Structural Damage Diagnosis-Oriented Impulse Response Function Estimation under Seismic Excitations

. 2019 Dec 9;19(24):5413. doi: 10.3390/s19245413

$C$	=	the $N \times N$ damping matrix
$D$	=	the mapping matrix relating the excitation forces to the corresponding DOFs of the structure
$e_{i}$ and $e_{r}$	=	denote the $i^{t h}$ , $r^{t h}$ excitation respectively
$f (t)$	=	a single excitation
$F$	=	implicitly expressed matrix form of excitation
$F^{D W T}$	=	$F^{D W T}$ is the force matrix $F$ after discrete wavelet transform
$h (t)$	=	IRF of displacement vector
$\dot{h} (t)$	=	IRF of velocity vector
$\ddot{h} (t)$	=	IRF of acceleration vector
${\ddot{h}}_{l}^{D W T}$	=	the unit impulse response vector ${\ddot{h}}_{l}$ after discrete wavelet transform
${\ddot{h}}_{l} (t)$	=	IRF of acceleration response from location $l$ at time $t$
${\ddot{h}}^{}_{r}$	=	the estimated IRF obtained from the measured acceleration in the damage state
${\ddot{h}}^{}_{a}$	=	the analytical IRF from the finite element model in the initial intact state
$H_{l}$	=	implicitly expressed matrix form of ${\ddot{h}}_{l} (t)$
$I$	=	the identity matrix
$K$	=	the $N \times N$ global stiffness matrix
$M$	=	the $N \times N$ mass matrix
$n$	=	the total number of time instants
$p$	=	the number of data points
$Q_{i}$	=	the first dimensionality reduction transformation matrix
$Q_{i}^{'}$	=	the second dimensionality reduction transformation matrix
$S$	=	the sensitivity matrix
$U$	=	the left singular matrix
$u_{1}, u_{2}, \dots, u_{m}$	=	the left singular vectors
$V$	=	the right singular matrix
$v_{1}^{}, v_{2}^{}, \dots, v_{n}^{}$	=	the right singular vector
$x (t)$	=	the displacement vector
$\dot{x} (t)$	=	the velocity vector
$\ddot{x} (t)$	=	the acceleration vector
${\ddot{x}}_{l} (t)$	=	the acceleration response from location $l$ at time $t$
$α_{i}$	=	the stiffness reduction factor of the $i^{t h}$ element
$Δ α$	=	the value of factional change of stiffness reduction
$β$	=	the normalized IRF estimation error
$δ (t)$	=	the Dirac delta function
$λ$	=	the regularization parameter
$ξ_{1}$ and $ξ_{2}$	=	the first two damping ratios
$σ_{1}, σ_{2}, \dots, σ_{r}$	=	singular values
$Σ$	=	the diagonal matrix of singular values
$φ (τ)$	=	the scaling function
$ψ (τ)$	=	the mother wavelet function
List of Abbreviations
CoC	=	covariance of covariance
DOFs	=	degrees-of-freedom
FRF	=	frequency response function
DWT	=	discrete wavelet transform
IDWT	=	inverse discrete wavelet transform
IRF	=	impulse response function
SHM	=	structural health monitoring
$std (•)$	=	standard deviation operation