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| موضوع: كتاب Dynamics - Theory and Applications الجمعة 20 أغسطس 2021, 10:55 am | |
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أخواني في الله أحضرت لكم كتاب 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
و المحتوى كما يلي :
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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