A Novel Analytical Model for Free Vibration of Irregular Stepped Orthotropic Cantilevered Plates

Multi-level thickness irregular plates have garnered widespread applications in diverse engineering fields, driven by the growing demands for lightweight structural designs, enhanced strength capabilities, and the reduction of welding necessities. In this paper, analytical solutions for free vibration of irregular stepped orthotropic cantilevered plates (SOCPs), including L-shaped, T-shaped, Z‑shaped, and cross-sharped plates, are derived within the framework of symplectic mechanics. The overall SOCP is regarded as a combination of multiple uniform-thickness rectangular plates, thereby its analytical solutions can be obtained by using the symplectic method and a boundary superposition technique. Therefore, accurate natural frequencies and analytical vibration mode shapes are obtained simultaneously. The derivation procedure is conducted in a step-by-step manner, ensuring each stage is accompanied by a rigorous definition and logical progression. In numerical examples, a comparison study is first given. The present results are compared with those reported in the literature and computed by finite element method, and excellent agreements are observed. Furthermore, the effects of cantilever boundary conditions and thickness ratios on the vibration characteristics of the L-shaped, T-shaped, Z-shaped, and cross-sharped plates are investigated and discussed in detail. The obtained natural frequencies and analytical vibration mode shapes of these irregular SOCPs can be served as benchmarks for future approximate and numerical methods.