Rolling Friction Coefficient Of An AWANA Kit Wheel

An experiment was run to determine the tread friction coefficient of an AWANA kit wheel on a rough track. Equations from an analysis of the rolling friction model were used to calculate the value and determine how much measurement error affected the result.

Geometric Measurements

The equation for the tread friction coefficient in terms of the coasting distance, yc, is:

u = -tanO/(yc/Lrcos3O+1) = -cos2OsinO/(yc/Lr+cos3O)

Coasting Measurements

After lots of practice, 10 trials were conducted. The coasting distance was measured. But in two of the trials, the release of the wheel was suspicious and the distance measured was too far because of it. These two values were ignored.

The average value was recorded below for the value of yc. The measurements were taken to the nearest 1/16 inch. The measurements had a (n-1) standard deviation of 0.5121 inches.

Measurements for the track are also recorded below to the nearest 1/16 inch.

Calculations

Using the geometric values and the average value for yc = 83.75, u is:

u = -cos2(0.07108)sin(-0.07108)/(83.75/22+cos3(0.07108))

u = 0.9949561366722*0.0710201614177/(3.806818181818+0.9924437532524)

u = 0.07066194542928/4.79926193507

u = 0.01472350256879

Using yc = 82.9375 the lowest value, we get

u = 0.07066194542928/(3.769886363636+0.9924437532524)

u = 0.01483768317083

And using yc = 84.3125 the highest value, we get

u = 0.07066194542928/(3.832386363636+0.9924437532524)

u = 0.01464547843497

Results

In summary,

How far off is the value we got for the tread friction coefficient?

Below, partial derivatives from the sensitivity analysis, are evaluated 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.

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

du/dyc = -0.07066194542928/22(3.806818181818+0.9924437532524)2

du/dyc = -0.07066194542928/22(4.79926193507)2

du/dyc = -0.07066194542928/(22*23.03291512141)

du/dyc = -0.07066194542928/506.724132671

du/dyc = -0.0001394485497593 1/in

Since the error in each measurement of yc was 1/32 in, we get a small error in the value of u. This small deviation is called "delta u in yc".

(du/dyc)Dyc = -0.0001394485497593 times 1/32 = 0.00000435775

So if in measuring the coasting distance we were off by 1/32 of an inch, the calculated value of the tread friction coefficient would be off by about 4 millionths!

But for each inch of deviation in measuring the coasting distance, the coefficient of rolling friction changes only by about a ten thousandth. This analysis shows that we only need to keep upto four significant digits for the value of the coefficient since the standard deviation was only half an inch.

To do a thorough job, we would find the partial derivative of u with respect to H, du/dH, and Lr, du/dLr and determine how much a deviation of 1/32 inch in these measurements affects the rolling coefficient. Adding all of the deviations together gives a better idea (an upper limit) of how far off the value for u might be.

For the deviation in Ramp height, H, we get from the analysis,

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

Using the measured and average values from the data, (* indicates multiplication)

du/dH = -22*0.9974748802212*(3*83.75*0.005043863327797-83.75-22*0.9924437532524)/(83.75+22*0.9924437532524)2\[22*22-1.5625*1.5625]

du/dH = 21.94444736486*(104.3164919106)/(105.5837625715)2\[481.55859375]

du/dH = 2289.167766019/11147.93091875*21.94444334564

du/dH = 2289.167766019/244635.1384676

du/dH = 0.009357477345072 1/in

Since the error in measureing H was 1/32 in, we get a small error in the value of u. This small deviation is called "delta u in H".

(du/dH)DH = 0.009357477345072 times 1/32 = 0.000292421

So if in measuring the ramp height we were off by 1/32 of an inch, the calculated value of the tread friction coefficient would be off by about 3 tenthousandths!

For the deviation in Ramp Length, Lr, we get from the analysis,

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

Before evaluating using the measured and average values from the data, we notice that du/dLr is the same as du/dH except for the Lr and H at the front! So we can borrow all the numbers except the initial Lr. We get,

du/dLr = 1.5625*0.9974748802212*(104.3164919106)/244635.1384676

du/dLr = 162.5829379275/244635.1384676

du/dLr = 0.000664593 1/in

Since the error in measureing Lr was 1/32 in, we get a small error in the value of u. This small deviation is called "delta u in Lr".

(du/dLr)DLr = 0.000664593 times 1/32 = 0.0000207685

So if in measuring the ramp length we were off by 1/32 of an inch, the calculated value of the tread friction coefficient would be off by about 2 hundredthousandths!

Total possible deviation

Adding the delta u's we get a total delta or deviation of

(du/yc)Dyc + (du/dH)DH + (du/Lr)DLr = 0.00000435775 + 0.000292421 + 0.0000207685

= 0.00031754725

This total means that our calculation of the value of the coefficient is off by at most 3 in ten thousand. Looking at the partial deltas of u that were added tells us something else that is important to know while setting up the experiment. Note that the delta of u in H is almost as large as the total.

This means that a small error in measuring the height of the ramp is 14 times more likely to show up as error in the calculated value of the coefficient than small errors in measuring the ramp length and 67 times more likely than errors in the coasting distance!