Estimation of the Grain Trajectory and Engaging on the Material

Abstract

In this chapter, a numerical estimation of the grain trajectory in the grinding process is introduced. The topic describes a fundamental of the numerical theory of the grain trajectory, the estimation of the grain trajectory in the hemispherical grinding stone in the micro grooving process as an application of the numerical estimation.

1. Introduction

In this chapter, a theoretical evaluation method of the grain trajectory and its engaging process onto the material in the grinding is introduced. In the first topic, the geometric relationship between the abrasive grain and the material in the cylindrical grinding process, that is, the movement of the abrasive grain and the cutting distance to the material will be described using a mathematical model. In the next topic, we will explain how to theoretically evaluate the resistance generated in the entire working surface of the grindstone, starting from the model of cutting with one abrasive grain for grinding force. In the next topic, to introduce the application of the above topics, a theoretical model of the grain trajectory and engaging depth on the material in the curve generation grinding by the spherical grinding stone will be described. Finally, an evaluation method using a statistical method for basic grinding parameters (Ex. length of successive grains, depth of grain engaging) will be described briefly.

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 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.

Figure 1.

A successive grains engaging in cylindrical grinding process.

$$t=\frac{a}{V}$$E1

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.

$$g=\mathit{UR}=\mathit{\text{vtsin}}\beta \cong \frac{v}{V}\mathit{a\beta }$$E2

In addition, the following relations can be described.

$$\overline{O_1^2{O}_2^2}={\overline{O_{2}C}}^2+{\overline{O_{1}C}}^2+2\overline{O_{2}C}·\overline{O_{1}C}\mathit{cos}\beta$$E3

$${\left(r+R-\Delta \right)}^2=R^2+r^2+2\mathit{\text{Rrcos}}\beta$$E4

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 β ∼ 1-β²/2, Δ² ∼ 0).

$${\beta }^2=\frac{2\Delta \left(r+R\right)}{\mathit{Rr}}$$E5

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

$$g=\frac{v}{V}a\sqrt{\frac{\left(r+R\right)}{\mathit{Rr}}}·\sqrt{2\Delta }$$E6

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 g_p and g_i can be expressed by following formulas, respectively.

$$\begin{array}{cc}{g}_p=\frac{v}{V}a\sqrt{\frac{2\Delta }{R}}& g_i=\frac{v}{V}a\sqrt{\frac{r-R}{\mathit{Rr}}}\end{array}$$E7

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′”, 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′ is considered in the PRU region. The PUC and PR are obtained by the following formula.

Figure 2.

Schematics of the grain engaging depth.

$$\begin{array}{l}\mathit{PUC}:x^2+y^2=R^2\\ \mathit{PR}:{\left\{x-\left(r+R-\Delta \right)\left(1-\mathit{cos}\gamma \right)\right\}}^2+{\left\{y-\left(r+R-\Delta \right)\mathit{sin}\gamma \right\}}^2=R^2\end{array}$$E8

In this region, two intersections between the arcs (PUC, PR) and the dashed line “y = -x tanϕ” (it is shown in Figure 2) are called H (x₁, y₁) and G (x₂, y₂). To approximate as cosγ ∼1-γ²/2, sinγ ∼ γ, tanϕ ∼ ϕ, the x coordinates of each intersection (x₁, x₂) are obtained by the following formulas.

$$\begin{array}{cc}\left(\begin{array}{c}{x}_1=R\left(1-\frac{{\varphi }^2}{2}\right)-\frac{{\left(r+R\right)}^2}{2R}{\gamma }^2+\frac{r+R}{2}\left({\gamma }^2+2\mathit{\gamma \varphi }\right)\\ x_2=R\left(1-\frac{{\varphi }^2}{2}\right)\end{array}\right.& \end{array}$$E9

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.

$$g_1\left(\varphi \right)=x_2-x_1=\frac{r\left(r+R\right)}{2R}{\gamma }^2-\left(r+R\right)\mathit{\gamma \varphi }$$E10

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

$$\begin{array}{cc}\mathit{RC}:& {\left\{x-\left(r+R-\Delta \right)\right\}}^2+y^2=r^2\end{array}$$E11

The x coordinate of the intersection of this arc and the line “y = -x tanϕ” x₃ which is the x coordinate of point H′ can be obtained by the following formula.

$$x_3=R-\Delta +\frac{R^2}{2r}{\varphi }^2$$E12

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

$$g_2\left(\varphi \right)=x_2-x_3=\Delta -\frac{R\left(r+R\right)}{2r}{\varphi }^2$$E13

If the average depth g′ is obtained using the above formulas (12) and (13), the following formula is obtained.

$$g\prime =\frac{1}{\alpha +\delta }\left[\frac{r\left(r+R\right)}{2R}{\gamma }^2{\int }_{-\epsilon }^{\delta }\mathit{d\varphi }-\left(r+R\right)\gamma {\int }_{-\epsilon }^{\delta }\mathit{\varphi d\varphi }\right.\begin{array}{cc}& +\end{array}\Delta {\int }_{-\alpha }^{\epsilon }\mathit{d\varphi }-\frac{R\left(r+R\right)}{2r}{\int }_{-\alpha }^{\epsilon }{\varphi }^2\mathit{d\varphi }$$E14

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′ and ε are assigned to formula (15) and integrated the formula, the g′ is obtained by the following formula.

$$\overline{g}=\frac{\delta }{\alpha +\delta }\left[2\Delta -\frac{R\left(r+R\right)}{r}\frac{{\delta }^2}{3}\right]$$E15

From of this formula (2):

$$\overline{O_{2}C}={\overline{O_1^2{O}_2^2}}^2+{\overline{O_{1}C}}^2-2\overline{O_1^2{O}_2^2}·\overline{O_{1}C}\mathit{cos}\alpha$$E16

the angle of α which is described in Figure 1 can be obtained by following formula, with approximation as “cosα ∼ 1 -α²/2”

$$\alpha =\sqrt{\frac{2\mathit{r\Delta }}{R\left(r+R-\Delta \right)}}\cong \sqrt{\frac{2\mathit{r\Delta }}{R\left(r+R\right)}}$$E17

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

$$\delta =\frac{a}{2R}\frac{v}{V}$$E18

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 A₁A_m and B₁B_m in parallel at the distance of w, and the same numbered abrasive grains on each line are arranged vertically. The grain A_m in this figure is feed to the prior grain B₁ to follow the line A_mB₁ by the relative motion during the grinding stone rotation and the material. In this situation, the line A_mB₁ coincides with the average of the length of the successive grains “a”. Before A_m passes through B₁, abrasive grains A₁ to A_m-1 on the line A₁A_m 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.

Figure 3.

Successive grain engaging and concerned parameters a, ω.

$$a=\frac{w^2}{b}$$E19

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 4.

An example of the (hemi) spherical grinding stone.

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:

Figure 5.

Changing of the stone attitude on its locus.

$$\begin{array}{cc}\left(\begin{array}{c}{X}_t=\mathit{sin}\theta \\ Y_t=\mathit{cos}\mathit{\text{θsin}}\varphi \\ Z_t=\mathit{cos}\mathit{\text{θcos}}\varphi \end{array}\right.& \end{array}$$E20

If the center of the ball nose moves from point A (X_A, Y_A, Z_A) to point B(X_B, Y_B, Z_B)as shown in Figure 5, the angles of grain locus (vector AB) α and β are described by the following formulas:

$$\begin{array}{cc}\left(\begin{array}{c}\alpha ={\mathit{tan}}^{-1}\left(\frac{Z_B-Z_A}{\sqrt{{\left(X_B-X_A\right)}^2+{\left(Y_B-Y_A\right)}^2}}\right)\\ \beta ={\mathit{tan}}^{-1}\left(\frac{Y_B-Y_A}{X_B-X_A}\right)\end{array}\right.& \end{array}$$E21

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:

$$\begin{array}{cc}\left(\begin{array}{l}X{"}_t=\mathit{cos}\alpha ·\mathit{cos}\beta ·X_t+\mathit{cos}\alpha ·\mathit{sin}\beta ·Y_t-\mathit{sin}\alpha ·Z_t\\ Y{"}_t=-\mathit{sin}\beta ·X_t+\mathit{cos}\beta ·Y_t\\ Z{"}_t=\mathit{sin}\alpha ·\mathit{cos}\beta ·X_t+\mathit{sin}\alpha ·\mathit{sin}\beta ·Y_t+\mathit{cos}\alpha ·Z_t\end{array}\right.& \end{array}$$E22

Based on the formula (22), a lead angle of θ_v and a tilt angle of ϕ_v of grinding stone in a coordinate of grain locus are described as follows:

$$\begin{array}{cc}{\theta }_v={\mathit{tan}}^{-1}\left(\frac{X{"}_t}{Z{"}_t}\right)& \end{array}$$E23

$${\phi }_v={\mathit{tan}}^{-1}\left(\frac{Y{"}_t}{\sqrt{X{"}_t^2+Z{"}_t^2}}\right)$$E24

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 (x_P, y_P, z_P) in the global coordinate system Y-Z can be described as the following formulas.

Figure 6.

Geometry of the grain locus in curve generation.

$$\begin{array}{cc}\left(\begin{array}{l}{x}_P=R_P·\mathit{cos}{\theta }_v·\mathit{cos}\left(\mathit{\omega t}-\gamma \right)+h_P·\mathit{sin}{\theta }_v+f·t\\ y_P=-R_P·\mathit{cos}{\phi }_v·\mathit{sin}\left(\mathit{\omega t}-\gamma \right)-R_P·\mathit{sin}{\phi }_v·\mathit{sin}{\theta }_v·\mathit{cos}\left(\mathit{\omega t}-\gamma \right)+h_P·\mathit{sin}{\phi }_v·\mathit{cos}{\theta }_v\\ z_P=R_P·\mathit{sin}{\phi }_v·\mathit{sin}\left(\mathit{\omega t}-\gamma \right)-R_P·\mathit{cos}{\phi }_v·\mathit{sin}{\theta }_v·\mathit{cos}\left(\mathit{\omega t}-\gamma \right)-A_d\\ \begin{array}{cccc}\begin{array}{cccc}\begin{array}{cccc}\begin{array}{ccc}& & \end{array}& & & \end{array}& & & \end{array}& & & \end{array}+h_P·\mathit{sin}{\phi }_v·\mathit{cos}{\theta }_v+\rho ·\left(1-\mathit{cos}{\phi }_v·\mathit{cos}{\theta }_v\right)\end{array}\right.& \end{array}$$E25

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 (x_Q, y_Q, z_Q) in the global coordinate can be expressed as follows:

Figure 7.

Schematics of the successive grain engaging by the spherical stone.

$$\begin{array}{l}\left(\begin{array}{l}{\mathrm{x}}_{\mathrm{Q}}={\mathrm{R}}_{\mathrm{Q}}·cos{\mathrm{\theta }}_{\mathrm{v}}·cos\left(\mathrm{\omega }\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)-{\mathrm{\gamma }}_{\mathrm{Q}}\right)+{\mathrm{h}}_{\mathrm{Q}}·sin{\mathrm{\theta }}_{\mathrm{v}}+\mathrm{f}\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)\\ {\mathrm{y}}_{\mathrm{Q}}=-{\mathrm{R}}_{\mathrm{Q}}·cos{\mathrm{\phi }}_{\mathrm{v}}·cos\left(\mathrm{\omega }\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)-{\mathrm{\gamma }}_{\mathrm{Q}}\right)-{\mathrm{R}}_{\mathrm{Q}}·sin{\mathrm{\phi }}_{\mathrm{v}}·sin{\mathrm{\theta }}_{\mathrm{v}}\\ ·cos\left(\mathrm{\omega }\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)-{\mathrm{\gamma }}_{\mathrm{Q}}\right)+{\mathrm{h}}_{\mathrm{Q}}·sin{\mathrm{\phi }}_{\mathrm{v}}cos{\mathrm{\theta }}_{\mathrm{v}}\\ {\mathrm{z}}_{\mathrm{Q}}={\mathrm{R}}_{\mathrm{Q}}·sin{\mathrm{\phi }}_{\mathrm{v}}·sin\left(\mathrm{\omega }\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)-{\mathrm{\gamma }}_{\mathrm{Q}}\right)-{\mathrm{R}}_{\mathrm{Q}}·cos{\mathrm{\phi }}_{\mathrm{v}}·sin{\mathrm{\theta }}_{\mathrm{v}}\\ ·cos\left(\mathrm{\omega }\left(\mathrm{t}+\mathrm{\Delta t}-\mathrm{d}{\mathrm{\delta }}_{\mathrm{ij}}/\mathrm{\omega }\right)-{\mathrm{\gamma }}_{\mathrm{Q}}\right)-{\mathrm{A}}_{\mathrm{d}}+{\mathrm{h}}_{\mathrm{Q}}·cos{\mathrm{\phi }}_{\mathrm{v}}·cos{\mathrm{\theta }}_{\mathrm{v}}\\ +\mathrm{\rho }·\left(1-cos{\mathrm{\phi }}_{\mathrm{v}}·cos{\mathrm{\theta }}_{\mathrm{v}}\right)\end{array}\right.\end{array}$$E26

Where γ_Q is the delay angle of Point Q to the bottom of the cutter, t is the cutting time, and dδ_ij is the lead angle of the prior grain of the recent engaging grain. According to this formula, the change of engaging depth of P t_P in the grinding process can be calculated by the following formula.

$$\begin{array}{cc}{t}_p& =\overline{\mathit{PQ}}\\ & =R_P-\left[\left(x_Q-f·t\right)·\mathit{cos}{\theta }_v-\left\{z_Q-\rho ·\left(1-\mathit{cos}{\theta }_v\right)\right\}\right]\mathit{sin}\left(\mathit{\omega t}-\gamma \right)+y_Q·\mathit{cos}\left(\mathit{\omega t}-\gamma \right)\end{array}$$E27

If t_P 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 small segments. The rotational radius R_p in each segment is determined with the actual (or theoretical) shape of the cutting edge. By calculating the engaging depth at each time step with calculated stone posture (inclination and tilting angles), the time series of the engaging depth can be calculated in any stone locus and attitude.

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′ (which is described by formula (15)). In Figure 8, considering the region OAB (of which area: “d_s”) inclined only by φ from the abrasive grain feed direction, the grinding force d_p applied on grain’s conical surface. Assuming that there is no friction between the abrasive grain and the material for simplicity, d_p acts perpendicularly on the conical surface, and the rubbing direction dt of the abrasive grain shown in the figure and its vertical component d_n can be decomposed into components as follows.

Figure 8.

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

$$\begin{array}{cc}\left(\begin{array}{c}\mathit{dt}=\mathit{\text{dpcosγcos}}\varphi \\ \mathit{dn}=\mathit{\text{dpsin}}\gamma \end{array}\right.& \end{array}$$E28

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, d_p is obtained by the following formula.

$$\begin{array}{cc}\mathit{dp}=\mathit{\text{σdscosγcos}}\varphi & \end{array}$$E29

The above formula shows that the distribution of d_p 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 d_s of the grain can be described as the following formula.

$$\begin{array}{cc}\mathit{ds}=\frac{{\rho }^2}{2}\mathit{sin}\mathit{\gamma d\varphi }& \end{array}$$E30

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

$$\begin{array}{cc}\mathit{dp}=\frac{{\rho }^2\sigma }{2}\mathit{sin}\mathit{\text{γcosγcos}}\mathit{\varphi d\varphi }& \end{array}$$E31

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

$$\begin{array}{cc}\left(\begin{array}{c}\mathit{dt}=\frac{{\rho }^2\sigma }{2}\mathit{sin}\gamma {\mathit{cos}}^2\gamma {\mathit{cos}}^2\mathit{\varphi d\varphi }\\ \mathit{dn}=\frac{{\rho }^2\sigma }{2}{\mathit{sin}}^2\gamma \mathit{cos}\gamma \mathit{cos}\mathit{\varphi d\varphi }\end{array}\right.& \end{array}$$E32

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.

$$\begin{array}{cc}\left(\begin{array}{c}\overline{t}={\int }_{-\pi /2}^{\pi /2}\frac{\mathit{dt}}{\mathit{d\varphi }}\mathit{d\varphi }=\frac{\pi {\rho }^2\sigma }{4}\mathit{sin}\gamma {\mathit{cos}}^2\gamma =\frac{\mathit{\pi \sigma }}{4}\overline{g}\mathit{sin}\gamma \\ \overline{n}={\int }_{-\pi /2}^{\pi /2}\frac{\mathit{dt}}{\mathit{d\varphi }}\mathit{d\varphi }={\rho }^2\sigma {\mathit{sin}}^2\mathit{\text{γcos}}\gamma =\sigma {\overline{g}}^2\mathit{sin}\mathit{\text{γtan}}\gamma \end{array}\right.& \end{array}$$E33

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

$$\begin{array}{cc}j=\frac{R\left(\alpha +\delta \right)f}{{\omega }^2}& \end{array}$$E34

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

$$\begin{array}{cc}\left(\begin{array}{c}\mathit{\text{Ft}}=j\overline{t}=\frac{\mathit{\pi \sigma }}{4}\frac{\mathit{Rf}}{w^2}\frac{{\delta }^2}{\alpha +\delta }{\left[2\Delta -\frac{R\left(r+R\right)}{r}\frac{{\delta }^2}{3}\right]}^2\mathit{sin}\gamma \\ \mathit{Fn}=j\overline{n}=\sigma \frac{\mathit{Rf}}{w^2}\frac{{\delta }^2}{\alpha +\delta }{\left[2\Delta -\frac{R\left(r+R\right)}{r}\frac{{\delta }^2}{3}\right]}^2\mathit{sin}\mathit{\text{γtan}}\gamma \end{array}\right.& \end{array}$$E35

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 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, 4, 5, 6, 7, 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.

6. Conclusions

In this chapter, a theoretical evaluation method of the grain trajectory and its engaging process onto the material in the grinding was introduced. The basic theory of the grain locus in grinding process and its example: spherical grinding stone for curve generation were introduced. And the numerical model of the grinding force applied on one grain and working surface is illustrated. Finally, the statistical approaches for evaluating the actual grinding stone were discussed.