Chapter 5 Tyre Modelling

Chapter 5

Tyre Model

Introduction• The tires are the primary element to carry the vehicle load and

guide it along its track. • The correct calculation of the contact forces between road and tire

are therefore crucial in any simulation of vehicle behavior. • The forces generated at tire contact patch are considered as

important factor for the dynamic behavior of a road vehicle.• Accuracy of tire models are necessary components of complete

vehicle models aimed at analyzing or simulating vehicle motion in real driving conditions.

• There are many previous models that describe the tire forces generated by the tires during braking, accelerating and cornering conditions.

• In vehicle dynamics, an accurate description of the tire characteristics is extremely important to the study of a wide range of vehicle behavior

• The primary forces during lateral maneuvering, acceleration, and braking are generated by tires as a function of the driver input.

• The longitudinal and lateral forces generated by a tire are a function of the slip angle and longitudinal slip of the tire relative to the road.

Longitudinal Force (Fx) Lateral Force (Fy)

• The longitudinal slip of the tire is defined as a difference between the tire tangential speed and the speed of the axle relative to the road, which is represented by the following equation:

• where s is the longitudinal slip, R is the radius of the wheel, ω is the angular velocity and u is the speed of the axle.

• The value of the longitudinal slip is limited such that |s|≤1. • For braking, axle speed is used in the denominator so that

longitudinal slip is 1 when ω is zero. • Slip has the opposite sign when tracking force is generated.

When the tire develops a sideslip velocity denoted by v , a lateral force will develop opposing the sideslip velocity. This lateral force is a function of slip angle, where slip angle is defined as:

• where v is the sideslip velocity, and u is the speed of the axle. The value of the slip angle is limited such that |α|≤ 90°.

Magic Formula Tire Model

• Magic Formula Tire Model produces Longitudinal and lateral forces, aligning moment

• During a cornering maneuver the tire contact patch ‘slips’ laterally while rolling such that its motion is no longer in the direction of the wheel plane.

• The angle between its direction of motion and the wheel plane is referred to as the slip angle, α.

• This lateral slip generates a lateral force, Fy at the tire ground interface. Because the force acts slightly behind the center of the wheel, it produces an aligning moment, Mz which tends to realign the wheel in the direction of rolling.

• The coefficients tire formula with load influence and coefficients for tire formula with camber influence parameters .

• For lateral force, the stiffness, shape, peak, and curvature factors are calculated as follows:

Parameter for Magic Formula

Calspan Tire Model

• The longitudinal and lateral forces are results of the inputs of the slip angle, longitudinal slip and normal forces.

• The normal force considered as a given quantity that results from the normal deflection of the tire.

• The functions can be obtained from measurements for given speed of travel and road and environment conditions.

• The previous theoretical developments lead to a complex, highly non-linear composite force as a function of composite slip. It is convenient to define a saturation function, f(σ), to obtain a composite force with any normal load and coefficient of friction values.

• The polynomial expression of the saturation function is presented by:

Parameter for Calspan Tire Model

Dugoff Tire Model

• Dugoff's model provides for calculation of forces under combined lateral and longitudinal tire force generation.

• It assumes a uniform vertical pressure distribution on the tire contact patch. This is a simplification compared to more realistic parabolic pressure distribution assumed in Pacejka and Sharp.

• However, the model offers one significant advantage it allows for independent values of tire stiffness in the lateral and longitudinal directions.

• Further, the lateral and longitudinal forces are directly related to the tire road friction coefficient in more transparent equations.

• Let σx be the longitudinal slip ratio of the tire under consideration and α be the side slip angle. Let the cornering stiffness of the tire be given by Cα, and the longitudinal tire stiffness by Cσ.

• Then the longitudinal tyre force is given by: