Dynamics – Theory and Applications

Dynamics – Theory and Applications
اسم المؤلف
David A. Levinson, David A. Levinson
التاريخ
20 أغسطس 2021
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Dynamics – Theory and Applications
Thomas R. Kane
Stanford University
David A. Levinson
Lockheed Palo Alto Research Laboratory
David A. Levinson
Lockheed Palo Alto Research Laboratory
CONTENTS
Preface ix
To the Reader xv
Chapter 1 Differentiation of Vectors 1
1.1 Vector Functions 2
1.2 Several Reference Frames 2
1.3 Scalar Functions 3
1.4 First Derivatives 5
1.5 Representations of Derivatives 7
1.6 Notation for Derivatives 8
1.7 Differentiation of Sums and Products 9
1.8 Second Derivatives 10
1.9 Total and Partial Derivatives 12
Chapter 2 Kinematics 15
2.1 Angular Velocity 15
2.2 Simple Angular Velocity 20
2.3 Differentiation in Two Reference Frames 23
2.4 Auxiliary Reference Frames 24
2.5 Angular Acceleration 26
2.6 Velocity and Acceleration 28
2.7 Two Points Fixed on a Rigid Body 30
2.8 One Point Moving on a Rigid Body 32
2.9 Configuration Constraints 34
2.10 Generalized Coordinates 37
2.11 Number of Generalized Coordinates 39
2.12 Generalized Speeds 40
2.13 Motion Constraints 43
2.14 Partial Angular Velocities, Partial Velocities 45
2.15 Acceleration and Partial Velocities 50Vi CONTENTS
Chapter 3 Mass Distribution
3.1 Mass Center
3.2 Curves, Surfaces, and Solids
3.3 Inertia Vector, Inertia Scalars
3.4 Mutually Perpendicular Unit Vectors
3.5 Inertia Matrix, Inertia Dyadic
3.6 Parallel Axes Theorems
3.7 Evaluation of Inertia Scalars
3.8 Principal Moments of Inertia
3.9 Maximum and Minimum Moments of Inertia
57
57
59
61
64
66
70
73
77
87
Chapter 4 Generalized Forces
4.1 Moment about a Point, Bound Vectors, Resultant
4.2 Couples, Torque
4.3 Equivalence, Replacement
4.4 Generalized Active Forces
4.5 Noncontributing Forces
4.6 Forces Acting on a Rigid Body
4.7 Contributing Interaction Forces
4.8 Terrestrial Gravitational Forces
4.9 Bringing Noncontributing Forces into Evidence
4.10 Coulomb Friction Forces
4.11 Generalized Inertia Forces
90
90
94
95
99
102
106
108
110
114
118
124
Chapter 5 Energy Functions
5.1 Potential Energy
5.2 Potential Energy Contributions
5.3 Dissipation Functions
5.4 Kinetic Energy
5.5 Homogeneous Kinetic Energy Functions
5.6 Kinetic Energy and Generalized Inertia Forces
Chapter 6 Formulation of Equations of Motion
6.1 Dynamical Equations
6.2 Secondary Newtonian Reference Frames
6.3 Additional Dynamical Equations
6.4 Linearization of Dynamical Equations
6.5 Systems at Rest in a Newtonian Reference Frame
6.6 Steady Motion
6.7 Motions Resembling States of Rest
Chapter 7 Extraction of Information from Equations
of Motion
7.1 Integrals of Equations of Motion
7.2 The Energy Integral
130
130
141
146
147
150
152
158
158
166
169
171
179
183
186
190
190
193CONTENTS Vll
7.3 Momentum Integrals 196
7.4 Exact Closed-Form Solutions 201
7.5 Numerical Integration of Differential Equations of Motion 204
7.6 Determination of Constraint Forces and Constraint Torques 217
7.7 Real Solutions of a Set of Nonlinear, Nondifferential Equations 222
7.8 Generalized Impulse, Generalized Momentum 225
7.9 Collisions 231
7.10 Motions Governed by Linear Differential Equations 241
Problem Sets 259
Problem Set 1 (Sees. 1.1-1.0) 259
Problem Set 2 (Sees. 2.1-2.5) 263
Problem Set 3 (Sees. 2.6-2.8) 269
Problem Set 4 (Sees. 2.9-2.15) 278
Problem Set 5 (Sees. 3.1-3.5) 285
Problem Set 6 (Sees. 3.6-3.9) 289
Problem Set 7 (Sees. 4.1-4.3) 294
Problem Set 8 (Sees. 4.4-4.11) 297
Problem Set 9 (Sees. 5.1-5.3) 310
Problem Set 10 (Sees. 5.4-5.6) 316
Problem Set 11 (Sees. 6.1-6.3) 320
Problem Set 12 (Sees. 6.4-6.7) 328
Problem Set 13 (Sees. 7.1-7.7) 335
Problem Set 14 (Sees. 7.8-7.10) 346
Appendix 361
I Inertia Properties of Uniform Bodies 361
Index 37
INDEX
Acceleration, 28
angular, 26, 27
Coriolis, 32
local gravitational, 101
of one point moving on a rigid body,
32
and partial velocities, 50
of two points fixed on a rigid body,
30
Action and reaction, law of, 97
Addition theorem for angular velocities,
24
Amplification factor, 357
Amplitude, 352
steady-state, 354
Angular acceleration, 26
scalar, 27
Angular momentum, 69, 196, 230, 289,
290, 346
central, 23, 69, 206, 318
principle of, 197, 230
Angular speed, 21
Angular velocity, 15, 16
addition theorem for, 24
definition of, 16
holonomic partial, 46
Angular velocity (Cont.):
nonholonomic partial, 46
partial, 15, 45, 46
simple, 20, 21
simplification of expressions for, 42
Approximations, 244
Astatic center, 296
Astronomical reference frame, 159
Attitude, 267
Auxiliary reference frames, 24
Balancing, static, 58
Ball-and-socket joint, 280
Beams, elastic, 301
Beats, 354
Bevel gear, 272, 302
Binormal, vector, 263
Bouncing, 240
Bound vector, 91
Buoyancy forces, 310
Cartesian coordinates, 35
Cavity, spherical, 303
373374 INDEX
Central angular momentum, 23, 69,
206, 318
Central axis, 295, 296
Central inertia dyadic, 69, 70
Central inertia scalars, 71, 308
Central principal axes, 77
Central principal moments of inertia, 77
Central principal planes, 77
Central principal radii of gyration, 77
Centroid, 60
Characteristic equation, 78
Circular frequency, 242, 352
Circular orbit, 266
Circular wire, 302
Closed-form solutions, 201
Coefficient:
inertia, 151, 319, 320
. of kinetic friction, 118, 233
of restitution, 232
of static friction, 118, 232
Collisions, 231
inelastic, 232
perfectly elastic, 232
Component of a vector, 3
Cone:
right-circular, 184, 274
truncated, 272
Configuration, 34
Configuration constraints, 15, 34
Conservation:
of angular momentum, 230
of energy, 230
of mechanical energy, 194
principle of, 194, 230
Constant:
damping, 108
gravitational, 93
spring, 100, 142, 334
Constraint equations:
holonomic, 35
nonholonomic, 43, 191
rheonomic, 35
scleronomic, 35
Constraint forces, 217
Constraint torques, 217
Contact forces, 99
Contributing interaction forces, 108
Contributions, potential energy, 141, 319
Control laws, 340
Coordinates:
Cartesian, 35
curvilinear, 53
cyclic, 196
generalized {see Generalized coordinates)
ignorable, 196
normal, 244
orthogonal curvilinear, 54
pseudo-generalized, 300, 332
Coriolis acceleration, 32
Coulomb friction forces, 118
Couple, 94
simple, 94
torque of, 94
Coupling, dynamic, 320
Critical damping, fraction of, 242
Critically damped free vibrations, 351
Cube, 314
Curvature, principal radius of, 264
Curvilinear coordinates, 53
orthogonal, 54
Cyclic coordinates, 196
Cylinder, 304, 325
and piston, 304, 325
Damped harmonically forced vibrations,
352
Damper:
nutation, 205
viscous fluid, 108
Damping, critical, 242
Damping constant, 108
Dashpot, 205, 315
Degrees of freedom, 43
Derivative:
first, 5
notation for, 8
ordinary, 6, 10, 12-14
of products, 9
representations of, 7
second, 10
of sums, 9
total, 12
Differential equation:
dynamical, 191
kinematical, 40, 191, 266
linear, 241INDEX 375
Differential equation (Cont.):
of motion, 204
numerical integration of, 204
Differentiation:
ordinary, 9
successive, 10, 11
of sums, 9
in two reference frames, 23
of vectors* 1
Direction cosine matrix, 289
Disk, sharp-edged circular, 265, 297
Displacement amplification factor, 357
Dissipation functions, 146, 316
Distance forces, 99
Door, 18
Double pendulum, 108, 278
Dyadic, 66, 67
central inertia, 69, 70
inertia, 68-70
scalar postmultiplication of, 67
scalar premultiplication of, 67
time-derivative of, 289
unit, 67
Dynamic coupling, 320
Dynamical differential equations, 191
Dynamical equations, 158, 324
additional, 169
Euler’s, 327
Kane’s, 159
linearization of, 171
Feedback control laws, 340
First derivatives, 5
Fluid damper, viscous, 108
Force amplification factor, 357
Forces:
acting on a rigid body, 106
buoyancy, 310
constraint, 217
contact, 99
contributing interaction, 108
Coulomb friction, 118
distance, 99
gravitational, 312
inertia, 124, 125
generalized (see Generalized inertia
forces)
noncontributing, 102, 114
terrestrial gravitational, 110
Forcing function, 242
Formulation of equations of motion, 158
Foucault pendulum, 159, 171
Fraction of critical damping, 242
Free vector, 91
Free vibrations (see Vibrations)
Freedom, degrees of, 43
Frequency:
circular, 242, 352
natural, 242
Friction, coefficients of, 118, 232, 233
Eigenvalues of a symmetric matrix, 79,
242
Eigenvectors of a symmetric matrix, 79,
242
Ellipsoid, inertia, 87, 293
Energy:
conservation of, 230
kinetic (see Kinetic energy)
mechanical, 194
conservation of, 194
potential, 130, 131, 180
Energy integrals, 193
Engine, reciprocating, 325
Equivalence, 95
Euler’s dynamical equations, 327
Exact closed-form solutions, 201
Extension of a spring, 142
Gains, 342
Gear, 272
bevel, 272, 302
pinion, 305
reduction, 302
Generalized active forces, 99
contribution of a rigid body to, 106
holonomic, 99
nonholonomic, 99
Generalized coordinates, 15, 37
number of, 39, 280
Generalized impulse, 225, 346
Generalized inertia forces:
contribution of a rigid body to, 124,
308
and Gibbs function, 310
holonomic, 124376 INDEX
Generalized inertia forces (Cont.):
and kinetic energy, 152
nonholonomic, 124
Generalized momentum, 225, 346
Generalized speeds, 15, 40
Gibbs function, 293, 310
contribution of rigid body to, 293
and generalized inertia forces, 310
Gimbal, 22, 328
Gimbal ring, 2, 22, 337
Gradient, 293
Gravitational acceleration, local, 101
Gravitational constant, 93
Gravitational forces, 312
terrestrial, 110
Gravitational torque, 312
Guldin’s rules, 286
Gyration, radius of (see Radius, of
gyration)
Gyroscope, 2, 4, 22, 328
Gyrostat, 197
Hamiltonian, 193, 335
Harmonic oscillator, 193
Helix, 287
Holonomic constraint equation, 35
Holonomic partial angular velocity, 46
Holonomic partial velocity, 46
Homogeneous kinetic energy functions,
150
Hooke’s joint, 95
Ignorable coordinates, 196
Impulse:
generalized, 225, 346
normal, 232
tangential, 232
Inelastic collision, 232
Inertia:
maximum moment of, 87, 288
minimum moment of, 87, 288
moment of, 62
principal moments of, 77
product of, 62, 287, 292
Inertia coefficient, 151, 319, 320
Inertia dyadic, 68-70
Inertia ellipsoid, 87, 293
Inertia forces, 124, 125
generalized (see Generalized inertia
forces)
Inertia matrix, 66
Inertia properties of uniform bodies, 361
Inertia scalar?, 61, 62
central, 71, 308
evaluation of, 73
Inertia torque, 125, 307, 308
and angular momentum, 308
Inertia vector, 61
Initial perturbations, 214
Integrals:
energy, 193
of equations of motion, 131, 190, 191
momentum, 196
Integration, numerical, 204
Interaction forces, 108
Kane’s dynamical equations, 159
Kinematical differential equations, 40,
191, 266
Kinetic energy, 147
and generalized inertia forces, 152
relative, 318
of a rigid body, 147, 317
of robot arm, 319
rotational, 147, 318
translational, 147
of a triangular plate, 317
Kinetic energy functions, 150
Kinetic friction, coefficient of, 118, 233
Kinetic potential, 196, 327, 337
Lagrange’s equations, 51
of the first kind, 327
of the second kind, 328
Lagrangian, 327, 335
Law of action and reaction, 97
Limit, 6, 16
Line of action, 90
Linear differential equations, 241
Linear momentum, 92, 196, 346
principle of, 197
Linear spring, 142
Linearization of dynamical equations,
171INDEX 377
Linkage, 279, 299
planar, 268
motion analysis of, 268
Local gravitational acceleration, 101
Logarithmic decrement, 351
Lumped-mass model, 249
Mass center, 57, 59
of a uniform body, 60
Mass distribution, 57
Mass matrix, 242
Matrix:
direction cosine, 289
mass, 242
modal, 243
stiffness, 242
symmetric, 79, 242
Maximum moment of inertia, 87, 288
Mean solar time, 159
Measure number of a vector, 3
Mechanical energy, 194
conservation of, 194
Minimum moment of inertia, 87, 288
Modal matrix, 243
Modal truncation, 244
Mode, normal, 244
Moment:
of inertia, 62
bounds, 62
maximum value of, 87, 288
minimum value of, 87, 288
principal, 77
of minimum magnitude, 295, 296
about a point, 90
Momentum:
angular (see Angular momentum)
generalized, 225, 346
linear, 92, 196, 346
Momentum integrals, 196
Motion constraints, 43
Motions resembling states of rest, 186
Natural frequency, 242
Newtonian reference frame, 158, 159,
321
secondary, 166, 167
systems at rest in a, 179
Newton’s second law, 327
Noncontributing forces, 102, 114
Nonholonomic constraint equations, 191
Nonholonomic partial angular velocity,
46
Nonholonomic partial velocity, 46
Nonholonomic system, 43
simple, 43, 51
Nonlinear, nondifferential equations, 222
Normal coordinates, 244
Normal impulse, 232
Normal mode, 244
Numerical integration, 204
Nutation damper, 205
Nutation generator, 217
Orbit:
circular, 266
of a satellite, 17
Ordinary derivative, 6, 10, 12, 13
evaluation of, 14
Orthogonal curvilinear coordinates, 54
Orthogonality, 246
Oscillator, harmonic, 193
Overdamped free vibrations, 351
Pappus, theorems of, 286
Parallel axes theorems, 70
Parallelepipeds, 259
Partial angular velocity, 15, 45
holonomic, 46
nonholonomic, 46
Partial derivatives:
first, 5
second, 10
Partial differentiation, 5
successive, 10, 11
Partial velocity, 15, 45
acceleration and, 50
holonomic, 46
nonholonomic, 46
Passerello-Huston equations, 51, 327
Path point, 266
Pendulum:
double, 108, 278
Foucault, 159, 171
simple, 314378 INDEX
Perfectly elastic collision, 232
Period, 349
Perturbations, 171
initial, 214
Phase angle, 352
Pinion gears, 305
Piston, 304, 325
Piston rings, 304
Planar linkages, 268
Postmultiplication, scalar, 67
Potential energy, 130, 131
principle of stationary, 180
Potential energy contributions, 141, 319
Precession, 334
Premultiplication, scalar, 67
Pressure, 119
Principal axis, 77, 291
central, 77
Principal moments of inertia, 77
Principal normal, vector, 263
Principal plane, 77
Principal radius:
of curvature, 264
of gyration, 77
Principle:
angular momentum, 197, 230
of conservation of mechanical energy,
194
energy conservation, 230
linear momentum, 197
of stationary potential energy, 180
Product of inertia, 62, 287
bounds, 292
Pseudo-generalized coordinates, 300,
332
Pseudo-generalized speeds, 300, 332
Pure rolling, 272
Rack-and-pinion drive, 276, 305
Radius:
of curvature, 264
of gyration, 62, 287
minimum, 290, 292
principal, 77
of a shell, 291
Reciprocating engine, 325
Reduction gear, 302
Reference frames, 12, 23
astronomical, 159
Reference frames (Cont.):
auxiliary, 24
inertial, 159
Newtonian, 158, 159, 321
secondary, 166, 167
systems at rest in a, 179
Replacement, 95
Resonance, 354
Rest:
motions resembling states of, 186
in a Newtonian reference frame, 179
Resultant, 90, 91
Rheonomic constraint equation, 35
Rigid body:
acceleration of one point moving on a,
32
acceleration of two points fixed on a,
30
angular acceleration of a, 26
angular velocity of a, 15
forces acting on a, 106, 124, 308
kinetiq energy of a, 147, 317
rotational motion of a, 15
time-derivative of vector fixed in a, 16
velocity of one point moving on a, 32
velocity of two points fixed on a, 30
Robot arm, 276, 284, 305, 308, 309
dynamical equations for, 324
feedback control laws, 340
kinetic energy of, 319
potential energy contribution of, 319
Rolling, 271, 303, 323
cone, 274
pure, 272
without slipping, 102, 271
Rotational motion of a rigid body, 15
Rotor, 2
of a gyroscope, 22
Routhian, 337
Satellite, 17
Scalar angular acceleration, 27
Scalar postmultiplication, 67
Scalar premultiplication, 67
Scalar triple product, 261
Scleronomic constraint equation, 35
Second derivatives, 10
Serret-Frenet formulas, 263
Shear, 119INDEX 379
Sidereal:
day, 159
time, 159
Simple angular velocity, 20, 21
Simple couple, 94
Simple nonholonomic system, 43, 51
Simple pendulum, 314
Slipping, 102, 271
Smooth surfaces, 102
Solar time, 159
Space curve, 263, 269
Spherical cavity, 303
Spring:
extension of, 142
linear, 142
torsion, 108, 142
Spring constant, 100, 142, 334
Spring modulus, 108, 142
Square of a vector, 10
Static balancing, 58
Static friction, coefficient of, 118, 232
Stationary potential energy, principle of,
180
Steady motion, 183, 334, 344
Steady precession, 334
Steady-state amplitude, 354
Steady-state response, 352
Stiffness matrix, 242
Successive differentiationr 10
in various reference frames, 11
Symmetric matrix:
eigenvalues of, 79, 242
eigenvectors of, 79, 242
Symmetry considerations, 78
Tangent, vector, 263
Tangential impulse, 232
Terrestrial gravitational forces, 110
Thrust bearing, 272
Time-derivative:
of a dyadic, 289
of a vector fixed in a rigid body, 16
Topspin, 239
Torque, 94
constraint, 217
gravitational, 312
inertia, 125, 307, 308
Torsion, 264
Torsion spring, 108, 142
Total derivative, 12
Transient response, 352
Translational motion of a point, 15
Triple product, scalar, 261
Truncation, modal, 244
Truss, 249
Undamped free vibrations, 349
Undamped harmonically forced vibrations, 354
Underdamped free vibrations, 349
Unit dyadic, 67
Unit vectors, mutually perpendicular, 3
Universal gravitational constant, 93
Variable-speed motor, 355
Vector:
bound, 91
free, 91
square of, 10
unit, 3
Vector binormal, 263
Vector functions, 2, 10
Vector principal normal, 263
Vector tangent, 263
Velocity, 28
angular (see Angular velocity)
of approach, 231
holonomic partial, 46
nonholonomic partial, 46
of one point moving on a rigid body,
32
partial (see Partial velocity)
of separation, 231
of two points fixed on a rigid body,
30
Vibrations:
critically damped free, 351
damped harmonically forced, 352
overdamped free, 351
undamped free, 349
undamped harmonically forced, 354
underdamped free, 349
Viscous fluid damper, 108
Wrench

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