Study on the cutting mechanism of randomly deflected truncated cone shape single abrasive grain

Based on the random distribution characteristic of abrasive grains for the grinding wheel circumferential surface during the manufacturing process, considering the grinding parameters and deflection parameters comprehensively, a randomly deflected truncated cone shape single abrasive grain cutting simulation model was established. Then, studying its cutting mechanism, i.e., the trajectory length value by numerical method and formula method was compared, the consequence indicates that the error is extremely small; this demonstrates the accuracy of the numerical method. The calculation and analysis of the workpiece cutting surface topography and the undeformed chip shape were carried out; it shows that the cutting surface topography and the undeformed chip shape are coincident. And the calculation results of the model were compared to verify. The influence laws of each grinding process parameter on the material removal volume (MRV) and material removal rate (MRR) were investigated; the results display that the grinding method affects the changing pattern of MRV and MRR; when the grinding method was up-grinding, a high wheel speed or a small wheel diameter is able to enhance the MRR; the MRV and MRR are increments with the increasing workpiece feed speed or abrasive grain cutting depth; and the MRV and MRR decrease with the raising of the absolute value of single abrasive grain deflection angle. The situation is a similar or opposite trend in down grinding. This work can provide a theoretical reference for the selection and optimization of parameters and further lay a foundation for the research in the grinding mechanism of the grinding wheel.


Introduction
Precision manufacturing equipment and its related frontier theory research have been a highlight area of concern in recent years. Grinding is an important precision and ultraprecision processing method in manufacturing; precision machining grinding process, generally, as the last process of machining, can effectively eliminate the processing defects that generated by the previous process, improve the quality of product processing, and play a very important role in ensuring the performance and service life of the product [1,2]. Grinding technology, as one of the important developments of modern manufacturing technology, is widely used in aerospace, precision instruments, automobiles, and other manufacturing fields [3][4][5][6][7]. The grinding process can be regarded as the comprehensive effect of slipping, plowing, and cutting by a large number of discrete abrasive grains, which are distributed on the grinding wheel surface, and is a multi-edge cutting process that involves a large number of abrasive grains on the grinding wheel surface [8]. The grinding process is essentially a course in which a large number of discrete abrasive grains distributed on the surface of the grinding wheel interact with the processed component to achieve material removal; each abrasive grain on the surface of the grinding wheel makes a microscopic cutting into the worked material. Due to the characteristics of the grinding wheel manufacturing process, abrasive grains are arranged randomly on the circumferential surface of the grinding wheel. The position and angle (the normal direction which is relative to the circumferential surface of the grinding wheel in contact with the abrasive grain) of the abrasive grains are irregular and varied. Moreover, abrasive grains are consolidated on the grinding wheel; the spatial attitude of abrasive grains is also random. The random characteristics of abrasive grains are one of the reasons for the complexity of the grinding process. Therefore, it is difficult to study the grinding mechanism through the complete grinding wheel, which is very unfavorable to the research promotion of the grinding process. The microscopic cutting of a single abrasive grain is a fundamental part of the grinding process, an effective means to understand the complex grinding processing mechanism, and its cutting action is the basis for material removal; thus, researchers often choose the cutting process of a single abrasive grain for analysis in order to achieve the research of grinding wheel grinding mechanism [9].
Anderson et al. [10] compared the cutting action of two different single abrasive grain geometries by using experimental observations and a validated finite element model and found that both tools required approximately the same energy to shear a chip from a workpiece when friction was subtracted from the specific energy for material removal. Gu et al. [11], who conducted single abrasive grain grinding experiments on SiCp/Al composites to determine the grinding forces at different grinding process parameters, established a prediction model for the single abrasive grain grinding force to study the influence of the grinding process parameters and grinding grain angle on the grinding force of SiCp/Al composite, and the PSO-SVM algorithm-based grinding force prediction model can accurately predict the grinding force of SiCp/Al composite and provide theoretical support for improved surface quality. Yin et al. [12] studied the effects of grinding speed on the material removal mechanism of SiCf/SiC by single grain grinding, results indicate that increasing speed grinding could embrittle the material and enhance the breakage of fibers, and both the groove surface quality and machining efficiency are improved by increasing speed grinding.
Grinding is a traditional precision machining method and is one of the most critical surface finishing processes to meet desired part requirements for difficult-to-machine materials in manufacturing industries, which are carried out after the rough machining and make the workpiece meet the requirements of high surface quality [13][14][15]. Grinding is a kind of abrasive processing technology, which means that abrasive plays an important role in grinding. There are thousands of abrasive grains on the grinding wheel surface, which makes the grinding mechanism complicated [16]. Abrasive grains on the grinding wheel are the main part involved in material removal. As abrasive grains directly participate in the material removal process, their location and angle are considered to be important factors affecting the grinding surface quality, as well as the sharpness and wear resistance of abrasive grains themselves [17,18]. In grinding, abrasive grains with rough and hard surfaces slide on the workpiece surface, make the surface material of the workpiece plastic deformation or fracture failure, and finally form the ground surface [19]. Accordingly, modeling and analyzing the abrasive model are conducive to grinding mechanism study and grinding quality prediction. In the previous grinding research, abrasive grains are usually considered as simple geometry to simplify the model. The spherical shape is the most widely used as an abrasive grain model [20,21]. Other simple geometries such as hexahedron and octahedron are often used as simplified models of abrasive grain [22,23]. However, there are little analyses in the relevant literature on the simplified model modeling method and their cutting mechanism for truncated cone shape abrasive grain. Because the simple single abrasive grain model lacks the random characteristics of abrasive grain, so, the random angle of the single abrasive grain model has obvious advantages in grinding research. In the field of grinding, due to the complexity of the model, few studies are using random angles in a single abrasive grain model.
At present, the methods for single abrasive grain cutting mechanism study are mainly focused on experimental research or finite element software simulation research, and the theoretical research on single abrasive grain cutting mechanism is relatively less. Based on the above, a theoretical model of cutting mechanism based on randomly deflected truncated cone shape single abrasive grain was established considering the grinding parameters and deflection parameters comprehensively. The fundamental theory research on the cutting behavior of a single abrasive grain was carried out and further laid a foundation for exploring the grinding mechanism of the grinding wheel.
2 Random deflection truncated cone shape single abrasive grain cutting model

Establish workpiece coordinate system
In order to describe the cutting process of the single abrasive grain with a randomly deflected angle, a coordinate system was established on the surface of the workpiece firstly, with the length of the workpiece as l w , the width as b w , and the height as z w ; the subscript "w" represents the workpiece. The x-axis direction is the length direction of the workpiece, the y-axis direction is the width direction of the workpiece, and the z-axis direction is the height direction of the workpiece; the origin of the coordinate system O w is located at the center of the upper surface of the workpiece. Δwx denotes the discrete step size for discretization in the length direction of the workpiece; Δwy denotes the discrete step size for discretization in the width direction of the workpiece. x w (i) denotes the x-coordinate of the location for the ith discrete point in the length direction of the workpiece, and y w (j) denotes the y-coordinate of the location for the jth discrete point in the width direction of the workpiece. The discretization process of x w (i) was expressed by the formula below: In the length direction of the workpiece, the range of values of x w (i) is the discrete point from the start point − l w /2 to the end point l w /2; discrete step is Δwx.
The discretization process of y w (j) was expressed by the formula below: In the width direction of the workpiece, the range of values of y w (j) is the discrete point from the start point − b w /2 to the end point b w /2, and discrete step is Δwy.
x w (i, j) was denoted to express the x-coordinate of all discrete points on the upper surface of the workpiece, y w (i, j) was denoted to express the y-coordinates of all discrete points on the upper surface of the workpiece, and z w (i, j) was denoted to express the z-coordinates of all discrete points on the upper surface of the workpiece, and: Up to now, the workpiece coordinate system constructed completely, as shown in Fig. 1.

Cutting process modeling
The diameter of the grinding wheel is d s , the radius is r s , the circumferential linear velocity of the grinding wheel is v s , the circumferential angular velocity of the grinding wheel is ω s , the center of the grinding wheel is O s , the grinding depth of the grinding wheel is a e , and the subscript "s" represents the grinding wheel; x so , y so , and z so denote the coordinates of the center of the grinding wheel circle on the x-axis, y-axis, and z-axis, respectively. After establishing the workpiece coordinate system, the relative motion of the abrasive grain and the workpiece was transformed into the workpiece that did not move, and the grinding wheel moved; the workpiece feed speed is v w ; at this time, the feed motion of the workpiece was transformed into the workpiece that did not move and the grinding wheel axis moved along the x-axis direction. The feed speed of the grinding wheel is v f , and v w = − v f . Consequently, the abrasive grain moving trajectory was a combination in the rotational motion of the grinding wheel and the feed motion of the workpiece. The grinding wheel grinds the workpiece as shown in Fig. 2. The abrasive grain was defined as a truncated cone shape, h max was represented to the maximum cutting depth of the abrasive grain, h g was represented to the height of the abrasive grain, and h gz was represented to the height of the conical geometry (hereinafter called cone) formed by the truncated cone abrasive grain as a reference; the subscript "g" indicates the abrasive grain, and "z" the subscript indicates the cone; and d gm was represented to the diameter of the abrasive grain's large circle, d gn was represented to the diameter of the abrasive grain's small circle, the subscript "m" indicates the large circular surface of the abrasive grain, and the subscript "n" indicates the small circular surface of the abrasive grain. According to the geometric relationship of the truncated cone shape abrasive grain, h gz could be expressed by h g , d gm , and d gn ; it was expressed by Eq. (4): The parameter β was denoted to express the half-top angle of the cone (in radians) and was expressed by the following formula and is shown in Fig. 3.
The parameter φ was indicated to express the cut-in angle or cut-out angle of the single abrasive grain for cutting workpiece process (Fig. 3); φ could be expressed in Eq. (6) and is shown in Fig. 4.
The parameter t was indicated to express the cutting time used by the abrasive grain from cut-in to cut-out for the workpiece, t min is the time when the abrasive grain starts cutting, t max is the time when the abrasive grain ends cutting, n t denotes the number of discrete step in cutting time, Δt denotes the discrete step size for discrete processing of the cutting time, and t i denotes the time corresponding to the ith discrete point in the cutting time of the abrasive grain, according to the cutting process of a single abrasive grain: At the cutting time t of the abrasive grain from cut-in to cutout of the workpiece, the range of values of t i is the discrete point from the start point t min to the end point t max , discrete step is Δt.
The v f is positive when it coincides with the positive direction of the x-axis, the length traveled by the center of the wheel in the x-axis direction at the ends cutting time was expressed by l sox , and it was expressed by the formula below: At the time of t i , the coordinates of the center of the grinding wheel circle on the x-axis, y-axis, and z-axis were as follows: The trajectory of the cutting motion of the abrasive grain is shown in Fig. 5.
The position of the abrasive grain was changed by the rotation of the grinding wheel; the center of the large circle of the grinding grain is O gm ; the top points of the cone are A gz , x gm , y gm , and z gm that denoted the x-, y-, and z-coordinates of O gm , respectively; and x gza , y gza , and z gza denoted the x-, y-, and z-coordinates of A gz , respectively. Then, at the time of t i , the coordinates of O gm and A gz were as follows: where " + " represents down-grinding, and " − " represents up-grinding.
The geometric relationship of the above formulas was as shown in Fig. 6: Fig. 6a shows up-grinding, and Fig. 6b shows down-grinding. O gn was defined to express the center of the small circle of the grinding grain.
The position of the abrasive grain would change under the deflection of its axis around O gm , and the abrasive grain was coaxial with the cone; δ 1 was defined to express the deflection angle of the central axis for the cone in the plane perpendicular to the plane xOz and passing through the line O s -O gm , and the subscript "1" indicates the plane that perpendicular to the plane xOz and passing through the line O s -O gm . The angle value was positive when the δ 1 deflection direction was the same as the positive direction of the y-axis, and the range of the angle is in [− π/2, π/2]; after deflection, the projection line of the axis in the xOz plane was co-linear with the line O s -O gm , as shown in Fig. 7.
After deflecting angle δ 1 , δ 2 was defined to express the deflection angle by the rotation of the abrasive grain axis projected straight line in the xOz plane around the y-axis, and the subscript "2" indicates the plane of rotation parallel to the xOz plane; the angle value was positive when the δ 2 deflection direction was the same as the positive direction of the x-axis, and the range of the angle is in [− π/2, π/2], as shown in Fig. 8.
The position of A gz was transformed to A gz1 when the deflected angle was δ 1 , x gza1 , y gza1 , and z gza1 were denoted to express the x-, y-, and z-coordinates of A gz1 , respectively. So, at the moment t i , the coordinates of A gz1 were as follows: where " + " represents down-grinding, and " − " represents up-grinding.
On the basis of a deflected angle δ 1 , the position of A gz1 was transformed to A gz12 , when the deflected angle was δ 2 ; x gza12 , y gza12 , and z gza12 were denoted to express the x-, y-, and z-coordinates of A gz12 , respectively. So, at the moment t i , the coordinates of A gz12 were as follows: where " + " represents down-grinding, and " − " represents up-grinding.
The workpiece topography could be represented by a matrix of points with different heights [24]. The number of discrete points in the length and width directions of the workpiece was denoted by n l and n b , respectively. The height of each point was stored in a two dimensional matrix Z[n l ,n b ]. Before grinding, the height of all the points was zero, i.e., all discrete points were on the surface of the workpiece. The distance between two adjacent points was 1/100 of the d gm : The point numbers were as follows: The square brackets indicate rounding. 5 The grain moving trajectory O gm was selected as the reference point in the cutting process. The coordinates of O gm could be expressed by (x gm,i ,y gm,i , z gm,i ). According to the abrasive grain position, the limit values X min , X max , Y min , and Y max of the coordinates on the workpiece surface points, for heights that might be affected by the abrasive grain, could be found in the matrix Z and could be easily determined by the following: The square brackets indicate rounding.

Cutting process calculation
In the cutting process of the abrasive grain, the workpiece was defined as an ideal body (without considering the effect of deformation, residual stress, etc.). And the cutting was simply regarded as the relative motion of the abrasive grain and the workpiece, i.e., the type of contacting abrasive grain in the grinding arc was only cutting abrasive grain, and the interference part between the abrasive grain and the workpiece in the process of movement was all regarded as the removal of material. After being cut by the abrasive grain, the z-coordinate of the cut point on the workpiece surface would change, while x-coordinate and y-coordinate stay the same; x w , y w , and z w denote the x-, y-, and z-coordinates of discrete points on the workpiece surface. According to the analysis above, it can solve the new z-axis coordinates of the cut point on the workpiece surface after abrasive grain cutting and use z wn to represent. The positions of discrete points on the workpiece surface after cutting were in two cases, the first situation was located on the conical surface of the abrasive grain, and the second situation was located on the plane of small circular surface of the abrasive grain.
In the first case, the axis of the cone after deflection δ 1 and δ 2 is denoted by r 1 and could be expressed as follows: The new position and its coordinates of the cut point were expressed in terms of C gz (x w , y w , z 1 ); the straight line C gz -A gz12 is denoted by r 2 and could be expressed as follows: The angle between the r 1 and r 2 was equal to the half-top angle of the cone, which could be expressed as follows: Organize Eq. (19) and obtain the following: Since the only unknown variable in Eq. (20) was z 1 ; thus, the constant part in Eq. (20) could define by other characters: The formula could be transformed as follows: The new z-axis coordinate values would be obtained after computing Eq. (22), which were named z 11 and z 12 , due to the cos 2 β having the following characteristics in Eq. (20), as shown in Fig. 9.
23) cos 2 = (−cos( − )) 2 = cos 2 ( − ) Fig. 9 The location of z 11 and z 12 on the conic surface From Eq. (23) and Fig. 9, the angle between r 1 and r 2 was either β or (π − β), corresponding to the two solutions z 11 and z 12 . Because the new position makes the angle β between r 1 and r 2 , thus, the unqualified solution z 12 should be discarded.
In the second case, as known from the previous case, r 1 was the normal vector of the large and small circular surfaces of the abrasive grain. C gm (x, y, z) were expressed at any point and its coordinates on the large circular surface of the abrasive grain. L(x, y, z) was the plane formula of the large circular surface. From the formula of the plane equation: Organize Eq. (24) and obtain the following: The new position and its coordinates of the cut point were expressed in terms of C gn (x w , y w , z 2 ). The distance from the point C gn to the large circular surface of the abrasive grain was equal to h g . So: From Eq. (25): Since the only unknown variable in Eq. (26) was z 2 , thus, the constant part in Eq. (26) could define by other characters: The formula could be transformed as follows: The new z-axis coordinate values would be obtained, after computing Eq. (29), which were named z 21 and z 22 , as shown in Fig. 10.
From Fig. 10, the planes with distance h g to the large circular surface of the abrasive grain were the small circular surface plane and its symmetrical mirror plane about Because the new location was on the small circular surface of abrasive grain, thus, the unqualified solution z 21 should be discarded.
Based on the above, the geometric relationship between z 11 and z 22 is as shown in Fig. 11.
According to the geometric relationship represented in Fig. 11, the positions of discrete points on the workpiece surface after cutting were in two cases, located either on the conical surface or on the plane of the small circular surface, and the new point must be located on the truncated conical surface and the small circular surface. Therefore, it was clear that: The discrete points on the workpiece surface can be divided into the cut and uncut points. After the abrasive grain cutting, the z-coordinate of the cut point on the (30) z wn = max z 11 , z 22 workpiece surface was become z wn , while the uncut point remains the same.

Case model calculation and study
According to Eqs. (7)-(13), the time interval [t min ,t max ] had been discretized; a series of discrete points on the abrasive grain cutting trajectory could be obtained. The adjacent time points could be expressed by t i and t i+1 , t i+1 = t i + Δt. The coordinates of the corresponding A gz12 positions were A gz12,i (x gza12,i ,y gza12,i ,z gza12,i ) and A gz12,i+1 (x gza12,i+1 ,y gza12,i+1 , z gza12,i+1 ); then, the distance between the two positions was used Δl i to express the following: The length of the abrasive grain cutting trajectory was used L g to express the following: (31)

Fig. 11 The geometric relationship between solutions
The following were calculation examples and verification: According to Eqs. (1)-(13), a calculation program was compiled and run with the parameters shown in Table 1. The calculation results are shown in Table 2. The calculation results in Table 2 represent verification of this work. The calculation results were compared with the theoretical model proposed by Malkin [14] for common plane surface grinding conditions. It should be noted that the grinding contact length obtained by Malkin's formula was for an abrasive grain moving from the lowest point to the endpoint of the cut-out stage, and the shape of the abrasive grain is spherical, taking the up-grinding as an Δl i example. Therefore, the abrasive grain cutting trajectory length calculated in this work was twice that of Malkin's formula.
Malkin's formula: where " + " represents down-grinding, and " − " represents up-grinding. Combining with Figs. 3 and 4, A gz was selected as the cut-in and cut-out reference position. The grinding depth a e in Eq. (33) needed to be replaced by the A gz12 's cutting depth a e + h gz − h g , because the cutting depths were different for different grains. Then, Eq. (33) becomes the following: The calculation results were compared under the parameters of group no. 1 and group no. 2, as shown in Table 2.
As shown in Table 2, the comparison errors were very small (< 0.16%). Thus, the proposed numerical method was verified.
According to the above analysis, after all of the points in the region were processed, the ground surface topography could be obtained. When each parameter was the same as Table 1 and the grinding method was up-grinding, Figs. 12 and 13 show the calculation results. Figure 12a, b show the front and top views of the A gz moving trajectory.
(34) Figure 13a, b show the top views of the simulated 3D ground surface: (a) δ 1 = 0 rad, (b) δ 1 = 0.1 rad. The deeper the color is, the greater the cut depth was. It can be seen from Fig. 13a, b that the deflection angle would affect the workpiece surface topography. When the deflection angle is zero, the top view of the workpiece surface topography is segmented symmetrical up and down along the x-axis, symmetrical left, and right along the y-axis, thick in the middle, and thin at both ends. When the deflection angle is not zero, the top view of the workpiece surface topography is segmented symmetrical left and right along the y-axis, thick in the middle and sharp at both ends, flat at the top, and round at the bottom.
The area of the workpiece's upper surface was denoted by p w , n w was denoted to express the number of all discrete points on the upper surface of the workpiece, and p we was denoted to express the area of each discrete point on the upper surface of the workpiece. Based on the cutting process model, the formula was as below: Material removal volume was denoted by MRV, which represented the sum of the cutting volume of all discrete points on the workpiece. Before grinding, the initial height value of all the points was zero; after grinding, the z-coordinate of the cut point on the workpiece surface becomes z wn , while the uncut point remains the same. Therefore, the MRV of the single abrasive grain cutting could be expressed as follows: The material removal rate was denoted by MRR, which represented the cutting volume of material removal per unit time. The MRR of the single abrasive grain cutting can be expressed as follows: When each parameter was the same as Table 1 (without value range) and the grinding method was up-grinding, the MRV was 9.0023 × 10 −3 mm 3 and the MRR was 20.9028 mm 3 /s by calculation.
In previous related studies, the cutting depth of the abrasive grain h, also known as the undeformed chip thickness, is a very important parameter in the modeling of grinding processes. Many researchers have used h to study grinding quality, removal, and chip formation mechanisms. For the single abrasive grain cutting studied in this paper, h max , i.e., the maximum unshaped chip thickness, is equal to a e , and in the process of abrasive grain cutting, the undeformed chip thickness had a large relationship with the grinding method, MRV, and MRR. Therefore, in order to investigate the cutting mechanism of single abrasive grain in depth and improve the surface quality of the workpiece, it is necessary to calculate the undeformed chip thickness and shape of the single abrasive grain cutting model. The workpiece was partitioned by a series of planes perpendicular to the x-axis, the distance between adjacent planes was equal to Δwx, and the resulting cross sections were the undeformed chip section shapes in which the thickness of the undeformed chip can be displayed. A Matlab calculation program was prepared; each parameter was the same as Table 1, and the grinding method was up-grinding; and then, the undeformed chip section shape could be obtained, as shown in Fig. 14. The undeformed chip shape was obtained by placing all the cross sections of the workpiece in one diagram, as shown in Fig. 15. Figure 15a, b show the z-axis positive direction views of the undeformed chip shapes, and Fig. 15c, d show the z-axis negative direction views of the undeformed chip shapes. It can be seen from Fig. 15a-d that the deflection angle would affect the undeformed chip shapes. When the deflection angle is zero, the shape of undeformed chip is segmented symmetrical up and down along the x-axis, symmetrical left, and right along the y-axis, thick in the middle, and thin at both ends. When the deflection angle is not zero, the shape of undeformed chip is segmented symmetrical left and right along the y-axis, thick in the middle and sharp at both ends, flat at the top, and round at the bottom. As can be seen from Figs. 13 and 15, for the single abrasive grain cutting, the undeformed chip shape and the workpiece surface topography were coincident.

Validation of the single abrasive grain cutting simulation model
Combining with the literature [25], the relevant data from the literature were substituted into the Matlab program of the single abrasive grain cutting simulation model for calculation, and the calculated results were compared with the available data in the literature. Since the shape of the abrasive grain in the literature [25] was conical, the value of d gn was set to zero to match the shape of the abrasive grain in the literature. Figure 16a was represented to the variation of the MRV during the process of grinding wheel speed from 10 to 150 m/s. Figure 16b was represented to the variation of the MRV during the process of half-top angle of the cone from 15 to 60°. The grinding method was up-grinding, and the results were shown below. As can be seen from Fig. 16, with the increase of the grinding wheel speed, the value of the MRV by model calculated and the existing value of the MRV in the literature [25] both gradually decreased, and the higher the grinding wheel speed, the smaller the change of the MRV. As the cone half-top angle of the grinding grain increases, the value of the MRV by model calculated and the existing value of the MRV in the literature [25] both gradually increased, and the larger the half-top angle, the greater the variation of the MRV. The calculated results were consistent with the data in the literature, and the error was small, which verified the accuracy of the calculated values for the single abrasive grain cutting model.

Influence of grinding process parameters on the MRV and MRR
Material removal has always been a big deal during the manufacturing process and the material removal rate has a direct connection with the period and cost of manufacturing. During the cutting process of randomly deflected truncated cone shape single abrasive grain, the grinding process parameters (wheel speed, wheel diameter, workpiece feed speed, cutting depth of the abrasive grain, δ 1 and δ 2 ) are very closely related to MRV and MRR and play a very important role in the cutting process. Therefore, it is meaningful and important to analyze the MRV and MRR under different grinding parameters and deflection parameters, discuss the rational selection and optimization of each grinding process parameter in depth, and study the influence law of each parameter in the cutting process of single abrasive grain in order to obtain higher grinding efficiency.
According to the previous subsection analysis, a Matlab calculation program was run under the parameters in Table 1, as shown in Fig. 17.
As can be seen from Fig. 17, the related results obtained were consistent with the findings in the relevant literature [25]. That is, when the grinding method is down-grinding, the MRV decreases with the increment of the wheel speed; however, the MRR increases with the wheel speed. The MRV and MRR are increment with the increasing grain's cutting depth of abrasive grain. The MRV and MRR have increased with the workpiece feed speed. All conclusions are as follows: 1. When the grinding method was down-grinding, the MRV increased with the increment of wheel speed, the growth rate decreased gradually, and the MRR also increased with the increment of wheel speed. When the grinding method was up-grinding, the changing way of MRV was the opposite of that in the down-grinding, while the MRR was the same. This indicated that a high wheel speed was able to enhance the MRV; however, when wheel speed increased to a certain value, its effect on MRV was not distinct. According to Eqs. (7) and (37), ω s = v s /r s , it can be seen that the MRR was both positively correlated with the wheel speed and the MRV, the variation range size of the MRV was not in the same order of magnitude as the variation range size of the wheel speed, which was negligible in comparison, so the MRR in the figure showed a linear increment with the increase of the wheel speed. 2. The MRV increased with the growth of wheel diameter; the larger the wheel diameter, the slower the  change rate, based on the same reason as the first one; the MRR was both positively correlated with the wheel diameter and the MRV; and thus, the MRR decreased linearly with the growth of wheel diameter. In addition, it was not affected by the pattern of the grinding method. So, increasing the grinding wheel diameter can increase the MRV, but it would decrease the MRR. 3. When the grinding method was down-grinding, the MRV and MRR decreased with the increment of workpiece feed speed, and when the grinding method was up-grinding, the MRV and MRR increased with the increment of the workpiece feed speed. Therefore, in the up-grinding condition, increasing the workpiece feed speed would help to improve the MRR. 4. The MRV and MRR increased with the increment of the grain's cutting depth, and the changing pattern was the same under different grinding methods. Therefore, increasing the grain's cutting depth would help to improve the MRR, but the depth of the grain's cutting cannot be increased indefinitely to avoid affecting the surface quality of the workpiece after cutting. 5. For the deflection angle δ 1 , regardless of the downgrinding or up-grinding, when the deflection angle δ 1 increased from small to large, the MRV and MRR increased first and then decreased, and there was a maximum value when the deflection angle was 0 rad. Thus, increasing the absolute value of the abrasive grain deflection angle δ 1 , the MRV and MRR would be reduced. 6. For the deflection angle δ 2 , when the grinding method was down-grinding, the MRV and MRR increased first and then decreased with the deflection angle δ 2 increased from small to large, and the increase rate was less than the decrease rate. When the grinding method was downgrinding, and the deflection angle δ 2 increased from small to large, the MRV and MRR increased first and then decreased; however, the increase rate was larger than the decrease rate. This indicates that decreasing the absolute value of the deflection angle δ 2 could help to improve the MRV and MRR, and its change pattern was related to the grinding method.

Conclusion
In the present study, based on the random distribution characteristic of abrasive grains on the grinding wheel circumferential surface during the manufacturing process, in order to reflect the random distribution for the abrasive grains' position and angle, considering the grinding parameters and deflection parameters comprehensively, a theoretical model of cutting mechanism based on randomly deflected truncated cone shape single abrasive grain was established.
The main contributions of this paper and the new findings are as follows: 1. Considering the grinding parameters and deflection parameters comprehensively, a mathematical case model of the moving motion trajectory of a randomly deflected truncated cone shape single abrasive grain was established. The trajectory length value calculated by the numerical method was verified, the obtained trajectory length error was extremely small between the numerical method and Malkin's formula, and the maximum error was less than 0.16%. 2. The single abrasive grain of a randomly deflected truncated cone shape cutting process model was also established; the grain moving trajectory, ground surface topography, and undeformed chip shape were obtained and found that the undeformed chip shape and the workpiece surface topography were coincident for single grain cutting. 3. The influence of the grinding parameters and deflection parameters on the MRV and MRR were studied. When the grinding method was up-grinding, the MRV decreases with the increasing wheel speed, while the MRR increases with wheel speed. The MRV increased with the growth of wheel diameter, while the MRR decreases with wheel diameter indicating that the high wheel speed and small wheel diameter are able to enhance the MRR. Both the MRV and MRR increase with the increment of cutting depth of abrasive grain or workpiece feed speed, while the MRV and MRR decrease with the raising of the absolute value of single abrasive grain deflection angle. The influence law on the MRV and MRR of down-grinding was similar or opposite trend to up-grinding.
In summary, the basic theory of cutting mechanism based on randomly deflected truncated cone shape single abrasive grain was established. The obtained conclusions have great significance for the selection and optimization of grinding parameters and deflection parameters and for further study in the grinding mechanism of the grinding wheel.