This paper presents a consistent formulation of the generalized-$α$ time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller dynamics, and kinematic constraints. The theoretical background relies on the analogy with linear multistep formulas, which leads to elegant results related to consistency, order conditions for constant and variable step-size methods, as well as global convergence. Those results are illustrated for a controlled spring-mass system, and the method is also applied for the simulation of a vehicle semi-active suspension.

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