An improved implicit method for mechanical systems with set-valued friction

  • Runsen Zhang
  • Yang Yu
  • Qi WangEmail author
  • Qingyun Wang


The computational procedures of higher-order implicit integrators for mechanical systems with friction are provided in this paper. The dynamic equations are established using the augmented Lagrangian formulation, and set-valued friction forces are described by projection functions. To reduce the accuracy loss caused by event transitions and to eliminate spurious oscillations in the acceleration, a new robust event-driven scheme, which accurately detects event transitions and corrects the friction forces and accelerations at switching points, is proposed. The numerical performance of the proposed scheme is demonstrated by solving several benchmark problems. Numerical results show that the newly developed scheme can approximately achieve second-order accuracy, and they are more accurate than the classical Moreau time-stepping scheme under close computational efforts. Finally, a slider-crank system is simulated to prove the validity of the developed method for nonlinear mechanical systems with friction.


Set-valued friction Implicit integrators Projection function Spurious oscillations 



We thank LetPub ( for its linguistic assistance during the preparation of this manuscript and we are grateful for the support of National Natural Science Foundation of China (11772021, 11702009).


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Aeronautics Science and EngineeringBeihang UniversityBeijingChina

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