Shock and Vibration

Volume 2016, Article ID 8618202, 24 pages

http://dx.doi.org/10.1155/2016/8618202

## Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification

State Key Laboratory for Manufacturing System Engineering, School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 11 November 2015; Revised 5 May 2016; Accepted 20 June 2016

Academic Editor: Nerio Tullini

Copyright © 2016 Xiaofeng Xue 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

A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF). It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI) finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD) method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.

#### 1. Introduction

For many practical engineering problems, such as structural strength analysis, health monitoring, fault diagnosis, and vibration isolation, the knowledge of the dynamic loads acting on the structure is always required and important [1]. The use of plate and shell in the industry becomes increasingly important [2]. As shear deformation Mindlin plate has a very wide range of application [3], load identification for Mindlin plate is very meaningful. However, it is difficult to directly measure the loads in a dynamic system, especially the interaction force in complex structures [4]. An alternative is to obtain these loads by the inverse force identification, which is an important topic in the identification of Mindlin plate under operating conditions. Therefore, load identification methods are necessary when the direct measurement is unfeasible [5]. The main idea of load identification is to use the measured structural response signals to determine the excitation load. However, it is an ill-posed inverse problem [6]. Different methods have been developed to make use of the response measurements of a structure to estimate the input excitations [4]. Frequency-domain method is one of the most commonly used load identification methods. Structure FRF can be constructed by numerical simulation. It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform [7, 8]. Through the experiment, some scholars studied load identification using the method of FRF [9]. As for an ill-posed inverse problem, regularization methods such as the SVD, Chebyshev-Based Method, cubic B-spline-based method [10], and Tiknohov regularization are often used to transform it into a well-posed problem for great identification accuracy. However, the frequency-domain requires a long stationary time to calculate the inversion of the FRF matrix at each frequency [11–13]. It should be noted that the load position or distribution are usually assumed to be known in the load identification procedure [6]. Liu analyzed the truncated SVD to overcome ill-posed FRF matrix inversion [14].

Many vibroacoustic studies use finite element method (FEM) model to predict the dynamic behavior of structures. The aim is to reverse the load identification of the model by calculating sources from the measurements of their effects. Current commercial software and computing resources allow engineers to create numerical models that can be applied for more complex structures [15]. Zhao [16] presented the sliding loads identification based on the FEM and the strain measurement point selection procedure without information on their initial positions and magnitudes. Bahra [17] adopted the frame frequencies and mode shapes in a FEM model and solved the axial loads until the difference between measured and model frequencies is minimized, thus inferring the member axial loads. Lage [18] proposed a two-step loads identification method: the first step identifies the number of forces and their locations based on the response transmissibility concept; the second step reconstructs the load vector. Berry [19] presented the theoretical developments based on the virtual fields method and processed numerically simulated data to validate the identification algorithm.

However, for many complicated problems, traditional FEM has some disadvantages, such as low efficiency, insufficient accuracy, and slow convergence to correct solutions. Load identification, an ill-posed inverse problem, needs multiresolution analysis to improve the accuracy and reduce the computational work. Due to the property of multiresolution analysis, wavelet provides a natural mechanism for decomposing the solution into a set of coefficients. Wavelet functions can be viewed as interpolating functions, which are similar to those used in signal and image processing. Basu [20] maintained that the finite difference and Ritz-type methods have been largely replaced by the FEM, the boundary element method, and the mesh-less method, and in the near future it might be the turn of the wavelet-based numerical method. Because wavelet finite element has the advantages of reasonable interpolating functions configuration and higher precision. One can use fewer elements to solve the load identification. HCSWI elements embody a prominent advantage of improving precision by adding the appropriate wavelet functions [21, 22]. By using the modified one-dimensional HCSWI elements, a multiscale wavelet-based numerical method can solve load identification in rod and Timoshenko beam. Compared with traditional FEM and B-spline wavelet on interval (BSWI) [23–29] finite element, the method of HCSWI rod and beam elements has the advantage of higher precision. The tensor product of the modified Hermitian wavelets expanded at each coordinate is used to construct two-dimensional Mindlin plate Hermitian wavelet interpolation function. The load identification based on FRF is restricted as the result is inaccurate when the frequency content is close to certain resonance frequency of the structure [30]. Hermitian wavelet finite element with high precision can alleviate this problem effectively.

In the present study, an effective new wavelet numerical method based on two-dimensional Mindlin plate Hermitian wavelets interpolation function is proposed to construct the FRF and recover the loads. The paper is organized as follows. Section 2 addresses Hermitian Mindlin plate wavelet element. Section 3 describes load identification of HCSWI. Next, Section 4 details the numerical implementation and analysis. Section 5 presents the experimental verification. Concluding remarks are included in Section 6.

#### 2. Hermitian Mindlin Plate Wavelet Element

A new Hermitian Mindlin plate wavelet element is constructed by the Hermitian cubic spline wavelets on interval interpolation functions. It can be substituted into finite element functions to solve the Hermitian wavelet stiffness and mass matrix for the purpose of obtaining FRF matrix. Through the inverse matrix of FRF matrix, one can solve the load identification. The method of SVD is adopted to solve the ill-posed inverse problem.

##### 2.1. Hermitian Wavelet Interpolation Functions

Hermitian scaling functions are shown in Figure 1 and the equations are where