Vehicle Dynamics - Longitudinal Load Transfer and Pitch Analysis
https://drive.google.com/file/d/1Tdd4FErC1Gzn3n71l895c-9CJ8TwzObg/view?usp=sharing
1) Calculate the following parameters
Front end deflection under load.
Rear end deflection under load.
Pitch angle.
Pitch gradient.
Without compensation - It's all physical properties and can't change the load transfer.(in simple terms without involvinf any links/arms)
1)Load transfer = (total weight*throttle acceleration*CG height)/(wheel base* acceleration due to gravity)
load transfer = (3750*16.25*17)/(98*32.2)
= 328.285 lbs
When acceleration
Front( rebound)= -328.285lbs
Rear(Jounce) = 328.285lbs
2)Front axle weight(static) = Static front axle weight - load transfer
(considered suring sudden acceleration - Load is transferred from front to rear, so load transfer is removed from front)
= (3750*55/100) - 328.285
= 1734.215 lbs
3)Rear axle weight(static) = Static rear axle weight - load transfer
(considered suring sudden acceleration - Load is transferred from front to rear, so load transfer is added to the rear)
= (3750*45/100) + 328.285
= 2015.785 lbs
4) Front wheel rate = front spring rate*(motion ratio)^2
= 450 * 0.76^2
= 259.92 lbs/in
Rear wheel rate = rear spring rate*(motion ratio)^2
= 650 * 0.72^2
= 336.96 lbs/in
5) Front end deflection under load = (load transfer*0.5)/front wheel rate
= 328.285*0.5/259.92
= 0.6315 in
6) Rear end deflection under load = (load transfer*0.5)/rear wheel rate
= 328.285*0.5/336.96
= 0.4871 in
7)Pitch angle = tan^-1((Front end deflection under load + Rear end deflection under load )/wheel base)*(180/pi)
= tan^-1((0.6315+0.4871)/98)*(180/pi)
= 0.6539 deg
8) Pitch gradient = (pitch angle* acceleration due to gravity)/ throttle acceleration
= (0.6539*32.2)/16.25
= 1.26 deg/g
9) Horizontal CG distance from front and rear axle
(additional info that can be evaluated)
a-distance from front axle to CG
b-distance from rear axle to CG
Wfront*l = total load*b
(3750*55/100)*98 = 3750 * b
b= 53.9 in
Wrear*l = total load*a
(3750*45/100)*98 = 3750*a
a = 44.1 in2) Use the IC location information to calculate the following
Front anti-lift angle.
Rear anti-squat angle.
Front swing arm angle for 100% anti-lift.
Rear swing arm angle for 100% anti-squat.
Front anti-lift %.
Rear anti-lift %
with compensation
( considering the arms, linkages for suspension travel)
Considering front and rear to be Solid axle
(torque is reacted by suspension links because differential is a part of unsprung mass< so we don't consider tire radius)
1) front anti-lift angle (theta_actual) = tan^-1(front IC vertical distance/front IC horizontal distance)
= tan^-1(13.22/28.5)
= 24.884 deg
2) rear anti-squat angle (theta_actual) = tan^-1(rear IC vertical distance/rear IC horizontal distance)
= tan^-1(16/42.5)
= 20.629 deg
3) front swing arm angle for 100% anti-lift(theta_ideal) = tan^-1(CG height/wheel base)*(180/pi)
= tan^-1(17/98)*(180/pi)
= 9.8411 deg
4) Rear swing arm angle for 100% anti-squat(theta_ideal) = tan^-1(CG height/wheel base)*(180/pi)
= tan^-1(17/98)*(180/pi)
= 9.8411 deg
5) Front anti-lift % = (front anti-lift angle (theta_actual)/front swing arm angle for 100% anti-lift(theta_ideal))* Drive ratio*100
= (24.88/9.844)*0
= 0
(Its a Rear wheel drive front drive ratio is 0)
6) Rear anti-squat % = (rear anti-squat angle (theta_actual)/rear swing arm angle for 100% anti-lift(theta_ideal))* Drive ratio*100
= (20.629/9.8411)*100
= 209.62%
rear anti squat with solid axle is not possible with 209%
Hence we go with independent suspension by removing torque
7) For independent suspension,
rear anti-squat angle (theta_actual) = tan^-1((rear IC vertical distance-tire radius)/rear IC horizontal distance)
= tan^-1((16-11.5)/42.5)
= 6.044 deg
Rear swing arm angle for 100% anti-squat(theta_ideal) = 9.8411 deg
Rear anti-squat % = (rear anti-squat angle (theta_actual)/rear swing arm angle for 100% anti-lift(theta_ideal))* Drive ratio*100
= (6.044/9.8411) * 100
= 61.41%
3) To eliminate the pitch angle (theoretically), keep the rear swing arm height fixed at 16 inches from the contact patch. Calculate the following
Front anti-lift % and rear anti-squat % for zero pitch angle.
Front end deflection under load, accounting for the new anti characteristics.
Rear end deflection under load, accounting for the new anti characteristics.
Pitch angle, accounting for the new anti characteristics (verify that it is indeed zero)
For Pitch compensation/ zero pitch
front deflection = - rear deflection
Kr/Kf = ((e/d)-(h/l))/(h/l)
e/d = Rear anti-squat angle (θ actual)
h/l = Rear swing arm angle for 100% anti-squat (θ ideal)
1) Rear swing arm angle for 100% anti-squat (θ ideal = h/l) = 9.8411 deg
2) Rear anti-squat angle (θ actual = e/d) = ((rear spring rate/front spring rate)*rear swing arm angle for 100% squat(theta_ideal) + rear swing arm angle for 100% squat(theta_ideal)
= ((650/450)*9.8411)+9.8411
= 24.056 deg
3) rear anti-squat % = (θ actual)/(θ ideal)* drive ratio*100
= 24.056/9.8411*100
= 244.44%
4) Front end deflection(with compensation) = front deflection(w/o compensation) * (100-anti-lift % /100)
= 0.6315 in
5) rear end deflection(with compensation) = rear deflection(w/o compensation) * (100-anti-squat % /100)
= 0.4871 *(100-244.44/100)
= -0.7035 in
6) Pitch angle = tan^-1((Front end deflection(with compensation)+ rear end deflection(with compensation)/wheel base)*(180/pi)
= tan^-1((0.6315-0.7035)/98)
= 0.04216 deg
7) Pitch gradient = pitch angle*acceleration due to gravity/throttle acceleration
= 0.04216*32.5/16.25
= 0.084 deg/g
8) % compensation = (pitch angle(w/o compensation) - pitch angle(with compensation))/pitch angle(w/o compensation)*100
= (0.6539+0.04216/0.6539)*100
= 93.5%Result:-
1) Calculate the following parameters
Front end deflection under load = 0.6315 in
Rear end deflection under load = 0.4871 in
Pitch angle = 0.6539 deg
Pitch gradient = 1.2958 deg/g
2) Use the IC location information to calculate the following
Front anti-lift angle = 24.88 deg
Rear anti-squat angle = 20.629 deg
Front swing arm angle for 100% anti-lift = 9.8411 deg
Rear swing arm angle for 100% anti-squat = 9.8411 deg
Front anti-lift % = 0
Rear anti-lift % = 61.416 %( for independent suspension)
3) To eliminate the pitch angle (theoretically), keep the rear swing arm height fixed at 16 inches from the contact patch. Calculate the following (12 points)
Front anti-lift % = 0
rear anti-squat % for zero pitch angle = 244.44 %
Front end deflection under load, accounting for the new anti characteristics = 0.6315 in
Rear end deflection under load, accounting for the new anti characteristics = -0.7036 in
Pitch angle, accounting for the new anti characteristics (verify that it is indeed zero) = 0.04 deg