The influence of various boundary conditions on dynamic stability of a beam-moving mass system

In this paper, the effect of different boundary conditions on dynamic stability of a beam located on a viscoelastic medium stimulated by moving masses and periodic axial force is studied. Partial differential equations governing the system are derived using Hamilton's method and based on Euler-Bernoulli beam theory. Then, equations are converted into a set of ordinary differential equation with time-varying coefficients using Galerkin method along with trigonometric shape functions. The time-varying position of moving loads causes these time-varying coefficients in the governing equations. By applying Floquet's theory to the obtained equations, the conditions of parametric resonance are analyzed for different values of mass and velocity of passing loads. The results obtained from this research show that the stiffness and viscosity of the elastic medium have positive effects on the stability of the beam under moving and fluctuating axial loads. So, with a suitable choice for these values in practical applications, it is possible to prevent unexpected vibrations of the structure. In addition, the use of fixed supports for the two ends of the beam exposed to the mentioned loadings has high reliability in the discussion of dynamic stability.