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Research Papers

# Development and Comparison of Laplace Domain and State-Space Models of a Half-Car With Flexible Body (ESDA2010–24518)

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

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

See Mierovitch [12] p. 347.

The correlation and time delay of the front and rear wheel road disturbances does not affect the full-order and infinite dimensional half-car models presented herein, but can effect (1) model reduction (e.g., [5] and [21]), and (2) control law synthesis. The model reduction examples in this paper are based on the assumption that the front and rear road disturbances are uncorrelated in order to simplify the presentation herein.

The exact G(s) and H(s) transfer functions can also be computed using transfer matrices as described in Refs. [6-8]. This can be accomplished by using three beam segments connected in series at x and x0. This approach requires the simultaneous solution to 12 equations, compared to the new method which requires the simultaneous solution to six equations (i.e., Eqs. 14,15). The differences in the resulting transfer functions using the two methods is very small ($J2<10-7, H∞<10-4$ ) and related to the computer numerical accuracy (EPS = 2.2 × 10 −16 ).

This model does not include the damping adjustment described in Ref. [5]. This is to be consistent with the undamped flexible body transfer functions in the infinite dimensional model. Therefore, the only source of damping in the example used in this paper is due to the suspension dampers, resulting in damping ratios less than 0.04.

The peak errors reported herein were determined using numerical methods and, therefore, may underestimate the true peak error values.

J. Dyn. Sys., Meas., Control 134(6), 061013 (Sep 24, 2012) (12 pages) doi:10.1115/1.4005501 History: Received July 24, 2010; Revised October 26, 2011; Published September 24, 2012

## Abstract

Math models of wheeled ground vehicle dynamics, including flexible body effects, have been the subject of research and development for many years. These models are typically based on a finite system of simultaneous ordinary differential equations (e.g., state-space models). Higher order models that include flexible body effects offer improved accuracy over a wider frequency range than lower order rigid body models; however, higher order models are typically more sensitive to uncertainties in the model parameters and have increased computational requirements. Lower order models with the desired accuracy may be achieved by model reduction of higher order models. A new more general infinite dimensional Laplace transfer function is derived for beam bending governed by a fourth order wave equation. The resulting infinite dimensional transfer functions for beam bending are then used to develop a transfer function model of a “half-car” with a flexible body. The infinite dimensional transfer function of the half-car model is then used to assess the accuracy of the state-space models. Differences between the models due to model reduction are compared to theoretical upper bounds.

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## Figures

Figure 4

Half-car model

Figure 5

Comparison of z··sm/z··smwfwf frequency responses for the rigid body, flexible body, and infinite dimensional models

Figure 6

Quadratic model error versus the number of assumed flexible body modes

Figure 7

Peak model error versus the number of assumed flexible body modes

Figure 8

Quadratic Jordan model reduction error versus reduced model order

Figure 9

Peak Jordan model reduction error versus reduced model order

Figure 10

Quadratic balanced model reduction error versus reduced model order

Figure 11

Peak balanced model reduction error versus reduced model order

Figure 1

Finite uniform slender beam

Figure 2

Frequency response of a free-free beam (EI=1, m=1, L=1)

Figure 3

Frequency response of a pinned–pinned beam (EI=1, m=1, L=1)

## Errata

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