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Research Papers

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

[+] Author and Article Information
R. Michael Van Auken

Dynamic Research, Inc.,
355 Van Ness Avenue, Suite 200,
Torrance, CA 90501
e-mail: rmv@dynres.com

Han et al. [12] Eq. (65) and Thomson [14] Eqs. (9.6)–(7).

Meirovitch [13] Eq. (5.1).

See Refs. [11] Eqs. (2)–(5) and [12] Eq. (44).

See Mierovitch [13], p. 347.

The signs for $σk$ were chosen herein as $σ0=σ3=1$ and $σ1=σ2=-1$ in order to be consistent with the signs in Ref. [11] for the special case were $a=0$ and $c=0$.

See Eqs. (31)–(45) in Ref. [11].

See Eqs. (14)–(22) in Ref. [11].

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received July 1, 2014; final manuscript received December 30, 2014; published online February 9, 2015. Assoc. Editor: Junmin Wang.

J. Dyn. Sys., Meas., Control 137(7), 071001 (Jul 01, 2015) (11 pages) Paper No: DS-14-1271; doi: 10.1115/1.4029528 History: Received July 01, 2014; Revised December 30, 2014; Online February 09, 2015

## 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.

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## Figures

Fig. 1

Finite uniform beam

Fig. 2

Frequency response of an exemplar clamped–free beam (x0 = 0.6 m, x = 0.4 m, L = 1 m, κ = 0.1097)

Fig. 3

Sensitivity of a hypothetical exemplar clamped–free beam frequency response to the slenderness index κ (x0 = 0.6 m, L = 1 m, EI = 2.34 × 107 N, ν = 0.29)

Fig. 4

Forced response of exemplar clamped–free beam at 31,600 rad/s (x0 = 0.6 m, L = 1 m, κ = 0.1097)

Fig. 5

Frequency response of an exemplar pinned–pinned beam (x0 = 0.6 m, x = 0.4 m, L = 1 m, κ = 0.1097)

Fig. 6

Half-car model

Fig. 7

Comparison of z··sm/wf frequency responses for the infinite dimensional, finite element, and rigid body models for a half-car with slender flexible body (κ = 0.0055)

Fig. 8

Comparison of z··sm/wf frequency responses for the infinite dimensional, finite element, and rigid body models for a half-car with nonslender flexible body (κ = 0.0409)

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