Roll Rate with Anti-Roll Bar Calculator

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Race Car Vehicle Dynamics

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Front Axle and Steering

Suspension Geometry

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Rates for Axle Suspension in Race Car

Ride Rate and Ride Frequency for Race Cars

Tire Behavior in Racing Car

Vehicle Cornering in Race Cars

Weight Transfer during Braking

Wheel Centre Rates for Independent Suspension

✖Tyre Vertical Rate is the spring rate controlled by the tyre compound, sidewall stiffness and operating pressure.ⓘ Tyre Vertical Rate [K_t]			+10% -10%
✖Rear Track Width is the distance between the centers of the rear wheels.ⓘ Rear Track Width [t_R]			+10% -10%
✖The Roll Rate of Anti-Roll Bar is the stiffness of the anti-roll bar in the roll mode.ⓘ Roll Rate of Anti-Roll Bar [R_arb]			+10% -10%
✖Wheel Centre Rate is the vertical force acting on the tire per unit vertical displacement at the location along the spindle corresponding to the wheel centerline, measured relative to the chassis.ⓘ Wheel Centre Rate [K_W]			+10% -10%
✖Spring Track Width is the distance between the springs on either side of the axle.ⓘ Spring Track Width [T_s]			+10% -10%

✖Roll Rate or Roll stiffness is the resistance moment generated per unit of roll angle of the sprung mass.ⓘ Roll Rate with Anti-Roll Bar [K_Φ]

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Formula

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Roll Rate with Anti-Roll Bar

Formula

`"K"_{"Φ"} = ("K"_{"t"}*("t"_{"R"}^2)/2*("R"_{"arb"}+"K"_{"W"}*("T"_{"s"}^2)/2))/("K"_{"t"}*("t"_{"R"}^2)/2+"R"_{"arb"}+"K"_{"W"}*("T"_{"s"}^2)/2)`

Example

`"20792.56Nm/rad"=("321300N/m"*(("1.5m")^2)/2*("4881.6Nm/rad"+"42419.8N/m"*(("0.9m")^2)/2))/("321300N/m"*(("1.5m")^2)/2+"4881.6Nm/rad"+"42419.8N/m"*(("0.9m")^2)/2)`

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Download Rates for Axle Suspension in Race Car Formulas PDF

Roll Rate with Anti-Roll Bar Solution

STEP 0: Pre-Calculation Summary

Formula Used

Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/(Tyre Vertical Rate*(Rear Track Width^2)/2+Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2)
K_Φ = (K_t*(t_R^2)/2*(R_arb+K_W*(T_s^2)/2))/(K_t*(t_R^2)/2+R_arb+K_W*(T_s^2)/2)
This formula uses 6 Variables

Variables Used

Roll Rate - (Measured in Newton Meter per Radian) - Roll Rate or Roll stiffness is the resistance moment generated per unit of roll angle of the sprung mass.
Tyre Vertical Rate - (Measured in Newton per Meter) - Tyre Vertical Rate is the spring rate controlled by the tyre compound, sidewall stiffness and operating pressure.
Rear Track Width - (Measured in Meter) - Rear Track Width is the distance between the centers of the rear wheels.
Roll Rate of Anti-Roll Bar - (Measured in Newton Meter per Radian) - The Roll Rate of Anti-Roll Bar is the stiffness of the anti-roll bar in the roll mode.
Wheel Centre Rate - (Measured in Newton per Meter) - Wheel Centre Rate is the vertical force acting on the tire per unit vertical displacement at the location along the spindle corresponding to the wheel centerline, measured relative to the chassis.
Spring Track Width - (Measured in Meter) - Spring Track Width is the distance between the springs on either side of the axle.

STEP 1: Convert Input(s) to Base Unit

Tyre Vertical Rate: 321300 Newton per Meter --> 321300 Newton per Meter No Conversion Required
Rear Track Width: 1.5 Meter --> 1.5 Meter No Conversion Required
Roll Rate of Anti-Roll Bar: 4881.6 Newton Meter per Radian --> 4881.6 Newton Meter per Radian No Conversion Required
Wheel Centre Rate: 42419.8 Newton per Meter --> 42419.8 Newton per Meter No Conversion Required
Spring Track Width: 0.9 Meter --> 0.9 Meter No Conversion Required

STEP 2: Evaluate Formula

Substituting Input Values in Formula

K_Φ = (K_t*(t_R^2)/2*(R_arb+K_W*(T_s^2)/2))/(K_t*(t_R^2)/2+R_arb+K_W*(T_s^2)/2) --> (321300*(1.5^2)/2*(4881.6+42419.8*(0.9^2)/2))/(321300*(1.5^2)/2+4881.6+42419.8*(0.9^2)/2)

Evaluating ... ...

K_Φ = 20792.5592230811

STEP 3: Convert Result to Output's Unit

20792.5592230811 Newton Meter per Radian --> No Conversion Required

FINAL ANSWER

20792.5592230811 ≈ 20792.56 Newton Meter per Radian <-- Roll Rate

(Calculation completed in 00.004 seconds)

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Created by Peri Krishna Karthik

National Institute of Technology Calicut (NIT Calicut), Calicut, Kerala

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< 10+ Rates for Axle Suspension in Race Car Calculators

Roll Rate with Anti-Roll Bar

Go Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/(Tyre Vertical Rate*(Rear Track Width^2)/2+Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2)

Rear Track Width given Roll Rate of Suspension with Anti-Roll Bar

Go Rear Track Width = sqrt(2*(Roll Rate*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width)^2/2))/((Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2-Roll Rate)*Tyre Vertical Rate))

Tyre Rate given Roll Rate of Suspension with Anti-Roll Bar

Go Tyre Vertical Rate = (Roll Rate*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/((Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2-Roll Rate)*(Rear Track Width^2)/2)

Spring Track Width given Roll Rate of Suspension with Anti-Roll Bar

Go Spring Track Width = sqrt(2*(((Roll Rate*Tyre Vertical Rate*(Rear Track Width^2)/2)/((Tyre Vertical Rate*(Rear Track Width^2)/2-Roll Rate))-Roll Rate of Anti-Roll Bar)/Wheel Centre Rate))

Vertical Tyre Axle Rate given Roll Rate of Suspension with Anti-Roll Bar

Go Wheel Centre Rate = ((Roll Rate*Tyre Vertical Rate*(Rear Track Width^2)/2)/(Tyre Vertical Rate*(Rear Track Width^2)/2-Roll Rate)-Roll Rate of Anti-Roll Bar)/((Spring Track Width^2)/2)

Roll Rate

Go Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*Wheel Centre Rate*(Spring Track Width^2)/2)/(Tyre Vertical Rate*Rear Track Width^2/2+Wheel Centre Rate*Spring Track Width^2/2)

Rear Track Width given Roll Rate

Go Rear Track Width = sqrt((Roll Rate*Wheel Centre Rate*Spring Track Width^2)/((Wheel Centre Rate*(Spring Track Width^2)/2-Roll Rate)*Tyre Vertical Rate))

Spring Track Width given Roll Rate

Go Spring Track Width = sqrt((Roll Rate*Tyre Vertical Rate*Rear Track Width^2)/((Tyre Vertical Rate*(Rear Track Width^2)/2-Roll Rate)*Wheel Centre Rate))

Tyre Rate given Roll Rate

Go Tyre Vertical Rate = (Roll Rate*(Wheel Centre Rate*(Spring Track Width^2)/2))/((Wheel Centre Rate*(Spring Track Width^2)/2-Roll Rate)*(Rear Track Width^2)/2)

Vertical Tyre Axle Rate given Roll Rate

Go Wheel Centre Rate = (Roll Rate*Tyre Vertical Rate*(Rear Track Width^2)/2)/(Tyre Vertical Rate*(Rear Track Width^2)/2-Roll Rate*(Spring Track Width^2)/2)

Roll Rate with Anti-Roll Bar Formula

What are the main functions of anti-roll bar?

The first function is the reduction of body lean which is dependent on the total roll stiffness of the vehicle. Increasing which does not change the steady-state total load (weight) transfer from the inside wheels to the outside wheels, it only reduces body lean. The total lateral load transfer is determined by the CG height and track width. The other function is to tune the handling balance of a car. Understeer or oversteer behaviour can be tuned out by changing the proportion of the total roll stiffness that comes from the front and rear axles. Increasing the proportion of roll stiffness at the front increases the proportion of total load transfer that front axle reacts to—and decreases proportion that rear axle reacts to. In general, this makes outer front wheel run at a comparatively higher slip angle, and outer rear wheel run at a comparatively lower slip angle, which is an understeer effect. Increasing the proportion of roll stiffness at the rear axle has the opposite effect and decreases understeer.

What are the drawbacks of anti-roll bar?

Because an anti-roll bar connects wheels on opposite sides of the vehicle, the bar transmits the force of a bump on one wheel to the opposite wheel. On rough or broken pavement, anti-roll bars can produce jarring, side-to-side body motions (a "waddling" sensation), which increase in severity with the diameter and stiffness of the sway bars. Other suspension techniques can delay or dampen this effect of the connecting bar. Excessive roll stiffness, typically achieved by configuring an anti-roll bar too aggressively, can make the inside wheels lift off the ground during hard cornering. This can be used to advantage: many front-wheel-drive production cars lift a rear wheel when cornering hard in order to overload the opposite wheel, limiting understeer.

How to Calculate Roll Rate with Anti-Roll Bar?

Roll Rate with Anti-Roll Bar calculator uses Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/(Tyre Vertical Rate*(Rear Track Width^2)/2+Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2) to calculate the Roll Rate, The Roll rate with Anti-roll bar formula used to find the stiffness of the car with an axle suspension and anti-roll bar in the roll mode. Roll Rate is denoted by K_Φ symbol.

How to calculate Roll Rate with Anti-Roll Bar using this online calculator? To use this online calculator for Roll Rate with Anti-Roll Bar, enter Tyre Vertical Rate (K_t), Rear Track Width (t_R), Roll Rate of Anti-Roll Bar (R_arb), Wheel Centre Rate (K_W) & Spring Track Width (T_s) and hit the calculate button. Here is how the Roll Rate with Anti-Roll Bar calculation can be explained with given input values -> 20792.56 = (321300*(1.5^2)/2*(4881.6+42419.8*(0.9^2)/2))/(321300*(1.5^2)/2+4881.6+42419.8*(0.9^2)/2).

FAQ

What is Roll Rate with Anti-Roll Bar?

The Roll rate with Anti-roll bar formula used to find the stiffness of the car with an axle suspension and anti-roll bar in the roll mode and is represented as K_Φ = (K_t*(t_R^2)/2*(R_arb+K_W*(T_s^2)/2))/(K_t*(t_R^2)/2+R_arb+K_W*(T_s^2)/2) or Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/(Tyre Vertical Rate*(Rear Track Width^2)/2+Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2). Tyre Vertical Rate is the spring rate controlled by the tyre compound, sidewall stiffness and operating pressure, Rear Track Width is the distance between the centers of the rear wheels, The Roll Rate of Anti-Roll Bar is the stiffness of the anti-roll bar in the roll mode, Wheel Centre Rate is the vertical force acting on the tire per unit vertical displacement at the location along the spindle corresponding to the wheel centerline, measured relative to the chassis & Spring Track Width is the distance between the springs on either side of the axle.

How to calculate Roll Rate with Anti-Roll Bar?

The Roll rate with Anti-roll bar formula used to find the stiffness of the car with an axle suspension and anti-roll bar in the roll mode is calculated using Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*(Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2))/(Tyre Vertical Rate*(Rear Track Width^2)/2+Roll Rate of Anti-Roll Bar+Wheel Centre Rate*(Spring Track Width^2)/2). To calculate Roll Rate with Anti-Roll Bar, you need Tyre Vertical Rate (K_t), Rear Track Width (t_R), Roll Rate of Anti-Roll Bar (R_arb), Wheel Centre Rate (K_W) & Spring Track Width (T_s). With our tool, you need to enter the respective value for Tyre Vertical Rate, Rear Track Width, Roll Rate of Anti-Roll Bar, Wheel Centre Rate & Spring Track Width and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.

How many ways are there to calculate Roll Rate?

In this formula, Roll Rate uses Tyre Vertical Rate, Rear Track Width, Roll Rate of Anti-Roll Bar, Wheel Centre Rate & Spring Track Width. We can use 1 other way(s) to calculate the same, which is/are as follows -

Roll Rate = (Tyre Vertical Rate*(Rear Track Width^2)/2*Wheel Centre Rate*(Spring Track Width^2)/2)/(Tyre Vertical Rate*Rear Track Width^2/2+Wheel Centre Rate*Spring Track Width^2/2)

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