Geometric error detection of curve-face gear pairs

Abstract

A new type of gear pair, the curve-face gear pair, composed of a curve-face gear and a non-circular gear with mutual engagement, was proposed to achieve variable ratio transmission of motion and power between the intersecting axes. Geometric error detection is required to evaluate the grade of the manufactured gears. Due to the complexity of the curve-face gear, direct detection has been conducted in a very limited way. In this article, a measurement method aimed at curve-face gear pair artifacts is presented based on the computer numerical control gear measuring center. By comparing the measured coordinate data of the surface points with the corresponding theoretical data, various errors such as tooth profile error and pitch deviation can be obtained, and accuracy grades are evaluated with reference to accuracy standards for cylindrical gear and bevel gear. The developed method is simple and robust without requiring a special measuring device; hence, it can be applied for the industrial practice as a means for measuring the tooth profile and pitch deviations which cannot be measured by conventional methods.

Introduction

Gears are crucial components for modern precision machinery as a means for power transmission mechanism. Face gear drive is composed of a cylindrical gear and a bevel gear meshing with each other, which has unique advantage in heavy-duty and high-speed occasion such as helicopter transmission system. ¹ Curve-face gear pair is proposed based on the face gear pair, generated by replacing cylindrical gear in face gear pair with a non-circular gear. It is composed of a non-circular gear and a curve-face gear in order to realize variable ratio transmission between the intersecting axes. ² The relevant researches of curve-face gear, especially the geometric error detections, mainly draw lessons from the existing cylindrical gear and face gear.

For gear drives, the requirements are different in various machineries, but main requirements such as working accuracy, transmission efficiency, and load distribution uniformity are all related to gear accuracy.^3,4 Pitch deviation and tooth profile error are important factors influencing gear accuracy. A lot of researches on measurement have been done, and two main methods are widely used to measure the errors, which are specialized measuring instruments, such as gear circular pitch measuring instrument to measure pitch deviation and involute tester to measure profile error, and universal measuring instruments, such as gear measuring center and coordinate measuring machine.^5,6 Nowadays, the researches on geometric error detection mainly focused on the following. (1) The development of novel measuring device using new principle and structure.^7–9 Some novel measuring device are developed, such as a transmission error tester for face gear based on single-flank rolling principle, which can inspect several errors such as pitch deviation and transmission error. ⁷ Tang et al. ⁸ developed a gear measuring using double-flank rack probe (DFRP) method, which can be used to measure pitch and profile deviations on the left and right flanks simultaneously. (2) New measurement and evaluation methods of geometric error using universal measuring instruments.^10–13 For example, Guenther ¹⁰ proposed a method to evaluate runout deviation at bevel gears based on pitch measurements using coordinate measuring machines. Sánchez et al. ¹¹ introduced geometric principles for analyzing hypoid gears in coordinate measuring machines and obtained a method to trace the tooth shape of hypoid gears. Suh et al. ¹² measured spiral bevel gear tooth using a coordinate measuring machine and obtained profile error applying profile error measurement algorithm. For detection of bevel gear and cylindrical gear, there exist a variety of equipment and standards around the world,^14–16 while there is not any specialized equipment or standard for the detection of curve-face gear. Thus, gear measuring center is used to measure the tooth flank, and a method to obtain and evaluate pith deviation and profile error using measured points is developed in this article.

Curve-face gear is a new type of gear; in order to verify the practicability and correctness of the design, the curve-face gear pair was processed by three different ways, that is, five-axis computer numerical control (CNC) machining, additive manufacturing, and composite machining (combining additive manufacturing and five-axis CNC machining).^17–18 Finally, a kind of pitch deviation, profile error detection, and evaluation method for curve-face gear pair artifacts are put forward based on the P26 automatic CNC-controlled gear measuring center from Klingelnberg of German, and related researches are carried out.

Design of curve-face gear pair

The curve-face gear pair drive is composed of a curve-face gear and a non-circular gear with mutual engagement. During the process of engagement, pitch curves of curve-face gear and non-circular gear do pure rolling with each other. As shown in Figure 1, at the initial moment, pitch curves of the curve-face gear pair contact at point $P_0$ ; after a certain time, pitch curves of the curve-face gear pair contact at point $P_1$ . During this time, non-circular gear turns ${\phi }_1$ and curve-face gear turns ${\phi }_2$ , where arc $\widehat{P_{0(1)}{P}_1}$ and arc $\widehat{P_{0(2)}{P}_1}$ are equal in length.

Figure 1.

Pitch curves of the curve-face gear pair. 1: non-circular gear; 2: curve-face gear.

Pitch curves of the curve-face gear pair

Non-circular gear is used to transmit inhomogeneous motion between two shafts, which has many advantages than other mechanism. The most used non-circular gears are ellipse gear, eccentric gear, and Pascal spiral gear. In this article, the pitch curve of non-circular gear is elliptic curve.

According to the spatial meshing theory and coordinate relation, the complex pitch curve of curve-face gear can be deduced from the readily available non-circular gear. ² The equation of the pitch curve of curve-face gear can be expressed as follows

$$\left[\begin{array}{c}x\hfill \\ \multicolumn{1}{c}{y}\\ \multicolumn{1}{c}{z}\end{array}\right]=\left[\begin{array}{c}-\frac{n_2}{2\pi }\cos \left({\phi }_2\right)\underset{0}{\overset{\frac{2\pi }{n_1}}{\int }}r\left({\phi }_1\right)d{\phi }_1\hfill \\ \multicolumn{1}{c}{\frac{n_2}{2\pi }\sin \left({\phi }_2\right)\underset{0}{\overset{\frac{2\pi }{n_1}}{\int }}r\left({\phi }_1\right)d{\phi }_1}\\ \multicolumn{1}{c}{r\left(0\right)-r\left({\phi }_1\right)}\end{array}\right]$$

(1)

where $r({\phi }_1)=a(1-e^2)/1-e\cos (n_1{\phi }_1)$ is the polar radius of pitch curve of non-circular gear, a is the semi-major axis of ellipse curve, e is the eccentricity of ellipse, $n_1$ is the order of ellipse, ${\phi }_1$ is the polar angle of ellipse, ${\phi }_2$ is the turned angle of curve-face gear when non-circular gear turns ${\phi }_1$ , and $n_2$ is alternation number of the teeth of curve-face gear in the range of $0-2\pi$ .

Tooth profile of the curve-face gear pair

Tooth profile of non-circular gear can be obtained by generating motion of cylindrical gear on pitch curve of non-circular gear. Similarly, tooth profile of curve-face gear can be obtained by generating motion of non-circular gear on pitch curve of curve-face gear. Since, each tooth profile of non-circular gear is different, and its surface equation is complex. It is very complicated to derive surface equation of curve-face gear from surface equation of non-circular gear. Assuming that there is a cylindrical gear engaging both of the non-circular gear and the curve-face gear at the same time, then cylindrical gear tooth profile can envelope the surface of curve-face gear tooth profile. The tooth profile of cylindrical gear is very simple; thus, this issue is simplified.

The tooth profile of curve-face gear was obtained by virtual machining with envelope method, while the gear shaper cutter is a standard involute cylindrical gear. The family of surfaces in the coordinate $f_2$ can be expressed as follows

$${\overrightarrow{r}}_2(u_k,{\theta }_k,{\phi }_1)=M_{2k}{\overrightarrow{r}}_k(u_k,{\theta }_k)$$

(2)

where ${\overrightarrow{r}}_k(u_k,{\theta }_k)$ is the equation of gear shaper cutter tooth profile, and $M_{2k}$ is the transformation matrix from coordinate $f_k$ of gear shaper cutter to coordinate $f_2$ of curve-face gear.

According to the conjugate relationship of tooth surfaces between the curve-face gear and the gear cutter, the projection at the normal line of the relative velocity vector of tooth surface in the mesh point equals to zero. Convert the mesh point from the curve-face gear’s coordinate system to the gear cutter; thus, the meshing equation of curve-face gear can be obtained. ¹⁹

The equation of meshing can be expressed as follows

$$\begin{array}{c}\multicolumn{1}{c}{f(u_k,{\theta }_k,{\phi }_1)={\overrightarrow{N}}^{\left(s\right)}·{{\overrightarrow{v}}_{2k}}^{\left(s\right)}}\\ \multicolumn{1}{c}{=\cos (-{\phi }_1-{\phi }_k+{\theta }_k+{\alpha }_k)}\\ \multicolumn{1}{c}{\times (u_k+R)-i_{12}\times (r_{\mathrm{bk}}+A\times }\\ \multicolumn{1}{c}{\cos (-{\phi }_1-{\phi }_k+{\theta }_k+{\alpha }_k+\lambda ))}\end{array}$$

(3)

where ${\overrightarrow{N}}^{(s)}$ is the normal vector of gear cutter tooth surface in coordinate S, ${\overrightarrow{v}}_{2k}^{(s)}$ is the relative velocity of gear cutter tooth surface and curve-face gear in coordinate S, ${\alpha }_u$ is the pressure angle of gear cutter, ${\alpha }_k$ is the pressure angle on any point k, $r_{\mathrm{bk}}$ is the radius of gear base circle of gear cutter, $L_1=\sqrt{r^2({\phi }_1)+r_k^2-2r_{k}r({\phi }_1)\sin \mu }$ , $r_k=r_{\mathrm{bk}}/\cos {\alpha }_k$ , $\lambda =\arccos (L_1^2+r^2({\phi }_1)-r_k^2/2L_{1}r({\phi }_1))$ , and ${\theta }_k=inv({\alpha }_k)$ .

Finally, according to the principle of gear engagement, the equation of the surface of curve-face gear can be expressed as follows

$${\overrightarrow{r}}_2\left({\theta }_k,{\phi }_1\right)=\left[\begin{array}{l}{L}_1\sin \lambda \sin {\phi }_2-A\cos {\phi }_2+r_{bk}\sin {\phi }_2\left(\sin {\varphi }_s+{\theta }_k\cos {\varphi }_s\right)\\ -L_1\sin \lambda \cos {\phi }_2-A\sin {\phi }_2-r_{bk}\cos {\phi }_2\left(\sin {\varphi }_s+{\theta }_k\cos {\varphi }_s\right)\\ r\left(0\right)-L_1\cos \lambda -r_{bk}\left(\cos {\varphi }_s+{\theta }_k\sin {\varphi }_s\right)\end{array}\right]$$

(4)

where $A=r_{\mathrm{bk}}+L_1\cos ({\varphi }_s-\lambda )/i_{21}\cos {\varphi }_s$ , ${\varphi }_s=\xi +{\phi }_1-{\theta }_k-{\theta }_{\mathrm{ok}}$ , and $\xi =\pi /2+\eta -{\phi }_1-\mu$ ; $\xi$ is the rotation angle of gear cutter which rotates around its own axis.

Measuring of curve-face gear pair

The geometric error detection of gear refers to the deviations of the theory model and measured points or curves which are obtained by the detecting instruments. For the normal cylindrical gears, bevel gears, and so on, there already exist mature detection methods, testing instruments, and quality evaluation standards, around the world.^20,21 For this curve-face gear pair, the contour scanning software of the German Klingelnberg P26 automatic CNC-controlled gear measuring center was used to measure the coordinates of tooth profile points of the gear pair. Three curve-face gear pairs processed by five-axis CNC machining, additive manufacture, and composite machining were measured in this article. The basic parameters of these curve-face gear pairs are shown in Table 1. The measurement coordinate system is shown in Figure 2.

Table 1.

Basic parameters of the curve-face gear pair.

Number of teeth of the non-circular gear, $z_1$	18
Number of teeth of the curve-face gear, $z_2$	36
Module, m (mm)	4
Order of ellipse, $n_1$	2
Alternation number of the curve-face gear in one cycle, $n_2$	4
Eccentricity of ellipse, e	0.1
Inner radius of the curve-face gear, $R_1$ (mm)	65
Outer radius of the curve-face gear, $R_2$ (mm)	78

Figure 2.

Measurement of the gear artifacts.

The method to establish measurement coordinate system is as follows:

1. Acquisition of more than three points on end face of non-circular gear and curve-face gear artifacts. Obtain a plane by these points as the datum plane $X_M{O}_M{Y}_M$ and select the normal direction of it as direction of $Z_M$ axis.

2. Acquisition of more than four points on cylindrical surface of shaft hole of non-circular gear and curve-face gear. Obtain a circle by these points and select the projection of the circle center on datum plane as the origin $O_M$ of the coordinate system.

3. Acquisition of a point on semi-major of non-circular gear and crest of curve-face gear, project this point on datum plane, and name it P. Select the direction of $O_{M}P$ as direction of $X_M$ axis. Then, $Y_M$ axis can be determined according to the right-hand rule.

In order to facilitate the measurement of the whole tooth flank, it is necessary to plan measurement path. To improve measurement efficiency, reduce alignment and calibration time; that is, keep the probe position invariant along tooth width direction and the probe moving along tooth depth direction relative to tooth surface in each measurement process. After one circle is completed, change probe position along tooth width direction and continue the next circle. The measurement path is shown in Figure 3.

Figure 3.

Measurement path.

As can be seen in Figure 6, during the measurement, the gear artifacts rotate around $Z_1$ axis with workbench, and then, the CNC system makes the probe move along the specified direction according to feedback signal from sensor in probe, so that the probe can remain in contact with the tooth surface. At the same time, the CNC system records the absolute position of each axis in real time and converts it to coordinate. For example, in the measurement process of curve-face gear, $X_2$ and $Y_2$ axes remain unchanged and the probe moves along $Z_2$ axis. After one circle is completed, the probe moves $\Delta X$ along $X_2$ axis and continues the next circle.

Pitch deviation

Single pitch deviation is the maximum absolute value of difference between the actual pitch and the theoretical pitch, which reflects the uniformity of teeth in the pitch curve and mainly affects the smoothness of gear running. Total accumulative pitch deviation is the maximum absolute value of the difference between actual pitch and nominal pitch of any two corresponding flanks, which mainly affect the kinematic accuracy of gear pair. The pitch of curve-face gear and the non-circular gear is defined as the arc length of the pitch curve between two adjacent corresponding flanks, as shown in Figure 4.

Figure 4.

Definition of pitch deviation of curve-face gear pair: (a) non-circular gear and (b) curve-face gear.

The theoretical pitch of the non-circular gear and the curve-face gear pair is the same

$$P_{t1}=P_{t2}=\pi \mathrm{m}=12.5664\mathrm{mm}$$

(5)

In accordance with the definition of pitch, the actual pitch can be confirmed as the arc length between two pitch points on corresponding flank, where pitch point is the intersection of pitch curve and the fitted curve of the measured coordinate points, as shown in Figure 5.

Figure 5.

Schematic representation of the calculation of pitch deviations.

It is easy to derive coordinates of pitch points based on measured points and equations of the pitch curves. With these coordinates, ${\phi }_{11}$ and ${\phi }_{12}$ can be determined according to the polar radius $r({\phi }_1)$ of pitch curve of non-circular gear. Similarly, ${\phi }_{21}$ and ${\phi }_{22}$ can be determined according to the pitch curve equation of curve-face gear, as shown in formula (1). The actual pitch can be obtained by calculation of curvilinear integral of pitch curve between the two adjacent corresponding pitch points, which can be expressed as follows

$$P{\prime }_{1i}=\underset{{\phi }_{11}}{\overset{{\phi }_{12}}{\int }}\sqrt{r^2({\phi }_1)+{r\prime }^2({\phi }_1)}d{\phi }_1$$

(6)

$$P{\prime }_{2i}=\underset{{\phi }_{21}}{\overset{{\phi }_{22}}{\int }}\sqrt{{x\prime }^2({\phi }_2)+{y\prime }^2({\phi }_2)+{z\prime }^2({\phi }_2)}d{\phi }_2$$

(7)

where i is the tooth number, $P{\prime }_{1i}$ is the actual pith of non-circular gear, $P{\prime }_{2i}$ is the actual pith of curve-face gear, $r({\phi }_1)$ is the polar radius of pitch curve of non-circular gear, $\mathrm{r}\prime ({\phi }_1)$ is the derivative of $r({\phi }_1)$ with respect to ${\phi }_1$ , ${\left[\begin{array}{c}x({\phi }_2)y({\phi }_2)z({\phi }_2)\hfill \end{array}\right]}^T$ is the equation of the pitch curve of curve-face gear, and $\begin{array}{ccc}x\prime ({\phi }_2)& y\prime ({\phi }_2)& z\prime ({\phi }_2)\end{array}$ is the derivative of $\begin{array}{c}x({\phi }_2)y({\phi }_2)z({\phi }_2)\hfill \end{array}$ with respect to ${\phi }_2$ .

According to the definitions in related standards, the value of pitch deviations can be calculated through equations in Table 2.

Table 2.

Equations of pitch deviation.

Individual pitch deviation	$f_{\mathrm{pi}}=P{\prime }_i-P_t$
Single pitch deviation	$f_p=max|f_{\mathrm{pi}}|$
Individual accumulative pitch deviation	$F_{\mathrm{pk}}=\sum _{i=1}^k{f}_{\mathrm{pi}}$
Total pitch accumulative error	$F_p=max{F}_{\mathrm{pi}}-min{F}_{\mathrm{pi}}$

Profile error

The profile error $f_f$ refers to the projection of the deviate distance between the measured tooth surfaces from the theoretical point on the unit normal vector at any given point of the tooth surface. The location of any point on the surface of tooth flank is to be measured and compared to the specified coordinate, and then, the deviation is calculated.

In order to facilitate the measurement of the whole tooth profile, grid division of the tooth surface is needed. The number relates to the sample accuracy of tooth flank, while too much grid will take excessive measurement time. A grid of $5\times 7$ points (5 places root-to-tip, 7 places toe-to-heel) is used in this article. The grid boundary was determined by the limitation of gear peak sharpening and undercut, that is, 10%–15% inward indentation, ²² as shown in Figure 6.

Figure 6.

Positions of grid points on one tooth.

The positions of reference points on different flanks vary with the spatial pitch curve of curve-face gear, but the same is that the reference point is at the tolerance diameter and it is the midpoint of the grid. For curve-face gear, tolerance diameter is the sum of inner radius and outer radius of the curve-face gear. In order to cover the whole tooth profile, Z coordinate of the reference point is the same as the intersection point of pitch curve and tooth flank. A point can be defined on an axial plane of the curve-face gear with the Z coordinate and radius of reference point, and then, define a grid with this point as the center. Each grid point is projected by rotation around the curve-face gear axis onto the theoretical tooth flank, see Figure 7. Thus, theoretical coordinates of grid points on tooth flank (including reference points) can be derived with Z coordinate, radius, and equation of tooth surface easily.

Figure 7.

Derivation of grid points.

The coordinates of actual measuring points were defined as the intersection of the normal vectors of the corresponding theoretical coordinates and the measured curve, as shown in Figure 8. Thus, in order to obtain the measured coordinates, the unit normal vectors should be worked out first. According to the equation of the surface of curve-face gear ${\overrightarrow{r}}_2$ , the unit normal vector ${\overrightarrow{e}}_n(x_i,y_i,z_i)$ of theoretical point $(x_i,y_i,z_i)$ was expressed as follows

$${\overrightarrow{e}}_n(x_i,y_i,z_i)=[a_i,b_i,c_i]=\frac{\frac{\partial {\overrightarrow{r}}_2}{\partial {\theta }_k}\times \frac{\partial {\overrightarrow{r}}_2}{\partial {\phi }_1}}{|\frac{\partial {\overrightarrow{r}}_2}{\partial {\theta }_k}\times \frac{\partial {\overrightarrow{r}}_2}{\partial {\phi }_1}|}$$

(8)

$$\frac{\partial {\overrightarrow{r}}_2}{\partial {\theta }_k}=\left[\begin{array}{l}\sin {\phi }_2[{r^{\prime }}_k\sin ({\phi }_1-\xi \pm {\theta }_k)\pm r_k{{\theta }^{\prime }}_k\cos ({\phi }_1-\xi \pm {\theta }_k)]\\ \cos {\phi }_2[{r^{\prime }}_k\sin ({\phi }_1-\xi \pm {\theta }_k)\pm r_k{{\theta }^{\prime }}_k\cos ({\phi }_1-\xi \pm {\theta }_k)]\\ -{r^{″}}_k\cos ({\phi }_1-\xi \pm {\theta }_k)\pm r_k{{\theta }^{\prime }}_k\sin ({\phi }_1-\xi \pm {\theta }_k)\end{array}\right]$$

(9)

$$\frac{\partial {\overrightarrow{r}}_2}{\partial {\phi }_1}=\left[\begin{array}{c}{{\phi }^{\prime }}_2\cos {\phi }_2[r_k\sin ({\phi }_1-\xi \pm {\theta }_k)+R\sin \lambda ]+\sin {\phi }_2[r_k(1-{\xi }^{\prime })\cos ({\phi }_1\\ \cdots -\xi \pm {\theta }_k)+R^{\prime }\sin \lambda +R{\lambda }^{\prime }\cos \lambda ]+{{\phi }^{\prime }}_2\sin {\phi }_2(u_k+R_2)\\ \cos {\phi }_2[r_k(1-{\xi }^{\prime })\cos ({\phi }_1-\xi \pm {\theta }_k)+R^{\prime }\sin \lambda +R{\lambda }^{\prime }\cos \lambda ]\\ \cdots -{{\phi }^{\prime }}_2\sin {\phi }_2[r_k\sin ({\phi }_1-\xi \pm {\theta }_k)+R\sin \lambda ]\\ r_k(1-{\xi }^{\prime })\sin ({\phi }_1-\xi \pm {\theta }_k)-R^{\prime }\cos \lambda +R{\lambda }^{\prime }\sin \lambda \end{array}\right]$$

(10)

Figure 8.

Schematic representation of the calculation of profile errors.

Finally, the coordinate $(x{\prime }_i,y{\prime }_i,z{\prime }_i)$ of corresponding measuring point can be derived from equation (9)

$$\{\begin{array}{c}\frac{x-x_i}{a_i}=\frac{y-y_i}{b_i}=\frac{z-z_i}{c_i}\hfill \\ \multicolumn{1}{c}{f(x,y,z)=0}\end{array}$$

(11)

where $f(x,y,z)=0$ represents the surface fitted from the measured points.

Thus, the profile error $f_f$ can be expressed as follows

$$f_f=\Delta d=\sqrt{{({x\prime }_i-x_i)}^2+{({y\prime }_i-y_i)}^2+{({z\prime }_i-z_i)}^2}$$

(12)

Calculation of deviations

The measurement coordinate system is established by the method introduced in section “Measuring of curve-face gear pair.” It is clear that the measurement coordinate system $S(X_M,Y_M,Z_M)$ does not coincide with the design coordinate system $S(X_D,Y_D,Z_D)$ . Thus, the rotating and shifting relation between the two systems cannot be avoided. The deviations are calculated by measured points and theoretical points, which should be expressed in the same coordinate system.

As shown in Figure 9, z represents the shift amount along $Z_D$ direction, and $\delta$ represents the rotation amount around $Z_D$ direction. The coordinate transformation from the measurement coordinate to design coordinate can be represented by matrix M as follows

$$M=\left[\begin{array}{cccc}\cos \delta \hfill & \sin \delta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-\sin \delta }& \cos \delta \hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & z\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

(13)

Figure 9.

Schematic representation of the coordinate systems.

The coordinate of measured points in design coordinate system can be expressed as follows

$${\left[\begin{array}{cccc}{X}_D\hfill & Y_D\hfill & Z_D\hfill & 1\hfill \end{array}\right]}^T=M·{\left[\begin{array}{cccc}{X}_M\hfill & Y_M\hfill & Z_M\hfill & 1\hfill \end{array}\right]}^T$$

(14)

Pitch deviation

Based on the calculation method of pitch deviation presented in Table 2, the values of pitch deviations of the curve-face gear pair artifacts can be worked out with the revised coordinate data. The single pitch deviation of non-circular gear artifacts is shown in Figure 10. The single pitch deviation of curve-face gear artifacts is shown in Figures 11–13.

Figure 10.

Pitch deviation of non-circular gear artifacts.

Figure 11.

Pitch deviation of five-axis CNC machined curve-face gear artifacts.

Figure 12.

Pitch deviation of additive manufactured curve-face gear artifacts.

Figure 13.

Pitch deviation of composite machined curve-face gear artifacts.

Since there are not any measurement and accuracy standards for non-circular gear and curve-face gear, ISO 1328-1:2013 (cylindrical gears-ISO system of flank tolerance classification), ISO 17485:2006 (bevel gears-ISO system of accuracy), and GB/T 11365:1989 (accuracy of bevel and hypoid gears (in Chinese)) are referred to evaluate the accuracy grades of the gear pairs.

With reference to ISO 1328-1:2013, single pitch tolerance of non-circular gear shall be calculated using equation (15). Total cumulative pitch tolerance of non-circular gear shall be calculated using equation (16) as follows

$$f_{pT}=\left(0.001d+0.4m_n+5\right){\left(\sqrt{2}\right)}^{(A-5)}$$

(15)

$$F_{\mathrm{pT}}=(0.002d+0.55\sqrt{d}+0.7m_n+12){\left(\sqrt{2}\right)}^{(A-5)}$$

(16)

where d is the reference diameter; for non-circular gear, choose the sum of major axis radius and minor axis radius of pitch curve as d; and A is the number of the required tolerance grade. The pitch deviation tolerance of non-circular gear in this size is presented in Table 3.

Table 3.

Pitch deviation tolerance of non-circular gear.

Accuracy grade	4	5	6	7	8	9	10	11
$f_{pT}$ (µm)	4.7	6.5	9	13	19	27	38	53
$F_{\mathrm{pT}}$ (µm)	14	20	28	39	55	78	111	157

So, values of pitch deviation and the corresponding accuracy grade of non-circular gear artifacts are presented in Table 4.

Table 4.

Accuracy grade of non-circular gear artifacts.

Machining method	Single pitch deviation $(\mathrm{\mu }\mathrm{m})$	Accuracy grade	Total cumulative pitch deviation $(\mathrm{\mu }\mathrm{m})$	Accuracy grade
Five-axis CNC machined	22.3	9	44.1	8
Additive manufactured	43.5	11	80.7	10
Composite machined	23.1	9	42.8	8

CNC: computer numerical control.

With reference to ISO 17485:2006, single pitch tolerance of curve-face gear shall be calculated according to equation (17). Total cumulative pitch tolerance shall be calculated according to equation (18)

$$f_{ptT}=\left(0.003d_T+0.3m_n+5\right){(\sqrt{2})}^{(B-4)}$$

(17)

$$F_{\mathrm{pT}}=(0.025d_T+0.3m_n+19){\left(\sqrt{2}\right)}^{(B-4)}$$

(18)

where $d_T$ is the tolerance diameter; for curve-face gear, the tolerance diameter is the sum of inner radius and outer radius of the curve-face gear; and B is the number of the required tolerance grade. The pitch deviation tolerance of curve-face gear in this size is presented in Table 5.

Table 5.

Pitch deviation tolerance of curve-face gear.

Accuracy grade	4	5	6	7	8	9	10	11
$f_{\mathrm{pT}}$ (µm)	6.5	9.5	13	19	27	37	53	74
$F_{\mathrm{pT}}$ (µm)	24	34	48	67	95	134	190	269

CNC: computer numerical control.

Same as non-circular gear, values of pitch deviation and corresponding accuracy grade are presented in Table 6.

Table 6.

Accuracy grade of curve-face gear artifacts.

Machining method	Single pitch deviation $(\mathrm{\mu }\mathrm{m})$	Accuracy grade	Total cumulative pitch deviation $(\mathrm{\mu }\mathrm{m})$	Accuracy grade
Five-axis CNC machined	39.8	10	79.9	8
Additive manufactured	119.7	− a	234.5	11
Composite machined	48.8	10	84.2	8

CNC: computer numerical control.

a

Pitch deviation beyond the maximum tolerance of grand 11.

Profile error

Similar to the pitch deviation, based on the calculation method of profile error introduced in equations (6)–(10), it is easy to work out the profile errors of curve-face gear artifacts with the revised coordinate data. One gear tooth is chosen as an example in this article. The profile errors on three curve-face gear artifacts are presented in Tables 7–9.

Table 7.

Profile error of five-axis CNC machined curve-face gear artifact.

		A	B	C	D	E	F	G
Right flank	1	14.2	13.5	16.4	8.2	12.6	8.0	14.8
	2	6.0	8.0	8.1	14.3	−4.0	5.9	8.5
	3	−7.1	11.2	−5.8	0.0	7.1	7.3	−7.7
	4	−4.7	6.5	8.1	−6.1	6.5	7.0	2.8
	5	7.5	−3.5	−4.8	6.3	−5.8	−6.7	1.9
Left flank	5	3.5	−4.3	5.3	−6.8	−5.7	4.6	−6.9
	4	5.6	5.8	12.6	10.3	2.8	−2.9	−4.4
	3	7.3	7.2	11.0	−5.9	4.3	−3.5	−2.9
	2	−3.5	−5.8	−6.6	9.3	−5.1	−1.9	5.4
	1	5.9	8.8	14.1	5.2	5.7	9.0	8.4

Table 8.

Profile error of additive manufactured curve-face gear artifacts.

		A	B	C	D	E	F	G
Right flank	1	74.3	73.6	69.4	34.9	−21.5	41.0	−54.8
	2	65.1	88.0	55.4	52.1	41.0	61.9	−46.6
	3	74.2	81.2	57.6	0.0	23.1	41.9	−32.2
	4	84.8	75.4	54.3	48.1	97.6	−48.2	12.8
	5	75.4	78.4	54.1	52.3	87.0	−22.6	−44.9
Left flank	5	77.9	77.3	77.1	58.2	52.6	−21.9	62.4
	4	83.3	67.5	78.9	38.7	−37.4	−47.7	−33.4
	3	82.3	65.1	81.8	74.4	−41.6	−45.0	−38.4
	2	70.7	70.2	59.3	−46.3	−34.4	42.2	80.7
	1	72.4	76.8	−36.9	63.1	−40.6	73.5	50.9

Table 9.

Profile error of composite machined curve-face gear artifacts.

		A	B	C	D	E	F	G
Right flank	1	19.7	19.1	18.5	18.4	17.6	18.4	17.4
	2	11.5	2.1	12.1	14.1	15.4	16.0	10.4
	3	10.3	1.1	8.5	0.0	12.3	9.2	−3.6
	4	8.1	−0.1	2.5	6.1	11.1	7.0	−12.2
	5	10.8	−2.2	8.7	−12.6	−5.8	−9.5	−16.8
Left flank	5	8.5	10.0	19.0	18.6	−12.5	−11.7	−16.1
	4	5.3	6.8	12.6	−10.6	6.1	5.8	4.8
	3	−0.2	−6.4	−4.8	−2.8	−9.4	−3.8	5.4
	2	2.0	4.4	−6.6	−13.4	−9.7	−4.9	4.5
	1	6.2	8.7	14.1	−16.4	8.4	11.5	9.6

In comparison with the numerical results, a simplified graphical output can provide a better overview of the course of deviations across the tooth flank. So, a kind of three-dimensional (3D) graph output of profile errors is plotted, as shown in Figures 14–16. In these figures, the theoretical tooth flank is represented by the grid in plane, and the nodes of grid represent theoretical points on tooth flank. The measured points deviate from the grid nodes, and deviations are illustrated by line segments vertical to grid plane, where the length of line segment represents the value of deviation. Thus, the grid connected by measured points can reflect actual tooth flank.

Figure 14.

3D graph output of five-axis CNC machined curve-face gear artifacts.

Figure 15.

3D graph output of additive manufactured curve-face gear artifacts.

Figure 16.

3D graph output of composite machined curve-face gear artifacts.

With reference to GB/T 11365:1989, profile error tolerance of curve-face gear is presented in Table 10. So, values of profile error and the corresponding accuracy grades of curve-face gear artifacts are presented in Table 11.

Table 10.

Profile error tolerance of curve-face gear.

Accuracy grade	7	8	9	10	11	12
A	0.84	1.34	2.1	3.35	5.3	8.4
C	6.7	8.4	13.4	21	34	53
$f_c=0.84(A{m}_n+\mathrm{Bd}+C)$ $B=0.0125A$	10	14	21	34	54	85

Table 11.

Accuracy grade of non-circular gear artifacts.

Machining method	Profile error $(\mathrm{\mu }\mathrm{m})$	Accuracy grade
Five-axis CNC machined	16.4	9
Additive manufactured	97.6	− a
Composite machined	19.7	9

CNC: computer numerical control.

a

Profile error beyond the maximum tolerance of grand 12.

For further analysis of profile error difference of curve-face gear processed by three methods, different parts of the tooth flank are selected for analysis. As shown in Figure 6, six lines are selected. Lines 1–3 are placed toe-to-heel, representing districts around tip, intermediate, and root. Lines 4–6 are placed root-to-tip, representing districts around outer flank, medial flank, and inner flank. Average profile errors of these lines are presented in Figures 17 and 18.

Figure 17.

Average profile error of lines on right flank.

Figure 18.

Average profile error of lines on left flank.

As can be seen from Figures 14–18, profile error of the curve-face gear processed by additive manufactured method is much larger than the other two methods; profile error of the curve-face gear artifact processed by composite machining method is slightly larger than the one processed by five-axis CNC machining. Since the tooth flank of curve-face gear is complex spatial surface, step effect will be produced in the process according to the basic principle of additive manufacturing. This leads to relatively large profile error in curve-face gear artifact processed by additive manufacturing. For composite machined curve-face gear artifact, accuracy is mainly guaranteed by five-axis CNC machining in the second step. The secondary clamping, which produces new error, leads to a slightly larger profile error than five-axis CNC machining.

Conclusion

In this article, a method based on gear measuring center has been developed to detect and evaluate geometric errors of curve-face gear artifacts processed by three different ways:

1. A method to find out the actual measured points that corresponds to a theoretical point has been put forward. By fitting the measured points into a curved surface, the normal vector of theoretical point intersects with the fitted curved surface, and the intersection can be regarded as the actual measured points.

2. Pitch deviations of curve-face gear pair artifacts processed by three methods are derived and accuracy grade of pith deviation is evaluated with reference to the accuracy standards for cylindrical gear and bevel gear.

3. Profile errors of curve-face gear artifacts are derived, 3D graph is plotted to provide a better overview of the course of deviations across the tooth flank, and accuracy grade is evaluated with reference to the accuracy standard for bevel gear.

4. Overall, geometric error of curve-face gear artifact processed by additive manufactured method is much larger than the other two methods because of step effect; and the geometric error of the curve-face gear artifact processed by composite machining method is slightly larger than the one processed by five-axis CNC machining owing to secondary clamping.