Dynamic stiffness matrix for free vibration analysis of flexure hinges based on non-uniform Timoshenko beam

As a typical non-uniform beam with complex varying cross-section and large curvature, free vibration analysis of flexure hinges in the state-of-the-art compliant mechanisms is challenging due to their complicated and even unsolvable governing differential equation. Investigation on the free vibration analysis of non-uniform beams was attractive in the past but still with a few types of specific beams or approximate solutions. The contribution of this paper is to formulate the dynamic stiffness matrix for generic non-uniform beams and applied for the first time to the free vibration analysis of flexure hinges. Firstly, the vibration solution of general non-uniform Timoshenko beam is derived by using a recursive integral in power series of circular frequency but not the traditional way in the well-known method of Frobenious. Then, the dynamic stiffness matrix of non-uniform Timoshenko beam is derived based on the variational principle. At last, the new method is applied for the free vibration analysis of flexure hinges and flexure-based compliant mechanisms. Several case studies demonstrate the advantage of the proposed approach in terms of high accuracy, generality and high efficiency (only one element is needed). The obtained results are useful for dynamic design of flexure hinges and can serve as a benchmark to examine the accuracy of other numerical solutions. Due to the closed form and universality of the proposed method, it is also of benefit for free vibration analysis of other non-uniform beams with all kinds of varying cross-section or functionally graded beams.