功能梯度材料梁自由振动问题的常微分方程求解器解

The solutions to free vibration problem of beams with functionally graded materials based on ordinary differential equation solver

建立具有连续分布参数的功能梯度材料Euler梁、Timoshenko梁自由振动的动力学方程,以常微分方程求解器为工具,分析计算这两种梁的自振频率;同时讨论Timoshenko梁的自振频率和振型随梁的参数而变化的规律,给出Timoshenko梁的弯曲振动弹性波和剪切振动弹性波的传播速度,分析弯曲和剪切耦合振动的特点和规律.结果表明:常微分方程求解器解和解析解几乎具有同样的精度;自振频率的大小取决于梁在振动时的弹性波的波速;Timoshenko梁在每个频率下的振动均为弯曲和剪切的耦合振动.

The dynamic equations of free vibration for Euler and Timoshenko beams with functionally graded materials and continuously distributed parameters were developed, respectively. The natural frequencies of the beams were computed and analyzed by employing ordinary differential equation （ODE） solver. The variation of the frequencies and modes of vibration of Tinoshenko beam with the changes of its parameters were discussed. The propagation velocities of elastic waves due to bending and shearing during free vibrations of the beams were given. The characteristics and pattern of coupled vibrations of bending with shearing were also analyzed. The results indicated that the accuracy of the solutions obtained with ODE solver were almost identical with those of analytical method, the magnitude of the natural frequency was dependent on the value of velocity of the elastic wave induced by the free vibration of the beam, and the vibration of Timoshenko beam under each natural frequency was the coupled vibration caused by bending and shearing deformatior.