Equations of Motion of Mechanical Systems with Switchable Constraints

Abstract

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted.