Mechanical systems with time-varying topology appear frequently in natural or human-made artificial systems. The nature of topology transitions is a key characteristic in the functioning of such systems. In this paper, a concept to decouple kinematic and kinetic quantities at the time of topology transition is used. This approach is based on the use of impulsive bilateral constraints and it is a useful tool for the analysis of energy redistribution and velocity change when these constraints are suddenly established. Two examples of systems with time-varying topology are analyzed: a bipedal walking system and a dual-pantograph robotic prototype making contact with a stiff environment. Detailed numerical and experimental analysis to gain insight into the dynamics and energetics of topology transitions is presented.