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Technical Paper

Analysis of Simple Rollover Metrics

Both tilt table testing and the calculation of so-called Critical Sliding Velocity (CSV) have the goal of determining conditions wherein a vehicle can be tripped by sideways impact with an obstacle and roll exactly one-quarter turn. This paper first reviews the mechanics associated with each of these metrics, verifying that (i) the tangent of the measured tilt table angle can be expected to yield a metric less than but closely related to T/2h, and (ii) CSV calculations are by and large dependent only on the vehicle's calculated cg height h and the ratio T/2h. The paper then addresses what we view to be an important related issue: How well do the mechanics of these measures and/or calculations carry over to calculations related to incidents which include more than one-quarter turn? We approach this question by extending the derivation of the CSV calculation to compute initial sideways velocities V2 needed to initiate a tripped roll of more than one-quarter turn.
Technical Paper

Tire Modeling for Low-Speed and High-Speed Calculations

Vehicle dynamics simulations typically use semi-empirical tire models. The input to these models are normal load, sideslip angle and longitudinal slip, and the output are shear forces, aligning moment, and overturning moment. Since the longitudinal speed is in the denominator of both sideslip angle and longitudinal slip, the calculation of sideslip angle and longitudinal slip at very low longitudinal velocities leads to numerical problems. This has not been a particular stumbling point in the past because vehicle dynamics calculations were largely concerned with high speed analysis. In situations wherein the vehicle was braked to a stop, patchwork techniques sufficed for calculations at low speeds. Now, however, with the advent of serious attention to driving simulators, low speed tire modelling has become more important.
Technical Paper

Validation of Computer Simulations of Vehicle Dynamics

This paper examines the validation process for computer simulations of ground vehicle dynamics. Validation in this context may be defined as the process of gaining confidence that the calculations yield useful insights into the behavior of the simulated vehicle. It is our view that this process requires three separate questions to be addressed: Is the model appropriate for the vehicle and maneuver of interest? Is the simulation based on equations that faithfully replicate the model? Are the input parameters reasonable? This paper addresses each of these questions, mainly from an analytical point of view. The paper then addresses strengths and weaknesses of vehicle testing as part of the validation process.
Technical Paper

The Relationship Between the Complexity of Linear Models and the Utility of the Computer Results

Linear analysis and corresponding vehicle tests have been used since the late 1950's to help understand the directional response of automobiles and commercial vehicles. This work is now well accepted, and linear terms such as understeer gradient and response time are descriptors routinely used to characterize vehicle performance in the linear range. This paper assesses the use of various levels of complexity in linear models. It verifies that, for steady state measures such as understeer gradient, all important effects can be handled quasistatically and a two degree of freedom model is adequate. The paper then illustrates situations in which the roll degree of freedom can be important for transient calculations, and assesses the changes in calculated transient results deriving from the addition to the model of time lags in lateral tire force buildup.
Technical Paper

Optimization of Dynamic Vehicle Response

The purpose of this paper is to demonstrate a computer program developed for interactive optimization of dynamic systems represented by ordinary differential equations. The program is capable of minimizing a cost function with respect to from one to five design variables. The program is command driven with on line help and has interactive graphics capability for displaying output. Three vehicle dynamics examples are used to demonstrate the capabilities of the program, and in doing so reveal some interesting vehicle handling results.
Technical Paper

Linear Analysis of a Vehicle with Four-Wheel Steering

Linear anaylsis is a frequently used tool to aid in the understanding of directional response. Understeer gradient, characteristic or critical speed, and yaw rate and lateral acceleration response times have been particularly helpful. This paper studies the use of these measures in the context of vehicles with four-wheel steering. The paper shows that, with only a slight change in SAE definitions, the understeer gradient retains its traditional meaning, but the characteristic speed will depend on the steering system if the steady state part of the steering control algorithm is speed sensitive. The paper also discusses testing for the understeer gradient, and the anticipated changes in response time due to four-wheel steering.
Technical Paper

Evaluation of Four-Wheel Steer Path Following Performance Using a Linear Inverse Vehicle Model

This paper presents linear first-order differential equations for a four-wheel steer vehicle which can be solved for yaw rate and sideslip angle as a function of lateral acceleration. These so-called inverse equations are useful for studying the steer angle needed to follow a given path. A root locus analysis of the inverse equations shows that the required frequencies of steer will decrease with increasing ratio of rear steer to front steer. Integration of the equations illustrates the phenomenon in the time domain. The analysis supports speculation that a driver will find it easier to track, closely to a desired path at high speeds with an appropriate ratio of rear to front steer.