vironment . The paper out l ines needs for

improved classroom learning, pedagogical

methods and underlying philosophy, how the

bicycle was introduced as a main portion of

the instruction, steps as to how the c lass was

managed, supporting materials used, and a

summary of major benefits achieved.

Introduction

A perennial task that plagues engineering

faculty is that of keeping students interestedin a rigorous introductory class in dynamic

systems modeling and automatic control. As

the students progress from sophomore-level

courses, such as differential equations, and

up to advanced-level systems classes, the

question before educators is when and how

to bring some relevancy to the typically the-

oretical and abstract mathematical treatment

associated with systems theory. Sophomore-

level courses introduce simple elements such

as spr ings and dashpots , however , the mo-

tivational value at that level is questionable.

Junior- and senior-level courses often use

somewhat more complex dev ices as physical

examples, such as liquid-level control of

tanks. Few students, in this author’s expe-

rience, seem to get excited about the rele-

vancy and scientific challenge presented by

a passive toilet-water closet where a float

controls the water flow in. Advanced courses

may utilize “high-tech” exam ples, such as

aircraft autopilots and fly-by-wire concepts,

but the students who continue on in the sys-

tems area to see these illustrations are usu-

ally a minority. Those students who acquire

an interest in systems theory often do so be -

cause of its mathematical elegance and not

necessarily because a mea ningful application

stimulated their interest.

This is a modified version of a paper presented atthe 1988 American Control Conference, Atlanta,

Georgia, June 15-17, 1988. Richard E . Klein is

with the Department of Mechanical and Industrial

Engineering at the University of Illinois, Urbana,

IL 61801.

In recent decades, the teaching of systems

theory has become increasingly based onmathematical underpinnings, and thus sys-

tems courses often resemble courses in

mathematical topology rather than an ele-

ments-of-machinery course. A relevant con-

crete example of a dynamic system that the

student can address and confront is needed.

The concrete example should encompass o r

use a full range of systems-theoretic tools so

a s to be challenging, but not yet overpower-

ing, to the student. Even the tuned mass

damper (TMD), in spite of its elegance and

sophistication, just doe s not serve a s a strong

motivation for students. A hardware dem-

ons trat ion of a laboratory TM D is examined,

equations are derived, Bode plots are dis-

cussed, the students place their hands on the

stationary mass when driven at the resonant

frequency of the tuned mass and are briefly

amazed, but then the experience appears to

be forgotten by the students as they pass o n

to their next class. Could the failure of the

TM D to ignite the students’ interest lie in

the fact that the students remain as passive

observers to a sterile problem that is neatly

packaged and solved, and presented as a

clinical dem onstration of established fact and

not as a current challenge’?

In the Department of Mechanical and In-

dustrial Engineering at the University of Il-

linois, the author uses bicycles to involve

and motivate students in a junior-level firstcourse in dynamic systems. The result is a

lively, hands-on course that produces an ar-

ray of unusual bicycles that prove, o r fail to

prove, a number of systems-theoretic ques-

tions related to the bicycle and its associated

dynamics and control. Questions examined

include open-loop stability, impulse re-

sponse behavior demonstrating the bicycle’s

nonminimum-phase characteristics, rider

control laws, parametric influences such as

altering mass distributions, augmented steer-

ing control laws, and the effect on dynamic

response due to variations in the steering

control law. A foundation concept in sys-

tems theory is that the design options in-clude: (1 ) s imple parameter changes , (2)

inclusion of dynam ic compensation or equal-

izers, and (3) alteration of loop structures in

a block diagram sense. Exp eriments with bi-

cycles permit all three levels of synthesis

methodology, which the students do natu-

rally, and which the professor later pointsout that three synthesis alternatives were ac-

tually used. T he bicycle is a dynamic system

in every sense of the word, and it follows

that any attempt to examine the bicycle ob-

jectively requires a simultaneous usage of

dynamic systems language and methodol-

ogy. Thus, by studying the bicycle, the stu-

dents ultimately are expo sed to systems-the-

oretic language and thought processes. The

pedagogical experience is enhanced because

the correlation, or lack thereof, betw een the-

oretical and physical is readily accomplished

and thus made real to the student. Mo reover,

the students embrace the abstract methods of

systems theory when complex and opaque

topics become clear as a result of application

of the system s-theoretic tools.

The bicycle has been in existence for over

a century but yet many mysteries presently

surround the bicycle. Issues that complicate

the bicycle include the nonholonomic con-

straints, algebraically coupled higher deriv-

atives, the vague nature of the lateral tire-

road forc es, the fact that different riders often

use different inputs (control laws) based on

some individualized combination of kine-

matic steering and body articulation, the

presence of both soft nonlinearities (such as

trigonometric terms associated with the in-

verted pendulum) and hard nonlinearities

(such as the sudden onset of slippage at thetire-road interface), and the often misunder-

stood (and frequently overstated) role of gy-

roscopic effects. Th e bicycle is dynam ically

coupled because of the kinematic constraints

present in its mechanical structure. The ki-

nematic coupling means that an acceleration,

in any given component of the bicycle (for

example, the front fork), is algebraically

coupled to accelerations in virtually all other

components of the bicycle. As an additional

subtlety, the rider is free to exert a torque

on the handlebars so as to accomplish steer-

ing, and yet a feedback loop internal to th e

rider usually monitors and dictates a kine-

matic (positional) input as opposed t o a forceor torque input .

The bicycle is not a trivial topic, as on e

might supp ose at first glance, but it is a rather

formidable subject worthy of study. Fortu-

nately, for pedagogical reasons, the students

0 2 7 2 1708/89f04000004 S O 1 00 d 1989 I E EE

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naively assume at first glance that the bicycle

must be "easy" because of their obvious

familiarity with the bicycle. From that point

on. the bicycle serves as an illusive target

that resists solution attempts and becomes

inore complex with each successive try. The

bicycle adopts a role much like the errant

broom in P. A . Dukas' n e Sorcwer' s Ap -

prenticx,. Each repeated attempt to grasp thebicycle usually results in a fractured problem

and for which the issues become more illu-

sive and multiplied. The students. as ap-

prentices. become increasingly motivated to

acquire more and niore theory so as to over-

come this common object. the bicycle. When

the students thirst for niore knowledge and

eagerly grab the body of systems-theoretic

material, then the pedagogical victory is as-

sured.

The usc of bicycles as a teaching vehicle

at the University of Illinois evolved over a

period of approximately five years. At first,

the introduction of a baseline or "generic"

bicycle was somewhat incidental and in-

volved only modest analysis and simulation.This stimulated student interest. but few

questions were really resolved. However.

within the last two years the students under-

took the building of prototypes capable of

parametric variation so as to test specific hy-

potheses. The students' interest and the ed-

ucational gains increased dramatically with

the introduction of hardware inquiry. Certain

fundamental issucs extant to the bicycle

tended to become resolved, as opposed to

remaining vague. For example. several

groups of students took on the construction

of zero-gyroscopic bicycles in an effort to

answer the assertion from certain well-in-

tended but misinformed scientists that it is

the gyroscopic effect of the rotating wheelsthat keeps a bicycle upright. Experiments

demonstrate that the zero-gyroscopic bicy-

cles are ridable. as shown in Fi g. 1. thus

refuting the assertion. After several senies-

ters of addressing parametric variations. the

result is a laboratory full of funny-looking

bicycles. all of which prove. o r disprove. a

range of theories or ideas concerning the dy -

namics of bicycles.

A s a consequence of experimenting with

modified bicycles. a connection emerges be-

tween the hardware prototypes and thc the-

oretical aspects and methodology of dynamic

systems and controls. A s an example, sev-

eral rear-steered bicycles were built and

tested. The first rear-steered bicycle incor-

porates a seat mounted over the steering

head, and then an extra steering head is

Fig. 1.

gyros copic bike 11, the "nai ve" bike Mith no feedback effects 011

th e ,front , fork. zero-gyroscopic bike I , mid th e front- and renr-

steered hikc.

Students t n i n g o u t f i ~ u r i kes , f ro m r ig ht to left: ,-cro-

to the steered wheel in the rear. Thus. the

bike is rcvcrscd completely: handlebars in

the front turn the rear wheel and the pedals

drive the front whccl. It is an annual chal-

lenge with students to ride this bicycle. but

nobody has mastered this particular rear-

steered bicycle. The author is depicted in

Fig. 2 as he attempts, u nsuccessfully, to ride

this bicycle.

The power of a nalysis and synthesis tech-

niques i s made vividly clcar to thc students

as related to the rear-steered bicycle. Spe-

cifically, analysis of the equations of motion,

based on an idealized rigid mass inverted

pendulum and an appropriate kinematic

ground track model. suggests several design

changes. including shifting of the center of

mass forward and upward. and use of a

shorter wheelbase. This design modification

yields conditions suitable fo r stable closed-

loop control assuming a first-order control-

ler. Students also have investigated the use

of pole-placement techniques in order to sta-

bilize the inherently unstable rear-steered bi-

cycles. The discussion of pole-placement

techniques then triggers a need for the dis-

cussion of observability and controllability

of state equation models. The class really

starts to believe in the power of systems-

theoretic techniques when class members re-

port that they experimentally modified a rear-steered bicycle, as suggested by analysis of

the equations. and that the bicycle is ridable.

although challenging to ride. See F ig. 3.

welded into the frame in the fomier position Fig. 2. The author demonsrrating the Fig. 3. University of Illinois

of the seat post, and a conventional bicycle

chain connects, via sprockets, the handlebars

futility o f a proportional steering control

/ U N $ on rear-steered bike 1.

undergraduate demonstrating his rear-

steered bike II.

ADrrl I989 5



Pedagogical Issues

A number of innovative teaching tech-

niques are employed, all of which are de-

signed to act in concert to bring about a

pedagogical environment conducive to

learning. Th e word ‘‘learning” is interpreted

in the sense of Bloom [l ], and thus involves,

as per Bloom, the six-level hierarchy of

knowledge (at the fact level), comprehen-

sion, application, analysis, synthesis, and

evaluation. In response to the pedagogical

evidence that “horizontal re levance” and

problem solving are needed (see [2]), th e

course in dynamic systems is applied such

that the bicycle is the central tangible ex-

ample. The learning is enhanced, in part,

because the student becomes an active par-

ticipant in an open-ended design, analysis,

and synthesis question. The method, where

the professor is hunting for the answer along

with the class, is commonly described as

“shared inquiry. ”

In addition to the bicycle, other hardware

applications are considered during the

course, including flyball governors, hy-

draulic servomechanisms, dynamics of raweggs, passive R-L-C circuits, a TMD, and

the task of backing up a vehicle with a trailer.

However, the bicycle application remains the

dominant thrust. In spite of the complexity

of the bicycle mathematically, the pedagog-

ical emphasis is to first address the bicycle

problem in physical terms, and then present

any essential theoretical tools and/or con-

cepts as the need dictates. Th e philosophy in

the introductory course is to ( 1 ) use and/or

introduce a minimum of tools and only tools

that are reliable and well understood, and ( 2 )

avoid muddying the waters with redundant

or nonessential theoretic methods. In es-

sence, only one method is initially intro-

duced to solve a given problem, and discus-

sion of the fact that various other methods

are available is deferred. Broad exposure to

parallel theoretic approaches is reserved for

later in the course and in elective courses in

systems theory.

At the University of Illinois in the Me-

chanical Engineering curriculum, the stu-

dents were deemed to have the following

prerequisite skills and understanding: (1)

matrix algebra up to the point of linear al-

gebraic equations including Cramer’s rule,

( 2 ) modular thinking in the use of subrou-

tines as a result of compu ter science courses

and general computer background, (3) the

calculus, (4) acceptance of and firm trust inHeaviside derivative operator (D = d/dt)

techniques, and (5) some basic exposure to

dynamics associated with Newton’s Second

Law. These five basic tools were identified

and are used as the foundation skills , al-

though students usually have, of course,

other skills in addition to these five.

At the start of the course, the students are

exposed in lecture to Lagrangian dynamics.

Next, they are introduced to the use of in-

tegrator (simulation) block diagrams as a

means of representing dynamic systems,

which rests on the recasting of dynamic

problems as integral equations rather than

differential equations. Additional instruction

includes simulation skills, block diagram

algebra, and state-variable representation

(taken directly from the integrator block dia-

grams and including systems w ith numerator

dynamics). The students are introduced to

the procedure of combining block diagram

models, Heaviside’s derivative operator,

transfer functions, and Cramer’s rule. A

FORTR AN subroutine called PERSEUS (see

[3]) is the computational tool that permits the

students to obtain transfer functions in a sin-

gle-step process from either raw equations

or any (linear) block diagram independent of

the degree of coupling, the number of equa-

tions, the respective orders of each of the

individual transfer-function blocks, the pres-ence of numerator dynam ics, the presence of

derivative blocks, and/or interlaced loops.

The student is first exposed to this approach

using hand calculation in order to obtain a

transfer function. The logic and methodol-

ogy of using PERSEUS rests solidly on the

five foundation skills cited earlier so that the

student readily accepts PERSEUS as a work-

ing tool and as a conceptual step.

As the semester progresses, the student

comes to rely on the subroutine PERSEUS

as its use eliminates the toil and abstractions

associated with block diagram algebra re-

duction and/or Mason’s gain rule. Block dia-

gram algebra reduction is a multistep pro-

cedure, and Mason’s rule is single step;

however, the task of identifying all of the

forward paths and all of the loop paths can

be formidable, even in problems of just

modest complexity. Also, the calculations

associated with application of Mason’s tule

may become tedious. It is true that state-of-

the-art programs such as MA TLA B will per-

form block diagram algebra reduction, but

in nonholonomic and kinematically coupled

problems like the bicycle, the task of coming

up with the initial block diagram represen-

tation directly from the kinematically cou-

pled equations of motion is immense. The

example used to introduce the student to

PERSEUS concepts is that of determiningthe ground tracking kinematics that relate

steer angle of a bicycle to the lateral dis-

placement of the ground contact points, and

then amving at a transfer function. This

transfer function is then augmented with two

additional transfer functions, one for the in-

verted pendulum dynamics, and another for

a steering control law, so as to achieve a

simplified closed-loop model of bicycle be-

havior. These procedures are detailed in th e

Appendix. State-variable models are then

shown to be obtainable directly from all

physically realizable transfer functions, both

those obtained by hand calculations and those

obtained with the aid of PERSEUS. The

methodology of using PERSEUS on more

complex systems, such as a set of four ki-

nematically coupled second-order bicycle

equations, and also for block diagram sys-

tems, is given in [3].

Once the understanding of state formula-

tions and transfer functions is steadfast, it

becomes a direct matter to cover root-locus

techniques as well as frequency-domain de-

sign. Students have access to a number of

different computing facilities, and they gain

exposure to state-of-the-art synthesis and de-

sign packages such as Program CC and

MATLA B. It is worthy to note that the tran-

sition into the frequency domain is preceded

by the study of R-L-C circuits using the La-place and Fourier domains where the use of

phasors is rigorized, and then the replace-

ment of the Laplace variable s with j w fol-

lows directly and in a pedagogically sound

manner. As such, the transition into the fre-

quency domain is reinforced by the students’

background in a prerequisite circuits course,

and, moreover, the concepts taught previ-

ously in the circuits course are also solidi-

fied.

Additional considerations, such as nonlin-

ear effects, are made evident to the student

because of exposure to the bicycle. As ex-

amples, the consideration of nonlinearities

arises with the need to linearize the trigo-

nometric terms in the inverted pendulum

equations; the recognition that excessive lat-

eral inertial loads will exceed the coulomb

friction bounds on the tire-to-road interface;

and that the steady-state value of the steering

control law for no-handed riding is a mono-

tonically diminishing (and thus nonlinear)

function of the amplitude of the bicycle’s

lean angle. Other advanced considerations

include the presence of unmodeled dynam-

ics, such as frame an d/or front fork flexure,

tire deformations and associated slip angles,

and aerodynamic considerations. In addi-

tion, students become exposed to the need

for process identification procedures and

adaptive control via the clinical study ofyoungsters learning to ride a bicycle. The

laboratory utilizes bicycles with variable pa-

rameter adjustments, and experimental ride

data are monitored with a SoMat 2000 field

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computer [ 4 ] .The SoMat 2000 permits the

acquisition of actual ride data, which are then

uploaded in a matter of minutes into a lap-top computer andlor a mainframe. Thus,

students are exposed to data acquisition

techniques and the subsequent issues, for

example, of model verification, spectral

analysis, and process identification.

Z(t) =

Course Specifics

The one-semester course, ME 24 0 (four

semester credit-hours), as taught at the Uni-

versity of Illinois, m eets three hours per wee k

for lecture, and the students also attend a

weekly 2-hr laboratory. The laboratory is

taught by teaching assistants, and the activ-

ities include digital simulation, use of pack-

ages such as Program CC and MATLAB,

hardware demonstrations andlor schematic

discussions of hardware, oral presentations

by students, bicycle testing and/or replay of

videotapes of bicycle tests, question-and-an-

swer periods on course material, and oral

defenses of student essays. The students are

provided prepared handouts for each of the

approximately 10 laboratory exercises each

semester. The students are required to re-

search and write some form of essay dealing,

typically, w ith some open-ended bicycle-re-

lated topic. Theoretical topics covered in

lecture, but which have a bearing on the bi-

cycle, include convolution, the Laplace

transform, open-loop vibrational problems,

stability tests, root locus, proportional-inte-

gral-derivative (PID) control laws, and fre-

quency-domain considerations. These theo-

retical topics follow in a logical manner, and

the student comes to view these topics as part

of systems theory. Once the notion of the

bicycle as a valid system is established in

the students’ minds, additional theoreticaltools such as root locus and PID control laws

are readily accepted.

Essay topics addressed in recent semesters

have included the following:

Explain how and why a bicycle works.

Investigate a bicycle in fractional gravity.

Analyze a rear-steered bicycle.

Explain how a child’s tricycle works.

Investigate gyroscopic influences

Explain a bicycle ridden with “n o hands.”

Design a single track stable trailer.

Design and build a robot for riding a bi-

Design an intuitive steering system.

Explain “critical velocity” of bicycles.

cycle.

The author makes an effort to have a sup-

ply of discarded or used bikes on hand. The

DIV 0 4 - 1

0 D 0 - V

+ 1AlAI L BIL 0 0

1 /L -1lL 0 -1

cost of raw materials is minimal as the bi-

cycles are often procured economically at

police auctions. Any fabrication, typically

welding, is performed in the departmental

machine shop, but only after the design and

drawings have been approved by the profes-

sor. Simple tasks such as drilling, hacksaw

cutting, and painting are done by the stu-

dents, often in a student shop that is equippedwith a bench, vise, and drill press.

Conclusions

In overview, the use of the bicycle as a

concrete but challenging example of a dy-

namic system permits students to improve

their mastery of the abstractions of systems

theory. The students have the experience of

working on a meaningful and challenging

open-ended design problem. Five years of

experimentation and refinement of the ap-

proach have been well received by both

administration and students. The results to

date show that ( 1 ) the notion of using the

bicycle in the classroom as a teaching tool

and research topic is feasible, (2) the asso-

ciated economics are attractive, (3) students

are able to apply the abstract notions of sys-

tems theory to a concrete problem, ( 4 ) the

professor is able to improve his or her ex-

pertise in a designated area (such as two-

wheeled vehicle dynamics), (5) the percent-

age and quality of students electing follow-

up courses in the systems area increase, and

(6) students improve their professional con-

fidence as a result of the substantially

broadened “horizontally relevant’ ’ educa-

tional experience.

Appendix

Consider the dynamics of the bicycle’s

ground track kinematics as shown in Fig.

A l, which depicts a projected top view of

the bicycle’s lateral motion with respect to a

centerline. The bicycle’s forward velocity is

V , L represents the wheelbase, A is the hor-

izontal distance from the rear contact point

to the center of gravity (CG), and B is the

distance from the C G to the forward tire con-

tact point. Thus,

L = A + B ( Al l

Kinematics and geometry yield the following

equations, where 4 is the steer angle and $

the yaw angle of the frame.

drldt = V sin (4 + $) (A2)

(A31

6 4 4 )

(A5)

Now, linearization of the trigonometric terms

utilizing sin 4 = 4, sin $ = $, cos 4 =

1 , and cos $ = 1 yields

dyldt = V sin $

z = (Ax + By) l L

$ = sin-’ [(x - y ) l L ]

drldt = V ( 4 + $) (A61

(A71

(AS)

(‘49)

dyldt = V $

z = (A x + B y ) / L

$ = (x - yV L

Introduction of the Heaviside derivative

operator ( D = dldt ) allows Eqs. (A6)-(A9)

to be cast in a matrix algebra form, where

the steer angle 4 is an assumed or given

input

1:1L -BI L -1 0

- 1 / L 0 -1

=[]By defining the 4 X 4 matrix in Eq. (A10)

as A , Cramer’s rule may now be invoked to

obtain Eq. (A1 1 ) fo r ~ ( t )s

Velocity

Reference/Centerline/g--

_ _ _ -Fig. Al . Ground contact projection

kinematics.

April 1989 7



Algebra yields the transfer function relat-

ing 4 an d z ( t ) as

(A 12)

Now consider the inverted pendulum of

the bicycle and the rider as shown in Fig.

A 2 . Using the method of Lagrange, or al -

ternately the method of Newton, the linear-

ized differential equation relating z ( t ) an d

p(t)-where M is the mass, H the CG height ,

I,, the mass moment of inertia about the

C G , g the gravitational constant, and p the

lean angle-is

z ( t ) (VlL) [ V + AD ]_ -

4(t) D ?

(MH’ + I,-,) ( d 2 p l d t 2 )- M g H p

= -M H ( d ’ z l d t 2 ) ( ~ 1 3 )

The necessity arises to consider a steering

control law. Simplicity of the mathematics,

combined with some empirical evidence,

suggests the initial use of a linear and pro-

portional steering control law, where K is the

proportionality constant

4(t) = fw ) (‘414)

As a consequenc e, the closed-loop bicyclesystem may now be expressed in block dia-

gram form as shown in F ig . A3 .

Introduction of simulation diagram tech-

niques, based on Eqs. (A6)-(A9 ), (A13 ), and

(A14), yields the system shown in F ig . A4 .

The effect of an impulse input, such as by a

jabbing forward thrust on one of the handle-

bars, yields an initial condition on the inte-

grator that produces the lateral displacement

of the front wheel, ~ ( t ) .alues for the nec-

essary constants may be assumed to be the

following: V = 6.7 rnisec, IC , = 20.0 kg -

m 2 , A = 0 . 4 m , g = 10.0 mls ec , B = 0.7

m , K = 0.4, H = 1. 2 m, and M = 70.0 kg .

The system shown in Fig. A4 lends itself todirect simulation, such as with ACSL, Pro-

Fig. A2 . Rigid idealized inverted

pendulum.

IC G

Fig. A3. Simplijed bicycle block

diagram.

Fig. A4 . Bicycle simulation block

diagram.

gr am CC, or MAT LAB, for example. The References

student is able to compute the bicycle’s non-

minimum-phase impulse response as shown

in Fig. A5. This response agrees with the

students’ experience, especially those stu-

dents who are exper ienced o n motorcycles ,

in that an impulse causing a momentary steer

to the left, for example, will result in a turn

or steer of the vehic le to the right of its initial

course. Thus, countersteering is demon-

strated in the simulation of the two -wheeled

vehicle.

[l] B. Bloom (ed.). Taxonomy of Educational

Objectives, New York: Longmans, Green

& Co., 1956.

L. G. Katz and S . C. C hard, Engaging Chil-

dren’s Minds: The Project Approach, Nor-

wood, N J : Ablex Publishing Co., 1989.

131 B . C. Mears, “Open Loop Aspects of TwoWheeled Vehicle Stability Characteristics.”

Ph.D. Dissertation, Dept. of Mechanical and

Industrial Engineering, Univ. of Illinois, Ur -

bana, 1988.

N. Miller, D. Socie, and S . Downing, “NewDevelopments in Field Computing Sys-

tems,” SA E Paper 870804, Apr. 1987.

[2]

141

Fig. AS.

to sudden JAB input.

Impulse response of bicycle due

Richard E. Klein was

born in Stratford, Con-

necticut, in 1939. He re-

ceived the B.S. and M.S.

degrees in mechanical en-

gineering in 1964 and1965, respectively, from

Th e Pennsylvania State

University. In 1968, he

received the Ph.D. degree

in mechanical engineering

from Purdue University.

His Ph.D. dissertation

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Bicycle Dynamics