Broadband Dynamic Sensitivity Analysis of a Euler–Bernoulli Beam Under Thermal Load Using the Efficient Wave-Based Method

Abstract

Beam-type structures used in aerospace applications may experience simultaneous broadband dynamic excitation and thermal loads. Design sensitivity, as a powerful tool for structural optimization and reliability analysis, is investigated in this work. The broadband dynamic response and its sensitivity to input parameters for a Euler–Bernoulli beam in a thermal environment are examined using an efficient wave-based method (WBM). First, the accuracy of the simulation for predicting the broadband dynamic response is validated. Then, the influence of thermal effects on the dynamic response is investigated. Further, the normalized sensitivities of the dynamic response with respect to thermal loads, material properties, and geometric parameters are studied. The simulation results highlight the critical role of thermally generated compressive forces in governing structural dynamics. The normalized sensitivities with respect to different input parameters can vary across the broadband frequency band. In the low-frequency ranges, the sensitivities with respect to thermal load, thermal expansion coefficient, the cross-section area, and moment of inertia are dominant. In the high-frequency ranges, the cross-section area, moment of inertia, elastic modulus, and density have major influence on the dynamic response. All the parameters investigated could significantly affect the mid-frequency dynamic response.

1. Introduction

The vibration problem for beam-type structures is frequently encountered in aerospace engineering [1,2,3,4,5]. Due to the complex aerodynamic load, beams used in aircraft can be subjected to a service environment consisting of thermal load and broadband dynamic loads, leading to the transverse vibration of the structure. To support the numerical simulations at the early stage of design, it is necessary to investigate the broadband vibration behavior and perform its sensitivity analysis for beam-like structures under thermal loads.

For the broadband dynamic problem, with the increase in analysis frequency, the wavelength decreases, and the modal density increases. Therefore, the broadband dynamic problem is commonly and separately solved in the low, medium, and high-frequency ranges. Different analysis methods are adopted to solve dynamic problems in these frequency ranges. In the low-frequency range, deterministic methods such as the Finite Element Method (FEM) are widely used. For high-frequency problems, statistical methods, such as the statistical energy analysis (SEA) and Energy Flow Analysis (EFA), are commonly used. Three kinds of methods are proposed to solve the mid-frequency problems, which include the extension of the deterministic method towards higher frequency, the extension of the statistical method to lower frequency, and the combination of the deterministic method and the statistical method. Among the mid-frequency methods, the wave-based method (WBM) developed by Desmet et al. [6] is an effective approach. Compared to the deterministic methods for low-frequency problems, the WBM achieves a higher convergence rate in the medium and high-frequency ranges. On the other hand, compared to the statistical methods for high-frequency problems, a more detailed local response can be obtained by the WBM. Through the development over two decades, the WBM has been applied to various engineering problems, such as structural dynamic analysis, acoustic simulation, and vibro-acoustic problems [6]. Lainer et al. [7] developed a novel extension of the WBM to treat poroelastic problems in the field of soil vibrations. Sun et al. [8] extended the WBM to anisotropic time-harmonic elastodynamic problems. Wu et al. [9] presented the topology optimization method of damping material layout in a coupled vibro-acoustic system using the hybrid finite element–wave-based method. Further, considering that the actual loads are stochastic excitations [10], Jonckheere develops [11] the novel particular solutions for the WBM to solve dynamic problems under random Turbulent Boundary Layer excitation. The above study mainly focuses on the structure under normal temperature conditions, without considering the coupling effect of thermal and mechanical loads.

For structures subjected to thermal loads, the temperature-dependent material properties and the thermal stresses caused by structural thermal expansion could affect the broadband structural dynamic responses [12,13]. Therefore, dynamic problems under the thermal environment have received much attention from researchers. Geng et al. [14] studied the low-frequency vibro-acoustic behavior of a plate under a thermal environment using theoretical methods and experimental research. Demir and Civalek [15] studied the thermal vibration behavior of nanobeams using the FEM with novel shape functions. For the high-frequency problems, Chen et al. [16] analyzed the SEA parameters and energy response of structures under the thermal environment. They concluded that the effect of thermal stress is dominant for the high-frequency statistical energy responses. Zhang et al. [17] investigated the energy response of thermal beams using EFA. Recently, the WBM has also been developed to solve mid-frequency dynamic problems for Euler–Bernoulli beams [18], the Timoshenko beam [19,20], plates [21], and vibro-acoustic systems [22] under thermal loads, where the corresponding wave functions and particular functions are derived to construct the dynamic response for these problems. Further, Chen et al. [23] extended the WBM to structures with fuzzy uncertainties. Existing applications of the WBM are mainly confined to low and medium-frequency regimes; the parameter competition effects across broadband spectra need further exploration.

Design sensitivity analysis (DSA) is a powerful tool for parameter identification, structural optimization, and reliability analysis. Information on the relative importance of input parameters could be provided through DSA. Methods to perform design sensitivity analysis include the finite difference method, adjoint method, and semi-analytical method [24,25,26,27]. Tortorelli and Michaleris [28] reviewed and compared different sensitivity analysis methods by using a linear dynamic system. Among these methods, the finite difference method is widely used because it is easy to implement. However, this method can suffer from computational inefficiency when the number of design variables is much larger than the number of constraint functions. To solve this problem, the adjoint method is proposed to compute the sensitivity efficiently and accurately, which makes it widely used in engineering practice [29]. Saltelli et al. [30] made a comprehensive and profound review of the applications of sensitivity analysis and discussed some existing problems.

DSA has also been conducted for various linear and nonlinear dynamic problems [31,32]. Most of the research to date has been in the low-frequency range. Jung et al. [33] combined the DSA with FEM software to optimize the frequency response of complex systems. Zhu et al. [34] combined the Sherman–Morrison–Woodbury formula and the finite difference method to analyze the sensitivity of frequency response functions with respect to structural parameters. Wang et al. [35] investigated the sensitivity of vibration-damping characteristics of sandwich panels. In the high-frequency range, the sensitivity of the statistical energy subsystems to the damping loss factor and the coupling loss factor is investigated. Manik et al. [36,37] presented the first and second-order sensitivity analysis of the system modeled by SEA, in which the direct method and the adjoint method are adopted. For the mid-frequency dynamic problem, the WBM has also been adopted to analyze the sensitivity of dynamic problems due to its efficiency. Koo et al. [38,39] used the WBM to investigate the design sensitivity of the acoustic cavity and vibro-acoustic system. The earlier work mainly investigated the sensitivity in the low, medium, and high-frequency ranges separately. However, the dynamic response of some engineering structures could present broadband characteristics. Therefore, it is necessary to investigate the dynamic sensitivity in the broadband frequency ranges for structural design and optimization.

This work investigates the broadband dynamic sensitivity of beams under the thermal environment using the wave-based method. The outline of this work is as follows: in Section 2, the theoretical basis on the dynamic problems for beams under thermal environment, the solution for the broadband dynamic response, and the sensitivity analysis are presented. In Section 3, the influence of thermal effects on the dynamic response is investigated, and numerical simulations are performed to investigate the global sensitivity for different parameters. Conclusions are drawn in Section 4.

2. Theoretical Basis

2.1. Problem Definition

The Euler–Bernoulli beam theory is widely adopted in the simplification of aerospace structures for the extraction of preliminary design rules [2,4]. Figure 1 illustrates a Euler–Bernoulli beam with simply supported boundary conditions subjected to a harmonic transverse excitation F = F₀e^jωt and a uniform thermal load T. Axial thermal stresses F_Tc are generated internally in the beam due to thermal expansion and boundary constraints.

The governing equation of flexural motion of a damped Euler–Bernoulli beam is given by

$$D_c{\displaystyle \frac{{\partial }^{4}w}{\partial x^4}}+F_{Tc}{\displaystyle \frac{{\partial }^{2}w}{\partial x^2}}-\rho S{\omega }^{2}w=F_0\delta (x-x_0)$$

(1)

where D_c = EI_b (1 + jη) is the complex bending stiffness, E is the elastic modulus, I_b is the moment of inertia of the beam cross-section, η is the damping factor, w denotes the transverse displacement, F_Tc = ESα(T − T₀)(1 + jη) denotes the complex in-plane axial force, S is the cross-section area, α is the thermal expansion coefficient, T₀ is the referenced temperature, ρ is the density, ω is the circular frequency, δ(x − x₀) is the Delta function, x₀ is the location of F, and F₀ is the amplitude.

Assumptions made in the following analysis mainly consist of three parts. First, the axial deformation due to thermal expansion is fully restrained, inducing compressive thermal stress. Secondly, thermal expansion stresses are computed under steady-state uniform temperature fields. Finally, linear elastic, small-amplitude vibration under harmonic excitation.

The general solution of the homogeneous part of Equation (1) is derived by the method of separation of variables

$$w(x)=C{\mathrm{e}}^{\mathrm{j}kx}$$

(2)

By substituting Equation (2) into the homogeneous part of Equation (1), the dispersion relation can be obtained as follows:

$$D_c{k}^4-F_{Tc}{k}^2-\rho S{\omega }^2=0$$

(3)

By solving Equation (3), the complex wavenumber k is given as

$$k=\pm k_{c1},\pm \mathrm{j}{k}_{c2}$$

(4)

where $k_{c1}=\sqrt{{\displaystyle \frac{F_{Tc}}{2D_c}}+\sqrt{{\left({\displaystyle \frac{F_{Tc}}{2D_c}}\right)}^2+{\displaystyle \frac{\rho S{\omega }^2}{D_c}}}}$, $k_{c2}=\sqrt{-{\displaystyle \frac{F_{Tc}}{2D_c}}+\sqrt{{\left({\displaystyle \frac{F_{Tc}}{2D_c}}\right)}^2+{\displaystyle \frac{\rho S{\omega }^2}{D_c}}}}$ Further, the general solution of Equation (1) can be obtained as

$$w(x)=C_1{\mathrm{e}}^{-\mathrm{j}{k}_{c1}x}+C_2{\mathrm{e}}^{\mathrm{j}{k}_{c1}x}+C_3{\mathrm{e}}^{-k_{c2}x}+C_4{\mathrm{e}}^{k_{c2}x}$$

(5)

where C₁, C₂, C₃, C₄ are constant determined from boundary conditions.

The particular function of Equation (1) could be derived based on the residue theorem. For a thermal beam under the point force excitation, the particular solution for Equation (1) has been derived in our previous work [13].

$$w_F(x)=-{\displaystyle \frac{\mathrm{j}{F}_0}{4EI}}{\displaystyle \frac{1}{\sqrt{{\left(\frac{F_{Tc}}{2D_c}\right)}^2+\frac{\rho S{\omega }^2}{D_c}}}}\left({\displaystyle \frac{{\mathrm{e}}^{-\mathrm{j}{k}_{c1}\left|x-x_F\right|}}{k_{c1}}}-\mathrm{j}{\displaystyle \frac{{\mathrm{e}}^{k_{c2}\left|x-x_F\right|}}{k_{c2}}}\right)$$

(6)

2.2. Dynamic Response Prediction Using the Wave-Based Method

In the WBM, the transverse displacement w(x) is approximated as the summarization of the general solutions and particular solutions

$$w(x)\approx \widehat{w}(x)={\displaystyle \sum _{s=1}^4{\varphi }_s(x)w_s+}{w}_F(x)=\Phi w+w_F(x)$$

(7)

where ${\varphi }_s(x)$ is the wave function, Φ is the wave function matrix, w_s is the participation factor of ${\varphi }_s(x)$, and w is the participation matrix.

The wave function ${\varphi }_s$ should satisfy the homogeneous part of the governing equation. Therefore, according to Equation (5), ${\varphi }_s$ in the matrix Φ are four linearly independent solutions

$${\varphi }_1{=\mathrm{e}}^{-k_{c1}x}; {\varphi }_2{=\mathrm{e}}^{\mathrm{j}{k}_{c1}x}; {\varphi }_3{=\mathrm{e}}^{-k_{c2}x}; {\varphi }_4={\mathrm{e}}^{k_{c2}x}$$

(8)

The commonly used boundary conditions, which include the clamped edge, simply supported edge, and free edge, are, respectively, given as

$$w(x)=0;{\displaystyle \frac{\mathrm{d}w(x)}{\mathrm{d}x}}=0$$

(9)

$$w(x)=0;D_c{\displaystyle \frac{{\mathrm{d}}^{2}w(x)}{\mathrm{d}{x}^2}}=0$$

(10)

$$D_c{\displaystyle \frac{{\mathrm{d}}^{2}w(x)}{\mathrm{d}{x}^2}}=0;D_c{\displaystyle \frac{{\mathrm{d}}^{3}w(x)}{\mathrm{d}{x}^3}}=0$$

(11)

The participation factors of wave functions can be obtained by using the weighted residual formulation in boundaries

Aw = f

(12)

where A and f are determined in terms of boundary conditions. For example, for the thermal beam with clamped boundaries, A and f are given as ${\left[\begin{array}{cccc}\Phi (0)& \Phi (l)& {\displaystyle \frac{\mathrm{d}\Phi (0)}{\mathrm{d}x}}& {\displaystyle \frac{\mathrm{d}\Phi (l)}{\mathrm{d}x}}\end{array}\right]}^{\mathrm{T}}$ and ${\left[\begin{array}{cccc}-{\widehat{w}}_F(0)& -{\widehat{w}}_F(l)& -{\displaystyle \frac{\mathrm{d}{\widehat{w}}_F(0)}{\mathrm{d}x}}& -{\displaystyle \frac{\mathrm{d}{\widehat{w}}_F(l)}{\mathrm{d}x}}\end{array}\right]}^{\mathrm{T}}$, respectively.

Therefore, Equation (7) could be rewritten as

$$w\approx \Phi {\left(A\right)}^{-1}f+w_F$$

(13)

Then, the broadband displacement response can be predicted. The computational efficiency of WBM mainly depends on the number of wave functions. For the vibration problems of the plate or acoustic cavity, more wave functions are needed to construct the dynamic responses in the high-frequency ranges, which could consume a lot of computational cost. Fortunately, for the vibration problem of the beam, as shown in Equation (8), only four wave functions are needed to construct the dynamic responses over the broadband frequency ranges. Therefore, the WBM is very efficient in solving the broadband vibration problems of beams.

2.3. Design Sensitivity Analysis

The finite difference method has three alternatives: the forward difference method, the backward difference method, and the central difference method. For the sensitivity analysis of the displacement response w with respect to input parameter v, the design sensitivity by using the forward difference method can be given as

$${\displaystyle \frac{\mathrm{d}w}{\mathrm{d}v}}={\displaystyle \frac{w(v_0+\Delta v)-w(v_0)}{\Delta v}}$$

(14)

For the backward difference method, the design sensitivity is calculated by

$${\displaystyle \frac{\mathrm{d}w}{\mathrm{d}v}}={\displaystyle \frac{w(v_0)-w(v_0-\Delta v)}{\Delta v}}$$

(15)

For the central difference method, the design sensitivity is given as

$${\displaystyle \frac{\mathrm{d}w}{\mathrm{d}v}}={\displaystyle \frac{w(v_0+\Delta v)-w(v_0-\Delta v)}{2\Delta v}}$$

(16)

The central difference method is more accurate than the forward difference method, but it also costs more computational effort.

As there are order-of-magnitude differences in the values of various input parameters, the normalized sensitivity is defined to analyze the effect of different parameters.

$${\displaystyle \frac{\mathrm{d}w}{w}}/{\displaystyle \frac{\mathrm{d}v}{v}}={\displaystyle \frac{\mathrm{d}w}{\mathrm{d}v}}{\displaystyle \frac{v}{w}}$$

(17)

Compared to the relative sensitivity introduced in Equations (14) and (16), the normalized sensitivity could describe the variability of displacement response to the change in proportion of the input parameter.

3. Numerical Simulations

A thermal beam made of titanium alloy with ρ = 4420 kg/m³, υ = 0.33, and η = 0.01 was taken as the simulation model in this work. The temperature-dependent material properties, which include E and α, are shown in Figure 2. The structural geometry parameters are given as follows: l = 1 m, S = 2 × 10⁻⁴ m², and I_b = 2 × 10⁻¹⁰ m⁴. The reference temperature was set to T₀ = 0 °C. A unit transverse harmonic force was applied at the middle of the beam.

3.1. Validation Cases

The accuracy of the WBM was first validated by the analytical method [5,17]. The boundary condition of the thermal beam was set to simply supported. Both the thermal effects of thermal stress and temperature-dependent material properties were considered in the following analysis. The environment temperature was set to T = 50 °C. Figure 3 presents the displacement frequency response at the middle point of the beam from 100 Hz to 10,000 Hz. Figure 4 gives the transverse displacement distributions at 5000 Hz. It can be observed from Figure 3 and Figure 4 that simulation results predicted by the WBM match well with the exact solutions obtained by analytical method.

3.2. Thermal Effect on the Broadband Dynamic Responses

In this section, numerical simulations were performed to systematically evaluate the thermal effects on bending vibration responses. Three distinct scenarios were analyzed: Case a: Incorporating only temperature-dependent material properties; Case b: Accounting for thermal stress alone; and Case c: Combining both temperature-dependent properties and thermal stress. The frequency response of the displacement at the beam’s midpoint was examined across the 100–10,000 Hz range. Figure 5 illustrates the comparative results, revealing how thermal coupling mechanisms influence vibrational behavior in each case.

As illustrated in Figure 5a, when only temperature-dependent material properties are considered, the dynamic response exhibits a minor shift toward lower frequencies with increasing environmental temperature. This softening behavior arises from the reduction in elastic modulus, which diminishes the bending stiffness of plates. However, the limited variation in elastic modulus under tested temperatures ranges results in negligible overall influence on vibrational characteristics.

In contrast, Figure 5b demonstrates the isolated effect of thermal stress. Here, resonant peaks shifted significantly to lower frequencies as temperatures rise, driven by thermally induced compressive forces from constrained thermal expansion. This compressive stress amplifies structural softening and leads to a decrease in resonance frequency. These findings are similar to those reported in the literature [40,41].

When both thermal effects are combined, as shown in Figure 5c, the resonant frequency reduction becomes more pronounced, synthesizing contributions from material property degradation and compressive thermal stress. Notably, the displacement amplitude oscillations observed in Figure 5c align closely with those in Figure 5b, underscoring that thermally induced stresses dominate the vibration response compared to temperature-dependent material softening. This dominance stems from boundary constraints amplifying stress-induced stiffness reduction, whereas material property variations exhibit secondary influence under the tested conditions.

3.3. Global Sensitivity Analysis

As emphasized by Saltelli et al. [30], it is very meaningful to explore the design sensitivity over a global space of input factors. Therefore, the global sensitivity analysis regarding different input parameters is presented in this section. It should be noticed that all the present sensitivities are calculated using the normalized sensitivity according to Equation (17).

The absolute global sensitivities for thermal load and thermal expansion coefficients are first presented in Figure 6 and Figure 7, respectively. It can be observed that both the thermal load and thermal expansion coefficient have a more significant effect on the low-frequency dynamic response. According to the governing equation introduced in Equation (1), the thermal load and thermal expansion coefficient are linearly related to the in-plane axial force F_Tc. As the sensitivity value becomes larger near the natural frequencies, the influence of the input factor on the natural frequencies can also be observed. As shown in Figure 6 and Figure 7, the natural frequencies decrease as the thermal load or thermal expansion coefficient increases. An increase in thermal load and thermal expansion coefficient leads to a higher in-plane axial force, which induces a softening effect on structural stiffness and results in the natural frequencies shifting towards lower frequency ranges.

The absolute global sensitivities of the displacement response with respect to the cross-section area and moment of inertia are given in Figure 8 and Figure 9, respectively. With the increase in analysis frequency, the influence of the moment of inertia and cross-section area on the displacement response becomes more significant. According to the governing equation, the moment of inertia is linearly related to the complex bending stiffness. An increase in the moment of inertia could strengthen the complex bending stiffness and lead to an increase in natural frequencies. Whereas the cross-section area is related to the complex in-plane axial force, the increase in cross-section area could result in higher in-plane axial force and lead to the natural frequencies shifting towards lower frequency.

Finally, the absolute global sensitivity of displacement response with respect to the elastic modulus and density of the material is given in Figure 10 and Figure 11, respectively. Simulation results indicate that both the elastic modulus and density could significantly affect dynamic responses across the broadband frequency ranges. The increase in elastic modulus could make the natural frequencies shift towards high-frequency ranges. This is due to the increase in structural stiffness, whereas the influence of density on natural frequencies is opposite.

According to the simulation results shown in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11, the following conclusions can be made. In the low-frequency ranges, the sensitivities with respect to thermal load, thermal expansion coefficient, the cross-section area, and moment of inertia are dominant. In the high-frequency ranges, the cross-section area, moment of inertia, elastic modulus, and density have a significant influence on the dynamic response. All the investigated parameters investigated could significantly affect the mid-frequency dynamic response.

4. Conclusions

Beam-type structures used in aerospace engineering can be subjected to a service environment consisting of thermal loads and broadband dynamic loads. The broadband dynamic sensitivity of beams in a thermal environment is investigated using the efficient wave-based method. First, the simulation accuracy for predicting the broadband dynamic response is validated. Then, the normalized sensitivities of the broadband dynamic response with respect to thermal loads, material properties, and geometric parameters are studied.

The simulation results indicate that the sensitivities could vary within the broadband frequency band. Thermal loads and thermal expansion coefficients have a more significant effect on low-frequency dynamic responses. As the analysis frequency increases, the influence of the moment of inertia and cross-section area on the displacement response becomes more significant. The elastic modulus and density could also affect the broadband dynamic response. These conclusions are helpful for structural design and optimization.

Future work will extend the proposed framework to Timoshenko beams under different boundary conditions, addressing the critical effects of shear deformation and rotational inertia for high-frequency ranges. This extension will leverage the current thermal coupling formulation while incorporating shear correction factors and inertia terms. Additionally, while the current finite difference method achieves robust sensitivity quantification, future extensions will integrate either the adjoint-WBM formulation or a semi-analytical derivation to reduce computational costs.