Selection of Anti-Roll bars and K&C parameters to meet the target understeer gradient for a Lightweight sporty vehicle
Introduction:
The main aim of this project is to select anti-roll bars and K&C parameters to meet the target understeer gradient using the specified vehicle level targets (roll gradient and understeer gradient) for a Lightweight sporty vehicle.
At first, the roll gradients were adjusted to get the dimensions of the Anti-roll bar. In this process we must try to maintain the target roll gradients and percentage auxiliary roll stiffness. The next step was to find the tire stiffness values from the tire input plot. Finally, with the help of tire data, roll gradient and the target K&C test parameters, the maximum possible understeer gradient was calculated.
Typical Vehicle Dynamics Target Cascade:
Selection of Anti-Roll bar:
When the vehicle is negotiating a turn (or) when one wheel of the vehicle hits a bump, the weight tends to transfer in the lateral direction, due to this the body tends to roll. With the help of Roll bars, this can be minimised.
Using the primary stiffness (stiffness of spring) and the target roll gradient, the auxiliary roll stiffness (Roll stiffness due to Anti-Roll bar) of front and rear were calculated.
Roll stiffness is the moment resisting body roll per degree of body roll. In simple words, it is the stiffness exerted by the body during vehicle roll. It is expressed in ft-lb/deg.
First, the distance between CG and the axles were calculated. Then the moment arm was calculated using the formula indicated below.
`h_(arm) =h_s−[(Z_(RCf)*b_s+Z_(RCr) *a_s)/L]`
Where, hs is the sprung mass CG height (in)
ZRCf and ZRCr are the height of front and rear roll centres respectively (in)
as and bs are the distance between CG and axles (front and rear) (in)
With the help of target roll gradient, the required roll moment was calculated. Here, the target roll gradient is 3.4~3.8°/g
`M_ϕ/A_y = (W_s*h_(arm))/12*(1+(ϕ/A_y*π/180))`
Where, Mϕ/Ay is the Roll moment (ft-lb/g)
Ws is the sprung mass weight (lb)
harm is the length of the moment arm (in)
ϕ/ Ay is the Roll gradient (deg/g)
The total roll stiffness (ft-lb/deg) was calculated using Roll moment and the target roll gradient.
`K_ϕ=M_ϕ/A_y*A_y/ϕ`
As a next step, the roll stiffness (ft-lb/deg) due to tire and wheel rate were computed using the formula (for both front and rear axle),
`K_(ϕtirea)=K_(tirea)*t^2/1375`
`K_(ϕwra)=K_(wra)*t^2/1375`
Where, t is the track width (in)
Ktirea is the stiffness of the tire for a particular axle (lb/in)
Kwra is the wheel rate for a particular axle (lb/in)
‘a’ represents axle (it can be front or rear)
Then, the Total load transfer (TLT in lb/g) was found,
`TLT = W_t*h/t`
Where, Wt is the total weight of the vehicle (lb)
h is the CG height (in)
t is the track width (in)
Then, the Total Load transfer in the front (lb/g) was determined. In order to get under steer effect, the load at front is set to be 5% greater than the rear.
`TLT_F=TLT*(Front_ _weight__distribution % + 5%)/100`
Since our major concern is to compute the total front roll stiffness (ft-lb/deg) due to lateral load transfer, load transfer due to sprung and unsprung masses can be subtracted.
`K_(ϕtf) = ((TLT_F − LT_(us f) - LT_(s f))*(t/12))/ ϕ`
Load transfer due to unsprung mass (lb/g) is given by,
`LT_(us f) = W_(us f)*R_t/t`
Where, Rt is the Loaded radius of the tire
Load transfer due to sprung mass (lb/g) is given by,
`LT_(s f) = Ws_f*Z_(RCf)/t`
Total roll stiffness in the rear (ft-lb/deg) is given by,
`K_(ϕtr)=K_ϕ−K_(ϕtf)`
In order to find the roll stiffness of Anti-roll bar, first we need to isolate the effect of tire, by using the formula,
`K_(ϕwca) = (K_(ϕtirea)*K_(ϕta))/(K_(ϕtirea)−K_(ϕta))`
Where, Kϕwca is the roll stiffness at wheel centre level for a particular axle (ft-lb/deg)
Kϕtirea is the roll stiffness due to the tire for a particular axle (ft-lb/deg)
Kϕta is the total roll stiffness for a particular axle (ft-lb/deg)
Springs and anti-roll bars are connected in parallel, which is connected with the tire in series. Refer the free body diagram below.
Roll stiffness due to Roll bar (ft-lb/deg) can be computed by,
`K_(ϕarba) = K_(ϕwca)−K_(ϕwra)`
Roll stiffness due to Roll bar (in-lb/deg) is given by,
`K_(ϕarba)*12`
Roll rate due to Auxiliary bar (Anti-roll bar) in lb/in,
`K_(arba) = (K_(ϕarba)*180/π)/(MR^2 * t^2)`
Allowed front auxiliary roll stiffness is 25~60%, in our case it is 61.48%
Allowed rear auxiliary roll stiffness is 0~20%, in our case it is 1.92%
Note: The % Front auxiliary roll stiffness front depends on the Spring rate. Since the values of spring rates were specified in this project this cannot be controlled. However, the value is 1.48% higher than the target, hence this can be considered.
Calculation for Outer Diameter of Anti Roll bar,
Assumption,
- Inner diameter is zero
- Material of the Anti-roll bar is steel
Outer diameter in inches
`OD = root (4) (frac{K_(arba)*((0.4422*A^2*B)+(0.2264*C^3))}{5*10^5})`
Outer diameter in mm = Outer diameter in inches * 25.4
Outer diameter of Anti roll bar
- Front – 22.98mm ~ 23mm
- Rear – 7.3mm ~ 7.5mm
Tire Stiffness computation:
The tire stiffness parameters play an important role in determining the behaviour of the car during cornering.
So, the next step was to determine the Cornering Stiffness, Camber Stiffness and Aligning Torque Stiffness parameters from the given tire data. These parameters depend on the normal load acting on a tire.
Cornering Stiffness:
During cornering, the tires develops lateral force and it experiences lateral slip as it rolls.
Lateral force is proportional to slip angle.
`F_(yα) prop α`
`F_(yα) = C_α*α`
Where
α is the slip angle
Fyα is the cornering force due to slip angle
Cα is the cornering stiffness.
It is one of the most important property of the tire and is crucial for the vehicle’s handling and stability performance.
Cornering Stiffness is defined as the rate of change of lateral force with respect to slip angle (usually evaluated at zero slip angle). Load and inflation pressure are the main parameters that influence cornering stiffness.
The below plot shows the effect of cornering stiffness with respect to normal load. From the graph, we can see that the cornering stiffness linearly increases with the normal load (Fz) till certain point peak (saturation) point and after reaching peak (saturation point), it starts to decrease, i.e., the variation of Cornering stiffness with respect to the normal load is non-linear. As the tire does not obey the conventional friction theory, the cornering force developed is not proportional to normal force. This phenomenon is known as tire load sensitivity. The non-linear behaviour of tire is due to this phenomenon.
Red color dot is the actual tire data and the line is from curve fitting.
Aligning Torque Stiffness:
During cornering, lateral force will be developed at the tire contact patch and the inertial force (centrifugal force) acts on the wheel center which pushes the tire outwards, thus the tire will no longer be rolling in one plane, the motion will be somewhat oblique. This causes the tread elements to deflect sideways, due to this lateral stress is developed at the contact patch. The shape of the stress distribution is triangular due to uneven stress distribution. The lateral force doesn’t act on the center of the contact patch, instead there will be some offset (foce act on the centroid of the triangle (behind the wheel centerline), this is due to the uneven stress distribution). This offset is known as pneumatic trail. Since, there is an offset, a moment/torque is generated at the contact patch, which is known as aligning torque. It tries to align the tire towards the direction of travel, hence called as Self-Aligning Torque.
The slope of the aligning torque versus slip angle curve at zero slip angle gives the aligning stiffness.
Camber Stiffness
Camber is the angle between the tire rotational plane and vertical axis. When the cambered wheel is forced to move in a straight line, it experiences a force which is called as camber thrust.
Camber thrust is directly proportional to camber angle.
`F_(yγ) prop γ`
`F_(yγ) = C_ γ * γ`
Where,
Fyγ is the cornering force due to camber angle
γ is the camber angle
Cγ is the camber stiffness
Camber stiffness is defined as the rate of change of camber thrust with respect to camber angle, evaluated at zero camber angle. Similar to cornering stiffness, the load and inflation pressure influence camber stiffness.
From the graph it is evident that the increases almost linearly with tire load. The value of camber stiffness will be normally less than one-tenth to one-fifth of the cornering stiffness.
The total lateral force acting on a cambered tire operating at a slip angle is given by,
`F_y = F_(yα) ± F_(yγ)`
If the direction of camber thrust and cornering force are same, then the lateral force contributed by individual parameter gets added up.
For small slip and camber angles, the relationship between the cornering force and the slip angle and that between the camber thrust and the camber angle are essentially linear. Thus, the total lateral force of a cambered tire at a slip angle can, therefore, be determined by
`F_y = C_α*α ± C_ γ * γ`
Since, the static tire loads lie on the linear zone of the stiffness plots, the slope and Y intercepts for the stiffnesses were calculated, from which the cornering stiffness, camber stiffness and aligning stiffness for the front and rear tires were calculated.
Understeer Budget:
The final step was to compute the maximum possible understeer gradient. Understeer is when the car steers less than the amount steered by the driver. Understeer is inherently stable - once the car reduces speed sufficiently grip will be restored, which is why almost all road cars are set up to understeer at the limit of adhesion.
Consider a car is negotiating a turn.
Let,
L be the wheelbase of the vehicle
a – Distance between CG and front axle
b – Distance between CG and rear axle
αf – Slip angle - front
αr – Slip angle – rear
u – Forward velocity of the vehicle
Cαf – Cornering stiffness on front axle = (Cαfr + Cαfl )
Cαr – Cornering stiffness on rear axle = = (Cαrr + Cαrl )
Wf – Weight on the front axle
Wr – Weight on the rear axle
R – Radius of the turn
g – acceleration due to gravity
`F_(yf) = W_f * frac{u^2}{Rg} = C_(αf) * α_f`
`F_(yr) = W_r * frac{u^2}{Rg} = C_(αr) * α_r`
`α_f = frac{W_f*u^2}{ C_(αf)*Rg}`
`α_r = frac{W_r*u^2}{ C_(αr)*Rg}`
Steering angle,
`δ = 57.3 frac{L}{R} + α_f - α_r`
`δ = 57.3 frac{L}{R} + [frac{ W_f}{C_(αf)} - frac{ W_r}{C_(αr)}]*frac{u^2}{Rg}`
`δ = 57.3 frac{L}{R} + K*a_y`
Where,
ay is the acceleration of the vehicle
K is the understeer gradient and its unit is deg/g
|
K |
Steer |
|
0 |
Neutral |
|
>0 |
Understeer |
|
<0 |
Oversteer |
But, the value of K is not only dependent on Weight and cornering stiffness. It also depends on other K&C parameters. The front and rear cornering compliance are the sum of individual contributors. Its given by
`K = D_f – D_r`
Df - Cornering compliance at front (deg/g)
Dr - Cornering compliance at front (deg/g)
Weight Distribution:
Ratio of the vertical load on the tire to the cornering stiffness of the tire. This parameter is the key contributor to understeer budget, but it is somewhat difficult to control.
`WD = frac{W_f}{2*C_( αf)} - frac{W_r}{2*C_( αr)}`
Where,
Wf - Weight on the front axle (lb/g)
Wr - Weight on the rear axle (lb/g)
Cαf – Cornering stiffness on front tire (deg/lb)
Cαr – Cornering stiffness on rear tire (deg/lb)
Rigid body Aligning Torque:
During cornering, the lateral force act behind the tire’s centerline, due to that a moment is experienced by the tire. This tries to align the tire straight. Hence, the effect of this is always understeer.
`RBAT = frac{W_f}{2*C_( αf)}*frac{p_r}{b} – (-frac{W_r}{2*C_( αr)}*frac{p_f}{a})`
where,
pf - pneumatic trail of front tire (m)
pr - pneumatic trail of rear tire (m)
a - distance between front axle and CG (m)
b - distance between rear axle and CG (m)
Note: Since, the effect of RBAT is understeer in both front and rear, a negative sign has been introduced in the equation for rear axle.
Roll Steer:
The roll steer is defined as undesirable and uncontrollable changes in the steering angle of the steered wheels during the rolling action of the vehicle body due to cornering manoeuvre or asymmetric bumps.
It is caused either by tie rod length error or tie rod angle error. In the tie rod length error case, the tire experiences toe out both in jounce and rebound. This is mostly avoided because the steering effect changes from understeer to oversteer depending on the wheel ride position. In the tie rod angle error case, during cornering the loaded front wheel experiences toe out (in jounce), the it is said to experience understeer effect. In the rear this is vice versa, during cornering, if the loaded rear wheel experiences toe out (in jounce), the it is said to experience understeer effect.
`RS = (Front__roll__steer__coefficient – Rear__roll__steer__coefficient) * K_ϕ`
Roll steer coefficient (deg of steer/deg of roll)
Kϕ - Roll gradient (deg/g)
Roll Camber:
Roll Camber is simply the amount that the camber changes due to the roll of the chassis as the vehicle goes around a corner.
When cornering, the vehicle rolls, due to the centrifugal force because of the load transfer. Due to this the outer wheel is loaded and the inner is unloaded. The centrifugal force, pushes the tires away, which induces camber. This depends on the length of the control arms.
`RC = (Front__roll__camber__coefficient*frac{C_(γf)}{C_(αf)} - Rear__roll__camber__coefficient*frac{C_(γr)}{C_(αr)}) * K_ϕ`
Roll camber coefficient (deg of camber/deg of roll)
Cγf is the camber stiffness of front tire (deg/lb)
Cγr is the camber stiffness of rear tire (deg/lb)
Lateral Force compliance steer:
This is caused by the relative stiffness of suspension and steering. The effect may be understeer or oversteer. In a rear mounted rack system, if the steering is more compliant than the suspension, the effect is understeer because the lateral forces causes the wheel to deflect away from the turn. If the steering is less compliant than the suspension, then the vehicle experiences oversteer.
`LFCS = LFCS__coefficient_(f ront) * F_(yf) - LFCS__coefficient_(rear) * F_(yr)`
LFCS coefficient (deg/kN)
Fy Lateral force in N/g
Lateral Force compliance camber:
The lateral force act on the contact patch below the suspension components causes the top of the tire to tilt outward (causing positive camber). The front always experiences understeer effect, while in the rear it is always oversteer effect.
`LFC C = LFC C__coefficient_(f ront) * F_(yf)* frac{C_(γf)}{C_(αf)} - LFC C__coefficient_(rear) * F_(yr)* frac{C_(γr)}{C_(αr)}`
LFCC coefficient (deg/kN)
Aligning Torque compliance steer:
The aligning torque generated always tries to straighten the wheel, hence it produces understeer effect in the front and oversteer effect in the rear.
`ATCS = ATCS__Coefficient_(front)*Aligni n g__t o rque_(f ront) - ATCS__Coefficient_(rear)*Aligni n g__t o rque_(rear)`
ATCS coefficient (deg/kN)
Aligning torque (Nm/g)
Inputs considered for Understeer gradient computation,
Total understeer effect,
In this case, the maximum understeer is, K = 2.06 deg/g
The plot shows the contribution of each parameters to understeer.
From the plot we can infer that the contribution Roll steer is maximum (26.8%). If we want to increase theundersteer gradient furtherr, either we can increase the roll steer coefficient or we can increase the roll gradient. For changing the roll steer coefficient the suspension tie rod hard points has to be varied.
The second highest contributor is Aligning Torque compliance steer (25.7%). If this has to be tuned, the compliance of suspension or steering system has to be varied.
The third highest contributor is Weight distribution (23.9%). This parameter is very hard to control.
The contribution of Rigid body aligning torque, Lateral force compliance steer and Roll camber are nearly 3~5 times lesser than the first three.
The least contributor of understeer in this case is Lateral force compliance camber. This can be further increased by changing the suspension hard points.
Conclusion:
The diameter of front and rear anti-roll bars were selected and the maximum possible understeer gradient was computed with the specified K&C parameters.
Link for Excel:
https://drive.google.com/file/d/16HG0h0kPaaKDSNWCIn0oRcHWr5wGlBYq/view?usp=sharing