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
NewtonEulerLagrange 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]
NewtonEuler 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: Gyros1868
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, p105120
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
Multibody dynamics
Robotics
Mechatronics
Multibody codes [ADAMS, DADS,etc]
Finite element codes
Nonlinear dynamicsand“chaos Theory”
Chaos Theory
Basic results: 19781998
•Dynamical systems can be very sensitive to initial
conditions.
•Periodic inputs can result in nonperiodic 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: 19301960.
•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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