ON TWO VARIATIONAL PRINCIPLES FOR THE NATURAL VIBRATIONS OF ELASTIC BODIES
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摘要: 对于几何形狀较复雜的彈性体,要嚴格地根据彈性体力学的全部方程來决定它的固有频率是有很大的困难的,因此在实际上常常采用了各式各样的近似解法,其中最有名的近似解法要算根据萊瑞原理的变分解法(称为萊瑞黎茲法)。在許多問題中,萊瑞黎兹法給出了滿意的結果。但是这个方法在实用上及理論上遺留下两个問題:第一、在許多問題中,我們往往能对应力分布的情況作适当的估計,可是这些估計在萊瑞黎兹法中不能加以充分利用;第二、在各种近似理論中,例如在梁、板、壳的近似理論中,虽然都各有自己的萊瑞原理,可是这些原理并不是彈性体力学中的萊瑞原理的直接的特殊形式。Abstract: Consider the natural vibrations of elastic bodies. For the natural frequencies, we have obtained two variational principles. In the first variational principle, an allowable state is defined as follows. The stress components and their variations have the formsand satisfy the stress-free boundary conditions. The corresponding displacement components are derived from equations of motion:Then it is proved that the natural frequencies ω of the body correspond to the extreme values of the following functional: where V is the strain energy per unit volume expressed in terms of stress components.In the second Variational principle, an allowable state is defined by any continuous stress and displacement distributions, not necessarily satisfying any equation and boundary condition. In this case, it is proved that the natural frequencies ω of the body correspond to the extreme values of the following functional:where H has the expressionPx,Py,Pz are the surface tractions, and Su is that part of
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