Descriptor Impulsive Modeling of a Constrained Mass–Spring System with Impact: Consistency-Preserving and Consistency-Violating Resets

This preprint presents a descriptor impulsive formulation of a constrained mass–spring system subject to rigid impact. The mechanical model retains the spring force as an algebraic state variable, so Hooke’s law defines the admissible descriptor constraint manifold. The study compares singular-system, reduced-order regular, impulsive-system, and descriptor impulsive-system formulations, and distinguishes between consistency-preserving and consistency-violating impact resets. Newton’s restitution law defines the mechanical velocity jump, while descriptor consistency requires the post-impact state to remain compatible with the constitutive relation between displacement and spring force. Numerical simulations, constraint-residual analysis, and geometric representations are used to demonstrate preservation and violation of the descriptor constraint manifold. The deposited archive contains the final manuscript, Figures 1–5, Python scripts reproducing the schematic and numerical results, a README file, and software requirements.