Wheel / Tire Weights and Rotational Inertia
Car and Wheel Dimensions:
Wheel size (inches) Tire width (mm) aspect ratio wheel weight (lbs) tire weight (lbs) Total wheel weight Car 0-60mph time Note: Units for inertia are slug feet^2 I1=Rotational Inertia 1, single uniform density disk I2=Rotational Inertia 2, Tire Disc + Wheel Disc wheel alone (solid disc) tire alone (hollow disc) I3=Rotational Inertia 3, Two Discs, Edge Weighted wheel alone tire alone 0.783 0.713 0.145 0.568 0.893 0.193 0.699 0.878 0.859 0.231 0.627 1.049 0.309 0.740 Formulas I=1/2 (MR^2) Implied Radius of Gyration Rgy = .71Rt wheel 1 16 205 60 21.0 23.0 44.0 6.0 wheel 2 18 235 40 26.5 24.0 50.5
Derived dimensions
Wheel Radius (feet) Tire sidewall height (feet) Tire radius (feet) wheel mass (slugs) tire mass (slugs) total mass (slugs) Linear Acceleration in ft/sec^2 Wheel rotat'l accel'n radians/sec^2
wheel 1 0.67 0.40 1.070 0.65 0.71 1.37 14.67 13.70
wheel 2 0.75 0.31 1.058 0.82 0.75 1.57
(~same)
I=Iw+It Iw=1/2(MR^2) Rgy = .71Rw It=1/2M (Ra^2+Rb^2) Rgy = Rw + .54(Rt-Rw) I=Iw+It Iw=2/3(MR^2) It=M (Rg^2) Tq difference Per Wheel -1.3 -2.0 -2.1 -3.2
approx
Rgy = .82Rw Rgy = Rw + .8(Rt-Rw) Torque Diff, rot'l+linear -17.9 -20.7 -21.3 Total Added Chassis Weight Equiv, W2 vs W1 -36.8 -42.4 -43.6 -26.0 lbs lbs lbs lbs ( linear only )
Rotational Torque G=IW Uniform density disk, Tire + Wheel Tire Disc + Wheel Disc Tire Disc+ Wheel Disc, Edge Weighted Torque devoted to linear acceleration
Torque Loss Per Wheel, Ft-lbs 10.7 9.8 12.2 21.4 12.0 11.8 14.4 24.6
Torque difference * 4 wheels -5.3 -8.0 -8.6 -12.7
LIMITATIONS: All rotational inertia models are approximations, but model 3 may be the most accurate. The two wheel / tire combos are assumed to have the same total radius (wheel1) when calculating the acceleration rates and torques. NOTES: This spreadsheet is meant to approximately calculate the effects of wheel and tire weight on a car's acceleration. It calculates how many ft lbs of torque are needed to accelerate a wheel at a certain rate. The rate is specified by the amount of time it takes the car to accelerate from 0 to 60 mph. The equivalent added weight reflects the change from wheel1 to wheel2. It does not vary with acceleration. This spreadsheet has 3 models for determining the rotational inertia of the wheel and tire combination. All of these are approximations, since individual wheel and tire sizes will have different weight distributions, even with the same mass and diameters. Model 1 assumes that the wheel/tire combination is a uniform density cylinder, like a hockey puck. Model 2 assumes that the wheel and tire are each uniform density, but that their rotational inertias are calculated separately Model 3 assumes that the wheel and tire are each more dense on the edge than in the center. ( see Rgy sheet ) As a physics refresher, a pound is a unit of force, and a slug is a unit of mass. In calculating the inertia of a wheel, I convert the weight of the wheel (mass * force of gravity) to it's mass in slugs. 1 pound force= 1 slug mass * 1 ft/sec^2 acceleration. 1 slug * gravity = 32.2 pounds. mass of wheel in slugs = weight of wheel / 32.2 ft/sec^2 revised 1-4-05 K Kelleher original 1st sheet by ?
RADIUS OF GYRATION
Rgy
Radius where mass can be assumed concentrated as a thin hoop
SOLD DISC Rgy = .707 Ro ( use Rgy = .82 Ro for wheel )
HOLLOW DISC Rgy = SQRT((Ro^2 + Ri^2)/2)
Rgy ~ Ri + .535 ( Ro-Ri ) for typical tire R's, 50 series or less
HOLLOW DISC Rgy = SQRT((Ro^2 + Ri^2)/2) Rgy ~ Ri + .535 ( Ro-Ri ) for typical tire R's, 50 series or less
TIRE Rgy = ????? Rgy ~ Ri + .8 ( Ro - Ri)
PARASITIC LOSSES DUE TO FLYWHEEL AND CLUTCH INERTIA
rpm1 rpm2 time (sec) Setup 1 FLYWHEEL OD Weight Inertia Average Engine Torque Loss Average Engine HP loss Maximum Engine HP loss OD Weight Inertia Average Engine Torque Loss Average Engine HP loss Maximum Engine HP loss 13.0 in 20.0 lbs 0.364 slug ft^2 15.0 in 11.5 lbs 0.279 slug ft^2 3000 6500 3.0 rpm change in gear
Setup 2
44.5 ft-lbs 40.3 hp 55.1 hp CLUTCH 11.0 in 15.0 lbs 0.196 slug ft^2
34.1 ft-lbs 30.8 hp 42.2 hp 11.0 in 15.0 lbs 0.196 slug ft^2
23.9 ft-lbs 21.6 hp 29.6 hp
23.9 ft-lbs 21.6 hp 29.6 hp
RPM1
assumes flat eng torque through rpm spread, with constant acceleration assumes discs, use od of clutch plate housing, not the flange
REDUCED HP AND TORQUE TO THE TRANSMISSION
RPM1
RPM2