粘弹性体的多体系统动力学建模

根据材料各向同性假设和分数阶导数的Grünwald定义,分析和研究了三维分数阶导数本构方程描述的粘弹性体的特性。基于虚功原理和连续介质力学理论推导了含分数阶导数本构的粘弹性体的多体系统动力学方程。由于在推导过程中采用完全非线性有限元法,因此该动力学方程能准确描述多体系统中柔性体的大变形和任意的刚体运动。采用隐式的BDF(backwards differentiation formulation)积分格式和Newton-Raphson迭代算法对动力学方程进行求解。最后通过数值算例对比研究了数值积分步长、分数阶导数的指数与截断级数等参数对系统动力学行为的影响规律。结果表明:对于粘弹性耗散多体系统,步长不应太大;分数指数越大,松弛时间越小,则材料阻尼越大;应合理地选择截断级数,这样既反映了材料特性又节省了计算时间。

Dynamic modeling of a viscoelastic body in a multibody system

The properties of viscoelastic materials described by the fractional derivative constitutive were analyzed using the isotropic hypothesis and the Grünwald definition for the approximation of the fractional derivative.The dynamics equations of the multibody system including the fractional constitutive are derived based on the virtual work principle and the continuum mechanics theory.The total nonlinear finite element method is used to solve the motion equations,so large deformations and arbitrary rigid body movement of the flexible solid body can be accurately predicted.The implicit BDF(backwards differentiation formulation) time integration scheme is used with the Newton-Raphson algorithm to solve the equations of motion.Several numerical examples are presented to analyze the effects of the time step,the fractional order and the truncation number of the Grünwald series on the system’s dynamic behavior.The results show that the integration step should not be too large for a multibody system with visoelatic dissipation and that the material damping increases with increasing fractional index and decreasing relaxation time.The truncated series should be reasonably selected to reflect the material’s property and reduce the calculating time.