Amount of Weight transfer on tires of the vehicle is very important parameter to for understanding vehicle dynamics. As when car accelerate or brakes then amount of load taken by the tire has to be measured because tires have the fixed load capacity and if you transfer load more than that then they will wear out and will effect the handling of car.

Let's first see the Coordinate system that is generally used in Vehicle dynamics.This is the ISO8555/DIN7000 Coordinate system.

X axis- Longitudinal axis Moment about x axis is called:- Roll

Y axis - Lateral axis Moment about y axis is called:- Pitch

Z axis - vertical axis Moment about z axis is called:- Yaw

Check the figure below

For weight transfer derivation ,

Assumption

Car is going up hill

Ground is making a small angle θ with the horizontal

Cosθ = 1

R= fr * W ( R- TOTAL ROLLING RESISTANCE)

h=ha=hd

Balancing all forces Parallel to ground we get:-

(W/g)*a = Ff+Fr-Ra-R-WSinθ-Fd (1)

Where, Ff- front wheel traction

Fr- rear wheel traction

Ra- aerodynamic force

R- total rolling resistance(R=Rf+Rr)

Fd- dragging or pulling force(if car is pulling a cart or trailer)

here ,Fd=0

Taking moment about point A we get:-

Wf*L+Ra*h+WSinθ*h+(W/g)*a*h - WCosθ*L2=0

Wf=(W*L2-Ra*h-WSinθ*h-(W/g)*a*h)/L (2)

Taking moment about point B we get:-

Wr=(W*L1+Ra*h+WSinθ*h+(W/g)*a*h)/L (3)

Solving 1,2 & 3 we get:-

Wf=L2*W/L - h/L[F-fr*W] (Load on front tire) (4)

Wr=L1*W/L + h/L[F-fr*W] (Load on rear tire) (5)

Where,

F= Ff +Fr (total tractive force)

R=fr *W (total rolling resistance)

Conclusion

At time of acceleration the weight shifts backward as you can see from the equation 4 & 5.

F is the total tractive force which is limited by the power of vehicle and also the load capability of tire.

Wf=L2*W/L - h/L[F-fr*W] (Load on front tire)

Wr=L1*W/L + h/L[F-fr*W] (Load on rear tire)