1. Creep Forces

Question 1

for a rolling tyre, what are the characteristics of the contact area with a small yaw angle δ (yaw angle = steering angle)

Answer 1

• 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

Question 2

how do the characteristics of the contact area change as the yaw angle is increased

Answer 2

• 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

Question 3

what does the deformation of the tyre contact patch result in and why

Answer 3

• lateral force Y and realigning moment N
• they are generated from the variation in the lateral stress along the contact patch

Question 4

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

Answer 4

• 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

Question 5

how does the analogy of the brush model help visualize what is going on in the contact patch

Answer 5

• 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

Question 6

what is the condition for no slip using the brush model

Answer 6

• the local surface traction is less than the available friction
• sqrt(σx^2 + σy^2) <= up
• u = coefficient of friction
• p = contact pressure

Question 7

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

Answer 7

• 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

Question 8

what are the solutions for the formulas of X, Y and N now that we have qx and qy in an integratable format

Answer 8

• X = 0
• Y = (4l^2h*Ky)δ
• N = -(4/3l^3h*Ky)δ

Question 9

what is the main takeaway on the relationship between the lateral force and realigning moment at small yaw angles

Answer 9

• the lateral force and realigning moment are directly proportional to the yaw angle at small angles

Question 10

when does microslip happen (worded)

Answer 10

• 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

Question 11

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

Answer 11

• p = Z / 4lh

Question 12

what is the formula for limit of slip, δ_lim

Answer 12

• δ_lim = uZ / 8l^2*hK_y

Question 13

what is the formula or the corresponding limiting lateral force Y_lim

Answer 13

• Y_lim = uZ/2

Question 14

what is the formula for lateral creep/sideslip α

Answer 14

• α = -tanδ

Question 15

what is the formula for lateral creep if the tyre has small lateral velocity v

Answer 15

• α = v/u - δ

Question 16

what is the formula for the coefficient of lateral creep, C_22

Answer 16

• C_22 = 4l^2*hK_y

Question 17

what is the formula for lateral force Y in terms of the coefficient for lateral creep C_22

Answer 17

• Y = -C_22*α
• the -ve sign is because the creep force is a restoring force in the opposite direction to the creep velocity

Question 18

what is the formula for longitudinal creep velocity v_x

Answer 18

• v_x = u - rw
• u = longitudinal (vehicle) velocity
• r = rolling radius
• w = angular velocity about axle

Question 19

what is the formula for the time taken for a wheel to pass over one of its theoretical bristles t_o

Answer 19

• t_o = 2l / U

Question 20

what is the formula for longitudinal creep ξ

Answer 20

• ξ = v_x / u

Question 21

what is the formula for the longitudinal displacement of bristles relative to the wheel rim at x =-l, q_x(x=-l)

Answer 21

• q_x(x=-l) = -v_x*t_o

Question 22

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)

Answer 22

• σ_x(x,y) = -K_x(l-x)*ξ
• σ_y(x,y) = 0

Question 23

what are the formulas for X, Y and N for longitudinal creep

Answer 23

• X = -(4l^2*hK_x)ξ
• Y = 0
• N = 0

Question 24

what is the formula for the coefficient of longitudinal creep C_11

Answer 24

• C_11 = (4l^2*hK_x)
• so X = -C_11*ξ

Question 25

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

Answer 25

• w_s = εw or φw

Question 26

what are the formulas for the corresponding spin creep velocities v_x and v_y

Answer 26

• v_x = w_s*y
• v_y = w_s*x

Question 27

what is spin creep Ψ

Answer 27

• Ψ = w_s / u

Question 28

what are the longitudinal and lateral creep displacements q_x and q_y

Answer 28

• q_x = -Ψ(l-x)y
• q_y = Ψ/2 * (l^2 - x^2)

Question 29

what are X, Y and N for spin creep

Answer 29

• X =0
• Y = -(4/3l^3*hK_y)Ψ
• N = (4/3l^2h^3K_x)Ψ

Question 30

what are C_11, C_22, C_23, C_32 and C_33 and why does remembering this matter

Answer 30

• 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

Question 31

what are 4 simplifying assumptions you can make when applying to vehicle dynamics

Answer 31

• 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

Question 32

what are the displacements q_x and q_y when the lateral creep α and the longitudinal creep ξ are small

Answer 32

• q_x = -ξ(l-x)
• q_y = -α(l-x)

Question 33

what is the total bristle displacement q in this case of simultaneous lateral and longitudinal creep

Answer 33

• q = sqrt(q_x^2 + q_y^2)

Question 34

what are the contact tractions σ_x and σ_y and why

Answer 34

• σ_x = -ξ(l-x)K
• σ_y =-α(l-x)K
• because K_x = K_y = K

Question 35

what are the lateral and longitudinal forces in the combined case

Answer 35

• the individual cases combined
• X = -C_11ξ
• Y = -C_22α

Question 36

what is the condition for the on-set of microslip in the combined case

Answer 36

• ξ^2 + α^2 = ξ_0^2

Question 37

what is ξ_0

Answer 37

• ξ_0 = uZ / 2C_11