Radial Stiffness and Damping of Mountain Bike Tires Subject to Impact Determined Using the Coefficient of Restitution
James Sadauckas, 
Mark Nagurka Mountain bikes are essentially ruggedized versions of a standard bicycle with geometry and features suited to off-road cycling. Their intended use case has them encountering jumps, bumps, drops, and impact. As such, their designs have evolved to include front and often rear suspension systems composed of springs and dampers in sometimes elaborate kinematic arrangements. Despite extensive design resources spent optimizing and tuning mountain bike suspension systems, very little has been published regarding the in-plane behavior of their tires or bicycle tires in general. In this work, a drop test bench is utilized to measure the dynamic radial stiffness and damping of numerous mountain bike tires including four common sizes spanning 29er, plus-sized, and fat tire variants, as well as various constructions ranging from trail, through enduro, and downhill. The tire is treated as a classical, lumped Kelvin-Voigt model with a parallel arrangement of a spring and damper. Identification of the system parameters is accomplished by treating the tire as a bouncing ball and using the coefficient of restitution from pre and post impact velocities to determine the dynamic stiffness and damping. The advantages of this approach in comparison to a classical logarithmic-decrement approach are discussed. Results suggest that due to its viscoelastic nature mountain bike tire dynamic radial stiffness is appreciably higher than its quasi-static value. Furthermore, although its damping is relatively low (spanning 2 to 5% of critical), it can affect subjective “trail feel” and can be perceptibly influenced by tire selection, size, construction, and inflation pressure.
Show LessType of the Paper: Extended Abstract
Radial Stiffness and Damping of Mountain Bike Tires Subject to Impact Determined Using the Coefficient of Restitution
J. Sadauckas1,*, N. Schoeneck2, and M. Nagurka3
1Trek Performance Research, Trek Bicycle Corporation, Waterloo, WI, 53594, USA; jim_sadauckas@trekbikes.com; ORCID 0000-0002-6055-9047
2Vehicle Measurements Group, Harley-Davidson Motor Company, Yucca AZ 86438, USA; noah.schoeneck@harley-davidson.com
3Professor Emeritus, Dept of Mechanical Engineering, Marquette University, Milwaukee, WI 53233, USA; mark.nagurka@marquette.edu
*corresponding author.
Name of Editor: Jason Moore
Submitted: 28/02/2023
Accepted: 13/04/2023
Published: 26/04/2023
Citation: Sadauckas, J., Schoeneck, N. & Nagurka, M. (2023). Radial Stiffness and Damping of Mountain Bike Tires Subject to Impact Determined Using the Coefficient of Restitution. The Evolving Scholar - BMD 2023, 5th Edition.
This work is licensed under a Creative Commons Attribution License (CC-BY).
Abstract:
Mountain bikes are essentially ruggedized versions of a standard bicycle with geometry and features suited to off-road cycling. Their intended use case has them encountering jumps, bumps, drops, and impact. As such, their designs have evolved to include front and often rear suspension systems composed of springs and dampers in sometimes elaborate kinematic arrangements. Despite extensive design resources spent optimizing and tuning these suspension systems, very little has been published regarding the in-plane behavior of their tires or bicycle tires in general (Knuit, 2014). In this work, a drop test bench is utilized to measure the dynamic radial stiffness and damping of numerous mountain bike tires including four common sizes spanning 29er, plus-sized, and fat tire variants (Dressel, 2020), as well as various constructions ranging from trail, through enduro, and downhill. The tire is treated as a classical, lumped Kelvin-Voigt model with a parallel arrangement of a spring and damper (Acosta, 2020).
Identification of the system parameters is accomplished by treating the tire as a bouncing ball and using the coefficient of restitution from pre and post impact velocities to determine the dynamic stiffness and damping (Nagurka, 2006). The advantages of this approach in comparison to a classical logarithmic-decrement approach (Wong, 2001; Cuong, 2013) are discussed.
Figure 1. Graphs of wheel acceleration, displacement, and velocity versus time (left) with system parameters derived per bounce (right) from coefficient of restitution during impact (green), contact (blue), and in contrast to log-decrement (magenta).
The coefficient of restitution \(\mathbf{\varepsilon}_{\mathbf{n}}\) at each tire bounce \(\mathbf{n}\) is expressed as a velocity ratio:
| \[\varepsilon_{n} = \left| \frac{{\dot{x}}_{n\ post}}{{\dot{x}}_{n\ pre}} \right|\] | (1) |
|---|
where \({\dot{\mathbf{x}}}_{\mathbf{n}}\) is the sled velocity and the subscripts indicate the pre- and post-contact velocities separated by contact time ΔT. Stiffness \(\mathbf{k}_{\mathbf{n}}\) can be determined given sled mass \(\mathbf{m}\):
| \[k_{n} = m\left( \frac{\pi}{{\Delta T}_{n}} \right)^{2}\left\lbrack 1 + \left( \frac{\ln\ \varepsilon_{n}\ }{\pi} \right)^{2} \right\rbrack\] | (2) |
|---|
The damping ratio \(\mathbf{\zeta}_{\mathbf{n}}\mathbf{\ }\)can be determined solely from the coefficient of restitution \(\mathbf{\varepsilon}_{\mathbf{n}}\):
| \[\zeta_{n} = \frac{- \ ln\ \varepsilon_{n}\ }{\pi}\left\lbrack 1 + \left( \frac{\ln\ \varepsilon_{n}\ }{\pi} \right)^{2} \right\rbrack^{- \frac{1}{2}}\] | (3) |
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Moreover, tire impact footprints are collected and compared to their much smaller static counterparts.
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| Figure 1. Tire impact footprints (black) for 29 × 2.3”, 27.5 × 2.8”, 29 × 3”, and 26 × 4” knobby tires with static footprints overlaid (color). Note “stadium” (rectangle with radiused corners) shape of former vs. more traditional ellipse of latter (1:5 scale). |
Results suggest that due to its viscoelastic nature mountain bike tire dynamic radial stiffness is appreciably higher than its quasi-static value. Furthermore, although its damping is relatively low (spanning 2 to 5% of critical), it can affect subjective “trail feel” and can be perceptibly influenced by tire selection, size, construction, and inflation pressure. The measured dynamic radial stiffness and damping values are critical for subsequent numerical and multibody modeling of the tire’s contribution to the suspension system (Schoeneck, 2023) with the goal of improving performance, safety, comfort, and control of two-wheelers. Additional attempts to isolate the contribution of contact patch friction, or “scrub,” to the dynamic radial stiffness and damping using ice, gel, or other low friction material proved challenging and may be explored in future work. Further implications relating tire damping to rolling resistance (Ejsmont, 2019) may also be the subject of future work.
References
Acosta, E. & Castillo, J., Cabrera, J., Velasco García, J. M., Fernández, J. & Alcazar, M. (2020). Modeling of tire vertical behavior using a test bench. IEEE Access. PP. 1-1. 10.1109/ACCESS.2020.3000533.
Cuong, D., Zhu, S., Hung, D., & Ngoc, N. (2013). Study on the vertical stiffness and damping coefficient of tractor tire using semi-empirical model. Hue University Journal of Science. 83. 5-15. 10.26459/jard.v83i5.3071.
Dressel, A., & Sadauckas, J. (2020). Characterization and modeling of various sized mountain bike tires and the effects of tire tread knobs and inflation pressure. Applied Sciences, 10(9), 3156. https://doi.org/10.3390/app10093156.
Ejsmont, J. & Owczarzak, W. (2019). Engineering method of tire rolling resistance evaluation. Measurement. 145. 10.1016/j.measurement.2019.05.071
Knuit, J., De Kok, F., Raaphorst, P. & Van der Spek, A. (2014). Bicycle tire stiffness and damping. TU Delft. http://bicycle.tudelft.nl/tiretestingvertical/
Nagurka, M.L., & Huang, S. (2006). A mass-spring-damper model of a bouncing ball. International Journal of Engineering Education, Vol. 22, No. 2, 393-401.
Schoeneck, N., Sadauckas, J., & Nagurka, M. (2023, Oct. 18-20). Identification and modeling of a mountain bike front suspension subsystem equipped with a telescopic fork and tire damping. Abstract submitted, BMD2023, Delft University, Netherlands.
Wong, J.Y. (1993). Theory of Ground Vehicles. John Wiley & Sons, New York, 2 ed.
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More details
- License: CC BY
- Review type: Open Review
- Publication type: Extended Abstract
- Submission date: 28 February 2023
- Conference: The Evolving Scholar - BMD 2023, 5th Edition
Citation
Sadauckas, J., Schoeneck, N. & Nagurka, M. (2023). Radial Stiffness and Damping of Mountain Bike Tires Subject to Impact Determined Using the Coefficient of Restitution. The Evolving Scholar - BMD 2023, 5th Edition. https://doi.org/10.24404/63fd8435fca7904adf915064
