Tooth surface Hobbing analysis for unconventional gears with ultrasonic-assisted motion

2.1. Tooth surface profile of normal gear

With the development and application of gear pair, various processing methods, such as milling, hobbing and so on, have been applied. Based on the simulation processing method proposed by F. L. Litvin [1], the surface hobbing process of normal cylindrical gear can be shown as Fig. 1.

Fig. 1 shows the conjugate generating process of tooth surface. $O_1$-$X_1{Y}_1{Z}_1$, $O_2$-$X_2{Y}_2{Z}_2$ and $O_3-X_3{Y}_3{Z}_3$ are fixed coordinates on cylindrical gear, hypothetical generating gear and gear hob respectively. $O_1^{\mathrm{\text{'}}}$-$X_1^{\mathrm{\text{'}}}{Y}_1^{\mathrm{\text{'}}}{Z}_1^{\mathrm{\text{'}}}$, $O_2^{\mathrm{\text{'}}}$-$X_2^{\mathrm{\text{'}}}{Y}_2^{\mathrm{\text{'}}}{Z}_2^{\mathrm{\text{'}}}$ and $O_3^{\mathrm{\text{'}}}$-$X_3^{\mathrm{\text{'}}}{Y}_3^{\mathrm{\text{'}}}{Z}_3^{\mathrm{\text{'}}}$ are movable coordinates on cylindrical gear, hypothetical generating gear and gear hob respectively. ${\beta }_3$ is the rotating angle of movable coordinate $O_3^{\mathrm{\text{'}}}$-$X_3^{\mathrm{\text{'}}}{Y}_3^{\mathrm{\text{'}}}{Z}_3^{\mathrm{\text{'}}}$ with different meshing point. In Fig. 1(b), $O_3\left(O_3^{\text{'}}\right)$ keeps a constant distance away from $O_2\left(O_2^{\text{'}}\right)$ with a distance $O_3{O}_2=E$. In the generating process of tooth surface of cylindrical gear, the gear hob is coupled rotating around $O_3{Z}_3$ and $O_1{Z}_1$, the tooth flank of gear hob is always the same with tooth flank of hypothetical generating gear (inner gear) and equivalent external gear at point $p$ with coordinate transformation by $O_3^{\mathrm{\text{'}}}$-$X_3^{\mathrm{\text{'}}}{Y}_3^{\mathrm{\text{'}}}{Z}_3^{\mathrm{\text{'}}}$ to $O_2^{\mathrm{\text{'}}}$-$X_2^{\mathrm{\text{'}}}{Y}_2^{\mathrm{\text{'}}}{Z}_2^{\mathrm{\text{'}}}$. The mathematical equation $r_p^{\left(3\mathrm{\text{'}}\right)}\left({\theta }_p\right)$ for tooth profile at point $p$ of gear hob can be derived by the tooth profile of hypothetical inner gear or equivalent external gear, shown as:

where $p$ is the common contact point on gear hob, equivalent external gear and hypothetical generating gear, ${\theta }_{op}$ is the starting angle of tooth profile ${\theta }_{op}=\pi /2z-inv{\alpha }_o$, ${\theta }_p$ is the range of tooth profile angle, $r_{bp}$ is the basic radius.

According to the pure rolling engagement of those three tooth profiles and conversion matrix $M_{1\mathrm{\text{'}}2\mathrm{\text{'}}}$ of the coordinate $O_2^{\mathrm{\text{'}}}$-$X_2^{\mathrm{\text{'}}}{Y}_2^{\mathrm{\text{'}}}{Z}_2^{\mathrm{\text{'}}}$ to the coordinate $O_1^{\mathrm{\text{'}}}$-$X_1^{\mathrm{\text{'}}}{Y}_1^{\mathrm{\text{'}}}{Z}_1^{\mathrm{\text{'}}}$, the tooth profile of target gear can be proposed as:

Eq. (3) illustrates the mathematical equation for target gear, $D_Y$ and $D_X$ are the translation distance of movable coordinate $O_2^{\mathrm{\text{'}}}$-$X_2^{\mathrm{\text{'}}}{Y}_2^{\mathrm{\text{'}}}{Z}_2^{\mathrm{\text{'}}}$ with meshing from $p$ (rotating angle of cylindrical gear is ${\theta }_1$) to $p\text{'}$ (rotating angle of cylindrical gear is ${\theta }_1+\mathrm{\Delta }{\theta }_1$). As described in Fig. 2(b), gear hob always keeps the distance $E$ away from $O_2\left(O_2^{\mathrm{\text{'}}}\right)$, so $D_Y$ and $D_X$ can be expressed by the component of arc length $E{\beta }_3$ in the $X$ or $Y$ directions. Compared with Eq. (2), Eq. (3) has the same form for tooth profile, the target gear is cylindrical gear with the constant radius $r_2$.

Fig. 1Theoretical presentation of normal cylindrical gear

2.2. Surface equation of non-circular gear

Based on the conventional Hobbing method of normal cylindrical gear with constant cutting steps, the simulation manufacturing process for non-circular gear tooth profile has been arranged as shown in Fig. 2.

Fig. 2Simulated hobbing of non-circular gear with conventional method

a)

b)

As shown in Fig. 2(b), the generating profile of gear hob is overlapped with the tooth profile of hypothetical external gear at point $p$. During the generating process of non-circular gear, the radius ${r_n}^{\left(1\mathrm{\text{'}}\right)}\left({\theta }_n\right)$ of non-circular gear is angle-varying, related parameters $L$, ${\gamma }_n$, $\sigma$ can be derived by external gear rolling on the pitch curve of non-circular gear as:

The coordinate transformation matrix $M_{12\mathrm{\text{'}}}$ from $O_2^{\mathrm{\text{'}}}$-$X_2^{\mathrm{\text{'}}}{Y}_2^{\mathrm{\text{'}}}{Z}_2^{\mathrm{\text{'}}}$ to $O_1$-$X_1{Y}_1{Z}_1$ can be established as:

Matrix $M_{11\mathrm{\text{'}}}$ is the coordinate transformation from $O_1$-$X_1{Y}_1{Z}_1$ to $O_1^{\mathrm{\text{'}}}$-$X_1^{\mathrm{\text{'}}}{Y}_1^{\mathrm{\text{'}}}{Z}_1^{\mathrm{\text{'}}}$ and can be established as:

In Fig. 2(b), non-circular gear is rotating clockwise while the gear hob is rotating counter-clockwise so the tooth surface of non-circular gear can be proposed as ${r_p}^{\left(1\mathrm{\text{'}}\right)}({\theta }_{op},{\theta }_p,{\theta }_1)=M_{1\mathrm{\text{'}}2\mathrm{\text{'}}}\cdot {r_p}^{\left(2\mathrm{\text{'}}\right)}({\theta }_{op},{\theta }_p)$, and the responded rotating angle ${\theta }_h$ and translation displacements $D_h$ of gear hob can be established as:

Taken the parameters of gear hob in Table 1 into Eq. (7), the responded displacement $D_h$ and rotating angle ${\theta }_h$ of equivalent external gear with rotating angle ${\theta }_n$ or $\mathrm{\Delta }{\theta }_n$ of non-circular gear are calculated by MATLAB, illustrated as Fig. 3.

Fig. 3Displacement Dh and rotating angle θh of equivalent external gear

a) Incremental displacement and angle

b) Responsible displacement and angle

c) Incremental displacement and angle

d) Responsible displacement and angle

In Fig. 3(a) and (b), the parameters of gear hob and step value are constant, while the value of translation parameters $D_{h|X}$, $D_{h|Y}$, ${\theta }_h$ and step increment $\mathrm{\Delta }{\theta }_h$, $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ are variable. In addition, with hobbing goes from short axis to long axis of non-circular gear, the change trend of $D_{h|X}$, $D_{h|Y}$, ${\theta }_h$ and $\mathrm{\Delta }{\theta }_h$, $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ are the same with radius of non-circular gear, as Eq. (3). The radius increases from short axis to long axis, the step increment of gear hob will increase with difference growth. Also, the change of translation parameters $D_{h|X}$, $D_{h|Y}$, ${\theta }_h$ will follow the regular pattern of axis. Additionally, $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ are range from –0.03 to 0.03 mm and the value of $\mathrm{\Delta }{\theta }_h$ is about 0.018 rad with $\mathrm{\Delta }{\theta }_n=$ 0.001 rad. Fig. 3(c) and (d) are values of step parameters with $\mathrm{\Delta }{\theta }_n=$ 0.01 rad, demonstrate $D_{h|X}$, $D_{h|Y}$, ${\theta }_h$ and $\mathrm{\Delta }{\theta }_h$, $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ are also varying periodically, the lager step value $\mathrm{\Delta }{\theta }_n$, the lager step parameters $D_{h|X}$, $D_{h|Y}$, ${\theta }_h$ and $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$, $\mathrm{\Delta }{\theta }_h$.

2.3. Surface Hobbing ultrasonic -assisted motion

As shown in Eq. (6) and Fig. 3, the surface roughness and accuracy of non-circular gear are different with different angle ${\theta }_n$, even the step angle $\mathrm{\Delta }{\theta }_n$ is controlled constant or step increment$\mathrm{\Delta }{\theta }_h$, $\mathrm{\Delta }{D}_{h|Y}$ and $\mathrm{\Delta }{D}_{h|X}$ are obtained accurately. The transmission performance (transmission error, noise, vibration and so on) is limited by surface roughness and accuracy. So the processing method should be improved, taken the range of step increment $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ and amplitude of ultrasonic-assisted processing method into consideration, ultrasonic-assisted processing will improve the step increment $\mathrm{\Delta }{D}_{h|Y}$, $\mathrm{\Delta }{D}_{h|X}$ to a certain degree. The manufacturing model with elliptical ultrasonic-assisted method is set up, shown as Fig. 4.

Fig. 4Ultrasonic-assisted hobbing of non-circular gear

Fig. 3 shows the step distance $\sqrt{{\left(\mathrm{\Delta }{D}_{h|X}\right)}^2+{\left(\mathrm{\Delta }{D}_{h|Y}\right)}^2}$ of any adjacent two hobbing points on tooth surface ranges from 0.0315 mm to 0.0355 mm with different ${\theta }_n$, which will cause machining accuracy unstable by influencing the feed of gear cutter. In order to ensure surface accuracy constant as much as possible, a compound cutting method with ultrasonic-assisted movement are presented. The ultrasonic motion of non-circular gear is established as an elliptical motion with $S_Y=A\mathrm{s}\mathrm{i}\mathrm{n}\left({\omega }_{u}t\right)$ at the $Y$ direction and $S_X=B\mathrm{c}\mathrm{o}\mathrm{s}\left({\omega }_{u}t\right)$ at the $X$ direction, shown as Fig. 4.

Ultrasonic-assisted motion of gear hob is an elliptical-like ultrasonic vibration with ultra-high frequency. Coupled with the relative rotation and movement between hob and non-circular gear, the trajectory of cutting point $p$ on tooth surface is actually compound motion. Compared with the rotation and movement of gear hob, the period of elliptical ultrasonic motion is extremely short, and the amplitude is small. It will not change the contact relationship between gear hob and target non-circular gear within a large range, but will improve the step distance as Fig. 4(b). While the high-frequency separation between hob and non-circular gear be ignored, the ultrasonic-assisted responded rotating angle ${\theta }_h^{\text{'}}$ and translation displacements $D_h^{\text{'}}$ are obtained as:

where $A=$ 0.005 mm,$B=$ 0.004 mm, ${\theta }_n^{\text{'}}$ is the rotating angle of non-circular gear with ultrasonic-assisted method, $\mathrm{\Delta }{\theta }_n^{\text{'}}$ is the step angle of non-circular gear and $t={\theta }_n^{\text{'}}/{\omega }_n^{\text{'}}$, ${\omega }_n^{\text{'}}$ is the angular velocity of non-circular gear.

3.1. Micro peak and groove on shaped surface

Compared Eq. (7) and Eq. (8), the responded rotating angle ${\theta }_h^{\text{'}}$, translation displacements $D_h^{\mathrm{\text{'}}}{|}_x$ and $D_h^{\text{'}}{|}_Y$ are affected by added ultrasonic motion. With the expansion and simplification of Eq. (7), the ultrasonic-assisted values ${\theta }_n^{\text{'}}$ and $D_h^{\text{'}}$ can be considered as Eq. (7) attached with some extra high frequency and small amplitude expressions.

In the normal forming process, the tooth profile of non-circular gear is created by material removal. Different amount and spacing of removed material make different surface quality as shown in Fig. 5.

Fig. 5Micro surface of shaped tooth profile

As shown in Fig. 5, the unsmooth surface is caused by the retained material between two adjacent cutting steps $N$th and $(N+1)$th on non-circular gear blank, making undulating continuous tooth surface as $\widehat{abcd}$ or $\widehat{a\text{'}b\text{'}c\text{'}d\text{'}}$ in Fig. 5(b) and (c). Based on the translation displacements Eq. (8), the upper cutting point $a$, which is shown as micro peak in Fig. 5(c), at $N$th cutting position can be derived as:

The mathematical expression for the other three points $\left(b,c,d\right)$ can be derived by ${r_a}^{\left(2\mathrm{\text{'}}\right)}({\theta }_h,{\theta }_2)$, and the area $\widehat{abcd}$ depits the material removed at the initial position with angle ${\theta }_n$. With the same method, the cutting area $\widehat{a\text{'}b\text{'}c\text{'}d\text{'}}$ with rotating angle ${\theta }_n+\mathrm{\Delta }{\theta }_n$ at $(N+1$)th cutting point can be built, so the remained material $\widehat{dmd\mathrm{\text{'}}}$ between $\widehat{abcd}$ and $\widehat{a\mathrm{\text{'}}b\mathrm{\text{'}}c\mathrm{\text{'}}d\mathrm{\text{'}}}$ is the unsmooth tooth surface for hobbing step increases from$N$th to $(N+1$)th. The value of micro peak $H_m$ and spacing $S_m$ of micro remained material can be derived as follows:

${r_f}^{\left(1\mathrm{\text{'}}\right)}\left({\theta }_m\right)$ is equation for the tip radius of gear hob and $r_f=r_{bp}+(2h_a^{\mathrm{*}}+C^{\mathrm{*}})m_f$, $m_f$ is equivalent modulus of non-circular gear, $r_m^{\left(1\mathrm{\text{'}}\right)}\left({\theta }_m\right)$ is obtained as an equation for a tooth profile $\widehat{cd}$ at point $m$ on non-circular gear. As a common point between arcs $\widehat{a\text{'}d\text{'}}$ and $\widehat{cd}$, the two mathematical expressions for cutting point $m$ should be consistent with each other:

With the solution of Eq. (11), the phase angle ${\theta }_m$ of $m$ on hypothetical equivalent external gear can be established by $r_{bh}$, $r_f$, ${\zeta }_{oh}$ and $\mathrm{\Delta }{\theta }_n$. Similarly, the micro groove $d\left(d\mathrm{\text{'}}\right)$ can be built by the tooth tip point $d\left(d\mathrm{\text{'}}\right)$ of gear hob.

3.2. Micro tooth surface with different parameters

As shown in Fig. 5 and Eq. (12), the height of micro peaks, spacing between micro grooves are related to some primary parameters, such as radius $r_{bh}$, $r_f$ of gear hob, starting angle ${\zeta }_{oh}$ of tooth profile, step length $\mathrm{\Delta }{\theta }_n$, radius ${r_n}^{\left(1\mathrm{\text{'}}\right)}\left({\theta }_n\right)$ and so on. Taken the processing parameters of gear hob and non-circular gear into Eq. (12), the microscopic micro tooth surface is shown in Fig. 6.

Fig. 6 contains the comparison of micro tooth surface with different methods. Compared Fig. 6(a) and (b), it can be seen that the tooth surface with ultrasonic-assisted vibration is more regular, the height $H_m$ of micro peak also decreases from 6.5 um to 2.7 um with much smaller spacing, making surface roughness much lower. The further variation of micro peak $H_m$ and spacing $S_m$ with different parameters are shown in Fig. 7.

As indicated in Fig. 6, the microfeatures of generated tooth surface can be reflected by the height and width of machining tracks, named as micro peak $H_m$ and spacing $S_m$ in this paper. Fig. 7(a)-(d) illustrate the trend of micro peak $H_m$ and spacing $S_m$ with different parameters.

Fig. 6Micro tooth surface with different methods

a) Micro surface without ultrasonic-assisted motion

b) Micro surface involved with ultrasonic-assisted motion

Fig. 7Micro surface with different control parameters

a) Height and spacing with different radius $r_{bh}$

b) Height and spacing with different step length $\mathrm{\Delta }{\theta }_n$

c) Height and spacing with different radius $a$

d) Height and spacing with different eccentricity $e$

According to the conjugate generating process in Eq. (11) and Fig. 5, spacing $S_m$ and height $H_m$ are primarily determined by the distance and depth of two adjacent cutting positions. Because of the variable transmission ratio between non-circular gear and equivalent generating gear with variable radius, the constant step angle $\mathrm{\Delta }{\theta }_h$ and depth $\left|r_{bh}\right|$ of equivalent generating gear will engender variable $\mathrm{\Delta }{D}_{h|Y}$ and $\mathrm{\Delta }{D}_{h|X}$, causing changeable spacing $S_m$ and micro peak $H_m$.

As shown in Eq. (2), the step distance $\mathrm{\Delta }d$ of two adjacent cutting positions on gear hob can be established as $\mathrm{\Delta }d=r_{bh}\cdot \mathrm{\Delta }{\theta }_h$, showing the same trend with radius $r_{bh}$, while the modulus $m_f$ and tooth number of gear is series values not arbitrary, so the height and thickness of tooth will change with different $r_{bh}$. Taken these decrease parameters into Eq. (12), the height $H_m$ and spacing $S_m$ decrease, too. In addition, $H_m$ is not only influenced by the step distance $\mathrm{\Delta }d$, but also the cutting depth of gear hob and non-circular gear, the thickness of tooth profile, the spacing $S_m$ is merely affected by the step distance, so the change of $S_m$ is more pronounced than it of height $H_m$ with different radius $r_{bh}$. Based on Eq. (4), with angle ${\theta }_n$ ranges from 0 to $2\pi$, radius increases first and then decreases, so the height $H_m$ on the tooth surface of non-circular gear should be added by the additional values of variable radius $\mathrm{\Delta }{H}_m=\left|\left|{r_n}^{\left(1\mathrm{\text{'}}\right)}\left({\theta }_n\right)\right|-\left|{r_n}^{\left(1\mathrm{\text{'}}\right)}({\theta }_n+\mathrm{\Delta }{\theta }_n)\right|\right|$ with much obvious range than spacing $S_m$ at the direction of rotating angle ${\theta }_n$ in Fig. 7(a).

The changeable height $H_m$ and spacing $S_m$ create different micro topography on generated tooth surface. $H_m$ mainly influence the height of peaks and spacing $S_m$ influence both the spacing and smoothness of peaks, with radius $r_{bh}$ decrease from 28 mm to 24 mm, the range of $H_m$ changes from –3.04 um-4.96 um to –1.81 um-2.71 um and the spacing of peaks decrease from 5 um to 3 um, causing more regular, compact and obvious peaks. Also, the average error of peaks is improved from 0.96 um to 0.45 um, which shows much better regularity and accuracy to the theoretical tooth surface.

Fig. 7(c)-(d) illustrate the further properties of height $H_m$ and width $S_m$ with different parameters: step length $\mathrm{\Delta }{\theta }_n$, radius parameter $a$ and eccentricity $e$ of non-circular gear, getting the similar trend as Fig. 7(a). With the decrease of step length $\mathrm{\Delta }{\theta }_n$, eccentricity$e$and parameter $a$, the range of height $H_m$ and spacing $S_m$ decrease with positive correlation. Combined Eq. (12) and Fig. 6-7, it can be seen that the micro topography of tooth surface is mainly affected by the spacing $S_m$ of adjacent cutting tracks and height $H_m$ of cutting peaks, these tracks and peaks are determined by the conjugate relationship between gear hob and non-circular gear, the tooth profile of gear hob, step angle and so on. With the decrease of radius $\left|r_{bh}\right|$, step angle $\mathrm{\Delta }{\theta }_n$, eccentricity $e$ and parameter $a$, the height $H_m$ and spacing $S_m$ decrease, also the average generating error decrease with more regular, compact and clear surface. So the micro topography of non-circular gear is variable and determined by any change of these parameters, showing the relationship between the trend of parameters and micro topography in appendix Table A1.