Coupled vibration of rotating Euler-Bernoulli beams with variable cross-sections in the plane

This paper investigates the in-plane coupled vibration characteristics of a rotating Euler–Bernoulli beam with a non-uniform cross-section. It is assumed that the cross-sectional radius of the beam varies exponentially along its longitudinal direction. Based on Hamilton’s principle, the governing equations for the in-plane coupled vibrations of the rotating variable-cross-section beam are established. The partial differential equations of motion are discretized numerically using the Galerkin method, and the complex frequencies of the system are determined. The effects of key parameters, such as hub radius, taper ratio, slenderness ratio, and rotational speed, on the frequency characteristics of the beam are further examined. Moreover, the complex modal motions of the rotating beam are investigated, with a detailed analysis of how these key parameters influence the complex modal behavior. The results indicate that the in-plane coupled vibration characteristics of the rotating variable-cross-section beam depend on both the structural parameters and the rotational speed. As the taper ratio and slenderness ratio increase, the natural frequencies and vibration amplitudes of the beam decrease. The effects of rotational speed and hub radius on axial and transverse vibrations are different, which shows distinct dynamic behaviors at different modal orders.