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.
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Notes
- 1.
Note 11.1.
- 2.
Problem 11.1.
- 3.
Problem 11.2.
- 4.
Note 11.2.
- 5.
Note 11.3.
- 6.
Note 11.4.
- 7.
Note 11.5.
- 8.
Note 11.6.
- 9.
Note 11.7.
- 10.
Note 11.8.
- 11.
Note 11.9.
- 12.
Note 11.10.
- 13.
Problem 11.3.
- 14.
See Footnote 13, Note 11.11.
- 15.
Note 11.12.
- 16.
See Footnote 15.
- 17.
Note 11.13.
- 18.
See Footnote 17.
- 19.
Note 11.14.
- 20.
Problem 11.4.
- 21.
Same as Eq. (11.68)
- 22.
Note 11.15.
- 23.
See Footnote 22.
- 24.
Note 11.16.
- 25.
Note 11.17.
- 26.
See Footnote 25.
- 27.
Note 11.19.
- 28.
Note 11.20.
- 29.
Same as Eq. (6.48).
References
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Notes
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 = eαx. 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
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
The equilibrium between the external force Fy and the force due to the spring of the sidewall yields
Hence, y is expressed by
Note 11.3 Eq. (11.41)
Using the relations e−λ ≅ 1 − λx, cos(λx) ≅ 1, sin(λx) ≅ λx for small λ, we obtain
Note 11.4 Eq. (11.50)
Using Eqs. (11.43) and (11.46) and the equation fy = μsqz at the sliding point x1 = lh, lh is given as the solution to
Note 11.5 Eq. (11.58)
The equation of the ellipse in Fig. 11.20 is
Because the ellipse passes the position (x1 = y1 = 0), b is given by
Suppose that yc is the trajectory of the tread base. Considering the relation r ≈ re, yc,max, the maximum value of yc, is expressed by
If we express yc as a parabola, yc is given by
The condition yc = yc,max at x1 = l/2 yields C = − l2sinγ/(2re).
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
where S is the shear force, ks 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:
where the shear force S and the moment M have the relation
The moment M is given by
where EIz is the flexural rigidity of the tread base. Substituting Eqs. (11.219) and (11.220) into Eq. (11.218), we obtain
In the case that w(x) = 0, referring to Eq. (3.225) in Appendix of Chap. 3, y is given as
where
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 Ky is expressed by the fundamental spring rate of the tire per unit length in the circumferential direction ksFootnote 29:
where r is the radius of the tire. Using Eq. (6.43), the torsion angle θ is given as
The out-of-plane torsional spring rate Rmz at radius rA per unit length in the circumferential direction is expressed as Rmz = πr3ks by substituting B = 0 into Eq. (6.38). Considering the relation
\(y_{\text{c}}^{ \hbox{max} }\) is expressed as
Suppose that yc is expressed by
The maximum value of yc is given by \(y_{\text{c}}^{ \hbox{max} } = A/4\). Comparing this equation with the above equation, the value of A is given as
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 σx expressed by
where Vsx is the longitudinal slip velocity, re 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)
lh is given by the solution to the following equations.
In the case of braking (s > 0),
Assuming Cx = Cy = C and considering pm = 3Fz/(2 lb) and CFα = Cl2b/2, we obtain
In the case of driving (s < 0),
where Cx = Cy = C and, in a manner similar to the case of braking, we obtain
Note 11.10 Eq. (11.99)
The following relations are obtained referring to Fig. 11.25.
In the case of braking,
In the case of driving,
Note 11.11 Eq. (11.104)
Using Eq. (11.99), we obtain
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
The substitution of the above equation into the first equation of Eq. (11.123) yields
In the case of driving, from Eq. (11.83), we obtain
Similarly, the substitution of the above equation into the first equation of Eq. (11.123) yields
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
In the case of driving (s < 0), using the relation VR/VB = (1 + s)/cosα and Eqs. (11.123) and (11.125), we obtain
Because the condition s < 0 is satisfied in the case of driving, we obtain
Note 11.14 Eq. (11.134)
The third equation of Eq. (11.119) is
Assume that fx 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 Mzx due to fx (component of Mz) is given by
Note 11.15 Eqs. (11.153) and (11.157)
Equation (11.153)
From Eqs. (11.119) and (11.129), we obtain
Equation (11.157)
Equation (11.156) can be rewritten as
One term of the above equation can be expressed as
where
Using Eq. (11.114) with a zero camber angle, the second term of the above equation is given by
where
The other terms of the above equation are given by
Note 11.16 Eq. (11.164)
Substituting Eq. (11.102) for \(y^{\prime}\), Eq. (11.115) for fy at a zero camber angle and Eq. (11.145) for qz into Eq. (11.156), we obtain
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.
Note 11.19 Eq. (11.183)
Considering the feedback loop in Fig. 11.39, Eq. (11.183) can be rewritten as
Applying Taylor series expansion and substituting δt = η/V to the above equation, we obtain
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 ωn, which are given by [35]
where CFα_f and CFα_r are, respectively, the cornering stiffnesses of the front and rear tires, lf and lr 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.
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Nakajima, Y. (2019). Cornering Properties of Tires. In: Advanced Tire Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-13-5799-2_11
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