Characteristics of steady vibration in a rotating hub-beam system

A rotating beam features a puzzling character in which its frequencies and modal shapes may vary with the hub׳s inertia and its rotating speed. To highlight the essential nature behind the vibration phenomena, we analyze the steady vibration of a rotating Euler-Bernoulli beam with a quasi-steady-state stretch. Newton׳s law is used to derive the equations governing the beam׳s elastic motion and the hub׳s rotation. A combination of these equations results in a nonlinear partial differential equation (PDE) that fully reflects the mutual interaction between the two kinds of motion. Via the Fourier series expansion within a finite interval of time, we reduce the PDE into an infinite system of a nonlinear ordinary differential equation (ODE) in spatial domain. We further nondimensionalize the ODE and discretize it via a difference method. The frequencies and modal shapes of a general rotating beam are then determined numerically. For a low-speed beam where the ignorance of geometric stiffening is feasible, the beam׳s vibration characteristics are solved analytically. We validate our numerical method and the analytical solutions by comparing with either the past experiments or the past numerical findings reported in existing literature. Finally, systematic simulations are performed to demonstrate how the beam׳s eigenfrequencies vary with the hub׳s inertia and rotating speed.