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Effect of Applied-Force Misalignments on Rolling Resistance Measurements Using the Force Method

This study demonstrates that force-based rolling resistance measurements are highly sensitive to geometric misalignments and validates a theoretical model capable of accurately predicting and correcting these alignment-induced errors across both flat and cylindrical testing surfaces.

Original authors: Anna-Mia Brulin, Mats Ainegren, David Sundström

Published 2026-07-13

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Anna-Mia Brulin, Mats Ainegren, David Sundström

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to measure how much a roller-ski wheel "squeaks" against the ground as it rolls. Scientists call this "rolling resistance," and it's a tiny, sneaky force that slows things down. To catch this force, researchers use a special setup where a wheel rolls over a surface while a super-sensitive scale (a load cell) pulls it, trying to measure exactly how hard it has to tug to keep the wheel moving.

But here's the twist: the paper by Anna-Mia Brulin and her team at Mid Sweden University discovered that if your measuring tool is even slightly crooked, your numbers go wild. It's like trying to weigh a feather on a bathroom scale while standing on a wobbly ladder; the result isn't just the weight of the feather—it's the feather plus your wobble.

The "Crooked Pull" Problem
The team built a mathematical model to predict what happens when the pulling force isn't perfectly straight up and down. They found that if the pulling device is tilted even a tiny bit—just 1 milliradian (which is about the angle of a tiny sliver of a degree)—the measured resistance jumps by about 0.001.

Think of it like this: Imagine you are pushing a heavy shopping cart. If you push perfectly straight, you feel the true weight of the cart. But if you push at a slight angle, you end up pushing down into the floor a little bit, too. The floor pushes back harder, and your muscles feel like the cart is heavier than it really is. That's exactly what happens to the roller-ski wheel. If the sensor is tilted, it thinks the wheel is fighting more resistance than it actually is.

The Drum vs. The Flat Road
The researchers tested this on two different surfaces: a flat rubber treadmill (like the ones in gyms) and a giant spinning metal drum (300 mm radius), which is like a giant rolling pin.

On the flat treadmill: The only thing that messed up the numbers was the tilt of the sensor. If the sensor was crooked, the numbers lied.
On the spinning drum: It was even trickier. Not only did the sensor tilt matter, but the position of the wheel's axle mattered too. If the axle was shifted sideways by just 1 millimeter, the error grew even bigger. The model predicted that for every 1 mm the axle was off-center, the resistance reading would jump by about 0.003.

The Proof is in the Pudding
The team didn't just guess; they tested it. They took four different roller-ski wheels (two for "classic" skiing and two for "skate" skiing) and deliberately messed up the alignment. They tilted the sensors by amounts like 10 milliradians (which is a 1% slope) and shifted the axles by 2 mm.

The result? The numbers they measured matched the numbers their math model predicted almost perfectly. The difference between what they measured and what the model said was so small it was barely noticeable—often just a few tiny decimal points. This proves that their math model is a solid tool for fixing these errors. If you know your sensor is tilted by 5 milliradians, you can use their formula to subtract that error and get the true resistance.

What They Didn't Find
It's important to note what this paper didn't say. While the math explained the errors caused by crooked sensors, the researchers noticed that the numbers on the spinning drum were still slightly different from the numbers on the flat treadmill, even after fixing the alignment. They argue that this isn't just a measurement error. Instead, they suggest it's because the wheel squishes differently against a round drum than a flat road. The paper explicitly states that geometry alone (the crooked sensor) isn't enough to explain this difference; the actual physics of the wheel squishing against the curve plays a role too.

The Bottom Line
The main takeaway is simple: If you want to measure how much a roller-ski wheel resists rolling, you have to be incredibly precise. A tiny tilt or a tiny shift in position can make your data look wrong. But thanks to this study, we now have a reliable map (the theoretical model) to correct those mistakes. It's like having a GPS that tells you exactly how much your crooked steering wheel is messing up your speedometer, so you can fix the reading and know the truth.

The study confirms that force-based measurements are highly sensitive to these small geometric misalignments, but with the right math, we can clean up the data and get a clear picture of how these wheels really perform.

Technical Summary: Effect of Applied-Force Misalignments on Rolling Resistance Measurements Using the Force Method

Problem Statement
Accurate quantification of the rolling resistance coefficient ( $C_{RR}$ ) is critical for optimizing wheeled systems, particularly in sports applications like roller skiing where small variations in resistive forces significantly impact performance and experimental validity. While the force method—measuring reaction forces at the wheel axle—is a standard approach in laboratory settings (e.g., drum testing and treadmills), it is highly sensitive to geometric misalignments. Unlike indirect methods (e.g., coast-down or power-based approaches), the force method directly measures the ratio of rolling resistance force to normal force. However, even minute angular deviations between the load-cell axis and the surface normal, or horizontal axle offsets, can introduce substantial relative errors because $C_{RR}$ is typically much smaller than the normal force. Despite strict alignment standards in the automotive industry (e.g., ISO 28580:2018), equivalent standardization and systematic quantification of these alignment errors for roller-ski wheels are lacking. This study addresses the gap by investigating how applied-force misalignments affect $C_{RR}$ measurements and developing a model to predict these errors.

Methodology
The study employed a three-pronged approach: theoretical modeling, experimental verification, and statistical analysis.

Theoretical Model: A geometric model was developed to predict measurement errors ( $C_{RR,lc}$ ) caused by two primary misalignment factors:
- Angular Deviation ( $\alpha_a$ ): The angle between the load-cell z'-axis and the normal direction of the rolling surface.
- Inclination Angle ( $\alpha_c$ ): Caused by horizontal axle offset ( $d_{Fz}$ ) and the displacement of the center of pressure ( $d_{AB}$ ) due to hysteresis, specifically relevant for cylindrical surfaces.
  The model derived equations relating the measured force components to the true $C_{RR}$ , approximating the error as $C_{RR,lc} \approx C_{RR} + \tan(\alpha)$ for small angles.
Experimental Setup: Verification was conducted using two distinct setups:
- Flat Surface: A motor-driven treadmill with a rubber mat.
- Cylindrical Surface: A rotating aluminum drum (radius 300 mm).
- Instrumentation: A six-axis load cell (ATI Industrial Automation) mounted on a screw jack with ball bearings to constrain motion and minimize unintended angular deviations.
- Subjects: Four types of Swenor roller-ski wheels (classic and skate, race and standard variants) were tested.
- Procedure: Measurements were taken at a constant speed (20 km/h) and normal load (300 N). On the cylinder, horizontal axle offsets were varied from -2 mm to +2 mm. On the treadmill, angular deviations were adjusted from -10 mrad to +10 mrad. Each condition was tested 10 times per wheel.
Statistical Analysis: Measured mean values were compared against theoretical predictions using one-sample t-tests. Repeated-measures ANOVA was used to assess the significance of misalignment levels on $C_{RR}$ , with Bonferroni post hoc tests for multiple comparisons.

Key Results

Sensitivity to Misalignment: The study confirmed that force-based $C_{RR}$ $C_{R R}$ measurements are highly sensitive to small geometric misalignments.
- Angular Deviation ( $\alpha_a$ ): On both flat and cylindrical surfaces, $C_{RR}$ increased by approximately 0.001 for every 1 mrad of angular deviation.
- Horizontal Axle Offset ( $d_{Fz}$ ): On the cylindrical surface, $C_{RR}$ increased by approximately 0.0029 to 0.0030 for every 1 mm of offset, depending on the wheel type.
Model Validation: The experimentally measured rolling resistance coefficients showed strong agreement with the values predicted by the theoretical model across all tested conditions. While statistical tests indicated significant differences between measured and predicted values (p < 0.05) in many cases, the magnitude of the residuals was negligible, suggesting the discrepancies were due to measurement precision rather than model failure.
Surface Differences: The study noted systematic differences between measurements on the cylindrical drum and the flat treadmill that the geometric misalignment model alone could not fully explain. The authors attribute this partly to "stress effects" (higher strain on the cylindrical surface) and differences in surface material (metal vs. rubber), suggesting that $C_{RR}$ measured on a 300 mm drum may need to be reduced by 5.6–8% to estimate flat-surface values.

Significance and Claims
The paper claims that the proposed theoretical model provides a practical tool for predicting and correcting alignment-induced measurement errors in laboratory-based rolling resistance testing. By demonstrating that the model adequately captures the dominant effects of force misalignment, the study highlights the critical importance of precise force-transducer alignment and wheel positioning.

The authors emphasize that while the model allows for the correction of known misalignments, the primary contribution is the validation that even sub-degree angular deviations (e.g., the 1 mrad standard in automotive testing) can produce relatively large errors in low $C_{RR}$ scenarios. Consequently, the study advocates for rigorous alignment protocols to ensure the validity and comparability of rolling resistance data across different sports engineering studies, particularly for roller skiing. The authors modestly note that factors such as surface roughness, thermal conditions, and dynamic loading remain areas requiring further investigation beyond the scope of this geometric analysis.

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