Dynamic modeling and simulation for the rigid flexible coupling system with a non-tip payload in non-inertial coordinate system

The rigid flexible coupling system with a payload at non-tip position of the flexible beam is studied in this paper. Using the theory about mechanics problems in a non-inertial coordinate system and the subsystem modeling principle, the dynamic equations of the rigid flexible coupling system with dynamic stiffening are established via material mechanics and the angular momentum principle. This paper elucidates that dynamic stiffening is produced by the coupling effect of the centrifugal inertial load distributed on the beam and the transverse vibration deformation of the beam. First, the payload at non-tip position is incorporated into the continuous dynamic equations of the system by means of the Dirac function and the Heaviside function. Then, the constrained modes of the flexible beam are analyzed. According to the boundary conditions of the constrained modes, a reasonable position interval of the payload is proposed. It corrects the inaccurate understandings in some papers. Finally, based on the conclusions of orthogonalization about the normal constrained modes, the finite dimensional state space model suitable for controller design is obtained. In addition, in order to make the finite dimensional dynamic model established more practical, based on the common constrained modes, the dimensionless variables and parameters are introduced, and the dimensionless finite dimensional dynamic model of the system is derived. The numerical simulation results show that: dynamic stiffening is included in the first-order dynamic model established in this paper, therefore, it can indicate the dynamic responses of the rigid flexible coupling system with large overall motion accurately; the payload has a softening effect on the dynamic behaviors of the flexible beam, and the effect would be more obvious when the payload has a larger rotary inertia or a larger mass, or lies closer to the tip of the beam.