اسم المؤلف
Dan B. Marghitu , Mihai Dupac
التاريخ
4 سبتمبر 2019
المشاهدات
380
التقييم

Advanced Dynamics – Analytical and Numerical Calculations with MATLAB
Dan B. Marghitu , Mihai Dupac
Contents
1 Vector Algebra . 1
1.1 Terminology and Notation 1
1.2 Position Vector . 10
1.3 Scalar (Dot) Product of Vectors . 11
1.4 Vector (Cross) Product of Vectors 13
1.5 Scalar Triple Product of Three Vectors . 15
1.6 Vector Triple Product of Three Vector 18
1.7 Derivative of a Vector Function . 18
1.8 Cauchy’s Inequality, Lagrange’s Identity,
and Triangle Inequality 20
1.9 Coordinate Transformation 22
1.10 Tensors . 26
1.10.1 Operations with Tensors . 34
1.10.2 Some Further Properties of Second-Order Tensor 35
1.11 Examples . 36
1.12 Problems . 65
1.13 Program 69
2 Centroids and Moments of Inertia . 73
2.1 Centroids and Center of Mass . 73
2.1.1 First Moment and Centroid of a Set of Points 73
2.1.2 Centroid of a Curve, Surface, or Solid 75
2.1.3 Mass Center of a Set of Particles 77
2.1.4 Mass Center of a Curve, Surface, or Solid 77
2.1.5 First Moment of an Area . 79
2.1.6 Center of Gravity . 82
2.1.7 Theorems of Guldinus–Pappus 82
2.2 Moments of Inertia 85
2.2.1 Introduction 85
2.2.2 Translation of Coordinate Axes . 88
viiviii Contents
2.2.3 Principal Axes 90
2.2.4 Ellipsoid of Inertia . 92
2.2.5 Moments of Inertia for Areas 93
2.3 Examples . 100
2.4 Problems . 130
3 Kinematics of a Particle . 143
3.1 Introduction 143
3.1.1 Position, Velocity, and Acceleration 143
3.1.2 Angular Motion of a Line . 144
3.1.3 Rotating Unit Vector . 145
3.2 Rectilinear Motion . 146
3.3 Curvilinear Motion 147
3.3.1 Cartesian Coordinates 147
3.3.2 Normal and Tangential Coordinates 148
3.3.3 Circular Motion 154
3.3.4 Polar Coordinates 155
3.3.5 Cylindrical Coordinates 157
3.4 Relative Motion 158
3.5 Frenet’s Formulas 159
3.6 Examples . 166
3.7 Problems . 201
4 Dynamics of a Particle . 209
4.1 Newton’s Second Law . 209
4.2 Newtonian Gravitation 210
4.3 Inertial Reference Frames . 211
4.4 Cartesian Coordinates . 211
4.4.1 Projectile Problem . 212
4.4.2 Straight Line Motion . 213
4.5 Normal and Tangential Components 213
4.6 Polar and Cylindrical Coordinates 214
4.7 Principle of Work and Energy . 215
4.8 Work and Power . 217
4.8.1 Work Done on a Particle by a Linear Spring . 218
4.8.2 Work Done on a Particle by Weight . 219
4.9 Conservation of Energy . 221
4.9.1 Exercise . 221
4.9.2 Exercise . 222
4.10 Conservative Forces . 227
4.10.1 Potential Energy of a Force Exerted by a Spring . 227
4.10.2 Potential Energy of Weight 228
4.10.3 Exercise . 229
4.10.4 Exercise . 232
4.11 Principle of Impulse and Momentum . 234
4.12 Conservation of Linear Momentum 235Contents ix
4.13 Principle of Angular Impulse and Momentum . 237
4.14 Examples . 238
4.15 Problems . 275
5 Kinematics of Rigid Bodies . 281
5.1 Introduction 281
5.2 Velocity Analysis for a Rigid Body . 282
5.3 Acceleration Analysis for a Rigid Body 285
5.3.1 Translation 286
5.3.2 Rotation 289
5.3.3 Helical Motion . 294
5.3.4 Planar Motion 298
5.4 Angular Velocity Vector of a Rigid Body 300
5.5 Motion of a Point that Moves Relative to a Rigid Body . 304
5.6 Planar Instantaneous Center . 309
5.7 Fixed and Moving Centrodes . 311
5.8 Closed Loop Equations 321
5.8.1 Closed Loop Velocity Equations 323
5.8.2 Closed Loop Acceleration Equations . 325
5.9 Independent Closed Loops Method . 328
5.10 Closed Kinematic Chains with MATLAB Functions 335
5.10.2 Position Analysis . 337
5.10.3 Complete Rotation of the Driver Link 344
5.10.4 Velocity and Acceleration Analysis . 348
5.11 Examples . 357
5.12 Problems . 400
6 Dynamics of Rigid Bodies . 411
6.1 Equation of Motion for the Mass Center . 411
6.2 Linear Momentum and Angular Momentum . 414
6.3 Spatial Angular Momentum of a Rigid Body 417
6.4 Kinetic Energy of a Rigid Body 421
6.5 Equations of Motion . 423
6.6 Euler’s Equations of Motion 425
6.7 Motion of a Rigid Body About a Fixed Point 426
6.8 Rotation of a Rigid Body About a Fixed Axis . 426
6.9 Plane Motion of Rigid Body 427
6.9.1 D’Alembert’s Principle 431
6.9.2 Free-Body Diagrams . 432
6.9.3 Force Analysis for Closed Kinematic Chains
Using MATLAB Functions 437
6.10 Examples . 446
6.11 Problems . 510x Contents
7 Analytical Dynamics . 521
7.1 Introduction 521
7.2 Equations of Motion . 525
7.3 Hamilton’s Equations 528
7.4 Poisson Bracket 531
7.5 Rotation Transformation 533
7.6 Examples . 537
7.7 Problems . 594
References . 601
Index .
Index
A
Absolute
angular acceleration, 325, 396, 400
angular velocity, 323, 324, 329, 395, 398
value, 1, 3, 406
Angle, 9, 13, 14, 36, 37, 45, 48, 49, 52, 65–68,
98, 99, 101, 102, 116, 144, 145, 150,
152, 154, 155, 161, 173, 189, 195, 200,
202, 204, 205, 208, 223, 238, 244, 246,
259, 267, 271, 289, 300, 329, 335, 338,
339, 342–344, 348, 350, 352, 356, 365,
372, 374, 376, 377, 379, 394, 397,
400–404, 429, 462, 465, 466, 470, 477,
482, 488, 495, 510–516, 518, 519, 533,
534, 536, 544, 552, 558, 567, 595,
596
Angular
acceleration, 144, 155, 157, 257, 269, 285,
291, 292, 306, 309, 325, 326, 329, 335,
352, 356, 381, 385, 391, 396, 400–402,
424, 429, 430, 437, 463–466, 472, 482,
489, 497, 498, 569, 570, 586
impulse, 237–238
momentum, 237, 238, 245, 414–426, 428,
532
position, 144
velocity, 144, 145, 156, 197, 202, 204,
205, 245, 267, 268, 276, 279, 284, 291,
292, 294, 296, 300–304, 306, 309–311,
318, 322–325, 329, 332–335, 350–357,
365, 381, 383, 389, 393–398, 400, 402,
404–406, 418–421, 424, 426, 428, 463,
464, 466, 471, 479, 482, 489, 497, 498,
552, 568–570, 572–574, 577, 579, 586
Associative, 4, 34
B
Base, 17
Bilinearity, 532
Binormal, 161, 162, 164, 165, 190, 192, 193,
197
Body centrode, 311
Body-fixed reference frame, 281, 286, 288,
296, 304–306, 365
Bound vector, 2
C
Cartesian, 6, 9, 10, 14, 20, 22, 23, 26–28, 31,
35, 48, 50, 53, 74, 75, 78, 87, 88, 92,
94, 127, 147–148, 152, 155, 156, 166,
203, 211–213, 219, 228, 229, 268, 271,
281, 282, 329, 339, 373, 381, 414, 427,
430, 521, 537, 544, 558, 565, 595
Cauchy, 20–22, 27, 474
Central principal moments, 91, 420
Centripetal acceleration, 157
Centrode, 311–321
Centroid, 73–141, 347, 348
Centroidal axis, 80, 94, 95, 420
Circular motion, 154–155, 256
Closed
kinematic chain, 322–328, 335–357,
436–446
loop equation, 321–328
loop method, 328–335
Commutative, 4, 11, 14, 34
Configuration space, 521–522, 530
Constraint
conditions, 344, 377
configuration, 522–524
D.B. Marghitu and M. Dupac, Advanced Dynamics: Analytical and Numerical
Calculations with MATLAB, DOI 10.1007/978-1-4614-3475-7,
603604 Index
Constraint (cont.)
equation, 522, 523
equations in velocity form, 523
force, 522
holonomic, 523, 524
non-holonomic, 524
relation, 523
Contour method, 392, 436, 452
Contraction, 35, 36
Coordinates
cylindrical, 157–158, 203, 214–215, 229
generalized, 462, 470, 477, 482, 488, 492,
497, 521–523, 525, 527, 537, 539, 544,
552, 555, 558, 567, 568, 577, 584
Coplanar, 56, 82, 86
Coriolis acceleration, 157, 269, 306, 307, 309,
322, 326, 328, 365, 368–370, 385,
386
Cross product, 13–15, 45, 49, 52, 53, 88, 161,
237, 465, 586
Curvature, 150–153, 155, 161, 164, 165, 177,
181, 182, 202, 203, 205, 311, 314
Curve, 75–79, 81–83, 104, 127, 130, 135, 149,
150, 161, 163, 164, 175, 193, 205, 206,
213, 254, 277, 279, 311, 314, 315, 468,
476, 595
D
D’Alembert’s principle, 431–432, 525
Decomposition, 74, 78, 106
Degrees of freedom, 295, 329, 470, 477, 487,
488, 497, 522, 525
Density, 78, 86, 100, 101, 112, 113, 122, 133,
141, 446, 449, 487, 510–515, 519, 520
Derivative, 18–19, 98, 143, 145, 146, 149, 150,
153, 154, 156, 158–163, 173, 182, 186,
187, 198, 224, 236, 237, 245, 268, 269,
282, 283, 285, 288, 300, 302, 305, 306,
349, 350, 352, 412, 416, 424, 465, 470,
471, 479, 488, 489, 492, 497, 499, 522,
530, 531, 540, 569, 571
Determinant, 14–16, 24, 30, 57, 58, 62
Diagonal elements, 15, 36
Differential, 75, 78, 86, 87, 94, 101, 102, 104,
109–111, 116, 122, 124, 127, 228, 240,
411, 412, 414, 465, 467, 468, 474, 476,
477, 495, 523, 529, 530, 543, 582, 583,
592
Direction
angle, 534
cosines, 9, 10, 22, 42–44, 49, 66, 67, 87,
91, 534
Displacement
infinitesimal, 216–218
relative, 277
vertical, 273
weighed average, 414
Distributive, 4, 12, 14, 35
Dot product, 11–13, 88, 159, 161, 216, 220,
524, 545, 573, 574, 586
Driver
link, 335–337, 339, 341, 344, 348, 353,
372, 377, 379, 393, 396, 400, 402–404,
445, 446, 452, 510–515
moment, 461, 595
Dyad, 335, 337–339, 342, 348–350, 432–435,
437–444, 450, 451, 573–575, 586, 597
Dynamical system, 521–523
Dynamics, 77, 196, 209–280, 411–520,
522–600
E
Elastic
constant, 277, 518, 537, 594, 595
force, 277
Ellipsoid of inertia, 92–93
Energy, 215–218, 221–230, 232, 233, 248,
250, 280, 421–423, 526–528, 530,
538–540, 545, 546, 549, 552, 553, 560,
561, 573–575, 593
Equilibrium
equations, 431, 436, 445
moment, 445, 446
position, 462
Euler equation, 430, 465, 472, 473, 480, 484,
491, 492, 515–517
External force, 216, 218, 236, 245, 307,
412–415, 428, 429, 431, 432, 436, 446,
456, 510, 512–514
F
First moment, 73–74, 79–81, 84, 95, 102,
104–106, 115, 116, 129
Fixed
axis, 289–291, 294, 299, 405, 426–427,
429, 464
centrode, 311, 320, 321
point, 37, 90, 168, 171, 244, 250, 281, 299,
319, 417, 420–422, 425, 426, 429–431,
465, 472, 480
reference frame, 197, 200, 232, 268, 269,
281, 283, 286, 317, 323, 325, 365–367,
397, 424, 426, 495, 497, 509, 568, 586Index 605
Force, 1–3, 67, 68, 77, 81, 82, 209–219, 221,
227–236, 245, 247, 250, 255–257, 259,
263, 267, 270, 277, 279, 280, 307, 309,
412–416, 423, 426, 428–446, 449–461,
464–466, 471, 474, 479, 480, 482, 486,
491, 492, 503, 504, 510–516, 522,
525–527, 540, 541, 546, 553, 557, 562,
563, 577–580, 585–593
Free
body diagram, 250, 432–436, 493
vector, 2, 17, 288, 302
Frenet, 159–166, 193–197
Friction, 217, 227, 246, 267, 277, 279, 431,
464, 477, 515, 517, 518, 544, 595
G
Generalized
active force, 525–528, 540, 557–580
coordinate, 462, 470, 477, 482, 488, 492,
497, 521–523, 525, 527, 537, 539, 544,
552, 555, 558, 564, 567, 568, 577, 584
inertia force, 525, 526, 585–593
velocities, 522, 528
Gravity, 82, 210–212, 247, 257, 462, 465, 469,
477, 482, 503, 515–519, 540, 546, 552,
553, 557
center, 82, 130, 586
Guldinus-Pappus, 82, 113, 115
H
Hamilton equations, 528–531, 551, 552, 555,
556
Helical motion, 294–298
I
Impulse, 234–238, 415
Impulsive force, 235
Independent
contour, 330, 392, 397
vector, 5
Inertial reference frame, 211, 237, 307–309,
462, 482, 525
Inertia matrix, 87, 91, 116–119, 126, 127, 129,
130, 419, 483, 504, 573, 574, 586
Initial conditions, 178, 179, 238, 252, 256–258,
261, 264, 271, 279, 467, 468, 474, 475,
477, 481, 482, 487, 495, 518–520,
555–557
Instantaneous
center, 309–311
radius of curvature, 151, 152, 155, 165
Invariant, 36, 91, 97, 127
J
Jacobian, 524
Jacobi identity, 532
Joint
rotational, 382, 392, 397, 597
translational, 392, 397, 453, 455, 477, 597
Joules, 217, 218
K
Kinematic chain, 322–328, 330, 355–357,
436–446, 477, 518, 565
Kinematics, 143–208, 281–409, 436–446, 477,
479, 489, 518, 522, 560, 565, 572
Kinetic energy, 216–218, 221–223, 248, 250,
280, 421–423, 526, 530, 538–540, 545,
552, 553, 560, 561, 573–575, 593
Kronecker
delta, 23, 30, 31
matrix, 23
L
Lagrange
equations of motion, 527, 558
identity, 20–22
Lagrangian, 528, 530, 549, 554
Left-handed, 14
Leibnitz property, 532
Linear
combination, 5, 17
independence, 5
momentum, 209, 234–237, 242, 243,
414–417
space, 35
spring, 218–219, 227, 595
Linearity, 532
Line of action, 1, 2, 291
Link, 166, 168, 171, 173, 197, 202, 279,
312–324, 326, 328–330, 335–337, 339,
341–348, 351–353, 355, 356, 365–368,
372, 376, 377, 379, 381–393, 395–398,
400–406, 432–446, 449–452, 455–459,
461–466, 469–474, 477, 479, 480,
487–493, 497–520, 552, 553, 558, 560,
562, 565, 567–579, 584, 587, 596, 597606 Index
Loop, 31, 32, 43, 169, 170, 196, 197, 321–335,
344, 377
M
Magnitude, 1, 3, 6, 8, 10, 13, 29, 36, 37, 39,
40, 42, 47, 48, 52, 65–67, 88, 143,
145–147, 149, 153, 156, 158, 159, 161,
165, 166, 168, 171, 175, 176, 181, 182,
186, 191, 194, 208–210, 214, 216, 227,
235, 241, 245, 272, 282, 284, 286, 288,
291, 292, 300, 318–320, 358–360, 412,
428, 430, 431, 510, 512–514, 545
Mass
center, 77–79, 88–91, 93, 100–102,
104–111, 114–116, 124, 129, 133, 134,
243, 411–414, 417, 419, 424, 427–431,
446, 464, 465, 469, 471, 477, 479, 482,
499, 500, 502, 517, 544, 552, 553, 558,
562, 567, 570–574, 584–586, 596
element, 86, 87, 91, 411
Maximum second moment, 98, 99
Method of decomposition, 78
Minimum second moment, 99
Mobile reference frame, 281, 288, 296, 298
Module, 1, 3, 92, 329
Moment of inertia, 85–89, 91–93, 100,
117–119, 122–124, 126, 128, 129, 133,
134, 136, 137, 141, 428–430, 437, 449,
452, 464, 465, 472, 477, 491, 517, 553,
560, 587
Momentum vector, 532
Monoloop, 322
Motion
angular, 144–145
circular, 154–155, 256
curvilinear, 147–158, 217
free fall, 259, 264
helical, 294–298
oscillatory, 166, 168, 171, 173
planar, 285, 298–299, 306, 311, 328, 367,
427–429
rectilinear, 146–147, 166
relative, 143, 158–159, 197, 304, 309, 322,
326, 431
rotational, 92, 284, 286, 289–295, 299, 335
spatial, 299
spherical, 299
straight line, 146, 213
three-dimensional, 165, 215, 422
translational, 286–289, 299
Moving centrode, 311–321
N
Newton
second law, 209–216, 234, 237, 250, 260,
270, 271, 307–309, 412, 414, 431,
525
third law, 235, 412, 413
Newton-Euler, 430, 465, 472, 473, 480, 484,
491, 492, 515, 517–519
Non
accelerating, 307, 308
centroidal axes, 95
holonomic, 524
homogeneous, 100
linear, 465, 474
parallel, 298, 300
rotating, 307, 308
zero, 5, 17, 20, 21, 28, 32
Norm, 1, 8, 44, 49, 361, 371, 372, 382,
388–390, 392
Normal
acceleration, 177, 292
coordinates, 148–155, 163, 250
direction, 159, 214, 250
plane, 193
unit vector, 150, 151, 160, 190, 192
vector, 161, 164, 192, 193
O
Orientation, 1–3, 10, 79, 91, 96, 99, 161, 165,
300, 302, 303, 394, 397, 429, 533
Orthogonal
axes, 80, 97, 99
component, 6, 161, 162, 200, 282, 535
matrix, 26
reference frame, 6
transformation, 36
Orthonormal
basis, 197
matrix, 535
Osculating plane, 160, 161, 165, 190, 193
P
Parallel, 1–4, 9–11, 13, 16, 20, 68, 88, 89,
91, 122, 146, 157, 159, 165, 186, 214,
237, 238, 286, 288, 289, 297, 301, 302,
311, 317, 328, 367, 383–386, 389–391,
404, 424, 428, 451, 495, 497, 524, 567,
572–574, 586, 597
axis theorem, 93–97, 117, 118, 125, 126,
129, 430
Parallelogram law, 3, 4Index 607
Partial
derivative, 165–166, 497, 499, 540, 571
solution, 335
Particle, 73, 77, 78, 82, 85, 143–280, 324, 326,
411–418, 421, 428, 495, 497, 501–505,
509, 510, 521–525, 578, 585, 586, 593,
595, 596
Period, 13, 166, 171, 182, 201, 223, 254, 415
Permutation
symbol, 29–34, 65, 69, 70
tensor, 34
Point of application, 2, 442, 455, 456, 479
Poisson
bracket, 531–533
formulas, 284, 288, 305
Polar
coordinates, 155–157, 186, 189, 203, 214,
219, 227
moment of area, 96–97, 141
moment of inertia, 119, 123, 124, 134, 136,
137
Position vector, 10–11, 39, 45, 73–75, 77, 78,
85, 143, 146–148, 155, 158, 163, 175,
177, 189, 198, 200, 204, 236, 237, 245,
247, 250, 268, 281, 284, 286, 304, 305,
325, 331, 341, 342, 359, 365, 366, 374,
375, 411, 414, 418, 421, 437, 439, 440,
442, 445, 450, 451, 462, 470, 471, 477,
479, 482, 486, 488, 489, 499, 501, 502,
521, 522, 524, 537, 544, 558, 570–572,
584, 585
Potential energy, 221, 227–233, 250, 527, 528,
530, 549, 552, 553
Power, 8, 217–220
Primary reference frame, 269, 281, 304–309
Principal
axes, 90–93, 97–100, 420–423, 425, 504,
572–574, 586, 597
axes of inertia, 420, 425, 572
centroidal moment, 420, 422, 425
direction, 116, 119, 120, 124, 127
ellipsoid of inertia, 93
moment of inertia, 91, 100
Product
of area, 94–96, 98, 99
of inertia, 90, 91, 94, 116, 117, 119, 125,
129, 137
Projectile, 212, 277
R
of curvature, 150–153, 155, 161, 164, 165,
177, 181, 182, 202, 203, 205, 314
of gyration, 87, 92
Rectangular component, 36, 52, 414
Rectifying plane, 190, 193
Reference frame, 6, 10, 19, 22–24, 26–28, 35,
36, 75, 79, 87, 100, 106, 109, 111, 116,
143, 146–148, 152, 158, 160, 165, 166,
171, 189, 193, 194, 197, 198, 200, 201,
211, 212, 219, 228, 232, 233, 237, 238,
268–271, 281–283, 285, 286, 288, 289,
296, 298, 300, 303–309, 317, 323–325,
339, 353, 365–367, 369, 373, 381, 393,
397, 424, 426, 427, 462, 482, 495, 497,
499, 504, 509, 521, 525, 526, 533–537,
552, 558, 565, 567–571, 586, 595
Relative
acceleration, 286, 306, 309, 326, 367, 383,
385
displacement, 277
motion, 143, 158–159, 197, 304, 309, 322,
326, 431
position, 283
velocity, 284, 306, 321, 322, 324, 367, 394
Resolution
of components, 5–8
of vectors, 5–8
Resultant, 4, 5, 42, 44, 45, 48, 49, 65–67, 81,
82, 277, 279, 414, 431, 525
Rheonomic, 524
Right-handed, 2, 13, 14, 31
Rigid body, 2, 86, 87, 89–93, 281–409,
411–520, 568, 573, 574, 578, 586, 593
Rolling motion, 430–431
Rotating unit vector, 145–146, 156
Rotation transformation, 533–536
S
Scalar
product, 11, 13, 20, 175
triple product, 15–18, 31, 56, 59–61, 422
Scleronomic, 524, 530
Second moment, 87, 93–99
Sense, 1–3, 9–11, 14, 161, 288, 291, 318, 404,
464, 516, 519
SI units, 210, 211, 217, 218
Sliding
direction, 338, 342, 356, 367, 383, 385,
389, 391, 434, 435, 440, 441, 443, 444,
453, 454, 456, 458, 459, 461, 492
line, 544
vector, 2
Solid, 38, 75, 77–78, 132, 313
Space centrode, 311
Spatial angular momentum, 417–421
Spring constant, 219, 229, 250, 276608 Index
Strength, 73, 74, 77
Surface, 1, 2, 75–79, 82–84, 92, 102, 113, 115,
116, 133, 205, 211, 246, 430, 431, 515,
522–524, 544
Symbolic, 7, 8, 10, 15, 17, 31, 37, 39, 40, 53,
57, 61, 101, 169, 189, 231, 240, 361,
466, 539, 540, 545
Symmetry
anti, 532
axis, 80, 94, 101, 102, 108, 109, 111, 122,
124
plane, 74, 80
skew, 35, 532
T
Tangential coordinates, 148–154
Temperature, 1
Tensor
alternating, 29, 30
antisymmetric, 30
completely antisymmetric, 30
inertia, 419, 420
magnitude, 29
permutation, 34
second order, 34–36
symmetric, 29, 35, 420
zero-order, 36
Tensorial product, 28
Time derivative, 19, 143, 145, 146, 149, 150,
153, 154, 156, 158, 159, 186, 187, 198,
236, 237, 269, 305, 306, 465, 531
Torsion, 162–164, 205
Transfer theorem, 94–96, 464
Translation, 88–90, 286–289, 295, 296, 298,
299, 311, 552
pure, 286, 287
Transmissible vector, 2, 3
Transpose, 19, 24, 29, 535, 539, 570, 578, 584,
589, 590
Triangle inequality, 20
U
Unconstrained dynamical system, 524
Unit vector, 3, 5, 6, 8–14, 22, 39–41, 65,
67, 75, 87, 91, 145–146, 149–152,
155–161, 163, 165, 186, 190, 192, 193,
195, 200, 215, 217, 219, 238, 245, 268,
281, 282, 288, 304, 495, 497, 524,
533–535, 537, 565, 567, 569–572
Unity tensor, 28, 29
V
Variation, 528
Varignon theorem, 14
Vector
(cross) product, 13–15, 52
Velocity, 1, 143, 209, 281, 414, 523
Volume, 17, 75, 76, 78, 82, 84, 86, 88,
109–111, 113, 114, 130, 133, 141
W
Work, 1, 8, 215–222, 224, 225, 227, 229–234,
247, 248, 422, 572
Z
Zero vector, 3, 6, 10
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