Research On Time-varying Characteristics Of Energy Model For Axially Travelling Continuum

A nonlinear governing equation describing the transverse vibration of an axially travelling string with constant and time-varying length is considered.The Lagrangian function and a finite element model comprising elements described by quadratic shape functions are applied to obtain the governing equation,and a nonlinear coordinate transform is used for the nonlinear terms in this equation.A new hybrid Newmark-Beta/time varying degree of freedom method is proposed to improve calculation accuracy for the case of a large variation in the string length.The commonly used numerical solution method of axially moving string system is introduced in detail,and the simulation results are compared with MATLAB software.It is found that the new hybrid method is similar to the Newmark-Beta method for the case of small variations in the string length,whilst it is superior in accuracy to the latter for the case of large variations in the length.When these methods are applied to the forced vibration of a travelling string with constant and variable length,the numerical results are in good agreement with each other.The complex modal technique is used to solve the problem that the real mode technology can not be solved by the gyroscope term caused by the axial movement in the system equation of the transverse vibration system of the string system.The modal matrix of the lateral vibration system of the axial moving string is obtained,and then the corresponding modal displacement components and the modal velocity components are obtained by the modal decomposition.Using the finite element method and the energy definition,the modal kinetic energy and the modal potential energy of each unit are calculated respectively.Finally,all the unit energy is superimposed to obtain the total modal energy of the whole order of the whole string system.A phenomenon is observed where the free vibration energy leaks from one mode to the others in a travelling string.The higher the speed of translation,the more energy that is leaked into the modes close to the initially excited mode.