This is great review on drum testing methodology, including this equation (11).
Data Structure and Quadratic Trend Fit
Rolling resistance test data consist of a single wheel test extrapolated to a four‑wheel total rolling power, versus speed, across multiple wheel–surface combinations. For each single wheel test, we fit a quadratic model to characterize the power curve across various speeds:
P(v) = a·v^2 + b·v + c (Eq. 2)
where P is power in watts and v is speed in mph. The coefficients [a, b, c] are fit by least‑squares regression. This model captures the nonlinear increase in rolling losses at higher speeds and allows interpolation.
The fit yields both optimal parameters and a covariance matrix. For speed v, the Jacobian vector is J = [v^2, v, 1]. The predictive variance of power is var(P) = J · pcov · J^T, and the uncertainty is σ_P = sqrt(var(P)). This yields 1‑σ confidence bands around the predicted power curve, which exhibits inconsistencies due to real-world noise and vibration in the test rig.
If uncertainty is above an unacceptable threshold (unlikely) the test is redone.
The coefficient of rolling resistance is described as:
C_rr = P_roll,1 / (N · v) (Eq. 3)
where N is the normal load on a single wheel, P_roll is the single wheel resistance, and v is speed in m/s. Tables of rolling resistance are then kept to apply to predictive estimates in the power model below, to quantify impacts of the wheels in real world scenarios.
Methods II — System Model for Power Calculations
Intro to Resistive Forces
An important part of power calculation is to understand that the longboard system is more complicated than wheel rolling resistance, with a major component being aerodynamic resistance, gravity on inclines, and bearing friction as well, to define our total resistive losses while coasting. An in depth review on how this is calculated can be found here: (resistive forces).
In cycling, running, rowing, our input force that goes toward forward movement is referred to as power. Best expressed as joules, the more commonly used term is watts, as joules normalized time. One should note that watts are the exact amount of energy applied towards motion, and Kilojoules (KJ’s) are the total energy expended towards motion in a workout, which is different than energy you burn, calories, which depend on bodily efficiency. Because power is the actual force applied towards work (movement) it is the best way to quantify performance in sports like running, where heart rate or speed are dependent on many factors such as efficiency on the day.
More on power here: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=38c07018c0636422d9d5a77316216efb3c10164f.
The power we apply, overcomes resistive power, forces such as aerodynamics or rolling resistance. Which is why power is the best term to describe the efficiency of a wheel in a relatable manner
Total resistive power is the sum:
P_total = P_roll + P_bearing + P_aero + P_grade (Eq. 4)
with terms:
P_roll = C_rr · N · v (Eq. 5)
P_bearing = τ_b · (v / r_roll) · n_b (Eq. 6)
P_aero = 0.5 · ρ · C_dA · v^3 (Eq. 7)
P_grade = m_sys · g · sinθ · v (Eq. 8)
Once most of the resistive forces are all quantified in terms of power, such as rolling resistance, that allows us to hone in and approve other mysteries, such as aerodynamics, as well as calculate the impact of equipment changes on your end performance in an event (which we calculate using our wheel data).
Applying Power to the Cyclical Nature of Longboarding.
Since power is a measurement of time-normalized work, it doesn’t capture the degree of work done by a longboarder while exerting themselves. Longboard pushing can be defined as an active “push” phase, and a “coast” phase, combined, which we refer to as a full push “cycle-average”. This is different from cycling, which is constant energy input.
A better way to quantify work done is expressing the average power during the push phase, which expresses the power output of the rider while working. This isn’t a useful metric across an entire activity, though, as it does not account for the lulls in between applying force.
Comparing Longboard Performances to Other Sports Using Normalized Power
Normalized Power was created to describe processes such as these, a metric designed to capture the equivalent work done by an athlete during inconsistent bursts of energy which place a disproportionate strain on the body (https://www.sciencedirect.com/science/article/pii/S2405896315002165).
This is not a perfect metric, but it is the best number we can use to try and relate longboard performances to other power-based sports performances.
