# A Semi-Analytical Approach for the Linearized Vibration of
## 1 Introduction  
During the operation of a hypersonic vehicle, it must endure a highly complex dynamic environment, including high thermal load, high static pressure, and acoustic pressure loads. Clamped-clamped structures demonstrate intricate geometric nonlinear response characteristics under combined loads. Although the strain does not exceed the elastic limit, the static equilibrium and vibration differential equations demonstrate nonlinear characteristics caused by thermal stress and significant deformation [1; 2; 3]. Structures are prone to experiencing large deformations under high-pressure load, and the geometric nonlinearity that arises can affect the mechanical characteristics of the structure. [PERSON] [4] clarified the impact of dead loads on the static characteristics of beams using the energy function and assumed modes method. Additionally, he investigated the effects of dead loads on the dynamic characteristics of both beams [5; 6] and plates [7]. [PERSON] et al. [8] developed a load-induced stiffness matrix that considers the stiffening effect of dead loads, and the natural frequencies of beams were calculated using finite-element methods. [PERSON] et al. [9] studied the large amplitude free vibration of plates based on static analysis through an iterative numerical scheme. [PERSON] et al. [10] utilized nonlinear shooting and Adomain decomposition methods to analyze the large deflection of beams. [PERSON] et al. [11] investigated the influence of static loads on the vibro-acoustic characteristics of plates. [PERSON] et al. [12] developed a geometrically nonlinear total Lagrangian formulation to analyze the vibration modes of beams in the nonlinear regime. The influence of high temperature on the mechanical characteristics of structures is mainly reflected in two aspects: on the one hand, high temperature can alter the mechanicalproperties of materials; on the other hand, high temperature can generate thermal stress within the structure, leading to changes in the stiffness characteristics of the structure [13; 14]. Numerous researchers work on the mechanical characteristics analysis of structures under thermal environment have been done in past years. [PERSON] et al. [15] studied the buckling and dynamic characteristics of isotropic plates under different thermal loads using the finite element method. The results showed that the thermal buckling point was closer to the hotter side of the structure. [PERSON] et al. [16] investigated the nonlinear dynamic response characteristics of composite sandwich panels under high-temperature environments based on higher-order shear deformation theory and the two-step perturbation method. [PERSON] et al. [17] studied the thermal buckling characteristics of functionally graded plates under thermal loads using classical plate theory. [PERSON] et al. [18] analyzed the thermal buckling and post-buckling behavior of rectangular composite plates subjected to local thermal heating using a semi-analytical method. [PERSON] et al. [19] studied the influence of different temperature distributions on the mode and buckling characteristics of a simply supported plate based on the Rayleigh-Ritz method. [PERSON] et al. [20] investigated the thermal buckling and vibration response characteristics of the beam with a stick-slip-stop boundary, taking into account assembly pre-tension and thermal expansion clearance. Nonlinear vibration of structures is an important area of research that deals with the dynamic behavior of beams when subjected to large amplitude vibrations. [PERSON] et al. [21] proposed a theoretical model for the geometrically nonlinear vibration analysis of functionally graded plates by precise series expansion method and perturbation method. [PERSON] et al. [22] studied the post-buckling and nonlinear vibration analysis of geometrically defective beams under nonlinear elastic constraints due to axial force. [PERSON] [23] investigated the large-amplitude thermal-mechanical coupling vibration of simply supported orthotropic rectangular plates.