Transverse random response of an axially moving beam

An axially moving beam is a simplified model for many aircraft structures. Its elasticity is patently more obvious with its slenderness ratio increased and mass reduced, and its velocity has a significant effect on its vibration characteristics at the same time. Here, the dynamic equation of transverse vibration of an axially moving beam subjected to a transverse excitation was derived with Hamilton's principle. At first, the dimensionless method and complex modal analysis method were applied to simplify the equation with an axial force or without an axial one. The frequency equation and modal functions were obtained, they were solved using the numerical method. Then, the decoupling method was used to simplify the control equation into a set of differential equations, the displacement responses of the beam were gained after solving those equations. Finally, the random response's correlation function was calculated by using the method of mathematical statistics, and the random response spectrum of the beam was achieved via Fourier transformation. The numerical example illustrated that the beam's moving velocity can affect its vibration characteristics and random responses significantly.