Nomenclature
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diagonal element of the modal damping matrix |
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terms in the expansion of approximate complex modes |
| α1, α2 | proportional damping constants |
| αj | coefficients in Caughey series, j = 0, 1, 2, ··· |
| 0j | a vector of j zeros |
| A | state-space system matrix |
| aj | a coefficient vector for the expansion of jth complex mode |
| α | a vector containing the constants in Caughey series |
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frequency response function of an SDOF system |
| B | state-space system matrix |
| bj | a vector for the expansion of jth complex mode |
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forcing vector in the Laplace domain |
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modal forcing function in the Laplace domain |
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effective forcing vector in the Laplace domain |
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response vector in the Laplace domain |
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Laplace transform of the state-vector of the first-order system |
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modal coordinates in the Laplace domain |
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Laplace transform of the internal variable yk(t) |
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positive real line |
| C | viscous damping matrix |
| C′ | modal damping matrix |
| C0 | viscous damping matrix (with a non-viscous model) |
| Ck | coefficient matrices in the exponential model for k = 0, …, n, where n is the number of kernels |
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non-viscous damping function matrix in the time domain |
| ΔK | error in the stiffness matrix |
| ΔM | error in the mass matrix |
| β | non-viscous damping factor |
| βc | critical value of β for oscillatory motion,  |
| βi(•) | proportional damping functions (of a matrix) |
| βk(s) | coefficients in the state-space modal expansion |
| βmU | the value of β above which the frequency response function always has a maximum |
| F | linear matrix pencil with time step in state-space, F = B –  A |
| F1, F2 | linear matrix pencils with time step in the configuration space |
| Fj | regular linear matrix pencil for the jth mode |
| f′(t) | forcing function in the modal coordinates |
| f(t) | forcing function |
| G(s) | non-viscous damping function matrix in the Laplace domain |
| G0 | the matrix G(s) at s → 0 |
| G∞ | the matrix G(s) at s → ∞ |
| H(s) | frequency response function matrix |
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real part of  |
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imaginary part of |
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jth measured complex mode |
| I | identity matrix |
| K | stiffness matrix |
| M | mass matrix |
| Oij | a null matrix of dimension i × j |
| Ω | diagonal matrix containing the natural frequencies |
| p | parameter vector (in Chapter 1 ) |
| Pj | a diagonal matrix for the expansion of jth complex mode |
| ϕj | eigenvectors in the state-space |
| ψj | left eigenvectors in the state-space |
| q(t) | displacement response in the time domain |
| q0 | vector of initial displacements |
| Qj | an off-diagonal matrix for the expansion of jth complex mode |
| r(t) | forcing function in the state-space |
| Rk | rectangular transformation matrices (in Chapter 4, [ADH 14] ) |
| Rk | residue matrix associated with pole sk |
| S | a diagonal matrix containing eigenvalues sj |
| T | a temporary matrix,  (Chapter 2) |
| Tk | Moore-Penrose generalized inverse of Rk |
| Tk | a transformation matrix for the optimal normalization of the kth complex mode |
| Θ | normalization matrix |
| u(t) | the state-vector of the first-order system |
| u0 | vector of initial conditions in the state-space |
| uj | displacement at the time step j |
| v(t) | velocity vector  |
| vj | a vector of the j-modal derivative in Nelson’s methods (in Chapter 1) |
| vj | velocity at the time step j |
| εj | error vector associated with jth complex mode |
| φk(s) | eigenvectors of the dynamic stiffness matrix |
| W | coefficient matrix associated with the constants in Caughey series |
| X | matrix containing the undamped normal modes xj |
| xj | undamped eigenvectors, j = 1, 2, ···, N |
| y(t) | modal coordinate vector (in Chapter 2, [ADH 14]) |
| yk(t) | vector of internal variables, k = 1, 2, ···, n |
| yk,j | internal variable yk at the time step j |
| Z | matrix containing the complex eigenvectors zj |
| zj | complex eigenvectors in the configuration space |
| ζ | diagonal matrix containing the modal damping factors |
| ζv | a vector containing the modal damping factors |
| χ | merit function of a complex mode for optimal normalization |
| χR, χI | merit functions for real and imaginary parts of a complex mode |
| Δ | perturbation in the real eigenvalues |
| δ | perturbation in complex conjugate eigenvalues |
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initial velocity (SDOF systems) |
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small error |
| η | ratio between the real and imaginary parts of a complex mode |
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dissipation function |
| γ | non-dimensional characteristic time constant |
| γj | complex mode normalization constant |
| γR, γI | weights for the normalization of the real and imaginary parts of a complex mode |
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frequency-dependent estimated characteristic time constant |
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estimated characteristic time constant for jth mode |
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an arbitrary independent time variable |
| κj | real part of the complex optimal normalization constant for the jth mode |
| λ | complex eigenvalue corresponding to the oscillating mode (in Chapter 3, [ADH 14]) |
| λj | complex frequencies MDOF systems |
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moment of the damping function |
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dissipation energy |
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non-viscous damping kernel function in an SDOF system |
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kinetic energy |
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potential energy |
| μ | relaxation parameter |
| μk | relaxation parameters associated with coefficient matrix Ck in the exponential non-viscous damping model |
| v | real eigenvalue corresponding to the overdamped mode |
| vk(s) | eigenvalues of the dynamic stiffness matrix |
| ω | driving frequency |
| ωd | damped natural frequency of SDOF systems |
| ωj | undamped natural frequencies of MDOF systems, j = 1, 2, ···, N |
| ωn | undamped natural frequency of SDOF systems |
| ωmax | frequency corresponding to the maximum amplitude of the response function |
| ωdj | damped natural frequency of MDOF systems |
| ρ | mass density |
| i | unit imaginary number,  |
| τ | dummy time variable |
| θj | characteristic time constant for jth non-viscous model |
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forcing function in the modal domain |
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normalized frequency ω/ωn |
| ςj | imaginary part of the complex optimal normalization constant for the jth mode |
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phase angle of the response of SDOF systems |
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phase angle of the modal response |
| ψ | a trail complex eigenvector (in Chapter 2, [ADH 14]) |
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asymmetric state-space system matrix |
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fitted damping matrix |
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fitted generalized proportional damping function (in Chapter 2) |
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state-space system matrix for rank-deficient systems |
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state-space system matrix for rank-deficient systems |
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integration of the forcing function in the state-space for rank-deficient systems |
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integration of the forcing function in the state-space |
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matrix containing the state-space eigenvectors for rank-deficient systems |
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eigenvectors in the state-space for rank-deficient systems |
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forcing function in the state-space for rank-deficient systems |
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the state vector for rank-deficient systems |
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vector of internal variables for rank-deficient systems, k = 1, 2, ···, n |
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internal variable yk at the time step j for rank-deficient systems |
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jth eigenvector corresponding to the kth the internal variable for rank-deficient systems |
| ξ | a function of ζ defined in equation [3.132] (Chapter 3, [ADH 14]) |
| ζ | viscous damping factor |
| ζc | critical value of ζ for oscillatory motion,  |
| ζj | modal damping factors |
| ζL | lower critical damping factor |
| ζn | equivalent viscous damping factor |
| ζU | upper critical damping factor |
| ζmL | the value of ζ below which the frequency response function always has a maximum |
| ak, bk | non-viscous damping parameters in the exponential model |
| B | response amplitude of SDOF systems |
| Bj | modal response amplitude |
| c | viscous damping constant of an SDOF system |
| ck | coefficients of exponential damping in an SDOF system |
| ccr | critical damping factor |
| dj | a constant of the j-modal derivative in Nelson’s methods |
| E | Young’s modulus |
| f (t) | forcing function (SDOF systems) |
| fd(t) | non-viscous damping force |
| G(iω) | non-dimensional frequency response function |
| G(s) | non-viscous damping kernel function in the Laplace domain (SDOF systems) |
| g(i) | scalar damping functions, i = 1, 2, ··· |
| h | constant time step |
| h(t) | impulse response function of SDOF systems |
| h(t) | impulse response function |
| Ik | non-proportionally indices, k1 = 1, 2, 3, 4 |
| k | spring stiffness of an SDOF system |
| L | length of the rod |
| le | length of an element |
| m | dimension of the state-space for non-viscously damped MDOF systems |
| m | mass of an SDOF system |
| N | number of degrees of freedom |
| n | number of exponential kernels |
| nd | number of divisions in the time axis |
| p | any element in the parameter vector p (in Chapter 1) |
| q(t) | displacement in the time domain |
| q0 | initial displacement (SDOF systems) |
| Qnck | non-conservative forces |
| R(x) | Rayleigh quotient for a trail vector x |
| R1, R2, R3 | three new Rayleigh quotients |
| rj | normalized eigenvalues of non-viscously damped SDOF systems (in Chapter 3, [ADH 14]) |
| rk | rank of Ck matrices |
| s | Laplace domain parameter |
| sj | eigenvalues of dynamic systems |
| t | time |
| Tn | natural time period of an undamped SDOF system |
| Tmin | minimum time period for the system |
| varrhoj | complex optimal normalization constant for the jth mode |
| x | normalized frequency-squared,  (in Chapter 3, [ADH 14]) |
| yj | modal coordinates (in Chapter 3, [ADH 14]) |
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forcing function in the Laplace domain |
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displacement in the Laplace domain |
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matrix containing |
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matrix containing |
| Φ | matrix containing the eigenvectors ϕj |
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vector of initial velocities |
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non-viscous proportional damping functions (of a matrix) |
| Y k | a matrix of internal eigenvectors |
| ykj | jth eigenvector corresponding to the kth the internal variable |
| PSD | power spectral density |
| 0 | a vector of zeros |
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Lagrangian (in Chapter 3, [ADH 14]) |
| δ(t) | Dirac-delta function |
| δjk | Kroneker-delta function |
| Γ(•) | gamma function |
| γ | Lagrange multiplier (in Chapter 3, [ADH 14]) |
| (•)* | complex conjugate of (•) |
| (•)T | matrix transpose |
| (•)–1 | matrix inverse |
| (•)–T | matrix inverse transpose |
| (•)H | Hermitian transpose of (•) |
| (•)e | elastic modes |
| (•)nv | non-viscous modes |
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derivative with respect to time |
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space of complex numbers |
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space of real numbers |
| ⊥ | orthogonal to |
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Laplace transform operator |
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inverse Laplace transform operator |
| det(•) | determinant of (•) |
| diag [•] | a diagonal matrix |
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for all |
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imaginary part of (•) |
| ∈ | belongs to |
| ∉ | does not belong to |
| ⊗ | Kronecker product |
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Laplace transform of (•) |
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real part of (•) |
| vec | vector operation of a matrix |
| O(•) | in the order of |
| ADF | anelastic displacement field model |
| adj(•) | adjoint matrix of (•) |
| GHM | Golla, Hughes and McTavish model |
| MDOF | multiple-degree-of-freedom |
| SDOF | single-degree-of-freedom |
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