Dynamic Behavior Analysis Of Beam-mass System

With the rapid development of civil engineering and mechanical engineering, it is important to study the vibration and stability of a beam carrying masses system, for both concentrated and axial moving masses. Based on Bernoulli-Euler beam theory and Timoshenko beam theory, the vibration characteristics, stability and nonlinear dynamics of a beam carrying masses are investigated respectively. Research include following sections: the transverse vibration, bending-torsional coupling vibration and nonlinear dynamic response of Bernoulli-Euler beam carrying single or multiple moving mass; the transverse vibration of Timoshenko beam carrying single moving mass. Considering the inertia effect of the masses, this work establishes effective dynamic equation and supplies as a reliable theory base for the analysis in engineering.Considering the inertia effect of single moving mass and concentrated dashpot, the linear transverse vibration of Bernoulli-Euler beam is studied and a general equation of the motion is derived under flexible constraints. Based on the motion equation, the natural frequencies of the beam with various boundary conditions are calculated. It shows that the frequencies of beam are affected by axial moving velocity, acceleration of single mass and mass ratio. Besides, under asymmetric constraints, certain axial velocity of moving mass leads to rapid change of beam frequency.Subjected to single moving mass and axial harmonic excitation, the nonlinear transverse vibration of Bernoulli-Euler beam is investigated. Considering the inertia effect of moving mass, the motion equation with the fifth-order nonlinear term is derived. Based on Galerkin method, the effect of Galerkin modal truncation on the dynamic behavior for the system is studied by various numerical examples. Numerical results show that, the disorder may occur at high order modes. For single mode model, the dynamic equation is solved by multi-scales method. In addition, the stability and local bifurcation of the system are analyzed for 1/2 sub harmonic resonance, which indicate simply-supported flexible beam has different transverse vibration in various system parameter regions.In engineering, the misalignment of mass centers and the shear centers may exist. The linear bending-torsional coupling vibration of Bernoulli-Euler beam carrying single moving mass is studied. Considering the inertia effect of single moving mass, the governing equations of bending-torsional coupling vibration are established. Meanwhile, the dynamic behavior of the beam by external torque is discussed. Numerical analysis shows, the mass per unit length of beam and the axial velocity of moving mass have significant influence on beam vibration; the frequencies of the beam are greatly affected by the moment of inertia.Transverse vibration of multi-channel beam carrying several toing-and-froing masses is analyzed. Considering the inertial effect of masses, the partial differential vibration equation of system is derived and simplified by using mode analysis and Galerkin integral. In numerical simulation examples, the motion forms of the beam are detected, the effect on the dynamic response of the system is investigated by various parameters as well. Furthermore, this work provides a theory base for multiple-lane high-speed bridge problem.According to Timoshenko beam theory, the governing equation of transverse vibration of the beam including shear deformation and rotator inertia is obtained. Considering the inertia effect of single moving mass, the effect of Galerkin modal truncation on the dynamic behavior of beam structure is investigated.For slender beam systems, quite similar behavior was observed in Bernoulli-Euler beam and Timoshenko beam theory. For short and wide beam, because of large effect of shear deformation and rotary inertia on dynamic behavior of beam, Timoshenko beam theory predicts smaller beam vibration frequencies than Bernoulli-Euler beam theory does.Comparing with Bernoulli-Euler beam, quite similar behavior was observed on these two slender beam systems. For short and wide beam, because of the large effect on shear deformation and rotary inertia of short and wide beam, the vibration frequencies of beam, which predict by Timoshenko beam theory, are smaller than the former.