Collision between bodies could occur in some applications of multibody systems. To include a precise and accurate representation of impact or contact in the equations of motion of a system, we must consider the deformation, shape, and possibly other features of the contacting bodies. However, in multibody dynamics, we need to combine all of these attributes into a very simple and therefore approximate representation. For such a simplified representation, two different approaches are mostly considered. In one approach, known as the piecewise or intermittent analysis, it is assumed that the impact results in an instantaneous change in the velocities. A classical method to determine the change in the velocities considers balancing the system’s momenta before and after an impact based on a given coefficient-of-restitution. In the other approach, known as the continuous analysis, it is assumed that the impact causes the contacting bodies to have local deformation in the contact region.

Either of the two methods is suitable for computational impact analysis in multibody dynamics. Either method requires accurate determination of the exact times of contact and loss of contact between impacting bodies. However, since the piecewise method requires special attention to several computational issues related to the discontinuities in the velocities, it is more common to apply the continuous method.

In the continuous analysis, it is assumed that when two bodies collide, although the contact period is very small, the change in the velocities is not discontinuous—the velocities vary continuously during the period of contact as the contacting bodies undergo local deformations. The deformation is represented as a logical linear or nonlinear spring-damper element that applies a pair of resistive forces on the two bodies during the period of contact. The parameters of this logical element that need to be determined are the effective mass, stiffness, damping coefficient, and the form of the nonlinearity.

In the past half-century, various models have been proposed that consider the force of the spring to be a nonlinear function of the deformation, where the stiffness could be adjusted based on the material properties of the contacting bodies. The models differ on whether the damping force, besides being a function of the deformation speed, should also be a function of the deformation or not. For these models, different formulas have been reported relating the damping coefficient to a desired value of the coefficient-of-restitution. Another parameter that needs to be determined for all of these models is the effective mass. This parameter can be determined for simple systems based on the kinetic energy of the bodies. However, for multibody systems containing kinematic joints or other constraints, and for systems having more than one degree-of-freedom, determination of the effective mass using the kinetic energy becomes more complicated.

In this presentation an overview of several continuous contact models is provided. Then simple formulas to compute the effective mass are derived based on the concept of impulse–momentum. The formulas are applicable to both constrained and unconstrained multibody equations of motion regardless of the number of degrees-of-freedom. Several examples are presented to clarify the use of these formulas.