[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)