Winkler地基上修正Timoshenko梁振动分析

利用Bernoulli-Euler梁理论建立的弹性地基梁模型应用广泛,但其在高阶频率及深梁计算中误差较大,利用修正的Timoshenko梁理论建立新的弹性地基梁振动微分方程,由于其在Timoshenko梁的基础上考虑了剪切变形所引起的转动惯量,因而具有更好的精确度。利用ANAYS beam54梁单元进行振动模态的有限元计算,所求结果与理论基本无误差,从而验证了该理论的正确性。基于修正Timoshenko梁振动理论推导出了弹性地基梁双端自由-自由、简支-简支、简支-自由、固支-固支等多种边界条件下的频率超越方程及模态函数。分析了弹性地基梁在不同理论下不同约束条件及不同高跨比情况下的计算结果,从而论证了该理论计算弹性地基梁的适用性。分析了不同弹性地基梁理论下波速、群速度与波数的关系。得到了约束条件和梁长对振动模态及地基刚度对振动频率有重要影响等结论。

Vibration analysis for a modified Timoshenko beam on Winkler elastic foundation

The elastic foundation beam model based on Bernoulli-Euler beam theory is widely used, but it has larger errors in calculations of higher order natural frequencies and deep beams.Here, based on the modified Timoshenko beam theory, a new vibration differential equation of an elastic foundation beam was established.Due to considering the moment of inertia caused by shear deformation based on Timoshenko beam theory, it has a better accuracy.The beam54 element of the finite element software ANSYS was used to do the finite element calculation for the beam’s vibration modes, and the results basically had no errors compared with the theoretical solution to verify the correctness of the proposed theoretical model.Based on the modified Timoshenko beam theory, the elastic foundation beam’s natural frequency transcendental equations and modal functions under various boundary conditions including free-free, pinned-pinned, pinned-free and clamped-clamped were deduced.The calculation results of an elastic foundation beam under different theories, different boundary conditions and different ratios of height to span were analyzed to verify the applicability of the proposed theoretical model.The relations among wave velocity, group velocity and wave number under different theories were analyzed.It was shown that constraint conditions and beam length have important effects on an elastic foundation beam’s vibration modes; foundation stiffness affects an elastic foundation beam’s natural vibration frequencies significantly.