references3a978-94-007-0335-3%2f… · references 1. o. bottema and b. roth. theoretical...
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References
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Index
Absolute acceleration, 58Absolute system of units, 60Absolute velocity, 58Acceleration
absolute, 58Coriolis, 168inertial, 58relative, 168
Acceleration level constraint, 364, 427, 438,442, 445, 446
Acceleration projection method, 442Active column solver, 486, 657Addition theorem, 135–136Admissible
momentum eld, 581stress eld, 581
Algebraic variable, 498Ampli cation matrix, 667Angular acceleration vector, 137Angular distortion, 582, 592Angular momentum, 75, 97, 202, 312Angular velocity vector, 129–133Augmented Lagrangian formulation, 477,
499Augmented Lagrangian term, 493Axial vector, 23
Base vector, 5893D space, 50derivatives, 44, 52of a surface, 41
Baumgarte’s method, 474–476Bi-quaternion, 546
algebra, 546–547Bilateral contact, 87Bound vector, 13Boundary conditions
displacement, 582force, 581geometric, 582natural, 581
Bushing element, 570
Calculus of variations, 284Cam-follower pair, 179Candidate contact point, 368Canonical basis, 118Cartesian basis, 6Cartesian coordinates, 31Cartesian rotation vector, 531, 537–538Cauchy-Green deformation tensor, 593Cayley’s
formula, 514motion parameters, 544–545rotation parameters, 513–514
Cayley-Gibbs-Rodriguesmotion parameters, 563–564rotation parameters, 532, 538–539
Center of mass, 95Central force, 66Change of
basis, 116, 193, 194frame, 155, 193, 194
Characteristic equation, 24Characteristic exponent, 338Compatible
O. A. Bauchau, Flexible Multibody Dynamics,DOI 10.1007/978-94-007-0335-3 © Springer Science+Business Media B.V. 2011
722 Index
strain eld, 582velocity eld, 582virtual strain eld, 585virtual velocity eld, 585
Component mode synthesis, 484–485Components of a vector, 6Composition of rotations, 124–127, 518,
523, 530Con guration constraint, 362Con guration space, 258Conformal rotation vector, 539Conservative force, 62–65Constrained problem, 359Constraint
acceleration level, 364, 427, 438, 442,445, 446
con guration, 362holonomic, 362–364, 392–394, 427kinematic, 362matrix, 357, 363, 426nonholonomic, 364–365, 402, 403, 427rheonomic, 363, 392, 402scleronomic, 362, 363, 392, 402velocity level, 364, 427, 428, 446violation, 432, 441, 444, 446
Continuous friction law, 89Contravariant component, 13Convected
basis, 588frame, 599
CoordinatesCartesian, 31curvilinear, 32cylindrical, 54orthogonal curvilinear, 53path, 39spherical, 55surface, 49, 369
Coriolis acceleration, 168Coulomb’s friction law, 88Covariant component, 13Cross product, 7Curvature
geodesic, 42normal, 42tensor, 143–145vector, 143
Curvebinormal vector, 33
curvature, 33, 143Frenet’s triad, 33intrinsic parameterization, 32natural parameterization, 32normal vector, 33osculating plane, 33planar, 34radius of curvature, 33radius of twist, 33tangent vector, 32twist, 33, 143
Curve sliding joint, 419Curvilinear coordinates, 32Cylindrical coordinates, 54Cylindrical joint, 415
D’Alembert’s principle, 295–303, 385Dashpot
constant, 70rectilinear, 70torsional, 70
Deformation gradient tensor, 592Deformed con guration, 589Degree of freedom, 259, 284, 362Determinant of tensor, 22Determination of
Euler angles, 110Euler parameters, 518the rotation parameter vector, 529
Differentialdisplacement vector, 263motion vector, 196position vector, 263rotation vector, 148work, 61
Differential-algebraic equation, 358, 395,427
Direction cosine matrix, 107, 108Director, 157, 632Displacement
boundary conditions, 582in nitesimal, 282interpolation matrix, 647virtual, 282
Displacement interpolation matrix, 641Dissipative force, 70Dot product, 5, 6Drift phenomenon, 432, 441, 444, 446, 463,
473
Index 723
Dynamic equilibrium, 295, 323
Eigenpair, 24Eigenvalue, 23Eigenvector, 23Elastic material, 586Elastodynamics, 579–583
constitutive laws, 583dynamic equilibrium equations, 581strain-displacement relationships, 582velocity-displacement relationships, 582
Energetically conjugate, 198Energy
kinetic, 62, 204strain, 64, 68, 69
Energy closure equation, 71Equilibrium
dynamic, 295static, 59, 295
Eulermotion parameters, 547–551Parameters, 516–519
Euler angle, 109, 111, 112, 125, 136–141attitude, 112, 139bank, 112, 139heading, 112, 139nutation, 109, 111, 138, 139precession, 109, 111, 138, 139spin, 109, 111, 138, 139
Euler’sequations, 213, 227rst law, 98, 287
second law, 99, 287theorem on rotations, 112
Euler-Lagrange equation, 257Euler-Rodrigues
motion parameters, 563rotation parameters, 532, 538
Exponential map of rotation, 531, 537–538Extended vectorial parameterization,
533–537External force, 94, 279, 285, 287Externally applied force, 58, 61
Finite element method, 639First-order tensor, 117Flexible joint, 570, 601–613Floating frame of reference, 482–484, 569Force
central, 66conservative, 62–65dissipative, 70external, 94, 279, 285, 287externally applied, 58, 61impressed, 61, 94impulse of, 75, 100inertial, 295internal, 94, 279, 285, 287natural, 279non-conservative, 65, 70normal contact, 86, 87tangential contact, 86, 87vector, 13viscous, 70
Force boundary conditions, 581Frame
inertial, 57Free vector, 3Frenet’s triad, 143, 148Friction coef cient
kinetic, 88static, 88viscous, 89
Gauss’ formula, 46Gauss’ principle, 457Gauss-Codazzi conditions, 145, 146Gauss-Legendre quadrature, 699–701
points, 699weights, 699
Gaussian, 457Generalized
constraint force, 393, 403coordinate, 258coordinates, 287, 308force, 268, 287inertial force, 323momentum, 308, 323speed, 428velocities, 260
Generating function, 526, 531Geometric boundary conditions, 582Geometric nonlinearity, 661Geometric notation, 29Geometrically exact beam theory, 617Gravitational constant, 61Green deformation tensor, 593Green-Lagrange strain tensor, 593
724 Index
Green-Saint Venant strain tensor, 593Gyroscopic moments, 227
Hamilton’s principle, 305–320, 392–394,402, 403, 586–588
Holonomicconstraint, 362–364, 392–394, 427system, 362
Hooke’s law, 583
Identity tensor, 23Impressed force, 61, 94Impulse of a force, 75, 100Independent quasi-acceleration, 460Independent quasi-velocity, 428Index notation, 29Index-1 formulation, 438Inertial
acceleration, 58force, 295frame, 14, 57velocity, 58
Infeasible direction, 277, 373In nitesimal
displacements, 282rotation vector, 526
Initial boundary value problem, 306Integrability conditions, 364Internal force, 94, 279, 285, 287Intrinsic
displacement of a rigid body, 163equations of motion, 314, 334
Invariantof a tensor, 24parameterization, 511
Jacobian, 50Jacobian matrix of the constraints, 357, 363Joint
curve sliding, 419cylindrical, 415planar, 416prismatic, 414revolute, 412screw, 416sliding, 421spherical, 417universal, 418
Kinematic
characteristic, 428constraint, 362constraints, 259parameter, 428
Kinematicallyadmissible direction, 277, 373admissible displacement eld, 582admissible virtual displacements, 263,
277, 374inadmissible direction, 277, 373
Kineticenergy, 62, 204energy density function, 586rotational energy, 204translational energy, 204
Kronecker’s symbol, 6
Lagrange’sequations of the rst kind, 426formulation, 322–334, 394, 403, 426multiplier, 359multiplier method, 358–361, 385
Lagrangian, 307, 587representation, 589strain tensor, 593
Levi-Civita symbol, 8Linear momentum, 74, 97, 312Linear parameterization, 531Lower pair joint, 405–418
constraints, 408–412kinematics, 406–408
Maggi’s formulation, 428–435Mass matrix, 311, 426Material
basis, 588compliance matrix, 583coordinates, 589elastic, 586frame, 599line, 589stiffness matrix, 583
Material nonlinearity, 661Matrix notation, 29Metric of a space, 450Metric tensor, 591Mixed product, 10Modal
analysis, 572
Index 725
expansion, 572Momentum
angular, 75, 97, 202, 312linear, 74, 97, 312
Moore-Penrose generalized inverse, 698Motion parameters
Cayley’s, 544–545Cayley-Gibbs-Rodrigues, 563–564Euler, 547–551Euler-Rodrigues, 563vector, 544, 545, 555Wiener-Milenkovic, 564
Motion tensor, 187–192Mozzi-Chasles’
axis, 163, 189theorem, 163
Natural boundary conditions, 581Natural force, 279Newton’s
rst law, 58second law, 59third law, 59
NewtonRaphson method, 658Non-conservative force, 65, 70Non-invariant parameterization, 511Non-vectorial parameterization, 511Nonholonomic
constraint, 364–365, 402, 403, 427quantity, 268vector, 130
Normal contact force, 86, 87Notation
geometric, 29index, 29matrix, 29
Null space, 429formulation, 442
Ordinary differential equation, 395Oriented line segment, 3Orthogonal
complement, 429curvilinear coordinates, 53parameterization, 51projection, 448vectors, 5
Orthonormal basis, 6Out-of-balance force array, 658
Parallel axis theorem, 205–206Parameterization
invariant, 511non-invariant, 511non-vectorial, 511vectorial, 511
Particle, 57path, 57speed, 39, 58
Path coordinates, 39Penalty method, 381Permutation symbol, 8Pfaf an form, 364Physical strain component, 592Pitch of a screw, 163Pivot equation, 213, 228Pivoting, 499, 500Plucker coordinates, 15, 187Planar
joint, 416motion, 227–237rotation, 108
Positive-de nite tensor, 23, 24Potential
de nition, 63function, 63, 66, 269of a conservative force, 63, 269of gravity force, 67of the body forces, 587of the constraint forces, 393of the surface tractions, 587
Principal axes of inertia, 207Principle of
angular impulse and momentum, 76, 101conservation of energy, 66D’Alembert’s, 295–303, 385Gauss, 458Hamilton, 305–320, 392–394, 402, 403,
586–588least action, 308linear impulse and momentum, 75, 101minimum total potential energy, 588virtual work, 271–287, 371–382, 583–585virtual work for a particle, 272virtual work for a particle system, 284work and energy, 62, 100, 214, 215
Prismatic joint, 414Product
cross, 7
726 Index
dot, 5, 6mixed, 10scalar, 5, 6tensor, 9vector, 7
Product of inertia, 206Projection, 5Projection tensor, 12Projector
image, 448, 450kernel, 448, 450null space, 448, 450
Quaternion, 514algebra, 514–516operators, 514orthogonal, 516scalar part, 514unit, 516vector part, 514
Radius of twist, 46Rayleigh damping, 660Reaction force, 580Reciprocal vector, 13Rectilinear
dashpot, 70spring, 68
Referencecon guration, 589frame, 14
Re ection tensor, 13, 27Relative
acceleration, 168elongation, 592velocity, 167
Relative elongation, 582Rescaling operation, 534–537Revolute joint, 412Rheonomic constraint, 363, 392, 402Right-hand basis, 8Rodrigues’ rotation formula, 114Rotation parameters
Cayley’s, 513–514Cayley-Gibbs-Rodrigues, 532, 538–539Euler-Rodrigues, 532, 538linear, 531vector, 512, 524Wiener-Milenkovic, 532, 539–541
Rotation tensor, 114properties, 115
Rotational kinetic energy, 204Rotationless formulation, 261
Scalar product, 5, 6Scaling
equations of motion, 490–504factor, 493
Scleronomic constraint, 297, 362, 363, 392,402
Screwaxis, 165joint, 416motion, 163, 165
Second-order tensor, 22Semi positive-de nite tensor, 23Shape function, 640Similarity transformation, 25Skew-symmetric part of tensor, 22Skew-symmetric tensor, 22Skyline solver, 499Sliding joint, 421Spectral radius, 667Spherical
coordinates, 55joint, 417
Springrectilinear, 68stiffness constant, 68stretch, 68torsional, 69un-stretched length, 68
Stability analysis, 336–344Staggered stabilization technique, 476State space, 260Static equilibrium, 59, 295Stationary point
of a de nite integral, 255of a function, 254
Straindirect, 582energy, 69shear, 582
Strain energydensity function, 586for spring, 68function, 68
Stress
Index 727
direct, 581shear, 581
Surfacebase vectors, 41coordinates, 49, 369dilatation, 595equilibrium equations, 581rst metric tensor, 41
Gaussian curvature, 44line of curvature, 44, 629mean curvature, 44principal curvature, 44principal radii of curvature, 44, 631radius of twist, 45second metric tensor, 42traction, 580
SymbolKronecker’s, 6Levi-Civita, 8permutation, 8
Symmetric part of tensor, 22Symmetric tensor, 22System of particles, 286
Tangent operator, 134, 138Tangential contact force, 86, 87Tensor
Cauchy-Green deformation, 593characteristic equation, 24curvature, 143–145deformation gradient, 592determinant, 22rst-order, 117
Green deformation, 593Green-Lagrange strain, 593Green-Saint Venant strain, 593identity, 23invariant, 24Lagrangian strain, 593metric, 591of mass moments of inertia, 203operation, 119positive-de nite, 23, 24product, 9projection, 12re ection, 13, 27rotation, 114second-order, 22semi positive-de nite, 23
skew-symmetric, 22skew-symmetric part, 22symmetric, 22symmetric part, 22trace, 22
Time integration schemes, 664–685Torsional
dashpot, 70spring, 69
Total kinetic energy, 586Total mechanical energy, 65Total strain energy, 586Trace of tensor, 22Translational kinetic energy, 204
Unconstrained problem, 359Unilateral contact, 87Unit vector, 4Universal constant of gravitation, 60Universal joint, 418
Variation ofthe position vector, 263
Vectorangular acceleration, 137angular velocity, 129–133axial, 23bound, 13component, 6curvature, 143differential rotation, 148force, 13free, 3nonholonomic, 130norm, 4null, 4orthogonal, 5product, 7reciprocal, 13unit, 4
Vectorial parameterization, 511of motion, 554–564of rotation, 524–537
Velocityabsolute, 58inertial, 58relative, 167
Velocity level constraint, 364, 427, 428, 446Virtual
728 Index
displacements, 272, 282rotation vector, 264
Virtual displacementdependent forces, 282rigid bodies, 283vector, 263
Virtual work, 268external, 285internal, 285
Viscous force, 70
Viscous friction law, 89Volumetric strain, 596
Warping, 619, 629Weingarten’s formula, 46Wiener-Milenkovic
motion parameters, 564rotation parameters, 532, 539–541
Workdifferential, 61