Chapter 1

Motorcycle Dynamics

Vittore Cossalter, Roberto Lot, and Matteo Massaro

University of Padova, Italy

This chapter aims at giving a basic insight into the two-wheeled vehicle dynamics to be applied to vehicle modelling and control. The most relevant kinematic properties are discussed in Section 1.1, the peculiarities of motorcycle tyres are reported in Section 1.2, the most popular suspension schemes are presented in Section 1.3, while Sections 1.4 and 1.5 are devoted to the analysis of the vehicle in-plane and out-of-plane vibration modes. Finally, Section 1.6 highlights the coupling between in-plane and out-of-plane dynamics.

1.1 Kinematics

From the kinematic point of view, every mechanical system consists of a number of rigid bodies connected to each other by a number of joints. Each body has six degrees of freedom (DOF) since its position and orientation in the space are fully defined by six parameters, such as the three coordinates of a point and three angles (yaw, roll, pitch). When a joint is included, the number of DOFs reduces according to the type of joint: the revolute joint (e.g., the one defining the motorcycle steering axis) inhibits five DOFs, the prismatic joint (e.g., the one defining the telescopic fork sliding axis) inhibits five DOFs, the wheel–road contact joint inhibits three DOFs when pure rolling is assumed (only three rotations about the contact point are allowed while no sliding is permitted), or one DOF when longitudinal and lateral slippage is allowed (the only constraint being in the vertical direction, where the compenetration between the wheel and the road is avoided).

1.1.1 Basics of Motorcycle Kinematics

Two-wheeled vehicles can be considered spatial mechanisms composed of six bodies:

• the rear wheel;
• the swingarm;
• the chassis (including saddle, tank, drivetrain, etc.);
• the handlebar (including rear view mirrors, headlamp, the upper part of the front suspension, etc.);
• the front usprung mass (i.e., the lower part of the front suspension, front brake calliper, etc.);
• the front wheel.

These bodies are connected each other and with the road surface by seven joints:

• a contact joint between the rear wheel and the road surface;
• a revolute joint between the rear wheel and the swingarm, to give the rear wheel spin axis;
• a revolute joint between the swingarm and the chassis, to give the swingarm pivot on the chassis;
• a revolute joint between the chassis and the handlebar, to give the steering axis;
• a prismatic joint between the handlebar and the front unsprung, to give the sliding axis of the telescopic fork;
• a revolute joint between the front unsprung and the front wheel, to give the front wheel spin axis;
• a contact joint between the front wheel and the road plane.

Therefore, the two-wheeled vehicle has nine DOFs, given the 20 DOFs inhibited by the four revolute joints, five DOFs inhibited by the prismatic joint and the two DOFs inhibited by the two contact joints (tyre slippage allowed), subtracted from the 36 DOFs related to the six rigid bodies. It is also common to include the rear and front tyre deformation due to the tyre compliance, and consequently the number of DOFs rises to 11.

Among the many different sets of 11 parameters that can be selected to define the vehicle configuration, it is common (e.g. Cossalter et al. 2011b, 2011c) to use the ones depicted in Figure 1.1: position and orientation of the chassis, steering angle, front suspension travel, swingarm rotation and wheel spin rotations.

Figure 1.1 Degrees of freedom of a two-wheeled vehicle

Finally, it is worth mentioning that these DOFs are related to the gross motion of the vehicle, while additional DOFs are necessary whenever some kind of vehicle structural flexibility is considered, e.g. Cossalter et al. (2007b).

Some geometric parameters such as the wheelbase , normal trail and caster angle , are very important when it comes to the vehicle stability, manoeuvrability and handling. In more detail, the wheelbase is the distance between the contact points on the road and usually ranges between 1.2 and 1.6 m, the normal trail is the distance between the front contact point and the steering axis (usually 80–120 mm) and the caster angle is the angle between the vertical axis and the steering axis (usually 19–35).

In general, an increase in the wheelbase, assuming that the other parameters remain constant, leads to an unfavourable increase in the flexional and torsional deformability of the frame (this may reduce vehicle manoeuvrability), an unfavourable increase in the minimum curvature radius, a favourable decrease in the load transfer during accelerating and braking (this makes wheelie and stoppie more difficult) and a favourable increase in the directional stability of the motorcycle.

The trail and the caster angle are especially important inasmuch as they define the geometric characteristics of the steering head. The definition of the properties of manoeuvrability and directional stability of two-wheeled vehicles depend on these two parameters, among others. Small values of trail and caster characterize sport vehicles, while higher values are typical of touring and cruiser vehicles. The trail and caster are related to each other by the following relationship:

1.1

where is the front tyre radius and is the fork offset; see Figure 1.2.

Figure 1.2 Wheelbase, caster angle and trail

Finally, it is worth noting that all these parameters are usually given for the nominal (standstill) trim configuration, while they change as the vehicle speed, longitudinal and lateral accelerations change.

1.1.2 Handlebar Steering Angle and Kinematic Steering Angle

While the driver operates the handlebar steering angle, the vehicle cornering behaviour is determined by the projection on the road surface of the angle between the rear and front wheel planes, the so-called kinematic steering angle. In two-wheeled vehicles, the relationship between the handlebar and kinematic steering angles varies appreciably with the roll angle. In particular, the steering mechanism is attenuated (i.e. the kinematic angle is lower than the handlebar angle) up to a certain value of the roll angle (close to the value of the caster angle), then it is amplified (i.e. the kinematic angle is higher than the handlebar angle); see Figure 1.3 for example.

Figure 1.3 Kinematic steering angle as a function of the handlebar steering angle δ for different values of the roll angle

The following simplified expression can be used to estimate the kinematic steering angle from the handlebar steering angle , the caster angle and the roll angle :

1.2

The local curvature of the vehicle trajectory (or the turning radius ) can be estimated from the kinematic angle and the wheelbase using the following expression:

1.3

Note that Equation 1.3 does not include the effect of tyre slippage, whose contribution will be described in Sections 1.2 and 1.5.2.

1.2 Tyres

The performance of two-wheeled vehicles is largely influenced by the characteristics of their tyres. Indeed, the control of the vehicle's equilibrium and motion occurs through the generation of longitudinal and lateral forces resulting from the rider's actions on the steering mechanism, throttle and braking system. The peculiarity of motorcycle tyres is that they work with camber angles up to 50 and even more, while car tyres rarely reach 10.

1.2.1 Contact Forces and Torques

From a macroscopic viewpoint, the interaction of the tyre with the road can be represented by a system composed of three forces and three torques, as in Figure 1.4:

• a longitudinal force (positive if driving and negative if braking);
• a lateral force ;
• a force normal to the road surface;
• an overturning moment ;
• a rolling resistance moment ;
• a yawing moment .

Figure 1.4 Tyre forces and torques

Experimental observations show that the force and torque generation is mainly related to the following input quantities:

• tyre longitudinal slip ;
• tyre lateral slip ;
• tyre camber angle ;
• tyre radial deflection ;
• tyre spin rate .

Therefore we can write:

1.4

with the longitudinal force mainly related to longitudinal slip , lateral force mainly related to the lateral slip and the camber angle , overturning moment mainly related to the camber angle , rolling resistance mainly related to the wheel spin rate and yawing moment mainly related to the lateral slip and camber angle .

The longitudinal slip (positive when driving and negative when braking) is defined as:

1.5

where is the tyre longitudinal velocity, is the tyre spin rate and is the tyre effective rolling radius. In particular, the effective rolling radius can be computed from the freely rolling tyre as

1.6

Note that the effective rolling radius does not coincide with either the tyre loaded radius or the the tyre unloaded radius ; see Figure 1.5. This should not be surprising since the tyre is not a rigid body. Experimental observations show that . However, a common assumption is .

Figure 1.5 Tyre radii

Sometimes a slightly different formulation of longitudinal slip is...