Application of the two-loop procedure in multibody dynamics with contact and constraint

This paper focuses on the performance of the two two-loop implicit sparse matrix numerical integration (TLISMNI) methods as the numerical solution of index-3 differential algebraic equations (DAEs) of motion arising in stiff multibody dynamics with contact and constraint. The original TLISMNI method proposed for the first time by Shabana and Hussein uses the Newmark method or Hilber/Hughes/Taylor (HHT) method as the implicit integration algorithm [TLISMNI (Newmark/HHT) method]. These two integration algorithms consider the accelerations and Lagrange multipliers as basic unknowns and their numerical accuracy is no more than order 2. In order to have a higher integral precision, we propose a new two-loop implicit procedure based on an extended backward differentiation formula scheme [TLISMNI (EBDF) method] which considers coordinates and velocities as basic unknowns. The whole structure of this proposed method is different from the traditional one due to the different unknowns. Furthermore, we apply these two kinds of methods, TLISMNI (Newmark/HHT) method and TLISMNI (BDF/EBDF) method, to the contact and constraint problems of flexible multibody dynamics while few studies of TLISMNI methods have been done on these non-smooth problems. The Baumgarte method and HHT-α method are also used to solve the same contact and constraint problem, and the advantages and disadvantages of all these methods above are compared in this study. Results show that the proposed method only needs to perform less iterations to satisfy the same tolerance of error than the TLISMNI (HHT) method does. In some cases the Baumgarte method and HHT-α method cannot give reasonable results while the TLISMNI methods perform well when the contact and constraint problems are involved. Moreover, the numerical experiment also indicates that the numerical damping property of the HHT method employed in both the TLISMNI method and the GGL formulation (used in the HHT-α method) can behave in a different way to damp out high frequency oscillations induced by impact.