**2. Basic theory of the grain engaging in the grinding process**

In this section, a geometric relationship between the abrasive grain and the material in the cylindrical grinding is introduced [1]. **Figure 1** shows the schematic of the cylindrical grinding process. In this figure, a cylindrical grinding stone which has a radius of R engages onto the side surface of the cylindrical workpiece of which the radius r with a depth of engaging of Δ. Also, the prior abrasive grain of Q on the workpiece processes a grinding mark of SPR, and the subsequent grain of P in the same cross-section of the axis which contains the Q processes the mark of PBC to follow the prior mark. And the grain P removes the region of "BRC" which is filled by the hatching in the figure. In general, the P is named the "successive grinding grain (or cutting point)", and the distance between QP is called "length of the

**Figure 1.** *A successive grains engaging in cylindrical grinding process.*

successive grains". In addition, generally, the rotational speed of grinding stone V is larger than that of the workpiece v (V > 100v), thus, the grinding marks can be regarded as the circle which is the same as the stone rotation. In this situation, the maximum depth of grain engaging onto the work surface can be obtained by formulas. The time t between the passing of the two grains, Q and P to the point R and C (they are described in **Figure 1**) can be expressed by the following formula using a length of successive grain "a" and the stone rotational speed V.

$$t = \frac{a}{V} \tag{1}$$

From this formula, g can be obtained by the following formula by the work rotational speed v and angle of α and β they are described in **Figure 1**.

$$\mathbf{g} = \mathbf{U}\mathbf{R} = v\mathbf{t} \sin \boldsymbol{\beta} \cong \frac{v}{V} \mathbf{a}\boldsymbol{\beta} \tag{2}$$

In addition, the following relations can be described.

$$
\overline{O\_1^2 O\_2^2} = \overline{O\_2 C}^2 + \overline{O\_1 C}^2 + 2\overline{O\_2 C} \cdot \overline{O\_1 C} \cos \beta \tag{3}
$$

$$(r+R-\Delta)^2 = R^2 + r^2 + 2Rr\cos\beta\tag{4}$$

Also, if the angle of *β* and depth of *Δ* in the formula (2) are small enough, the b can be described by following formula (cos <sup>β</sup> � 1-β<sup>2</sup> /2, <sup>Δ</sup><sup>2</sup> � 0).

$$
\beta^2 = \frac{2\Delta(r+R)}{Rr} \tag{5}
$$

When the above formula is assigned onto the formula (2), g can be obtained by the following.

$$\mathbf{g} = \frac{v}{V} \mathcal{a} \sqrt{\frac{(r+R)}{Rr}} \cdot \sqrt{2\Delta} \tag{6}$$

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

From this formula, since r = ∞ when replaced in the case of planner grinding and "r < 0 "in the case of internal cylindrical grinding, the maximum depth of each cases gp and gi can be expressed by following formulas, respectively.

$$\mathbf{g}\_p = \frac{v}{V} \mathbf{a} \sqrt{\frac{2\Delta}{R}} \quad \mathbf{g}\_i = \frac{v}{V} \mathbf{a} \sqrt{\frac{r-R}{Rr}} \tag{7}$$

In general, in various grindings, the maximum depth g is an indicator for the grinding force applied to an abrasive grain. If the binding degree (intensity) of the abrasive grain is constant during grinding, when g becomes large, excessive grinding force applies to the abrasive grain, the abrasive grain is easy to leave out from the working surface of stone ("shedding"). And conversely, when the g is small, the abrasive grain is difficult to leave out, and the chips are deposited in the vacancy of the working surface ("loading") and abrasive grains are worn out and the escape surface land (in cutting) expands ("glazing"). In general, the binding degree does not only depend on the material properties, but also depends on the conditions of stone and process including g.

In the next topic, the average depth of grain engaging is introduced. In this section, the average depth of cut is called "g<sup>0</sup> ", it may be considered to the average in the diagonal region that conscloses the removal region PRCU which is described in **Figure 2**. However, in this discussion, the region is divided into curved triangles PRU and CRU, respectively. In the first step, the g<sup>0</sup> is considered in the PRU region. The PUC and PR are obtained by the following formula.

$$\begin{aligned} \text{PUC}: &\mathbf{x}^2 + \mathbf{y}^2 = \mathbf{R}^2\\ \text{PR}: &\left\{\mathbf{x} - (r + R - \Delta)(\mathbf{1} - \cos \gamma)\right\}^2 + \left\{\mathbf{y} - (r + R - \Delta)\sin \gamma\right\}^2 = \mathbf{R}^2 \end{aligned} \tag{8}$$

In this region, two intersections between the arcs (PUC, PR) and the dashed line "y = -x tan*ϕ*" (it is shown in **Figure 2**) are called H (x1, y1) and G (x2, y2). To approximate as cos<sup>γ</sup> �1-γ<sup>2</sup> /2, sinγ � γ, tan*ϕ* � *ϕ,* the x coordinates of each intersection (x1, x2) are obtained by the following formulas.

$$\begin{cases} \varkappa\_1 = R\left(1 - \frac{\phi^2}{2}\right) - \frac{\left(r + R\right)^2}{2R}\gamma^2 + \frac{r + R}{2}\left(\chi^2 + 2\chi\phi\right) \\\\ \varkappa\_2 = R\left(1 - \frac{\phi^2}{2}\right) \end{cases} \tag{9}$$

In **Figure 2**, the length of HG is a grinding depth at the angle of *ϕ*. However, therefore the angle of α + β is small enough in this figure, HG can be approximated as "x2 - x1", The engaging depth g1 (ϕ) at the angle of ϕ can be expressed as follows.

$$\log\_1(\phi) = \infty - \infty = \frac{r(r+R)}{2R}\gamma^2 - (r+R)\gamma\phi \tag{10}$$

In the next step, g in the CRU region is considered. The RC can be expressed by the following formula.

$$RC: \quad \left\{\mathbf{x} - (r + R - \Delta)\right\}^2 + \mathbf{y}^2 = r^2 \tag{11}$$

The x coordinate of the intersection of this arc and the line "y = -x tan*ϕ*" x3 which is the x coordinate of point H<sup>0</sup> can be obtained by the following formula.

**Figure 2.** *Schematics of the grain engaging depth.*

$$\mathbf{x}\_3 = \mathbf{R} - \boldsymbol{\Delta} + \frac{\mathbf{R}^2}{2r} \boldsymbol{\phi}^2 \tag{12}$$

Therefore, the engaging depth g2 (ϕ) in this region is obtained by the following formula.

$$\log\_2(\phi) = \mathbf{x}\_2 - \mathbf{x}\_3 = \Delta - \frac{R(r+R)}{2r} \phi^2 \tag{13}$$

If the average depth g<sup>0</sup> is obtained using the above formulas (12) and (13), the following formula is obtained.

$$\mathbf{g}' = \frac{1}{a+\delta} \left[ \frac{r(r+R)}{2R} \gamma^2 \right]\_{-\varepsilon}^{\delta} d\phi - (r+R)\gamma \int\_{-\varepsilon}^{\delta} \phi d\phi \qquad + \Delta \int\_{-a}^{\varepsilon} d\phi - \frac{R(r+R)}{2r} \int\_{-a}^{\varepsilon} \phi^2 d\phi \tag{14}$$

However, in **Figure 2**, if it can be approximated as PB � SB/2, and γ = (2R/r)*δ*, and PBC=SPR + SB + PBU, the angle of ε can be placed with "ε = α -2δ". When the g<sup>0</sup> and ε are assigned to formula (15) and integrated the formula, the g<sup>0</sup> is obtained by the following formula.

$$\overline{\mathbf{g}} = \frac{\delta}{a + \delta} \left[ 2\Delta - \frac{R(r+R)}{r} \frac{\delta^2}{\mathfrak{Z}} \right] \tag{15}$$

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

From of this formula (2):

$$
\overline{O\_2C} = \overline{O\_1^2O\_2^2}^2 + \overline{O\_1C}^2 - 2\overline{O\_1^2O\_2^2} \cdot \overline{O\_1C} \cos a \tag{16}
$$

the angle of α which is described in **Figure 1** can be obtained by following formula, with approximation as "cos<sup>α</sup> � 1 -α<sup>2</sup> /2"

$$a = \sqrt{\frac{2r\Delta}{R(r+R-\Delta)}} \cong \sqrt{\frac{2r\Delta}{R(r+R)}}\tag{17}$$

In addition, δ can be obtained from the following formula because (av / V) / 2 � Rδ.

$$
\delta = \frac{a}{2R} \frac{v}{V} \tag{18}
$$

Finally, a formula of the length of the successive grains "a" is introduced. Although, grains on the working surface of the stone are arranged randomly, it is considered that they are arranged regularly at the average abrasive grain pitch w for simplifying discussion. In this case, since there are 1/w2 grains per unit area, w can be obtained if the number of particles per unit area is measured on the grinding wheel work surface. **Figure 3** shows an example of the arrangement of a grinding stone. In this figure, grains are regularly arranged on the lines A1Am and B1Bm in parallel at the distance of w, and the same numbered abrasive grains on each line are arranged vertically. The grain Am in this figure is feed to the prior grain B1 to follow the line AmB1 by the relative motion during the grinding stone rotation and the material. In this situation, the line AmB1 coincides with the average of the length of the successive grains "a". Before Am passes through B1, abrasive grains A1 to Am-1 on the line A1Am are pass through the same point. In this case, if the average width of the scrape marks generated in the workpiece finishing surface is b (= (w/m) cosθ). On the other hand, because a = m w/cosθ, the "a" can be obtained by the following simple formula with b and w.

$$a = \frac{w^2}{b} \tag{19}$$

**Figure 3.** *Successive grain engaging and concerned parameters a, ω.*

Based on this formula, if w and b can be obtained by actual measurement, also the a can be obtained.

## **3. Example: curve generation by the spheric grinding stone**

In this section, to show the application of the previous section, a theoretical grain engaging model of the curve generation process by the spheric grinding stone is introduced. For example, an electro-plated micro-spherical grinding stone is shown in **Figure 4**. The spherical grinding stones are widely used for grooving or free form shapes of glass materials, optical lenses, and in recent years for processing joint parts (bone heads, etc.) of implants such as hip joints and knee joints. A trajectory of the abrasive grain draws a trochoid during the grinding process. However, its movement is changed by the tool posture and grinding conditions (Ex. rotational speed, feed rate, etc.). In this topic, to simplify the discussion, the theoretical model of a grain trajectory is illustrated. However, it is assumed that grains have the same profile (Ex. cone) and size are arranged with the periodic length of successive grains a and pitch "w" on the working surface.

**Figure 5** shows the motion of a grinding stone in the grinding process. In this situation, the stone is fixed on the coordinate system of the machine tool (described as X, Y, Z) at any posture and the tool attitude changes by the grain locus. As shown in **Figure 5**, the fixed stone inclines to the +X direction of the coordinate system of the machine tool Y-Z at a lead angle of θ and to +Y direction at a tilt angle of ϕ, respectively. The grinding stone with the nose radius ρ is then fed to the +X direction of the machine tool coordinate system at a feed rate of f and rotational speed of ω, respectively. In this situation, the tool attitude in machine coordinate (Xt, Yt, Zt) is described by the following formulas:

$$\begin{cases} X\_t = \sin \theta \\ Y\_t = \cos \theta \sin \phi \\ Z\_t = \cos \theta \cos \phi \end{cases} \tag{20}$$

If the center of the ball nose moves from point A (XA, YA, ZA) to point B(XB, YB, ZB)as shown in **Figure 5**, the angles of grain locus (vector AB) α and β are described by the following formulas:

$$\begin{cases} a = \tan^{-1}\left(\frac{Z\_B - Z\_A}{\sqrt{\left(X\_B - X\_A\right)^2 + \left(Y\_B - Y\_A\right)^2}}\right) \\\\ \beta = \tan^{-1}\left(\frac{Y\_B - Y\_A}{X\_B - X\_A}\right) \end{cases} \tag{21}$$

**Figure 4.** *An example of the (hemi) spherical grinding stone.*

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

**Figure 5.** *Changing of the stone attitude on its locus.*

According to these formulas, the tool attitude in the coordinate of tool locus (X″, Y″, Z″) as shown in **Figure 5** can be described as follows:

$$\begin{cases} X''\_t = \cos a \cdot \cos \beta \cdot X\_t + \cos a \cdot \sin \beta \cdot Y\_t - \sin a \cdot Z\_t \\\\ Y''\_t = -\sin \beta \cdot X\_t + \cos \beta \cdot Y\_t \\\\ Z''\_t = \sin a \cdot \cos \beta \cdot X\_t + \sin a \cdot \sin \beta \cdot Y\_t + \cos a \cdot Z\_t \end{cases} \tag{22}$$

Based on the formula (22), a lead angle of *θ<sup>v</sup>* and a tilt angle of *ϕ<sup>v</sup>* of grinding stone in a coordinate of grain locus are described as follows:

$$\theta\_v = \tan^{-1} \left( \frac{X''\_t}{Z''\_t} \right) \tag{23}$$

*Tribology of Machine Elements - Fundamentals and Applications*

$$\varphi\_v = \tan^{-1} \left( \frac{Y\_t''}{\sqrt{X''\_t^2 + Z''\_t^2}} \right) \tag{24}$$

If the grain locus in the global coordinate system can be divided by time step t and its profile can be approximated as a line, the change in stone attitude can be calculated with the above formulas. And the change in engaging thickness in the grinding process can be described. This model can calculate a change in the engaging thickness by locus of prior and successive grains at any attitude. **Figure 6** shows the schematics for the grain locus in the grinding process which is viewed along axis (a), Y-axis (b), and center axis of grinding stone (c), respectively. In this figure, P is an engaging point on the grain profile in which removes the material. In this situation, the location of the P (xP, yP, zP) in the global coordinate system Y-Z can be described as the following formulas.

$$\begin{cases} \mathbf{x}\_P = \mathbf{R}\_P \cdot \cos \theta\_v \cdot \cos \left(\alpha t - \gamma \right) + h\_P \cdot \sin \theta\_v + f \cdot t \\\\ \mathbf{y}\_P = -\mathbf{R}\_P \cdot \cos \varphi\_v \cdot \sin \left(\alpha t - \gamma \right) - \mathbf{R}\_P \cdot \sin \varphi\_v \cdot \sin \theta\_v \cdot \cos \left(\alpha t - \gamma \right) + h\_P \cdot \sin \varphi\_v \cdot \cos \theta\_v \\\\ \mathbf{z}\_P = \mathbf{R}\_P \cdot \sin \varphi\_v \cdot \sin \left(\alpha t - \gamma \right) - \mathbf{R}\_P \cdot \cos \varphi\_v \cdot \sin \theta\_v \cdot \cos \left(\alpha t - \gamma \right) - A\_d \\\\ \quad + h\_P \cdot \sin \varphi\_v \cdot \cos \theta\_v + \rho \cdot (\mathbf{1} - \cos \varphi\_v \cdot \cos \theta\_v) \end{cases} \tag{25}$$

In formula (25), γ is the delay angle of Point P to the bottom of the cutter, t is the cutting time. **Figure 7** illustrates the schematic of a chip removing process by the inclined grinding stone. In this figure, point Q is the crossing point between the rotational radius of the P and the previously machined surface by the prior grain. The location of Q (xQ, yQ, zQ) in the global coordinate can be expressed as follows:

xQ <sup>¼</sup> RQ � cos <sup>θ</sup><sup>v</sup> � cos <sup>ω</sup> <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � � <sup>γ</sup><sup>Q</sup> � � <sup>þ</sup> hQ � sin <sup>θ</sup><sup>v</sup> <sup>þ</sup> f t <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � yQ ¼ �RQ � cos <sup>φ</sup><sup>v</sup> � cos <sup>ω</sup> <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � � <sup>γ</sup><sup>Q</sup> � � � RQ � sin <sup>φ</sup><sup>v</sup> � sin <sup>θ</sup><sup>v</sup> � cos <sup>ω</sup> <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � � <sup>γ</sup><sup>Q</sup> � � <sup>þ</sup> hQ � sin <sup>φ</sup><sup>v</sup> cos <sup>θ</sup><sup>v</sup> zQ <sup>¼</sup> RQ � sin <sup>φ</sup><sup>v</sup> � sin <sup>ω</sup> <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � � <sup>γ</sup><sup>Q</sup> � � � RQ � cos <sup>φ</sup><sup>v</sup> � sin <sup>θ</sup><sup>v</sup> � cos <sup>ω</sup> <sup>t</sup> <sup>þ</sup> <sup>Δ</sup><sup>t</sup> � <sup>d</sup>δij*=*<sup>ω</sup> � � � <sup>γ</sup><sup>Q</sup> � � � Ad <sup>þ</sup> hQ � cos <sup>φ</sup><sup>v</sup> � cos <sup>θ</sup><sup>v</sup> þρ � 1 � cos φ<sup>v</sup> ð Þ � cos θ<sup>v</sup> 0 BBBBBBBBBBBBBBB@ (26)

Where γ<sup>Q</sup> is the delay angle of Point Q to the bottom of the cutter, t is the cutting time, and dδij is the lead angle of the prior grain of the recent engaging grain. According to this formula, the change of engaging depth of P tP in the grinding process can be calculated by the following formula.

$$\begin{split} t\_p &= \overline{PQ} \\ &= R\_P - \left[ \left( \mathbf{x}\_Q - f \cdot \mathbf{t} \right) \cdot \cos \theta\_v - \left\{ \mathbf{z}\_Q - \rho \cdot (\mathbf{1} - \cos \theta\_v) \right\} \right] \sin \left( at - \gamma \right) + y\_Q \cdot \cos \left( at - \gamma \right) \end{split} \tag{27}$$

If tP is a negative value, the grain does not engage onto the material in formula (27). The simulation can be performed by dividing the cutting edge into *Estimation of the Grain Trajectory and Engaging on the Material DOI: http://dx.doi.org/10.5772/intechopen.104519*
