Shock and Vibration

Volume 2016 (2016), Article ID 6461427, 9 pages

http://dx.doi.org/10.1155/2016/6461427

## Identification of Dynamic Loads Based on Second-Order Taylor-Series Expansion Method

^{1}School of Astronautics, Beihang University, 37 XueYuan Road, Haidian District, Beijing 100191, China^{2}School of Astronautics, Beihang University, Beijing 100191, China

Received 19 May 2015; Revised 2 August 2015; Accepted 31 August 2015

Academic Editor: Miguel Neves

Copyright © 2016 Xiaowang Li and Zhongmin Deng. 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 method based on the second-order Taylor-series expansion is presented to identify the structural dynamic loads in the time domain. This algorithm expresses the response vectors as Taylor-series approximation and then a series of formulas are deduced. As a result, an explicit discrete equation which associates system response, system characteristic, and input excitation together is set up. In a multi-input-multi-output (MIMO) numerical simulation study, sinusoidal excitation and white noise excitation are applied on a cantilever beam, respectively, to illustrate the effectiveness of this algorithm. One also makes a comparison between the new method and conventional state space method. The results show that the proposed method can obtain a more accurate identified force time history whether the responses are polluted by noise or not.

#### 1. Introduction

With the continuous development of science and technology nowadays, the mechanical structure becomes more and more complex. To ensure the structure’s reliability and safety, it is essential to get the dynamic loads acting on the structure. Generally speaking, directly measuring the external load through a load transducer is preferred. However, in many cases of engineering, such as the missile subjected to the wind load and the tall building suffering the seismic load, direct measurement of dynamic loads exerted on the structure is difficult to achieve. To overcome this problem, force identification, as the second class of inverse problem in structural dynamics, is put forward. Along with the increase of complex engineering problems, force identification technology has become a crucial issue in structural dynamics. It is devoted to providing effective load reference for the design of the structure and plays a significant role in mechanical vibration analysis, structural reliability analysis, mechanical fault diagnosis, and other fields. Thanks to the rapid advancement of computer technology, the establishment of mathematical model becomes easier and the accuracy of the finite element simulation is improved obviously. Force identification technology, therefore, has made a great progress.

There are mainly two categories of methods for dynamic load identification: the frequency domain method and the time domain method [1]. The frequency method has developed fast because of its simple principle. It is focused on searching for a system’s frequency response function (FRF) and response spectrums to calculate load spectrums [2]. Some classical frequency domain methods are introduced in [3–5]. Nevertheless, the frequency domain method has the drawback of low accuracy and being sensitive to the response noise [6], which makes its application limited in some cases.

In recent years, a series of time domain methods have been studied. Compared with the frequency domain method, force identification technology in time domain is able to achieve relatively higher accuracy. Law and Fang [7] adopted the state space method in the control theory to perform force identification. This method is commonly utilized to identify stationary periodic loads. Yan [8] used Bayesian approach in statistics to calculate impact loads. The author first transformed the impact load identification problem to a parameter identification problem by representing the impact load using a set of parameters. Sun et al. [9] proposed a new improved regularization method for load identification, which can overcome the ill-condition of load reconstruction to some extent. In [10], the authors transformed the conventional implicit algorithm into an explicit form for linear analysis of the structure. Liu et al. [11] presented a shape function method of moving least square fitting, by which the time domain of load is discretized and the local load is approximated by SFM under LSF. An analytical method was proposed in [12] to identify dynamic loads acting on stochastic structures, based on the Gegenbauer polynomial expansion theory and regularization method. In [13], the authors put forward an inverse method that combines the interval analysis with regularization. This algorithm is able to stably identify the bounds of dynamic load acting on the uncertain structures. Besides, the authors in [14] utilized support vector regression to identify nonlinear systems represented by Wiener models. Simulation results show that the method gives accurate models of systems. Polynomial interpolated Taylor series method was studied in [15]. It advances the technique in parameter identification of structures with significant nonlinear response dynamics.

Taylor formula [16, 17], as an indispensable math tool in mathematical analysis, plays a key role on approximate calculation. It aims to transform a complex function into a concise polynomial function on the premise of maintaining a high approximation precision. Due to the distinguished advantages, Taylor-series expansion has been used in numerous areas to deal with sophisticated mathematical problems.

In this paper, one proposes a new approach for the identification of dynamic loads, utilizing the formula of Taylor-series expansion. The proposed method expresses the response vectors as Taylor-series approximation in the neighborhood of time and then deduces a series of formulas for force identification. Finally, a simple explicit equation which links system characteristic, system response, and input excitation together is established. The full use of outstanding feature of Taylor formula makes this approach reach a high identification accuracy. Complex iteration calculation is eliminated because of the establishment of explicit equation. In addition, this method is an implicit integration in essence; hence it has the merits of remaining unconditional stable. Multi-input-multi-output (MIMO) numerical simulations are carried out to illustrate this method. State space method is adopted to make a contrast with this method. The results indicate that the proposed method can obtain more satisfactory identified force time histories even in the case that noise is added into the responses.

#### 2. Force Identification Based on Taylor-Series Expansion

For a general linear elastic structure with multiple dofs [18], the equation of motion is expressed aswhere , , and denote the mass, damping, and stiffness matrices. denotes the vector of forces exerted on the structure. , , and denote the vectors of acceleration, velocity, and displacement responses.

Assuming Rayleigh damping [19], the damping matrix iswhere and are the damping coefficients.

##### 2.1. Method Deduction

The nature of this method is to express the acceleration vector as a Taylor-series approximation in the neighborhood of time ; namely,where and are vectors to be determined. Also . Then the vectors of velocity and displacement responses are written as

For the purpose of removing the integral symbol in (4) and carrying out follow-up deduction, one must set as a specific number. Meanwhile, considering the process of force identification in this paper is completed by second-order Taylor-series expansion method, one lets , and then

The equation of motion at time is

Replace with and substitute (5) into (6) to getwhere

Replace with and substitute (5) into (6) to getwhere

is obtained by linear interpolation between and ; namely,

Combining (7) and (9), one getsLetthen and are written aswhereSubstituting (14) into (3) and (4) one obtainswhere

Combine (16), (17), and (18) to get

Then the responses at time are expressed as

Zero initial responses are assumed and (21) is simplified towhere

Equation (22) can be rewritten aswhere the time duration is from to and

Considering there may exist conditioning issues when using the Moore-Penrose inverse, one adopts the Tikhonov regularization method [20] to calculate . Systematic error is written as

To find out the minimum of , a penalty function is introduced; namely,

As the first-order derivative of is zero, reaches a minimum. Then the solution of is

is the regularization parameter. In this work, one applies the discrepancy principle [21] to determine .

##### 2.2. Method Implementation Sequence

The Taylor-series expansion method for force identification is implemented in the sequence as follows:(i)Determine parameter and appropriate time step . In principle the range of is 1~2. However, as is replaced with , then , , and are expressed as , , and . This is to ensure the subsequent deduction carried out at the cost of a certain calculation error. In order to make the calculation error as small as possible, must be close to 1. But if , the matrix in (12) will be singular. This situation should be avoided. Weighing the above two aspects, one sets . depends on the frequency of the excitation. In this work the excitation frequency is 20 Hz, so .(ii)Compute matrices , , , , , , , , , , , , , , and in (16), (17), and (18).(iii)Compute matrices and .(iv)Assemble matrix in (24).(v)Calculate the vector by (28). In addition, the relative error between identified forces and real forces can be computed as

##### 2.3. Selection of Expansion Order

In the process of method deduction in Section 2.1, the acceleration vector is expanded to a second-order Taylor-series approximation. In order to verify the accuracy of second-order approximation, method [22] is compared with the proposed method. method is a classical stepwise integral method which expresses the acceleration, velocity, and displacement vectors as follows:

From (30), (31), and (32) it can be seen that the truncation errors of acceleration, velocity, and displacement are , , and , respectively. As for the new method, the truncation errors are , , and in (5). The truncation error of the new method is one order higher than that of method, so two-order Taylor-series expansion method can achieve higher accuracy than method in the case of choosing the same time step.

In addition, one makes a comparison between second-order approximation and first-order approximation. To implement a first-order approximation, let in (3). After a series of similar deductions, a formula as (24) is founded.

A two-storey shear structure in Figure 1 is set up. A sinusoidal force is subjected to the structure, , , , and . The displacement responses of two storeys are calculated by modal decomposition method. The dynamic load is identified and the identification results are shown in Figure 2.