Axle shaft design

Axle Function:

The axle function is to transmit the engine torque to the wheel.

1-    Wheel torque (from engine):

Maximum limit for the wheel torque (due to engine torque) is: Where:

(Tw max)engine = limit of wheel torque produce by the engine

Te max = maximum engine torque (Nm)

ig = gearbox ratio (maximum torque at 1st gear ratio)

if  = final drive ratio

kd = dynamic factor

kl = the coefficient of differential locking (kl = 0 for 4x2 cars with no differential locking) Where:

(Tw max)road = limit of road torque that can support wheel TE

rw = dynamic radius of the tire

Rw = vertical wheel reaction = ½ (Wr + wt)

Where:

Wr = weight on the rear axle

wt = transferred weight

2-    Wheel reaction Rw:  Where:

Wr = static weight on the rear = rear weight ratio x W

= (r.w.r x W) = (wr/W) W

= 45% W (for front engine RWD)

= 40% W (for front engine FWD)

Wta = transferred weight due to (acceleration- TE)

= ½ m x a x (h/L) = ½ W/g x  a x (h/L)

= ½ (TEmax – RR) (h/L)

Limit of wt due to acceleration

wtmax = ½ f W (r.w.r)

Wtg = transferred weight due to (climbing a gradient)

= ½ w sin q (h/L) =  ½ (TE – RR) =  ½ (Te max ig if / rw - RR)

W sin q = (W cos q b/L) f

tan q = (b/L) f ……… (slip)

W sin q  h = W cos q b

Tan q = b/h ………(turn over)

wtc = transferred weight due to (cornering)

= W/g x (r.w.r) x w2 R x (h/t)

The corner limit:

(roll over)

Rr = ½ (r.w.r) W

(skid)

wtc = W (r.w.r) f (h/t)

Assumptions:

-         The rear axle has no differential lock

-         The car is a rigid body.

-         Neglect car motion resistance

-         The max transferred weight is:

(Normal force x coefficient of adhesion) x (dimension ratio)

-         Limit of TE = Normal force x coefficient of adhesion

-         Limit of Fb = Normal force x coefficient of adhesion

-         Limit of Fc = Normal force x coefficient of adhesion

·       Where normal force = static weight ± wt

·       Wr = W (r.w.r)

Force in Z direction:

Case i (car moving with max acceleration):

* Limit of maximum acceleration: (ma = W (r.w.r) f)

(Rr)i = W (r.w.r) + wt = Wr+ Wr f (h/L)

= Wr (1 + f (h/L))

Case ii (car moving up a gradient):

(Rr)ii = Wr cos q + wt = Wr  cos q + W sin q (h/L)

* Limit of maximum gradient (RWD):

TE = w sin q …… Engine torque

sin q = TE/ W  …… q = sin-1 (TE/W) or

Rr f = [W  cos q (r.w.r) + W sin q (h/L)] f = W sin q ……. Wheel slip

W cos q (r.w.r) f = W sin q - W sin q (h/L) f = W sin q (1-  (h/L) f)

tan q = (r.w.r) f / (1-  (h/L) f)

q = tan-1 [(r.w.r) f / (1-  (h/L) f)]

Overcome road irregularities (rear wheel dynamic reaction Rrd):

Rrd = Rr Kdr

Where

kdr = the dynamic factor of road:

= 1.75 for cars, = 2.50 for trucks.

Case iii (Cornering):

(Rr)outer wheel = (Wr + wt) = Wr + m v2/R  (h/t)

* skid limit
(Rr)outer wheel = (Wr + wt) = ½ Wr + W (r.w.r) µ  (h/t)

* roll over limit

(Rr)outer wheel = [(W (t/2) + (W/g  v2/R) h) / t] x (r.w.r)

Force in X direction:

Case i (max acceleration):

TE = Te max  ig if  kd

Where:

Te max = maximum engine torque

ig = gear box ratio at 1st gear

if = final drive ratio

kd = dynamic factor (= 1.2-1.6)

* Limit of maximum TE: (TE = Rr f)

(TEmax)i = (Wr + wt) f = (Wr + Wr (h/L) f) f = Wr (1 + (h/L) f) f

(TEmax)ii = W sin q  [q = sin-1 (TE/W)] …….. Engine torque

or
q = tan-1 [(r.w.r) f / (1-  (h/L) f)] ……….wheel slip

TE h = W cos q  b …… (car turn over)

cos  q  = (TE/W) h/b …… q = cos-1 (TE/W) h/b

Force in Y direction:

Case i (Cornering):

Fc = ½ mv2/R = ½ W (v2/R) (r.w.r)

* skid limit:

(Fc)out wheel = ½ W (r.w.r) µ

* rollover limit:

m (v2/R) h =  W/g (v2/R) h = W t/2

(Fc)out wheel = W (r.w.r)/g (v2/R) = W (t/2h)