Creep forces in rolling contact

Question 1

How do you find the overall forces and realigning moment acting on the wheel?

Answer 1

Integrate the surface tractions (Stresses) over the contact area A (surface/double integral)

Question 2

What is the purpose of the Brush model?

Answer 2

It is a model to estimate unknown surface tractions (stresses), using a mathematical model for the stress caused by deflection/elastic deformation of a wheel/guideway system during contact

Question 3

What are the 4 assumptions in the Brush model?

Answer 3

• The bristles are attached to a rigid wheel rim
• The tips of the bristles are not connected
• Only elastic effects considered (no damping/permanent deformation)
• Friction - bristles do not slip until the local surface traction exceeds he available friction)

Question 4

For small yaw angles δ, what is the expression for the lateral bristle tip displacement qy?

Answer 4

qy = δ(l-x) 
l = half length of contact patch (placing origin at centre of contact patch)

Question 5

What condition is met at small yaw angles δ?

Answer 5

There is no slip

Question 6

How do you find the limiting yaw angle before microslip occurs

Answer 6

Use the equation limiting traction = Pμ
P-contact pressure
μ- coefficient of friction

and equate to the brush model equations for lateral surface tractions

Question 7

How does the method for finding lateral force change when δ > δlim?

Answer 7

Some of the contact area will be in microslip so the integral will have to be evaluated in two parts, for the sticking and slipping regions (the slipping region will have lateral traction equal to the limiting frictional stress Pμ)

(Tip: use geometry to simplify the integration of these areas, as can just use shapes to fine the area - same as integrating)

Question 8

What is the definition of lateral creep?

Answer 8

Velocity normal to wheel plane / Velocity parallel to wheel plane

OR
(lateral velocity / forward velocity) - steer angle (data sheet)

Question 9

What is the equation for longitudinal creep velocity?

Answer 9

Vx = U - rω
U- Forward velocity
r- rolling radius
ω- angular velocity about axle

Question 10

Why does a spin creep arise when a wheel is at a camber angle, or a railway wheelset has a cone angle?

Answer 10

As a component of the wheel’s angular velocity is normal to the contact patch.

Question 11

What is the equation for the spin component of the angular velocity (small camber angles φ?

Answer 11

ωs = φω

ωs- slip angular velocity
ω- wheel angular velocity

Question 12

How do you find the longitudinal and lateral creep displacements for the case of spin creep?

Answer 12

• Creep velocity= slip angular velocity x distance to centre of contact patch
• Integrate the spin creep velocities wrt time to get distance qx/qy (remember that the bristles move through contact patch x direction at dx/dt= -U, use this to substitute for dt in the integrals)
• Find the integrating factors by using the boundary conditions that bristles are undeformed at the entry to the contact patch (at x=l)

Question 13

What is the resultant longitudinal force during spin creep?

Answer 13

0

Question 14

For an n degree of freedom system, how many equations of motion are there?

Answer 14

n

Question 15

What is a way to increase the cornering stiffness C of a tyre?

Answer 15

Increase the inflation pressure

Question 16

If asked to find some form of behaviour for a vehicle, what should the first step be if unsure?

Answer 16

Create the equations of motion, often you can plug in the relevant conditions to the question into this and it will help solve for the vehicle’s behaviour.