The Tire-Force Ellipse (Friction Ellipse) and Tire Characteristics

The tire-force ellipse and tire-force circle (more frequently referred to as the friction ellipse and the friction circle, respectively) have been used for many years to qualitatively illustrate the concept of tire-road force interaction, particularly the force-limiting behavior for combined braking and steering (combined tire forces). Equations of the tireforce circle/ellipse, or, more specifically, the force limit envelope, in its idealized form have also been used in the development of quantitative models of combined tire forces used in vehicle dynamic simulation software. Comparisons of this idealized tire-force circle/ellipse using a simple bilinear tire force model and using actual tire data show that it provides only a limited, simplified notion of combined tire forces due to its lack of dependence upon the slip angle and traction slip. Furthermore, these comparisons show that the idealized tire-force circle/ellipse does not represent actual tire behavior, even approximately, since it is incapable of modeling the nonlinear behavior of tires. For this reason, the idealized tire-force circle/ellipse should not be used as a quantitative tire-force model particularly because superior validated models of nonlinear behavior of tires exist and are widely available. Here a development is presented of a more realistic version of the tire-force circle/ellipse which incorporates slip angle, traction slip and the actual nonlinear tire-force. Because of the complexity of nonlinear tire force behavior the F y F x force relationship is not a true ellipse and the force limit is dependent on the kinematic slip angle and traction slip variables, a and s , respectively. INTRODUCTION Published 04/12/2011 Copyright C 20 II SA£ International doi: 10 4271 flO I Hl! -oo94 The tire force developed over, and acting tangent to, the tire contact patch plane provides directional control of a vehicle as well as braking and acceleration traction. For analysis and modeling, this force is usually broken into its components, F::r and F,. The former is the force component along the heading axis of the tire and supplies traction, whereas the latter is the force component perpendicular (normal) to the heading axis of the tire and controls steering. The resultant of these two forces is limited in magnitude by the tire characteristics, the tire-surface sliding friction and the normal force. Precipitated by friction concepts, the relationship of these two force components is often depicted as a circle or ellipse in (Fx, Fy) space. Often called a friction ellipse, it actually is a tire-force ellipse. Though not an exact ellipse, the force relationship in the mathematical form of an ellipse appears to be used for at least two purposes. The first is to portray the limit of the resultant of the tire force components, F.f and F1 , as the wheel slip and the slip angle change. A second purpose is to use the equation of the tire-force ellipse for modeling tire force components, particularly under the condition of combined braking and steering. In general, it has been found that the longitudinal (traction) sliding friction coefficient, J.lx, and the lateral (steering) sliding friction coefficient, liy• for tires can differ. This effect is observable based on measurements of racing vehicles where lateral vehicle accelerations can exceed longitudinal accelerations. The threshold of the resultant tire force from control (partial slip) to sliding (full slip) varies with wheel slip, s, and slip angle, a . The limit of the resultant of the tire force components for any combination of s and a is 11Fz, where ~ is a combined sliding coefficient of friction and Fz is the nonnal force. If the friction coefficients for longitudinal sliding and lateral sliding are equal, the tire-force ellipse becomes a tire-force circle. In what follows, the tenn tireforce circle is used for brevity, but the results apply to a tireforce ellipse when llx :f. lly· In addition to the tire force components, Fx and Fy, moments also develop over the contact area between a tire and the roadway. Although they fonn an important part of vehicle control, they are not discussed here. In practice, tire forces can be longitudinally and laterally asymmetrical w. Such asymmetry is also not discussed in the following. The tenn adhesion is often used in the tire literature to indicate non-slip of a tire over a portion of the tire-roadway contact region. In some cases it is even used synonymously with friction. However, current use of the tenn adhesion in the scientific literature refers to the presence of a tensile force over a contact area due to molecular attraction forces [2]. Since no significant tensile force is present between a tire and a road surface, this term is avoided herein. A detailed glossary of tenns and acronyms is presented in the following section to provide consistent tenninology within the paper. A history of the friction circle is covered thoroughly in the book by Milliken and Milliken ll]. lt should be pointed out that a related concept called the g-g diagram exists. The g-g diagram takes the form of the lateral acceleration of a vehicle (usually a racing vehicle) plotted versus its longitudinal acceleration as the vehicle undergoes cyclical motion (such as over a racing oval). Though similar in concept to a tire-force circle, the g-g diagram is a plot of vehicle kinematics and depends on vehicle effects (such as aerodynamics and suspension characteristics), not simply tire characteristics. It is common practice to start an analysis of tire forces by relating the traction force component, Fx to the longitudinal slip, s and the lateral force component, Fy, to the slip angle, a . The relationship of the two force components is then established for combined braking and steering. This is the approach taken here. First, tire patch kinematics is covered. Experimental data are subsequently presented for use later in the paper to introduce the behavior of realistic tire force components, Fy(a) and Fx(s). This is followed by presentation of an analytical model that allows the use of the components Fy(a) and F.-.:(s) for a combined steering and traction force. Then the concept of the idealized tire-force circle is covered. Following that, the behavior of tire forces in [F~a,s), Fx(a,s)] space, where a and s are parameters, is examined. Finally, the concept and use of the idealized tire-force circle for realistic tire behavior is examined. This paper is not intended to cover tire models. Discussions of the many models of tire forces can be found in the references in the paper by Sandu and Umrsrithong [ll] and in the introduction of the papers by Brach and Brach {2, 1]. Properties and characteristics of the tire-force circle are examined for simple bilinear fonns of the steering and traction components, Fy and Fx, and using the BNP tire force equations (Magic Fonnula) [1] with experimental data. In both cases, the tire forces for combined traction and steering are modeled using the Modified Nicolas-Comstock model [i, ~· This model is chosen because of its efficiency and accuracy, but also because it can be used with any pair of traction and steering model equations. Tire Kinematics Two kinematic variables typically are used with tire force models and with the measurement of tire forces. These are the slip angle, a, and the longitudinal wheel slip, s. The slip angle is illustrated in fl&...l and is defined as a= tan-(Vy /Vx) (I) The wheel slip can have different definitions [i). The one used here is such that 0 ~ lsi ~ I (for both positive and negative traction), where for braking V -R(J) s=~x __ vx (2) Figures I and 2 show the tire slip velocity, Vp, has components V P.x = Vx Rw and VPJ' VY' Note that, in general, the vector velocity, V, at the wheel hub and the slip velocity, VP, at the contact patch center differ both in magnitude and direction. The slip velocity, V P' is the velocity of a contact point P of the tire relative to the road surface. Figure 1. WheeVtire velocity componenls The directions of the resultant force, F, and the slip velocity, VP' can differ. For no steering, (a = 0) the longitudinal (traction) tire force component typically is expressed mathematically as a function of the wheel slip alone, FJ..s). Similarly, for no braking, (s 0), the lateral (cornering, steering) force component typically is expressed mathematically as a function of the sideslip angle alone, FJ_a). Figure 1. Tire patch velocity and force components. Experimentally Measured Tire Forces Experimental tire data is presented here because some of the results given later in the paper use tire parameters corresponding to values developed from these tests. One set of data used for illustrations in this paper is obtained from a cooperative research sponsored by NHTSA (National Highway Traffic Safety Administration) [lQ] for a 295/75R22.5 truck tire. Figure 3 shows the measured traction force Fx(s) for different normal forces for zero slip angle, a = 0. It shows that the slip stiffitess, Cs, depends on the normal force. The slip angle stiffuess coefficient, Ca (not shown), also depends on the normal force. Most importantly, the curves show that the longitudinal tire force, F.x(s), over a large range of traction slip significantly exceeds the locked wheel force value FJ..s)ls. I• at least for higher normal forces.