Study On Some Issues Of The New Version Of Transfer Matrix Method For Mulitbody Systems

The transfer matrix method for multibody system dynamics(MSTMM)is a relatively new method of multibody system dynamics(MSD)proposed and developed in the recent twenty years.MSTMM mainly consists of three parts,wherein the first part is the transfer matrix method for linear multibody system(LMSTMM),which applies for the dynamics of linear multibody systems,the second part is the discrete time transfer matrix method for multibody system dynamics(MSDTTMM),which applies for the dynamics of general multibody systems undergoing nonlinear large motion,and the third part is the new version of transfer matrix method for multibody system dynamics,in which the accelerations and internal forces are collected as the new state vectors.When dealing with a multibody system dynamics problem by using the MSTMM,the dimension of the matrices involved in the overall transfer equation is independent from the number of degrees of freedom of the multibody system.The computational efficiency of MSTMM could be very high,which may contribute to the optimal design of a complex multibody system.In the MSDTTMM,the state variables in the state vector of an input or output end of an element consists of displacements,rotational angles,internal torques and internal forces.The linear relationship between the state vectors of the input and output end of an element,namely the transfer equation of the element,could be deduced by the linearization of the kinematics and kinetic equations of the element based on a numerical integration procedure.In 2012,a new version of MSTMM was proposed by the author's supervisor and the author as a co-author,where the translational and angular accelerations as well as the internal forces and torques were taken as new state variables.The key advantage of the new version of MSTMM compared with MSDTTMM lies in that the time-based linearizations will be not needed any more when deducing the transfer equations of elements or the transfer equation of the whole system.For a multi-rigid-body system dynamics problem,it has been demonstrated that the new version of MSTMM belongs to the accurate methods of MSD in principle.The main idea and the solving strategy of the new version of the transfer metrix method of multibody system are introduced in detail in this paper.The transfer equations and the cooresponding transfer matrices of rigid body elements,hinge elements(such as a revelute hinge,a prismatic hinge,as well as a ball and socket hinge)are deduced based on the Newton's law of motion and Euler's rotation theorem.By using the principle of virtual power,floating frame of reference method and the modal synthesis method,the transfer equation of a flexible body undergoing large motion and relatively small deformation are deduced.The dynamics equation,as well as the transfer equation,of a curved beam is deduced in this paper,in which the axial shortening effect of the curved beam is taken into consideration.The stress stiffening effect and the rotational softening effect of a curved beam are analysed by using the finite element method of floating frame of reference,where an additional stress stiffening matrix derived from the nonlinear relationship between deformation and strain of the curved beam is taken into account.In order to reduce further the dimension of the matrices involed in the new version of MSTMM and improve the stabilization of the version of MSTMM,the Riccati transfer matrix method for a multibody system with a tree topology is developed based the Riccati transfer matrix method for a chain multibody system.