Shock and Vibration

Volume 2015 (2015), Article ID 143254, 15 pages

http://dx.doi.org/10.1155/2015/143254

## Experimental Studies on Finite Element Model Updating for a Heated Beam-Like Structure

^{1}State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China^{2}Shanghai Institute of Satellite Engineering, Shanghai 200240, China^{3}MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

Received 22 September 2014; Revised 6 December 2014; Accepted 11 December 2014

Academic Editor: Tony Murmu

Copyright © 2015 Kaipeng Sun 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

An experimental study was made for the identification procedure of time-varying modal parameters and the finite element model updating technique of a beam-like thermal structure in both steady and unsteady high temperature environments. An improved time-varying autoregressive method was proposed first to extract the instantaneous natural frequencies of the structure in the unsteady high temperature environment. Based on the identified modal parameters, then, a finite element model for the structure was updated by using Kriging meta-model and optimization-based finite-element model updating method. The temperature-dependent parameters to be updated were expressed as low-order polynomials of temperature increase, and the finite element model updating problem was solved by updating several coefficients of the polynomials. The experimental results demonstrated the effectiveness of the time-varying modal parameter identification method and showed that the instantaneous natural frequencies of the updated model well tracked the trends of the measured values with high accuracy.

#### 1. Introduction

Hypersonic vehicles are subject to very tough aerodynamic load and heating during their missions in the Earth’s atmosphere. The aerodynamic heating is extremely important because high temperature can affect the structural behavior in several detrimental ways [1]. The elevated temperature not only degrades the ability of structure materials to withstand loads, but also produces the thermal stresses in heated structures. In addition, the thermal stresses increase deformation, change buckling loads, and alter flutter behavior. For example, the studies [2–5] reported that even a minor change of temperature would result in a significant alteration in natural frequencies of a beam because of the thermal stresses when the beam was constrained. Thermal modal testing techniques can provide quantitative analysis for the effect of the thermal load [6]. Since the 1950s, many experiments have been done in NASA Langley and Dryden research centers for the metal and the composite panels [7, 8], the X-15 wing [9], and the X-34 FASTRAC Composite Rocket Nozzle [10]. The aforementioned studies have primarily focused on the dynamical properties of thermal structures in steady high temperature environments (SHTEs).

As a matter of fact, thermal structures in unsteady high temperature environments (UHTEs), for example, hypersonic vehicles subjected to unsteady aerodynamic heating, have the characteristic of time-varying multiphysics fields. The identification of time-varying modal parameters is the forefront of inverse problems in structural dynamics. There have been numerous theoretical and experimental studies on the identification of time-varying modal parameters for engineering structures. The special issue [11] of* Mechanical Systems and Signal Processing* on the identification of time varying structures and systems in 2014 offered a survey of the field in its current state, reflected recent developments, and also pointed out into the future, in various aspects of the theories and applications. Although recent years have witnessed successful identifications of time-varying modal parameters of many engineering systems, such as vehicle-bridge systems [12, 13], machine condition monitoring systems [14], flexible manipulators [15], and civil structures [16, 17], their applications to thermal structures in temperature-varying environment are still not available. Yu et al. [18] proposed an undetermined blind source separation method to investigate the thermal effect on the modal parameters of a TC4 titanium-alloy column in a temperature-varying environment. They [19] also developed a time-varying modal parameter identification algorithm based on finite-data-window PAST and used it to investigate the effect of varying temperature and heating speed on the natural frequencies of a trapezoidal TA15 titanium-alloy plate. To the best knowledge of authors, considerably less attention has been paid to the finite element model updating (FEMU) for thermal structures, especially based on the identified time-varying modal parameters in UHTEs.

Finite element (FE) modelling has received widespread acceptance and witnessed applications in various engineering disciplines. Hence, the FE model updating (FEMU) has become a useful tool to improve the modelling assumptions and parameters until a correlation between the analytical predictions and experimental results satisfies practical requirements. Mottershead and Friswell [20] comprehensively reviewed the model updating methods of structural models. In Marwala’s work [21], numerous computational intelligence techniques were introduced and applied to FE model updating with in-depth comparisons. As a most widely used method in FEMU, the iterative method has been formulated as an optimization problem and often based on sensitivity analysis [22] and computational intelligence techniques [23–25]. Generally, the optimization procedure is nonlinear and complex for a complicated FEMU problem. In recent years, the particle swarm optimization (PSO) technique has been developed to implement the optimization procedure.

The structural FE models with many geometric and physical parameters to be updated may involve a large number of computations and need to be constructed by one of commercial finite element analysis packages, such as COMSOL, ANSYS, and NASTRAN. Hence, higher time consumption may be the disadvantage for the optimization-based algorithms, due to their iterative strategy and repeated analysis in simulation models during the optimization process. One way to overcome the difficulty of time consumption and FE package-related problems during the optimization-based model updating is to replace the FE model by an approximate surrogate/replacement meta-model that is fast-running and has fewer parameters involved. Simpson [26] made a comparison of response surface and Kriging models, which are the two commonly used meta-models, and drew a conclusion that both approximations predict reasonably well with the Kriging models having a slight overall advantage because of the lower root mean squared error values.

The meta-model method for damage detection and reliability analysis has a long history. However, the Kriging meta-model (KMM) method for structural FEMU is somewhat new, especially for thermal structures in temperature-varying environment. As a continuation of authors’ work [27, 28], this paper presents KMM-PSO based FE model updating and selection based on the experimentally identified time-invariant and time-varying modal parameters of a thermal structure.

The remainder is organized as follows. In Section 2, the FE model updating and selection for a thermal structure is formulated, and two objective functions are given for the PSO. An overview of PSO and a brief introduction to the time-varying autoregressive method for output only identification are also presented. The Kriging meta-model is introduced to employ the PSO based FE model updating and selection. In Section 3, four groups of experiments are discussed for the modal parameters of a beam-like structure in room temperature environment, SHTE, and UHTE. In Section 4, the Kriging meta-model based FE model updating and selection procedure is carried out to update the software-based FE model. Finally, some conclusions are drawn in Section 5.

#### 2. Formulation of FEMU for Thermal Structures

##### 2.1. Problem Description

A linear thermal structure subject to unsteady heating can be described in terms of the distributed mass, damping, and stiffness matrices of the structure in time domain via the following differential equation: The real eigen-value problem for the th mode reads where the detailed meanings of the physical quantities in the above two equations can be found in [28].

The traditional FE model updating process is achieved by identifying the correct mass and stiffness matrices, which are generally time-invariant. This study, however, has to deal with a time-variant FE model updating process because the system matrices change over time when the temperature-dependent material and temperature-dependent boundary conditions are taken into consideration. Afterwards, using the measured data, the correct mass and stiffness matrices can be obtained by identifying the correct material parameters of the structure and the appropriate boundary conditions under different temperature conditions.

For simplicity, the thermal structure of concern is a kind of slender or thin-walled structures, such as beams and plates, subject to uniform heating with a uniformly distributed temperature field . In other words, the heat transfer process is neglected in the study. Let denote the temperature increase in the thermal structure as follows: where is the reference temperature. For the FEMU problem of the thermal structure subject to unsteady heating, the system parameters to be identified depend on the transient temperature and can be expressed as where is the parameter vector including the temperature-dependent material and boundary parameters to be corrected, and is the general function of the temperature increase . In general, can be expressed as a low-order polynomial, for example, a linear, quadratic, or cubic function, in which represents the highest order of the polynomial. Then, the target parameters to be corrected change from temperature-dependent to a constant parameter vector , .

##### 2.2. Objective Function for FEMU at Reference Temperature

In a reference temperature environment (RTE), to correctly identify the moduli of elasticity that gives the updated FE model, the following objective function, which measures the distance between the measured modal data and the modal data predicted by FE model, should be minimized: where is the error function or objective function, is the Euclidean norm, is the weighted factor for the th mode, and is the number of measured modes, respectively. In (5), is the th instantaneous natural frequency obtained from the meta-model, and is the th instantaneous natural frequency identified from the measured responses in a thermal-structural experiment based on the identification method for the time-invariant modal parameters.

Thus, the process of FEMU may be viewed as an optimization problem as follows: where and represent the lower and upper bounds of the parameter coefficient vector , respectively.

##### 2.3. Objective Function for FEMU in a UHTE

After an updated model in the RTE is achieved, the FEMU can be performed in a UHTE, which contextually means that the parameters to be corrected in this subsection become the constant parameter vector in (4). It should be noted that here represents several special parameters that cannot be identified in an RTE. The constant coefficient of thermal expansion, for example, has no effect on the stiffness matrix of the FE model and, thus, cannot be identified in an RTE, but must be updated in a UHTE.

Furthermore, a new problem arises when the temperature-dependent parameter is approximated as a low-order polynomial. Because it is impossible to know a priori the exact order of polynomial in (4), a parameter vector expressed by polynomials of different orders should be considered. In general, it is known that the higher the order is, the smaller the deviations between test and analysis are. However, the purpose of FEMU is to predict the structural response, to predict the effects of structural modifications, or to serve as a substructure model to be assembled as part of a model of the overall structure. From this viewpoint, for simplicity, one may use a low order expansion for the material properties. If the order is fixed to the maximal value, one would manually accept or reject the high-order terms by identifying whether the coefficient is close enough to zero. If any terms are neglected, the deviations between test and analysis should be reexamined. Therefore, the potential problems in the FEMU procedure are not only how to determine those parameters, but also how to select the updated model. To avoid establishing many models for FEMU and manually selecting the updated model, it is preferable to have an all-in-one procedure for both updating and selection. The PSO framework allows this simultaneous updating of all competing models and selection of the best model. Hence, the two aforementioned problems can be solved by minimising an integrated objective (or fitness) function.

A number of fitness or objective functions have been available so far. Most previous studies have sought a model with the fewest updating parameters needed to produce FE model results that are closest to measured results. In this study, the Akaike information criterion (AIC) was used to represent the integrated objective function with an additional term to treat ill-conditioned and noisy systems. AIC can be described by the following equation: with where is the number of samplings, and are frequencies at time , and is the initial estimate of parameter , respectively. The second term in the bracket represents regularization, where the parameter weighting matrix should be chosen to reflect the uncertainty in the parameter estimation, and represents the classical Tikhonov regularization. Link [29] suggested that the factor lies in the range of . High values are used if there are many insensitive parameters and if the inverse problem is strongly ill-conditioned. The second term of the above equation is known as the model complexity penalty term, in which is a weighting factor, and represents the highest order of the th polynomial. is the dimension of the parameter vector.

Similarly to the FEMU in an RTE, the optimization process here may be expressed as follows:

Figure 1 illustrates the detailed flow chart of the Kriging meta-model and the particle swarm optimization based FE model updating and selection. In the left dashed block, a Kriging meta-model is introduced to overcome the difficulty of time consumption because of the quantities of iterations during the updating process. In the right dashed block, modal parameters analyzed from Kriging meta-model and experimentally identified by time-varying autoregressive (TVAR) method are used to establish the objective function for the PSO based FE model updating and selection. Therefore, the next 3 subsections briefly introduce the stochastic optimization technique, the Kriging meta-modeling and the time-varying autoregressive method, and their relevance to the FE model updating and selection for thermal structures.