预应力梁等效刚度与固有频率的近似解析解

Approximate analytical solution of equivalent stiffness and natural frequency for prestressed beams

预应力梁被广泛应用于微传感器和执行器中,其刚度、固有频率与预应力的关系是非常重要的物理关系.针对纯弯曲问题容易得到静态变形,而纯张力问题容易得到振动频率的特点,基于张力不影响弯曲振型和振型不影响张紧频率的假设,本文提出了高精度近似求解新方法,获得了两端简支、两端固定、一端固定一端简支三种典型边界条件下的等效刚度与固有频率的高精度近似解.与有限元分析结果相比,最大刚度相对误差为1.6%,频率相对误差为2.21%.本文研究为预应力梁的等效刚度和频率计算提供了一种简单有效的方法.

Prestressed beams are widely used in micro-sensors and actuators, the relationship between the stiffness, natural frequency and prestress is an important physical relationship. It is easy to obtain static deformation for pure bending, and vibration frequency for pure tension. In this paper, based on assumption that the tension force does not affect the bending deformation, and mode shape does not affect the frequency of tension, we proposed a new approximate solution, and find the solution for three different boundary conditions including clamped-clamped, hinged-hinged and clamped-hinged. Compared with the finite element analysis results, the relative error is less than 2.21%. The presented method is simplify but efficient for estimate the equivalent stiffness and natural frequency of prestressed beams.