Development and Comparison of Laplace Domain Models for Nonslender Beams and Application to a Half-Car Model With Flexible Body

Abstract

Math models of flexible dynamic systems have been the subject of research and development for many years. One area of interest is exact Laplace domain solutions to the differential equations that describe the linear elastic deformation of idealized structures. These solutions can be compared to and complement finite order models such as state-space and finite element models. Halevi (2005, “Control of Flexible Structures Governed by the Wave Equation Using Infinite Dimensional Transfer Functions,” ASME J. Dyn. Syst., Meas., Control, 127(4), pp. 579–588) presented a Laplace domain solution for a finite length rod in torsion governed by a second-order wave equation. Van Auken (2012, “Development and Comparison of Laplace Domain and State-Space Models of a Half-Car With Flexible Body (ESDA2010–24518),” ASME J. Dyn. Syst., Meas., Control, 134(6), p. 061013) then used a similar approach to derive a Laplace domain solution for the transverse bending of an undamped uniform slender beam based on the fourth-order Euler–Bernoulli equation, where it was assumed that rotary inertia and shear effects were negligible. This paper presents a new exact Laplace domain solution to the Timoshenko model for an undamped uniform nonslender beam that accounts for rotary inertia and shear effects. Example models based on the exact Laplace domain solution are compared to finite element models and to slender beam models in order to illustrate the agreement and differences between the methods and models. The method is then applied to an example model of a half-car with a flexible body.