# Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## Featured Application

**The presented Magic Formula model of a bicycle tire for cargo trailers can be used for the mathematical modeling and simulation of a bicycle trailer or vehicle without camber angle. Further, the model can be used in linearized or original form for the implementation of a vehicle dynamics controller on an ECU to consider the tire behavior.**

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Measurements

#### 2.1.1. Mobile Tire Testing Laboratory

- Longitudinal tire force ${F}_{t,x}$;
- Lateral tire force ${F}_{t,y}$;
- Vertical tire force ${F}_{t,z}$;
- Torque around x-axis ${T}_{t,x}$;
- Torque around z-axis, also referred to as aligning torque ${T}_{t,z}$.

- Spinning the test wheel faster than the truck’s constant speed ${v}_{v,x}$ and recording ${F}_{t,x}$ respective to the longitudinal slip $\kappa $ (driving);
- Spinning the test wheel slower than the truck’s constant speed ${v}_{v,x}$ and recording ${F}_{t,x}$ respective to the longitudinal slip $\kappa $ (braking);
- Steering the freewheeling test wheel with a slip angle $\alpha $ from $-{2}^{\xb0}$ to ${18}^{\xb0}$ with the truck driving in a straight line to record the Force ${F}_{t,y}$ and aligning torque ${T}_{t,z}$ (steering);
- Combined measurements of spinning the wheel faster or slower than the truck’s constant speed ${v}_{v,x}$ while a slip angle $\alpha $ from $-{2}^{\xb0}$ to ${18}^{\xb0}$ is applied and the truck is driving in a straight line to record the Force ${F}_{t,y}$ and aligning torque ${T}_{t,z}$ (steering with combined slip—not covered in this research).

- Normal Force ${F}_{N,z}$, which depends on the mass distribution of the vehicle. ${F}_{N,z}$ results from the average value of ${F}_{t,z}$ in ${F}_{N,z}={\overline{F}}_{t,z}$.;
- Longitudinal speed ${v}_{v,x}$ at which the measurements are to be carried out;
- Slip angle $\alpha $;
- Camber angle $\gamma $.

- Offset correction;
- x-, y- and z-axis crosstalk correction;
- If needed, smoothing the data.

#### 2.1.2. Tire Parameters and Test Matrix

^{®}, which was specially developed for use in cargo bikes or trailers. The tested size of 55–406 ETRO (20 × 2.15 in) is a common tire choice for these applications. According to [17], this tire can be inflated with an air pressure of 2.5–4.5 bar (35–65 psi) and should mainly be used on asphalt or light gravel roads. In addition, the tire has a double wire carcass and puncture protection. The weight is listed as 0.790 kg. The tire mounted with an inner tube on the adapter rim is shown in Figure 4a. A view of the profile before tests were carried out can be seen in Figure 4b.

#### 2.2. Tire Model

- Y as the output, which can be ${F}_{t,x}$, ${F}_{t,y}$ or ${T}_{t,z}$;
- x as the input, which can be $\kappa $ or $\alpha $ depending on the input;
- B as the stiffness factor, which affects the slope;
- C as the shape factor, which affects the peak and horizontal asymptote;
- D as the peak value respective to the central x-axis, as well as for C ≥ 1;
- E as the curvature factor, which affects the curvature at the peak point as well as the horizontal position of the peak.

## 3. Results

#### 3.1. Post-Processing

#### 3.2. Observations during the Measurement

- Loose spokes of the adapter rim after steering measurements;
- High tire abrasion after steering measurements;
- Thin high-temperature line on the tire surface.

#### 3.3. Tire Modeling

- x for MF coefficients used to model ${F}_{t,x}\left(\kappa \right)$;
- y for MF coefficients used to model ${F}_{t,y}\left(\alpha \right)$;
- z for MF coefficients used to model ${T}_{t,z}\left(\alpha \right)$.

^{2}value > 0.9. Therefore, the choice of coefficients results in a suitable model quality.

#### 3.4. Impact of Tire Pressure and Normal Force

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MoReLab | Mobiles Reifen Labor (engl. mobile tire testing laboratory) |

BFH | Bern University of Applied Sciences |

MF | Magic formula |

RLOWESS | Robust locally weighted scatterplot smoothing |

RMSE | Root mean square error |

NRMSE | Normalized root mean squared error |

MBSE | Model based systems engineering |

RWU | Ravensburg-Weingarten University of Applied Sciences |

## References

- Carla Cargo: Products—eCarla. Available online: https://www.carlacargo.de/de/produkte/ecarla/ (accessed on 14 July 2022).
- Nüwiel: ProductP. Available online: https://www.nuwiel.com/etrailer/ (accessed on 14 July 2022).
- Kyburz: Products—DXP. Available online: https://kyburz-switzerland.ch/en (accessed on 14 July 2022).
- Heißing, B.; Schimmel, C. Fahrverhalten. In Fahrwerkhandbuch; Ersoy, M., Gies, S., Eds.; Springer Vieweg Wiesbaden: Wiesbaden, Germany, 2017; pp. 202–203. [Google Scholar]
- Korayem, A.H.; Khajepour, A.; Fidan, B. A Review on Vehicle-Trailer State and Parameter Estimation. IEEE T-ITS
**2022**, 23, 5993–6010. [Google Scholar] - Vempaty, S.; He, Y. A Review of Car-Trailer Lateral Stability Control Approaches; SAE Technical Paper 2017-01-1580. In Proceedings of the WCX™ 17: SAE World Congress, Detroit, MI, USA, 4 April 2017. [Google Scholar]
- Isermann, R. Tire Traction and Force Transfer. In Automotive Control; ATZ/MTZ-Fachbuch; Springer: Berlin/Heidelberg, Germany, 2022. [Google Scholar]
- Guiggiani, M. Mechanics of the Wheel with Tire. In The Science of Vehicle Dynamics; Springer: Cham, Switzerland, 2023. [Google Scholar]
- Vempaty, S.; Lee, E.; He, Y. Model-Reference Based Adaptive Control for Enhancing Lateral Stability of Car-Trailer Systems. In Proceedings of the ASME 2016 International Mechanical Engineering Congress and Exposition, Phoenix, AZ, USA, 7 March 2016. [Google Scholar]
- Shamim, R.; Islam, M.M.; He, Y. A Comparative Study of Active Control Strategies for Improving Lateral Stability of Car-Trailer Systems. In Proceedings of the SAE 2011 World Congress & Exhibition, Detroit, MI, USA, 14 April 2011. [Google Scholar]
- Frey, M.; Gnadler, R.; Guenter, F. Untersuchung der Verlustleistung an Pkw-Reifen. Proc. Reifen Fahrwerk Fahrb. Tag.
**1995**, 10, 101–128. [Google Scholar] - Eckert, M. Energieoptimale Fahrdynamikregelung Mehrmotoriger Elektrofahrzeuge; KIT Scientific Publishing: Karlsruhe, Germany, 2015. [Google Scholar] [CrossRef]
- Dressel, A.E. Measuring and Modeling the Mechanical Properties of Bicycle Tires. Master’s Thesis, University of Wisconsin-Milwaukee, Milwaukee, WI, USA, May 2013. [Google Scholar]
- Dressel, A.; Sadauckas, J. Characterization and Modelling of Various Sized Mountain Bike Tires and the Effects of Tire Tread Knobs and Inflation Pressure. Appl. Sci.
**2020**, 10, 3156. [Google Scholar] [CrossRef] - Doria, A.; Tognazzo, M.; Cusimano, G.; Bulsink, V.; Cooke, A.; Koopman, B. Identification of the mechanical properties of bicycle tyres for modelling of bicycle dynamics. Veh. Syst. Dyn.
**2012**, 51, 405–420. [Google Scholar] [CrossRef] - Pacejka, H.B.; Besselink, I. Tire and Vehicle Dynamics, 3rd ed.; Butterworth-Heinemann: Oxford, UK, 2012. [Google Scholar]
- Schwalbe Pick-Up. Available online: https://www.schwalbe.com/en/tour-reader/schwalbe-pick-up (accessed on 20 July 2022).
- Miller, M.; Pfeil, M.; Reick, B.; Murri, R.; Stetter, R.; Kennel, R. Measurement Data of Longitudinal and Lateral Behaviour of a Bicycle Tire for Cargo Trailers. Dataset; Technical University of Munich: Munich, Germany, 2022. [Google Scholar] [CrossRef]
- Bakker, E.; Nyborg, L.; Pacejka, H.B. Tyre Modelling for Use in Vehicle Dynamics Studies. SAE Trans.
**1987**, 96, 190–204. [Google Scholar] - Besselink, I.J.M.; Schmeitz, A.J.C.; Pacejka, H.B. An improved Magic Formula/Swift tyre model that can handle inflation pressure changes. In Proceedings of the 21st Symposium of the International Association for Vehicle System Dynamics (IAVSD 09), Stockholm, Sweden, 17–21 August 2009; pp. 337–352. [Google Scholar] [CrossRef]
- Blundell, M.; Harty, D. Multibody Systems Approach to Vehicle Dynamics, 2nd ed.; Elsevier: Amsterdam, The Netherlands, 2015. [Google Scholar] [CrossRef]
- Schrand, D. Cross-Talk Compensation Using Matrix Methods. Sens. Transducers J.
**2007**, 5, 1157–1163. [Google Scholar] - Cossalter, V.; Doria, A.; Lot, R.; Ruffo, N.; Salvador, M. Dynamic Properties of Motorcycle and Scooter Tires: Measurement and Comparison. Veh. Syst. Dyn.
**2003**, 39, 329–352. [Google Scholar] [CrossRef] - Massaro, M.; Cossalter, V.; Cusimano, G. The effect of the inflation pressure on the tyre properties and the motorcycle stability. Proc. Inst. Mech. Eng. Part J. Automob. Eng.
**2013**, 10, 1480–1488. [Google Scholar] [CrossRef] - VDI/VDE 2206 Entwurf. In Entwicklung Cyber-Physischer Mechatronischer Systeme (CPMS); Beuth: Berlin, Germany, 2020.
- Gräßler, I.; Hentze, J. The new V-Model of VDI 2206 and its validation. At-Automatisierungstechnik
**2020**, 68, 312–324. [Google Scholar] [CrossRef] - Holder, K.; Zech, A.; Ramsaier, M.; Stetter, R.; Niedermeier, H.P.; Rudolph, S.; Till, M. Model-based requirements management in gear systems design based on graph-based design languages. Appl. Sci.
**2017**, 7, 1112. [Google Scholar] [CrossRef] [Green Version] - Eisenbart, B.; Gericke, K.; Blessing, L.T.; McAloone, T.C. A DSM-based framework for integrated function modelling: Concept, application and evaluation. Res. Eng. Des.
**2017**, 28, 25–51. [Google Scholar] [CrossRef] [Green Version] - Elwert, M.; Ramsaier, M.; Eisenbart, B.; Stetter, R.; Till, M.; Rudolph, S. Digital Function Modeling in Graph-Based Design Languages. Appl. Sci.
**2022**, 12, 5301. [Google Scholar] [CrossRef] - Stetter, R. Approaches for Modelling the Physical Behavior of Technical Systems on the Example of Wind Turbines. Energies
**2020**, 13, 2087. [Google Scholar] [CrossRef] - Shaked, A.; Reich, Y. Using Domain-Specific Models to Facilitate Model-Based Systems-Engineering: Development Process Design Modeling with OPM and PROVE. Appl. Sci.
**2021**, 11, 1532. [Google Scholar] [CrossRef] - Laing, C.; David, P.; Blanco, E.; Dorel, X. Questioning integration of verification in model-based systems engineering: An industrial perspective. Comput. Ind.
**2020**, 114, 103163. [Google Scholar] [CrossRef] - Lu, J.; Chen, D.; Wang, G.; Kiritsis, D.; Törngren, M. Model-Based Systems Engineering Tool-Chain for Automated Parameter Value Selection. IEEE Trans. Syst. Man Cybern. Syst.
**2021**, 52, 2333–2347. [Google Scholar] [CrossRef] - Stetter, R. Fault-Tolerant Design and Control of Automated Vehicles and Processes, 1st ed.; Springer: Cham, Germany, 2020. [Google Scholar] [CrossRef]

**Figure 1.**MoReLab truck and measurement chamber with mounted bicycle tire and corresponding coordinate system.

**Figure 3.**Measurement hub with offsets of bicycle rim and tire (

**a**) yz-plane view (

**b**) xy-plane view (

**c**) xy-plane view with $\alpha ={18}^{\xb0}$.

**Figure 6.**Measured and post-processed data of test no. 3.1 where (

**a**) shows ${F}_{t,x}\left(\kappa \right)$, (

**b**) shows ${F}_{t,y}\left(\alpha \right)$ and (

**c**) shows ${T}_{t,z}\left(\alpha \right)$.

**Figure 8.**Tire wear according to the measurements where (

**a**) has a replaceable but acceptable tire condition, while (

**b**) has been overstressed.

**Figure 9.**Thermal image of a tire after measuring the steering case with approx. 18 measurements in xy-plane view.

**Figure 10.**Wheel fixture deformation according to lever arm where (

**a**) shows the optimal case with no error of ${\alpha}_{0}$ and (

**b**) shows an elastic deformation with ${\alpha}_{0}\ne {0}^{\xb0}$.

**Figure 11.**Post-processed data and MF Model of test no. 3.1, where (

**a**) shows ${F}_{t,x}\left(\kappa \right)$, (

**b**) shows ${F}_{t,y}\left(\alpha \right)$ and (

**c**) shows ${T}_{t,z}\left(\alpha \right)$.

**Figure 12.**MF Models of Test no. 1.1, 2.1 and 3.1 where (

**a**) shows ${F}_{t,x}\left(\kappa \right)$, (

**b**) shows ${F}_{t,y}\left(\alpha \right)$ and (

**c**) shows ${T}_{t,z}\left(\alpha \right)$.

**Figure 13.**MF Models of Test no. 1.2, 2.2 and 3.2 where (

**a**) shows ${F}_{t,x}\left(\kappa \right)$, (

**b**) shows ${F}_{t,y}\left(\alpha \right)$ and (

**c**) shows ${T}_{t,z}\left(\alpha \right)$.

Test No. | Tire Pressure p in psi | Tire Pressure p in bar | ${\mathit{F}}_{\mathit{N},\mathit{z}}$ in N | ${\mathit{v}}_{\mathit{v},\mathit{x}}$ in m/s | Range $\mathit{\alpha}$ in ${}^{\xb0}$ | $\mathit{\gamma}$ in ${}^{\xb0}$ |
---|---|---|---|---|---|---|

1.1 | 43.511 | 3.0 | 625 | 5.556 | −2–18 | 0 |

1.2 | 43.511 | 3.0 | 765 | 5.556 | −2–18 | 0 |

2.1 | 50.763 | 3.5 | 625 | 5.556 | −2–18 | 0 |

2.2 | 50.763 | 3.5 | 765 | 5.556 | −2–18 | 0 |

3.1 | 58.015 | 4.0 | 625 | 5.556 | −2–18 | 0 |

3.2 | 58.015 | 4.0 | 765 | 5.556 | −2–18 | 0 |

Test Case | Force Channel | Force before Test in N | Force after Test in N |
---|---|---|---|

drive | ${F}_{x}$ | −5.875 | −4.491 |

${F}_{y}$ | 5.702 | −106.491 | |

${F}_{z}$ | −2.891 | 11.616 | |

brake | ${F}_{x}$ | −11.230 | −5.473 |

${F}_{y}$ | −0.397 | −27.325 | |

${F}_{z}$ | −8.052 | 14.288 | |

steer | ${F}_{x}$ | −5.180 | −0.403 |

${F}_{y}$ | 3.108 | −85.590 | |

${F}_{z}$ | −0.890 | 32.541 |

Input | ||||||
---|---|---|---|---|---|---|

${\mathit{F}}_{\mathit{x}}$ in N [1203] | ${\mathit{F}}_{\mathit{y}}$ in N [1222] | ${\mathit{F}}_{\mathit{z}}$ in N [1200] | ${\mathit{T}}_{\mathit{x}}$ in Nm [305] | ${\mathit{T}}_{\mathit{z}}$ in Nm [−25] | ||

Output | ${F}_{x}$ in N | 1165.781 | 22.869 | −51.702 | 46.202 | −10.975 |

${F}_{y}$ in N | −32.988 | 1124.557 | 16.104 | 28.154 | 19.211 | |

${F}_{z}$ in N | −38.956 | −13.866 | 1235.548 | −63.938 | 2.544 | |

${T}_{x}$ in Nm | 1.665 | 13.150 | 3.433 | 292.221 | 0.599 | |

${T}_{z}$ in Nm | 8.318 | 0.058 | −4.131 | −14.675 | −25.366 |

Test No. | ${\mathit{B}}_{\mathit{x}}$ | ${\mathit{C}}_{\mathit{x}}$ | ${\mathit{D}}_{\mathit{x}}$ | ${\mathit{E}}_{\mathit{x}}$ | ${\mathit{S}}_{\mathit{H}},\mathit{x}$ | ${\mathit{S}}_{\mathit{V}},\mathit{x}$ | R^{2} | NRMSE |
---|---|---|---|---|---|---|---|---|

1.1 | 0.133 | 1.364 | 678.7 | 0.000 | 0.000 | −36.080 | 0.996 | 0.026 |

1.2 | 0.094 | 1.700 | 892.4 | 0.665 | 0.000 | −17.260 | 0.998 | 0.018 |

2.1 | 0.108 | 1.640 | 663.4 | 0.680 | 0.000 | −42.600 | 0.997 | 0.023 |

2.2 | 0.105 | 1.436 | 826.9 | 0.122 | 0.000 | −57.670 | 0.997 | 0.023 |

3.1 | 0.121 | 1.611 | 675.2 | 0.713 | 0.000 | −17.170 | 0.998 | 0.018 |

3.2 | 0.067 | 2.146 | 831.1 | 0.982 | 0.000 | −12.100 | 0.999 | 0.015 |

Test No. | ${\mathit{B}}_{\mathit{y}}$ | ${\mathit{C}}_{\mathit{y}}$ | ${\mathit{D}}_{\mathit{y}}$ | ${\mathit{E}}_{\mathit{y}}$ | ${\mathit{S}}_{\mathit{H}},\mathit{y}$ | ${\mathit{S}}_{\mathit{V}},\mathit{y}$ | R^{2} | NRMSE |
---|---|---|---|---|---|---|---|---|

1.1 | 0.210 | 1.412 | 774.0 | 0.633 | 0.000 | 0.000 | 0.995 | 0.017 |

1.2 | 0.127 | 1.373 | 1045.0 | 0.409 | 0.000 | 0.000 | 0.999 | 0.009 |

2.1 | 0.250 | 1.119 | 789.7 | −0.461 | 0.000 | 0.000 | 0.994 | 0.019 |

2.2 | 0.123 | 1.542 | 922.1 | 0.388 | 0.000 | 0.000 | 0.998 | 0.011 |

3.1 | 0.174 | 1.561 | 788.1 | 0.618 | 0.000 | 0.000 | 0.993 | 0.020 |

3.2 | 0.161 | 1.216 | 912.0 | 0.397 | 0.000 | 0.000 | 0.997 | 0.015 |

Test No. | ${\mathit{B}}_{\mathit{z}}$ | ${\mathit{C}}_{\mathit{z}}$ | ${\mathit{D}}_{\mathit{z}}$ | ${\mathit{E}}_{\mathit{z}}$ | ${\mathit{S}}_{\mathit{H}},\mathit{z}$ | ${\mathit{S}}_{\mathit{V}},\mathit{z}$ | R^{2} | NRMSE |
---|---|---|---|---|---|---|---|---|

1.1 | 0.185 | 7.715 | 3.600 | 1.340 | 1.150 | 0.000 | 0.941 | 0.061 |

1.2 | 0.082 | 9.000 | 4.397 | 1.717 | 2.200 | 0.000 | 0.969 | 0.047 |

2.1 | 0.189 | 7.442 | 3.500 | 1.339 | 1.160 | 0.000 | 0.912 | 0.065 |

2.2 | 0.101 | 8.381 | 4.000 | 1.607 | 1.910 | 0.000 | 0.937 | 0.058 |

3.1 | 0.126 | 8.611 | 3.700 | 1.627 | 1.490 | 0.000 | 0.902 | 0.074 |

3.2 | 0.128 | 7.572 | 4.200 | 1.441 | 1.680 | 0.000 | 0.905 | 0.072 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Miller, M.; Pfeil, M.; Reick, B.; Murri, R.; Stetter, R.; Kennel, R.
Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation. *Appl. Sci.* **2023**, *13*, 2542.
https://doi.org/10.3390/app13042542

**AMA Style**

Miller M, Pfeil M, Reick B, Murri R, Stetter R, Kennel R.
Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation. *Applied Sciences*. 2023; 13(4):2542.
https://doi.org/10.3390/app13042542

**Chicago/Turabian Style**

Miller, Marius, Markus Pfeil, Benedikt Reick, Raphael Murri, Ralf Stetter, and Ralph Kennel.
2023. "Measurement and Modeling of a Cargo Bicycle Tire for Vehicle Dynamics Simulation" *Applied Sciences* 13, no. 4: 2542.
https://doi.org/10.3390/app13042542