## 5.1.1 Analysis of contact arc length

Due to the influence of the groove of grinding wheel, the actual contact arc length between grinding wheel and workpiece always changes periodically. As shown in Fig. 8, the structured grinding wheel with the groove width of *l*g, the continuous arc length of *l*w, the diameter of *D* and the thickness of *b* is analyzed. Under the conditions that the grinding depth is *a*p, the linear speed of the grinding wheel is *v*s, and the workpiece feed speed is *v*w, according to the relationship between the length of one intermittent cycle of grinding wheel (*l*w+*l*g) and the theoretical contact arc length (*l*c), the following five cases are discussed respectively.

**case1**

*l* w+*l*g>*l*c and *l*g< *l*c, *l*w*<l*c

**case2**

*l* w+*l*g>*l*c and *l*g< *l*c, *l*w*>l*c

**case3**

*l* w+*l*g>*l*c and *l*g> *l*c, *l*w*<l*c

**case4**

*l* w+*l*g>*l*c and *l*g> *l*c, *l*w*>l*c

**case5**

*l* w+*l*g<*l*c

Figure 9 shows the relationship between the actual contact arc length and time in one cycle under the first four cases. In order to simplify the analysis, the average contact arc length(\(\overline {{{l_c}}}\)) is used to instead of the actual contact arc length, that is, it is considered that the contact arc length is constant value during the grinding process of structured grinding wheel. The relationship between the average contact arc length and the actual contact arc length in the four cases can be expressed as follows:

\({\mathop l\limits^{ - } _{\text{c}}} \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}=({l_{\text{c}}} - {l_{\text{g}}}) \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}+\left( {\frac{{{l_{\text{c}}}}}{{{v_{\text{s}}}}} - \frac{{{l_{\text{c}}} - {l_{\text{g}}}}}{{{v_{\text{s}}}}}} \right) \cdot \left( {{l_{\text{w}}} - {l_{\text{c}}}+{l_{\text{g}}}} \right)\) (case1) (13)

\({\mathop l\limits^{ - } _{\text{c}}} \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}=({l_{\text{c}}} - {l_{\text{g}}}) \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}+\left( {\frac{{{l_{\text{w}}}}}{{{v_{\text{s}}}}} - \frac{{{l_{\text{c}}} - {l_{\text{g}}}}}{{{v_{\text{s}}}}}} \right) \cdot {l_{\text{g}}}\) (case2) (14)

\({\mathop l\limits^{ - } _{\text{c}}} \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}=\frac{{{l_{\text{c}}}}}{{{v_{\text{s}}}}} \cdot {l_{\text{w}}}\) (case3) (15)

\({\mathop l\limits^{ - } _{\text{c}}} \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{v_{\text{s}}}}}=\frac{{{l_{\text{w}}}}}{{{v_{\text{s}}}}} \cdot {l_{\text{c}}}\) (case4) (16)

By solving the above four equations respectively, the average contact arc length can be expressed uniformly as:

$${\mathop l\limits^{ - } _{\text{c}}} \cdot =\frac{{{l_{\text{w}}} \cdot {l_{\text{c}}}}}{{{l_{\text{w}}}+{l_{\text{g}}}}}$$

17

When *l*w + *l*g < *l*c, the continuous segment of grinding wheel is always in contact with the workpiece, and the actual contact arc length hardly changes with time. It can be considered that the average contact arc length is equal to the actual contact arc length, and the average contact arc length can also be expressed by Eq. (17).

## 5.1.2 Analysis of maximum undeformed chip chickness

Due to the discontinuity of the structured grinding wheel, the maximum undeformed chip thickness of a single abrasive grain during grinding will be affected by the groove width and the intermittent ratio. On the premise that the chip shape is a cuboid, for ordinary continuous grinding wheel, the chip volume generated by each abrasive grain can be expressed as [2]:

$${V_{\text{c}}}=\frac{{{a_{{\text{gmax}}}}}}{2} \cdot \frac{{r{a_{{\text{gmax}}}}}}{2} \cdot {l_{\text{c}}}$$

18

$${l_{\text{c}}}=\sqrt {{a_{\text{p}}}D}$$

19

Where *V*c is the chip volume, *a*gmax is the maximum undeformed chip thickness, *r* is the chip width-thickness ratio, *r* = tan*θ* [25, 26]. For structured grinding wheel, the average contact arc length (\({\mathop l\limits^{ - } _{\text{c}}}\)) is used instead of the theoretical contact arc length (*l*c). It can be known that the chip volume produced by each abrasive grain on the surface of the structured grinding wheel can be expressed as:

$$V_{{\text{c}}}^{{\text{'}}}=\frac{{{a_{{\text{gmax}}}}}}{2} \cdot \frac{{r{a_{{\text{gmax}}}}}}{2} \cdot {\mathop l\limits^{ - } _{\text{c}}}$$

20

Figure 10 is the schematic of material volume removed by structured grinding wheel per unit time under given grinding parameters. Assuming that there is no material residue on the gound surface, the volume of material removed from the workpiece per unit time should be equal to the sum of volumes of chips removed by all abrasive grains per unit time, which is:

$$Cb{v_{\text{s}}} \cdot V_{{\text{c}}}^{'}={a_{\text{p}}}{v_{\text{w}}}b$$

21

Where *Cbv*s is the number of abrasive grains acting on the wheel width *b* of the grinding wheel per unit time, *C* is the number of dynamic effective abrasive grains per unit area of the wheel, *a*p*v*w*b* is the volume of workpiece material removed per unit time. Through Equations (19), (20) and (21), the maximum undeformed chip thickness *a*gmax can be obtained as:

$${a_{{\text{gmax}}}}=\sqrt {\frac{4}{{rC}} \cdot \frac{{{v_{\text{w}}}}}{{{v_{\text{s}}}}} \cdot \sqrt {\frac{{{a_{\text{p}}}}}{D}} \cdot \frac{{{l_{\text{w}}}+{l_{\text{g}}}}}{{{l_{\text{w}}}}}} {\text{ }}$$

22

For any abrasive grain *i*, its undeformed chip thickness *a**gi* can be expressed as:

$${a_{gi}}={a_{{\text{gmax}}}} - \left( {{h_{\hbox{max} }} - {h_i}} \right)$$

23

Where *h*max is the maximum protruding height of abrasive grains, *h**i* is the protruding height of abrasive grain *i*.