Greenwood D.T. Advanced Dynamics

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419 Answers to problems

6.11. (a)

x = v,

θ = ω, ˙v =

−lω

sin θ + g sin θ cos θ

2 −cos

˙ω =

−lω

sin θ cos θ + 2g sin θ

l(2 − cos

θ)

(b)

tx θvω

0.2 0.506533 0.113307 0.099539 0.703590

0.4 0.554881 0.314942 0.420204 1.409748

0.6 0.676823 0.712392 0.740709 2.618152

(d)

= 0.741316,

= 2.741575

(e)

x = 1.333450,

θ = 1.843889,

λ = 1.833450

(f)

x = g sin θ cos θ −l

sin θ,

θ =

sin θ

(g) Leaves wall at t = 0.697577 when

x = 0.801676 θ = 0.930094

x = v = 1.446778

θ = ω = 2.420338

Index

acatastatic system, 38

accelerating reference frame, 20, 21

acceleration of a point

Cartesian coordinates, 2

centripetal, 7, 9

Coriolis, 9, 104

cylindrical coordinates, 5

normal, 7

relative, 7, 9

rotating frame, 9

spherical coordinates, 6

tangential, 7, 9

accessibility, 38, 111, 112

Adams–Bashforth predictors, 343, 344

free sinusoidal response, 362–4

truncation error, 344–6

Adams–Moulton correctors, 345

truncation error, 344, 346

admittance matrix, 198

algorithm, 336

analytical dynamics, deﬁnition, 73

angle of rotation, 147, 148, 151

angular equation of motion, 19, 20, 54

angular impulse, 54

angular momentum

absolute, 18, 52

deﬁnition, 17

error correction, 390, 391

rigid body, 161, 167, 196

system of particles, 17–20

angular velocity, 3, 4, 143, 152–6

angular velocity coefﬁcient, 180, 181, 275

applied force, 3, 43, 73

augmented Lagrangian, 293

axis and angle variables, 147

axis of rotation, 147, 151

backward difference, 335

Baumgarte, J. W., 364, 369, 396

Baumgarte method

holonomic constraints, 364–6

nonholonomic constraints, 369–70

stability, 366, 367

Bjorck, A., 396

body axes, 140, 147

Boltzmann–Hamel equation, 227

generalized form, 229

transpositional form, 305

Burden, R. L., 396

Cartesian coordinates, 4, 140

catastatic constraint, 37, 115, 125, 263

catastatic system, 38, 45

center of curvature, 6

center of mass, 16, 21

center of percussion, 189

central difference, 334, 347

central equation, 308

explicit form, 310

centripital acceleration, 7

characteristic equation, 170, 350, 360, 361, 367

Chasles’ theorem, 157

Chetaev equation, 112

coefﬁcient of restitution, 56

coefﬁcient of sliding friction, 31

collision of particles, 56, 57

cone of friction, 31, 57, 110

conﬁguration, 34, 35, 110, 140

conﬁguration point, 35, 38, 110

conﬁguration space, 35, 110

coning motion, 183, 186–88

conservation

of angular momentum, 54, 196, 197

of energy, 24, 57, 269

of generalized momentum, 96, 97, 389

of linear momentum, 17, 54

conservative system, 24, 92–4, 264, 269

constraint

catastatic, 37, 126

conservative, 262, 263

Euler parameter, 150

holonomic, 35, 36, 110, 111

421

422 Index

constraint (cont.)

ideal, 43

impulsive, 120, 121, 125, 126

instantaneous, 41, 73, 112, 190, 219

knife-edge, 43, 83

nonholonomic, 36, 37, 218

rheonomic, 37

scleronomic, 37

virtual 41, 73, 112, 190, 219

workless, 43

constraint damping, 366

constraint force, 43, 73, 74, 78

constraint impulse, 60, 120, 127

constraint surface, 110, 111, 114

contemporaneous variation, 290, 297

coordinates

Cartesian, 4, 140

cylindrical, 4

generalized, 35

ignorable, 97

polar, 5

spherical, 5

tangential and normal, 6, 7

Coriolis acceleration, 9, 104

Coulomb friction, 31, 57, 105

cuspidal motion, 174, 175

Dahlquist, G., 396

D’Alembert’s principle, 74, 76, 77

damping coefﬁcient, 100

degrees of freedom, 35, 140

Desloge, E. A., 65, 130, 278

difference equation, 336, 349

differential displacement, 41

differential form, 37, 40, 217

direction cosine, 141, 149, 152

disk problem, 222–5, 240–2, 250–2, 256–8, 312–15,

320–2

divided difference, 331

driving-point mass, 198, 199

dumbbell problem, 83, 84, 116, 117, 220–2, 232–4,

249, 250, 255, 305–7, 318–20

dyadic

conjugate, 160

deﬁnition of, 159

inertia, 160

operations of, 160–1

rotation, 162

skew-symmetric, 160

symmetric, 160

time derivative of, 161

unit, 161

eigenvalue problem, 169, 350

elastic impact, 57

ellipsoid of inertia, 168, 169, 183–5

energy

kinetic, 21, 22, 46, 47, 51, 52, 126, 162, 163,

168

potential, 24–6, 48

energy correction, 383–5, 394, 396

energy ellipsoid, 392

energy integral, 92, 93, 383

energy rate, 23, 100, 264–9

equilibrium point, 113

error function, 331, 332, 335

Euler angles, 142–7

Euler equations, 166

generalized form, 166

Euler 3–vector, 150

Euler–Lagrange equation, 292, 293

Euler parameter rate, 154

Euler parameters, 149–51

Euler’s integration method, 336

Euler’s theorem, 147, 265

event space, 112

exactness, 39, 40

explicit integration, 341

extended conﬁguration space, 112

extraneous roots, 355, 356

extrapolation, 335

Faires, J. D., 396

force

applied, 3, 43, 73

collinearity of, 2

components of, 3

conservative, 24

constraint, 43, 73, 74, 78

contact, 1

dissipative, 100

external, 15, 16

ﬁeld, 1

generalized, 43, 44, 48, 181

gravitational, 25

gyroscopic, 103

inertia, 20, 21, 77, 236, 237

interaction, 2, 16

internal, 15, 16

linear spring, 26

monogenic, 24

total, 1, 16

forward difference, 333–5

free sinusoidal response, 360–4

friction

Coulomb, 31, 57, 105

linear, 30, 99, 100

sliding 31, 57, 105

friction coefﬁcient, 30, 31, 104

frictional stress, 105

423 Index

Frobenius’ theorem, 300

Fufaev, N. A., 278, 323

fundamental dynamical equation, 247, 248

momentum form, 248

general dynamical equation, 236, 238

momentum form, 238

generalized coordinate, 35

generalized force, 43, 44, 48, 181

generalized impulse, 61, 118, 120

generalized mass, 47, 48, 61, 91, 194

generalized momentum, 51, 61, 84, 91

Gibbs–Appell equation, 254, 256

Gibbs–Appell function, 254, 256

Ginsberg, J. H., 65, 130, 205

gradient, 184

Greenwood, D. T., 65, 130, 205, 278, 323

gyroscopic coefﬁcient, 102

gyroscopic force, 103

gyroscopic system, 102

Hamel coefﬁcient, 227, 229

Hamiltonian function, 85–7

scleronomic system, 86

Hamilton’s equations

canonical form, 86

nonholonomic form, 87

Hamilton’s principle, 291, 294, 322

Hamming, R. W., 396

herpolhode, 185

Heun’s method, 341

holonomic constraint, 35, 36, 110, 111

holonomic system, 36

homogenous function, 263

ideal constraint, 43

ignorable coordinate, 97

impact

elastic, 57

inelastic, 57

implicit integration, 340, 345, 348

impulse

angular, 54

constraint, 60, 120, 127

friction, 57

generalized, 61, 118, 120

linear, 53

impulse response, 53, 118, 194, 198

constrained, 120, 190, 193

coupled systems, 199

planar motion, 188–90

reciprocity of, 200

inelastic impact, 57

inertia (mass) coefﬁcient, 47, 48, 61, 91, 194

inertial reference frame, 1, 2

inﬁnitesimal rotation, 155

input mass, 198, 199

instantaneous axis of rotation, 157

instantaneous center, 10, 157

instantaneous constraint, 41, 73, 112, 190,

219

integrability, 39, 40

integral of motion, 92, 94, 97, 329, 389

integrating factor, 39, 40

integrator, 357, 358

interpolating polynomial, 330, 331

invariable plane, 184

iteration, 340

Jourdain’s principle, 116

Kane, T. R., 205, 236, 278

Kane’s equation, 236

Kelvin’s theorem, 205

kinetic energy

constrained, 247, 254, 316

coupling, 52

of relative motion, 22, 52, 126

rigid body, 162–4, 168

system of particles, 21, 22, 46–8, 91

Koenig’s theorem, 22

Lagrange interpolation formula, 330

Lagrange multiplier, 79, 293

impulsive, 120, 191

Lagrange’s equation

derivation, 76–8

fundamental holonomic form, 77

fundamental nonholonomic form, 79

standard holonomic form, 78

standard nonholonomic form, 79, 80

Lagrange’s principle, 77

Lagrangian function, 78

lamina, 9

Lanczos, C., 323

Laplace transform, 358

law of action and reaction, 2

law of motion, 1, 16, 53, 73

Levinson, D. A., 205, 278

linear damper, 30, 31

linear impulse, 53

linear momentum, 2, 16, 53

Maggi’s equation, 220

mass (inertia) coefﬁcient, 47, 48, 61, 91, 194

mass matrix, 91

mean value theorem, 333

midpoint method, 348

modiﬁed Euler equations, 171, 172

modiﬁed Euler integration, 341

424 Index

moment

applied, 19

inertial, 21, 238

moment of inertia, 160

principal, 169

rotation of axes, 162

translation of axes, 167, 168

moment of momentum, 17

momentum

angular, 17, 18, 161, 166, 167

generalized, 51, 61, 84, 91

linear, 2, 16, 53

momentum ellipsoid, 186, 391

monogenic force, 24

multiple-pass method, 341–3

multiplier rule, 292, 293

Neimark, Ju. I., 278, 323

Newtonian reference frame, 1, 2

Newton’s interpolation formula, 331

Newton’s law of gravitation, 25, 26

Newton’s laws of motion, 1

nonholonomic constraint, 36, 37, 218

noninertial frame, 20, 21

numerical stability, 349–55

nutation rate, 175, 182, 240

oblate spheroid, 183, 187

one-step method

angular momentum correction, 391, 394

energy correction, 384, 394

Euler parameter constraint, 382, 383

holonomic constraint, 372–6

nonholonomic constraint, 379, 380

orthogonal matrix, 142

Papastavridis, J. G., 278, 323

Pars, L. A., 130, 278, 323

particle deﬁnition, 1

perturbation equation, 349, 350

Pfafﬁan differential form, 40

phase point, 112

phase space, 86, 112–14

pitch rate, 143

planar motion, 9–12

Poinsot method, 183–6

Poisson equations, 152

polar coordinates, 5

polhode, 185, 392

potential energy, 24, 48, 49

gravitational, 25, 26

linear spring, 26

velocity-dependent, 85

precession rate, 175, 182, 187, 188

predictor–corrector methods, 341, 346, 347

principal axes, 166, 169

principle

of angular impulse and momentum, 54

of conservation of angular momentum, 54

of conservation of energy, 24

of conservation of linear momentum, 54

d’Alembert’s, 74, 76, 77

of generalized impulse and momentum, 61

Jourdain’s, 116

Lagrange’s, 77

of least constraint, 258–61

of linear impulse and momentum, 53

of relative motion, 21

of virtual work, 45, 48

of work and kinetic energy, 23

product of inertia, 160

rotation of axes, 162

translation of axes, 168

prolate spheroid, 183, 186

pure integrator, 357, 358

quasi-coordinate, 217

quasi-velocity, 177, 193, 218, 275

radius of curvature, 7

radius of gyration, 189

Rayleigh’s dissipation function, 100

reciprocity of impulse response, 200

reference frame

body-axis, 140, 147

inertial, 1, 2

noninertial, 20, 21

relative acceleration, 7, 9

relative spin, 171, 182, 183, 188

relative velocity, 7–9

remainder term, 332, 335

rheonomic constraint, 37

rheonomic system, 38

rigid body

angular momentum, 161, 167, 196

conﬁguration, 140

deﬁnition, 140

degrees of freedom, 140

equations of motion, 165–7, 171, 173, 179

free rotational motion, 181–8

impulse response, 188–91, 193, 194, 198, 199

inertia dyadic, 160

kinetic energy, 162–4, 168

oblate, 183, 187

prolate, 183, 186

roll rate, 143

rotation of axes, 140–2

rotation matrix, 141, 142, 144, 146, 148–50

roundoff error, 338–40

Routhian function, 97, 98

425 Index

Runge–Kutta methods, 342, 343

Runge phenomenon, 347

sampling polynomial, 330

scleronomic system, 38

signum function, 31

singular point, 113

spherical coordinates, 5

spiral motion, 13–15

stability

of rotational motion, 186

of static equilibrium, 49, 50, 113

state space, 113, 114

static equilibrium, 45, 48, 113

stability of, 49, 50, 113

stationarity, 291–3

Suslov’s principle, 317

explicit form, 318

system of particles

angular momentum, 17–20

center of mass, 16–21

collisions, 56, 57

equations of motion, 16, 17, 19, 20, 74, 77–80, 86,

kinetic energy, 21, 22, 46–8, 91

momentum, 16, 53

tangential acceleration, 7, 9

tangential and normal coordinates, 6, 7

Taylor series, 338

tilde matrix, 152

top motion, 173–5, 239, 240

torsion of a curve, 7

total energy, 24, 86

total spin, 171, 183

transfer function, 356, 357

transfer operator, 357

transformation equations, 35, 217, 218

scleronomic, 226

transpositional equation, basic forms,

297–9

transpositional quadrilateral, 301–3

trapezoidal method, 340

truncation error, 337, 338, 340–8

unit dyadic, 161

unit matrix, 142

unit vector, 2, 3

vector, derivative of, 3, 4

velocity

Cartesian coordinates, 2

cylindrical coordinates, 5

relative, 7–9

rotating frame, 8

spherical coordinates, 6

tangential, 7

virtual, 114, 219

velocity coefﬁcient, 42, 181, 275

velocity space, 114

virtual constraint, 41, 73, 112, 190, 219

virtual displacement, 41, 111, 219, 220

virtual power, 276

virtual velocity, 114, 219

virtual work, 43, 74, 180, 236

principle of, 45, 48

Volterra’s equation, 252, 253

Voronets’ equation, 324

weight, 25

work, 23, 40, 43, 63, 125, 200

work rate, 23, 103, 264

yaw rate, 143