High-Precision Approximate Analytical Solutions for Free Bending Vibrations of Thin Plates and an Improvement
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摘要: 对于薄板弯曲自由振动问题,已有如下方法:在Hamilton(哈密顿)体系下基于分离变量法得到挠度的解析形式,并建立自振频率联立方程组,给出求解振动频率和振型函数的方法.笔者指出该方法中所用挠度函数的解析式实际上是一种满足位移边界条件的高精度近似解,基于Rayleigh-Ritz(瑞利-里茨)法再次求近似频率后发现,原方法的近似解的精度很高.另外,对于含有固支、简支等不同的边界形式,恰当地选取不同位置作为坐标系的原点,得到含有频率的方程组的统一形式,且较为简洁.这些形式可基于四边固支、四边简支等边界条件的矩形板研究,依照板变形的对称性可验证频率方程组形式的正确性,并得到不同边界条件下频率方程形式之间的联系与转化.Abstract: For free bending vibrations of thin plates, based on the variable separation method, the analytical solutions of deflection functions were obtained in the Hamiltonian system, and the eigenvalue equations about the 2 coordinate axes were established. Then the vibration frequencies were solved as parametrical variables, and the mode shapes of different orders were got. The analytical form of the deflection function was in fact a high-precision approximate solution satisfying the displacement boundary conditions. From the approximate frequency values calculated with the Rayleigh-Ritz method, it is found that the previous analytical method had great precision, thus the effectiveness was demonstrated. In addition, for clamped or simply supported boundary conditions, different points were properly selected as the coordinate origins. The unified forms of the frequency equations were given. These forms were used to discuss the rectangular plates with 4 edges clamped, or 4 edges simply supported, or some clamped and the other(s) simply supported, and so on. The correctness of the frequency equation forms was verified with the symmetry of the plates’ deformations. The linkage and transformation between the frequency equations under different boundary conditions were obtained.
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Key words:
- Hamiltonian system /
- thin plate /
- free vibration /
- deflection /
- frequency /
- symmetry
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