Modeling and simulation of multi-body systems with multi-friction and fixed bilateral constraint

Modeling and simulating the dynamics of the multibody systems with bilateral constraints and dry friction are important in mechanical system and robotics. For smooth bilateral constraints, it is easy to solve the dynamical equations numerically. The dynamic equations of the multibody systems with the friction of constraint are the discontinuous differential-algebraic equations (DAE) and the equations cannot be expressed as being linear with respect to the generalized accelerations and the Lagrange multipliers directly. In the present paper, modeling of planar multi-rigid-body system with multi-friction and fixed-bilateral constraints is proposed. It is assumed that the system has sliding joints with Coulomb's dry friction and smooth hinge joints, while the sliding joints move along the fixed-slots. Firstly, the motion equations of the system are derived from Lagrange's equations of the first kind in Cartesian coordinate system, and constraint equations are expressed by local approach. A one-to-one map between the normal constraint forces and the Lagrange multipliers is established to analysis and compute the friction forces. Secondly, using the constraint equations and the principle of virtual work, the generalized forces of the friction forces are derived in the matrix form. The absolute value of Lagrange multiplier |λ| in the motion equations is given as λsgn(λ) by sign function. Therefore, the sign function, sgn(λ), sgn(ṡ) and sgn(s̈), included in the motion equations, correspond to Lagrange multipliers, the velocity and tangential acceleration of the slider, respectively. Thirdly, the DAE are transformed into ordinary differential equations (ODE) by means of the augmentation approach. An improved trial-and-error method is proposed according to the characteristics of the piecewise smooth of the systems, which can improve the efficiency of computation. Finally, an example of one degree of freedom mechanism is given by improved trial-and-error method and R-K method.