## VEHICLE DYNAMICS

• ### Vehicle Critical Speed Formula - Values for the Coefficient of Friction - A Review

Abstract
This paper covers briefly the theory of tire-road friction, coefficient of friction measurement techniques, and the vagaries of tire-road friction as they relate to critical speed estimation. A literature review of tire-road friction studies was conducted to identify the primary factors effecting the tire-road coefficient of friction. Background information is presented covering general definitions and the connection between the basic critical speed formulas and the coefficient of friction. The primary components of tire-road friction, adhesion and hysteresis, are discussed along with minor effects such as tearing, wear, waves, and roll formation. Common coefficient of friction field measuring techniques are described, including the skid-to-stop test and drag sled. Influential factors such as tire characteristics, tire inflation pressure, road conditions, and dynamic factors are reviewed. Important dynamic factors are listed and the connection between longitudinal and lateral friction is discussed. Overwhelmingly, the literature indicates that the coefficient of friction is the function of many variables and that it is the most ubiquitous factor affecting speed estimates when the critical speed formula is used. Unfortunately, there appears to be no consensus regarding the appropriate value or measurement technique for the tire-road coefficient of friction used to estimate critical speed.
• ### Energy Formulas for Estimating Vehicle Critical Speed from Yaw Marks - A Review

Abstract
This report presents a review and clarification of two work-energy methods that can be used to estimate vehicle critical speed. Both methods divide a known vehicle trajectory, as represented by yaw marks, into unequal intervals, and sum the vehicle's energy expended in each interval by slip angle-induced resistance forces in the direction of travel. The work-energy equation is then used to determine, on a piecemeal basis, the total amount of energy expended over the entire trajectory. The change in vehicle velocity from the beginning of the trajectory to the point of rest (POR) is determined using the work-energy equation. The first method treats the vehicle as a point mass and considers both longitudinal and lateral vehicle resistance forces induced by the vehicle side-slip angle. The second method differentiates between front and rear axle resistance forces, employs the combined-speed formula, and considers the effect of assumed optimum front axle steering angle.
• ### Formulas for Estimating Vehicle Critical Speed From Yaw Marks - A Review

Abstract
This paper provides an exposition of the basic and some refined inertial critical speed estimation formulas. A literature review of existing inertial formulas for estimating critical cornering speed were identified for the ultimate purpose of developing a useful, compact, and more accurate speed estimation formula. Background information is presented covering the general definitions and utility of critical speed formulas. First, as a point of reference, the basic critical speed formulas are derived. Included is a list of the key assumptions on which the basic formulas are based. It is shown that the basic formulas are founded on the fundamental principles of physics and engineering mechanics; namely, Newton's Second Law and centrifugal force. Then refined formulas are presented which account for the effects of many important kinematic and dynamic factors ignored in the basic formulas such as: road grade, vehicle weight distribution, vehicle side-slip angle, axle and tire slip angles, superelevation, lateral and longitudinal drag factors, wheelbase, front steering angle, cornering stiffnesses, lateral load shift, friction dependency on load, aerodynamic forces, and anti-lock brake effectiveness
• ### An Optimized Vehicle Lane-Change Trajectory

Abstract
Functional analysis is employed to determine the ideal path of a vehicle undergoing a limit lane-change maneuver, given the tire-road coefficient of friction and either vehicle velocity or geometric constraints. Vehicle velocity is assumed to be constant. The problem is formulated using the calculus of variations. The solution technique relies on elliptic functions to achieve a closed-form solution. The synthesis of an ideal lane-change trajectory is treated as a minimal-energy-curve optimization problem with prescribed continuity and boundary conditions. The concept of critical speed is employed to limit the maximum curvature of any specified lane-change, thereby ensuring that the trajectory function determined describes a path that actually can be traversed. The analytical solution is confirmed by comparison to a numerical solution.
• ### Comparison of Ideal Vehicle Lane-Change Trajectories

Abstract
This paper seeks to compare various models of desired, or ideal, vehicle lane-change trajectories or paths and to determine which is best based on selected criteria. Background information is presented covering the utility of lane-change maneuvers, lane-change terminology, and known desired open-loop trajectories. The performance of a vehicle may be assessed by measuring its deviation from an ideal path during simulated or actual limit lane-change maneuvers. Therefore, several techniques for assessing vehicle performance against an ideal trajectory are demonstrated. Similarly, in the present effort, performance indices such as integral penalty (cost) functions are used for assessing candidate lane-change trajectories. Therefore, ideal trajectory comparison is essentially treated as an optimization problem with prescribed continuity and boundary conditions, where a path's length, curvature, and rate of change of curvature are taken as its costs. The maximum constant velocity or critical speed is employed as an additional discriminator between the candidate paths. The results of side-by-side comparison of candidate ideal trajectories are presented and discussed.
• ### Vehicle Critical Speed Formula - Determining the Path Radius of Curvature - A Review

Abstract
Techniques are presented for measuring and computing the radius of curvature of vehicle yaw marks. The radius of curvature is a key factor for the estimation of critical speed. The accuracy of speed estimates is sensitive to the portion and length of yaw mark chosen. Various methods for measuring and computing the radius of curvature are necessary because of variability in the quality of yaw marks deposited on roadways. The computational methods have their origins in analytical geometry and the principles of surveying. The techniques for determining the center of gravity path radius of curvature are also applicable to measurement of the radius of curvature of roadway curves.
• ### Vehicle Critical Speed Formula - Measuring Superelevation - A Review

Abstract
This paper explains the fundamental concept of roadway superelevation and its effect on critical speed. Different methods of measuring superelevation are described in brief. Vehicle Yaw Marks - A Review Nathaniel H. Sledge, Jr. and Kurt M. Marshek University of Texas at Austin Abstract A literature review of vehicle yaw marks is presented. A yaw mark is defined to distinguish it from similar curved tire marks deposited on roadways. Yaw marks must meet stringent criteria before ted inertial critical speed estimation formulas. A literature review of existing inertial formulas for estimating critical cornering speed were identified for the ultimate purpose of developing a useful, compact, and more accurate speed estimation formula. Background information is presented covering the general definitions and utility of critical speed formulas. First, as a point of reference, the basic critical speed formulas are derived. Included is a list of the key assumptions on which the basic for also discussed. The measurement of yaw marks is covered briefly. Derivations of the basic critical speed formulas are available .

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Professor Kurt M. Marshek
Department of Mechanical Engineering
The University of Texas at Austin

email Kurt. M. Marshek :

kmarshek@mail.utexas.edu