1. Creep Forces Flashcards
for a rolling tyre, what are the characteristics of the contact area with a small yaw angle δ (yaw angle = steering angle)
- the whole contact area is fixed to the road so theres no micro-slip
- meaning the whole contact area is parallel to the rolling direction
how do the characteristics of the contact area change as the yaw angle is increased
- the rear of the contact area starts to slide towards the wheel plane
- when the yaw angle approaches 15 degrees, the whole contact area slides
what does the deformation of the tyre contact patch result in and why
- lateral force Y and realigning moment N
- they are generated from the variation in the lateral stress along the contact patch
for the contact area between a wheel and the road, what are the directions that the longitudinal force X, lateral force Y, normal force Z and realigning moment N act in ON the wheel
- they can be figured using both right hand rules
- the index finger is X and points in the direction of the wheel’s motion
- the middle finger is Y
- the thumb pointing upwards is Z
- the realigning moment N is in the direction of the curled fingers
how does the analogy of the brush model help visualize what is going on in the contact patch
- the unstressed bristles of the brush enter the contact patch at their leading edges
- they progressively bend over as they pass through, progressively building up stress
- they spring back to their unstressed position when they leave the contact area
what is the condition for no slip using the brush model
- the local surface traction is less than the available friction
- sqrt(σx^2 + σy^2) <= up
- u = coefficient of friction
- p = contact pressure
if a tyre with a contact patch of length 2l and height 2h is rolling at a small yaw angle δ, what are the lateral and longitudinal displacements of the ‘bristles’ relative to the unstressed wheel, qy and qx, in an x-y coordinate system and why
- qy = δ(l-x)
- qx = 0
- its triangle geometry (look at notes)
- the bristle tips move in the y-direction but not in the x-direction
what are the solutions for the formulas of X, Y and N now that we have qx and qy in an integratable format
- X = 0
- Y = (4l^2h*Ky)δ
- N = -(4/3l^3h*Ky)δ
what is the main takeaway on the relationship between the lateral force and realigning moment at small yaw angles
- the lateral force and realigning moment are directly proportional to the yaw angle at small angles
when does microslip happen (worded)
- when there in insufficient surface friction available to deflect the bristles at the rear of the contact area any further
- so they start to slip towards to wheel rim
- this happens at larger yaw angles
what is the formula for the mean contact pressure p with a contact patch of length 2l and heigh 2h where Z is the total vertical load
- p = Z / 4lh
what is the formula for limit of slip, δ_lim
- δ_lim = uZ / 8l^2*hK_y
what is the formula or the corresponding limiting lateral force Y_lim
- Y_lim = uZ/2
what is the formula for lateral creep/sideslip α
- α = -tanδ
what is the formula for lateral creep if the tyre has small lateral velocity v
- α = v/u - δ
what is the formula for the coefficient of lateral creep, C_22
- C_22 = 4l^2*hK_y
what is the formula for lateral force Y in terms of the coefficient for lateral creep C_22
- Y = -C_22*α
- the -ve sign is because the creep force is a restoring force in the opposite direction to the creep velocity
what is the formula for longitudinal creep velocity v_x
- v_x = u - rw
- u = longitudinal (vehicle) velocity
- r = rolling radius
- w = angular velocity about axle
what is the formula for the time taken for a wheel to pass over one of its theoretical bristles t_o
- t_o = 2l / U
what is the formula for longitudinal creep ξ
- ξ = v_x / u
what is the formula for the longitudinal displacement of bristles relative to the wheel rim at x =-l, q_x(x=-l)
- q_x(x=-l) = -v_x*t_o
what is the formula for the surface tractions at the trailing edge of this contact area, σ_x(x,y) and σ_y(x,y)
- σ_x(x,y) = -K_x(l-x)*ξ
- σ_y(x,y) = 0
what are the formulas for X, Y and N for longitudinal creep
- X = -(4l^2*hK_x)ξ
- Y = 0
- N = 0
what is the formula for the coefficient of longitudinal creep C_11
- C_11 = (4l^2*hK_x)
- so X = -C_11*ξ
what is the formula for spin angular velocity w_s with a coned railway wheelset of cone angle ε, and a cambered pneumatic tyre with a camber angle φ
- w_s = εw or φw
what are the formulas for the corresponding spin creep velocities v_x and v_y
- v_x = w_s*y
- v_y = w_s*x
what is spin creep Ψ
- Ψ = w_s / u
what are the longitudinal and lateral creep displacements q_x and q_y
- q_x = -Ψ(l-x)y
- q_y = Ψ/2 * (l^2 - x^2)
what are X, Y and N for spin creep
- X =0
- Y = -(4/3l^3*hK_y)Ψ
- N = (4/3l^2h^3K_x)Ψ
what are C_11, C_22, C_23, C_32 and C_33 and why does remembering this matter
- C_11 = 4l^2*hK_x
- C_22 = 4l^2hK_y
- C_23 = 4/3l^3*hK_y
- C_32 = 4/3l^3*hK_y
- C_33 = 4/3l^2*h^3K_x
- remembering these means you can apply them to the linear creep equations in the datasheet
what are 4 simplifying assumptions you can make when applying to vehicle dynamics
- neglect spin creep Ψ because its small
- neglect realigning moment N due to lateral creep
- for road vehicle handling and steady speed, X= 0
- For rail vehicles, assume K_x = K_y so C_11 = C_22
what are the displacements q_x and q_y when the lateral creep α and the longitudinal creep ξ are small
- q_x = -ξ(l-x)
- q_y = -α(l-x)
what is the total bristle displacement q in this case of simultaneous lateral and longitudinal creep
- q = sqrt(q_x^2 + q_y^2)
what are the contact tractions σ_x and σ_y and why
- σ_x = -ξ(l-x)K
- σ_y =-α(l-x)K
- because K_x = K_y = K
what are the lateral and longitudinal forces in the combined case
- the individual cases combined
- X = -C_11ξ
- Y = -C_22α
what is the condition for the on-set of microslip in the combined case
- ξ^2 + α^2 = ξ_0^2
what is ξ_0
- ξ_0 = uZ / 2C_11
