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 |  موضوع: كتاب Dynamics of Structures الإثنين 22 نوفمبر 2021, 10:39 pm | |
| 
أخواني في الله
أحضرت لكم كتاب
Dynamics of Structures
Third Edition
Jagmohan L. Humar
Carleton University, Ottawa, Canada
و المحتوى كما يلي :
Contents
Preface xv
Preface to Second Edition xvii
List of symbols xxi
1 Introduction 1
1.1 Objectives of the study of structural dynamics 1
1.2 Importance of vibration analysis 2
1.3 Nature of exciting forces 3
1.3.1 Dynamic forces caused by rotating machinery 3
1.3.2 Wind loads 4
1.3.3 Blast loads 4
1.3.4 Dynamic forces caused by earthquakes 5
1.3.5 Periodic and nonperiodic loads 7
1.3.6 Deterministic and nondeterministic loads 8
1.4 Mathematical modeling of dynamic systems 9
1.5 Systems of units 11
1.6 Organization of the text 12
PART 1
2 Formulation of the equations of motion:
Single-degree-of-freedom systems 19
2.1 Introduction 19
2.2 Inertia forces 19
2.3 Resultants of inertia forces on a rigid body 21
2.4 Spring forces 26
2.5 Damping forces 29
2.6 Principle of virtual displacement 31
2.7 Formulation of the equations of motion 35
2.7.1 Systems with localized mass and localized stiffness 35
2.7.2 Systems with localized mass but distributed stiffness 37
2.7.3 Systems with distributed mass but localized stiffness 38
2.7.4 Systems with distributed stiffness and distributed mass 42
2.8 Modeling of multi-degree-of-freedom discrete parameter system 51
2.9 Effect of gravity load 53vi Contents
2.10 Axial force effect 57
2.11 Effect of support motion 62
Selected readings 63
Problems 63
3 Formulation of the equations of motion:
Multi-degree-of-freedom systems 69
3.1 Introduction 69
3.2 Principal forces in multi-degree-of-freedom dynamic system 71
3.2.1 Inertia forces 71
3.2.2 Forces arising due to elasticity 74
3.2.3 Damping forces 76
3.2.4 Axial force effects 78
3.3 Formulation of the equations of motion 79
3.3.1 Systems with localized mass and localized stiffness 80
3.3.2 Systems with localized mass but distributed stiffness 81
3.3.3 Systems with distributed mass but localized stiffness 83
3.3.4 Systems with distributed mass and distributed stiffness 89
3.4 Transformation of coordinates 102
3.5 Static condensation of stiffness matrix 106
3.6 Application of Ritz method to discrete systems 109
Selected readings 112
Problems 112
4 Principles of analytical mechanics 119
4.1 Introduction 119
4.2 Generalized coordinates 119
4.3 Constraints 124
4.4 Virtual work 127
4.5 Generalized forces 132
4.6 Conservative forces and potential energy 137
4.7 Work function 142
4.8 Lagrangian multipliers 145
4.9 Virtual work equation for dynamical systems 148
4.10 Hamilton’s equation 154
4.11 Lagrange’s equation 155
4.12 Constraint conditions and Lagrangian multipliers 162
4.13 Lagrange’s equations for multi-degree-of-freedom systems 163
4.14 Rayleigh’s dissipation function 165
Selected readings 168
Problems 168
PART 2
5 Free vibration response: Single-degree-of-freedom system 175
5.1 Introduction 175
5.2 Undamped free vibration 175
5.2.1 Phase plane diagram 177Contents vii
5.3 Free vibrations with viscous damping 186
5.3.1 Critically damped system 186
5.3.2 Overdamped system 188
5.3.3 Underdamped system 189
5.3.4 Phase plane diagram 191
5.3.5 Logarithmic decrement 192
5.4 Damped free vibration with hysteretic damping 197
5.5 Damped free vibration with coulomb damping 199
5.5.1 Phase plane representation of vibrations under
Coulomb damping 202
Selected readings 205
Problems 205
6 Forced harmonic vibrations: Single-degree-of-freedom system 211
6.1 Introduction 211
6.2 Procedures for the solution of the forced vibration equation 212
6.3 Undamped harmonic vibration 214
6.4 Resonant response of an undamped system 218
6.5 Damped harmonic vibration 219
6.6 Complex frequency response 232
6.7 Resonant response of a damped system 237
6.8 Rotating unbalanced force 239
6.9 Transmitted motion due to support movement 244
6.10 Transmissibility and vibration isolation 249
6.11 Vibration measuring instruments 253
6.11.1 Measurement of support acceleration 253
6.11.2 Measurement of support displacement 255
6.12 Energy dissipated in viscous damping 258
6.13 Hysteretic damping 260
6.14 Complex stiffness 265
6.15 Coulomb damping 265
6.16 Measurement of damping 268
6.16.1 Free vibration decay 268
6.16.2 Forced-vibration response 269
Selected readings 275
Problems 275
7 Response to general dynamic loading and transient response 281
7.1 Introduction 281
7.2 Response to an Impulsive Force 281
7.3 Response to general dynamic loading 283
7.4 Response to a step function load 284
7.5 Response to a ramp function load 287
7.6 Response to a step function load with rise time 288
7.7 Response to shock loading 293
7.7.1 Rectangular pulse 293
7.7.2 Triangular pulse 297viii Contents
7.7.3 Sinusoidal pulse 301
7.7.4 Effect of viscous damping 304
7.7.5 Approximate response analysis for short-duration
pulses 306
7.8 Response to ground motion 307
7.8.1 Response to a short-duration ground motion pulse 313
7.9 Analysis of response by the phase plane diagram 315
Selected readings 317
Problems 317
8 Analysis of single-degree-of-freedom systems: Approximate and
numerical methods 323
8.1 Introduction 323
8.2 Conservation of energy 325
8.3 Application of Rayleigh method to multi-degree-of-freedom
systems 330
8.3.1 Flexural vibrations of a beam 335
8.4 Improved Rayleigh method 339
8.5 Selection of an appropriate vibration shape 345
8.6 Systems with distributed mass and stiffness: analysis of
internal forces 349
8.7 Numerical evaluation of Duhamel’s integral 352
8.7.1 Rectangular summation 353
8.7.2 Trapezoidal method 354
8.7.3 Simpson’s method 355
8.8 Direct integration of the equations of motion 359
8.9 Integration based on piece-wise linear representation of the
excitation 360
8.10 Derivation of general formulas 364
8.11 Constant-acceleration method 365
8.12 Newmark’s β method 368
8.12.1 Average acceleration method 370
8.12.2 Linear acceleration method 372
8.13 Wilson-θ method 375
8.14 Methods based on difference expressions 377
8.14.1 Central difference method 377
8.14.2 Houbolt’s method 380
8.15 Errors involved in numerical integration 381
8.16 Stability of the integration method 382
8.16.1 Newmark’s β method 384
8.16.2 Wilson-θ method 387
8.16.3 Central difference method 390
8.16.4 Houbolt’s method 390
8.17 Selection of a numerical integration method 390
8.18 Selection of time step 393
Selected readings 394
Problems 395Contents ix
9 Analysis of response in the frequency domain 399
9.1 Transform methods of analysis 399
9.2 Fourier series representation of a periodic function 400
9.3 Response to a periodically applied load 402
9.4 Exponential form of Fourier series 405
9.5 Complex frequency response function 407
9.6 Fourier integral representation of a nonperiodic load 408
9.7 Response to a nonperiodic load 410
9.8 Convolution integral and convolution theorem 411
9.9 Discrete Fourier transform 413
9.10 Discrete convolution and discrete convolution theorem 416
9.11 Comparison of continuous and discrete fourier transforms 419
9.12 Application of discrete inverse transform 426
9.13 Comparison between continuous and discrete convolution 432
9.14 Discrete convolution of an infinite- and a finite-duration
waveform 437
9.15 Corrective response superposition methods 442
9.15.1 Corrective transient response based on initial conditions 444
9.15.2 Corrective periodic response based on initial conditions 448
9.15.3 Corrective responses obtained from a pair of force pulses 456
9.16 Exponential window method 459
9.17 The fast Fourier transform 464
9.18 Theoretical background to fast Fourier transform 465
9.19 Computing speed of FFT convolution 469
Selected readings 469
Problems 470
PART 3
10 Free vibration response: Multi-degree-of-freedom system 477
10.1 Introduction 477
10.2 Standard eigenvalue problem 478
10.3 Linearized eigenvalue problem and its properties 479
10.4 Expansion theorem 483
10.5 Rayleigh quotient 484
10.6 Solution of the undamped free vibration problem 488
10.7 Mode superposition analysis of free-vibration response 490
10.8 Solution of the damped free-vibration problem 496
10.9 Additional orthogonality conditions 506
10.10 Damping orthogonality 509
Selected readings 518
Problems 519
11 Numerical solution of the eigenproblem 523
11.1 Introduction 523
11.2 Properties of standard eigenvalues and eigenvectors 524x Contents
11.3 Transformation of a linearized eigenvalue problem to the
standard form 526
11.4 Transformation methods 527
11.4.1 Jacobi diagonalization 529
11.4.2 Householder’s transformation 534
11.4.3 QR transformation 538
11.5 Iteration methods 542
11.5.1 Vector iteration 543
11.5.2 Inverse vector iteration 546
11.5.3 Vector iteration with shifts 556
11.5.4 Subspace iteration 562
11.5.5 Lanczos iteration 564
11.6 Determinant search method 571
11.7 Numerical solution of complex eigenvalue problem 576
11.7.1 Eigenvalue problem and the orthogonality relationship 576
11.7.2 Matrix iteration for determining the complex
eigenvalues 579
11.8 Semidefinite or unrestrained systems 586
11.8.1 Characteristics of an unrestrained system 586
11.8.2 Eigenvalue solution of a semidefinite system 587
11.9 Selection of a method for the determination of eigenvalues 595
Selected readings 596
Problems 597
12 Forced dynamic response: Multi-degree-of-freedom systems 601
12.1 Introduction 601
12.2 Normal coordinate transformation 601
12.3 Summary of mode superposition method 604
12.4 Complex frequency response 608
12.5 Vibration absorbers 615
12.6 Effect of support excitation 616
12.7 Forced vibration of unrestrained system 626
Selected readings 631
Problems 631
13 Analysis of multi-degree-of-freedom systems: Approximate and
numerical methods 635
13.1 Introduction 635
13.2 Rayleigh–Ritz method 636
13.3 Application of Ritz method to forced vibration response 653
13.3.1 Mode superposition method 654
13.3.2 Mode acceleration method 658
13.3.3 Static condensation and Guyan’s reduction 663
13.3.4 Load-dependent Ritz vectors 668
13.3.5 Application of lanczos vectors in the transformation
of the equations of motion 676Contents xi
13.4 Direct integration of the equations of motion 679
13.4.1 Explicit integration schemes 681
13.4.2 Implicit integration schemes 685
13.4.3 Mixed methods in direct integration 694
13.5 Analysis in the frequency domain 702
13.5.1 Frequency analysis of systems with classical mode shapes 702
13.5.2 Frequency analysis of systems without classical mode
shapes 707
Selected readings 712
Problems 713
PART 4
14 Formulation of the equations of motion: Continuous systems 719
14.1 Introduction 719
14.2 Transverse vibrations of a beam 720
14.3 Transverse vibrations of a beam: variational formulation 722
14.4 Effect of damping resistance on transverse vibrations of a beam 729
14.5 Effect of shear deformation and rotatory inertia on the flexural
vibrations of a beam 731
14.6 Axial vibrations of a bar 734
14.7 Torsional vibrations of a bar 736
14.8 Transverse vibrations of a string 738
14.9 Transverse vibrations of a shear beam 739
14.10 Transverse vibrations of a beam excited by support motion 742
14.11 Effect of axial force on transverse vibrations of a beam 746
Selected readings 748
Problems 749
15 Continuous systems: Free vibration response 753
15.1 Introduction 753
15.2 Eigenvalue problem for the transverse vibrations of a beam 754
15.3 General eigenvalue problem for a continuous system 757
15.3.1 Definition of the eigenvalue problem 757
15.3.2 Self-adjointness of operators in the eigenvalue problem 759
15.3.3 Orthogonality of eigenfunctions 760
15.3.4 Positive and positive definite operators 761
15.4 Expansion theorem 762
15.5 Frequencies and mode shapes for lateral vibrations of a beam 763
15.5.1 Simply supported beam 763
15.5.2 Uniform cantilever beam 766
15.5.3 Uniform beam clamped at both ends 767
15.5.4 Uniform beam with both ends free 768
15.6 Effect of shear deformation and rotatory inertia on the
frequencies of flexural vibrations 772
15.7 Frequencies and mode shapes for the axial vibrations of a bar 774xii Contents
15.7.1 Axial vibrations of a clamped–free bar 776
15.7.2 Axial vibrations of a free–free bar 777
15.8 Frequencies and mode shapes for the transverse
vibration of a string 785
15.8.1 Vibrations of a string tied at both ends 786
15.9 Boundary conditions containing the eigenvalue 787
15.10 Free-vibration response of a continuous system 792
15.11 Undamped free transverse vibrations of a beam 794
15.12 Damped free transverse vibrations of a beam 796
Selected readings 797
Problems 798
16 Continuous systems: Forced-vibration response 799
16.1 Introduction 799
16.2 Normal coordinate transformation: general case of an
undamped system 800
16.3 Forced lateral vibration of a beam 803
16.4 Transverse vibrations of a beam under traveling load 805
16.5 Forced axial vibrations of a uniform bar 809
16.6 Normal coordinate transformation, damped case 819
Selected readings 825
Problems 825
17 Wave propagation analysis 827
17.1 Introduction 827
17.2 The Phenomenon of wave propagation 828
17.3 Harmonic waves 830
17.4 One dimensional wave equation and its solution 833
17.5 Propagation of waves in systems of finite extent 839
17.6 Reflection and refraction of waves at a discontinuity in the
system properties 847
17.7 Characteristics of the wave equation 851
17.8 Wave dispersion 855
Selected readings 860
Problems 860
PART 5
18 Finite element method 865
18.1 Introduction 865
18.2 Formulation of the finite element equations 866
18.3 Selection of shape functions 869
18.4 Advantages of the finite element method 870
18.5 Element Shapes 870
18.5.1 One-dimensional elements 870
18.5.2 Two-dimensional elements 871
18.6 One-dimensional bar element 872Contents xiii
18.7 Flexural vibrations of a beam 880
18.7.1 Stiffness matrix of a beam element 883
18.7.2 Mass matrix of a beam element 884
18.7.3 Nodal applied force vector for a beam element 886
18.7.4 Geometric stiffness matrix for a beam element 886
18.7.5 Simultaneous axial and lateral vibrations 887
18.8 Stress-strain relationships for a continuum 900
18.8.1 Plane stress 902
18.8.2 Plane strain 903
18.9 Triangular element in plane stress and plane strain 904
18.10 Natural coordinates 911
18.10.1 Natural coordinate formulation for a uniaxial
bar element 911
18.10.2 Natural coordinate formulation for a constant
strain triangle 915
18.10.3 Natural coordinate formulation for a linear
strain triangle 921
Selected readings 926
Problems 926
19 Component mode synthesis 931
19.1 Introduction 931
19.2 Fixed interface methods 932
19.2.1 Fixed interface normal modes 932
19.2.2 Constraint modes 933
19.2.3 Transformation of coordinates 933
19.2.4 Illustrative example 933
19.3 Free interface method 940
19.3.1 Free interface normal modes 941
19.3.2 Attachment modes 941
19.3.3 Inertia relief attachment modes 942
19.3.4 Residual flexibility attachment modes 943
19.3.5 Transformation of coordinates 944
19.3.6 Illustrative example 945
19.4 Hybrid method 951
19.4.1 Experimental determination of modal parameters 952
19.4.2 Experimental determination of the static
constraint modes 957
19.4.3 Component modes and transformation of
component matrices 960
19.4.4 Illustrative example 961
Selected readings 971
Problems 972
20 Analysis of nonlinear response 975
20.1 Introduction 975
20.2 Single-degree-of freedom system 977xiv Contents
20.2.1 Central difference method 979
20.2.2 Newmark’s β Method 981
20.3 Errors involved in numerical integration of nonlinear systems 985
20.4 Multiple degree-of-freedom system 990
20.4.1 Explicit integration 990
20.4.2 Implicit integration 995
20.4.3 Iterations within a time step 999
Selected readings 1000
Problems 1000
Answers to selected problems 1003
Index 101
List of symbols
The principal symbols used in the text are listed below. All symbols, including those
listed here, are defined at appropriate places within the text, usually at the time of their
first occurrence. Occasionally, the same symbol may be used to represent more than
one parameter, but the meaning should be quite unambiguous when read in context.
Throughout the text, matrices are represented by bold face upper case letters while
vectors are generally represented by bold face lower case letters An overdot signifies
differential with respect to time and a prime stands for differentiation with respect to
the argument of the function
a acceleration; constant; linear dimension
a decay parameter in exponential window method
an coefficient of Fourier series cosine term
aij flexibility influence coefficient
am real part of mth eigenvector
A constant; cross-sectional area
Aa amplitude of dynamic load factor for acceleration
Ad amplitude of dynamic load factor for displacement
Av amplitude of dynamic load factor for velocity
A amplification matrix; flexibility matrix; square matrix
A˜ transformed square matrix
b constant; linear dimension; width of beam cross section
bn coefficient of Fourier series sine term
b vector of body forces per unit volume
bm imaginary part of mth eigenvector
B constant; differential operator
B square matrix
c damping constant; velocity of wave propagation
ccr critical damping constant
c
g velocity of wave group
cn coefficient of Fourier series term, constant
cs internal damping constant
cij damping influence coefficient
c¯ damping constant per unit length
c∗ generalized damping constant
c vector of weighting factors in expansion theoremxxii List of symbols
C constant
Cn modal damping constant for the nth mode
C damping matrix; transformation matrix
C∗, C˜ transformed damping matrix
d diameter
dn constant
D dynamic load factor
D diagonal matrix; dynamic matrix; elasticity matrix
e eccentricity of unbalanced mass
E modulus of elasticity
Em remainder term in numerical integration formula
EA axial rigidity
EI flexural rigidity
E dynamic matrix = D−1
f undamped natural frequency in cycles per sec
f (x) eigenfunction of a continuous system
fd damped natural frequency
fD damping force
fG force due to geometric instability
fI inertia force
fS spring force
fSt total of spring force and damping force for hysteretic damping
f0 frequency of applied load in cycles per sec
f vector representing spatial variation of exciting force
f vector of forces acting on element nodes
fD
vector of damping forces
fG vector of geometric instability forces
fI
vector of inertia forces; vector of global inertia forces
fi
I vector of inertia forces in element i
fS
vector of spring forces; vector of global spring forces
fi
S vector of spring forces in element i
F force
Fx, Fy, Fz components of force vector along Cartesian coordinates
F force vector
Fa vector of applied forces
Fc vector of constraint forces
g acceleration due to gravity
g(t) forcing function
gˆ scaled forcing function e−atg(t)
G constant; modulus of rigidity
G1, G2 constants
G() Fourier transform of g(t)
Gˆ () Fourier transform of gˆ(t)
GJ torsional rigidity
G flexibility matrix
Gd residual flexibility matrix
Gf inertia relief flexibility matrixList of symbols xxiii
Gk flexibility matrix of retained modes
h height; time interval
h(t) unit impulse response
h¯(t) periodic unit impulse response
hˆ(t) scaled unit impulse function h(t)e−at
H(ω0), H() complex frequency response, Fourier transform of h(t)
H¯ () periodic complex frequency response, Fourier transform of h¯(t)
Hˆ (t) Fourier transform of hˆ(t)
H matrix of frequency response functions
i imaginary number; integer
i unit vector along x axis
I impulse; moment of inertia
IA mass moment of inertia for rotation above point A
I0 functional; mass moment of inertia for rotation about the mass
center
I identity matrix
j integer
j unit vector along y axis
J polar moment of inertia
k spring constant; stiffness; integer; wave number
kG geometric stiffness
kT tangent stiffness
kij stiffness influence coefficient
k shape constant for shear deformation
k¯ spring constant per unit length
k∗ generalized stiffness
k unit vector along z axis
K differential operator
Kn modal stiffness for the nth mode
K stiffness matrix
Ke stiffness matrix of an element
Kˆ i augmented stiffness matrix for element i
KG geometric stiffness matrix
K∗, K˜ transformed stiffness matrix
K˜ cc, K˜ ss effective constrained coordinate stiffness matrix
l length
L Lagrangian; length; length of an element
L operator matrix; vector of interpolation functions
LK lower triangular factor of stiffness matrix
LM lower triangular factor of mass matrix
m integer; mass; mass per unit length
m0 mass; unbalanced mass
mij mass influence coefficient
m¯ mass per unit length; mass per unit area
m∗ generalized mass
M concentrated mass, differential operator; momentxxiv List of symbols
MI inertial moment
Mn modal mass for the nth mode
Ms moment due to internal damping forces
M0 concentrated mass
M mass matrix
Me mass matrix of an element
Mˆ i augmented mass matrix for element i
M∗, M˜ transformed mass matrix
n integer
N normal force; number of degrees of freedom
Ni interpolation function
N transformation matrix
p integer; force
pn modal force in the nth mode
p¯ force per unit length
p∗ generalized force
p(λ) characteristic polynomial
p left eigenvector; force vector; global force vector
p vector of generalized coordinates
pe equivalent forces at element nodes
P axial force; concentrated applied load
PI inertial force; inertia relief projection matrix
P0, p0 amplitude of applied force
P matrix of left eigenvectors
q integer
qi generalized coordinate
q right eigenvector; global nodal parameters
q vector of generalized coordinates
q˜ transformed eigenvector
qe vector of displacements at element nodes
qi nodal parameters for element i
Q applied force
Qi generalized force
Q matrix of eigenvectors, orthogonal transformation matrix
r common ratio; constant; integer; radius of gyration
r rank of a matrix; radius vector
r(t) response due to unit initial displacement
r¯(t) response due to periodic unit displacement changes
r vector of applied forces per unit volume
R Rayleigh dissipation function; reaction; remainder term
Ra inertance
Rd receptance
Rv mobility
Ri magnitude of ith corrective force impulse
Ra inertance matrix
Rd receptance matrix
Rv mobility matrixList of symbols xxv
RS vector of support reactions
s complex eigenvalue
s¯ conjugate of complex eigenvalue s
s(t) response due to initial unit velocity
s¯(t) response due to a periodic unit velocity changes
S axial force
S matrix of complex eigenvalues; transformation matrix
Sn matrix for sweeping the first n eigenvectors
t time
tp
time at peak response
t surface forces per unit area
T torque
T kinetic energy; tensile force; undamped natural period
Td damped natural period
TR transmission ratio
T0 period of applied load
T transformation matrix; tridiagonal matrix
u(t), u displacement
u
g ground displacement
ui constrained coordinate; displacement along degree-of-freedom i
ux displacement along x direction
u
y displacement along y direction
u0 initial displacement
ut absolute displacement
us static displacement
u¯(t) periodic displacement response
u displacement vector
U complex frequency response; strain energy
U() Fourier transform of u(t)
U upper triangular matrix, complex frequency response matrix
v velocity
v(x) comparison function
v0 initial velocity
v complex eigenvector
v¯ conjugate of complex eigenvector
V potential energy; shear force
V0 base shear
V matrix of complex eigenvectors
w(x) comparison function
WD energy loss per cycle in viscous damping
We work done by external forces
Wi energy loss per cycle; work done by internal forces
Ws work done by elastic force
x Cartesian coordinate
x¯ coordinate of the mass center
X Lanczos transformation matrix
y Cartesian coordinatexxvi List of symbols
y0n initial value of the nth normal coordinate
y vector of normal coordinates
y0 vector of initial values of the normal coordinates
z generalized coordinate
α angular shear deformation; coefficient; constant; parameter
αi generalized coordinate
α vector of generalized coordinates
β constant; frequency ratio; parameter
γ angle; inverse eigenvalue; parameter
δ deflection; eigenvalue; eigenvalue measured from a shifted origin
δ logarithmic decrement
δ(x) delta function
δij Kronecker delta
δu vector of virtual displacements in an element
δz virtual displacement
δε vector of virtual displacements in an element
δθ, δφ virtual rotation
δqe vector of virtual displacements at element nodes
δr virtual displacement vector
δu virtual
δWe virtual work done by external forces
δWi virtual work done by internal forces
δWei virtual work done by forces acting on internal elements
δWS virtual work done by axial force
displacement
st static deflection
t increment of time
increment of frequency
vector of displacements
ε strain; quantity of a small value
ε strain vector; real part of complex eigenvector
η hysteretic damping constant, angle
η(t) corrective response
ηk imaginary part of eigenvector
η imaginary part of complex eigenvector
θ angular displacement; flexural rotation; polar coordinate; parameter
κ curvature
λ eigenvalue; Lagrangian multiplier; wave length
matrix of eigenvalues
µ coefficient of friction; eigenvalue; eigenvalue shift
µ(t) unit step function
µm real part of mth eigenvalue
νm imaginary part of mth eigenvalue
ξ damping ratio; spatial coordinate
ξh equivalent hysteretic damping ratio
ξk real part of eigenvector
ρ root of difference equation, mass per unit volumeList of symbols xxvii
ρ amplitude of motion; Rayleigh quotient
ρh amplitude of motion for hysteretic damping
ρ(A) spectral radius of A
σ stress
σD damping stress
σ stress vector
τ time
φ angle; normalized eigenvector or mode shape; phase angle
φ potential function; spherical coordinate
φ(x) normalized eigenfunction
φh phase angle for hysteretic damping
φ mode shape, mass-normalized mode shape
modal matrix, matrix of mass-normalized mode shapes
k matrix of retained modes
f flexible body modes
r rigid body modes
χ response amplitude
ψ shape vector
ψ(x) shape function
a matrix of attachment modes
c matrix of static constraint modes
s matrix of static constraint modes
ω undamped natural frequency in rad/s
ωd damped natural frequency
ω0 frequency of applied load in rad/s
ω vector of natural frequencies
frequency of the exciting force
∇ gradient vector
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