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2022 | OriginalPaper | Buchkapitel

2. Tyre-Road Contact Mechanics Equations

verfasst von : Luigi Romano

Erschienen in: Advanced Brush Tyre Modelling

Verlag: Springer International Publishing

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Abstract

The brush theory constitutes the simplest, full-physical approach to model the tyre-road interaction. It describes the tyre-wheel system as a rigid body equipped with bristles, which undergo tangential deformations in their attempt to stick to the ground. This chapter introduces the governing equations of the brush models. These may be divided into four separate sets: the friction model, the constitutive relationships, the tyre-road kinematic relationships and the equilibrium equations. The fundamental assumptions behind the proposed formulation are outlined, and the boundary and initial conditions are stated in a general manner.

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Vorheriges Kapitel The Tyre as a Nonlinear System

Nächstes Kapitel Steady-State Brush Theory

There is no well-established theory for the case in which the distribution \(q_z(\boldsymbol{x},t)\) is affected by friction-induced phenomena. See [2, 3].

When \(\partial \mathscr {P}\) is only \(C^0\), the solution may be not well defined on the trajectories that originate at the corners. However, this does not introduce any complication in the computation of the tyre forces and moment, since these lines have the Lebesgue measure zero. See also [21].

As also discussed in [21], the assumption \(\boldsymbol{u}_{\boldsymbol{t}0}^\text {(a)}(\boldsymbol{x}) \in C^1(\mathring{\mathscr {P}}^\text {(a)}_0;\mathbb {R}^2)\) is not obvious, and often only weak solutions may be found to Eqs. (2.10) and (2.15).

When integrating over \(\mathscr {P}\), it may be written, with some abuse of notation,

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-98435-9_2/517912_1_En_2_IEq112_HTML.gif

, since z is fixed.

Pacejka HB (2012) Tire and vehicle dynamics, 3rd edn. Elsevier/BH, Amsterdam

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Kalker JJ (1997) Variational principles of contact elastostatics. J Inst Math Its Appl 20:199–219MathSciNetCrossRef

Guiggiani M (2018) The science of vehicle dynamics, 2nd edn. Springer International, Cham (Switzerland)CrossRef

Canudas de Wit C, Olsson H, Astrom KJ, Lischinsky P (1995) A new model for control of systems with friction. IEEE Trans Autom Control 40(3):419–425. https://doi.org/10.1109/9.376053

Sharifzadeh M, Timpone F, Farnam A, Senatore A, Akbari A (2017) Tyre-road adherence conditions estimation for intelligent vehicle safety applications. In: Advances in Italian mechanism science. Adv Ital Mech Science Mech Mach Sci, vol 47. Springer, Cham. https://doi.org/10.1007/978-3-319-48375-7_42

Sharifzadeh M, Senatore A, Farnam A, Akbari A, Timpone F (2019) A real-time approach to robust identification of tyre–road friction characteristics on mixed-\(\rm \mu \) roads. Vehicle Syst Dyn 57(9):1338–1362. https://doi.org/10.1080/00423114.2018.1504974

Canudas-de-Wit C, Tsiotras P, Velenis E, Basset M, Gissinger G (2003) Dynamic friction models for road/tire longitudinal interaction. Veh Syst Dyn 39(3)189–226. https://doi.org/10.1076/esd.39.3.189.14152

Deur J, Ivanovic V, Troulis M, Miano C, Hrovat D, Asgari J (2005) Extensions of the LuGre tyre friction model related to variable slip speed aong the contact patch length. Veh Syst Dyn 43(supp):508–524. https://doi.org/10.1080/00423110500229808

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Deur J, Asgari J, Hrovat D (2004) A 3D brush-type dynamic tire friction model. Veh Syst Dyn 42(3):133–173. https://doi.org/10.1080/00423110412331282887

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Velenis E, Tsiotras P, Canudas-de-Wit C, Sorine M (2005) Dynamic tyre friction models for combined longitudinal and lateral vehicle motion. Veh Syst Dyn 43(1):3–29. https://doi.org/10.1080/00423110412331290464

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Liang W, Medanic J, Ruhl R (2008) Analytical dynamic tire model. Veh Syst Dyn 46(3):197–227. https://doi.org/10.1080/00423110701267466

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Persson BNJ (2001) Theory of rubber friction and contact mechanics. J Chem Phys 115(8):3840–3861. https://doi.org/10.1063/1.1388626

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Persson BNJ (2006) Contact mechanics for randomly rough surfaces. Surf Sci Rep 61(4):201–227. https://doi.org/10.1016/j.surfrep.2006.04.001

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O’Neill A, Gruber P, Watts JF, Prins J (2019) Predicting tyre behaviour on different road surfaces. In: Klomp M, Bruzelius F, Nielsen J, Hillemyr A (eds) Advances in dynamics of vehicles on roads and tracks. IAVSD. Lecture notes in mechanical engineering. Springer, Cham. https://doi.org/10.1007/978-3-030-38077-9_215

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O’Neill A, Prins F, Watts JF, Gruber P (2021) Enhancing brush tyre model accuracy through friction measurements. Veh Sys Dyn. https://doi.org/10.1080/00423114.2021.1893766

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Kalker JJ (1990) Three-dimensional elastic bodies in rolling contact. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7889-9

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Pacejka HB (2005) Spin: camber and turning. Veh Syst Dyn 43(1):3–17. https://doi.org/10.1080/00423110500140013

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Limebeer DJN, Massaro M (2018) Dynamics and optimal control of road vehicle. Oxford University Press

21.

Romano L, Timpone F, Bruzelius F, Jacobson B. Analytical results in transient brush tyre models: theory for large camber angles and classic solutions with limited friction. Meccanica. https://doi.org/10.1007/s11012-021-01422-3

22.

Romano L, Bruzelius F, Jacobson B. Brush tyre models for large camber angles and steering speeds. Vehicle Syst Dyn. https://doi.org/10.1080/00423114.2020.1854320

Titel: Tyre-Road Contact Mechanics Equations
verfasst von: Luigi Romano
Verlag: Springer International Publishing
Buch: Advanced Brush Tyre Modelling
Print ISBN: 978-3-030-98434-2

Electronic ISBN: 978-3-030-98435-9

Copyright-Jahr: 2022
DOI: https://doi.org/10.1007/978-3-030-98435-9_2

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