悬臂梁结构的应变模态积分法与灵敏度分析

应变模态振型决定了结构动应变分布规律。基于梁应变与曲率的正比关系，提出一种直接求解悬臂梁结构曲率模态的积分方法，适用于任意形式的刚度和质量函数。通过与数值和解析解比较模态频率与振型，验证了该方法的准确性。此外，该方法将连续体模态求解问题转化为有限自由度的广义特征值形式，进一步证明可以应用Nelson法求解该形式的特征灵敏度，获得了曲率模态振型对于结构设计参数的灵敏度显式表达。为了更加明显地表现动应变变化，提出了曲率梯度模态的概念，其参数灵敏度可由曲率模态灵敏度获得。算例结果表明，该方法可以有效地优化结构参数、避免动应变集中。该方法同样也适用于固支—固支和固支—简支这两种边界条件。

The Integral Method and Sensitivity Analysis for Strain Modes of Cantilever Beam-Like Structures

The spatial distribution of dynamic strain is determined by the modal shape and hence the strain at a certain area can be reduced by modal shape design. Based on the proportional relation between strain and curvature, this paper proposes an integral method to solve curvature modal frequencies and shapes directly of cantilever beam-like structures with variable stiffness and mass functions. The method is validated by comparing the computation results with both numerical and analytical solutions. Furthermore, the presented method transforms the problem solving of continuous systems modes into a generalized eigenvalue form of finite degrees of freedom. The explicit expressions of curvature eigen-sensitivity have been established with Nelson''s method, and the curvature gradient modes are also calculated to show the change of dynamic strain more visibly. An example of parameter design is presented with the proposed sensitivity analysis method. Moreover, this method is proved capable of treating the other two boundary conditions of clamped-simply supported and clamped-clamped.