Relaxation in contact patch

As for longitudinal, there is a delay in how fast the steady state conditions can be reached in contact patch, which is sometimes important to consider. A similar model, as for relaxation in longitudinal direction Section 2.3.3.4.2, is to add a first order delay of the force:

̇ = 𝐴 ∙ ( (𝑠 … ) );

where (𝑠 … ) is the force according to a steady state model and 𝐴 = 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡

𝐿 =

|𝑅∙𝜔| 𝐿 or 𝐴 =

| _𝑥|

𝐿 and is the relaxation length, which is a

VEHICLE INTERACTIONS

2.4.6 Other lateral effects

The deformations of the tyre during cornering are quite complex when compared to the case of pure longitudinal motion, see Figure 2-26. Hence, there are more effects than simply a lateral force. Some of these will be discussed in the following.

2.4.6.1 Overturning moment

The contact patch is deflected laterally from the centre of the carcass. This creates an overturning moment 𝑀 due to the offset position of the normal force. The lateral force also contributes to the overturning moment.

2.4.6.2 Tyre aligning moment

(This section has strong connection with Section 4.2.3.2 Steering system forces.)

In the lowest diagram in Figure 2-27, one can see that shear stress is concentrated to the outlet side of the contact patch for small slip angles. So, the equivalent lateral force acts behind the centre of wheel rotation for small slip angles. As seen in Figure 2-26 b) it acts at a position 𝑡 behind the wheel’s y axis. The distance 𝑡 is called the pneumatic trail, see also Figure 4-5, and the resulting yawing moment around a vertical axis through centre of the contact patch will be = 𝑡 , which is often called the aligning moment, 𝑀 . The aligning moment is named after that is normally acts to steer the wheel to zero side slip.

If the tyre is on a steered axle, the aligning moment influence on steering wheel torque is important. When finding that influence, the moment around the steering axis intersection with ground is the important moment = (𝑡 + ), which can be called the steering moment. The distance is the mechanical trail, which is built into the suspension linkage design. One typically designs the suspension so that > , which makes the whole steering moment act in the same direction as the aliging moment. If driver takes his hands from steering wheel in a curve, and/or if steering power assistance is lost, the steering will tend to steer in the direction of body motion above the steered axle, which is normally relatively smooth and safe.

side slip angle lateral tyre force = Fy (for a tyre with peak)

total trail = caster trail + pneumatic trail = + 𝑡

steering moment = + 𝑡

steering moment peaks at lower slip than lateral tyre force.

tyre contact patch steering axis intersection with ground sh ea r str es s, 𝜏 𝑡 𝑡 + 𝑡

Figure 2-34: General Response of Steering torque to Side slip angle. Tyre aligning moment = ∙ 𝑡 is one part of the steering moment=

Figure 2-34 shows the combined response of lateral force and slip angle. It is interesting to note that the steering torque reaches a peak before the maximum lateral force capacity of the tyre is reached. It can be used by drivers to find, via steering wheel torque, a suitable steering angle which gives a large lateral force but still does not pass the peak in lateral force. The reason why pneumatic trail can become slightly negative is because pressure centre is in front of wheel centre, see Figure 2-9.

2.4.6.2.1 Aligning Torque Model

A model for (yawing) aligning moment around a vertical axle through centre of contact point, 𝑀 , will now be derived. Any model for lateral shear stress can be used, but we will here only use the uniform pressure distribution and independent bristles in Section 2.4.1.1. A corresponding expression as Eq [2.33] is derived, but for 𝑀 instead of .

𝑀 = 𝑊 ∙ ∫ 𝜏 ∙ (𝜉 ) 𝜉 𝐿 0 = ⋯ = { = Cy ₆∙ 𝑠 ; 𝑜𝑟 < ⇔ 𝑠 < 𝐶 = ∙ 8 ∙ 𝐶 ∙ 𝑠 (1 3 ∙ 𝐶 ∙ 𝑠 ) ; 𝑙𝑠 𝑤ℎ 𝑟 𝐶 =𝐺 𝑊 𝐻 ; [2.40]

The curve of Eq [2.40] is plotted in Figure 2-35.

The lateral force and the aligning torque can be used to calculate the steering forces. If also the steering assistance is known, the steering wheel torque can be calculated. It can be noted that the model does not include the moment from steering rotation itself, i.e. the torque counteracting ̇.

Small and increasing , and constant non-zero lever

Large , but lever approaches zero

Figure 2-35: Aligning moment (𝑀 ) around contact patch center for uniform pressure distribution.

2.4.6.3 Camber force

Camber force (also called Camber thrust) is the lateral force caused by the cambering of a wheel. One explanation model is shown in Figure 2-36. It is that Camber force is generated when a point on the outer surface of a leaning (cambered) and rotating tyre, that would normally follow a path that is elliptical when projected onto the ground, is forced to follow a straight path due to friction with the ground. This deviation towards the direction of the lean causes a deformation in the tyre tread and

VEHICLE INTERACTIONS

model is that the tyre “climbs” sideways as the inlet edge is directed, because there is more stick and less slip in the inlet edge as compared to the outlet edge; see brush model.

Camber thrust is approximately linearly proportional to camber angle for small angles: Camber thrust= _𝑐= 𝐶𝛾∙ 𝛾. The camber stiffness, 𝐶𝛾, is typically 5-10 % of the cornering stiffness and opposite sign. The tyre lateral forces due to lateral slip and due to camber are superimposable for small lateral slip and small camber angle. The approximate Eq [2.36], can then be developed to:

= 𝑆 𝑑𝑒𝑆 + 𝐶 𝑚 𝑒 = 𝐶 ∙ 𝑠 + 𝐶𝛾∙ 𝛾; or = 𝐶𝛼∙ 𝛼 + 𝐶𝛾∙ 𝛾 [2.41] There is also a rotation perpendicular to the ground due to the camber. The total rotation vector of the wheel is directed along the wheel rotation axis. The component of this vector that is perpendicular to ground creates friction moment which steers the wheel towards the side it tilts; on a cambering vehicle like a bicycle, the wheel is steered “into the fall”, which counteracts falling.