# Research into the Peculiarities of the Individual Traction Drive Nonlinear System Oscillatory Processes

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## Abstract

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## 1. Introduction

## 2. Methods and Materials

#### 2.1. Criterion for the Occurrence of the Auto-Oscillating Mode

#### 2.2. The Mathematical Model of Frictional Interaction of an Elastic Wheel with a Solid Support Base

#### 2.3. The Mathematical Model of Elastic Wheel Friction against a Solid Support Base

- − for the wheel rolling braking mode:

- − for the wheel rolling traction and driven modes:

#### 2.4. Analysis of the Occurrence of the Auto-Oscillation Mode for the Traction and Slave Modes of Wheel Rolling on a Solid Support Base

#### 2.5. Analysis of the Occurrence of Auto-Oscillation for the Braking Mode of the Wheel Rolling on a Solid Support Base

## 3. Results

#### 3.1. Research of the Occurrence of the Auto-Oscillation Mode in Individual Traction Electric Drive by Simulation Mathematical Modeling

#### 3.2. Determination of the Auto-Oscillation Frequency in a Traction Electric Drive

#### 3.3. Experimental Investigation of the Occurrence of the Auto-Oscillation Modes in an Individual Traction Electric Drive

#### 3.4. Investigation of Auto-Oscillation Modes in the Actual Traction Electric Drive of an Electric Bus

## 4. Conclusions

- The developed mathematical model describing the process of auto-oscillations of an electric vehicle of high reliability (confirmed by test results).
- Based on the results of simulation modeling, the frequencies of tire oscillations are determined, which are 6–7 Hz and coincide with the frequency of auto-oscillations to realize the frequency of rotation of the electric motor shaft.
- Correctness of the results of theoretical research and the results of simulation modeling are confirmed by experimental studies of the process of the occurrence of auto-vibration phenomena in the movement of the vehicle to the support base.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Vilke, V.G.; Shapovalov, I.L. Auto Oscillations in the Process of Car Braking; Series 1: Mathematics. Mechanics; Bulletin of Moscow University: Moscow, Russia, 2015; pp. 33–39. [Google Scholar]
- Svetlitsky, V.A. Random Vibrations of Mechanical Systems; Mashinostroenie: Moscow, Russia, 1976; 216p. [Google Scholar]
- Kruchinin, P.A.; Magomedov, M.K.; Novozhilov, I.V. Mathematical Model of an Automobile Wheel on Antiblock Motion Modes; MTT Series; Izvestiya RAN: Moscow, Russia, 2001; pp. 63–69. [Google Scholar]
- Awrejcewicz, J.; Dzyubak, L.; Grebogi, C. Estimation of Chaotic and Regular (Stick–Slip and Slip–Slip) Oscillations Exhibited by Coupled Oscillators with Dry Friction. Nonlinear Dyn.
**2015**, 42, 383–394. [Google Scholar] [CrossRef] - Pascal, M. Dynamics and Stability of a Two Degree of Freedom Oscillator with an Elastic Stop. J. Comput. Nonlinear Dyn.
**2006**, 1, 94–102. [Google Scholar] [CrossRef] - Shin, K.; Brennan, M.J.; Oh, J.E.; Harris, C.J. Analysis of disc brake noise using a two-degree-of-freedom model. J. Sound Vib.
**2002**, 254, 837–848. [Google Scholar] [CrossRef] - Kotiev, G.O.; Padalkin, B.V.; Kartashov, A.B.; Dyakov, A.S. Designs and development of Russian scientific schools in the field of cross-country ground vehicles building. ARPN J. Eng. Appl. Sci.
**2017**, 12, 1064–1071. [Google Scholar] - Ergin, A.A.; Kolomejtseva, M.B.; Kotiev, G.O. Antiblocking control system of the brake drive of automobile wheel. Prib. Sist. Upr.
**2004**, 9, 11–13. [Google Scholar] - Moaaz, A.O.; Ali, A.S.; Ghazaly, N.M. Investigation of Anti-Lock Braking System Performance Using Different Control Systems. Int. J. Control Autom.
**2020**, 13, 137–153. [Google Scholar] - Sun, C.; Pei, X. Development of ABS ECU with Hardware-in-the-Loop Simulation Based on Labcar System. SAE Int. J. Passeng. Cars-Electron. Electr. Syst.
**2014**, 8, 14–21. [Google Scholar] [CrossRef] - Sabbioni, E.; Cheli, F.; D’alessandro, V. Analysis of ABS/ESP Control Logics Using a HIL Test Bench; SAE International: Warrendale, PA, USA, 2011. [Google Scholar] [CrossRef]
- Hart, P.M. Review of Heavy Vehicle Braking Systems Requirements (PBS Requirements); Draft Report. Available online: https://trid.trb.org/View/1407753 (accessed on 24 April 2003).
- Marshek, K.M.; Cuderman, J.F.; Johnson, M.J. Performance of Anti-Lock Braking System Equipped Passenger Vehicles Part I: Braking as a Function of Brake Pedal Application Force. In Proceedings of the SAE 2002 World Congress, Detroit, MI, USA, 4–7 March 2002. [Google Scholar]
- Kuznetsov, A.P.; Kuznetsov, S.P.; Ryskin, N.M. Nonlinear Oscillations; Fizmatlit: Moscow, Russia, 2002; 292p. [Google Scholar]
- Pacejka, H.B. Tyre and Vehicle Dynamics, 2nd ed.; Butterworth Heinemann: Oxford, UK, 2006; 672p. [Google Scholar]
- Wellstead, P.E.; Pettit, N.B. Analysis and Redesign of an Antilock Brake System Controller. IEE Proc. Control Theory Appl.
**1997**, 144, 413–426. [Google Scholar] [CrossRef] - Zhileikin, M.M. Investigation of Autocoletive Processes in the Zone of Interaction of an Elastic Tire with a Solid Support Base. Izvestiya vysshee obrazovaniya vysshee obrazovaniya. Mashinostroenie
**2021**, 10, 3–15. [Google Scholar] [CrossRef] - Chelomey, V.N. Vibrations in Engineering; Volume 2: Fluctuations of Nonlinear Mechanical Systems; Blekhman, I.I., Ed.; Mashinostroenie: Moscow, Russia, 1979; 351p. [Google Scholar]
- Kryukov, B.I. Forced Vibrations of Essentially Nonlinear Systems; Mashinostroenie: Moscow, Russia, 1984; 216p. [Google Scholar]
- Nekorkin, V.I. Lectures on the Fundamentals of Vibration Theory: Textbook; Nizhny Novgorod University: Nizhny Novgorod, Russia, 2011; 233p. [Google Scholar]
- Babakov, I.M. Theory of Vibrations, 4th ed.; Drofa: Moscow, Russia, 2004; 591p. [Google Scholar]
- Strelkov, S.P. Introduction to the Theory of Vibrations; Nauka: Moscow, Russia, 1964; 438p. [Google Scholar]
- Yablonsky, A.A.; Noreiko, S.S. Course of the Theory of Vibrations; Lan: Moscow, Russia, 2003; 256p. [Google Scholar]
- Moiseev, N.N. Asymptotic Methods of Nonlinear Mechanics; Izd. “Nauka”, Main Editorial Office of Physical and Mathematical Literature: Moscow, Russia, 1969; 380p. [Google Scholar]
- Bogolyubov, N.N.; Mitropolsky, Y.A. Asymptotic Methods in the Theory of Nonlinear Oscillations; Nauka: Moscow, Russia, 2005; Volume 3, 605p. [Google Scholar]
- Gorelov, V.A.; Komissarov, A.I.; Miroshnichenko, A.V. 8 × 8 wheeled vehicle modeling in a multibody dynamics simulation software. Procedia Eng.
**2015**, 129, 300–307. [Google Scholar] [CrossRef] - Keller, A.V.; Gorelov, V.A.; Anchukov, V.V. Modeling truck driveline dynamic loads at differential locking unit engagement. Procedia Eng.
**2015**, 129, 280–287. [Google Scholar] [CrossRef] - Volskaya, V.N.; Zhileykin, M.M.; Zakharov, A.Y. Mathematical model of rolling an elastic wheel over deformable support base. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 315, 012028. [Google Scholar] [CrossRef] - Belousov, B.; Ksenevich, T.; Vantsevich, V.; Komissarov, D. 8 × 8 Platform for Studing Terrain Mobility and Traction Performance of Unmanned Articulated Ground Vehicles with Steered Wheels. SAE Tech. Pap.
**2013**, 9. [Google Scholar] [CrossRef] - Belousov, B.; Shelomkov, S.; Ksenevich, T.; Kupreyanov, A. Experimental verification of a mathematical model of the wheel-supporting surface interaction during nonstationary rolling motion. J. Mach. Manuf. Reliab.
**2009**, 38, 501–505. [Google Scholar] - Wong, J.Y. Theory of Ground Vehicles; Wiley: New York, NY, USA, 2001; 560p. [Google Scholar]
- Antonyan, A.; Zhileykin, M.; Eranosyan, A. The algorithm of diagnosing the development of a skid when driving a two-axle vehicle. In Proceedings of the Design Technologies for Wheeled and Tracked Vehicles (MMBC) 2019, Moscow, Russia, 1–2 October 2019; Volume 820. [Google Scholar] [CrossRef]
- Polungyan, A.A.; Fominykh, A.B.; Staroverov, N.N. Dynamics of Wheeled Machines; Polungyan, A.A.A., Ed.; Bauman Moscow State Technical University Publishing House: Moscow, Russia, 2013; 118p, ISBN 978-5-7038-3706-1. [Google Scholar]

**Figure 1.**Calculation scheme of interaction of an elastic wheel with a solid support base: 1—mass m1 of sprung parts of the car falling on the wheel; 2—mass m2 of the wheel; 3—rollers; 4—elastic element characterizing the pliability of the tire in the longitudinal direction; 5—support base; 6—rotating wheel; 7—traction motor; c—spring stiffness.

**Figure 2.**Dependence of friction force F on relative sliding velocity ${V}_{2sk}$ for the Coulomb dry friction model (

**a**) and for Paseika’s “magic formula” (

**b**).

**Figure 6.**Angular velocities of the driving wheels in a turn on dry asphalt: 1—left rear wheel; 2—right rear wheel.

**Figure 7.**Process of time variation in torques on the driving wheels in a turn on dry asphalt: 1—left rear wheel; 2—right rear wheel.

**Figure 10.**Angular velocities of the driving wheels in a turn on ice with snow: 1—left rear wheel; 2—right rear wheel.

**Figure 11.**Process of change in time of torques on the driving wheels in a turn on ice with snow: 1—left rear wheel; 2—right rear wheel.

**Figure 14.**The process of change in time of regenerative moments given to the driving wheels of an electric bus during braking in a turn on ice with snow: 1—left rear wheel; 2—right rear wheel.

**Figure 15.**Process of change in time of angular velocities of the driving wheels during braking in a turn on ice with snow: 1—left rear wheel; 2—right rear wheel.

**Figure 16.**Fragments of the traction motor torque (

**a**), the driving wheel rotation angular velocity (

**b**) and the radial tire deformation (

**c**) observed during the electric bus straight-on acceleration on dry asphalt.

**Figure 17.**Spectral energy densities of traction motor torque (

**a**), angular velocity of the driving wheel rotation (

**b**) and radial tire deformation (

**c**) during the straight-on acceleration of an electric bus on dry asphalt.

**Figure 19.**Characteristic fragment of the realization of traction electric torque, obtained during testing of the electric bus.

**Figure 20.**Characteristic fragment of the realization of traction electric torque, obtained as a result of modeling of electric bus motion.

**Figure 21.**Spectral energy density of the traction motor torque energy for the realization obtained experimentally.

**Figure 22.**A characteristic fragment of the wheel angular velocity obtained during the electric bus tests.

**Figure 23.**A characteristic fragment of the wheel angular velocity, obtained as a result the electric bus motion modeling.

**Figure 25.**Realizations of the position of the travel pedal (

**a**), electric torque of the rear left driving wheel, reduced to the wheel (

**b**); angular speed of rotation of the rotor of the rear left electric motor (

**c**).

**Figure 26.**Spectral energy density for torque realization of Figure 25b.

**Figure 27.**Spectral energy density for realization of angular speed of rotor rotation of Figure 25c.

**Figure 28.**Variation in torques on the shaft of electric motors, angular velocities of wheels and rotors, as well as motor currents in time during the acceleration up to 20 km/h and subsequent complex braking.

**Figure 29.**Variation in traction motor torques, angular velocities of wheels and rotors, and motor currents in time during the acceleration to maximum speed with increased wheel slippage.

**Figure 30.**Dependences of torques and currents of traction motors on time at slippage of the driving wheels.

**Figure 31.**Fragment of torque realization of traction electric motors when driving on an urban route.

№ | Character of Movement | Wheel Rolling Modes | Traction Properties of the Supporting Surface | Initial Speed, km/h |
---|---|---|---|---|

1 | Acceleration in a left turn with fully depressed accelerator pedal | Traction | Dry asphalt, ${\mathsf{\mu}}_{sxmax}={\mathsf{\mu}}_{sxmax}=0.8$ | 10 |

2 | Acceleration in a left turn with fully depressed accelerator pedal | Traction | Ice with snow, ${\mathsf{\mu}}_{sxmax}={\mathsf{\mu}}_{sxmax}=0.35$ | 10 |

3 | Regenerative braking in a left turn | Braking | Ice with snow, ${\mathsf{\mu}}_{sxmax}={\mathsf{\mu}}_{sxmax}=0.35$ | 70 |

Feature | Significance |
---|---|

GVW of electric bus, kg | 18,000 |

GVW distribution by axles, kg | 6400/11,600 |

Dimensions D × W × H, mm | 12,350 × 2550 × 2770 |

Wheelbase, mm | 6170 |

Front track, mm | 2120 |

Rear track, mm | 1845 |

Tires | 275/70 R22.5 |

Front suspension | independent, pneumatic |

Rear suspension | dependent, pneumatic |

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**MDPI and ACS Style**

Klimov, A.V.; Ospanbekov, B.K.; Keller, A.V.; Shadrin, S.S.; Makarova, D.A.; Furletov, Y.M.
Research into the Peculiarities of the Individual Traction Drive Nonlinear System Oscillatory Processes. *World Electr. Veh. J.* **2023**, *14*, 316.
https://doi.org/10.3390/wevj14110316

**AMA Style**

Klimov AV, Ospanbekov BK, Keller AV, Shadrin SS, Makarova DA, Furletov YM.
Research into the Peculiarities of the Individual Traction Drive Nonlinear System Oscillatory Processes. *World Electric Vehicle Journal*. 2023; 14(11):316.
https://doi.org/10.3390/wevj14110316

**Chicago/Turabian Style**

Klimov, Alexander V., Baurzhan K. Ospanbekov, Andrey V. Keller, Sergey S. Shadrin, Daria A. Makarova, and Yury M. Furletov.
2023. "Research into the Peculiarities of the Individual Traction Drive Nonlinear System Oscillatory Processes" *World Electric Vehicle Journal* 14, no. 11: 316.
https://doi.org/10.3390/wevj14110316