Grand Prix Racing - The Science of Fast Pinewood Cars

Measuring Rolling Friction Coefficient

Measuring the rolling friction coefficient of a wheel is not difficult. As you will see, it can be done knowing nothing about the wheel itself! You don't need to know how much it weighs, what its moment of inertia is, or even what its radius is! The method to determine its value is an experiment you can do yourself. But read on if you want to see how the rolling friction model was used to get an expression for calculating the coefficient in the experiment. You will also read about how the measurements you take affect the calculated value of the tread friction coefficient.

In related reading, see how tread friction is rolling friction, and look at the results of the experiment preformed with an AWANA kit wheel.

Equations from the Rolling Friction Model

Starting from the rolling friction model, we pick expressions for the variables we want to measure in the experiment. In particular, the coasting distance. But to do this, we need to know what some of the model symbols (variables) are.

Model Variables
Symbol Description
H Height of the staring line
Lr Length of ramp surface
O 'Theta', angle the ramp forms with horizon
yc Coasting distance on the flat measured from the join
u Rolling drag coefficient
m Mass
g Acceleration by gravity
v0 Speed of the wheel at the bottom of the ramp
Df Tread drag on the flat
Dr Tread drag on the ramp
M Inertial mass, the effective mass of a spinning object

We know yc and the geometry of the track, H, Lr and O. So we find the model equation that relates the coasting distance to tread friction.

yc = v02M/2Df

Now we need to find a model expression for v0 that introduces as few other symbols as possible.

v0 = \[2Lr(-mgsinO-Dr)/M]cosO

Now find a model expression for Dr and Df.

Dr = DfcosO

Df = umg

What about M? After solving equations like these for some time and having a "feeling" for the physics they represent, you get an idea about which variables are important and which ones will "drop" out. m, g and M will drop out - just watch!

Now by expanding symbols an inverse relation between coasting distance and tread friction can be found.

Plug Df into Dr

Dr = umgcosO

Plug Dr into v0 and simplify

v0 = \[-2mgLr(sinO+ucosO)/M]cosO

Plug v0 and Df into yc

\[-2mgLr(sinO+ucosO)/M]2cos2OM/2umg

Simplify noting that \[X]2 = X (for positive X) and so g, m and M end up above and below the division bar and become 1, that is, they drop out. We get:

yc = -Lrcos2O(sinO/u+cosO)

Solve for u: distribute

yc = -Lrcos2OsinO/u-Lrcos3O

"Move" terms so the tread coefficient, u, is on the left

u(yc+Lrcos3O) = -Lrcos2OsinO

Isolate u by dividing through

u = -cos2OsinO/(yc/Lr+cos3O)

For purposes of computation, one of the trigonomic functions can be shuffled away by dividing by cos3O and replacing sinO/cosO by tanO:

u = -tanO/(yc/Lrcos3O+1)

Tread Friction Coefficient
u = -tanO/(yc/Lrcos3O+1)

Sensitivity of results to measurements

Extending the analysis more rigorously, take partial derivatives of u to determine the sensitivity of the model equation to measured values. For example, we can see how sensitive the equation is to measurement of the coasting distance.

Sensitivity to coasting distance measurement

du/dyc = d(-cos2OsinO/(yc/Lr+cos3O))/dyc

Take out the constants

du/dyc = -cos2OsinOd(1/(yc/Lr+cos3O))/dyc

differentiate according to the pattern -Ad(1/U)/dX = A(dU/dX)/U2

du/dyc = cos2OsinOd(yc/Lr+cos3O)/dyc)/(yc/Lr+cos3O)2

differentiate according to the pattern Ad(X/B+C)/dX = Ad(X/B)/dX + AdC/dX = A/B + 0

du/dyc = cos2OsinO/Lr(yc/Lr+cos3O)2 1/in

To find how much error in the measurement of coasting distance can affect the tread friction coefficient, we have to know how good the measurement was. In the experiment, the coasting distance was measured to the nearest 1/16 of an inch. That means it was no farther off than 1/32 of an inch! It is customary to call this tiny error "delta". The measurement delta is 1/32 inch. If we use our experimental measurements to evaluate the expression for du/dyc and multiply the result by "delta yc" (=1/32), then we can know how much our value of u might be off by because of the small error in yc. We write delta yc as Dyc. Then, the maximum error or deviation in u caused by measuring yc is

(du/dyc)Dyc (deviations are conventionally evaluated as positive values)

Continuing in this way, we find the all the partials of u with respect to the measured variables and all of the deltas of the measurements. Adding all of the resulting deviations together gives us a total value for the deviation in u that can be caused by measurement. This upper bound on the error introduced by the various measurments is written as:

Du = (du/yc)Dyc + (du/dO)DO

But we'd like to express Du directly in terms of the partials and deviations in yc, Lr and H. To do this, we use the chain rule of differentials. It tells us that

(du/dLr)DLr = (du/dO)(dO/dLr)DLr and (du/dH)Dh = (du/dO)(dO/dH)DH

Since the total deviation in the ramp angle is DO = (dO/dLr)DLr + (dO/dH)DH, the total deviation, Du, can be written,

Du = (du/dyc)Dyc + (du/dLr)DLr + (du/dH)DH

All we have to do to find Du is find expressions for the partial derivatives and use our measurement errors for the deviations, Dyc, DLr and DH.

Sensitivity of ramp angle to ramp height measurement

Because the ramp angle, O, depends on two measurements, it has to be treated the same way to get a total deviation. It's total deviation is it's delta!

From O = arcsin(H/Lr) we get

dO/dH = darcsin(H/Lr)/dH

Differentiate according to the pattern darcsinU/dX = (dU/dX)/\[1-U2]

dO/dH = (d(H/Lr)/dH)/\[1-H2/Lr2]

Differentiate according to the pattern Ad(X/B)/dX = A/B and simplify

dO/dH = 1/Lr\[1-H2/Lr2] = 1/\[Lr2-H2] rad/in

Sensitivity of ramp angle to ramp length measurement

From O = arcsin(H/Lr) we get, similar to the previous case,

dO/dLr = darcsin(H/Lr)/dLr

Differentiate according to the pattern darcsinU/dX = (dU/dX)/\[1-U2]

dO/dLr = (d(H/Lr)/dLr)/\[1-H2/Lr2]

Differentiate according to the pattern Ad(B/X)/dX = -AB/X2 and simplify

dO/dLr = -H/\[1-H2/Lr2]Lr2 = -H/Lr\[Lr2-H2] rad/in

Sensitivity to ramp angle calculation

Using u = -tanO/(yc/Lrcos3O+1) this time because O appears fewer times to make things a bit easier,

du/dO = d(-tanO/(yc/Lrcos3O+1))/dO

Apply the chain rule to get

du/dO = -(dtanO/dO)/(yc/Lrcos3O+1) + -tanOd(1/(yc/Lrcos3O+1))/dO

We will differentiate these two terms seperatly then add them back together.

Differentiate the first using the pattern Adtan(O)/dO = A/cos2O

-(dtanO/dO)/(yc/Lrcos3O+1) = -1/(yc/Lrcos3O+1)cos2O

Differentiate the second according to the pattern -Ad(1/U)/dX = A(dU/dX)/U2

-tanOd(1/(yc/Lrcos3O+1))/dO = -tanO(d(yc/Lrcos3O+1)/dO)/(yc/Lrcos3O+1)2

Differentiate again according to the pattern -Ad(B/U+C)/dX = AB(dU/dX)/U2

-tanOd(1/(yc/Lrcos3O+1))/dO = -yctanO(d(Lrcos3O)/dO)/(yc/Lrcos3O+1)2Lr2cos6O

Differentiate according to the pattern -Ad(BU3)/dX = 3ABU2(dU/dX)

-tanOd(1/(yc/Lrcos3O+1))/dO = -3ycLrtanOcos2O(d(cosO)/dO)/(yc/Lrcos3O+1)2Lr2cos6O

Differentiate according to the pattern -AdcosX/dX = AsinX

-tanOd(1/(yc/Lrcos3O+1))/dO = 3ycLrtanOcos2OsinO/(yc/Lrcos3O+1)2Lr2cos6O

Simplify by recognizing tanO = sinO/cosO, etc.

-tanOd(1/(yc/Lrcos3O+1))/dO = 3ycsin2O/(yc/Lrcos3O+1)2Lrcos5O

Now add the two terms back together

du/dO = -1/(yc/Lrcos3O+1)cos2O + 3ycsin2O/(yc/Lrcos3O+1)2Lrcos5O

Make the denominator of the first term the same as the second and combine

du/dO = (3ycsin2O-Lrcos3O(yc/Lrcos3O+1))/(yc/Lrcos3O+1)2Lrcos5O

Simplify by distributing Lrcos3O above and below the division line

du/dO = LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2

Sensitivity to ramp length mesurement

This one is really already done! Using the chain rule,

du/dLr = (du/dO)(dO/dLr)

Substituting from the results above,

du/dLr = (LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2) (-H/Lr\[Lr2-H2])

Simplifying

du/dLr = -HcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2\[Lr2-H2] 1/in

Sensitivity to ramp height mesurement

This one is also really already done! Using the chain rule,

du/dH = (du/dO)(dO/dH)

Substituting from the results above,

du/dH = (LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2) (1/\[Lr2-H2])

Simplifying

du/dH = -LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2\[Lr2-H2] 1/in

Sensitivity of Calculated Ramp Angle to Measurements

Measurment Expresson Description
H (in) dO/dH = 1/\[Lr2-H2] rad/in Height of the staring line
Lr (in) dO/dLr = -H/Lr\[Lr2-H2] rad/in Length of ramp surface

Sensitivity of Tread Friction Coeficient to Measurements

Measurment Expresson Description
O (rad) du/dO = LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2 1/rad 'Theta', angle the ramp forms with horizon
Lr (in) du/dLr = -HcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2\[Lr2-H2] 1/in Length of ramp surface
H (in) du/dH = -LrcosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2\[Lr2-H2] 1/in Height of the staring line
yc (in) du/dyc = cos2OsinO/Lr(yc/Lr+cos3O)2 1/in Coasting distance on the flat measured from the join

Total Sensitivity of Tread Friction Coeficient to Measurements

Substituting the partials from the table into the total deviation expression gives us:

Du = cos2OsinODyc/Lr(yc/Lr+cos3O)2 - (HDLr + LrDH)cosO(3ycsin2O-yc-Lrcos3O)/(yc+Lrcos3O)2\[Lr2-H2]

where it was recognized that the partials of u in Lr and H had a long common factor and so simplified. By factoring out Lr in the denominator of the first term, it can be written as,

Lrcos2OsinODyc/(yc+Lrcos3O)2

The denominator in this term is also found in the other term, so we can simplify to

Du = (Lrcos2OsinODyc - (HDLr + LrDH)cosO(3ycsin2O-yc-Lrcos3O)/\[Lr2-H2])/(yc+Lrcos3O)2

Finally, by recognizing one more common factor, cosO, between the two terms,

Du = (du/dyc)Dyc + (du/dLr)DLr + (du/dH)DH
Du = (LrcosOsinODyc - (HDLr + LrDH)(3ycsin2O-yc-Lrcos3O)/\[Lr2-H2])cosO/(yc+Lrcos3O)2
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Grand Prix Racing - The Science of Fast Pinewood Cars
Copyright © 1997, 2004 by Michael Lastufka, All rights reserved worldwide.