An Orthotropic Vibration Analysis Method for Simply Supported Rectangular Thin Plates

Abstract

The invention discloses a method for analyzing the vibration of an orthotropic simply supported rectangular thin plate with opposite sides. This method is based on the free vibration equation of an orthotropic rectangular thin plate. Firstly, the mode differential equation is established, and the Hamiltonian canonical equation is constructed. Secondly, according to the symplectic geometry method, the separation of variables method is used to solve the problem, and the corresponding The expressions of the eigenvalues ​​of the two groups of eigenvalues ​​were analyzed; the general solution form of the mode shape function with undetermined constants was constructed, and the expressions of the eigenvalues ​​and their eigenvectors were derived by combining the simply supported boundary conditions on opposite sides , and proved the symplectic orthogonality and completeness of the eigenvector system; using the symplectic eigenvector expansion, the expression of the state vector was obtained, and then the general solution of the mode shape function was obtained; finally, for the six common Boundary conditions, the corresponding frequency equations were derived respectively, so as to realize the accurate solution of the free vibration frequency, which can lay the foundation for the development of its dynamic analysis and related applications.

Description

technical field [0001] The invention belongs to the technical field of plate free vibration analysis and solution, and particularly relates to a vibration analysis method for an orthotropic simply supported rectangular thin plate on opposite sides. The rational derivation and accurate solution of the free vibration frequency of the rectangular thin plate provide a theoretical reference for further applications such as dynamic analysis and performance evaluation, fault diagnosis, and vibration and noise reduction of the rectangular thin plate. Background technique [0002] Vibration is a common physical phenomenon in engineering and widely exists in aerospace, civil engineering and other fields. Vibration problems have caused many aircraft and bridge accidents, and vibrations also exist during take-off, landing and the entire flight process of aircraft. Therefore, it is of great engineering significance to reasonably analyze and accurately solve vibration problems. [0003]...

AI Technical Summary

Problems solved by technology

Although the superposition method is helpful to solve this problem, because it is still based on the semi-inverse method, when dealing with some complex rectangular plate problems, the selection of superposition items is irregular and often makes people feel at a loss.

[0004] Over the years, many scholars have used numerical methods such as the Rayleigh-Leeds method, finite element method, and differential quadrature method to directly solve the problem. Although there is no need to assume in advance, there are calculation errors, and the high-precision requirements often require an astonishing amount of calculation.

There are also scholars who directly separate the variables of the mode shape function. Although they have obtained accurate results, they have not proved its rationality from a mathematical point of view. This method of separating variables will be limited by the spectrum of self-conjugate operators. and completeness are theoretically not guaranteed

Images