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

Modeling and simulating the dynamics of the multibodysystems with bilateral constraints and dry friction are important inmechanical system and robotics. For smooth bilateral constraints, it is easyto solve the dynamical equations numerically. The dynamic equations of themultibody systems with the friction of constraint are the discontinuousdifferential-algebraic equations (DAE) and the equations cannot be expressedas being linear with respect to the generalized accelerations and theLagrange multipliers directly. In the present paper, modeling of planarmulti-rigid-body system with multi-friction and fixed-bilateral constraintsis proposed. It is assumed that the system has sliding joints with Coulomb'sdry friction and smooth hinge joints, while the sliding joints move alongthe fixed-slots. Firstly, the motion equations of the system are derivedfrom Lagrange's equations of the first kind in Cartesian coordinate system,and constraint equations are expressed by local approach. A one-to-one mapbetween the normal constraint forces and the Lagrange multipliers isestablished to analysis and compute the friction forces. Secondly, using theconstraint equations and the principle of virtual work, the generalizedforces of the friction forces are derived in the matrix form. The absolutevalue of Lagrange multiplier | \lambda | in the motion equationsis given as \lambda \rm sgn ( \lambda ) by sign function. Therefore,the sign function, \rm sgn(\lambda ), \rm sgn(\dot s) and\rm sgn(\ddot s),included in the motion equations, correspond to Lagrange multipliers, thevelocity and tangential acceleration of the slider, respectively. Thirdly,the DAE are transformed into ordinary differential equations (ODE) by meansof the augmentation approach. An improved trial-and-error method is proposedaccording to the characteristics of the piecewise smooth of thesystems, which can improve the efficiency of computation. Finally, anexample of one degree of freedom mechanism is given by improvedtrial-and-error method and R-K method.