Shock and Vibration

Volume 2018, Article ID 6476783, 16 pages

https://doi.org/10.1155/2018/6476783

## Damping Identification with Acceleration Measurements Based on Sensitivity Enhancement Method

^{1}Key Lab of Structures Dynamics Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China^{2}Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China^{3}School of Civil Engineering and Architecture, Heilongjiang University, Harbin 150080, China^{4}School of Civil Engineering and Architecture, University of Jinan, Jinan 250022, China

Correspondence should be addressed to Kun Liu; nc.ude.tih@uil.nuk

Received 12 February 2018; Accepted 15 May 2018; Published 12 June 2018

Academic Editor: Emanuele Reccia

Copyright © 2018 Xusheng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The damping is important for forward and inverse structural dynamic analysis, and damping identification has become a hot issue in structural health monitoring recently. The dynamic responses of the structure can be measured in practice, and the structural parameter usually can be identified by inverse response sensitivity analysis. To reduce the measurement noise effect and enhance the effectiveness of the response sensitivity method, an enhanced sensitivity analysis method was proposed to identify the structural damping based on the Principal Component Analysis (PCA) method. The measured acceleration responses were analyzed by PCA method, and the updated analytical responses and the response sensitivities were projected into the subspace determined by the first-order principal component. The projection equations were adopted to identify the parameters of damping model. The proposed damping identification method was numerically validated with a planar truss structure at first, and then the experimental study was conducted with a steel planar frame structure. It shows that the proposed method is effective in identifying the parameters of damping model with better accuracy compared with the conventional acceleration response sensitivity method, and it is also robust to the sensor placement and measurement noise.

#### 1. Introduction

The damping ratio is a dimensionless measure and a measure of describing how rapidly the oscillations of a structural system decay from one bounce to the next, which is a significant factor when analyzing the structural dynamic behavior dominated by energy dissipation [1]. Unlike the mass or stiffness that can be measured or determined by static test, the damping cannot be determined by measurement or static test [2], but the damping characteristic is very important in structural health monitoring. Accurate damping matrix construction is a determining factor in analyzing structural dynamic responses and predicting energy dissipation behavior, which makes the damping estimation become a key issue for the structural design, dynamic response analysis, structural health monitoring, etc.

In order to construct accurate structural damping matrix, damping identification has been studied by some researchers for different systems, including rotor systems [3], mistuned blisks [4] and monopile foundation [5]. For the structural system, the damping identification method can be classified into frequency domain method [6–8], time domain method [9–11], and time-frequency domain method [12, 13]. The half-power bandwidth method [6, 7] in frequency domain is usually adopted in the dynamic test, and Wang [8] studied the errors of calculating damping effect between the classical and the third-order half-power method. Li and Law [9] proposed a time domain damping ratio identification method based on the acceleration response sensitivity, and the proposed method is validated by numerical study and experimental study [11]. The time domain response sensitivity method was combined with iterative regularization method to identify damping ratios [10]. The wavelet analysis method [12, 13] and Hilbert–Huang Transform method [14] have also been adopted to identify the damping ratio.

Sensitivity analysis can be used to estimate the system output variation due to a perturbation in the system parameters by means of partial derivatives [15, 16], and forward sensitivity analysis has been used in many applications [17–19]. Inverse sensitivity-based method with model updating is usually based on a first-order Taylor series that minimizes an error function to assess the system parameter perturbation, and time domain sensitivity method has been adopted in structural parameter identification widely [20–22]. The time domain response sensitivity method has advantages including no requirement of computing the higher order system model parameters, obtaining the responses easily, and providing more identification equations, which makes it a good tool for damping identification. Time domain response sensitivity method has also been studied and gained significant attention in damping identification [9–11].

The time domain response sensitivity method also has the disadvantage of being sensitive to the measurement noise [23, 24], and it is also important to enhance the response sensitivity for the structural parameter identification [24–26]. The Principal Component Analysis (PCA) technique, also known as Karhunen-Loeve transform or proper orthogonal decomposition [27], decomposes data series through orthogonal linear transformation to get the principal components, the first few of which contain more information of the parameter variation and less random noise information. The relationship between the system parameters and its output has been broadly studied with a combination of sensitivity analysis and Principal Component Analysis (PCA) [28, 29], and it has been proved that PCA method can improve the response sensitivity for structural damage identification [25, 26] with subspace projection method.

With the rapid development of measurement technique, the time domain responses of the structure can be obtained, and the acceleration responses of the structure can be measured easily. This paper will propose a damping identification method based on the acceleration response measurement. The inverse acceleration response sensitivity method for damping ratio identification is revisited, and the model updating method is also briefly reviewed in this paper at first. The measured acceleration responses are decomposed by PCA method, and time domain response sensitivity equations of damping ratio identification are projected into the finer subspace. With the iterative procedure of model updating and subspace projection, the enhanced sensitivity method is proposed to identify the parameters of damping model. The proposed method is described in detail, and it is validated with simulation studies on a plane truss structure, in which different sensor placements and different measurement noise levels are studied. A seven-storey steel frame was designed and manufactured in the laboratory, and hammer test was performed. The proposed acceleration response sensitivity enhancement method is used to identify the damping ratio of the steel frame structure.

#### 2. Methodology

##### 2.1. Damping Identification Based on Acceleration Response Sensitivity

The equation of motion of a damped linear structure can be written aswhere , , and are the mass, damping, and stiffness matrices of the structural system, respectively. is the vector of excitations on the structure and is the mapping matrix for the excitations. , , and are vectors of the acceleration, velocity, and displacement responses, respectively.

Performing differentiation to both sides of (1) with respect to damping ratio , which is the critical parameter in structural damping model, we havewhere is the th damping ratio of the structural system, and , , and are the acceleration, velocity, and displacement sensitivity vectors, respectively, which can be determined by* Newmark*- method solving (2).

The acceleration sensitivity vector corresponding to th damping ratio can be rewritten as . All the sensitivity vectors are assembled asThe identification equation for all the damping ratios of a structure can be represented asThe higher order term can be omitted in (4). With an iterative method the damping ratio perturbations can be determined from (4), and Tikhonov regularization is used for optimizing the following objective function in the th iteration aswhere is the regularization parameter in the th iteration obtained with the L-curve method [30].

After is solved, the damping matrix is updated with where is the assumed initial damping ratio vector.

Then after recalculating the structural responses and the sensitivity matrix, the vector for the next identification iteration is obtained until the given convergence criterion is met:*Tol* values are selected to meet the challenge in convergence of the identified results with measurement noise effect, and then the final value of the damping ratio can be obtained.

##### 2.2. Acceleration Response Sensitivity Enhancement by PCA

The initial structural damping ratio vector is assumed as , and the initial analytical acceleration responses corresponding to in the initial stage can be determined from (1) with the* Newmark*-*β* method. The measured acceleration responses can be obtained with the accelerometers in the structure. Based on the initial analytical acceleration response vectors and measured acceleration response vectors , the identification equation for the first identification iteration can be written asThe PCA is applied to the measured acceleration responses , and the covariance matrix of is , when written in the form of spectral decomposition as where is the vector of eigenvalues of the covariance matrix; is the corresponding eigenvector matrix which defines an orthogonal subspace;* p* is the number of measured acceleration responses here.

The principal components of measured acceleration responses are then obtained as The initial analytical responses can be projected into the subspace constructed by the eigenvector matrix asThe same projection can be conducted to the sensitivity vectors, so the projection of (8) with the same subspace can be written asThe orthogonal vectors insure each equation in (12) is independent in the subspace. The dimension of the projection sensitivity matrix for each principle component is , where is the length of the measured acceleration and is the number of the damping ratios. If there are principal components selected for determining the damping ratios, the projection sensitivity matrix dimension becomes .

The principal components, which contain less measured noise information and most of the structural damping perturbation information, will be selected, and the selection of principal components will be discussed in next section. Then the damping ratio perturbation can be obtained by Tikhonov regularization method as (5) with the projection sensitivity equation in (12).

Similar to the conventional response sensitivity method [9], the model updating procedure is conducted when each damping ratio perturbation is determined in the th iterative step. After each iteration step, the damping ratio is updated with (6), and the analytical acceleration and sensitivity matrix are updated. The procedure of the proposed method for damping ratios identification is presented as the following 5 steps.

*Step 1. *The measured acceleration responses are analyzed with the PCA method, and the principal components, which contain less measured noise information and more damping ratio perturbation information, are selected.

*Step 2. *Based on the finite element model and initial damping ratio, the corresponding analytical acceleration responses and sensitivity matrix are calculated with the* Newmark*- method, and then the identification equation in (12) is determined based on (1)-(12).

*Step 3. *The identification equation is solved with Tikhonov regularization method to obtain the damping ratio perturbations.

*Step 4. *The damping ratio vector is updated, and with the new damping ratios, the damping matrix of the structural model is updated and the analytical responses and corresponding response sensitivity matrix are recalculated. The identification equation in (12) is determined based on (1)-(12) with the updated damping ratios.

*Step 5. *Step 3 to Step 4 are repeated, and the iterative process will stop until the convergence criterion defined in (6) is satisfied.

The flowchart of the proposed damping identification method is shown in Figure 1.