Theoretical and experimental study on the dynamic characteristics of an axially moving nested clamped-hinged beam

The present paper is concerned with the transverse vibration properties of an axially moving nested clamped-hinged beam, which can be regarded as a stepped beam. The transverse vibration equation for the axially moving nested clamped-hinged beam is derived by D’Alembert’s principle. The modified Galerkin’s method, which uses the instantaneous modal function of the clamped-hinged stepped beam as a trial function, is used to solve the vibration equation. An axially moving nested clamped-hinged beam model is designed for the vibration test. The theoretical model is updated by calculating the flexural rigidity values of the first segment of the nested beam based on the measured first-order vibration frequencies, which are tested for different lengths in the main beam. The first order decay coefficients are identified by the logarithmic decrement method. Then, the functional relationship between the flexural rigidity and beam length, as well as the decay coefficient and beam length, is established using the polynomial fitting method. 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 model updating and damping determination. The theoretical and experimental results demonstrate that the change law of the frequency with the main beam length increasing is a low-high-low-high trend. Further investigations into the non-damping free vibration properties of the nested clamped-hinged beam during extension and retraction of the main beam are performed. It is determined that there is no obvious change of the dynamic response amplitude of the nested structure during different axial moving rates in the main beam. Furthermore, as the length of the main beam increases, the vibration displacement decreases gradually, and the total mechanical energy increases constantly; therefore, the extension movement of the main beam becomes unstable. Moreover, the numerical results indicate that the non-damping free vibration characteristics of the nested clamped-hinged beam during extension and retraction of the main beam are inversely related