Development of simplified dynamic models using optimization: Application to crushed tubes Original Research Article

Optimization formulations and algorithms for development and verification of simplified dynamic models for dissipative dynamic systems are presented using axially crushed tubes as examples. The crushing of a tube is simplified to a single degree of freedom lumped mass model whose motion is resisted by nonlinear force elements. The crushing behavior is captured in the nonlinear force–displacement (F–D) curve for the force elements. The shape of the force curve is interpreted as a progressive elastoplastic-collapse behavior. In the first proposed formulation, the F–D curve is mathematically represented using several elastoplastic-collapse elements that are defined with piecewise linear functions of the displacement. Other representations for the F–D curve are also presented using splines and the moving least squares approximation. To identify the F–D curve for a crushed tube, the error between the acceleration calculated using the mathematical representation of the force element and the given numerical acceleration is minimized with constraints on the response of the system. Optimization problems defined for simplified model development and redesign of an automotive component are solved and the results are discussed. The developed optimization formulations are general where any error function can be minimized subject to constraints on the dynamic response and optimization variables. The formulations can be useful in other practical applications that require verification of analytical models using the experimental data.