Chapter 2

Tire Mechanics

Abstract

The tire cornering characteristics, lateral force and self-aligning torque to side slip angle, are dealt with in this chapter. Following the theoretical investigation into the effects of the tire parameters on the cornering characteristics based on Fiala's theory, more specific explanations of the effects of side slip angle, vertical load, road condition, tire pressure, tire shape, braking and traction forces, etc. are given based on previous experimental data. In order to study the tire cornering characteristics during combined slip of lateral and longitudinal directions, the brush tire model is introduced. After giving the explanation of the model, detailed descriptions of the lateral and longitudinal forces are given as functions of basically lateral slip, longitudinal slip, and vertical load. The transient characteristics of lateral force and self-aligning torque are also dealt with using simplified tire model.

Keywords

Brush model; Combined slip; Cornering characteristics; Fiala's theory; Lateral force; Self-aligning torque; Side slip angle; Transient characteristics; Vertical load

2.1. Preface

Chapter 1 discussed how this book deals with the independent motion of the vehicle, in the horizontal plane, without restrictions from a preset track on the ground. The force that makes this motion possible is generated by the relative motion of the vehicle to the ground. The contact between the vehicle and the ground is at the wheels. If the wheel possesses a velocity component perpendicular to its rotation plane, it will receive a force perpendicular to its traveling direction. In other words, the wheel force that makes the vehicle motion possible is produced by the relative motion of the vehicle to the ground, and is generated at the ground. This is similar to the lift force acting vertically on the wing of a body in flight and the lift force acting perpendicularly to the direction of movement of a ship in turning (for the ship, this becomes a force in the lateral direction). The wheels fitted to the object vehicle not only support the vehicle weight while rotating and produce traction/braking forces, but they also play a major role in making the motion independent from the tracks or guide ways. This is the essential function of our vehicle. In dealing with the dynamics and control of a vehicle, it is essential to have knowledge of the forces that act on a wheel. Consequently, this chapter deals mainly with the mechanism for generating the force produced by the relative motion of the wheel to the ground and an explanation of its characteristics.

2.2. Tires Producing Lateral Force

2.2.1. Tire and Side-Slip Angle

Generally, when a vehicle is traveling in a straight line, the heading direction of the wheel coincides with the traveling direction. In other words, the wheel traveling direction is in line with the wheel rotational plane. However, when the vehicle has lateral motion and/or yaw motion, the traveling direction can be out of line with the rotational plane. Figure 2.1 is the wheel viewed from the top, where (a) shows the traveling direction in line with the rotation plane, and (b) shows it not in line. The wheel in (b) is said to have side slip. The angle between the wheel traveling direction and the rotational plane, or its heading direction, is called the side-slip angle. The wheel is also acted on by a traction force if the wheel is moving the vehicle in the traveling direction, or braking force if braking is applied. Also, a rolling resistance force is always at work. If the wheel has side slip, as in (b), a force that is perpendicular to its rotation plane is generated. This force could be regarded as a reaction force that prevents side slip when the wheel produces a side-slip angle. This is an important force that the vehicle depends on for its independent motion. Normally, this force is called the lateral force, whereas the component that is perpendicular to the wheel rotation plane is called the cornering force. When the side-slip angle is small, these two are treated as the same. This force corresponds to the lift force, explained in fluid dynamics, which acts on a body that travels in a fluid at an attack angle, as shown in Figure 2.2.
Figure 2.1 Vehicle tire in motion, (a) without side slip and (b) with side slip. There are many kinds of wheels, but all produce a force perpendicular to the rotation plane when rotated with side slip. Figure 2.3 shows the schematic comparison of the lateral forces at small side-slip angles for a pneumatic tire wheel, a solid-rubber tire wheel, and an iron wheel. From here, it is clear that the magnitude of the force produced depends on the type of wheel and is very different. In particular, the maximum possible force produced by an iron wheel is less than one-third of that produced by a rubber tire wheel. Compared to a solid-rubber tire wheel, a pneumatic tire wheel produces a larger force. For independent motion of the vehicle, the force that acts on a wheel with side slip is desired to be as large as possible. For this reason, the traveling vehicle that is free to move in the plane without external restrictions is usually fitted with pneumatic tires. These are fitted for both the purpose of vehicle ride and for achieving a lateral force that is available for vehicle handling.
Figure 2.2 Lifting force.
Figure 2.3 Lateral forces for several wheels. In the following text, the pneumatic tire is called a tire, and the mechanism for generating a lateral force that acts on a tire with side slip is explained.

2.2.2. Deformation of Tire with Side Slip and Lateral Force

Generally, forces act through the contact surface between the tire and the road. A tire with side slip, as shown by Figure 2.4, is expected to deform in the tire contact surface and its outer circumference: (a) shows the front and side views of the tire deformation; (b) shows the tire contact surface and outer circumference deformation viewed from the top. At the front of the surface, the deformation direction is almost parallel to the tire's traveling direction. In this part, there is no relative slip to the ground. When the tire slip angle is small, the whole contact surface is similar to this and the rear end of the contact surface has the largest lateral deformation.
Figure 2.4 Tire deflection with side slip, (a) front and side view and (b) plane view. When the tire slip angle gets bigger, the front of the surface remains almost parallel to the tire traveling direction. The deformation rate reduces near the center of the contact patch, and the lateral deformation becomes largest at a certain point between the front and rear of the surface. After this maximum point, the tire contact surface slips away from the tire centerline, and the lateral deformation does not increase. As tire slip angle gets even larger, the point where lateral deformation becomes maximum moves rapidly toward the front. When the slip angle is around 10 to 12°, the contact surface that is parallel to the tire travel direction disappears. The contact surface deformation is nearly symmetric around the wheel's center and consists of nearly all the slip regions. The lateral deformation of the tire causes a lateral force to act through the contact surface, which is distributed according to the deformation. This lateral force is sometimes called the cornering force when the side-slip angle is small. Looking at the tire lateral deformation, the resultant lateral force may not act on the center of the contact surface. Thus, the lateral force creates a moment around the tire contact surface center. This moment is called the self-aligning torque and acts in the direction that reduces the tire slip angle.

2.2.3. Tire Camber and Lateral Force

As shown in Figure 2.5, the angle between the tire rotation plane and the vertical axis is called the camber angle. If a tire with a camber angle of ? is rotated freely on a horizontal plane, as shown in Figure 2.5, the tire makes a circle with the radius of /sin? and has its origin at O. If the circular motion is prohibited for a tire with camber angle, and the tire is forced to travel in a straight line only, a force will act on the tire as shown in the figure. This force, due to the camber between the tire and the ground, is called camber thrust.
Figure 2.5 Tire with camber angle and camber thrust.

2.3. Tire Cornering Characteristics

The characteristics of the tire that produces lateral force and moment, as elaborated in Section 2.2, are defined as the cornering characteristics. In this section, the tire cornering characteristics will be examined in more detail.

2.3.1. Fiala's Theory

The mathematical model proposed by E. Fiala [1] is widely accepted for the aforementioned analysis of the lateral force due to side slip of the tire. It is commonly called Fiala's Theory and is related to the tire cornering characteristics. It is one of the fundamental theories used by many people for explaining tire cornering characteristics [2]. Here, based on Fiala's theory, the...