299 - bibliography [1] caughey, t.k. o'kelly, m.e.j. [2] clough, r.w.978-94-011-3514... ·...
TRANSCRIPT
[1] Caughey, T.K.
O'Kelly, M.E.J.
[2] Clough, R.W.
Penzien, J.
299 -
BIBLIOGRAPHY
Classical Normal Modes in Damped Linear Dynamic
Systems,
J. Appl. Mech., sept. 1965, 583-588
Dynamic of Structures,
McGraw-Hill, 1975
[3] Den Hartog, J.P. Vibrations mecaniques (2nd ed.),
Dunod, Paris, 1%0
[4] Dimentberg, F.M. Flexural Vibrations of Rotating Shafts,
Butterworth, London, 1961
[5] Fertis, D.G.
[6] Filipov, A.P.
[ 7] Frazer, R.A.
Duncan, W.J.
Collar, A.R.
[8] Harker, R.J.
[ 9] Harris, C.M.
Crede, C.E.
Dynamics and Vibrations of Structures,
Krieger, Melbourne FL, 1983
Vibrations of Mechanical Systems,
(translated by Z.F. Reif), National Lending library,
Boston Spa, 1971
Elementary Matrices,
Cambridge Univ. Press. N.Y., 1957
Generalized Methods of Vibration Analysis,
J. Wiley and Sons, N.Y., 1983
Shock and Vibration Handbook (2nd ed. 3 vol.),
McGraw Hill, 1976
[10] Hurty, W.C. Dynamics of Structures
Rubinstein, M.F. Prentice-Hall, Englewood Cliffs, N. J., 1964
[11] Klostermann, A.L. On the Experimental Determination and Use of Modal
Representations of Dynamic Characteristics,
University of Cincinnati, Thesis, 1971
[ 12 J Klutter, K.
[13J Kulsky, H.
[14J Lalanne, M.
Berthier, P.
Der Hagopian, J.
[15 J Lur' e, 1..
[ 16 J Mathey, R.
[17J Mazet, R.
[18] Meirovitch, L.
[19J Meirovitch, L.
[20] Milller, P.C.
[21] Nashif, A.D.
Jones, D. I.G.
Henderson, J.P.
[22J Parlett, B.N.
[23] Preumont, A.
[24J Rao J.S.
Gupta, K.
- 300
Technische Schwingungslehre (2 vol.),
Springer, Berlin, Heidelberg, N.Y., 1980-81
Stress Waves in Solids,
Dover, N.A., 196:3
Mecan i que; des vi brations lineaires (2nd ed.),
Masson, Paris, 1986
Mecanique analytique, (volumes I and II)
Masson, Paris, 1968
Physique des vibrations mecaniques,
Dunod, Paris, 1963
Mecanique vibratoire (2nd ed.),
Duood, Paris, 1966
Analytical Methods in Vibrations,
Macmillan Cumpany, N.Y., 1967
Elements of Vibration Analysis,
McGraw-Hill, 1975
Stabilitat und Matrizen
Springer. Verlag, Berlin, 1977
Vibration Damping,
J. Wiley and Sons, N.Y., 1985
The Symmetric Eigenvalue Problem,
Prentice-Hall, 1980
Vibrations aleatoires et analyse spectrale,
Presses poly techniques romandes, Lausanne 1983
Theory and Practice of Mechanical Vibrations,
Wiley East, Ltd, 1984
[25] Rayleigh, Lord
[26] Rocard, Y.
[27] Roseau, M.
[28] Salles, F.
Lesueur, c.
[29] Sneddon, I.N.
[30] Soutif, M.
[31] Steidel, R.F.
[32] Thomson, W.T.
[ 33] Timoshenko, S.
Young, D.H.
Weaver, W.
[ 34] Tse, F.S.
Morse, 1. E.
Minkle, R.T.
[ 35] Wylie, R.C.
Barrett, L.C.
- 301 -
The Theory of Sound, Vol. 1/
Dover, N.Y., 1945
Dynamique generale des vibrations (4th ed.),
Masson, Paris, 1971
Vibrations des systemes mecaniques,
Masson, Paris, 1984
Les vibrations mecaniques,
ed. Masson, Paris, 1972
Fourier Transforms,
McGraw-Hill, N.Y., 1951
Vibrations, propagation, diffusion,
Dunod, Paris, 1970
An Introduction to Mechanical Vibrations,
(2nd ed.) /
J. Wiley and Sons, 1980
Theory of vibrations with Applications (2nd ed.),
Prentice-Hall, 1981
Vibration Problems in Engineering (4th ed.),
J. Wiley and Sons, N.Y., 1974
Mechanical Vibrations (2nd ed.),
Allyn & Bacon, 1978
Advanced Engineering Mathematics (5th ed.),
MCMillan, 1982
- J02 -
I N 0 E X
The numbers which follow each entry refer to the corresponding pages in
the book.
When several numbers are given, the main reference(s) is (are) indicated
by bold type.
When it helps one to understand the terms in the Index, the nature of
the system is indicated by means of the following abbreviations :
02 oscillator with two degrees of freedom
OG generalized oscillator with n degrees of freedom
Proper Names
d'Ale.bert (history of vibrations), 1, 2
Bernoulli (history of vibrations), 1, 2
Betti, theorem of, 179
carson, transform of, 100
caughey, condition of, 235, 238, 255, 256, 267, 280, 282
Clapeyron, formula of, 179
Couloab, friction, 45, 170
Dirac, del ta function 105, 294
Dirichlet, condi tion of, 117, 123
integral, 104 DuhaJlel,
Duncan, transformation of, 235, 248, 261
- 303 -
Fourier
- series, 2, 80, 84, 115
- transformation, 98, 115, 117, 121, 296
- integral, 115
Frahm,
Galileo
damper, 138 156, 162, 166, 167
(hist.ory of vibrations),
Gibbs, phenomenon, 96
Baailton
- equations, 242, 245, 246, 261
- function (Hamiltonian), 245
Relmloltz, resonator, 22
Rooke (history of vibrations), 2
Kirchhoff (history of vibrations), 2
Kronecker, delta symbol, 200, 256
Lagrange
- equations of, 174, 175, 209, 243
- function of (Lagrangian), 242, 246
Laplace, transformation, 98, 100, 119, 283, 289
Lanchester,
Legendre,
lliDVell,
damper, 169
dual transformation of,
theorem, 179
243, 245
lIevton, law (equation), 6, 14, 23, 45, 138, 176
Nyquist, graph, 64, 65
Poisson (history of vibrations), 2
Pythagorus (history of vibrations),
Rayleigh
- quotient, 204, 207, 222
- 304 -
- dissipation function, 174, 178, 245, 267
Sauveur (history of vibrations),
Silvester, criteria of. 173
Stodola
Taylor
(history of vibrations),
(history of vibrations),
Acceleration (s)
- resonance,
- vector of,
AcD.ittance
62, 63
138, 172
- complex, 57, 58, 86
- operational matrix, 290
- operational (transfer function),
- temporal,
AIIplitude
102, 106
- as a function of frequency, 49
- of the nth harmonic, 81
1, 2
2
101, 103, 106
- reference (normalization, OG), 187
- resonance, 51
Analysis
- classical modal, 283
- experimental modal, 288
- harmonic, 80
Axes, main, of a trajectory, 270
Bandwidth (at half-power), 72
- 305 -
Basis
change of. 192, 193, 248, 251
for the phase space, 255
modal,
Beaa
199, 292
mass concentrated along, 145, 218
natural angular frequenciers of a continuous beam, 221
Beats
- in the free state (02), 153, 154
- in the steady state,
cable of a service lift,
88, 89, 90
149, 150
capacity, electrical,
Circle
129, 131
- of the phase plane, 39
- of angular frequencies, 148, 152
Circuits of force, 133, 135, 136
Clapeyron's foraula, 179
Coefficient (s)
- complex Fourier, 85
- damping,
- Fourier,
- inertia,
80, 94
182
- influence, 180, 188, 215, 216, 219, 262
- modal damping, 237
Compression, alternating, 71
Condition(s)
- Caughey, 235, 238, 255, 256, 267, 280, 282
- 306
- Dirichlet, 117 , 123
- initial, 6, 9, 27, 30, 31
- initial (02) , 142, 268, 269, 272
- initial (OG) , 191, 202, 239, 257,
Constant(s)
- internal damping of a polymer, 43
- of integration,
Constraints
- additional, 190
- hulonomic, 180
9, 27, 30, 31, 191
259
Container, rigid (Helmholtz resonator), 22
Coordinate(s)
- cartesian, 181
- generalized, 172, 181, 192, 225, 264, 289
- modal, 193
- nllrmal, 192, 193, 198
Core (or core matrix of a system),
Coupling,
elastic, 137, 145, 146, 213
inertial, 137
resistive, 137
DaJlper
184, 194, 209, 250, 273
- for elementary oscillator (resistance), 5
- Frahm's, 138, 156, 162
- Frahm's optimal, 162, 166, 167
- Lanchester' s, 169
Damping
- coefficient of,
- critical, 26, 29, 103, 106, 109, 114, 115
- 307 -
- factor (relative damping), 7, 26, 50, 72
- internal (of a polymer bar), 43
- matrix, 138, 172
- proportional, 235
- modal damping factor (relative modal damping, OG),
- sub-critical, 26, 31, 104, 106, 110, 114
- super-critical, 26, 27, 103, 106, 108, 114, 115
- viscous (1 inear) , 7, 236
Decreaent, loqarithaic, 33, 34, 44
Degree(s) of freedo., 5, 135, 137, 172, 180, 289
Diagonalization of a .atrix,
Diagra.
194, 234, 250
- electric, 130, 132, 133, 134, 136
- Frahm's damper, 156
- of a milling table, 225
- principle of elementary oscillator,
- principle of oscillator with 2 degrees of freedom,
Differentiation
- of a quadratic form, 174
- Lagrange's equations, 22, 174, 175, 209, 243
Directions, .ain, of a .ode (02),
Disk on a .achine shaft, 75
Displace.ent(s), static and dyna.ic
- beam, 218, 222
- complex, 57, 86
- cord, 214
- elastic, 8, 59, 112
- generalized, 172
- imposed, 8, 133
- initial,
271, 280
237, 255, 274
139, 177
J08 -
limit (oscillator with dry friction) I 48
- 5tatic r 50, 59, 82, 87, 159
Eigenvalue problea, 194, 250
Eigenvalue (5) , i 86 I 194, 197 , 250, 288
Eigenvector, approxiaate fundamental,
Electrical analogue
207, 222
- force-current (of mobility),
force-voltage, 129
Element(5)
129, 130, 134, 135, 136, 156
- 5i~ple (of the operational impedance matrix), 291
- of a vehicle suspenSIon, 41
Energy
- conservation of, 11
- dissipative oscillator, 35
- electromagnetic, 132
- electrostatic, 132
- kinetic, 12, 15, 21, 35, 107, 132
- kinetic (02) , 177 , 265
- kinetic (OG) , 173, 180, 197, 208, 226
- lost by oscillator, 37
- mechanical, total, 12, 36, 185
- potential, 12, 15, 21, 35, 111 , 132
- potential (02), 145, 177, 265
- potential (OG), 173, 178, 197, 208, 227
Envelope
- displacement, 32, 36
- oscillating (beats), 89, 153
- 309 -
Equation(s), relation(s)
- characteristic (02), 140, 143, 146, 151
- characteristic (OG), 186, 188, 189, 210, 220, 241, 262
- complex differential, 57
- decoupled (independant) differential, 192, 196, 252, 282, 285
- differential (matrix form) (02), 138
- differential (matrix form) (OG), 172, 176
- differential, of motion, 6, 21, 98, 137
- Euler, 84
- Hamilton, 242, 245, 246, 261
- Lagrange, 22, 174, 175, 243
Equilibrium, static, of an oscillator, 14
Excitation
- periodic (saw tooth), 94
- rectangular, non-periodic, 121
- rectangular, periodic, 90
Factor
damping (relative damping), 7, 26, 50, 72
- dynamic amplification, 50, 52, 58, 82, 159
- modal damping (rela ti ve modal dampin<1),
- quality, 53
Fatigue of materials,
Filter, high frequency, 83, 93
Flux in a self-inductance, 130
Force(s)
- random,
- complex, 56, 85
- coupling, 137
- dissipative (viscous resistance),
- elastic, 7, 55
7, 55
237, 255, 274
- 310 -
- external, exciting, 6, 49, 54, 80, 94, 112, 282, 284
- dry friction, 45
- generalized, 137
- harmonic, 6, 132
- impulse, 6, 105
- inertia, 7, 54
- modal, 282, 283, 285
- periodic, 6, 49, 80
- static, 19, 51
Fourier series, 2, 80, 84, 115
Fourie:r spectrllll, 80, 82, 125, 128
Fo:tllS,
- pnergetic (02), 177
- energetic (OG), 196
- symmetrical quadratic, 173, 174, 175
Frequencies, angular
- circle of (02), 148, 152
- for zero coupling (02), 146, 141, 148, 151
- fundamental (of generalized oscillator), 191
- natural, approximate (Rayleigh quotient, OG), 201, 223
- natural, for conservative oscillator, 7, 18, 22, 42, 50
- natural, for dissipative (damped) oscillator, 31, 42
- natural, for generalized conservative oscillator, 186
- natural, for generalized dissipative oscillator, 237
- at acceleration resonance, 62, 63
- at amplitude resonance, 51, 66
- at phase resonance, 54, 62, 66
- at power resonance, 62, 66
- at velocity resonance, 62, 63, 66
- parabola of (02), 147
- relative, of the external force, 50
- 311 -
Frequency, natural [see frequencies, angular],
Friction, dry (Coulomb),
Function(s)
45, 170
- aperiodic, 27, 115, 241
- complex exponential, 84
- even, 81, 90
- frequency, 189, 220
- Hamiltonian, 245
- hyperbolic, 32
- Lagrangian, 242, 246
- odd, 81
- periodic, 80, 115
- Rayleigh dissipation, 174, 178, 245, 267
- transfer, 101, 294, 297
- trigonometric (harmonic), 9, 32, 84
Fundaaental of a Fourier series, 81, 93
Generator
- current, 1.32
- force, 132, 135
- tension, 133
- velocity, 133
Graph
- of Nyquist, 64, 65
- of rotating vectors, 54, SS
- of the complex plane, 56
- of the phase plane, 38
Baraonic (of a Fourier series), 81, 83, 91, 93
Bydropulser (to test for fatigue), 93
10, 16, 19, 22
- 312 -
IJIPeClance(s)
- complex, 57
- matrix, operational (OG), 289
- mechanical, 116
- operational, 101
IlIPlllse
- Dirac, 105, 294
- elastic displacement, 112
- force, 105
Independence, linear, (of the p.igen- and modal vectors),
Inertia
- coefficients of, 138, 182
- moment of, 20, 150, 224
-- mc>mfmt (bending) of,
Integral
19, 77
- of convolution, 102, 283
- of Duhamel, 104
- of Fourier, 115
Isentropic exponent, 24
Isocline in the phase plane, 40
143, 199,255
Isolation of a .ade, 203, 259, 260, 268, 274, 276, 277, 278
Line, polar (of phase plane), 40
llass
- at the end of a wire, 13
- generalized, 182
- matrix, 138, 172, 183, 209, 273
- modal, 200
- on a massless cord, 214
- rigid, 5
- on a massless beam,
313 -
145, 218
- point, in a plane (02), 264, 272
- point, system of (OG), 180
- rigid,
- unit mndal, 201
Matrices
- change of basis, 193, 230, 249, 251
- core, 184, 194, 209, 250, 273
- damping (loss), 138, 172, 233,273
- diagonalization of, 194, 234
- flexibility (influence coefficients),
- inverse of core, 189, 217
180, 215, 217, 220, 262
- mass (inertia), 138,172,183,209,273
- modal, 199
- of residues at the poles, 291
- stiffness, 138, 172, 209, 266, 273
- transfer functions,
Recbanics
290, 292, 294, 296
- Hamiltonian, 235, 242, 243
- Lagrangian, 243
Milling table, 224
Mode(s) shape(s), natural
cuwplcx (damped, non-classical, OC;) , 254, 255, 261, 264
- directions, pr incipal of (02), 271, 280
- fundamental, 191
- isolation of, 203, 259, 260, 268, 274, 276, 277, 278
- of conservative system (02),
- of conservative system (OG),
139, 142, 144, 145
187, 198
~ 314 ~
~ orthogonality of, 143, 199, 200, 212, 255, 273
~ real (damped, classical, OG), 235, 236, 261
~ zero (or rigid body, OG), 199, 229, 231
19, 41, 77, 150, 215 Nodulus of elasticity,
Ro.ent of inertia, 20
Ro.ent of inertia, bending, 19, 77
Ro.entua (OG), 231, 243
Kewton's law (equations), 6, 14, 23, 45, 138, 176
Koraalization of eigenvectors and natural .ode shapes,
Optiaization of Fraha daaper, 162
Orthogonality
~ of modal vectors, 199, 200,238, 255, 256
201, 255, 268
~ of natural modes and modes shapes, 143, 199, 212, 273
Oscillator
~ electric, 53, 129,
~ elementary conservative (harmonic), 9, 13, 22, 25
~ elementary dissipative (damped), 26, 33, 41
~ elementary linear,
~ elementary non~linear,
~ elementary with dry friction, 45
~ generalized conservative, 184, 208
~ generalized dissipative, 233, 281
~ symmetric (02), 144, 145
~ with two degrees of freedom, 131, 177
Parabola of natural angular frequencies, 147
- 315 -
Penclullm
- double symmetric, 144
- triple symmetric, 208
Period [see natural angular frequencies]
- of conservative oscillator, 10
- of dissipative oscillator, 33
Phase(s)
- as function of frequency, 49
- opposition, 10, 54
- quadrature, 10, 54 ,55
- resonance, 54
- space, 235, 255
Phase shift(s)
different, in a mode (OG), 255
- of the displacement of an external force,
- of the nth harmonic, 82
Phenomenon of Gibbs, 96
Plane
- complex, 56, 66, 67
- inertial reference, 180
- phase, 38, 39
Point, inflexion, 28, 30
Polyaer (bar, damping of), 43
Power
53, 57
- 316 -
- active, 60
- consumed in the steady state, 59
- dissipated by systems (dissipation function, OG),
- dissipated in damper by free state,
- instantaneous, 59
- mean (effective) , 60
- reactive, 60
- relative, 61
Pressure (in a Helmholtz resonator) ,
Radius of convergence, 99
204, 207,222 Rayleigh quotient,
Rea1ease (initial),
elementary oscillator, 9, 27, 31
22
oscill~tor with 2 degrees of freedom,
oscillator with n degrees of freedom,
Residues at the poles,
Resistance
291, 295
- electric, 129, 131, 161
- linear mechanical (viscous),
- dry friction (Coulomb), 170
5, 131
16
268
202, 239, 257
174, 178, 245
Resonances [see frequency, natural angular, of resonance]
Resonator of Helaholtz, 22
Response
- complex, 86, 118
- complex frequency, 51, 56, 58, 59, 64, 67, 86, 121, 296
- impulse, 105, 106, 112, 113
- indicial, 105, 107, 112, 114
- operational, 113
- time, 94, 119
- to an initial excitation (OG), 202, 239, 257
- 317 -
Rheological aodel, 44
Roots of characteristic equation, 241, 251
Scalar product (of natural mode shapes), 200, 201
Self-inductance, 129, 131
Service lift (natural frequencies of), 149
Shaft
- of a machine, 18, 19, 75, 76
- of a service lift,
Silvester's criterion,
Solution(s)
149, 150
173
generalized (see states)
- specific, 98, 141, 185, 261
Source of vibrations, 4
Space
- configuration, 199
- Euclidian, 264
- phase,
Spiral
235, 255
- elliptical, 270, 275, 276
- of displacement, 32
- of phase plane, 39
State
- forced, of elementary oscillator, 6, 98, 101, 103, 104, 119
- forced, of generalized oscillator with real modes, 281
- 318
- forced, of generalized oscillator with complex modes, 284, 288
- free, of conservative elementary oscillator, 6, 9
- free, of dissipative elementary oscillator, 26
- free, of generalized conservat.ive oscillator, 184, 212, 217
- free, of generalized dissipative oscillator with complex modes, 254
- free, of generalized dissipat.i ve oscillator with real modes,
- free, of osci lla tor with 2 degrees of freedom, 139
- harmonic steady, of elementary oscillator, 49, 68
- harmonic steady, of Frahm's oscillator, 157
- periodic steady, of elementary oscillator, 6, 49, 80, 88
Step
- of elastic displacement, 112
- of force,
Stiffness
105, 107
- matrix, 138, 172, 180, 209, 266, 273
- modal, 200
- of system, 5, 14, 58, 227
- reciprocal, 179
Stress, bending, 75, 79
Structure
- behaviour of J 298
- dynamic characteristics of, 288
Suspension for vehicles, 41
SyRbol of Kronecker, 200, /56
Syste.s [see also oscillators and states]
236
319
- deformable continuous, 172
- discrete general linear oscillating, 172
- linearly elastic, energy of, 178
- of three equal masses on a string, 216
- pendulum, 20
- rigid solid, 172
- semi-definite, 199
- with several degrees of freedom, 133
- with two degrees of freedom,
Test(s)
137, 149, 177, 264
- fatigue, 69, 93
- free state, 75
TheoreJI
- Clapeyron, 179
- Maxwell-Betti, 179
- spectral, 194
Trajectory of a mass,
Transformation
270, 272, 275, 276, 279
- Carson-Laplace, 100
- Duncan, 235, 248, 261
- Fourif~r , 98, 115, 117 , 121,
- Laplace, 98, 100, 119, 283,
- Legendre, 243, 245
Transmission of vibrations,
Unbalanced mass or weight, 68,
variable of Laplace, 106, 115
Vectors
294, 296
289
69, 71, 75, 76
- 320 .
- modal, 198, 199, 201, 241
- of accelerations, 138, 172
- of displacements, 138, 172
- of external forces, 172
- of modal forces, 285
- of velocities, 138, 172
- rotating, 9, 10, 11, 32, 55
Velocities
- angular, 9, 68
- initial [see initial conditions),
- resonance, 63
- rotation, 71, 75
- vector of, 138, 172
Vibrator (for fatigue tests), 68
9, 28
- 321 -
SYMBOL LIST
We have used latin and greek letters only and we have resorted to the
least possible number of indices. Consequently, it is inevitable that
certain symbols have several meanings. Thus, for example, T represent.s
the kinet.ic energy of a system, the tension in a wire or the period of an
oscillating quantity. However, confusion is highly unlikely.
Symbols appearing only briefly, in particular in the applied examples,
are not mentioned in this list.
When it appears to be necessary to identify the symbols below more
clearly (but not in an exhaustive way), the following abbreviations are
added :
02 oscillator with two degrees of freedom
OG generalized oscillator (with n degrees of freedom)
OGA conservative generalized oscillator
OGB dissipative generalized oscillator with real modes
(ie the Caughey condition is satisfied)
OGC dissipative generalized osrillator with complex modes
(ie the general case where the Caughey condition is not
sati:;(ied).
General symbols
It. is useful to define the meaning, in this book, of several more general
symbols.
- 322 -
u (t) function of time
u(t) first derivative of u(t) with respect to time
u(t) second derivative of u(t) with respect to time
v.. --I vv': n1 }
.. II vII
vector having n components
.. transposed vectur of v
magnitude of the vector .. v
=~ symbol for the square root of minus one
N complex number
N* complex conjugate of N
Re N real part of the complex number N
Tm N imaginary part of the complex nllmber N
Note: complex numbers are no longer underlined from section 12.4,
page 189
i,j,k,l,m,n,r,s,t,m indices, whole numbers
rr
IT
[
[Mj
[Mja
ratio of the circumference to the radius (= 3,1416)
symbol for a product
symbol for a summation
square matrix M of orcler n
ilt'ijoint matrix of [Mj
diagonal matrix of order n
Latin alphabet
a speed of propagation of elastic waves in a solid
a real part of the complex frequency response
A area of a section
- 323 -
A real constant (in particular, the constant of integration)
An coefficient of the nth cosine term of a Fourier series
A complex constant
[A] core matrix (OGA, order n)
b imaginary part of the frequency response .. b eigenvector
B real constant (in particular, the constant of integration)
Bn coefficient of the nth sine term of a Fourier series
~ complex constant
Bp eigenvector of the matrix [B] (OGA, B, order n; OGC, order 2n)
[B] change of basis matrix (OGA, B, order n; OGC, order 2n)
c linear ViGCOU5 damping factor (or linear resistance)
c' critical damping factor
Cj jth resistance acting on a point mass (coplanar system, 02)
c~ modal damping factor of the pth natural mode (OGB)
C electrical capacity
Co half the mean value of a periodic force (:1/2 Fo)
~n complex amplitude of the term ejnwt of a complex Fourier
series (~~ : conjugate of ~n)
[C] damping matrix (or loss matrix) (OG, order n)
t' C°-J diagonal modal damping matrix (OGB, order n)
- 324 -
d (t) impulse response (response of an oscillator to an impulse excitation)
d~ complex term of the pth row of the diagonal matrix ~ DO~ (d~ ,same
but the rth row)
DO p
D(s)
[OJ
~ D'~
e(t)
E
E(s)
[Ej
f(t)
fc(t)
fi (t)
fk(t)
fm(t)
1(t)
.. f(t) .. fO(t)
F
F(s)
modulus of the complex term dO p
Laplace transform of Dirac delta function (=1)
complex amplitude of the nth harmonic of a complex Fourier series
matrix for a dissipative oscillator (OGC, order 2n)
diagonal matrix for a dissipative oscillator (OGC, order 2n)
indicial response (response of an oscillator to a unit step force)
modulus of elasticity (Young's modulus)
Laplace transform of a step function
static displacement caused by a step function
inverse core matrix (= [Aj-1, OGA, order n)
frequency of oscillating quantity (f = 2~/w , see letter w)
external force applied to a system; function of time
viscous frictional force
external force associated with the generalized coordinate xi(t)
elastic restoring force
force of inertia
complex external force
vector of external forces applied to a system
vector of modal forces (OGA,B order n)
amplitude of an harmonic force
Laplace transform of a function f(t)
- 325 -
F' (s) Carson-Laplace transform of a function f(t)
amplitude of the nth harmonic of a periodic force
mean value of a periodic force
I(w) Fourier transform of a function f(t)
[F] core matrix for a dissipative oscillator (OGC, order 2n)
g terrestrial gravitational constant (: 9.81 m/s2)
g(t) response of an oscillator to a step displacement
term of the nth row of the diagonal matrix f' GO,J
matrix for a dissipative oscillator (OGC, order 2n)
diagonal matrix for a dissipative oscillator (OGC, order 2n)
h(t) response of an oscillator to a unit impulse displacement
H total energy of an oscillator (of a system)
H Hamiltonian function (or Hamiltonian, : T+V)
H(s) operational response of an oscillator
energy lost during one period of a dissipative oscillator
initial total energy of a dissipative oscillator
transfer function between the ith and jth displacements
frequency response between the ith and jth displacements
mean total energy of a dissipative oscillator
H complex frequency response
Ho complex response at the natural frequency of a conservative
oscillator, as well as at the velocity, power and phase resonances
Hl complex response at the natural frequency of a dissipative oscillator
H2 complex response at the amplitude resonance
RJ complex response at the acceleration resonance
Hn complex frequency response of the nth harmonic to a periodic
displacement
[H(s) 1
[H' (s) 1
- 326 -
transfer function matrix (OGC, order n)
expanded matrix of transfer functions (or operational admittance
matrix) (OGC, order 2n)
itt) electric current
amplitude of an alternating electric current
I moment of inertia of a plane area about an axis (bending moment of
inertia)
Ip polar moment of inertia of a plane area (torsional moment of
inertia)
~I~ unit matrix (or identity matrix)
J moment of inertia of a solid (mass moment of inertia)
k stiffness of an elastic element
ke equivalent stiffness of an elastic system
ki stiffness of the ith spring acting on a point mass (coplanar system,
02)
kij coupling stiffness between the generalized coordinates xi and Xj
L
L
L
L(f(t) )
modal stiffness of the pth natural mode (kr , same but rth mode)
stiffness matrix (OG, order n)
diagonal modal stiffness matrix (OGA,B, order n)
distance, length of a beam
self-inductance
Lagrange function (or Lagrangian), (= T-V)
Laplace transform of a function f(t), (= F(s))
m
Il1u
[M]
n
p
p
- 327 -
mass of an oscillator (of a system)
generalized mass (or coefficient of inertia), (: mji)
nth mass of a system of point masses
mass matrix (or inertia matrix) (OG, order n)
diagonal modal mass matrix (OGA, B, order n)
determinant of the mass matrix
speed of rotation (in revolutions per minute)
pressure
root of the characteristic equation (02)
p(t) instantaneous power consumed by an oscillator, in the natural or
steady state
generalized motion variable
pth complex modal force
mean power consumed by an oscillator in a steady state
Po mean power consumed by an oscillator in a steady state, at the power
resonance .. p(t) vector of generalized variables of motion .. p(t) vector of external forces (OGe, order 2n) .. pO(t) vector of modal forces (aGe, order 2n)
p(s) vector of Laplace transforms of the external forces (OGe, order 2n)
pth normal (or modal, or decoupled) coordinate
vector of normal coordinates (OGA, B, order n; aGe, order 2n)
Q quality factor of an oscillator
- 328 -
Qi generalized force acting on an elastic system
Qp amplitude of the harmonic function qp(t)
r
R
R
R
R(s)
R(t)
P Rik
[~]
[R'P]
s
[5 I
t
T
T
T
T,
u
u
Laplace transform of qp(t)
vector of generalized forces Qi acting on an elastic system
root of a characteristic equation
gas constant
radius of a circle (of a cylinder, of a pulley, etc.)
electric resistance
radius of convergence of a Laplace transform
radius of the phase plane spiral
Rayleigh quotient
residue at the pole op
matrix of the residues at the pole op (OGC, order n)
expanded matrix of the residues at the pole op (OGC, order 2n)
Laplace variable
element of a symmetric matrix (= Sji)
symmetric matrix
time
kinetic energy of an oscillator (of a system)
tension in a wire or cord
period of an oscillating quantity (= 2w/w, see letter w)
period of a conservative oscillator
period of a dissipative oscillator
auxiliary variable of a convolution integral
electric tension
- 329 -
~ approximate eigenvector (OG, order n)
U amplitude of an alternating electric voltage
[uJ inverse core matrix of a dissipative oscillator (=[FJ-l, OGe, order
2n)
v velocity
v~ velocity of a point mass m~
V volume
V amplitude of an harmonic velocity
V potential energy of an oscillator (of a system)
Vo initial velocity
V. potential energy transmitted to an oscillator by a unit-step force
w
x(t)
x(t)
'x(t)
Xl(t) }
X2 (t)
xa(t)
xe(t)
xi(t)
Xip(t)
(indicial response)
vector of initial velocities (OG, order n)
inverse of the eigenvalue 6p
dissipation function (half the power dissipated)
displacement of an oscillator
velocity
acceleration
generalized displacement of an oscillator with 2 degrees of freedom
(including the Frahm damper)
displacement due to an external force (inverse of VIs) F(s»
displacement corresponding to the initial conditions Xo, Vo
elastic displacement
ith generalized coordinate (generalized displacement)
ith generalized coordinate (generalized displacement) in the pth
natural mode
- 330 -
x' (t) specific solution (displacement) of the differential equation without
right-hand side
x" (t) general solution (displacement) of the differential equation without
right-hand side
complex displacement
Ke(t) complex elastic displacement
~(t) vector of displacements (OG, order n)
vector of velocities (OG, order n)
vector of accelerations (OG, order n)
pth natural mode
amplitude of an harmonic displacement
XIs) Laplace transform of the displacement x(t)
initial displacement
alnplitudes of the harmonic displacements of an oscillator with 2
degrees of freedom (including the Frahm damper)
reference amplitude for the pth natural mode
amplitude of the nth harmonic of a periodic displacement
static displacement
static displacement of the nth harmonic of a periodic
displacement
X' reference amplitude of the mode Xr(t) caused by the initial .. .. conditions Xo' Vo (aGe)
Xe (s) Laplace transform of the elastic displacement xe(t)
complex amplitude of the displacement
K(w) Fourier transform of the displacement x(t)
complex static displacement of the nth harmonic of a periodic
displacement .. XIs) vector of Laplace transforms of the displacements (OG, order nl
- 331 -
vector of the Laplace transforms of the displacements xik(t)
due to an impulse following the displacement xk (OG, order n)
Xo vector of initial displacements (OG, order n)
Xp pth natural mode shape (vector of amplitudes of the pth
natural mode( )OGA, B, order n)
yet) temporal admittance
yet) vector of velocities and displacements (OGe, order 2n)
yes) operational admittance
1 complex admittance
Yn complex admittance of the nth harmonic of a periodic
displacement ~
Yo vector of initial velocities and of initial displacements (OGe,
order 2n) ~
yes) vector of the Laplace transforms of the velocities and displacements
z
z (s)
Z~
Z ~a
Z k
[Z(5)]
(OGe, order 2n)
auxiliary variable to calculate convolution integrals (= t-u)
operational impedance
cartesian coordinate of a point mass ma
complex impedance
column vector of the adjoint matrix [Z(S=6 p)], (OGe, order 2n)
operational impedance matrix (OGe, order 2n)
Greek alphabet
angle, phase, phase shift
- 332 -
ratio of the natural angular frequencies in a Frahm damper
ai any real coefficient
Ui angle defining the direction of the ith spring acting on a point
mass (coplanar system, 02)
Uij influence coefficient between the coordinates xi and Xj
argument of the complex term
[aJ flexibility matrix (matrix of influence coefficients)
angle, phase, phase shift
relative angular frequency of an harmonic external force (= wfwo)
~:lp relative amplitude of the coordinate xip(t) of the pth
natural mode (~ir' the same but rth mode)
~1 }
~2
eigenvectors of an oscillator with 2 degrees of freedom
.. Pp pth natural mode shape (vector of the relative amplitudes of .. ..
the pth natural mode), (P r , ~s' the same but rth and 5th
modes)
isentropic exponent in the (''Illation of state of a gas
angle, in the phase plane, between the tangent to the spiral and the
normal to the radius
1p multiplicative constant, real or complex .. 'Y vector of the constant 1p
6 deflection, static displacement
6p pth eigenvalue
6rs Kronecker delta (= 1 if r = 5, = 0 if r ~ s)
to II ..J diagonal matrix of the eigenvalues 6p (OGA,B order n,
--~-- ""_\
- 333 -
relative deformation (strain) in a material
relative power in a steady state (= p/Po)
ratio of the masses in a Frahm damper (=m2/m,)
"max relative power at a power resonance
n damping factor (or relative damping)
np modal damping factor of the pth natural mode (or relative modal
damping)
e~ } 8 "
P
~
~ , ~o
~max
~n
angle, angle of rotation of a system
argument of the complex modal force pO(t)
angles of the principal axes of the trajectory of a mass of a
coplonar system of springs and linear resistances
damping coefficient
modal damping coefficient of the pth natural mode
logarithmic decrement
diagonal modal damping matrix
coefficient of dry friction (Coulomb friction)
dynamic amplification factor
linearly distributed mass (mass per unit length)
dynamic amplification factor at the phase resonance
dynamic amplification factor at the amplitude resonance
dynamic amplification factor of the nth harmonic of a periodic
displacement
- 334 -
~j angle defining the direction of the jth resistance acting on a point
mass (coplanar system)
density (of matter)
o normal stress (tensile, bending, ... )
damping constant of a polymer
frequency function (= 1/6)
angle, phase shift between two oscillating quantities
'Pl }
'P2
phase shifts of the displacements of an oscillator with two degrees
of freedom
phase shift of the nth harmonic of a periodic displacement
'Pp phase shift of a pth normal mode (OG)
'Por phase shift of the mode ~r(t) stimulated by the initial conditions .. .. vo ' Xo (OGC)
magnetic flux
~n phase shift of the nth harmonic of a periodic force
~ip complementary phase shift of the displacement xip(t) of a pth
complex normal mode (OGC), (~ir' the same but rth mode)
U/ angular frequency of an oscillating quantity
{ M'",,1 angular frequency of a conservative oscillator
angular frequency at the phase resonance
angular frequency at the power resonance
angular frequency at the velocity resonance
W1
W3
W1 }
W2
Q4 12
~ Q2 ~ o
- 335 -
natural angular frequency of a dissipative oscillator
angular frequency at the amplitude resonance
angular frequency at the acceleration resonance
natural angular frequencies of an oscillator with two degrees of
freedom
natural angular frequency of the pth mode
angular frequencies for zero coupling for an oscillator with 2 degrees
of freedom
term representing the elastic coupling of an oscillator with two
degrees of freedom
diagonal matrix of the natural angular frequencies of an
oscillator with real modes (OGA,B, order n)