Output-Only Modal Parameter Estimation Using a Continuously Scanning Laser Doppler Vibrometer System with Application to Structural Damage Detection

Abstract

Spatially dense vibration measurement can be obtained by use of a continuously scanning laser Doppler vibrometer (CSLDV) system that sweeps its laser spot along a scan path. For a linear, time-invariant, viscously damped structure undergoing free vibration, a new type of vibration shapes called free response shapes was defined and obtained by the authors using a CSLDV system with the demodulation method. To date, application of free response shapes is limited to structural damage identification, and they cannot be directly used for model validation while mode shapes can be. This paper extends the concept of free response shapes by proposing a new output-only modal parameter estimation (OMPE) method using a CSLDV system to estimate modal parameters of the structure undergoing free vibration, including natural frequencies, modal damping ratios, and mode shapes. Advantages of the proposed method are: (1) modal damping ratios and mode shapes can be accurately estimated from obtained free response shapes, (2) the scanning frequency of the CSLDV system can be relatively low, and (3) estimated mode shapes can be used for structural damage identification as if they were measured by stepped scanning of a scanning laser Doppler vibrometer. A baseline-free method is applied to identify structural damage using mode shapes estimated by the proposed OMPE method. The analytical expression of free response shapes of the structure is derived, based on which the OMPE method is proposed and presented as a step-by-step procedure. In the proposed OMPE method, natural frequencies of the structure are identified from free response of certain fixed points on the structure; its modal damping ratios and mode shapes are simultaneously estimated using free response shapes measured by a CSLDV system. A numerical investigation is conducted to study the OMPE method and its application to baseline-free damage identification with mode shapes estimated by the OMPE method.

Appendix: Short-Time Fourier Transform

The short-time Fourier transform of $\tilde {z}$, denoted by $\tilde {V}_{w}\left (t,f\right )$, can be expressed by

$$\displaystyle \begin{aligned} \tilde{V}_{w}\left(t,\omega\right)=\int_{-\infty}^{\infty}\tilde{z}\left(\tau\right)g_{s}^{*}\left(\tau-t\right)\mathrm{e}^{-\mathrm{j}\omega\tau}\mathrm{d}\tau{} \end{aligned} $$

(13.28)

where g _s is a window function with a scale s, the superscript ∗ denotes complex conjugation, and $\mathrm {j}=\sqrt {-1}$. The scale s determines the width of g _s in the time domain, which should be smaller than that of a half-scan period. When $\tilde {V}_{w}$ at the i-th natural frequency of the structure becomes almost zero at an instant t _i,0, the amplitude of $\tilde {y}_{i}$ is considered to be zero. Note that in Eq. (13.28), $\tilde {V}_{w}\left (t,\omega \right )$ is visualized by use of a spectrogram whose intensity denotes the power spectral density associated with $\tilde {V}_{w}\left (t,\omega \right )$; g _s is a Hamming function that can be expressed by

$$\displaystyle \begin{aligned} g_{s}\left(t\right)=\left\{ \begin{array}{lll} 0.54-0.46\cos\left(\frac{2\pi t}{s}\right) & , & 0\leq t\leq s\\ 0 & , & \mathrm{otherwise} \end{array}\right. \end{aligned} $$

(13.29)