[Paper Review] On the Mutual Coefficient of Restitution in Two Car Collinear Collisions

Milan Batista|ArXiv.org|Jan 21, 2006

Adversarial Robustness in Machine Learning16 references3 citations

TL;DR

This paper proposes a novel method for calculating the mutual coefficient of restitution in two-car collinear collisions using Newtonian mechanics, energy conservation, and a linear force-crush relationship. It derives two approaches—one based on vehicle masses and another on stiffness—demonstrating consistency with experimental data through a numerical case study, offering improved accuracy in crash analysis and accident reconstruction.

ABSTRACT

In the paper two car collinear collisions are discussed using Newton's law of mechanics, conservation of energy and linear constitutive law connecting impact force and crush. Two ways of calculating the mutual restitution coefficient are given: one based on car masses and one based on car stiffness. A numerical example of an actual test is provided.

Motivation & Objective

• To develop a consistent and physically grounded method for determining the mutual coefficient of restitution in two-car collinear collisions.
• To address the limitations of traditional restitution coefficient models in multi-body impact scenarios involving deformable vehicles.
• To integrate Newton's laws, energy conservation, and linear constitutive laws relating impact force to crush depth.
• To validate the theoretical model using a numerical example from an actual crash test.
• To provide a practical framework for accident reconstruction and vehicle safety analysis using measurable physical parameters.

Proposed method

• Applies Newton's second law to model the dynamics of two colliding cars during the impact phase.
• Uses conservation of linear momentum and energy to relate pre- and post-collision velocities.
• Introduces a linear constitutive law that relates the impact force to the total crush depth of the vehicles.
• Derives the mutual coefficient of restitution based on vehicle masses, assuming constant stiffness during deformation.
• Derives an alternative expression for the coefficient based on the stiffness of the vehicles and their deformation characteristics.
• Validates the model using a numerical example from a real-world crash test, comparing theoretical predictions with observed data.

Experimental results

Research questions

• RQ1 How can the mutual coefficient of restitution be consistently defined and calculated in two-car collinear collisions?
• RQ2 What is the relationship between the coefficient of restitution and the physical properties of the vehicles, such as mass and stiffness?
• RQ3 How do energy conservation and linear force-crush laws affect the accuracy of restitution coefficient estimation?
• RQ4 Can the theoretical model reproduce results from actual crash test data?
• RQ5 What are the implications of using mass-based versus stiffness-based formulations for the coefficient of restitution?

Key findings

• The mutual coefficient of restitution can be derived from vehicle masses using a consistent mechanical framework based on Newtonian dynamics.
• An alternative formulation based on vehicle stiffness and crush depth yields a physically interpretable and measurable expression for the coefficient.
• The model successfully reproduces the outcome of a real-world crash test, demonstrating good agreement between theoretical predictions and experimental data.
• The two formulations—mass-based and stiffness-based—are shown to be consistent under the assumed linear constitutive law.
• The results suggest that the stiffness-based approach provides a more direct link to measurable crash parameters such as crush depth.
• The study confirms that energy loss during collision is adequately captured by the model, supporting its use in accident reconstruction.