Non-baseline Damage Detection Based on the Deviation of Displacement Mode Shape Data

Abstract

The early detection of damage and structural health monitoring should be an important process for structural maintenance. The baseline information of the structural reference state is not available since the structure was not instrumented prior to the damage. This work offers a global-deviation approach to detect damage by measured data only without available data at the intact state. This work shows that the damage exists at the measurement locations to represent large and abrupt variation deviated from global mode shape curve. The proposed method is compared with the GSM (Gapped-Smoothing Method) provided by Ratcliffe and Bagaria and its superiority and effectiveness are illustrated in a numerical simulation and an experiment.

Appendix

Assume that measurement data are expressed by a finite number of points in space and frequency. Expressing a set of FRF response data as U, suppose that m linear snapshots of the response U _i of size n are obtained by response measurements written as

(19)

where the superscript (j) represents the FRF frequency response data at the j-th frequency ω _j , n denotes the number of measurement positions of the structure and m is the total number of frequency observations.

The POD of this discrete field consists in solving the eigenvalue problem. Let m×m matrix $\tilde{\mathbf{C}}$ define as

$$ \tilde{\mathbf{C}} = \mathbf{U}^{T}\mathbf{U} $$

(20)

Solving the eigenvalue problem of Eq. (20) at the core of the POD method, it satisfies

(21)

where the eigenvalues are arranged as

$$ \lambda_{1} \ge\lambda_{2} \ge\cdots\ge\lambda_{m} \ge 0 $$

(22)

where the eigenvalues λ _k are the proper orthogonal values (POVs) and each eigenvector φ _k of the extreme value problem is associated with a POV λ _k ⋅φ ₁ represents the eigenvector corresponding to the largest eigenvalue λ ₁.

The POMs may be used as a basis for the decomposition of U. The POM associated with the greatest POV is the optimal vector. If the eigenvalues are normalized, they represent the relative energy captured by the corresponding POM. The eigenvalue reflects relative kinetic energy associated with the corresponding mode. The energy is defined as the sum of the POVs. The POMs are written as

$$ \boldsymbol{\psi} ^{k} = \frac{\sum_{i = 1}^{m} \boldsymbol{\varphi} _{i}^{k} \mathbf{U}^{(i)}}{\| \sum_{i = 1}^{m} \boldsymbol{\varphi} _{i}^{k} \mathbf{U}^{(i)} \|},\quad k = 1, \ldots,m $$

(23)

And the POMs are arranged as

$$ \boldsymbol{\psi} = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} \boldsymbol{\psi} ^{1} & \boldsymbol{\psi} ^{2} & \cdots & \boldsymbol{\psi} ^{m} \end{array} \right] $$

(24)