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Physical Model of Tire-Road Contact Under Wet Conditions

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Abstract

A physical model to describe the contact between rubber and a rough surface with water as intermediate medium is presented. The Navier–Stokes equations are simplified and surface properties are approached by the Abbott–Firestone curve to generate an approximative description of the water squeeze out between a visco-elastic rubber block and a macro-rough surface. The model is used to describe the pattern dependent wet grip performance of vehicle tires at moderate water heights between pure wetgrip and full hydroplaning. Influence of surface macro-roughness, water height, tire pattern, and vehicle speed on braking performance is considered in particular. For validation purpose, braking tests on two different surfaces were done at an inner drum test bench. Test results show good agreement with the theory presented.

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Abbreviations

\(A\) :

Surface area of tread block

\(A_{\rm F}(h)\) :

Area of contact between fluid and rubber

\(A_q(h)\) :

Free area between track and tread block

\(A_{q_x}(h)\) :

Free area between track and tread block parallel to yz-plane

\(A_{q_y}(h)\) :

Free area between track and tread block parallel to xz-plane

\(A_{\rm R}(h)\) :

Area of contact between track and rubber

\(B\) :

Width of tread block

\(C\) :

Geometric factor

\(E\) :

Modulus of elasticity of Kelvin–Voigt element

\(\eta\) :

Viscosity of Kelvin–Voigt element

\(f_{L}(h)\) :

Water height dependent fluid velocity coefficient in x-direction

\(f_{B}(h)\) :

Water height dependent fluid velocity coefficient in y-direction

\(F_z(t)\) :

Load on tread block

\(\gamma _{\rm T}\) :

Factor for churning losses

\(G_1(h)\) :

Track parameter

\(G_2(h)\) :

Track parameter

\(G_3(h)\) :

Track parameter

\(G_4(h)\) :

Track parameter

\(h(t)\) :

Water height

\({\dot{h}}(t)\) :

Initial change of water height

\(h_0\) :

Initial water height

\({\bar{h}}(h)\) :

Equivalent water height

\(K(h)\) :

Track parameter

\(\kappa (z)\) :

Correction factor for control volume \(V_{\rm C}\)

\(L\) :

Length of tread block

\({\dot{m}}_{{\rm out}}\) :

Mass flow density over the control volume boundaries

\(\mu\) :

Dynamic viscosity of fluid

\(p_{\rm F}(t)\) :

Mean fluid pressure

\(p_{\rm m}(t)\) :

Mean pressure acting on tread block

\(p_{\rm R}(t)\) :

Mean contact pressure at interface \(A_{\rm R}(h)\)

\(\psi\) :

Relation between \(f_{L}\) and \(f_{B}\)

\(\rho\) :

Density of water

\(\rho _{\rm R}\) :

Density of rubber

\(s(x,y,h(t))\) :

Local rubber deformation at interface \(A_{\rm R}(h)\)

\(\bar{s}(h)\) :

Mean rubber deformation at interface \(A_{\rm R}(h)\)

\(s^*(t)\) :

Additional rubber deformation at interface \(A_{\rm F}(h)\)

\(\dot{s}^*(t)\) :

Time derivative of additional rubber deformation at interface \(A_{\rm F}(h)\)

\(\mathbf {v}_{\rm F}\) :

Fluid velocity

\(u(x)\) :

Fluid velocity in x-direction

\(v(y)\) :

Fluid velocity in y-direction

\(w(z)\) :

Fluid velocity in z-direction

\(t_{\rm c}\) :

Contact time

\(u(x,z')\) :

Fluid velocities x-direction in equivalent coordinate system

\(v(y,z')\) :

Fluid velocities y-direction in equivalent coordinate system

\(w(x,y,z')\) :

Fluid velocities z-direction in equivalent coordinate system

\(v_{{\rm out}}(x,y)\) :

Fluid velocity at boundary surfaces

\(v_{{\rm out}, A_{q_x}}(y)\) :

Fluid velocity at boundary surface \(A_{q_x}\)

\(v_{{\rm out}, A_{q_y}}(x)\) :

Fluid velocity at boundary surface \(A_{q_y}\)

\(v_{{\rm out}_{\perp }, A_{q_x}}(y)\) :

Perpendicular fluid velocity at boundary surface \(A_{q_x}\)

\(v_{{\rm out}_{\perp }, A_{q_y}}(x)\) :

Perpendicular fluid velocity at boundary surface \(A_{q_y}\)

\(V_{\rm C}(h)\) :

Control volume

\({\dot{V}}_{{\rm in}}\) :

Volume flow in \(V_{{\rm in}}\)

\(V_{\infty }\) :

Infinite volume under tread block

\({\dot{V}}_{{\rm out}}\) :

Volume flow out \(V_{{\rm out}}\)

\({\dot{W}}_{{\rm in}}\) :

Engery flow into \(V_{\rm C}\)

\({\dot{W}}_{{\rm kin}}\) :

Change of kinetic energy inside \(V_{\rm C}\)

\(W_{{\rm kin},F}\) :

Initial kinetic energy of fluid

\({\dot{W}}_{{\rm out}}\) :

Energy flow out of \(V_{\rm C}\)

\({\dot{W}}_{{\rm visc}}\) :

Viscous losses inside \(V_{\rm C}\)

\({\mathbf{x}}\) :

State vector

\(x_a\) :

Material share

\(x,y,z'\) :

Coordinate system for equivalent water height

\(\dot{y}_{\perp }\) :

Initial vertical velocity of tread block on an undisturbed circular path

\(Z(x,y)\) :

Track profile

\(z_a\) :

Track height

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Correspondence to J. Löwer.

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Appendices

Appendices

Appendix A: Pattern Geometries

Block geometries of the patterns used, void volume is equal to surface void. BB-tire contains 3 rows with the block geometry described, SB and SBv contain 5 rows, including the tire shoulder

Geometry Abbrv. Block length (mm) Block width (mm) Void volume (%) Nr. of rows
Big blocks BB 36.6 53.2 24.8 3
Small blocks SB 23.2 25.6 25.0 5
Small blocks SBv 20.8 25.6 35.3 5

Appendix B: Additional Information for Change of Kinetic Energy

Steps after Eq. 23

$${\dot{W}}_{{\rm kin}} = \frac{{\rm d}}{{\rm d}t} \left( \int _{-\infty }^{h(t)} \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} 2 \rho \kappa (z) \left( f_{B}^2 y^2 + f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y\,{\rm d}z \right)$$
(63)
$${\dot{W}}_{{\rm kin}} = 2 \rho \frac{{\rm d}}{{\rm d}t} \left( \int _{-\infty }^{h(t)} \kappa (z) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( f_{B}^2 y^2+ f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y\,{\rm d}z \right)$$
(64)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ \int _{-\infty }^{h(t)} \frac{d}{dt} \left( \kappa (z) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( f_{B}^2 y^2+ f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y \right) {\rm d}z + {\dot{h}} \kappa (h) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( f_{B}^2 y^2 + f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y \right ]$$
(65)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ \int _{-\infty }^{h(t)} \left( \kappa (z) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} 2 \left( f_{B} \dot{f}_{B} y^2+ f_{L} \dot{f}_{L} x^2 \right) {\rm d}x\,{\rm d}y \right) {\rm d}z + {\dot{h}} \kappa (h) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( f_{B}^2 y^2 + f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y \right ]$$
(66)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} 2 \left( \psi ^2 f_{L} \dot{f}_{L} y^2 + f_{L} \dot{f}_{L} x^2 \right) {\rm d}x\,{\rm d}y + {\dot{h}} \kappa (h) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( \psi ^2 f_{L}^2 y^2 + f_{L}^2 x^2 \right) {\rm d}x\,{\rm d}y \right ]$$
(67)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} 2 f_{L} \dot{f}_{L} \left( x^2 +\psi ^2 y^2 \right) {\rm d}x\,{\rm d}y + {\dot{h}} \kappa (h) \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( f_{L}^2 \left( x^2 + \psi ^2 y^2\right) \right) {\rm d}x\,{\rm d}y \right ]$$
(68)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ 2 f_{L} \dot{f}_{L} \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) f_{L}^2 \right ] \int _{0}^{\frac{B}{2}} \int _{0}^{\frac{L}{2}} \left( x^2 + \psi ^2 y^2 \right) {\rm d}x\,{\rm d}y$$
(69)
$${\dot{W}}_{{\rm kin}} = 2 \rho \left [ 2 f_{L} \dot{f}_{L} \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) f_{L}^2 \right ] \left[ \left[ \frac{1}{3} x^3y + \frac{1}{3} \psi ^2xy^3 \right] _0^{\frac{B}{2}} \right] _0^{\frac{L}{2}}$$
(70)
$${\dot{W}}_{{\rm kin}}= 2 \rho \left [ 2 f_{L} \dot{f}_{L} \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) f_{L}^2 \right ] \left[ \frac{1}{3} \frac{L^3}{8}\frac{B}{2} + \frac{1}{3} \psi ^2 \frac{L}{2}\frac{B^3}{8} \right]$$
(71)
$${\dot{W}}_{{\rm kin}}= 2 \rho \left [ 2 f_{L} \dot{f}_{L} \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) f_{L}^2 \right ] \left[ \frac{1}{48} B L \left( L^2 + \psi ^2 B^2 \right) \right]$$
(72)
$${\dot{W}}_{{\rm kin}} = \frac{1}{24} \rho B L \left( L^2 + \psi ^2 B^2 \right) \left [ 2 f_{L} \dot{f}_{L} \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) f_{L}^2 \right ]$$
(73)
$${\dot{W}}_{{\rm kin}} = \frac{1}{24} \rho B L \left( L^2 + \psi ^2 B^2 \right) \left [ 2 \left( 2C{\dot{h}}\frac{1}{1+\psi } \frac{A_F}{A_q} \right) \left( 2 C \frac{1}{(1+\psi )} \left( {\ddot{h}}\frac{A_F}{A_q} + {\dot{h}}^2\frac{\partial }{\partial h}\left( \frac{A_F}{A_q} \right) \right) \right) \cdot \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) \left( 2C{\dot{h}}\frac{1}{1+\psi } \frac{A_F}{A_q} \right) ^2 \right ]$$
(74)
$${\dot{W}}_{{\rm kin}} = \frac{1}{24} \rho B L \left( L^2 + \psi ^2 B^2 \right) \frac{4 C^2}{(1+\psi )^2} \left [ 2 {\dot{h}} \frac{A_F}{A_q} \left( {\ddot{h}}\frac{A_F}{A_q} + {\dot{h}}^2\frac{\partial }{\partial h}\left( \frac{A_F}{A_q} \right) \right) \cdot \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z + {\dot{h}} \kappa (h) \left( {\dot{h}} \frac{A_F}{A_q} \right) ^2 \right ]$$
(75)
$${\dot{W}}_{{\rm kin}} = \frac{1}{6} \rho C^2 \frac{L^2 + \psi ^2 B^2}{(1+\psi )^2} \left [ 2 {\dot{h}} \frac{A_F}{A_q} \left( {\ddot{h}}\frac{A_F}{A_q} + {\dot{h}}^2\frac{\partial }{\partial h}\left( \frac{A_F}{A_q} \right) \right) \cdot \overbrace{\int _{-\infty }^{h(t)} A_F(z) {\rm d}z}^{K(h)} + {\dot{h}} A_F \left( {\dot{h}} \frac{A_F}{A_q} \right) ^2 \right ]$$
(76)
$${\dot{W}}_{{\rm kin}} = \frac{1}{6} \rho C^2 \frac{L^2 + \psi ^2 B^2}{(1+\psi )^2} \left [ 2 K(h) \left( {\dot{h}} {\ddot{h}} \overbrace{\left( \frac{A_F(h)}{A_q(h)} \right) ^2}^{G_1(h)} + {\dot{h}}^3 \overbrace{\frac{A_F(h)}{A_q(h)} \frac{\partial }{\partial h} \left( \frac{A_F(h)}{A_q(h)} \right) }^{G_2(h)} \right) + {\dot{h}}^3 \overbrace{\left( \frac{A_F^3}{A_q^2} \right) }^{G_3(h)} \right ]$$
(77)
$${\dot{W}}_{{\rm kin}} = \frac{1}{6} \rho C^2 \frac{L^2 + \psi ^2 B^2}{(1+\psi )^2} \left [ 2 K(h) \left( {\dot{h}} {\ddot{h}} G_1(h) + {\dot{h}}^3 G_2(h) \right) + {\dot{h}}^3 G_3(h) \right ]$$
(78)

Appendix C: Additional Information for Change of Energy Outflow

Steps after Eq. 31

$${\dot{W}}_{{\rm kin}} =2 \rho \int _{-\infty }^{h(t)} \kappa (z) \int _0^{\frac{B}{2}} f_{L}^2 \left( \frac{L^2}{4} + \psi ^2 y^2 \right) \cdot f_{L} \frac{L}{2} {\rm d}y + \int _0^{\frac{L}{2}} f_{L}^2 \left( x^2 + \psi ^2 \frac{B^2}{4} \right) \cdot \psi f_{L} \frac{B}{2} {\rm d}x \, {\rm d}z$$
(79)
$${\dot{W}}_{{\rm kin}}=2 \rho \int _{-\infty }^{h(t)} \kappa (z) \left( \left[ f_{L}^3 \left( \frac{L^3}{8}y + \psi ^2 \frac{L}{2} \frac{1}{3} y^3 \right) \right] _0^{\frac{B}{2}} + \left[ f_{L}^3 \psi \left( \frac{B}{2} \frac{1}{3} x^3 + \psi ^2 \frac{B^3}{8} x \right) \right] _0^{\frac{L}{2}} \right) {\rm d}z$$
(80)
$${\dot{W}}_{{\rm kin}}=2 \rho \int _{-\infty }^{h(t)} \kappa (z) \frac{f_{L}^3}{16} \left( L^3B + \frac{1}{3} \psi ^2 B^3L + \frac{1}{3} \psi BL^3 + \psi ^3 B^3L \right) {\rm d}z$$
(81)
$${\dot{W}}_{{\rm kin}}= \frac{1}{8} \rho \left( L^3B + \frac{1}{3} \psi ^2 B^3L + \frac{1}{3} \psi BL^3 + \psi ^3 B^3L \right) f_{L}^3 \int _{-\infty }^{h(t)} \kappa (z) {\rm d}z$$
(82)
$${\dot{W}}_{{\rm kin}} = \frac{1}{8} \rho \left( L^3B + \frac{1}{3} \psi ^2 B^3L + \frac{1}{3} \psi BL^3 + \psi ^3 B^3L \right) \left( - 2 C {\dot{h}}\frac{1}{(1+\psi )} \frac{A_F}{A_q} \right) ^3 \int _{-\infty }^{h(t)} \frac{A_F(z)}{A} {\rm d}z$$
(83)
$${\dot{W}}_{{\rm kin}} = - \frac{1}{8} \frac{ 8 C^3 {\dot{h}}^3 }{(1+\psi )^3 \underbrace{BL}_{=A}} \rho \left( L^3B + \frac{1}{3} \psi ^2 B^3L + \frac{1}{3} \psi BL^3 + \psi ^3 B^3L \right) \left( \frac{A_F}{A_q} \right) ^3 \int _{-\infty }^{h(t)} A_F(z) {\rm d}z$$
(84)
$${\dot{W}}_{{\rm kin}} = - \frac{\rho C^3 }{(1+\psi )^3 } {\dot{h}}^3 \left( L^2 + \frac{1}{3} \psi ^2 B^2 + \frac{1}{3} \psi L^2 + \psi ^3 B^2 \right) \overbrace{\left( \frac{A_F(h)}{A_q(h)} \right) ^3 }^{G_4(h)} \overbrace{\int _{-\infty }^{h(t)} A_F(z) {\rm d}z}^{K(h)}$$
(85)
$${\dot{W}}_{{\rm kin}} = - \frac{\rho C^3 }{(1+\psi )^3} {\dot{h}}^3 \left( L^2 + \frac{1}{3} \psi ^2 B^2 + \frac{1}{3} \psi L^2 + \psi ^3 B^2 \right) G_4(h) K(h)$$
(86)

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Löwer, J., Wagner, P., Unrau, H. et al. Physical Model of Tire-Road Contact Under Wet Conditions. Tribol Lett 68, 25 (2020) doi:10.1007/s11249-019-1264-6

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Keywords

  • Wet braking
  • Tribology
  • Friction
  • Road texture
  • Tire-road contact
  • Water height