| Axially travelling string system widely used in engineering equipment is concerned with the vibration problem,especially the transverse vibration.The transverse vibration of the system is diverse for different boundary conditions.In this paper,the transverse free vibration of the axially travelling string system is studied for four different typical boundary conditions.Firstly,the physical model of axially travelling string system with four kinds of boundary conditions,such as fixed-fixed,fixed-free,damping-fixed and fixed-moving are established respectively.The mathematical models of the system are also established by two different methods.The first is the finite element equation based on Lagrange equation and the shape function;the second is the partial differential equation which is derived using the Hamilton's principle.Secondly,in view of the traditional finite element method for the transverse vibration of the high-speed moving string calculation error and the divergence problem,a calculation method for the transverse vibration of axially moving string system with finite length is presented based on traveling wave in different boundary reflection superposition technology.Thirdly,combined with initial and different boundary conditions,expression of reflected wave can be obtained.Transverse free vibration is equivalent to the superposition of different speed of the travelling wave and reflected wave in order to derive its theoretical formula.According to the motion regularity of the traveling wave,the motion period of the axially travelling string system with constant length and varying length are obtained respectively.Finally,the periodicity of energy variation is analyzed with the energy reflection coefficient as the starting point,and the energy variation law of the system with different boundary conditions and the energy conversion mechanism of the rope passing through the boundary are also analyzed. |
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