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Unsteady-State Brush Theory

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Abstract

Many transient phenomena concerning the tyre-road 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 time-varying 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.

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Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 4.5

Notes

  1. 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. 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. 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 micro-sliding velocity vanishes, that is \(\bar{\boldsymbol{v}}_\text {s}(\boldsymbol{\xi },s) = \boldsymbol{0}\).

  4. 4.

    It should be noticed that in Eqs. (4.19) the adhesion solution \(u_y^\text {(a)}(\boldsymbol{\xi },s) \) has been extended analytically over the whole contact patch \(\mathscr {P}\). This makes it possible to restate \(\mathscr {P}^\text {(a)}\) and \(\mathscr {P}^\text {(s)}\) as in Eqs. (4.20).

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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 \(\xi \in (0,\xi _{\mathscr {T}}(\eta )]\):

(4.52a)
(4.52b)
(4.52c)
(4.52d)

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. 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. 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

$$\begin{aligned} u_{y}^\text {(s)}(\boldsymbol{\xi })&= \frac{\mu }{k}{q_z(\boldsymbol{\xi })}{{\,\mathrm{sgn}\,}}\sigma _y, \end{aligned}$$
(4.57a)

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 steady-state solution \(q_y^{+}(\boldsymbol{\xi },s)\) is always concordant with the slip itself, and hence the result follows trivially. Instead, for \(\xi \in [s, \xi _{\mathscr {T}}(\eta )]\):

(4.58)

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 \((\xi + \delta s, \delta s) \in (0, \xi _\mathscr {T}(\eta )] \times \mathbb {R}_{>0}\):

$$\begin{aligned}&\sigma _y \ge 0 , \, q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) > \mu q_z(\xi +\delta s, \eta ),\end{aligned}$$
(4.59a)
$$\begin{aligned}&\sigma _y \ge 0 , \, q_y^\text {(a)}(\boldsymbol{\xi },s) \le -\mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) < -\mu q_z(\xi +\delta s, \eta ), \end{aligned}$$
(4.59b)
$$\begin{aligned}&\sigma _y < 0, \, q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) > \mu q_z(\xi +\delta s, \eta ), \end{aligned}$$
(4.59c)
$$\begin{aligned}&\sigma _y< 0, \, q_y^\text {(a)}(\boldsymbol{\xi },s) \le -\mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s)< - \mu q_z(\xi +\delta s, \eta ) . \end{aligned}$$
(4.59d)

Proof

Again, the Lemma is only proved for \(\sigma _y \ge 0\); the cases for \(\sigma _y<0\) are specular.

  1. 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. 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 \((\boldsymbol{\xi },s) \in \mathscr {P} \times \mathbb {R}_{>0}\):

(4.62a)
(4.62b)
(4.62c)
(4.62d)

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

$$\begin{aligned} u_{y}^\text {(s)}(\boldsymbol{\xi })&= \frac{\mu }{k}{q_z(\boldsymbol{\xi })}{{\,\mathrm{sgn}\,}}\sigma _y, \end{aligned}$$
(4.63)

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 \( (\xi + \delta s, \delta s) \in (0, \xi _\mathscr {T}(\eta )] \times \mathbb {R}_{>0}\):

$$\begin{aligned}&\sigma _y \ge 0 ,\, q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) \ge \mu q_z(\xi +\delta s, \eta ),\end{aligned}$$
(4.64a)
$$\begin{aligned}&\sigma _y \ge 0 , \, q_y^\text {(a)}(\boldsymbol{\xi },s) \le -\mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) \le -\mu q_z(\xi +\delta s, \eta ), \end{aligned}$$
(4.64b)
$$\begin{aligned}&\sigma _y < 0, \, q_y^\text {(a)}(\boldsymbol{\xi },s) \ge \mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s) \ge \mu q_z(\xi +\delta s, \eta ), \end{aligned}$$
(4.64c)
$$\begin{aligned}&\sigma _y < 0, \, q_y^\text {(a)}(\boldsymbol{\xi },s) \le -\mu q_z(\boldsymbol{\xi })&\implies&q_y^\text {(a)}(\xi +\delta s, \eta , s+\delta s)\le - \mu q_z(\xi +\delta s, \eta ) . \end{aligned}$$
(4.64d)

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 non-supercritical 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 \(\xi \in [s, a]\)

(4.65)

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 \((\xi ,s) \in [s, 2a] \times \mathbb {R}_{>0}\):

$$\begin{aligned} \varphi&= \varphi ^\text {cr}&\implies&q_y^\text {(a)}(\xi ,s) \le \mu q_z(\xi ), \end{aligned}$$
(4.66a)
$$\begin{aligned} \varphi&=- \varphi ^\text {cr}&\implies&q_y^\text {(a)}(\xi ,s) \ge - \mu q_z(\xi ). \end{aligned}$$
(4.66b)

Proof

For implication (4.66a):

$$\begin{aligned} \begin{aligned} q_y^\text {(a)}(\xi , s)&= q_y^{+}(\xi , s) = \dfrac{1}{2} k\varphi ^\text {cr}s (2a-2\xi +s) + k u_{y0}(\xi -s) = \mu \frac{q_z^*}{a^2}s (2a-2\xi +s) +q_{y0}(\xi -s) \\&\le \mu \frac{q_z^*}{a^2}s\bigl (2a-2\xi +s) + \mu \frac{q_z^*}{a^2}(\xi -s)(2a-\xi +s) = \mu q_z(\xi ). \end{aligned} \end{aligned}$$
(4.67)

For implication (4.66a):

$$\begin{aligned} \begin{aligned} q_y^\text {(a)}(\xi , s)&= q_y^{+}(\xi , s) = -\dfrac{1}{2}k\varphi ^\text {cr}s(2a-2\xi +s) + k u_{y0}(\xi -s) = - \mu \frac{q_z^*}{a^2}s (2a-2\xi +s) +q_{y0}(\xi -s) \\&\ge - \mu \frac{q_z^*}{a^2}s(2a-2\xi +s) - \mu \frac{q_z^*}{a^2}(\xi -s)(2a-\xi +s) = -\mu q_z(\xi ). \end{aligned} \end{aligned}$$
(4.68)

Lemma 4.A.7

Consider non-supercritical 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 \(q_y^\text {(a)}(\xi ,s) \ge \mu q_z(\xi )\):

$$\begin{aligned} \begin{aligned} q_{y}^\text {(a)}(\xi +\delta s,s + \delta s)&= \dfrac{1}{2} k\varphi \delta s (2a-2\xi -\delta s) + q_y^\text {(a)}(\xi ,s) \ge \mu \frac{q_z^{*}}{a^2}\delta s (2a-2\xi -\delta s) + \mu q_z(\xi ) \\&= \mu \frac{q_z^{*}}{a^2}(\xi +\delta s)(2a-\xi -\delta s) = \mu q_z(\xi +\delta s), \end{aligned} \end{aligned}$$
(4.69)

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

$$\begin{aligned} \begin{aligned} q_{y}^\text {(a)}(\xi +\delta s,s+\delta s)&= \dfrac{1}{2} k\varphi \delta s (2a-2\xi -\delta s) + q_y^\text {(a)}(\xi ,s) \le - \mu \frac{q_z^{*}}{a^2}\delta s (2a-2\xi -\delta s) - \mu q_z(\xi ) \\&= -\mu \frac{q_z^{*}}{a^2}(\xi +\delta s)(2a-\xi -\delta s) = -\mu q_z(\xi +\delta s). \end{aligned} \end{aligned}$$
(4.70)

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 \((\xi , s) \in (0,2a]\times \mathbb {R}_{>0}\):

(4.71a)
(4.71b)
(4.71c)
(4.71d)

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. 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)

    Combining (4.72) and (4.73), (4.71a) is deduced.

  2. 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

$$\begin{aligned} u_y^\text {(s)}(\xi ) = \dfrac{\mu }{k}q_z(\xi ){{\,\mathrm{sgn}\,}}\sigma _y, \end{aligned}$$
(4.76)

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 steady-state 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 \(\xi \in [s,2a]\):

(4.77)

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:

$$\begin{aligned}&\sigma _y> 0 , \, q_y^\text {(a)}(\xi ,s) \ge \mu q_z(\xi )&\implies&q_y^\text {(a)}(\xi +\delta s, s+\delta s) > \mu q_z(\xi +\delta s), \end{aligned}$$
(4.78a)
$$\begin{aligned}&\sigma _y > 0 , \, q_y^\text {(a)}(\xi ,s) \le -\mu q_z(\xi )&\implies&q_y^\text {(a)}(\xi +\delta s, s+\delta s) < -\mu q_z(\xi +\delta s), \end{aligned}$$
(4.78b)
$$\begin{aligned}&\sigma _y < 0, \, q_y^\text {(a)}(\xi ,s) \ge \mu q_z(\xi )&\implies&q_y^\text {(a)}(\xi +\delta s, s+\delta s) > \mu q_z(\xi +\delta s), \end{aligned}$$
(4.78c)
$$\begin{aligned}&\sigma _y< 0, \, q_y^\text {(a)}(\xi ,s) \le -\mu q_z(\xi )&\implies&q_y^\text {(a)}(\xi +\delta s, s+\delta s)< - \mu q_z(\xi +\delta s) . \end{aligned}$$
(4.78d)

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. 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. 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 outward-pointing unit normal to \(\mathscr {S}\) is thus given by

(4.81)

Furthermore, differentiating (4.13) with respect to the travelled distance following a point on the sliding edge yields [19, 20]

(4.82)

for some representation of the velocity \(\bar{\boldsymbol{v}}_{\mathscr {S}}^{(\boldsymbol{\rho })}(\boldsymbol{\rho },s) \). Therefore,

(4.83)

In particular, the partial derivative \(\partial \gamma _{\mathscr {S}}(\boldsymbol{x},s)/\partial s\) reads

(4.84)

A particular representation of the velocity of a sliding edge that is oriented as the unit normal is thus given by

(4.85)

Analogously, for a travelling edge, similar equations yield the following expression for the unit normal:

(4.86)

whilst a nondimensional velocity vector which is oriented as the unit normal may be computed as

(4.87)

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Romano, L. (2022). Unsteady-State Brush Theory. In: Advanced Brush Tyre Modelling. SpringerBriefs in Applied Sciences and Technology. Springer, Cham. https://doi.org/10.1007/978-3-030-98435-9_4

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  • Print ISBN: 978-3-030-98434-2

  • Online ISBN: 978-3-030-98435-9

  • eBook Packages: EngineeringEngineering (R0)