REFERENCES

REFERENCES

REFERENCES Ackermann, J. (1972), “Der Entwurf linearer Regelsysteme im Zustandsraum”, Regelungstechnik, Vol. 20, 297-300. Bathe, K.J. (1982). Finite ...

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LIST OF SYMBOLS

The symbols we use in this book are related to the corresponding ISO-Norm. As far as possible, they represent common abbreviations (with English (and Latin, respectively) origin such as v: velocity, a: acceleration, T: transformation, or German origin such as A: Abbildung, Z: (Gaussian) Zwang, F : F¨uhrung). Vectors and matrices are denoted by bold-faced letters (capital: matrices, small: vectors, in general). A special case is given for the zero element(s): Introducing a zero matrix (0) would consequently need a zero vector (o) which can easily be confused with the letter bold o. Using the slash for a zero matrix (0/ ) and for a zero vector representation (o/) may lead to confusion with the boldfaced Greek “phi”. We therefore define the non-bold zero symbol 0 to generally represent scalar, vector or matrix values where the corresponding dimension follows from the context. 0 a b c d e f

g h i j k l, or L

zero element (∈ IRm,n , m, n ∈ ZZ) (absolute) acceleration (∈ IR3 ) (nonlinear) state function (∈ IRf +g ) inhomogeneous state input, or disturbance (∈ IRf +g ) coefficient vector (∈ IRn ), transposed measurement matrix (SISO) (outer) differential symbol, distance unit vector (∈ IR3 ) force (∈ IR3 ), number of positional d.o.f. (∈ IR1 ), function (∈ IR1 ) disturbance (∈ IRf +g ) gravity vector (∈ IR3 ), | g | = 9.81 for Middle Europe (see p.67), number of velocity d.o.f. (∈ IR1 ) height √ imaginary unit ( −1), counting index counting index counting index, spring coefficient, feedback coefficient length 437

438 m n m

p

q q r s˙ s

t u

u v w we x x x ˆ x, ˆx y ˆ y y˙ i y˙ n y z

List of Symbols number of nonholonomic constraints counting index, “geometrical nonlinearity” index, state dimension (n = f + g) mass, counting index, number of holonomic constraints momentum (∈ IR3 ), generalized momentum (Hamilton, ∈ IRg ), Rodrigues parameter (∈ IR3 ), predecessor index (∈ IR1 ) vector of minimal coordinates (∈ IRf ), Euler parameter (∈ IR4 ) f eigenvector (∈ IR  )  position vector ∈ IR3 vector of minimal velocities (∈ IRg ) and s˙ i (∈ IRgi ), resp. quasi-coordinate (∈ IRg , assigned to s˙ ), tip body mass center (∈ IR3 ), successor index (∈ IR1 ) time rotation axis (∈ IR3 ), neutral axis displacement: (u1 , u2 , u3 )T or (u, v, w)T , vector of beam shape functions (∈ IRn ), (electrical) voltage (∈ IR1 ) displacement of an arbitrary point: (u1 , u2 , u3 )T or (u, v, w)T (absolute) velocity (∈ IR3 ), vector of beam shape functions ∈ IRn vector of beam shape functions (∈ IRn ), disturbance vector (∈ IRf +g ) eigenfunction (plate) spatial variable vector of independent spatial variables (right) eigenvector, augmented state vector (observer) state estimations (observer) left eigenvector estimated measurement (observer) vector of “intermediate”, or auxiliary, velocity variables describing velocity variables (subsystem n) spatial variable (non-minimal) position variables, y˙ = H(z)˙z, vector of basic disturbance functions (observer), spatial variable (∈ IR1 )

List of Symbols A

A B B B Bi C C

D D Dir D D E F F F

Gi Gn G G H

I ID J J

transformation matrix (∈ IR3,3 ), area (∈ IR1 ), area (= normal) vector (∈ IR3 ), system matrix (∈ IRn,n , n = f + g) or Jx : moment of inertia, x-axis control input matrix (state space) control input matrix (configuration space) or Jy : moment of inertia, y-axis operator, yields boundary conditions measurement matrix or Jz : moment of inertia, z-axis, longitudinal stiffness (plate or disk), mass center, (electrical) capacity damping matrix bending stiffness (plate) spatial operator (∂ i /∂xir ) operator, defines variables operator, yields partial differential equations unit matrix, Young’s modulus (∈ IR1 ) (intermediate) functional matrix (single body) (intermediate) functional matrix (subsystem) (general) functional matrix, deformation gradient, disturbance input matrix (F = WU) gyroscopic matrix (single body) gyroscopic matrix (subsystem) gyroscopic matrix (∈ IRg,g ), shear modulus (∈ IR1 ) Green-Lagrange strain tensor ˙ H ∈ IRg,f ), minimal velocity coefficient matrix (˙s = H(q)q, 6,6 Hooke’s matrix (∈ IR ), Hamilton function (∈ IR1 ) tensor of area moments of inertia (∈ IR3,3 ), (electrical) current (∈ IR1 ) torsional area moment of inertia tensor of mass moments of inertia (∈ IR3,3 ), Jacobian (e.g. ∈ IR6N,g ) Jordan matrix

439

440 K dKn L

Mi Mn MRi M N N P Pij P P Q Qn Q

R

R(x) Si T

T∗ Tip T∗ip U

V

List of Symbols restoring matrix (∈ IRg,g ) state feedback matrix (∈ IRn,n , n = f + g) dynamical stiffening matrix momentum of momentum (∈ IR3 ), observer gain matrix, inductance (∈ IR1 ) mass matrix (single body) mass matrix (subsystem) reduced mass matrix (see p. 95) moment (∈ IR3 ), mass matrix (∈ IRg,g ) number of bodies in MBS nonconservative restoring matrix (∈ IRg,g ), recursion matrix (page 95) linearization: velocity dependent matrix (∈ IRg,g ) power (∈ IR1 ) Trefftz (or 2nd Piola-Kirchoff) stresses Trefftz (or 2nd Piola-Kirchoff) stress tensor strain operator (∈ IR6,3 ) generalized force, single body generalized force (subsystem) linearization: position dependent matrix (∈ IRg,g ), generalized force (∈ IRg ), weighting matrix (∈ IRn,n , n = f + g) rotation matrix (∈ IR3,3 ), weighting matrix (∈ IRn,n , n = f + g), radius (∈ IR1 ), (electrical) resistance (∈ IR1 ) Rayleigh quotient estimation matrices (observer, i = 1, 2) transformation matrix (general, e.g. ∈ IR6,6 ), kinetic energy (∈ IR1 ) maneuver time (∈ IR1 ) kinetic co-energy (∈ IR1 ) kinematical chain: y˙ i = Tip y˙ p + Fi s˙ i = NTi Tip (see p. 105) basic disturbance function coefficient matrix (Uz = w), Lyapunov function (∈ IR1 ), (electrical) voltage potential, volume (also Vol )

List of Symbols Vi X W Z

α α, β, γ γ δ − δ 

 ij ij ϑ κ

λ ν π ρ σ σij σ τ ϕ ψ, θ, ϕ ψ˙

441

= FTi M∗i Fi (see p. 106) modal matrix disturbance input matrix, work (∈ IR1 ) disturbance matrix (˙z = Z z), Gaussian constraint (“Zwang”, ∈ IR1 ) (absolute) angular acceleration (∈ IR3 ), degree of nonlinearity (∈ IR1 ) Cardan angles universal gravity constant variation symbol Dirac-distribution Cauchy strain vector (∈ IR6 ), variational parameter (∈ IR1 ), rotor unbalance (∈ IR1 ) Green-Lagrange strain vector (∈ IR6 ) Cauchy strains Green-Lagrange strains vector of beam shape functions (torsion) (∈ IRn ), torsion (∈ IR1 ) (vector of) curvatures, shear correction factor (∈ IR1 ), rotor to shaft inertia relation (∈ IR1 ) vector of Lagrangean multipliers, eigenvalue (∈ IR1 ) Poisson number, frequency quasi-coordinates (∈ IR3 , assigned to ω) mass density, spring elongation stress vector (∈ IR6 ), Cauchy stresses Cauchy stress tensor tangential plane vector of bending angles (∈ IR3 ), rotation angle (∈ IR1 ) Euler angles rotation vector (∈ IR3 ) column matrix of Cardan and Euler angles, resp. (∈ IR3 ), motor angular velocity (∈ IR1 )

List of Symbols

442 ω

angular velocity (∈ IR3 ), frequency (∈ IR1 )



difference, Laplace operator Nabla operator = Θ(f1 , f2 ) defines rotation vector quantities (∈ IR3,3 ) diagonal matrix of eigenvalues = 0 (holonomic) constraint (∈ IRm ), fundamental (or transition) matrix (∈ IRn,n , n = f + g), (matrix of) shape functions magnetic flux (∈ IR1 ) := D ◦ Φ(x): (matrix of) shape function derivatives = 0 (nonholonomic) constraint (∈ IRm ) angular velocity

∇ Θ Λ Φ

Ψ ˙ Ψ Ω Indices:

upper (˙) time derivation f ( ) spin matrix (∈ IR3,3 )

upper right ( )0 spatial derivation e “impressed” c “constrained”, or “w.r.t. mass center” T transposed

middle right ◦ “applied to’

lower right free index, p e.g. “point”, e.g. “predecessor” xy relativity, e.g. between x and y

lower left: Basis, e.g. R reference (general) B body-fixed frame I, o inertial frame

List of Symbols

443

Vectors are generally defined as column vectors. Row vectors are thus obtained by transposition (upper right index T ). Their component representation is characterized by the lower left index (indicating the chosen base). A single lower right index remains free for actual considerations, while an index pair in general indicates relativity. (For instance: vector R rAB connects points A and B (and the origins of frame A and frame B, resp.) in a component representation of frame R). For transformation matrices ∈ IR3,3 , the lower right index pair is to be read from right to left in the sense of “transforming from – to”. (For instance: AAB characterizes the mapping of a vector determined in frame B into its representation in frame A). One obtains thus simple connection rules such that, for transformations, the indices appear in pairs: AAB AB C AC D D r transforms vector r from component representation in frame D via C and B into components of frame A. Using the spin matrix representation e ab for vector product a × b yields the following useful relationships.

(R3)

e =b eT a e ab = −ba e +b ee e−b ee e abc ca + ece ab = 0 = (e ab a−e afb)c e = abT − aT b E e ab

(R4)

e afb = abT − baT

(R1) (R2)

The “dot” for time derivation follows the definition f˙ =

df lim =∆t→0 [f (t + ∆t) − f (t)]/∆t dt

for any function f . This holds also in case that f is a vector component.

INDEX

calculus of variations, 237, 383 canonical equation, 387 Cardan angles, 35, 38 Cauchy, 60, 167, 169 Cayley, 399, 421 central equation, 4, 20–22, 26, 59, 69, 71, 126, 426, 427 change-over gear, 253 Chetaev, 404 closed-loop control, 389 comparison function, 10, 385 constraint force, 2, 4, 8, 9, 64, 71, 73, 80, 81, 85, 100–102, 104, 114, 115, 207, 216, 221, 298, 307, 369 constraints, 9, 10, 15–19, 24–26, 32, 34, 59, 60, 68, 72, 75, 77, 81, 83, 84, 100–102, 104, 105, 107, 108, 114–118, 123, 124, 126, 214, 215, 221, 237, 245, 253, 257, 306, 307, 322, 323, 369, 385, 386 continuity condition, 63 control, 151, 235, 277, 284, 286, 288, 308– 310, 337, 348, 369–371, 373– 375, 377–382, 387–390, 401, 403, 407, 409, 410, 413, 414, 417, 419–421, 423–425, 428, 429 controllability, 235, 286, 372, 375, 381, 401–404, 424 convergence, 110, 111, 221, 235, 242, 243, 250, 251, 253, 322–324, 413 Coriolis, 14, 72, 81, 84, 274, 277, 283, 324, 395, 409

acceleration, 49, 97, 107, 276, 280, 281, 283, 311, 315, 320, 325, 377 Ackermann, 406 Almansi, 171 analytical methods, 2, 3, 71, 126, 130, 131, 214, 322, 327 Appell, 71, 72 Aristotle, 15 auxiliary equation, 12, 18, 19, 21, 70, 72, 427 auxiliary function, 242, 246, 251, 253, 323, 324 auxiliary variables, 79, 81, 113, 214, 297, 315, 321, 334 axioms, 7, 71, 116 backlash, 410 Baumgarte, 107, 108, 114 Bernoulli, 3, 15, 117, 121, 134, 138, 144–146, 150, 151, 153, 164, 233 Boltzmann, 116 Bolza, 384 Born, 60 boundary conditions, 5, 124, 129, 131, 133, 136–139, 142, 143, 145, 151, 155, 158, 161, 163, 164, 166, 187, 188, 196, 210–212, 215, 217, 220, 223, 224, 233, 234, 237, 238, 242, 245, 246, 253, 259, 265, 312, 321–323, 326, 332, 339, 377 Bryan, 35 Bubnov, 219

445

446 d’Alembert, 2, 8, 150 decentralized control, 423, 424 ˆ pital, 3, 47 de l’Ho Descartes, 2 describing (velocity) coordinates, 85 describing variables, 85, 89, 98, 104, 113, 114, 124–126, 132, 134, 153, 190, 191, 194, 198, 220, 223, 247, 301, 321, 328, 330, 345, 348 Dirac, 176, 206, 207, 249, 256, 295 disk, 120, 253, 255, 262, 263, 269, 326 double pendulum, 8, 204, 207, 209, 364 dynamical stiffening, 167, 183–185, 201, 206, 211, 216, 219, 257, 294, 303, 315, 324, 339, 344, 352, 360, 361, 367 dynamical stiffening matrix, 177–180, 185, 190, 195, 201, 295, 302–304, 319, 325, 326, 344, 359 eigenfunction, 111–113, 140, 142–144, 146, 147, 150, 151, 164, 165, 220, 222, 223, 227, 228, 234, 235, 237, 242, 246, 249, 250, 253, 261, 262, 267, 268, 271, 273, 286, 323, 324 eigenvalue, 31, 32, 37, 112, 113, 139, 140, 142, 143, 227, 228, 230, 231, 236, 238, 242, 245, 247, 261, 263, 265–268, 274, 279, 280, 284, 285, 288, 395–398, 403, 404, 406, 414–416, 422 eigenvector, 31, 112, 140, 143, 227–233, 235, 243, 245, 274, 279, 284, 285, 288, 395–397, 403 eigenvector problem, 31, 37, 139, 227, 229, 230, 236, 261, 274, 284, 285, 396, 402, 403 elastic double pendulum, 198, 212, 217, 332, 363, 364 elastic potential, 115, 128, 135, 151, 152, 154, 167, 171, 175, 200, 205, 214, 216, 226, 237, 305, 320, 326, 356 elastic robot, 6, 291, 324, 348, 349, 369, 371 elastic rotor, 127, 159, 255, 273, 283–285, 288, 323, 324, 380 elastic TT-robot, 296, 325

Index Euler, 3, 7, 22, 24, 26, 36, 38, 39, 41, 43, 72, 114, 126, 134, 138, 144, 145, 150–153, 164, 171, 188, 189, 234, 259, 264–268, 292, 296, 308, 313, 316–318, 358, 384, 385, 387, 388 exact linearization, 410, 423 excess function, 385 F¨ oppl, 2, 7, 273 feedback, 67, 374, 375, 380, 393, 426 finite elements, 5, 322 finite segmentation, 109 FORTRAN, 98 Fourier, 150, 238 Frobenius, 400, 406, 409, 411 fundamental lemma, 384 Funk, 17 Galerkin, 68, 164, 188, 216, 217, 219, 221, 233, 234, 322 Gauss, 11, 12, 24, 26, 68, 72, 95, 352 generalized inverse, 18, 87, 307, 373, 391 Gibbs, 71, 72 gravitation potential, 66 Green, 169, 170 guidance motion, 55, 124, 134, 194, 335 guidance velocities, 55, 91, 190, 191, 193, 335, 345, 346, 366 Hamel, 1–3, 12, 21, 69, 72, 75, 126, 130, 176 Hamilton, 17, 18, 22, 26, 72, 124–126, 130, 216, 385, 388, 390, 399, 421 Hautus, 235, 402, 403 helicopter blade, 162, 183, 291 Helmholtz, 12, 18, 19, 21, 70, 72, 427 Hermann, 3, 12, 22, 44 Hertz, 2, 25 Heun, 3, 8, 21 holonomic, 18, 19, 22, 25, 26, 72, 76, 124– 126, 131, 133, 190, 214, 307, 321, 371, 389, 395 Hooke, 14, 116, 120, 122, 171 Hurwitz, 405, 416, 422 Huygens, 80 impact, 146, 147, 149, 267, 364, 370

Index impressed forces, 9, 13, 52, 87, 115, 135, 185, 199, 205, 219, 294, 295, 302, 303, 325, 329, 331, 337, 340, 344, 357, 361, 395 intermediate variables, 81, 84, 85, 124, 132, 214, 315, 321, 325 Jacobi, 2, 7, 23, 24, 26, 72, 102, 107 Jacobian, 18, 71, 85, 86, 89, 124, 192, 193, 299, 309, 332, 333 Jeffcott, 272, 278, 283, 284, 324 Jordan, 397, 398 Jourdain, 12, 25 Kalman, 403 Kane, 13, 73 Kappus, 168, 171 Kepler, 14, 29 kinematic chain, 92, 93, 104, 114, 190, 191, 194, 215, 333, 335, 345, 347, 352, 362, 363, 368, 369 kinetic energy, 13, 20, 21, 29, 65, 69, 70, 123, 124, 127, 130, 214, 229, 237, 238 Kirchhoff, 2, 117, 120, 154, 155, 157, 169, 176, 177, 236, 245 Klein, 1, 22 Kronecker, 239 Lagrange, 2, 3, 7–9, 11–13, 15–17, 20, 22, 44, 63, 64, 72, 73, 100, 102, 103, 126, 130, 131, 169–171, 219, 221, 234, 385, 386, 426– 428 Lamb, 15, 155 Laval, 82, 272, 273, 276–278, 282–284, 324 Legendre, 72, 323 Leibniz, 2, 3, 22, 23 Lie, 20, 52, 281 linear mechanical systems, 389 Luenberger, 404 Lur’e, 20 Lyapunov, 381, 382, 390, 392, 402, 413 Mach, 15 Maggi, 72, 75, 126, 130 magnetic bearing, 288, 381 magnetic levitation vehicle, 428 Magnus, 3, 36, 280

447 mass element, 20, 63–66, 167, 199, 214, 228, 321, 331, 338, 343, 426 mass point, 3, 59, 60, 230, 278 Maupertuis, 22, 26, 72 minimal coordinates, 4, 15–17, 19, 126, 135, 300 minimal representation, 17, 18, 93–95, 98, 103, 107, 114, 125, 220, 333, 337, 352, 370 minimal velocities, 4, 15, 16, 18, 19, 21, 69, 70, 81, 92, 98, 101, 102, 104, 114, 132, 153, 155, 193, 194, 215, 220, 221, 255, 272, 301, 321, 324, 327, 332, 333, 348, 371, 427 mobile robot, 348 modal matrix, 31, 227, 229, 261, 396, 403 modeling, 4, 7, 19, 25, 29, 38, 174, 215, 217, 278, 296, 298, 327, 419 moments of inertia, 65, 80, 123, 127, 133, 134, 157, 159, 174, 246, 298, 308, 329, 331 momentum theorem, 2, 3, 7, 9, 71, 281 moving reference frame, 39, 49, 81, 214, 357 M¨ uller, 404 multiplier, 18, 103 Newton, 2, 14, 107, 247, 269 non-holonomic, 3, 17, 19, 20, 22, 25, 26, 69, 72, 75, 77, 82, 83, 85, 93, 94, 114, 124–126, 177, 190, 214, 321, 327, 395 nutation, 36, 280, 282, 284–286, 324, 397 observability, 235, 401, 403, 411, 415, 421, 422 observer, 14, 370, 375, 401, 410, 412–415, 417, 419, 422–424 one-link elastic robot, 291 optimal control, 388, 389 optimality, 385 optimization, 17, 238, 382, 383, 391 order-n-recursion, 114 ordinary differential equation, 144, 233 output control, 286 partial differential equation, 61, 136, 137, 144, 165, 228, 236, 246 particle, 59

448 pervasive damping, 108, 151, 235, 277, 282, 369, 404 Petrov, 219 Piola, 167, 169 plane motion, 63, 76, 77, 82, 83, 86, 93, 98, 109, 120, 192, 198, 201, 206, 212, 216, 230, 318, 337, 352, 353, 363, 371, 393, 429 plate, 117, 118, 120, 121, 144, 154, 155, 157, 216, 220, 237, 238, 242, 243, 245, 322, 323 ´, 245 Poincare Poinsot, 2, 13, 15, 20, 72 point mechanics, 60 Poisson, 117 Pontryagin, 387 potential, 13, 14, 60, 66–69, 112, 115, 124, 151, 152, 155, 175, 186, 219, 230, 235, 238, 294, 297, 300, 309, 326 potential energy, 229, 237, 391 power, 10, 219, 280, 323, 399 precession, 36, 280–282, 285 predecessor, 50, 53, 55, 58, 87, 91, 92, 95, 96, 98, 105, 107, 190, 192, 194, 196, 215, 254, 335, 339, 345, 362 principle, 2–4, 8, 9, 11–13, 15–18, 20, 22– 26, 63, 64, 68, 72, 98, 100, 114, 115, 124–126, 150, 168, 207, 216, 219, 221, 230, 234, 369, 388, 426 prismatic joint, 142, 180, 296, 338 Projection Equation, 4, 5, 70, 71, 77, 78, 114, 116, 131, 159, 177, 214, 309, 318, 321, 327, 329, 337, 353, 354 quasi-coordinates, 18, 220 Rayleigh, 68, 138, 150, 151, 153, 229, 230, 232, 233, 237, 238, 253, 265, 322 recursive representation, 4, 95, 100, 104, 114, 329, 333, 352, 373 reduced mass matrix, 96 reference frame, 5, 33, 44, 45, 55, 57, 58, 65, 66, 70, 71, 77–79, 81, 82, 85, 87, 91, 123, 127, 133, 202, 214,

Index 296, 299, 301, 321, 323, 327– 331, 338, 355, 356, 395 Resal, 36, 38 residual, 220 revolute joint, 57, 91, 142, 180, 346, 347 rheonomic, 25, 257, 323, 369, 392 Riccati, 380, 388–390, 414, 415 Riemann, 47 rigidity condition, 59, 61 Ritz, 150, 153, 164, 219, 221, 226, 229, 233–238, 245, 253, 255, 269, 286, 290, 292, 293, 297, 300, 301, 304, 308, 312, 319, 322, 323, 325, 326, 332, 333, 336, 337, 344, 345, 348, 365, 371, 375, 377, 380, 382 Rodrigues, 41 rotating beam, 164, 183, 226, 291, 324 rotation angle, 34, 37–39 rotation axis, 33, 34, 37, 39, 275, 281, 338 Schwarz, 44 scleronomic, 15, 25, 72, 102, 391 second-order displacements, 166, 167, 172, 175, 178, 195, 337 shape functions, 150, 164, 188, 217, 220, 221, 224, 228, 233–236, 238, 242, 243, 245, 248–251, 253, 255, 257, 259–264, 266–268, 283–286, 299, 308, 310, 322– 324, 326, 357, 359, 369, 377, 379, 380, 382 shell, 153, 154 SISO, 401 sorting matrices, 210 spatial operators, 64, 132, 153, 193, 196, 215, 301, 332 spillover, 286, 326, 379, 382 spring potential, 67 stability, 67, 277, 278, 281, 282, 305, 312, 324, 375, 380, 391, 392, 401– 405, 407, 413, 416 state equation, 146, 380, 389, 395–397, 399, 400, 404, 411, 413, 414, 419, 421 state feedback, 375, 380, 389, 410, 414, 423 Steiner, 80 Stodola, 405, 422 strain, 63, 116, 117, 119, 121, 167, 170, 171, 177, 348, 375

Index strain tensor, 167, 169–171 stress, 115, 117, 167, 169, 172 stress tensor, 115, 116, 167, 169, 170 Stribeck, 419 structurally variant systems, 104 subsystem, 78, 79, 83–95, 98, 100, 104, 105, 113, 114, 124, 131, 214, 215, 297, 327–331, 335–339, 341, 344, 345, 348, 354, 363, 364, 366, 367, 371, 377, 379 successor, 53, 91, 92, 95, 96, 100, 101, 107, 190, 193, 195, 196, 212, 335, 339 synthetic procedure, 73 Szabo, 22 Tait, 35, 404 Taylor, 16, 17, 67, 158, 372, 373, 383, 384, 392, 403 telescoping arm, 57, 93, 325 Timoshenko, 151, 153 tip body, 339, 340, 346, 356 topological chain, 100 topological tree, 53, 100 topology matrix, 50–53 torsional shaft, 246, 323, 324 transitivity equation, 20, 69, 72, 75

449 Trefftz, 167, 169, 170 variation, 9, 16, 17, 20, 22, 24, 69, 125, 126, 128, 144, 176, 195, 384– 386, 389, 399, 424 variational calculus, 16–18, 20, 24, 68, 126 Varignon, 3 virtual acceleration, 11 virtual displacements, 9, 11, 13, 15–17, 24, 25, 194, 311, 372 virtual velocities, 9, 11, 13 virtual work, 15, 87, 94, 116, 126, 130–132, 137, 144, 146, 166, 178, 180, 215, 219, 221, 224, 234, 248, 295, 297, 300, 332, 365, 427 Voigt, 157 walking machine, 417 wave equation, 146, 149, 150 Weierstrass, 384, 385, 388 whirl, 276, 281, 283–286, 324 work, 8–10, 13, 14, 64–66, 72, 100, 104, 108, 151, 172, 174, 177, 207, 286 Young, 117

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