InLine Skate Wheel Power Loss
and "Rolling Resistance"

c. by
P. J. Baum
August 1997

1.: INTRODUCTION

On starting this page I ran across the following quote:

According to Engineering Mechanics: Vol. 1, Statics (2nd ed), JL Meriam and LG Kraige, Wiley and Sons: New York, 1986, the coefficient of ROLLING RESISTANCE, while analogous to the coefficient of static or kinetic friction, is really an entirely different beast. It would be most difficult to describe fully without a free body diagram, but is a function of many factors, including, but not limited to: road and tire deformation and the resultant pressure over the area of contact, elastic and plastic properties of the mating materials, wheel radius, speed of travel, and roughness of the surfaces. Meriam and Kraige state, "... depends on many factors which are difficult to quantify, so that a comprehensive theory of rolling resistance is not available."

Being forewarned I have not attempted too produce a comprehensive theory. Instead this is a simplified model which treats a skate wheel as a radial array of linear springs. The springs are plastic and their spring constant is defineable in terms of the plastic compound's modulus of elasticity or Young's Modulus: Y. The problem is broken down into two parts: first an estimate of the power lost to wheel compression and deformation under the assumption that all the deformation energy is rapidy converted to heat. Having this estimate of the upper limit to power loss an attempt is made to see what portion of this power is actually returned to the wheel in "undeforming" it rather than being lost. In spite of the naivete of the model, a number of interesting scalings are produced which can be tested.

2.: UPPER LIMIT TO WHEEL POWER LOSS AND MODEL PARAMETER DEPENDENCE

The image below illustrates the basic concept. The wheel's tire is considered to be an array of radial springs separated by the blue lines. The number of springs shown is schematic only . The number of springs used is not necessarily the number shown. One spring is sketched in the bottom segment of the tire. The addition of a weight (Mg) produces a deformation from the unloaded case on the left.

The next drawing shows more detail of the wheel geometry. It is presumed to have an outer radius Ro and an inner radius Ri (Ri is the inner radius of the tire or the hub's outer radius, not the bearing radius). The wheel is also described by a smaller radius r where 2*r is the wheel thickness. The tire deformation (dRo) leads to a flat arc length dSx in the direction of travel and a flat arc length dsy in the wheel-axis direction (perpendicular to the travel direction).

From geometry, the arc length in the direction of travel of the wheel is

dSx ~ 2*x = 2*Ro*[(2*dRo/Ro) -(dRo/Ro)**2]**0.5 ~ 2*[2*Ro*dRo]**0.5
While the arc length in the axial direction is
dSy ~ 2*[2*r*dRo]**0.5
Then the area of the spring is estimated from
area ~ dSx*dSy ~ 8*dRo*[r*Ro]**0.5

Next I Use the Young's modulus formula relating force (Force = Mg/N; N = number of wheels on the ground) and deformation:

dRo = Force * (Ro-Ri)/(area*Y)

So, using the earlier formula for area we arrive at the final deformation equation:
dRo = [M*g*(Ro-Ri)/(8*N*Y*{r*Ro}**0.5)]**0.5
Finally I estimate the maximum rolling power as Power ~ Force*dRo*v/dSx (v is the skater's forward speed) to obtain the rolling power upper limit (in watts if you use mks units-- meter, kilogram, second):

All other things being equal, the power needed to elastically compress the wheel while rolling is low for:


Of these parameters, one with significant promise for inline skates I think is the last one: reduce the tire thickness by increasing the hub diameter. It seems to me that this is the direction which CORE Wheels took and may have been more important than the single bearing innovation. In section 4 the wheel width r becomes more important than it seems here.

Now if wheels really achieved this worst case power limit you might expect your wheels to heat up like a light bulb. This is not the case as only some fraction of this power is dissipated while the rest is returned as in a good elastic spring. Just what determines the material loss properties of the wheel compound seems to be a molecular chemistry problem. Good wheels lose only a small fraction of the power limit I calculated. That fraction is estimated in section 4. below.

3.: WHEEL COMPOUND LOSS DATA

So far the closest I have come to actual wheel compound loss data is for SORBOTHANE which is a filled urethane compound developed for its ability to convert vibration and deformation into heat. So an estimate based on this data can be considered a realistic but very bad (high loss) wheel. The graph below is adapted from the site linked above.

For an 80 millimeter diameter wheel travelling at top speed (~30mph or 15 meters/second) the wheel rotates at close to 60 Hz. So, I consider the ground-point of the wheel to be driven with a gravity (weight) force at a frequency of about 60 Hz. Consequently a black line is drawn across the sorbothane data corresponding to 60 Hz on the right. Near room temperature the damping factor or phase angle Delta has a tangent shown on the left as tan(Delta)~0.6 or the sorbothane gives rise to a phase shift of about 30 degrees between the stress and strain in the compound. A lossless compound would have no phase shift and for it tan(Delta)~0.0 This phase shift will be used in section 4 for a simple estimate of the wheel power loss due to viscoelastic damping for a wheel made of sorbothane.

Notice that the Young's (Dynamic) Modulus and the loss curves cross at a temperature just below -15 degrees Celsius. For temperatures below -15 degrees the wheel losses get smaller as the compound becomes harder. However, as the compound is heated above the intersection point the wheel losses get smaller as the compound gets softer. The wheel compression or elastic properties depend on the hardness or Young's Modulus while the losses depend on an effective viscosity and are measured by the phase shift between the driving stress and the resulting strain. The compound is said to be "viscoelastic"and it exhibits high elasticicty as well as an ability to convert vibrational energy to heat .

4.: WHEEL POWER DAMPING ESTIMATE

In the next figure a dashpot representing friction or damping has been added to the wheel's spring. The spring has a spring constant "k" which depends on the elastic properties (Young's Modulus == Y) of the compound while the losses depend on a damping constant (b).

The figure below sets up the equation of motion for the viscoelastic wheel model as driven by an oscillating gravitational force representing the frequency with which a given point on the wheel rotates to hit the ground. N is the number of wheels on the ground. The solution is given for the compression (x) as well as for the damping factor which is is the fraction of vibrational power which is lost to heat.

The following figure rexpresses the damping factor and shows the compression (x==dRo) now expressed in the model terms of section 1 above.

The last figure below shows the final form of the damping ratio and the total power loss of the (N) skate wheels. For typical parameters and using the loss factor tan(Delta)~0.6 I got a wheel power loss for 70 durometer Sorbothane of about 60 watts. Presumably an unfilled urethane compound would have a lot less damping and power loss.

5.: CONCLUSIONS

A formalism for estimating inline skate wheel power losses was developed and applied for the case of a highly lossy filled urethane compound (Sorbothane). I am still searching for loss data for better compounds. Some interesting points result from the parameter scaling of the power:

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