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In this paper, dynamic stability of a simply supported beam excited by a sequence of moving masses is investigated by preserving nonlinear terms in the analysis. This type of loading is important in problems such as motion of vehicles on bridges, high-speed transportation on rails, machining processes, conveying pipelines and barrel dynamics, so its investigation is important from practical viewpoint. The intermittent loading across the beam results in a periodic time-varying equation system. The effects of convective mass acceleration beside large deformation beam theory are both taken into account in the derivation of governing equations which is performed through adopting Hamilton's principle for mass-varying systems. In order to deal with the coupling between longitudinal and transversal deflections, the inextensibility assumption is implicitly introduced into the Hamiltonian formulation and an appropriate interpretation is presented to maintain this approximation reasonable. The method of multiple scales is implemented to find the domains of stability and instability of the problem in a parameter space. The results of applying the method forecast a qualitative change in beam behavior due to nonlinear terms. Results of different numerical simulations show the validity of the analytical approach obtained by the applied perturbation method.

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In this paper, instability due to occurrence of parametric resonance in transverse vibration of a rectangular plate on an elastic foundation under passage of continuous series of moving masses is examined as a model of bridge-moving loads interaction. The extended Hamilton’s principle is employed to derive the partial differential equation of motion. Subsequently, the governing partial differential equation is transformed into a set of ordinary differential equations by the Galerkin procedure. Considering local, Coriolis and centripetal acceleration components of the moving masses in the analysis leads to appearance of time-varying mass, damping and stiffness matrices in the coefficients of the governing equation. The passage of continuous series of moving masses along the rectilinear path results in a parametrically excited system with periodic coefficients. Applying incremental harmonic balance method as a semi-analytical method to the governing equations, stability of the system is investigated for a wide range of masses and velocities of the passing loads and different boundary conditions of the plate. Moreover, effect of the foundation stiffness on stability of the plate is examined. Results indicate that using clamped supports for the edges of entrance and departure of masses over the plate’s surface leads to formation of an instability tongue in the parameters plane which does not appear for the case of using simply supports. Also, it is observed that critical velocities of the moving masses will be increased by escalation the foundation stiffness. Numerical simulations confirm the accuracy of the semi-analytical results.

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