**4. Estimation of the grinding force**

In this section, a theoretical model of the grinding force in the grinding process is introduced. Generally, the profile of abrasive grains varies on the working surface with the nonperiodic arrangement. However, in this topic, to simplify the illustration, it is assumed that the grinding force is applied on only one abrasive grain (which is arranged on the work surface as periodic successive length and pitch) during grinding, and the conical grain which has the vertex angle of "2γ" (= 120 deg.) as shown in **Figure 8**. In addition, each grain tip faces the radial direction of the cylindrical grinding stone. Although, a strict solution cannot be obtained, the above assumption can make to approximate the grinding force which is applied on one conical abrasive grain when the engaging on the material surface at the average abrasive depth of g<sup>0</sup> (which is described by formula (15)). In **Figure 8**, considering the region OAB (of which area: "ds") inclined only by *φ* from the abrasive grain feed direction, the grinding force dp applied on grain's conical surface. Assuming that there is no friction between the abrasive grain and the material for simplicity, dp acts perpendicularly on the conical surface, and the rubbing direction dt of the abrasive grain shown in the figure and its vertical component dn can be decomposed into components as follows.

$$\begin{cases} dt = dp \cos \chi \cos \phi \\ dn = dp \sin \chi \end{cases} \tag{28}$$

The force per area (per perpendicular) to the abrasive grain's feed direction is placed as the specific grinding force σ, and assuming that this is constant, dp is obtained by the following formula.

$$dp = \sigma d \text{cos} \chi \cos \phi \tag{29}$$

*Estimation of the Grain Trajectory and Engaging on the Material DOI: http://dx.doi.org/10.5772/intechopen.104519*

#### **Figure 8.**

*Grinding force which apply on an ideal (conical) grain.*

The above formula shows that the distribution of dp is circular as shown by the wavy lines of **Figure 8b**. As shown in **Figure 8a**, when a conical generatrix ρ in the grain profile, the conical surface ds of the grain can be described as the following formula.

*Tribology of Machine Elements - Fundamentals and Applications*

$$ds = \frac{\rho^2}{2} \sin \chi d\phi \tag{30}$$

To assign this formula onto formula (29), also dp can be obtained by the following formula.

$$dp = \frac{\rho^2 \sigma}{2} \sin \chi \cos \chi \cos \phi d\phi \tag{31}$$

And, to substitute this formula onto the formula (28) the two components of the grinding force dt, dn can be obtained by the following formula.

$$\begin{cases} dt = \frac{\rho^2 \sigma}{2} \sin \chi \cos^2 \chi \cos^2 \phi d\phi \\\\ dn = \frac{\rho^2 \sigma}{2} \sin^2 \chi \cos \chi \cos \phi d\phi \end{cases} \tag{32}$$

And, to integrate these two formulas by angle *ϕ*, the tangential component t and its vertical component n of the grinding force are obtained from the following formulas.

$$\begin{cases} \overline{t} = \int\_{-\pi/2}^{\pi/2} \frac{dt}{d\phi} d\phi = \frac{\pi \rho^2 \sigma}{4} \sin \chi \cos^2 \chi = \frac{\pi \sigma}{4} \overline{\mathfrak{g}} \sin \chi \\\\ \overline{n} = \int\_{-\pi/2}^{\pi/2} \frac{dt}{d\phi} d\phi = \rho^2 \sigma \sin^2 \chi \cos \chi = \sigma \overline{\mathfrak{g}}^2 \sin \chi \tan \chi \end{cases} \tag{33}$$

The number of grains j in the grinding region PBC (described in **Figure 1**) can be obtained by the following formula.

$$j = \frac{\mathcal{R}(a+\delta)f}{a^2} \tag{34}$$

By three formulas, (15), (33), and (34), the tangential component Ft and its vertical component Fn of the total grinding force applied on the working surface can be obtained by the following formula.

$$\begin{cases} Ft = j\overline{t} = \frac{\pi \sigma}{4} \frac{R\overline{f}}{w^2} \frac{\delta^2}{a+\delta} \left[ 2\Delta - \frac{R(r+R)}{r} \frac{\delta^2}{3} \right]^2 \sin \chi \\\\ Fn = j\overline{n} = \sigma \frac{R\overline{f}}{w^2} \frac{\delta^2}{a+\delta} \left[ 2\Delta - \frac{R(r+R)}{r} \frac{\delta^2}{3} \right]^2 \sin \chi tany \end{cases} \tag{35}$$

If the above components of the grinding force are measured with experiments, and the half vertex angle γ is assumed of 60 deg., the experimental formula of σ can be obtained with a converse solution of the formula (35) [2].

#### **5. Remarks: Statistical approaches**

In the final section of this chapter, an evaluation of the grain locus by a statistical approach is discussed. During the previous sections, it was obtained based on (like a) "fly-cutter" model in which abrasive grains are arranged with "equal heights" at

*Estimation of the Grain Trajectory and Engaging on the Material DOI: http://dx.doi.org/10.5772/intechopen.104519*

equal length and pitch. However, this model cannot duplicate a profile in the actual grinding stone correctly, also the length of the successive grains and pitch are different from that on the actual working surface. Generally, these parameters have a statistical distribution. Therefore, to understand this phenomenon, an attempt to evaluate grinding parameters by statistical considerations has been reported from about 1960's [3–8]. For example, Matsui and Shoji, they proposed the statistically model for length and engaging depth of the successive grains (for more details, refer to their reports [6, 7]). However, generally, it is required the highly calculation cost in the numerical simulation with the statistical method [8, 9]. In addition, to obtain the reasonable solutions of the simulation, the optimal statistical model must be chosen for evaluation of the actual problems. If an inadequate model is selected, the incorrect solution is obtained in the numerical simulation. Therefore, it is necessary to understand the problems to solve, and chose the optimal calculation method to solve the problem with consideration of the calculation cost in numerical simulations.
