Abstract
Many transient phenomena concerning the tyreroad interaction are effectively explained within the theoretical framework of the brush theory. The analysis in vanishing sliding conditions is relatively simple and may be conducted with respect to any timevarying slip input. The case of limited friction available inside the contact patch is rather involving. In this context, the investigations proposed in this chapter are limited to small spin slips under the assumption of a thin tyre. The situation further complicates when considering a flexible tyre carcass, but it may be still approached using some intuition from Chap. 3. A rather general formulation of the transient problem is proposed, which allows to gain some preliminary insights about the relaxation behaviour of the tyre.
This is a preview of subscription content, access via your institution.
Buying options
Notes
 1.
It should be observed that if the initial conditions are described by a function \(\boldsymbol{u}_{\boldsymbol{t}0}(\boldsymbol{x})\) using the original coordinates \(\boldsymbol{x}\), then in the coordinate system \(\boldsymbol{\xi }\) they should be represented by a new function \(\boldsymbol{u}_{\boldsymbol{t}0}^\prime (\boldsymbol{\xi }) \triangleq \boldsymbol{u}_{\boldsymbol{t}0}(\boldsymbol{x}(\boldsymbol{\xi }))\). Similar considerations also hold for other functions. However, the same notation is used in the remaining of the chapter for the sake of simplicity.
 2.
Therefore, the transient brush theory may be seen as a weak one, in the sense that the solutions are always \(C^0(\mathscr {P}\times \mathbb {R}_{\ge 0};\mathbb {R}^2)\), but higher regularity cannot be required.
 3.
The coordinates for which and correspond to the points where the slope of the lateral shear stress equals that of the friction bound, and for which the microsliding velocity vanishes, that is \(\bar{\boldsymbol{v}}_\text {s}(\boldsymbol{\xi },s) = \boldsymbol{0}\).
 4.
References
Schlippe B von, Dietrich R (1941) Das flattern eined bepneuten rades. Bericht 140 der Lilienthal Gesellschaft. NACA TM 1365
Force and moment response of pneumatic tires to lateral motion inputs. Trans ASME, J Eng Ind 88B (1966)
Pauwelussen JP (2004) The local contact between Tyre and road under steady state combined slip conditions. Veh Syst Dyn 41(1):1–26. http://doi.org/10.1076/vesd.41.1.1.23406
Takács D, Orosz G, Stèpán G (2009) Delay effects in shimmy dynamics of wheels with stretched stringlike tyres. Eur J Mech A Solids 28(3):516–525
Takács D, Stèpán G (2012) Microshimmy of towed structures in experimentally uncharted unstable parameter domain. Veh Syst Dyn 50(11):1613–1630
Takács D, Stèpán G, Hogan SJ (2008) Isolated large amplitude periodic motions of towed rigid wheels. Nonlinear Dyn 52:27–34. https://doi.org/10.1007/s110710079253y
Takács D, Stèpán G (2009) Experiments on quasiperiodic wheel shimmy. ASME J Comput Nonlinear Dynam 4(3):031007. https://doi.org/10.1115/1.3124786
Takács D, Stèpán G (2013) Contact patch memory of tyres leading to lateral vibrations of fourwheeled vehicles. Phil Trans R Soc A:37120120427. http://doi.org/10.1098/rsta.2012.0427
Besselink IJM (2000) Shimmy of aircraft main landing gears [doctoral thesis]. Delft
Ran S (2016) Tyre models for shimmy analysis: from linear to nonlinear [doctoral thesis]. Eindhoven
Pacejka HB (1966) The wheel shimmy phenomenon: a theoretical and experimental investigation with particular reference to the nonlinear problem [doctoral thesis]. Delft
van Zanten A, Ruf WD, Lutz A (1989) Measurement and simulation of transient tire forces. SAE Technical Paper 890640. https://doi.org/10.4271/890640
Mavros G, Rahnejat H, King PD (2004) Transient analysis of tyre friction generation using a brush model with interconnected viscoelastic bristles. In: Wolfson school of mechanical and manufacturing engineering, Loughborough University, Loughborough, UK. https://doi.org/10.1243/146441905X9908
Kalker JJ (1997) Survey of wheelrail rolling contact theory. Vehicle Syst Dyn 84:317–358. https://doi.org/10.1080/00423117908968610
Romano L, Bruzelius F, Jacobson B (2020) Unsteadystate brush theory. Veh Syst Dyn:129. https://doi.org/10.1080/00423114.2020.1774625
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/s11012021014223
Cattaneo C (2008) Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Rendiconti dell’Accademia Naturale dei Lincei. Serie 6, 227, 342–348, 434–436, 474–478
Guiggiani M (2018) The science of vehicle dynamics, 2nd edn. Springer International, Cham(Switzerland)
Truesdell C, Toupin RA (1960) The classical field theories. In: Flügge S (ed) Handbuch der Physik, vol 3/1. Berlin, Springer, p 226
Truesdell C, Rajagopal KR (2000) An introduction to the mechanics of fluids. Birkhäuser Boston . https://doi.org/10.1007/9780817648466
Author information
Authors and Affiliations
Corresponding author
Appendices
4.A Lemmata and Propositions
4.A.1 Results for Pure Lateral Slip
The results for pure lateral slip are proved under the assumption that the pressure distribution \(q_z(\boldsymbol{\xi })\) satisfies Assumption 3.1.2 and that \(u_{x0}(\boldsymbol{\xi }) = 0\).
Lemma 4.A.1
Consider pure lateral slip conditions, i.e. \(\sigma _x = 0\), \(\sigma _y \not = 0\), \(\varphi = 0\) and assume that \(q_z(\boldsymbol{\xi })\) satisfies Assumption 3.1.2 with \(q_z^{(\eta )}(\cdot )\) strictly concave. Then, if for all \(\boldsymbol{\xi } \in \mathscr {P}\), the following implications hold for all \((\boldsymbol{\xi },s) \in \mathscr {P} \times \mathbb {R}_{>0}\) such that \(\xi \in (0,\xi _{\mathscr {T}}(\eta )]\):
Proof
The proofs for \(\sigma _y \ge 0\) and \(\sigma _y < 0\) are mirrored, and thus only the analysis for \(\sigma _y \ge 0\) is conducted.

1.
Consider the case . For \( \xi \in (0, s)\), Eq. (4.19) gives
(4.53)where the last inequality follows from Assumption 3.1.2. For \(\xi \in [s,\xi _\mathscr {T}(\eta )]\):
(4.54)the last inequality following from Assumption 3.1.2. Combining (4.53) and (4.54), (4.52a) is deduced.

2.
Consider the case . For \(\xi \in (0, s)\), Eq. (4.19) gives
(4.55)For \( \xi \in [s, \xi _\mathscr {T}(\eta )]\):
(4.56)the last inequality following from Assumption 3.1.2. Combining (4.55) and (4.56), (4.52b) is deduced.
Proposition 4.A.1
Consider a vertical pressure distribution \(q_z(\boldsymbol{\xi })\) satisfying Assumption 3.1.2 with \(q_z^{(\eta )}(\cdot )\) strictly concave. Then, if for all \(\boldsymbol{\xi } \in \mathscr {P}\), the sliding solution is given by
for all \((\boldsymbol{\xi },s) \in \mathscr {P}\times \mathbb {R}_{>0}\) such that \(\xi \in (0,\xi _{\mathscr {T}}(\eta )]\) and .
Proof
The proof may be split into two parts depending on whether \(\sigma _y \ge \sigma ^*(\eta )\) or \(\sigma _y \le \sigma ^*(\eta )\). Only the part for \(\sigma _y \ge \sigma ^*(\eta )\) will be proved; the proof for \(\sigma _y \le \sigma ^*(\eta )\) is analogous. First, it is observed that the steadystate solution \(q_y^{+}(\boldsymbol{\xi },s)\) is always concordant with the slip itself, and hence the result follows trivially. Instead, for \(\xi \in [s, \xi _{\mathscr {T}}(\eta )]\):
Lemma 4.A.2
Consider pure lateral slip conditions, i.e. \(\sigma _x = 0\), \(\sigma _y \not = 0\), \(\varphi = 0\), and a vertical pressure distribution \(q_z(\boldsymbol{\xi })\) satisfying Assumption 3.1.2 with \(q_z^{(\eta )}(\cdot )\) strictly concave. Then, the following implications hold for all \((\boldsymbol{\xi },s) \in \mathscr {P}\times \mathbb {R}_{>0}\) such that \((\xi + \delta s, \delta s) \in (0, \xi _\mathscr {T}(\eta )] \times \mathbb {R}_{>0}\):
Proof
Again, the Lemma is only proved for \(\sigma _y \ge 0\); the cases for \(\sigma _y<0\) are specular.

1.
Consider the case \(\sigma _y \ge 0, q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })\). First, it is observed that, owing to (4.52a), it must necessarily be to have \(q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })\). Thus, recalling Assumption 3.1.2, it holds that
(4.60) 
2.
Consider the case \( \sigma _y \ge 0, q_y^\text {(a)}(\boldsymbol{\xi },s) \le \mu q_z(\boldsymbol{\xi })\). First, it is observed that, owing to (4.52b), it must necessarily be to have \(q_y^\text {(a)}(\boldsymbol{\xi },s) \le  \mu q_z(\boldsymbol{\xi })\). Thus, recalling Assumption 3.1.2, it holds that
(4.61)
Lemma 4.A.3
Consider pure lateral slip conditions, i.e. \(\sigma _x = 0\), \(\sigma _y \not = 0\), \(\varphi = 0\). Then, if \(q_z(\boldsymbol{\xi })\) satisfies Assumption 3.1.2 and for all \(\boldsymbol{\xi } \in \mathscr {P}\), the following implications hold for all \((\boldsymbol{\xi },s) \in \mathscr {P} \times \mathbb {R}_{>0}\):
Proposition 4.A.2
Consider a vertical pressure distribution \(q_z(\boldsymbol{\xi })\) satisfying Assumption 3.1.2. Assume that for some \(\eta \in \mathscr {P}\) it holds that . Then, if for all \(\boldsymbol{\xi } \in \mathscr {P}\), the sliding solution is given by
for all \((\boldsymbol{\xi },s) \in \mathscr {P}\times \mathbb {R}_{>0}\) such that \(\xi \in (0,\xi _{\mathscr {T}}(\eta )]\) and .
Lemma 4.A.4
Consider pure lateral slip conditions, i.e. \(\sigma _x = 0\), \(\sigma _y \not = 0\), \(\varphi = 0\), and a vertical pressure distribution \(q_z(\boldsymbol{\xi })\) satisfying Assumption 3.1.2. Then, the following implications hold for all \((\boldsymbol{\xi }, s) \in \mathscr {P}\times \mathbb {R}_{>0}\) such that \( (\xi + \delta s, \delta s) \in (0, \xi _\mathscr {T}(\eta )] \times \mathbb {R}_{>0}\):
Remark 1
If Assumption 3.1.2 is only satisfied with \(q_z^{(\eta )} \in C^1(\mathring{\mathscr {P}}^{(\eta )};\mathbb {R})\) for some or every fixed \(\eta \), the results advocated in Lemmata 4.A.1, 4.A.3 and Propositions 4.A.1, 4.A.2 are only valid for \((\boldsymbol{\xi },s) \in \mathscr {P} \times \mathbb {R}_{>0}\) such that \(\xi \in \mathring{\mathscr {P}}^{(\eta )}\).
4.A.2 Results for Pure Spin Conditions
The following results for pure spin slip conditions are proved assuming that the pressure distribution \(q_z(\xi )\) is as in Eq. (3.22) and \(u_{x0}(\xi ) = 0\).
Lemma 4.A.5
Consider nonsupercritical pure spin slip conditions, i.e. \(\boldsymbol{\sigma }=\boldsymbol{0}\), . Then, if for all \(\xi \in [0,2a]\), it holds that for all \((\xi ,s) \in [0,a]\times \mathbb {R}_{>0}\). Additionally, if , then for all \((\xi , s)\in (0,a] \times \mathbb {R}_{>0}\).
Proof
The case for \(\xi \in [0,s)\) is trivial and follows directly by the assumption . Instead, when \(\xi \in [s, a]\)
The case for may be proved similarly.
Lemma 4.A.6
Consider pure critical spin slip conditions, i.e. \(\boldsymbol{\sigma }=\boldsymbol{0}\), . Then, if for all \(\xi \in [0,2a]\), the following implications hold for all \((\xi ,s) \in [s, 2a] \times \mathbb {R}_{>0}\):
Proof
For implication (4.66a):
For implication (4.66a):
Lemma 4.A.7
Consider nonsupercritical pure spin slip conditions, i.e. \(\boldsymbol{\sigma }=\boldsymbol{0}\), . Then, if for some \(\xi \in [a,2a]\), it holds that for all \((\xi ,s) \in (a,2a] \times \mathbb {R}_{>0}\) such that \((\xi + \delta s, \delta s) \in (a, 2a]\times \mathbb {R}_{>0} \). Additionally, if , then for all \((\xi ,s) \in (a,2a] \times \mathbb {R}_{>0}\) such that \((\xi + \delta s, \delta s) \in (a, 2a)\times \mathbb {R}_{>0} \).
Proof
For \(q_y^\text {(a)}(\xi ,s) \ge \mu q_z(\xi )\):
since \(\xi> a \implies \xi > a\delta s/2\) for all \(\delta s >0\). With the same reasoning, it is easy to show that, for \(q_y^\text {(a)}(\xi ,s) \le  \mu q_z(\xi )\), it is
Combining (4.69) and (4.70) the first result follows automatically. The case for may be proved similarly.
4.A.3 Results for Combined Lateral Slip and Subcritical Spin
The results for combined lateral and spin slip conditions are proved assuming that the pressure distribution \(q_z(\xi )\) is as in Eq. (3.22) and \(u_{x0}(\xi ) = 0\).
Lemma 4.A.8
Consider combined lateral and spin slips conditions with subcritical spin, i.e. \(\sigma _x=0\), \(\sigma _y \not = 0\), . Then, if for all \(\xi \in [0,2a]\), the following implications hold for every \((\xi , s) \in (0,2a]\times \mathbb {R}_{>0}\):
Proof
Only the cases for \(\sigma _y > 0\), i.e. implications (4.71a) and (4.71b), will be proved; the cases for \(\sigma _y<0\) may be proved similarly.

1.
To prove (4.71a), coordinates \(\xi \in (0,2a]\) such that need to be considered. For \(\xi \in (0,s)\):
(4.72)For \( \xi \in [s, 2a]\):
(4.73) 
2.
To prove (4.71b), coordinates \(\xi \in (0,2a]\) such that need to be considered. For \(\xi \in (0,s)\):
(4.74)For \( \xi \in [s, 2a]\):
(4.75)Combining (4.74) and (4.75), (4.71b) is deduced. Finally, the proofs for \(\sigma _y < 0\) may be easily obtained by noticing that it is always possible to write and when \(\sigma _y < 0\).
Proposition 4.A.3
Consider combined lateral and spin slips conditions with and . Then, if for all \(\xi \in [0,2a]\), the lateral component of the sliding solution is given by
for all \((\xi ,s)\in [0,2a]\times \mathbb {R}_{>0}\) such that \(\xi \in (0,2a]\).
Proof
Only the case for \(\sigma _y > 0\) is proved; the proof for \(\sigma _y <0\) is analogous. First, it is observed that the steadystate shear stress is always concordant with the slip itself, and thus the result follows trivially for \(\xi \in (0,s)\). On the other hand, for \(\xi \in [s,2a]\):
Since \(q_y^\text {(a)}(\xi ,s)\) is always greater than the friction parabola which has the opposite sign to the slip, the lateral component of the sliding solution must be concordant with the lateral slip itself.
Lemma 4.A.9
Consider combined lateral and spin slips conditions with subcritical spin, i.e. \(\sigma _x=0\), \(\sigma _y \not = 0\), . Then, for every \((\xi ,s) \in (0,2a] \times \mathbb {R}_{>0}\) such that \((\xi + \delta s, \delta s) \in (0,2a] \times \mathbb {R}_{>0}\), the following implications hold:
Proof
Again, only the cases for \(\sigma _y > 0\), i.e. implications (4.78a) and (4.78b), will be proved; the cases for \(\sigma _y < 0\) may be proved similarly.

1.
To prove (4.78a), it may be observed that, owing to (4.71a), it must necessarily be to have \(q_y^\text {(a)}(\xi ,s) \ge \mu q_z(\xi )\). Therefore,
(4.79) 
2.
To prove (4.78b), it may be observed that, owing to (4.71b), it must necessarily be to have \(q_y^\text {(a)}(\xi ,s) \le  \mu q_z(\xi )\). Therefore,
(4.80)The cases for \( \sigma _y < 0\) may be proved analogously by noticing that it is always possible to write when \(\sigma _y < 0\).
4.B Sliding and Travelling Edge Dynamics
To derive an expression for the velocity of a sliding edge, some basic notions from differential geometry are required. To start, it should be noted that, for a generic \(\mathscr {S}\), the product \(\bar{\boldsymbol{v}}_{\mathscr {S}}(\boldsymbol{x},s)\cdot \hat{\boldsymbol{\nu }}_{\mathscr {S}}(\boldsymbol{x},s)\) represents the normal component of the velocity of the sliding edge. This may be represented in implicit form as in Eq. (2.21). The outwardpointing unit normal to \(\mathscr {S}\) is thus given by
Furthermore, differentiating (4.13) with respect to the travelled distance following a point on the sliding edge yields [19, 20]
for some representation of the velocity \(\bar{\boldsymbol{v}}_{\mathscr {S}}^{(\boldsymbol{\rho })}(\boldsymbol{\rho },s) \). Therefore,
In particular, the partial derivative \(\partial \gamma _{\mathscr {S}}(\boldsymbol{x},s)/\partial s\) reads
A particular representation of the velocity of a sliding edge that is oriented as the unit normal is thus given by
Analogously, for a travelling edge, similar equations yield the following expression for the unit normal:
whilst a nondimensional velocity vector which is oriented as the unit normal may be computed as
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Romano, L. (2022). UnsteadyState Brush Theory. In: Advanced Brush Tyre Modelling. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/9783030984359_4
Download citation
DOI: https://doi.org/10.1007/9783030984359_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 9783030984342
Online ISBN: 9783030984359
eBook Packages: EngineeringEngineering (R0)