计及非线性变形的刚—柔耦合动力学建模

DYNAMICS MODELING FOR A RIGID-FLEXIBLE COUPLIING SYSTEM WITH NONLINEAR DEFORMATION FIELD

考虑变形产生的几何非线性效应对运动柔性梁的影响,在柔性梁的纵向、横向变形位移中均考虑横向弯曲以及轴向伸缩的耦合作用,从非线性应变—变形位移的原理出发,说明增加耦合变量后,剪应变为零,由此得出的变形模式更符合工程实际和简化需要。并采用有限元离散,通过Lagrange方程导出系统的动力学方程。最后对一带有中心体的柔性梁,在大范围运动为自由和大范围运动为已知两种情况下进行仿真计算,结果表明,在结构有初始变形的情况下,仅在纵向变形中计及变形二次耦合量的一次动力学模型,与考虑完全几何非线性变形的文中模型具有一定的差异。

A moving flexible beam, which incorporates the effect of the geometrically non-linear kinematics of deformation, is investigated. Considering the second-order coupling terms of deformation in the longitudinal and transversal deflections, the exact none-linear strain-displacement relations for a beam element is described. The shearing strain formulated is zero. So, it is reasonable to use geometrically nonlinear deformation fields to demonstrate and simplify a flexible beam undergoing large overall motions. Then, finite element shape functions of a beam element and Lagrange＇ s equations are employed for deriving the coupling dynamical formulations. A model consisting of a rotating base and a flexible beam is presented. Considering two cases： 1） the beam with an unprescribed base movement; 2） the beam with a prescribed base movement, a numerical example is given in the end. The simulation illustrates that geometrically nonlinear dynamical model are different to one-order coupling dynamical model which only considers the second coupling terms of deformation in the longitudinal deflection, especially, when the model has initial deformation. For case 1, the joint trajectory of the beam is unknown, because of the coupling effect of rotation and deformation, the difference between two models is not distinct. For case 2, the joint trajectory of the beam is known, the simulation illustrates that the beam tip deflection of geometrically nonlinear dynamical model in the present model is larger than that of one-order coupling dynamical model, and when the movement reaches the steady state, the frequency of the present model is lower than that of the one-order model. So, when the more coupling terms of deformation are added to the longitudinal and transversal deformation field, a ＇ softer beam＇ can be obtained.