Nonlinear methods of vehicle velocity determination based on inverse systems and tensor products of Legendre polynomials in compact car class

• Published in: Forensic science international Vol. 295; pp. 19 - 29
• Main Authors: Kubiak, P., Mierzejewska, P., Krukowski, M.
• Format: Journal Article
• Language: English
• Published: Ireland Elsevier B.V 01.02.2019; Elsevier Limited
• Subjects: Car accidents; Coefficient of restitution; Collision Deformation classification; Compact cars; Crashworthiness; Deformation ratio; Energy; Equivalent energy speed; Forensic sciences; Highway safety; Mathematical models; Methods; Polynomials; Tensors; Traffic accidents & safety; Traffic safety; Vehicle deformation index; Vehicle speed; Velocity; Car accidents; Collision Deformation classification; Vehicle speed; Vehicle deformation index; Deformation ratio; Equivalent energy speed; Coefficient of restitution
• ISSN: 0379-0738; 1872-6283
• DOI: 10.1016/j.forsciint.2018.11.023

•A nonlinear approach to finding velocity before car crash is proposed.•Approach relies on nonlinear dependence between precrash velocity, deformation coefficient and vehicle mass.•Approach shows great improvement in comparison to linear methods. The article presents research on the methods of crash velocity determination based on the Compact car class. The database used in the research is provided by the NHTSA (National Highway Traffic Safety Administration) and includes numerous frontal crash tests, which allowed the determination of the mathematical model parameters. Two methods are presented that enable the determination of vehicle velocity before the collision. The first researched method is the so-called inversed system method and is based on the assumption that the relationship between bk coefficient Cs is an inverse function. The second line of research focuses on the tensor product method, which is grounded in the Legendre polynomials, orthogonal on the interval [–1, 1] (Axler, 1997; Cheney and Kincaid, 2002). The article presents the calculation algorithm for both cases and the results with reference to the NHTSA database (Sharma et al., 2007; Siddall and Day, 1996). The application of the least square method provides more precise results in both cases than in previously researched solutions, with a slight advantage of the tensor product method. The obtained mean relative error of the velocity determination using the inverse system method is approximately 16,22% for the linear model and 10,58% for the nonlinear model. In the case of the tensor product method the errors for linear and nonlinear models are respectively 6,74% and 6,3%.