Free vibration of rotating pre-twisted thin plates with elastic boundary considering nonlinear centrifugal stiffening effect

Abstract

The blade root is typically connected to the hub via tongue-and-groove joint for aeroengine blade structures. This connection form exhibits limited rotational and translational constraint characteristics at the blade root, not completely rigid, and can be regarded as elastic support. At high rotation speed, the centrifugal load induces substantial deformation, significantly altering the structural stiffness distribution, thereby affecting the vibration characteristics. Consequently, a dynamic analytical framework for rotating pre-twisted curved plates with elastic boundary is established. The nonlinear governing equation for rotating pre-twisted plate is derived based on Novozhilov nonlinear shell theory and Hamilton's principle. The stiffening effect of the centrifugal load is characterized through nonlinear quasi-static equilibrium analysis and linearization of the nonlinear terms surrounding equilibrium solutions. According to penalty method, the springs are introduced to simulate boundary condition. This means that the admissible functions of displacement field need not satisfy geometric boundary conditions, but only ensure the linear independence and completeness of functions set. By applying Chebyshev orthogonal polynomials for mode discretization, the natural frequencies and mode shape are obtained through eigenvalue analysis. The effects of boundary conditions, geometric parameters, and rotation speed on free vibration characteristics are investigated in detail. The results indicate that the geometric pre-twist induces bending-in-plane coupled mode within the system, and that each mode frequency under elastic boundary condition is positively correlated with the corresponding spring stiffness, with coupled modes governed by in-plane springs. Complex mode crossover and “crossover avoidance” phenomena occur with parameter variations.