Shock and Vibration

Volume 2015 (2015), Article ID 157208, 14 pages

http://dx.doi.org/10.1155/2015/157208

## High-Frequency Dynamic Analysis of Plates in Thermal Environments Based on Energy Finite Element Method

^{1}State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China^{2}School of Mechanical and Electrical Engineering, Xi’an University of Architecture and Technology, Xi’an 710055, China

Received 20 November 2014; Accepted 18 January 2015

Academic Editor: Tai Thai

Copyright © 2015 Di 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 energy density governing equation to analyze the high-frequency dynamic behavior of plates in thermal environments is derived in this paper, in which the thermal effects are considered to change the membrane stress state and temperature dependent material properties of plates. Then the thermal effects on the energy reflection and transmission coefficients are dealt with hereof. Based on the above, an EFEM (energy finite element method) based approximate approach for the energy analysis of coupled plates under nonuniform thermal environments is proposed. The approach could be conducted by three steps: (1) thermal analysis, (2) thermal stress analysis, and (3) forming element matrixes, joint matrixes, and the whole EFEM formulation for the energy analysis. The same mesh model is used for all the three steps. The comparison between EFEM results and classical modal superposition method results of simply supported plates in various uniform thermal environments and coupled plates in nonuniform thermal environments demonstrated that the derived energy governing equation and the proposed approach described well the smooth time- and locally space-averaged energy density. It is found that the distributions and levels of energy density are affected by thermal effects, and the variation trends are related to exciting frequency.

#### 1. Introduction

With the great development of hypersonic crafts which are usually subjected to extremely aerodynamic heating and high-frequency exciting during working, there is a great need for the high-frequency dynamic analysis of structures in thermal environments.

Thermal environments have a series of effects on material properties, geometry shapes, stress state, and so on. Great efforts have been made to analyze the dynamic characters of the structure in thermal environments at low frequencies. Ganesan and Dhotarad [1] developed a numerical method for the vibration analysis of thermally stressed plates in which the thermal stresses were evaluated by the finite element method and these stress values were then used in a dynamic analysis of the plate performed by either the finite difference method or variation methods. Jeyaraj et al. [2, 3] used combined FEM/BEM to analyze the vibration and acoustic radiation characters of an isotropic plate and a composite plate with inherent material damping in a thermal environment, finding out that the natural frequencies decrease when the temperature increases and the overall sound radiation of the plate reduces marginally only due to the interaction between reduced stiffness and enhanced damping. Geng and Li [4] analyzed the acoustic radiation and vibration of a flat plate in thermal environments through both the theory method and combined FEM/BEM, and the two results correspond to each other. Liu and Li [5] investigated the vibration and acoustic response of rectangular sandwich plate in thermal environments.

With the frequency increase, more and more elements are needed to describe the vibration of structures, and also small uncertainties will have more and more effects on the calculating results. Thus the FEM and BEM calculation will not only cost much time, but the error is much bigger. Though SEA (statistical energy analysis) is widely used to predict the space- and frequency-averaged behavior of built-up structures at high frequencies where the modal density of structures is high [6], it could only approximate a single acoustic or vibrational energy value for each subsystem of structures.

Due to the disadvantages of the FEM, BEM, and SEA at high frequency, EFEM (energy finite element method) which could predict the time- and space-averaged far field vibrational energy of structures appeared. Belov et al. [7] first proposed the power flow model. Nefske and Sung [8] developed the finite element formulation of the power flow model and applied it to beams. Wohlever and Bernhard [9] made a further research on the vibration energy response of rods and beams. Bouthier and Bernhard extended the work to two-dimensional structures and analyzed the energy density distribution of a membrane [10] and a Kirchhoff plate [11]. Wang et al. [12] presented the high-frequency energy boundary element method combined with energy finite element to describe the energy density distribution of structure-acoustic coupled system. Park et al. [13] developed the power flow model of in-plane waves in thin plates and flexural waves in finite orthotropic plates. Then they derived the energy governing equation of flexural waves in Timoshenko beam [14, 15], Mindlin plate [16], and Rayleigh-Bishop rod [17] which take shear distortion and rotatory inertia acting an important role in high-frequency range into consideration. Zhang et al. developed an alternative energy finite element formulation for interior acoustic spaces and thin plates considering the wave response as a summation of incoherent orthogonal waves [18]. They also expanded the EFEM to analyze the energy distribution of stiffened plates under heavy fluid loading [19]. Xie et al. applied EFEM to high-frequency structural-acoustic coupling of an aircraft cabin with truncated conical shape [20] and proposed the transient vibrational energy response analysis of a rod under high-frequency excitation [21]. Park made an introduction of the developed EFEM based software–EFADS, and the results of one part of a real automobile derived from EFADS matched well with the experimental results [22].

Though EFEM has been developed well since 1980s, only Zhang et al. [23] researched on the thermal effects on the high-frequency vibration. They derived the energy density governing equation of beams and verified the accuracy. In this paper, we derived the energy governing equation of plates in thermal environments. The thermal environments are considered to affect the material property and thermal stress condition. We then study the thermal effects on power transmission and reflection coefficient and develop an approximate approach to analyze the energy density distribution of coupled plates in nonuniform thermal environments: first, the thermal analysis and thermal stress analysis are conducted; then based on the temperature and thermal stresses derived above, the energy distribution of the structure could be obtained therefore. The accuracy of the derived equation is verified by comparing the developed EFEM results with classical modal superposition results of a simply supported plate in uniform thermal environments. The thermal effects on the energy distribution and level are also analyzed. Finally, the numerical example of the coupled plate in nonuniform thermal environments shows that the approach could describe well the thermal effects on the level and distribution of energy density.

#### 2. Energy Governing Equation for a Plate in Thermal Environments

##### 2.1. Derivation of Wavenumber and Group Velocity with Thermal Effects

For a plate with the temperature uniform throughout the thickness, assume that the plate is stress-free at the reference temperature . With the temperature increasing, the changes of thermal stress and material properties should be taken into consideration. So when the temperature changes to , the transversely vibrating governing equation will be [4] where is the complex bending stiffness of the plate, is the structural damping loss factor, is the density, is the thickness, is the transverse mechanical load applied on plate, and is the imaginary unit. , , and are the thermally induced membrane forces.

Use the separation of variables, the general solution can be expressed as where and are the wavenumbers in and directions and is the circular frequency. At small damping, the complex wavenumbers in and directions above could be approximate as By substituting (2) into (1), the dispersion relation can be derived as

Assume that is the total wave numbers in the wave propagation angle , so and could be expressed as [25] Substituting (5) into (4) Assume that Thus we could derive the expression of in terms of : From (8) and (7) we could find that the total wavenumber varies with the wave propagation angle . In EFEA method, the plate is diffuse wave fields, so averaging and over , we could obtain the averaged diffuse wavenumber in and directions:

The group velocity in terms of could be expressed as Equation (10) shows that the group velocity also depends on the direction of the wave propagation. So the same as (9), the group velocity should also be averaged over [26]: From (9) and (11), we can see that the wavenumber and group velocity will be changed in thermal environments.

Because, in high-frequency range, the wavelength is quite small and the near field waves usually decay in half wavelength, we only utilize the far field solution of (1) to simplify the energy analysis [11]. So the general form of the far field travelling plane wave solution can be expressed as where the unknown constants , , , and are the amplitudes of the waves in and directions.

##### 2.2. Derivation of the Energy Governing Equation with Thermal Effects

As discussed by Bouthier and Bernhard [11], the time-averaged energy density which includes both the kinetic and potential energy densities of the plate across the thickness can be expressed in terms of transverse displacement as

As above, the time-averaged energy intensity across the thickness can be expressed in terms of transverse displacement as [11]

Substituting (12) into (13)-(14), we can obtain the complete time-averaged far field energy density and energy intensity expressions in terms of transverse displacement. For we could not find obvious relations between them, they are averaged spatially over a half wavelength for small damping in the following manner [11]: Thus the time- and space-averaged density can be written as The time- and space-averaged intensities and can be written as

Obviously the relationship between the time- and space-averaged energy density and intensity is

For an elastic medium, the differential steady energy balance equation in terms of time- and space-averaged energy density can be written as [8]

Combining (18) and (19), we can obtain the steady far field energy governing equation of plates in uniform thermal environments:

It is obvious that thermal environments affect the first coefficient of (20) by changing wavenumbers of the plate; thus the energy distribution state of the plate is changed thereof. Compared to the energy governing equation derived by Bouthier and Bernhard [11], (20) becomes the same when the temperature becomes reference temperature.

The finite element formulation is introduced here to solve (20) [19]: where is the system matrix for element including thermal effects, is the vector of nodal energy density for the corresponding element, is the vector of input power at the nodal locations, and is the vector of power flow across element boundaries including thermal effects, which will be introduced in the next sections.

In high-frequency range, for the wavelength is small, the input power could be expressed as [27] or where is the amplitude of the external point force, is the conjugate of the velocity of the exciting point and is the impedance of the corresponding infinite plates with thermal effects at the driving point expressed as follows: The wavenumber will be anisotropic when the thermal stresses are different in different directions, and the structure is diffused field, so the wavenumber is averaged over .

#### 3. An Approximate Method for the Energy Density Analysis of Coupled Plates in Arbitrary Thermal Environments

Usually, the thermal environments are nonuniform, and the structures are made up of coupled plates, which means the thermal stresses and temperature dependent material properties are not the same everywhere. Thus the group velocities will be different everywhere. And also the power transmission and reflection will happen everywhere. Therefore the derived energy governing equation (20) cannot be used directly for structures as traditional method. So it is of great importance to develop a method to analyze the high-frequency vibration energy density of coupled plates in arbitrary thermal environment.

##### 3.1. Derivation of Power Transmission and Reflection Coefficients in Thermal Environments

As we know, it is not easy and takes a lot of time to consider all the energy transmission and reflection. So we assume that the temperature and thermal stresses are uniform in every element and just calculate the energy transmission and reflection coefficients between every two adjacent elements. The method introduced by Langley and Heron [28] is expended here to take the thermal effects into consideration for the plates.

As shown in Figure 1(a), the model of plate junction in global coordinate is introduced here. The tractions (), displacements (), thermal stresses (), and temperature of either single plate are defined in local coordinate as shown in Figure 1(b). For the membrane stress will not affect the in-plane vibration characteristics, the thermal effects will change the in-plane energy density distribution and level only by the temperature dependent material properties. So the governing equations of out-of-plane and in-plane deformation of the th plate defined in local coordinate are