Abstract
Analysis of impact problem in the presence of any tangential component of impact velocity requires a friction model capable of correct detection of the impact modes such as sliding, sticking, and reverse sliding. A survery of literature has shown that studies on the impact analysis of multibody systems have either been limited to the direct impact type with only a normal component of impact velocity (no frictional effect) or the ones that include friction have shown energy gains in the results due to the inherent problem in the use of Newton’s hypothesis. This paper presents a formulation for the analysis of impact problems with friction in constrained multibody mechanical systems. The formulation recognizes the correct mode of impact, i.e., sliding, sticking, and reverse sliding. The Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of Newton’s hypothesis are avoided. The formulation is developed by using a canonical form of the system equation of motion using joint coordinates and joint momenta. The use of canonical formulation is a natural way of balancing the momenta for impact problems. The joint coordinates reduces the equations of motion to a minimal set, and eliminate the complications arised from the kinematic constraint equations. The canonical form of equations are solved for the change in joint momenta using Routh’s graphical method. The velocity jumps are then calculated balancing the accumulated momenta of the system during the impact process. The impact cases are classified based on the pre-impact positions and velocities, and mass properties of the impacting systems. Analytical expressions for normal and tangential impulse are derived for each impact case. The classical problem of impact of a falling rod with the ground (a single object impact) is solved with the developed formulation, and the results are compared and verified by the solution from other studies. Another classical problem of a double pendulum striking the ground (a multibody impact) is also solved. The results obtained for the double pendulum problem confirms that the energy gain in impact analysis can be avoided by considering the correct mode of impact and using Poisson’s instead of Newton’s hypothesis.
Modeling Flexible Bodies in Multibody Systems in Joint-Coordinates Formulation Using Spatial Algebra