Abstract
Wear is phenomenologically characterized by not only physical factors, such as fracture, but also chemical factors, such as oxidization.
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Notes
- 1.
Problem 14.1.
- 2.
See Footnote 1.
- 3.
Same as Eq. (11.4).
- 4.
Same as Eq. (11.5).
- 5.
Note 14.1.
- 6.
See Footnote 5.
- 7.
Note 14.2.
- 8.
See Footnote 7.
- 9.
Note 14.3.
- 10.
Note 14.4.
- 11.
Note 14.5.
- 12.
Note 14.6.
- 13.
Note 14.7.
- 14.
See Footnote 13.
- 15.
See Footnote 13.
- 16.
See Footnote 13.
- 17.
Note 14.8.
- 18.
Note 14.9.
- 19.
Same as Eq. (11.72).
- 20.
Note 14.10.
- 21.
Note 14.11.
- 22.
Note 14.12.
- 23.
See Footnote 22.
- 24.
Note 14.13.
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Notes
Notes
Note 14.1 Eqs. (14.34) and (14.35)
-
(i)
In the case that lh > l/2,
$$\begin{array}{*{20}l} {S_{y} = 0\quad 0 \le x < l_{h} } \hfill \\ {S_{y} = x\tan \alpha - S_{h} \quad l_{h} \le x < x_{h} } \hfill \\ {S_{y} = x\tan \alpha - \mu q_{z} (x)/C_{y}^{\text{down}} \quad x_{h} \le x < l}. \hfill \\ \end{array}$$(14.190)
The force equilibrium at the sliding point gives
We introduce a new parameter:
lh is expressed by
xh is expressed by
It follows that
where
Wear energy \(E_{y}^{\text{w}}\) is given by
The substitution of Eq. (14.195) into Eq. (14.197) yields
The first term of Eq. (14.198) is the wear energy without hysteresis loss while the second term is the wear energy due to hysteresis loss. Because (1 − ρ) ≥ 0 is satisfied, the wear energy is reduced by hysteresis loss.
-
(ii)
In the case that lh ≤ l/2,
$$\begin{array}{*{20}l} {S_{y} = 0\quad 0 \le x < l_{h} } \hfill \\ {S_{y} = x\tan \alpha - \mu q_{z} (x)/C_{y}^{\text{up}} \quad l_{h} \le x < l/2} \hfill \\ {S_{y} = x\tan \alpha - S_{l/2} \quad l/2 \le x < x_{h} } \hfill \\ {S_{y} = x\tan \alpha - \mu q_{z} (x)/C_{y}^{\text{down}} \quad x_{h} \le x < l}, \hfill \\ \end{array}$$(14.199)where lh is given by Eq. (14.191) and xh is given by
$$S_{l/2} = \mu p_{m} /C_{y}^{\text{up}} = \mu q_{z} (l/2)/C_{y}^{\text{up}} = \mu q_{z} (x_{h} )/C_{y}^{\text{down}} .$$(14.200)
It follows that
The wear energy \(E_{y}^{\text{w}}\) is given by
The substitution of Eq. (14.201) into Eq. (14.202) yields
The first term of Eq. (14.203) is the wear energy without hysteresis loss, and the second term is the wear energy due to hysteresis loss. Because (1 − ρ) ≥ 0 is satisfied, the wear energy is reduced by hysteresis loss.
Note 14.2 Eqs. (14.45) and (14.46)
Equation (14.45)
Referring to Fig. 14.92, Fy is given as
where G and γ are the shear modulus and shear strain of rubber and y is shear displacement. Considering that Cy is the shear spring rate per unit area of the tread rubber, we obtain Cy = G/H. Cy cannot be expressed by the shear modulus of rubber when the tread pattern is considered. The block rigidity discussed in Sect. 7.1 is therefore used.
Equation (14.46)
From Eq. (11.53), we obtain
Substituting the relation \({\lambda} = \root 4 \of {{{k_{s} /\left( {4EI_{z} } \right)}}}\) into the above equation, we obtain
Note 14.3 Eq. (14.50)
Referring to Fig. 14.93, we have d2 = PL3/(48EI) from formulae of the mechanics of materials. Using the relation τ = Gγ and noting that the in-plane bending problem of contact patch is approximated by the bending problem of cantilever beam with length L/2 and load P/2, we obtain
Note 14.4 Eq. (14.58)
The reason for the value of 2 in the third term is that when the variable of integration is changed from L to α, the integral is taken twice in the integration region from α0 − αm to α0 + αm.
Note 14.5 Eq. (14.64)
Using the above relation, 〈Ew〉 is obtained as
When lateral forces on left and right sides have similar amplitudes, \(\alpha_{m} \gg \alpha_{0}\) is satisfied. Applying Taylor series expansion to the above equation, we obtain
Note 14.6 Eq. (14.66)
where
Using the relation
the above equation for 〈Ew〉 can be simplified as
Assuming \(\sigma \gg \alpha_{0}\) and applying Taylor series expansion to the second term of the above equation, we obtain
Note 14.7 Eqs. (14.77), (14.78), (14.81) and (14.82)
-
(i)
The case that 0 ≤ lh ≤ l:
In braking (s > 0),
In driving (s < 0),
For simplicity, assuming Cx = Cy = C and considering the relation pm = 3Fz/(2lb), we obtain
-
(ii)
In the case that lh < 0:
In braking (s > 0),
In driving (s < 0),
Assuming the relation Cx = Cy = C, we obtain
Note 14.8 Eq. (14.92)
The substitution of Eq. (14.85) into Eq. (14.89) yields
The solution to the above differential equation is
Considering initial conditions, we obtain
Using Eqs. (14.85) and (14.86), at t = tc, Eq. (14.89) can be rewritten as
The above equation can be rewritten as
The substitution of the above equation into equation for ξ yields
The substitution of ξ and t = T + tc into equation for y yields
Note 14.9 Shear Spring Rate Between Adjacent Blocks
Fujikawa et al. [23] estimated the intra-shear stiffness of a block. As shown Fig. 14.94, the shear strain γ at longitudinal position x and vertical position z is expressed by
where lE is the distance between adjacent elements. The shear force f(x) acting between the tread elements is therefore expressed by
From the above equation, the intra-shear stiffness k2 is obtained as k2 = bhG/(2lE), where b is the width of the block. Meanwhile, the shear stiffness of a block element \(t_{\text{tread}}^{\prime }\) is expressed as \(t_{\text{tread}}^{\prime } = bl_{E} G/h\). For a block with dimensions of 20 mm (length) × 20 mm (width) × 8 mm (height), considering that h = 8 mm, lE = 20/3 mm and b = 20 mm, the relation between the intra-shear stiffness and shear stiffness of a block element is \(k_{2} = 0.72t_{\text{tread}}^{\prime }\). Therefore, Fujikawa’s assumption of \(k_{2} = 2t_{\text{tread}}^{\prime }\) may not be appropriate for this block.
Note 14.10 Eq. (14.141)
Using Eqs. (14.136), (14.139) and (14.140), we obtain
Using Eqs. (14.138), (14.139) and the above equation, we obtain the equation for the side slip angle of the vehicle β.
Note 14.11 Time-Delay System
When there is a time delay in the spring term for the equation of motion with one degree of freedom, the governing equation is \(m\ddot{x}(t) + c\dot{x}(t) + kx(t - \tau ) = 0\). When the time delay is small, using the relation \(x(t - \tau ) = x(t) - \tau \dot{x}(t)\), the above equation can be rewritten as \(m\ddot{x}(t) + (c - k\tau )\dot{x}(t) + kx = 0\). Unstable vibration occurs when the relation c − kτ < 0 is satisfied. If the time delay is not small, by applying the Laplace transform to the original equation, we obtain the characteristic equation ms2 + cs + ke−τs = 0. Because this characteristic equation contains the term e−τs, the number of solutions is infinite. This is because the relation \({\text{e}}^{ - j(\theta + 2n\pi )} = {\text{e}}^{j\theta } (j = \sqrt { - 1} )\) is satisfied for an arbitrary n. This is a feature of a time-delay system.
Note 14.12 Eqs. (14.160) and (14.162)
Equation (14.160)
From the relations W1(t) = 0 and W2(t) = 0, we obtain X0(t) = V0(t) = U0(t − T). Using Eq. (14.157), we obtain
From the relation \(\ddot{X}(t) = 0\) and the initial condition X0(0) = 0, we obtain X0(t) = At, where A is a constant. From the relation X0(t) = U0(t − T), U0(t) is found to be a linear function with respect to t. Using the above equation, U0(t) and U0(t − T) are given as
Equation (14.162)
From Eq. (14.152), we obtain
where
The substitution of Eqs. (14.152) and (14.161) into Eq. (14.157) yields
where
From the above equation, we obtain
Note 14.13
Schallamach’s equation, given as Eq. (14.30), corresponds to n = 2.
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Nakajima, Y. (2019). Wear of Tires. In: Advanced Tire Mechanics. Springer, Singapore. https://doi.org/10.1007/978-981-13-5799-2_14
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