A numerical method for the dynamics of planar rigid multibody system with friction-affected translational joints based on HLCP

In the study of the multibody dynamics problems in the area of aerospace, machinery and vehicles, the influence of friction as well as other non-smooth factors on the system has to be taken into account. In order to reduce the difference between numerical simulation and experimental results, friction models should be embedded in the dynamical equations of multibody systems. Non-smooth multibody system dynamics with friction becomes one of the hot issues in the research of multibody system dynamics in recent years. In this paper, we present the numerical method for the dynamical equations of planar rigid multibody system with translational joints in which the friction is considered. Each translational joint in the system is composed of a slider and a slot. The slider is treated as a particle and the slot is treated as a bilateral constraint. The constraint equations of the sliders of translation joints in the system can be expressed as the functions of generalized coordinates. Using Coulomb's friction law and the first kind of Lagrange equation, we obtain the dynamical equations of the system. These equations include the absolute values of Lagrange multipliers which are the magnitude of normal constraint forces. In this case, the Lagrange multipliers in the dynamical equations are not always positive. The key problem of solving these equations is how to determine the transitions of the normal constraint forces of bilateral constraints as well as the stick-slip transitions of the transitional joints in the system. To solve the above problem, firstly, we establish the complementary relationship between two normal constraint forces of bilateral constraints by using the property of translational joint. Through the complementary relationship, the absolute values of the Lagrange multipliers can be expressed as the linear combination of the magnitude of normal constraint forces in positive and negative directions, which will redound to computing the generalized forces of the tangential friction forces. Secondly, we present the complementary formulation for Coulomb's friction law by using the concept of the friction saturation in positive and negative directions, and then, using event-driven method, we formulate and solve the state transition problem of normal forces of bilateral constraints and sticking-slipping as a horizontal linear complementarity problem (HLCP), and use the pivoting algorithm to solve the horizontal linear complementary problem above. Finally, we consider the planar rigid multibody system with two translational joints and two degrees of freedom as a demonstrative application example, and present the numerical results. The method presented in this paper can be used to simulate the dynamical equations of the planar rigid multibody system which has n degree of freedom and n* translational joints with friction numerically.