Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.