## 4 Conclusions  
In this paper, a unified analysis model of the vibro-acoustic characteristics of the coupled system between the composite laminated rotationally stiffened plate and cavities is developed, encompassing three distinct system types. This model was established using a combination of the finite element method, the modified Fourier series, and the Rayleigh-Ritz method. First, utilizing the modified Fourier series method, we determined the allowable functions of sound pressure and displacement associated with rotationally cavities, laminated plates, and laminated beams. Second, we introduced the potential energy of coupling among the plate-cavity and plate-beam interfaces, thereby formulating the functional of the total energy of the entire coupling system. Third, to solve the energy equation that was formulated, the Rayleigh-Ritz method was employed. After conducting numerical analyses on various examples, a study was conducted on the vibro-acoustical characteristics of the coupled system between composite laminated rotationally stiffened plate and cylindrical-cylindrical, spherical-cylindrical, and conical-cylindrical cavities. The selection of the truncation value and convergence was discussed and the correctness of  
Figure 19: (**a**) Sound pressure response of observation point 1 under the point sound source. (**b**) Sound pressure response of observation point 1 under the point sound source. (**c**) Sound pressure response of observation point 1 under the point force. (**d**) Sound pressure response of observation point 2 under the point force. Sound pressure response of the coupling system among the composite laminated rotationally stiffened plate and the conical-cylindrical cavity under the excitation of a single monopole point source or a point force with different acoustic wall impedance values. the unified analysis model was confirmed. The effects of various parameters on natural frequency and steady-state response were studied. The specific conclusions are as follows:  
(1) When the truncation value of the allowable functions is set to \(M_{\rm C}\times N_{\rm C}\times Q_{\rm C}=4\times 4\times 4\) and \(M_{\rm S}\times N_{\rm S}=18\times 18\), the analytical model developed in this paper of the coupling system between the composite laminated rotationally stiffened plate and cavities demonstrates excellent convergence. The maximum error in natural frequencies compared to the finite element method results in the examples of this paper is 3.51%. Additionally, the spring stiffness values also converge when they reach \(10^{12}\). (2) In the validation of the model's correctness through the pressure response test, the error between the experimental results of specific frequency sound pressure values and the results of the present method is 7.49%, confirming the correctness of the unified analysis model established in this paper. (3) Under free vibration conditions, the natural frequencies of the coupling system among the composite laminated rotationally stiffened plate and cavities decrease with increasing rotation angle, decrease with increasing height of the cylindrical cavity, decrease with increasing height of the cylindrical cavity and the apex angle of the spherical cavity, decrease with increasing height of the cylindrical cavity and decreasing cone-apex angle of the conical cavity, and increase with increasing boundary spring stiffness values. (4) In the coupled system between the composite laminated rotationally stiffened plate and cavities, the composite laminated rotationally stiffened plate has a certain impact on noise reduction, and this impact increases with the thickness of the stiffened plate. The sound pressure response amplitude increases with the increase in the amplitude of point sound sources and point force excitation. The sound pressure response amplitude decreases with decreasing impedance values, but it does not affect the response waveform. Conceptualization, L.H. ; methodology, C.S. ; software, C.J. ; validation, Y.D. ; formal analysis, Y.D. ; investigation, Y.D.