History of Dynamics of

Mechanisms & Machines

From Ctesibius to Chaos Theory

Francis C. Moon

Cornell University

Ithaca, New York, USA

Bangalore: 2007

Dynamics ofMechanisms

Kinematic analysis demands geometric,

velocity and acceleration calculations.

Dynamic analysis requires the use ofthe

Newton-Euler-Lagrange equations of motion

either to

Predict torques, power, forces and stresses.

Or

Determine the motion of the mechanism.

Premise of paper:

The advances in dynamic analysis in the 18th and

19th centuries were not absorbed into the theory of

machines and mechanisms until the early 20th

century.

1.The reasons for this delayinclude lack of common interests

between applied mathematics community and the machine

theorists and practitioners.

2.Different paradigms between dynamicists and mechanism

theorists.

3.Failures of early machines were not speed or dynamics related.

Greek and Roman interests in

dynamics ofmachines

Ctesibius

Archimedes

Hero

Vitruvius

Clock of Ctesibius

Leonardo da Vinci and the Dynamics of

Machines

Clock escapements

Trebuchets

Flywheels

Perpetual motion wheel

Flywheels in Codex Madrid

Trebuchet: Codex Atlanticus:

Double Pendulum

Perpetual Motion Wheel

“such a wheel is

sophistical”

L. Da Vinci

Codex Madrid

Besson’s

Pendulum

(c. 1569)

Human resonance

excitation drives the

pendulum whose

inertia moves the

mangle and the

saw.

Böckler’s Pump

Escapement

(c. 1616)

Verge and foliot

escapement used to

control the weight driven

torque and the speed of

the chain of pots pump.

Automata: 18th & 19th Centuries

Japanese Tea Serving Doll

Cam and

escapement

controlled: Spring

Driven

Leonardo’s Automata?

Codex Madrid:

circa 1490

See Rosheim;

Leonardo’s

Lost Robots

2007

Analytical Dynamics History

Galileo (1634) [Falling bodies, projectiles]

Newton (1686) [Orbital mechanics of planets]

L. Euler (1760) [Continuum mechanics]

Lagrange (1788) [Spinning tops]

Hamilton (1834) [Optics]

Newton-Euler Equations of Motion

ΣF

i

=

d

dt

mv

c

{Linear Momentum Law}

ΣM

ci

=

d

dt

H

c

{Angular Momentum Law}

H

c

=I

1

ω

1

e

1

+I2

ω

2e2

+I

3

ω

3e

3

Euler’s Equations [1760]

P=A

d

dt

p+(C−B)qr

Q=B

d

dt

q+(A−C)pr

R=C

d

dt

r+(B−A)qp

ω

=[p,q,r]

T

{Angular Velocity Vector}

Lagrange’s Equations (c. 1788)

d

dt

∂

∂

Ý

q

L−

∂

∂

q

L=0 (Holonomic Constraints)

L=T(

Ý

q ,q)−V(q)

T= [Kinetic Energy]

V= [Potential Energy]

v=J(q)

Ý

q

Technical Dynamics of Mechanisms

in the Age of Machines

Watt’srotating ball governor

Clock escapements

Gyroscope

Regulators & Servomechanisms

Rotating ball

governor

James Watt,

circa 1790

Brown: Gyros-1868

501 Mechanical Movements

Kinematics of Machines:

19th Century

Borgnis

Willis

Rankine

Redtenbacher

Laboulaye

Galon

Reuleaux

Kennedy

Burmester

Grübler

James Clerk Maxwell:

“On Governors”, 1868

Proc. Royal Society, No. 100, 1868, p105-120

Experiments of Mr. Fleeming Jenkins;

“By alternating these adjustments the regulation could

be made more and more till at last a dancing motion

of the governoraccompanied with a jerking motion

of the main shaftshewed that an alteration had taken

place among the impossible roots of the equation.”p.

107.

James Clerk Maxwell:

“On Governors”, 1868

MBn

3

+(MY

+

䙂)n

2

+

FYn

+

䙇

=

0

Characteristic equation of the governor

x=Ae

nt

Maxwell applied stability theory to steam

enginerotating ball mechanisms.

Gyroscopes

Foucault, 1853

H. Anschütz,

1908

Elmer Sperry,

1911

Dynamics ofClocks

Cylinder

escapement:

Impact forces,

Friction,

Nonlinear Limit

cycles.

Components of Mechanical Clocks:

1. Pendulum

2. Anchor arm of escapement

3. Escape wheel

4. 5. 7. Parts of gear train

8. Drive weight

From Martinek and Rehor

Mechanische Uhren, 1996

Dynamical Theory of Machines:

Clock Escapements

Huygens’Clock

(1657)

Verge and Foliot

Escapement

Pendulum with

isochronous support

Dynamics of Mechanisms:

Early 20th Century

Balancing of engines

Gyroscopes

Wheel shimmy

Turbine dynamics

Self excitation: valve chattering andengine hunting

Elastic vibrations of engine components

Servomechanisms

Vibration absorbers

Two Types of Problems in

Dynamicsof Machines

(i)Dynamics of entire machine system:

e.g. clocks, regulators, flight stability of

aircraft

(ii)Dynamics of individual machine

components:e.g. vibration of machine

components

Russian School: Mertzalov

Early 20th Century

Thanks toProf.

Alexander Golovin,

Moscow

Dynamics of Machines: Mertzalov

Dynamics of

Mechanisms;

Moscow

1914

Dynamics of Engine Slider Crank

Mertzalov; 1914

Vibrations in Machines:

Mertzalov, 1914

Balancing of Engines

From:

Ham &Crane

c.1927

S. Timoshenko (1928) and

J.P. Den Hartog (1934)

Fig. 232;

Mechanical

Vibrations,

1934

Den Hartog

Late 20th Century Developments in

Dynamics of Machines

Multi-body dynamics

Robotics

Mechatronics

Multi-body codes [ADAMS, DADS,etc]

Finite element codes

Nonlinear dynamicsand“chaos Theory”

Chaos Theory

Basic results: 1978-1998

•Dynamical systems can be very sensitive to initial

conditions.

•Periodic inputs can result in non-periodic vibrations

•Chaotic vibrations exhibit fractal properties in phase

space {Poincare maps}

•Chaotic systems can sometimes be controlled with

small inputs.

Chaos Theory:

Examples

Gear rattling chaos

F. Pfeiffer: TU München

c. 1982.

Chaotic Dynamics:

Examples

Impact Printer Chaos

IBM: c. 1980

Chaotic Dynamics in Clocks

James Bloxam, 1854, Mem. Royal Astronomical Society:

“On the Mathematical Theory and Practical Defects of Clock

Escapements”

P 121: “It is well known that the force transmitted by the clock-

trains is very far from constant”

P128”the theory of this escapement, however perfect it may be

by itself, must be rendered practically imperfect by the

mechanical imperfections which we cannot estimate.”

P136 “How it happens that disturbing causes, ---, occasion so

much irregularityin the rate of the clock is not immediately

obvious”

The Role of Chaos and Noise

in Clock Escapements:

* Chaos or Noise (dither) may break frictionor

stiction in gears and other components.

* Noise may help lubrication

* May help remove wear particles out of key parts

Model: Oscillator with Noise triggered

impulse

Ý Ý

x

1

+

γ

1

Ý

x

1

+

ω

1

2x

1

=

tq(x3

)sign(

Ý

x

1); (1

Ý Ý

x

3

+

γ

2

Ý

x

3

+

ω

2

2

x

3

+

β

x

3

3

=

α

x

1; (

2

tq=T

0, if (Ý x

3)

2

≥

ε

; (3)

tq=0, otherwise.

Pendulum

Structure

‘Noise’

Trigger

Pendulum/Structural noise triggered

escapement dynamics:

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

-2

-1

0

1

2

-2

-1

0

1

2

0

500

1000

1500

2000

-2

-1

0

1

2

0

500

1000

1500

2000

-1

-0.5

0

0.5

1

Pendulum Structure

Chaos in a Clock Model

[F.C. Moon, 2003, IUTAM Chaos:Rome

Moon & Steifel 2006, Trans. Royal Soc.]

PendulumStructure

Poincare Mapof chaotic clock noise in model. (FCM 02)

Fractalsignature of strange attractor.

Lessons from the History of ClockDesign:

* With each new design generation, some noiseand

unpredictability remains in the next generationof clocks.

* With each new generation, the number of parts orcomplexity

increaseswhile the overall size decreases.

* The primary function, the rate of the pendulum,is a projection

of the total dynamic state space vector, onto a very low

dimensional subspace.

* Chaos/ noise may serve todecrease frictionin gear train.

Control

Scheme for

Elasticarm

Elastic robot manipulator arm:

dynamics plays an integral role in

design.

Compliant Mechanisms: Dynamics

Essential to Design

QuickTime™ and a

YUV420 codec decompressor

are needed to see this picture.

Walking Robots: Prof. Andy Ruina,

Cornell Univ. Dynamics Essential to Design

QuickTime™ and a

Sorenson Video 3 decompressor

are needed to see this picture.

Two

Coupled

Double

Pendula

.

Summary

•Dynamic problems in machines reach back to antiquity.

•Analytical dynamicsmatured by the early 19thC.

•Kinematics of machines in 19thC.rarely discussed

dynamics problems in mechanisms.

•A systematic use of dynamics and vibrations in

machines developed in early to mid 20th C. with

Timoshenko, Den Hartog and others: 1930-1960.

•Systematic use of dynamic principles standard in

robotics but still generally lacking in teaching of

mechanism design.

Reuleaux in India: 1884

Rx in India

Rx in India

Rx in India

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