Cornering Properties of Tires

Abstract

The motion of a vehicle is determined by the force and moment generated by tires, which are the only points of contact between a vehicle and the road.

Notes

Note 11.1

From Eqs. (11.87) and (11.88), the displacement in the circumferential direction $\Delta x^{{\prime }}$ is given by the relation $\Delta x^{{\prime }} = sx_{1}$, where s is the slip ratio. Meanwhile, from Eq. (11.3), the lateral displacement v is given by v = αx. Comparing both expressions, the slip ratio s corresponds to the slip angle α (unit: radian), if both are small.

Note 11.2 Eq. (11.39)

The characteristic equation for Eq. (11.38) is obtained through the substitution of w(x) = 0 and y = e^αx. Introducing the parameter $\uplambda = \sqrt[4]{{k_{\text{s}} /\left( {4{{EI}}_{z} } \right)}}$, the characteristic equation is obtained as α⁴ + 4λ⁴ = 0. The solutions to this equation are α = (1 ± i)λ, (−1 ± i)λ. Hence, y is expressed by

$$y = {\text{e}}^{\lambda x} \left( {C_{1} \cos \,\lambda x + C_{2} \sin \,\lambda x} \right) + {\text{e}}^{ - \lambda x} \left( {C_{3} \cos \,\lambda x + C_{4} \sin \,\lambda x} \right).$$

Considering the boundary condition y(∞) = 0, we obtain C₁ = C₂ = 0. The symmetry condition $y^{{\prime }}$(0) = 0 yields C₃ = C₄. Hence, y is expressed by

$$y = C_{3} {\text{e}}^{ - \lambda x} \left( {\cos \,\lambda x + \sin \,\lambda x} \right).$$

The equilibrium between the external force F_y and the force due to the spring of the sidewall yields

$$F_{y} = 2k_{\text{s}} \int\limits_{0}^{\infty } y (x){\text{d}}x = 2k_{\text{s}} C_{3} {\text{e}}^{ - \lambda x} \int\limits_{0}^{\infty } {\left( {\cos \lambda x + \sin \lambda x} \right)} {\text{d}}x = 2k_{\text{s}} C_{3} /\lambda .$$

Hence, y is expressed by

$$y = \frac{{\lambda F_{y} }}{{2k_{\text{s}} }}{\text{e}}^{ - \lambda x} \left( {\cos \,\lambda x + \sin \,\lambda x} \right).$$

Note 11.3 Eq. (11.41)

Using the relations e^−λ ≅ 1 − λx, cos(λx) ≅ 1, sin(λx) ≅ λx for small λ, we obtain

$$y = \frac{{\lambda F_{y} }}{{2k_{\text{s}} }}{\text{e}}^{ - \lambda x} \left( {\cos \,\lambda x + \sin \lambda x} \right) \cong \frac{{\lambda F_{y} }}{{2k_{\text{s}} }}\left( {1 - \lambda^{2} x^{2} } \right).$$

Note 11.4 Eq. (11.50)

Using Eqs. (11.43) and (11.46) and the equation f_y = μ_sq_z at the sliding point x₁ = l_h, l_h is given as the solution to

$$x_{1} \tan \,\alpha - \frac{\delta }{{C_{y} }}F_{y} \frac{{x_{1} }}{l}\left( {1 - \frac{{x_{1} }}{l}} \right) = 4\mu_{\text{s}} p_{\text{m}} \frac{{x_{1} }}{l}\left( {1 - \frac{{x_{1} }}{l}} \right).$$

Note 11.5 Eq. (11.58)

The equation of the ellipse in Fig. 11.20 is

$$\left( {x_{1} - l/2} \right)^{2} /r^{2} + \left( {y - r_{\text{e}} \sin \,\gamma } \right)^{2} /b^{2} = 1.$$

Because the ellipse passes the position (x₁ = y₁ = 0), b is given by

$$b = r_{\text{e}} \sin \,\gamma /\sqrt {1 - \left( {l - 2r_{\text{e}} } \right)^{2} } .$$

Suppose that y_c is the trajectory of the tread base. Considering the relation r ≈ r_e, y_c,max, the maximum value of y_c, is expressed by

$$y_{\text{c,max}} = - \left( {b - r_{\text{e}} \sin \,\gamma } \right) = - \left( {\frac{{r_{\text{e}} \sin \,\gamma }}{{\sqrt {1 - \frac{{l^{2} }}{{4r^{2} }}} }} - r_{\text{e}} \sin \,\gamma } \right) \cong - \frac{{l^{2} r_{\text{e}} \sin \,\gamma }}{{8r^{2} }} \cong - \frac{{l^{2} \sin \,\gamma }}{{8r_{\text{e}} }} \cong - \frac{{l^{2} \gamma }}{{8r_{\text{e}} }}.$$

If we express y_c as a parabola, y_c is given by

$$y_{\text{c}} = C\frac{{x_{1} }}{l}\left( {1 - \frac{{x_{1} }}{l}} \right).$$

The condition y_c = y_c,max at x₁ = l/2 yields C = − l²sinγ/(2r_e).

Note 11.6 Eq. (11.64)

Equation (11.64) is derived by adding the term related to the circumferential tension T to Eq. (11.38). Figure 11.67 shows the force equilibrium of the belt element including the tension. The force equilibrium at the element with length dx along the y-axis is given by

Fig. 11.67

Force equilibrium in the circumferential direction for an element of the belt

$$- T\frac{\partial y}{\partial x} + \left( {T + {\text{d}}T} \right)\left( {\frac{\partial y}{\partial x} + \frac{{\partial^{2} y}}{{\partial x^{2} }}{\text{d}}x} \right) + S - \left( {S + {\text{d}}S} \right) - k_{\text{s}} y{\text{d}}x + w(x){\text{d}}x = 0,$$

(11.217)

where S is the shear force, k_s is the fundamental lateral spring rate of the sidewall, and w(x) is the distributed external force. Because y is a function of only x, the partial differential equation changes to an ordinary differential equation:

$$T \cdot {\text{d}}^{2} y/{\text{d}}x^{2} = {\text{d}}S/{\text{d}}x - k_{\text{s}} y + w(x) = 0,$$

(11.218)

where the shear force S and the moment M have the relation

$${\text{d}}M - S{\text{d}}x = 0.$$

(11.219)

The moment M is given by

$$M = {{EI}}_{z} \cdot {\text{d}}^{2} y/{\text{d}}x^{2} ,$$

(11.220)

where EI_z is the flexural rigidity of the tread base. Substituting Eqs. (11.219) and (11.220) into Eq. (11.218), we obtain

$${{EI}}_{z} \cdot {\text{d}}^{4} y/{\text{d}}x^{4} - T \cdot {\text{d}}^{2} y/{\text{d}}x^{2} + k_{\text{s}} y = w(x).$$

(11.221)

In the case that w(x) = 0, referring to Eq. (3.225) in Appendix of Chap. 3, y is given as

$$y = \frac{{\delta F_{y} }}{{4k_{\text{s}} }}{\text{e}}^{{ - \lambda_{1} x}} \left( {\cos \,\lambda_{2} x + \frac{{\lambda_{1} }}{{\lambda_{2} }}\sin \,\lambda_{2} x} \right),$$

(11.222)

where

$$\delta = \left( {\lambda_{1}^{2} + \lambda_{2}^{2} } \right)/\lambda_{1} ,$$

(11.223)

$$\begin{aligned} & \lambda_{1} = {{\root 4 \of {{{\frac{{k_{y} }}{{4{{EI}}_{z} }}}}} \sqrt {1 + \frac{T}{{\sqrt {4{{EI}}_{z} k_{\text{s}} } }}} }} \\ & \lambda_{2} = {{\root 4 \of {{{\frac{{k_{y} }}{{4{{EI}}_{z} }}}}} \sqrt {1 - \frac{T}{{\sqrt {4{{EI}}_{z} k_{\text{s}} } }}} }}. \\ \end{aligned}$$

(11.224)

The value of the parameter ${\text{T}}/\sqrt {4{{EI}}_{z} k_{\text{s}} }$ of Eq. (11.224) is about 0.02 for a passenger-car belted radial tire, and the term for the circumferential tension of the tread ring T can thus be neglected.

Note 11.7 Eq. (11.65)

The belt is twisted around the x-axis by the side force. The lateral displacement of the belt is generated at the contact patch by twisting deformation, and it is expressed by a function similar to that for the belt deformation under camber thrust shown in Fig. 11.20. Referring to Note 11.5, when the twisting angle of a tire is θ, the maximum displacement in the lateral direction $y_{\text{c}}^{ \hbox{max} }$ is given as $y_{\text{c}}^{ \hbox{max} } = l^{2} \theta /\left( {8r} \right)$. The lateral spring rate of the tire K_y is expressed by the fundamental spring rate of the tire per unit length in the circumferential direction k_s²⁹:

$$K_{y} = 2\pi rk_{s} /3,$$

where r is the radius of the tire. Using Eq. (6.43), the torsion angle θ is given as

$$\theta = rF_{y} /R_{mz} .$$

The out-of-plane torsional spring rate R_mz at radius r_A per unit length in the circumferential direction is expressed as R_mz = πr³k_s by substituting B = 0 into Eq. (6.38). Considering the relation

$$\theta = \frac{{rF_{y} }}{{R_{mz} }} = \frac{{rF_{y} }}{{\pi r^{3} k_{s} }} = \frac{{F_{y} }}{{\pi r^{2} }}\frac{2\pi r}{{3K_{y} }} = \frac{{2F_{y} }}{{3rK_{y} }},$$

$y_{\text{c}}^{ \hbox{max} }$ is expressed as

$$y_{\text{c}}^{\hbox{max} } = l^{2} \theta /(8r) = F_{y} l^{2} /(12r^{2} K_{y} ).$$

Suppose that y_c is expressed by

$$y_{\text{c}} = A\frac{{x_{1} }}{l}\left( {1 - \frac{{x_{1} }}{l}} \right).$$

The maximum value of y_c is given by $y_{\text{c}}^{ \hbox{max} } = A/4$. Comparing this equation with the above equation, the value of A is given as

$$A = F_{y} l^{2} /(3r^{2} K_{y} ).$$

Note 11.8 Eq. (11.83)

Pacejka [1] introduced two slip ratios, namely the practical longitudinal slip ratio s and the theoretical longitudinal slip ratio σ_x expressed by

$$\begin{aligned} & s = - V_{sx} /V_{\text{R}} = - \left( {V_{\text{R}} - V_{\text{B}} } \right)/V_{\text{R}} = - \left( {V_{\text{R}} - r_{\text{e}}\Omega } \right)/V_{\text{R}} \\ & \sigma_{x} = - V_{sx} /V_{\text{B}} = s/(1 + s), \\ \end{aligned}$$

where V_sx is the longitudinal slip velocity, r_e is the effective rolling radius, and Ω is the angular velocity of the wheel. s is negative for the braking condition and positive for the driving condition. The value of s ranges from −1 to ∞. Meanwhile, this book defines s according to Eq. (11.83), the definition of s is thus different between braking and driving conditions, and the quantities related to s are not continuous at s = 0. If the definition of s proposed by Pacejka is used, this discontinuity does not occur, but s becomes infinite under the driving condition. The range of s proposed by Sakai is from −1 to +1, and the sign of s is positive for the braking condition and negative for the driving condition. Readers need to be careful which definition is used for the slip ratio.

Note 11.9 Eq. (11.98)

l_h is given by the solution to the following equations.

In the case of braking (s > 0),

$$\begin{aligned} & 4\mu_{\text{s}} p_{\text{m}} \frac{x}{l}\left( {1 - \frac{x}{l}} \right) = x\sqrt {C_{x}^{2} s^{2} + C_{y}^{2} \sin^{2} \alpha } \\ & x = l\left( {1 - \frac{l}{{4\mu_{\text{s}} p_{\text{m}} }}\sqrt {C_{x}^{2} s^{2} + C_{y}^{2} \sin^{2} \alpha } } \right). \\ \end{aligned}$$

Assuming C_x = C_y = C and considering p_m = 3F_z/(2 lb) and C_Fα = Cl²b/2, we obtain

$$l_{h} = l\left( {1 - \frac{{C_{F\alpha } }}{{3\mu_{\text{s}} F_{z} }}\sqrt {s^{2} + \sin^{2} \alpha } } \right) \cong l\left( {1 - \frac{{C_{F\alpha } }}{{3\mu_{\text{s}} F_{z} }}\sqrt {s^{2} + \tan^{2} \alpha } } \right).$$

In the case of driving (s < 0),

$$x = l\left( {1 - \frac{l}{{4\mu_{\text{s}} p_{\text{m}} }}\sqrt {C_{x}^{2} s^{2} + C_{y}^{2} (1 + s)^{2} \tan^{2} \alpha } } \right),$$

where C_x = C_y = C and, in a manner similar to the case of braking, we obtain

$$l_{h} = l\left( {1 - \frac{{C_{F\alpha } }}{{3\mu_{\text{s}} F_{z} }}\sqrt {s^{2} + (1 + s)^{2} \tan^{2} \alpha } } \right) \cong l\left( {1 - \frac{{C_{F\alpha } }}{{3\mu_{\text{s}} F_{z} }}\sqrt {s^{2} + \tan^{2} \alpha } } \right).$$

Note 11.10 Eq. (11.99)

The following relations are obtained referring to Fig. 11.25.

In the case of braking,

$$s = \frac{{V_{\text{R}} \cos \alpha - V_{\text{B}} }}{{V_{\text{R}} \cos \alpha }} = \frac{a}{{a + V_{\text{B}} }},\;\tan \theta = \frac{d}{a},\;\tan \alpha = \frac{d}{{a + V_{\text{B}} }} \Rightarrow \frac{a}{{a + V_{\text{B}} }}\frac{d}{a} = \frac{d}{{a + V_{\text{B}} }} \Rightarrow s\tan \,\theta = \tan \,\alpha .$$

In the case of driving,

$$s = \frac{{V_{\text{R}} \cos \alpha - V_{\text{B}} }}{{V_{\text{B}} }} = \frac{a}{{V_{\text{B}} }},\quad \tan \theta \approx \tan \theta^{\prime} = \frac{d}{a},\;\tan \alpha = \frac{d}{{V_{\text{B}} }} \Rightarrow \frac{a}{{V_{\text{B}} }}\frac{d}{a} = \frac{d}{{V_{\text{B}} }} \Rightarrow s\tan \theta = \tan \alpha .$$

Note 11.11 Eq. (11.104)

Using Eq. (11.99), we obtain

$$\begin{aligned} & \sin \theta = \tan \,\alpha /\sqrt {s^{2} + \tan^{2} \alpha } = h\tan \,\alpha \\ & \cos \theta = s/\sqrt {s^{2} + \tan^{2} \alpha } = hs. \\ \end{aligned}$$

Equation (11.104) is expressed as a function of α and s.

Note 11.12 Eqs. (11.124) and (11.125)

In the case of braking, from Eq. (11.83), we obtain

$${\text{Braking}}\quad s = \left( {V_{\text{R}} \cos \,\alpha - V_{\text{B}} } \right)/\left( {V_{\text{R}} \cos \,\alpha } \right) > 0 \to s = 1 - V_{\text{B}} /\left( {V_{\text{R}} \cos \,\alpha } \right) \to V_{\text{B}} /V_{\text{R}} = \left( {1 - s} \right)\cos \,\alpha .$$

The substitution of the above equation into the first equation of Eq. (11.123) yields

$$\begin{aligned} V^{{\prime \prime 2}} & = V_{{\text{R}}}^{2} + V_{{\text{B}}}^{2} - 2V_{{\text{R}}} V_{{\text{B}}} \cos \alpha = V_{{\text{R}}}^{2} \left\{ {1 + \left( {\frac{{V_{{\text{B}}} }}{{V_{{\text{R}}} }}} \right)^{2} - 2\frac{{V_{{\text{B}}} }}{{V_{{\text{R}}} }}\cos \alpha } \right\} \\ & = V_{{\text{R}}}^{2} \left\{ {1 + \left( {1 - s} \right)^{2} \cos ^{2} \alpha - 2\left( {1 - s} \right)\cos ^{2} \alpha } \right\} \\ & = V_{{\text{R}}}^{2} \left\{ {1 + \left( {1 - 2s + s^{2} } \right)\cos ^{2} \alpha - 2\left( {1 - s} \right)\cos ^{2} \alpha } \right\} \\ & = V_{{\text{R}}}^{2} \left\{ {1 + \left( {s^{2} - 1} \right)\cos ^{2} \alpha } \right\}. \\ \end{aligned}$$

In the case of driving, from Eq. (11.83), we obtain

$${\text{Driving}}\quad s = \left( {V_{\text{R}} \cos \alpha - V_{\text{B}} } \right)/V_{\text{B}} < 0 \to s = - 1 + V_{\text{R}} \cos \alpha /V_{\text{B}} \to V_{\text{R}} /V_{\text{B}} = \left( {1 + s} \right)/\cos \alpha .$$

Similarly, the substitution of the above equation into the first equation of Eq. (11.123) yields

$$\begin{aligned} V^{{\prime \prime 2}} & = V_{{\text{R}}}^{2} + V_{{\text{B}}}^{2} - 2V_{{\text{R}}} V_{{\text{B}}} \cos {\mkern 1mu} \alpha = V_{{\text{B}}}^{2} \left\{ {1 + \left( {\frac{{V_{{\text{R}}} }}{{V_{{\text{B}}} }}} \right)^{2} - 2\frac{{V_{{\text{R}}} }}{{V_{{\text{B}}} }}\cos {\mkern 1mu} \alpha } \right\} \\ & = V_{{\text{B}}}^{2} \left\{ {1 + \frac{{\left( {1 + s} \right)^{2} }}{{\cos ^{2} \alpha }} - 2\left( {1 + s} \right)} \right\} = \frac{{V_{{\text{B}}}^{2} }}{{\cos ^{2} \alpha }}\left\{ {1 + 2s + s^{2} - \cos ^{2} \alpha - 2s\cos ^{2} \alpha } \right\} \\ & = \frac{{V_{{\text{B}}}^{2} }}{{\cos ^{2} \alpha }}\left\{ {\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} } \right\} = V_{{\text{B}}}^{2} \left\{ {\tan ^{2} \alpha + 2s\tan ^{2} \alpha + \frac{{s^{2} }}{{\cos ^{2} \alpha }}} \right\}. \\ \end{aligned}$$

Note 11.13 Eqs. (11.126) and (11.127)

In the case of braking (s > 0), from Eqs. (11.123) and (11.124), we obtain

$$\begin{aligned} & \sin \,\theta = \frac{{V_{\text{R}} }}{{V^{\prime\prime}}}\sin \,\alpha = \frac{{V_{\text{R}} \sin \alpha }}{{V_{\text{R}} \sqrt {1 + \left( {s^{2} - 1} \right)\cos^{2} \alpha } }} = \frac{\sin \alpha }{{\sqrt {\sin^{2} \alpha + s^{2} \cos^{2} \alpha } }} = \frac{1}{{\sqrt {1 + \frac{{s^{2} }}{{\tan^{2} \alpha }}} }} \\ & \cos \,\theta = \sqrt {1 - \sin^{2} \theta } = \sqrt {1 - \frac{1}{{1 + \frac{{s^{2} }}{{\tan^{2} \alpha }}}}} = \frac{{\frac{s}{\tan \,\alpha }}}{{\sqrt {1 + \frac{{s^{2} }}{{\tan^{2} \alpha }}} }} \\ & \tan \,\theta = \frac{\tan \alpha }{s}. \\ \end{aligned}$$

In the case of driving (s < 0), using the relation V_R/V_B = (1 + s)/cosα and Eqs. (11.123) and (11.125), we obtain

$$\begin{aligned} \sin \theta & = \frac{{V_{{\text{R}}} }}{{V^{{\prime \prime }} }}\sin \alpha = \frac{{V_{{\text{R}}} \sin \alpha }}{{V_{{\text{B}}} \frac{{\sqrt {\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} } }}{{\cos {\mkern 1mu} \alpha }}}} \\ & = \frac{{1 + s}}{{\cos \alpha }}\frac{{\sin \alpha }}{{\frac{{\sqrt {\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} } }}{{\cos \alpha }}}} = \frac{{\left( {1 + s} \right)\sin \alpha }}{{\sqrt {\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} } }}, \\ \cos \theta & = \sqrt {1 - \sin ^{2} \theta } = \sqrt {1 - \frac{{\left( {1 + s} \right)^{2} \sin ^{2} \alpha }}{{\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} }}} = \frac{{s\cos \alpha }}{{\sqrt {\sin ^{2} \alpha + 2s\sin ^{2} \alpha + s^{2} } }} \\ \tan \theta & = \frac{{\left( {1 + s} \right)\tan \alpha }}{s}. \\ \end{aligned}$$

Because the condition s < 0 is satisfied in the case of driving, we obtain

$$\theta = \pi + \tan^{ - 1} \left( {\frac{1 + s}{s}\tan \,\alpha } \right).$$

Note 11.14 Eq. (11.134)

The third equation of Eq. (11.119) is

$$M_{z} = b\int\limits_{0}^{l} {\left\{ {f_{y} \left( {x - \frac{l}{2}} \right) - f_{x} y} \right\}{\text{d}}x} .$$

Assume that f_x is uniform in the width direction and y is expressed by Eq. (11.95). Using the relation $\alpha \ll 1$, y₀ in Eq. (11.95) is neglected. The moment M_zx due to f_x (component of M_z) is given by

$$M_{zx} = - b\int\limits_{0}^{l} {f_{x} y{\text{d}}x} = - b\int\limits_{0}^{l} {f_{x} x\tan \alpha {\text{d}}x} = f_{x} l^{2} b/2 \cdot \tan \alpha .$$

Note 11.15 Eqs. (11.153) and (11.157)

Equation (11.153)

From Eqs. (11.119) and (11.129), we obtain

$$\begin{aligned} & F_{y} \left( {s,\alpha _{0} ,V} \right)= F_{y}^{\prime } + F_{y}^{{\prime \prime }} = b\int\limits_{0}^{{l_{h} }} {f_{y} {\text{d}}x_{1} } + b\int\limits_{{l_{h} }}^{l} {\mu _{{\text{d}}} q_{z} (x_{1} )\sin \theta {\text{d}}x_{1} } \\ & \quad= bC_{y} \int\limits_{0}^{{l_{h} }} {\left\{ {x_{1} \tan \alpha - \varepsilon l^{2} F_{y} (1 - s)\frac{{x_{1} }}{l}\left( {1 - (1 - s)\frac{{x_{1} }}{l}} \right)} \right\}{\text{d}}x_{1} } + b\int\limits_{{l_{h} }}^{l} {\mu _{{\text{d}}} \frac{{n + 1}}{n}\frac{{F_{z} }}{{wl}}D_{{{\text{gsp}}}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\sin \theta {\text{d}}x_{1} } \\ & \quad= bC_{y} l^{2} \int\limits_{0}^{{r_{h} }} {\left\{ {t\tan \alpha - \varepsilon lF_{y} (1 - s)t\left( {1 - (1 - s)t} \right)} \right\}{\text{d}}t} + \frac{{n + 1}}{n}\mu _{{\text{d}}} \left( {s,\alpha _{0} ,V} \right)F_{z} \sin \theta \int\limits_{{r_{h} }}^{1} {D_{{{\text{gsp}}}} \left( {t;n,\zeta } \right){\text{d}}t} . \\ \end{aligned}$$

Equation (11.157)

Equation (11.156) can be rewritten as

$$M_{z} \left( {s,\alpha_{0} ,V} \right) = b\int\limits_{0}^{{l_{h} }} {\left\{ { - f_{x} y + f_{y} \left( {x_{1} - x_{c} } \right)} \right\}{\text{d}}} x_{1} + b\int\limits_{{l_{h} }}^{l} {\mu_{\text{d}} q_{z} \left\{ { - y^{\prime}\cos \theta + \left( {x_{1} - x_{c} } \right)\sin \theta } \right\}{\text{d}}x_{1} } .$$

One term of the above equation can be expressed as

$$b\int_{0}^{{l_{h} }} {f_{x} y{\text{d}}} x_{1} = b\int\limits_{0}^{{l_{h} }} {C_{x} sx_{1} \cdot y{\text{d}}} x_{1} = bC_{x} s\int\limits_{0}^{{l_{h} }} {x_{1}^{2} \tan \alpha {\text{d}}} x_{1} = \frac{{bC_{x} l_{h}^{3} s\tan \alpha }}{3} = \frac{{bC_{x} l^{3} r_{h}^{3} s\tan \alpha }}{3} = 4C_{Ms\_0} r_{h}^{3} s\tan \alpha ,$$

where

$$C_{Ms\_0} = bC_{x} l^{3} /12.$$

Using Eq. (11.114) with a zero camber angle, the second term of the above equation is given by

$$\begin{aligned} & b\int\limits_{0}^{{l_{h} }} {f_{y} \left( {x_{1} - x_{c} } \right){\text{d}}} x_{1} = bC_{y} \int\limits_{0}^{{l_{h} }} {\left\{ {x_{1} \tan \alpha - \frac{\delta }{{C_{y} }}F_{y} (1 - s)\frac{{x_{1} }}{l}\left( {1 - (1 - s)\frac{{x_{1} }}{l}} \right)} \right\}\left( {x_{1} - x_{c} } \right){\text{d}}} x_{1} \\ & = bC_{y} l^{3} \int\limits_{0}^{{r_{h} }} {\left\{ {t\tan \alpha - \varepsilon lF_{y} (1 - s)t\left( {1 - (1 - s)t} \right)} \right\}\left( {t - \frac{{x_{c} }}{l}} \right){\text{d}}} t \\ & = 12C_{M\alpha \_0} \int\limits_{0}^{{r_{h} }} {\left\{ {t\tan \alpha - \varepsilon lF_{y} (1 - s)t\left( {1 - (1 - s)t} \right)} \right\}\left( {t - \frac{{x_{c} }}{l}} \right){\text{d}}} t, \\ \end{aligned}$$

where

$$C_{M\alpha \_0} = bC_{y} l^{3} /12.$$

The other terms of the above equation are given by

$$b\int\limits_{{l_{h} }}^{l} {\mu_{\text{d}} q_{z} \left( {x_{1} - x_{c} } \right)\sin \theta {\text{d}}x_{1} } = b\mu_{\text{d}} \sin \theta l^{2} \int\limits_{{r_{h} }}^{i} {\frac{n + 1}{n}\frac{{F_{z} }}{wl}D_{\text{gsp}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\left( {t - \frac{{x_{c} }}{l}} \right){\text{d}}t} = \frac{n + 1}{n}\mu_{\text{d}} lF_{z} \sin \theta \int\limits_{{r_{h} }}^{i} {D_{\text{gsp}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\left( {t - \frac{{x_{c} }}{l}} \right){\text{d}}t}$$

$$\begin{aligned} b\int\limits_{{l_{h} }}^{l} {\mu _{{\text{d}}} q_{z} y^{\prime } \cos \theta {\text{d}}x_{1} } & = b\mu _{{\text{d}}} \cos \theta \int\limits_{{l_{h} }}^{l} {\frac{{n + 1}}{n}\frac{{F_{z} }}{{wl}}D_{{{\text{gsp}}}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\left\{ {\frac{{\left( {x_{1} - l} \right)l_{h} \tan \alpha }}{{l_{h} - l}} + y_{0} } \right\}{\text{d}}x_{1} } \\ & = \frac{{n + 1}}{n}\mu _{{\text{d}}} F_{z} l\cos \theta \int\limits_{{r_{h} }}^{1} {D_{{{\text{gsp}}}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\left\{ {\frac{{\left( {t - 1} \right)r_{h} \tan \alpha }}{{r_{h} - 1}} + \frac{{y_{0} }}{l}} \right\}{\text{d}}t} \\ & = \frac{{n + 1}}{n}\mu _{{\text{d}}} F_{z} l\cos \theta r_{h} \tan \alpha \int\limits_{{r_{h} }}^{1} {D_{{{\text{gsp}}}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\frac{{t - 1}}{{r_{h} - 1}}{\text{d}}t} \\ & \quad + \frac{{n + 1}}{n}\mu _{{\text{d}}} F_{z} l\cos \theta \int\limits_{{r_{h} }}^{1} {D_{{{\text{gsp}}}} \left( {\frac{{x_{1} }}{l};n,\zeta } \right)\frac{{y_{0} }}{l}{\text{d}}t.} \\ \end{aligned}$$

Note 11.16 Eq. (11.164)

Substituting Eq. (11.102) for $y^{\prime}$, Eq. (11.115) for f_y at a zero camber angle and Eq. (11.145) for q_z into Eq. (11.156), we obtain

$$M_{z} \left( {s,\alpha_{0} ,V} \right) = M^{\prime}_{z} + M^{\prime\prime}_{z} = b\int\limits_{0}^{{l_{h} }} {\left\{ { - f_{x} y + f_{y} \left( {x_{1} - x_{c} } \right)} \right\}{\text{d}}} x_{1} + b\int\limits_{{l_{h} }}^{l} {\mu_{\text{d}} q_{z} \left\{ { - y^{\prime}\cos \theta + \left( {x_{1} - x_{c} } \right)\sin \theta } \right\}{\text{d}}x_{1} }$$

$$\begin{aligned} b\int\limits_{0}^{{l_{h} }} {f_{y} \left( {x_{1} - x_{c} } \right){\text{d}}} x_{1} & = bC_{y} \int\limits_{0}^{{l_{h} }} {\left\{ {(1 + s)x_{1} \tan \alpha - \varepsilon l^{2} F_{y} \frac{{x_{1} }}{l}\left( {1 - \frac{{x_{1} }}{l}} \right)} \right\}\left( {x_{1} - x_{c} } \right){\text{d}}} x_{1} \\ & = bC_{y} l^{3} \int\limits_{0}^{{r_{h} }} {\left\{ {(1 + s)t\tan \alpha - \varepsilon lF_{y} t\left( {1 - t} \right)} \right\}\left( {t - \frac{{x_{c} }}{l}} \right){\text{d}}} t \\ & = 12C_{{M\alpha \_0}} \int\limits_{0}^{{r_{h} }} {\left\{ {(1 + s)t\tan \alpha - \varepsilon lF_{y} t\left( {1 - t} \right)} \right\}\left( {t - \frac{{x_{c} }}{l}} \right)} {\text{d}}t. \\ \end{aligned}$$

Derivations of other terms are the same as the derivation of Eq. (11.157) in Note 11.15.

Note 11.17 Eqs. (11.169) and (11.177)

Iida [53] discussed time constants T₁, T₂ and T₃ of the dynamic properties of a tire using Euqations (11.169) and (11.177).

Time constant T₁:

• (i) T₁ of steel-belted radial tires is larger than that of bias tires.

• (ii) T₁ decreases a little with increasing inflation pressure.

• (iii) T₁ decreases with increasing wheel width.

• (iv) T₁ increases with increasing load.

• (v) T₁ decreases with increasing slip angle.

Time constants T₂ and T₃:

• (i) T₂ has the same properties as T₁.

• (ii) The properties of T₃ differ from those of T₁ such that T₃ does not depend on the inflation pressure and slip angle and T₃ is smaller for the radial tire than for the bias tire.

Note 11.18 Gyro-moment at high speed

See Fig. 11.68.

Fig. 11.68

Mechanism of gyro-moment at high speed

Note 11.19 Eq. (11.183)

Considering the feedback loop in Fig. 11.39, Eq. (11.183) can be rewritten as

$$F_{y} (t) = C_{F\alpha \_0} \left( {\alpha_{0} - \frac{1}{3}\varepsilon lF_{y} (t + \delta t) - \frac{{M_{Z} \left( {t + \delta t} \right)}}{{R_{mz} }}} \right).$$

Applying Taylor series expansion and substituting δt = η/V to the above equation, we obtain

$$\begin{aligned} F_{y} (t) & = C_{F\alpha \_0} \left[ {\alpha_{0} - \frac{1}{3}\varepsilon l\left\{ {F_{y} (t) + \frac{{{\text{d}}F_{y} (t)}}{{{\text{d}}t}}\delta t)} \right\} - \left\{ {\frac{{M_{Z} (t)}}{{R_{mz} }} + \frac{{{\text{d}}M_{Z} (t)}}{{R_{mz} {\text{d}}t}}} \right\}\delta t} \right] \\ & = C_{F\alpha \_0} \left\{ {\alpha_{0} (t)\frac{\varepsilon l}{3}F_{y} (t) - \frac{\varepsilon l}{3}\frac{{{\text{d}}F_{y} (t)}}{{{\text{d}}t}}\frac{\eta }{V} - \frac{{M_{z} (t)}}{{R_{mz} }} - \frac{1}{{R_{mz} }}\frac{{{\text{d}}M_{Z} (t)}}{{R_{mz} {\text{d}}t}}\frac{\eta }{V}} \right\}. \\ \end{aligned}$$

Note 11.20 Stability of the vehicle and tire properties [55]

The vehicle maneuverability is classified by the stability factor SF and the yawing resonance frequency ω_n, which are given by [35]

$$\begin{aligned} & {\text{SF}} = - \frac{2m}{l}\frac{{l_{\text{f}} C_{{F\alpha \_{\text{f}}}} - l_{\text{r}} C_{{F\alpha \_{\text{r}}}} }}{{C_{{F\alpha \_{\text{f}}}} C_{{F\alpha \_{\text{r}}}} }} \\ & \omega_{n} = \frac{{2\left( {C_{{F\alpha \_{\text{f}}}} + C_{{F\alpha \_{\text{r}}}} } \right)}}{mV}\sqrt {\frac{{l_{\text{f}} l_{\text{r}} }}{{k^{2} }}} \sqrt {1 + {\text{SF}} \cdot V^{2} } \\ & I = mk^{2} , \\ \end{aligned}$$

where C_{Fα_f} and C_{Fα_r} are, respectively, the cornering stiffnesses of the front and rear tires, l_f and l_r are, respectively, the distances between the center of gravity of the vehicle and the front and rear axles, m is the vehicle mass, l is the distance between the front and rear axles, I is the moment of inertia of the vehicle, k is the yawing radius of the vehicle, and V is the velocity of the vehicle.

The stability of the vehicle is improved by increasing the stability factor, while the response of the vehicle is improved by increasing the yawing resonance frequency. Okano et al. [55] studied the relationship between the subjective evaluation and the stability factor/yawing resonance frequency for various tires. Figure 11.69 shows that the subjective evaluation of maneuverability is improved by increasing both the stability factor and yawing resonance frequency. In other words, the subjective evaluation of maneuverability is improved by increasing the yawing resonance frequency when the stability factor has values within some range. Figure 11.70 shows the effect of the cornering stiffnesses of front and rear tires on the stability factor/yawing resonance frequency. The effect of the cornering stiffness of the rear tires is larger than that of the front tires; e.g., the lane-change performance can be improved by increasing the cornering stiffness of rear tires.

Fig. 11.69

Reproduced from Ref. [55] with the permission of JSAE

Relationship between points in terms of the subjective evaluation and stability factor/yawing resonance frequency.

Fig. 11.70

Reproduced from Ref. [55] with the permission of JSAE

Effect of cornering stiffnesses of front and rear tires on the stability factor/yawing resonance frequency.