Dynamics of Double-Beam System with Various Symmetric Boundary Conditions Traversed by a Moving Force: Analytical Analyses

Abstract

Dynamics of the double-beam system under moving loads have been paid much attention due to its wide applications in reality from the analytical point of view but the previous studies are limited to the simply supported boundary condition. In this study, to understand the vibration mechanism of the system with various boundary conditions, the double-beam system consisted of two general beams with a variety of symmetric boundary conditions (fixed-fixed, pinned-pinned, fixed-pinned, pinned-fixed and fixed-free) under the action of a moving force is studied analytically. The closed-form frequencies and mode shapes of the system with various symmetric boundary conditions are presented by the Bernoulli-Fourier method and validated with Finite Element results. The analytical explicit solutions are derived by the Modal Superposition method, which are verified with numerical results and previous results in the literature. As found, each wavenumber of the double-beam system is corresponding to two sub-modes of the system and the two sub-modes associated with the first wavenumber of the system both contribute significantly to the vibration of the system under a moving force. The analytical solutions indicate that the mass ratio, the bending stiffness ratio, the stiffness ratio of contact springs and the speed ratio of the moving force are the factors influencing the vibrations of the system under a moving force. The relationships between these dimensionless parameters and the displacement ratio of the system are investigated and presented in the form of plots, which could be referred in the design of the double-beam system.

The Behavior of the Synchronous and Asynchronous Natural Frequencies for Asymmetric Double Beams

“…Using the differential transformation approach, the dynamic response of the beams was estimated. Yang et al [4] explored analytically the double-beam system, which consists of two generic beams with an assortment of symmetric boundary conditions and found the double beam mode shapes are similar to those of a single at identical boundary conditions and the amplitude of its for a double-beam system is doubled that of a single beam. He and Feng [5] developed a formula for the dynamic response of an elastically coupled multiple beam system under a moving oscillator using the finite sine-Fourier inverse transform.…”