Research on Generating Gear Grinding Machining Error Based on Mapping Relationship between Grinding Wheel Surface and Tooth Flank

Abstract

This paper proposes a method to study the machining error when one is generating gear grinding based on the mapping relationship between the grinding wheel surface and the tooth flank. The profile of the grinding wheel after grinding is collected, and the theoretical tooth flank is derived according to a model of generating gear grinding. By comparing the normal deviation of the theoretical tooth flank with the measured tooth flank, the machining error characteristics are identified. Firstly, the mathematical model of generating gear grinding is established according to a multi-body theory and coordinate transformation. Secondly, the experiments of the lead modification for helical gear were performed on a YW7232 CNC gear grinding machine. The tooth flank twist compensation function was turned off in experiments No. 1–No. 3, and it was turned on in experiments No. 4–No. 6, while the other process parameters kept the same. Then, the corresponding theoretical tooth flank data were derived from the grinding wheel surface which were measured using the laser profilometer LJ-V7000. Finally, the machining error characteristics were obtained by analyzing the difference between the theoretical and measured tooth flank deviations. The result of this paper can provide theoretical and practical references for the improvement of the gear grinding accuracy and gear modification design.

1. Introduction

With the rapid development of industrial technology, gears are increasingly used in new energy vehicles, aerospace, and industrial robots as important transmission components. At the same time, the requirements for gear accuracy and transmission performance (noise, vibration, transmission errors, etc.) are also increasing in every industry. The heat treatment process of the gears after rough machining results also in tooth surface deformation, and gear finishing (gear grinding, gear honing, etc.) is required to improve the surface quality of the gears [1]. Gear grinding technology has great advantages in gear manufacturing companies by virtue of its high efficiency [2]. According to the form of meshing between the grinding wheel and the workpiece gear, the gear grinding process can be divided into generating grinding and profile grinding. In practical applications, due to the complexity of the grinding wheel dressing and the large contact area between the grinding wheel and the gear surface, which can easily cause tooth surface burns, spreading grinding is more widely used [3].

In recent years, the research on gear grinding technology has focused on gear grinding errors, tooth flank topological modification, tooth flank twist, and processing parameter optimization. The gear tooth flank is enveloped by the grinding wheel according to generating motion. The change of the grinding wheel position, profile, and the change of the helix angle due to the decreasing diameter due to the dressing process will directly affect the geometric profile of the gear tooth flank. Hence, this paper presents a method to investigate the machining error of a test generating gear grinding based on the mapping relationship between the grinding wheel surface and the tooth flank.

Regarding the research on gear grinding errors, scholars mainly paid attention to the effect of the grinding machine geometric errors on the tooth profile. Yoshino et al. [4] investigated the effects of grinding machine positioning errors (center distance error, grinding wheel inclination error, and grinding wheel axial position error, etc.) on the helical gear tooth profile, and they proposed an interactive analysis method to compensate for the grinding errors by eliminating more than two types of positioning errors. Xia et al. [5] established a spatial error model of a profile of a grinding machine in terms of homogeneous transformation matrices (HTMs) and the machine kinematic chain, and they proposed a novel approach to improve the geometric accuracy of a worktable based on single-axis motion measurements and an actual inverse kinematic model (IKM). The method avoids the interference errors of the non-targeted axes and improves the error identification accuracy. Zhou et al. [6] developed an open kinematic chain from the workpiece gear to the worm grinding wheel of a five-axis gear grinding machine based on the theory of multi-body topological systems. With the analysis of 37 geometric error components, the error propagation relationship from the grinding wheel to the workpiece was obtained. Litvin [7] calculated the workpiece tooth profile by the enveloping motion of given tool profile, and they optimized the initial machine settings on this basis.

Concerning the study of tooth profile modification, Shih et al. [8,9,10] represented the motion of each axis of a five-axis gear grinding machine as a fourth-order polynomial function, and they established the sensitivity matrix of the polynomial coefficients. By adjusting the polynomial coefficients, the machined tooth flank could approximate the ideal tooth flank. Wang et al. [11] proposed a tooth lead modification method based on the compound modification curve with uncertain parameters and a modification effect evaluation method. Orthogonal experiments were carried out using the modified gears to achieve the ideal modification effect under specific working conditions. Fong et al. [12] suggested a novel method of tooth flank crowning modification for cylindrical gears. This method uses a variable lead worm gear wheel (VLGW) with a diagonal feed (combination of tangential feed and axial feed) to reduce the tooth flank twist. Yang et al. [13] proposed an indirect method of applying a virtual rack tool (VRC) to calculate the grinding wheel surface for the difficult problem of solving the three-dimensional meshing equation that exists in the modified tooth flank.

In terms of the research on tooth flank twists, Guo et al. [14] established a tool profile sensitivity matrix based on the mathematical model of gear shaping. The tooth flank twist was reduced by adjusting the tool profile, which was fitted by the B-line curve in this method. Tran et al. [15,16] proposed a novel additional rotation angle for the workpiece gear during the hobbing process to obtain the anti-twist helical gear tooth flank with a lead modification. Zhang et al. [17] developed a mathematical model for the gear grinding of spiral bevel gears and hypoid gears, and they analyzed the sensitivity of the tooth flank to the change of the motion axis. Finally, the motion coefficients of each axis were optimized to achieve a twist-free tooth flank of a gear with a lead modification.

For the investigation of the process parameter optimization, Kang et al. [18] conducted experiments on vacuum pump gear machining with different grinding parameters. The effects of the parameters such as grinding speed, grinding depth, and feed on the grinding force, surface quality, and residual stress distribution were also investigated by using piezoelectric force sensors, a scanning electron microscope, and an X-ray diffractometer. Ming et al. [19] established a theoretical model of face gear roughness based on the grinding trajectory. Taking the tooth surface roughness as an optimization target, the orthogonal experiment method was used to optimize the machining parameters. The calculated results of the theoretical model were compared with the actual results, and the maximum error did not exceed 13.5%, which verified the rationality of the model. Guerrini et al. [20] proposed a two-step finite element simulation method for predicting the thermal damage during gear grinding. The first step is to calculate the grinding force and thermal energy generated by the interaction with the workpiece gear according to the actual shape of a single abrasive particle. The second step was to apply the calculation results and the motion model of the tooth flank grinding for determining the temperature distribution during gear grinding. Finally, the accuracy of the temperature model was verified by the grinding experiments, and the process parameters were improved.

In summary, the gear grinding accuracy can be improved by compensating for the machine tool errors. The results of the software and numerical simulations indicate that the modification of the grinding wheel surface can suppress the tooth flank twist. However, the above methods have not established a direct relationship between the real grinding wheel surface and the machined tooth flank. Therefore, this paper aims to present a method to study the machining error which occurred by generating a gear grinding operation based on the mapping relationship between the grinding wheel surface and the tooth flank.

Firstly, the mathematical model of continuous generating gear grinding is established based on the motion relationship between the grinding wheel and the workpiece gear. Secondly, when the compensation function of the tooth flank twist was closed and opened, three gears were successively ground, respectively. The true profile of the grinding wheel was measured using a laser profilometer, and the corresponding tooth flank was calculated. Finally, by comparing the normal deviation of the measured tooth flank with the ideal tooth flank, the error characteristics and the twist compensation effect were investigated.

2. Principle of Generating Gear Grinding

2.1. Mathematical Model of Gear Grinding

As shown in Figure 1a, the generating gear grinding machine mainly contains four linear axes (the grinding wheel radial-feed axis X1, the grinding wheel tangential-feed axis Y1, the grinding wheel axial-feed axis Z1, and the outer bracket moving axis Z2) and five rotating axes (the grinding wheel rotation axis B1, the workpiece gear rotation axis C1, the diamond wheel rotation axis B2, the grinding wheel swing axis A1, and the diamond wheel swing axis A2). The grinding wheel is installed on axis B1, while the workpiece gear is installed on C1 axis. The generating process of the cylindrical gear is controlled by the electronic gear box (EGB) in the CNC system for the machine axes movements. The tooth surface meshing motions between the grinding worm and the workpiece gear is accomplished along six axes: A1 axis, B1 axis, C1 axis, X1 axis, Y1 axis, and Z1 axis. Among them, the swing angle of A1 axis is determined by the grinding wheel lead angle and the gear helix angle, which is kept unchanged during the machining process. Figure 1b shows the coordinate transformations used to describe, mathematically, the generating gear grinding process. $S_{1w} \left(O_{1w}-x_{1w}, y_{1w}, z_{1w}\right)$ and $S_{4g} \left(O_{4g}-x_{4g}, y_{4g}, z_{4g}\right)$ are rigidly connected to the grinding wheel and the gear, respectively, and they rotate with them. $S_2 \left(O_2-x_2, y_2, z_2\right)$ and $S_3 \left(O_3-x_3, y_3, z_3\right)$ are auxiliary coordinate systems. ${\Sigma }_{gw}$ is the installation angle of grinding wheel $\left({\Sigma }_{gw}={\beta }_g\pm {\beta }_w\right)$. ${\beta }_g$ and ${\beta }_w$ is the helix angle of the gear and the lead angle of the grinding wheel, respectively. There are five motions in the continuous generating gear grinding process: the grinding wheel rotation ${\phi }_{B1}$, gear rotation ${\phi }_{C1}$, the grinding wheel axial-feed motion $F_{Z1}$, the grinding wheel radial-feed motion $F_{X1}$, and the grinding wheel tangential-feed motion $F_{Y1}$. Among them, $F_{X1}$ does not participate in EGB linkage control when the gear is not modified. The linkage relationship between the remaining axes is as follows [21]:

$$\begin{array}{c} {\phi }_{C1}=\frac{Z_w}{Z_g}{\phi }_{B1}+\frac{tan{\beta }_g}{r_g}{F}_{Z1}+\frac{Z_{w}tan{\beta }_w}{Z_g{r}_w}{F}_{Y1}\end{array}$$

(1)

where $Z_w$ and $Z_g$ denote the grinding wheel thread number and gear tooth number, respectively. $r_w$ and $r_g$ represent the pitch circle radius of the grinding wheel and gear, respectively.

When the tooth flank lead-crowning modification is performed (secondary parabolic curve), the relationship between the additional motion of $F_{X1}^{\prime }$ and $F_{Z1}$ is [22]:

$$\begin{array}{c}{F}_{X1}^{\prime }=-\frac{4\delta }{B^2}{F}_{Z1}^2+\frac{4\delta }{B}{F}_{Z1}\end{array}$$

(2)

where $\delta$ denotes crowning amount, and $B$ represents tooth width.

The position vector and normal vector of tooth flank can be obtained from grinding the wheel surface by coordinate transformation as follows:

$$\begin{array}{c}{r}_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)=M_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)·r_w\left(u,\theta \right)\end{array}$$

(3)

$$\begin{array}{c} n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)=L_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)·n_w\left(u,\theta \right)\end{array}$$

(4)

$$M_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)=M_{g3}{M}_{32}\left(F_{Z1}\right)M_{2w}\left({\phi }_{B1},F_{Y1}\right)$$

$$M_{2w}\left({\phi }_{B1},F_{Y1}\right)=\left[\begin{array}{cccc}cos{\phi }_{B1}& 0& -sin{\phi }_{B1}& 0\\ 0& 1& 0& F_{Y1}\\ sin{\phi }_{B1}& 0& cos{\phi }_{B1}& 0\\ 0& 0& 0& 1\end{array}\right]$$

$$\begin{array}{c} M_{32}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& cos{\Sigma }_{gw}& -sin{\Sigma }_{gw}& 0\\ 0& sin{\Sigma }_{gw}& cos{\Sigma }_{gw}& F_{Z1}\\ 0& 0& 0& 1\end{array}\right] M_{g3}\left({\phi }_{C1,}{F}_{Y1}\right)=\left[\begin{array}{cccc}cos{\phi }_{C1}& -sin{\phi }_{C1}& 0& -F_{X1}\\ sin{\phi }_{C1}& cos{\phi }_{C1}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$$

(5)

where $r_w\left(u,\theta \right)$ and $n_w\left(u,\theta \right)$ denote the position vector and normal vector of grinding wheel, respectively. $u$ and $\theta$ are tooth surface parameters of the grinding wheel. $r_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)$ and $n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)$ denote the position vector and normal vector of the workpiece gear, respectively. $M_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)$ is the coordinate transformation matrix from $S_{1w}\left(O_{1w}-x_{1w}, y_{1w}, z_{1w}\right)$ to $S_{g3} \left(O_{g3}-x_{g3}, y_{g3}, z_{g3}\right)$ and $L_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)$ is the upper-left $3\times 3$ submatrix of $M_{gw}\left({\phi }_{B1},F_{Y1},F_{Z1}\right)$.

$$\begin{array}{c} f_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)=n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)·\frac{\partial r_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)}{{\phi }_{B1}}=0\end{array}$$

(6)

$$\begin{array}{c} f_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)=n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)·\frac{\partial r_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)}{\partial F_{Y1}}=0\end{array}$$

(7)

$$\begin{array}{c} f_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)=n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)·\frac{\partial r_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)}{\partial F_{Z1}}=0\end{array}$$

(8)

Combining Equations (1)–(8), the position vector $r_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)$ and the normal vector $n_g\left({\phi }_{B1},F_{Y1},F_{Z1},u,\theta \right)$ of the tooth flank can be derived.

2.2. Tooth Flank Normal Deviation

Since there are some errors in the gear grinding process, the gear tooth flank will also have deviations. The helical gear tooth flank is a spatial surface, which needs to be meshed to obtain more comprehensive information [23]. The mesh density is determined by the accuracy of the data of the instrument. Referring to the relevant requirements of Gleason’s gear detection, the sampling area of the tooth flank is the area where the indent of the tip and root are taken as 5% of the working tooth height, respectively, and the indent of both of the ends are taken as 5% of the working face width, respectively. Generally, the lead direction is taken as nine columns, and the profile direction is taken as five rows, with a total of forty-five measurement points [24], as shown in Figure 2a. The standard involute tooth flank is projected to the plane as a planar grid, as shown in Figure 2b. The black plane is the standard tooth flank $r_g$, and the blue surface is the machined tooth flank $r_g^{\prime }$. By comparing the position vector of the machined tooth flank with the standard tooth flank, the deviation value ${\epsilon }_{ij}\left(i=A~\mathrm{E}, j=1~9\right)$ of each measurement point can be calculated as follows:

$$\begin{array}{c}{\epsilon }_{ij}=\mathsf{\Delta }{r}_g·n_g=\left(r_g^{\prime }-r_{\mathrm{g}}\right)·n_g\end{array}$$

(9)

where $n_g$ is the unit normal vector at the measurement point.

2.3. Tooth Flank Twist

The tooth flank twist refers to the phenomenon that the tooth profile symmetry line presents a certain deviation from its theoretical position along the gear width. When the grinding wheel grinds the workpiece gear, numerous contact traces are generated, and the grinding amount is equal on the same contact trace. In addition, the helical gear is grinding evenly at all of the points on the tooth flank without lead crowning modification. If the helical gear tooth flank with a lead crowning modification to ground, the grinding amount of the flank at different radii of the same cross section along lead direction is not equal, and the tooth flank twist is generated [25].

As shown in Figure 3, the tooth flank is a lead crowning modified flank. Since the grinding amount is the equal on the same contact trace, the grinding amount of $A_1$ is equal to $P_2$, and the grinding amount of $B_1$ is equal to $P_1$. Therefore, the grinding amount of each point on the cross section ($A_{1}P{B}_1$) is different. Similarly, the grinding amount of each point on the same cross section along the lead direction is also different, so the tooth flank is twisted. The smaller the number of teeth is and the larger the crowning amount is, the more prominent the tooth flank twist becomes.

3. Experimental Setup

The generating gear grinding experiments were conducted on a YW7232 CNC gear grinding machine (produced by Chongqing Machine Tool Co., Ltd., Chongqing, China), and its main structure is shown in Figure 4a. The maximum spindle speed of the machine is 8000 $\mathrm{rpm}$, the maximum module of machined gear is 8 $\mathrm{mm}$, and the batch gear grinding accuracy can reach level five (IS0 1328-1: 2013). In addition, this machine has additional functions: a tooth lead modification and tooth flank twist compensation. When the tooth flank twist compensation function is turned on, the grinding wheel is dressed to adjust its profile. The main parameters of the diamond wheel, grinding wheel and workpiece gear are shown in

Table 1. Parameters of diamond wheel, grinding wheel, and workpiece gear.

Diamond Wheel		Grinding Wheel		Workpiece Gear
Normal module ($m_{nd}$)	4 mm	Thread number ($Z_w$)	3	Number of teeth ($Z_g$)	35
Normal pressure angle (${\alpha }_{nd}$)	20°	Lead angle (${\beta }_w$)	2.559°	Normal module ($m_n$)	4 mm
Outside diameter ($D_d$)	123.50 mm	Normal pressure angle (${\alpha }_{nw}$)	20°	Normal pressure angle (${\alpha }_{ng}$)	20°
Effective diameter ($D_{nutz}$)	93 mm	Outside diameter ($D_w$)	279 mm	Helix angle (${\beta }_g$)	30°

, and the process parameters are shown in

Table 2. Gear griding parameters.

Rotation Speed of Grinding Wheel ($\mathit{n}$)	Radial Cutting Depth (${\mathit{a}}_{\mathit{p}}$)	Axial Feed Rate ($\mathit{f}$)
2800 $\mathrm{rpm}$	0.05 $\mathrm{mm}$	60 $\mathrm{mm}/\min$

. The amount of lead crowning is set as 20 $\mathsf{\mu }\mathrm{m}$, and the experimental gears are shown in Figure 5. The material of these gears is 45 # steel. The gears were subjected to high frequency quenching at temperatures of 800–840 °C, and the hardness of the gears after heat treatment is HRC40–45. Gears No. 1–No. 3 were continuously machined when the tooth flank twist compensation function was turned off. Additionally, Gears No. 4–No. 6 were continuously machined when the tooth twist compensation function was turned on.

The tooth flank deviation was detected using the gear measurement center JE32 (produced by Harbin Jingda Measuring Instruments Co., Ltd., Harbin, China), as shown in Figure 4b. The detection points of both the left and right tooth flanks were $9\times 5=45$, and the detected results of NO. 1 gear are shown in Figure 5b.

After each gear was ground, the grinding wheel was removed and its profile was measured using a laser profilometer LJ-V7000 (produced by Keyence Co., Ltd., Osaka, Japan), as shown in Figure 4c.

4. Results and Discussion

4.1. Machining Error a Test Generating Gear Grinding

Figure 6a shows the coordinate systems of the grinding wheel surface measurement. $S_4 \left(O_4-x_4, y_4, z_4\right)$ and $S_5 \left(O_5-x_5, y_5, z_5\right)$ are the fixed coordinate systems of the laser profilometer and the grinding wheel, respectively. $H$ is the transverse distance between $S_4 \left(O_4-x_4, y_4, z_4\right)$ and $S_5 \left(O_5-x_5, y_5, z_5\right)$. Using the grinding wheel profile data corresponding to Gears No. 1 and No. 4, the single tooth profiles were plotted, as shown in Figure 6b. It can be seen that the pressure angle of the grinding wheel had been altered after the second dressing.

Figure 6c illustrates the forming principle of helicoidal surface for the grinding wheel. The point $M$ is on the axis cross-section $YOZ$, and $OM$ is the starting line of right tooth flank. $u$ is the parameter to determine the position of point $M$, and $\alpha$ is the angle between $OM$ and the end cross-section. Therefore, the position vector and normal vector of the right tooth flank for the grinding wheel can be expressed as follows:

$$\begin{array}{c}{r}_w^{\prime }\left(u,\theta \right)={\left[-ucos\alpha sin\theta , ucos\alpha cos\theta ,usin\alpha +p\theta ,1\right]}^{\mathrm{T}}\end{array}$$

(10)

$$\begin{array}{c} n_w^{\prime }\left(u,\theta \right)={\left[pcos\alpha cos\theta +ucos\theta sin\theta sin\theta ,pcos\alpha sin\theta -ucos\alpha sin\alpha cos\theta ,uco{s}^2\left(\alpha \right)\right]}^{\mathrm{T}}\end{array}$$

(11)

where $p$ is the spiral parameter of the grinding wheel. By substituting the grinding wheel surface data collected in experiment No. 1 and the flank parameters into Equations (9) and (10), the complete grinding worm surface can be derived, as shown in Figure 6d.

According to the mathematical model of generating gear grinding in Section 2.1, the theoretical tooth flank of Gear No. 1 can be calculated, as shown in Figure 7a. The deviation of each point is shown in

Table 3. Theoretical deviations.

Profile Direction	Lead Direction
	1	2	3	4	5	6	7	8	9
Right tooth flank normal deviation (unit: $\mathsf{\mu }\mathrm{m}$)
A	−18.53	−11.92	−6.14	−2.26	−0.42	−0.11	−1.69	−4.63	−9.78
B	−14.56	−8.07	−3.95	−1.03	0.00	−0.66	−3.21	−7.64	−13.08
C	−10.28	−4.98	−1.90	−0.16	−0.33	−3.03	−5.76	−11.39	−18.91
D	−6.11	−2.24	−0.42	−0.11	−1.70	−5.20	−9.81	−16.87	−25.83
E	−2.24	−0.27	−0.21	−1.71	−5.21	−10.61	−16.89	−25.85	−36.72
Left tooth flank normal deviation (unit: $\mathsf{\mu }\mathrm{m}$)
E	−37.64	−26.63	−17.52	−11.11	−5.56	−1.91	−0.28	−0.20	−2.01
D	−26.61	−17.50	−10.29	−5.55	−2.91	−0.16	−0.32	−2.02	−5.74
C	−19.58	−11.90	−6.13	−2.25	−0.42	−0.11	−1.69	−4.64	−9.79
B	−13.63	−8.06	−3.48	−0.79	0.00	−0.88	−3.66	−7.65	−13.99
A	−10.26	−4.97	−1.89	−0.16	−0.33	−2.03	−5.77	−11.40	−17.88

. The measured tooth flank of Gear No. 1 is shown in Figure 7b, and the deviation of each point is shown in

Table 4. Measured deviations.

Profile Direction	Lead Direction
	1	2	3	4	5	6	7	8	9
Right tooth flank normal deviation (unit: $\mathsf{\mu }\mathrm{m}$)
A	−33.34	−28.52	−24.08	−16.89	−9.52	−8.17	−9.42	−10.64	−13.36
B	−27.87	−23.32	−18.35	−13.05	−7.45	−8.25	−9.95	−11.07	−14.66
C	−22.26	−17.63	−12.73	−7.26	−5.41	−6.31	−10.34	−11.27	−17.78
D	−14.46	−10.89	−8.88	−5.97	−1.83	−9.53	−14.27	−21.41	−27.02
E	−7.31	−6.52	−5.81	−4.51	−3.72	−12.37	−17.58	−26.03	−36.51
Left tooth flank surface normal deviation (unit: $\mathsf{\mu }\mathrm{m}$)
E	−45.18	−32.16	−21.08	−18.17	−1.45	−0.82	−2.43	−5.31	−6.11
D	−37.12	−28.05	−19.91	−16.58	−0.31	−3.25	−4.50	−6.12	−8.88
C	−31.26	−25.65	−15.63	−12.56	−7.74	−5.92	−6.89	−7.78	−13.13
B	−28.07	−21.46	−14.37	−13.22	−10.11	−9.36	−11.65	−13.82	−15.62
A	−26.68	−19.37	−13.89	−12.96	−12.36	−11.03	−12.17	−15.35	−19.51

.

As can be seen from Table 3 and Table 4, the difference of the values between the left and right tooth flanks are within ±15 $\mathsf{\mu }\mathrm{m}$, and there is a certain regularity. In order to investigate the cause, a further analysis of the deviation values was required. The points on the reference circle along the lead direction were selected, and the difference between the theoretical tooth flank and the measured tooth flank were calculated, as shown in Figure 8a. The points on the middle of tooth width along profile direction were selected, and the difference between the theoretical tooth flank and the measured tooth flank were calculated, as shown in Figure 8b. In the same way, the grinding wheel surface and the corresponding theoretical tooth flank corresponding to Gear No. 4 are calculated. The difference curves are shown in Figure 9.

In Figure 8 and Figure 9, the deviation difference decreases gradually from the top to the bottom of the gear, and they are basically positive. The main reason for this is that the cutting force during initial grinding is large and it generates a cutting vibration which causes a larger machining error. As grinding wheel cuts into the gear, the cutting state tends to be stable, and the machining error reduces. Moreover, the deviation difference decreases from the tooth tip to the root, and they are negative near the tooth root. Thus, it can be concluded that the small curvature radius of tooth root leads to insufficient grinding.

4.2. Effect of Grinding Wheel Wear on Tooth Flank Deviation

The grinding wheel is a wearable part. As the number of grinding times increases, the abrasive particle on grinding wheel surface will fall off, resulting in surface passivation and profile changes. For precision gear machining, the grinding wheel wear could significantly affect the gears’ accuracy. The manufacturers usually dress the grinding wheel regularly based on the production experience. Frequent grinding wheel dressing not only reduces the processing efficiency, but it also accelerates the grinding wheel wear. In the gear grinding experiment, three gears were machined continuously with the tooth flank twist compensation function being turned off (Gears No. 1–No. 3) and turned on (Gears No. 4–No. 5). The grinding wheel was dressed twice in total, namely, when the tooth flank twist compensation function was off and when the tooth flank twist compensation function was on. The tooth flank detection results are shown in Figure 10. The maximum value and root mean square (RMS) of tooth flank deviation are shown in Figure 11.

As can be seen from Figure 11a, the maximum and RMS values of the tooth flank deviation for Gears No. 1 to No. 3 both increased. The increased rate of the maximum and RMS values are 2.6% and 3.2% (left tooth flank) and 2.7% and 4.1% (right tooth flank), respectively.

At the same time, the maximum and RMS values of tooth flank deviation for Gears No. 4–No. 6 also increased as shown in Figure 11b. The increased rate of the maximum and RMS values are 2.6% and 3.9% (left tooth flank) and 2.8% and 4.2% (right tooth flank), respectively.

The grinding wheel wear leads tooth flank deviation to increase the uniformly, and the effect on the left and right tooth flanks is the same. It means that the dressing cycle of the grinding wheel can be estimated according to the tooth flank accuracy requirements.

4.3. Effect of Tooth Flank Twist Compensation

Nowadays, the tooth flank twist compensation method of most of the gear grinding machine is to set compensate for the amount of it according to the tooth profile angle error. Then, it is achieved by adjusting the pressure angle of the grinding wheel to achieve a certain degree of tooth flank twist compensation effect. The gear grinding machine in this study also adopted this method. Tian et al. [26] established a tooth flank twist compensation method based on a flexible EGB. The motion compensation amount for each axis (X axis, Y axis, Z axis, B axis, and C axis) was calculated based on the generating gear grinding model and inverse kinematic theory, and then, it compensates for them in the EGB. The EGB compensation principle is shown in Figure 12, and the motion compensation amount can be calculated as follows:

$$\begin{array}{c}\{\begin{array}{c}\mathsf{\Delta }{\phi }_B={\phi }_{BI}-{\phi }_{BT}\\ \mathsf{\Delta }{\phi }_C={\phi }_{CI}-{\phi }_{CT}\\ \mathsf{\Delta }{F}_X=F_{XI}-F_{XT}\\ \mathsf{\Delta }{F}_Y=F_{YI}-F_{YT}\\ \mathsf{\Delta }{F}_Z=F_{ZI}-F_{ZT}\end{array}\end{array}$$

(12)

where ${\phi }_{BI}$, ${\phi }_{CI}$, $F_{XI}$, $F_{YI}$, $F_{ZI}$ denote the motion of each axis for the ideal modified tooth flank; ${\phi }_{BT}$, ${\phi }_{CT}$, $F_{XT}$, $F_{YT}$, $F_{ZT}$ represent the motion of each axis for the twisted tooth flank.

There are two evaluation indicators of the tooth flank twist: (1) the maximum twist amount $M_t\left(=\delta {\epsilon }_{max}-\delta {\epsilon }_{min}\right)$; (2) the crowning evenness ratio $C_r\left(=\delta {\epsilon }_{min}/\delta {\epsilon }_{max}\right)$. $\delta {\epsilon }_{max}$ and $\delta {\epsilon }_{min}$ indicate the largest and smallest tooth flank deviations, respectively. If the maximum twist amount tends to 0 and the crowning evenness ratio tends to 1, it means that the gear has a twist-free tooth flank. The left tooth flank of No. 4 gear is shown in Figure 13a. The maximum twist amount is $M_{t1}=11.6$, and crowning evenness ratio is $C_{r1}=0.71$. The left tooth flank calculated by the EGB compensation method is shown in Figure 13b. The maximum twist amount is $M_{t2}=3.6$, and crowning evenness ratio is $C_{r2}=0.91$. It is obvious that $M_{t1}>M_{t2}$ and $C_{r1}<C_{r2}$, indicating the EGB compensation method, is better than the traditional compensation method is.

5. Conclusions

A novel method is presented in this paper to investigate the machining error a test generating gear grinding based on the mapping relationship between the grinding wheel surface and the machined tooth flank. The mathematical model for generating gear grinding is established. The laser profilometer is used to collect the grinding wheel surface after gear grinding, which avoids interference by the other error factors. In addition, the tooth flank twist compensation effect of the traditional compensation method and the EGB compensation method are analyzed. The contributions and advantages are concluded as follows:

(1)

Compared with the theoretical tooth flank, the deviation difference from the gear top to the bottom is positive, and it decreases gradually, indicating that the cutting state tends to be stable as the grinding wheel cuts in. The deviation difference also decreases from the tooth tip to the root, but it is negative at the tooth root, suggesting that the cutting volume at the tooth root is insufficient.

(2)

With the increase in the grinding wheel grinding times, the maximum and RMS values of the tooth flank deviation increase by nearly 3% and 4%, respectively.

(3)

The two indexes of the tooth flank twist show that the compensation effect of the EGB compensation method is better than that of the traditional compensation method.

The significance of this paper is to develop the direct relationship between the grinding wheel surface and the machined tooth flank. For future research, the error characteristics of gear grinding can be considered for adjusting the position of the grinding wheel in the CNC system to improve gear accuracy.