# A Semi-Analytical Approach for the Linearized Vibration of
## 1 Introduction
The governing equations are derived by the [PERSON] method and simplified into three nonlinear ordinary differential equations. [PERSON] [24] investigated the large-amplitude vibration of rectangular plates under harmonic excitation. The influence of geometric imperfections on the nonlinear response trend and natural frequency was studied by numerical and experimental methods. [PERSON] et al. [25; 26] compared the classical [PERSON] theory, the first-order shear deformation theory, and the higher-order shear deformation theory to study the nonlinear forced vibration of isotropic and laminated composite rectangular plates. [PERSON] et al. [27] proposed a finite element formulation to determine the large-amplitude free and steady-state forced vibration response of plates. The linearized updating-mode method with nonlinear time function approximation was used to solve the nonlinear eigenvalue equation of the system. [PERSON] et al. [28] proposed a unified theory of beams that incorporates geometric nonlinearity by utilizing the Carrera Unified Formula. The Newton-Raphson linearization scheme and path-tracking method are employed to solve the problem of geometric nonlinearity. Large amplitude vibration analysis of composite beams is studied by [PERSON] et al. [29] by using the Rayleigh-Ritz method. The nonlinear vibration of beams is characterized by the presence of higher-order terms in the equations of motion. The linearization of a nonlinear vibration equation can be done using different methods, including the Taylor series expansion method [30] and the Jacobian matrix method [31]. Both methods involve approximating the nonlinear function with a linear function and then solving the resulting linear equation. [PERSON] et al. [32] proposed a perturbation method based on Ritz expansion to analyze the dynamic characteristics of a modified structure in the reduced modal subspace. [PERSON] et al. [33] studied the free vibration of non-uniform Euler-Bernoulli beams by using the Adomian decomposition method. The pressure load for the hypersonic vehicle is the combined load of static pressure and acoustic pressure. And the significant deformation of the structure is primarily influenced by static loads. By considering the large deformation as an initial deformation term and incorporating its impact with thermal effects into the dynamic equations of the structure, the response under acoustic pressure loads can be analyzed more efficiently in the frequency domain by the mode-superposition technique. This paper presents a semi-analyticalapproach to illustrate the linearized vibration of clamped-clamped beams in the nonlinear regime due to the effect of static load and thermal load. The paper is organized as follows. In Section 2, the nonlinear governing equation of the beam, considering the impact of static load and thermal load, is briefly introduced. Section 3 demonstrates the linearization method of the nonlinear equation. The vibration equation is a variable coefficient partial differential equation. It is solved by the transfer-matrix method and local homogenization technique. The random response analysis is determined using the mode-superposition methods. The accuracy of the proposed method is validated by commercial software ABAQUS 2020 and literature in Section 4. The effect of static load and thermal load on the mechanical characteristics of structures is studied in this section as well. Finally, the main conclusions are reported in Section 5.