Local coordinate systems-based method to analyze high-order modes of n -step Timoshenko beam

Abstract

High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes (k ≤ 12) are solvable for i-step (i ≥ 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory.