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Frequency equation and mode shape formulae for composite Timoshenko beams

机译：Timoshenko复合梁的频率方程和模态公式

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摘要

Exact expressions for the frequency equation and mode shapes of composite Timoshenko beams with cantilever end conditions are derived in explicit analytical form by using symbolic computation. The effect of material coupling between the bending and torsional modes of deformation together with the effects of shear deformation and rotatory inertia is taken into account when formulating the theory (and thus it applies to a composite Timoshenko beam). The governing differential equations for the com- posite Timoshenko beam in free vibration are solved analytically for bending displacements, bending rotation and torsional ro- tations. The application of boundary conditions for displacement and forces for cantilever end condition of the beam yields the frequency equation in determinational form. The determinant is expanded algebraically, and simplified in an explicit form by extensive use of symbolic computation. The expressions for the mode shapes are also derived in explicit form using symbolic computation. The method is demonstrated by an illustrative example of a composite Timoshenko beam for which some published results are Available.
机译：通过符号计算，以明确的解析形式，导出了具有悬臂端条件的复合Timoshenko梁的频率方程和模态的精确表达式。制定该理论时，要考虑到弯曲和扭转变形模式之间的材料耦合效应以及剪切变形和旋转惯性的影响（因此，它适用于复合Timoshenko梁）。对于自由位移下的复合Timoshenko梁，其控制微分方程可通过弯曲位移，弯曲旋转和扭转旋转进行解析求解。施加梁的位移的边界条件和悬臂梁端部条件的力得出确定形式的频率方程。行列式通过代数展开，并通过广泛使用符号计算而以显式形式简化。模式形状的表达式也使用符号计算以显式形式导出。通过复合Timoshenko光束的说明性示例证明了该方法，有关该结果的某些公开结果是可用的。

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