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

. As an atypical conventional gear pair, non-circular gear has a variable transmission ratio, which can be used to improve the structure and transmission characteristics in some applications, such as variable speed grinding mechanism, wool blending machine guide mechanism, continuously variable transmission structure and so on. According to the friction theory of conjugate surfaces, the transmission characteristics will be affected by the micro topography of tooth surface, but the shaped tooth surface of non-circular gear with normal hobbing method is not uniform, because of its variable radius and flank orientations of tooth surface. The research on the relationship between micro topography and parameters of non-circular gear is necessary, thus an improved manufacturing method with ultrasonic-assisted motion was presented in this paper, the mathematical equations for the theoretical tooth surface of non-circular gear with and without ultrasonic-assisted motion have been derived in this paper, also the equations for cutting parameters (step displacement, step rotating angle) have been proposed, shown that all the parameters (initial parameters of gear and gear hob, cutting parameters in manufacturing process) will affect the formed micro surfaces. In detail, the height 𝐻 𝑚 and width 𝑆 𝑚 of micro undulating tooth surface periodically increase and decrease, getting the maximum value at 𝜃 𝑛 = 𝜋 , where is the maximum radius. The height 𝐻 𝑚 and width 𝑆 𝑚 of micro tooth surface will increase with any control parameter increase, the height 𝐻 𝑚 of most micro peaks is near 6.5um in normal hobbing process. While, the tooth surface with ultrasonic-assisted motion is more regular and clearer with the maximum height 𝐻 𝑚 decrease from 6.5 um to 2.7 um, getting more compact cutting tracks and lower surface roughness. The experimental results shows that ultrasonic-assisted hobbing has obviously positive influence for micro tooth surface, provides the theoretical support for a further analysis and better application of unconventional gear pair.


𝜃 𝑝
The angle of tooth profile at point P of gear hob  2 The angle between movable coordinate  2 ′ - The modulus of vector   ∆ The step distance of adjacent cut position on gear hob The starting angle of tooth profile of gear hob ∆ 1 Difference of rotating angle  1 between two adjacent cutting steps  ℎ Rotating angle of equivalent external gear hob at cutting point h  Vector of translation distance between coordinate  3 - 3  3  3 and fixed  2 - 2  2  2  3 The rotating angle of movable coordinate Radius of gear hob at point P in coordinate  3 - 3  3  3  1 The angle between movable coordinate  1 ′ - 1 ′  1 ′  1 ′ and fixed  1 - 1  1  1 a Scale factor for the radius of non-circular gear e Eccentricity for the radius of non-circular gear

Introduction
As a gear drive with variable transmission characteristics, non-circular gear is different from conventional ones in tooth profiles and pitch curves, so the geometric, transmission and load characteristics of unconventional gear forms are variable with different transmission functions [1].
Based on the meshing theory of conjugate surface, many scholars did the analysis and optimization for the characteristics of different gear drives.Tang focused on the tooth contact analysis of bevel gear with SGM method revealed the influence properties of different errors (misalignment, fabrication offsets) on the contact results of gear pair [2,3].Wang and Sun focused on the computerized research and simulation of epicycloids hypoid gear, proposing the characteristics of contact area and transmission errors, found the epicycloid tooth profile can improve the sensitivity of errors [4,5].Taken the relative sliding friction of meshed tooth surface into consideration, the improved contact analysis of spiral bevel gears was made by Tian [6].F. L. Litvin and Fan presented the shaving computation methods for both spur and helical gears, revealed the trend of contact ellipse for different gear types [7].The common manufacturing methods for conventional gear types are based on the generating motion of conjugate surface with constant hobbing parameters, the discrete processing traces on target tooth surfaces are even and the roughness is uniform.However, non-circular gear usually rotates with variable radius, causing irregular cutting steps, processes traces and mutative roughness, the conventional machining methods with constant cutting parameters cannot be directly applied on non-circular gears for uniform micro surface.
In order to reduce roughness, improve cutting force and surface precision, a compound generating method with ultrasonic-assisted motion was presented.Yin and Zhao discussed the process of surface grinding with ultrasonic-assisted motion or not, and established a mathematical model for the prediction of grinding force and surface quality [8][9][10].Venkatesh considered the surface finishing of bevel gear, presented a finite element simulation for bevel gear with different input parameters [11].Harpreet proved that the average and maximum surface roughness of bevel gear could be improved to 91.04 % and 71.98 % in an experiment on bevel gear with AISI 1040 carbon steel [12].Wei and Yang focused on the surface lapping with ultrasonic-assisted motion, found that the material removal rate of ultrasonic lapping is much higher and the surface quality is much better than conventional lapping method [13,14].Su investigated the SEM topography of two different Ti6Al4V with the method of conventional milling and ultrasonic-assisted milling, shows the SEM surface roughness can be reduced up to 23.3 % and 19.1 % with UAM method than CM method, and indicates the positive influence of UAM method on surface roughness [15].An focused on the material removal mechanism of FRCMCs-SiC in machining processes, discussed the influence of process parameters on the machined surface quality, proved that ultrasonic-assisted machining method can reduce cutting force and tool wear, improve machining quality and efficiency [16].In addition, the experimental turning performance of ultrasonic-assisted turning with textured cutting insert was carried out by Sofuoglu, revealed the reason for the 35.89 % improvement during ultrasonic-assisted turning as compared to conventional turning [17].Liu analyzed the wear state of tool in ultrasonic-assisted milling and conventional milling, found that the burr produced by ultrasonic-assisted milling was not obvious and the curl angle of chip generated from ultrasonic-assisted milling was smaller than that from conventional milling in the same wear time, and better surface roughness with lower cutting force and temperature [18].Due to the benefits of ultrasonic-assisted machining, Amini designed an elliptical ultrasonic-assisted turning model, the better surface finish and lower cutting forces on the turning process of copper with elliptical ultrasonic-assisted motion has been verified by experiment [19].
These results confirmed that the new generating methods with ultrasonic-assisted motion could improve tooth surface roughness obviously.But the basic parameters of non-circular gear are complex and should be derived for surface hobbing with ultrasonic-assisted motion.
Figliolini established the base curve of non-circular gear by the Aronhold theorem, derived the constant pressure angle of this kind of non-circular gear pair [20].Furthermore, the mathematical model, transmission principle and theoretical simulation model of non-circular gear was presented by F. Y. Zheng et al. [21].Fuentes proposed two different methods for shaped teeth by facemilling cutters, and analyzed the advantages and disadvantages of the proposed two versions by numerical examples [22].
The above results demonstrate that ultrasonic-assisted processing method is better for surface roughness than normal cutting method, and the specific parameters of non-circular gear can be derived.
This paper compared the two forming methods of non-circular gear with or without ultrasonic-assisted motion, including the mathematical equations of theoretical surface, incremental offset distance and rotating angle of step, micro height and grooves of adjacent cutting points with different parameters.Also, the discussion of the relationship between finished surfaces and cutting methods is covered in this paper.The results provide the theoretical support for further analysis and application of unconventional gear type with complex tooth surface.

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. 1 - where  is the common contact point on gear hob, equivalent external gear and hypothetical generating gear,   is the starting angle of tooth profile   =  2 ⁄ −   ,   is the range of tooth profile angle,   is the basic radius.
According to the pure rolling engagement of those three tooth profiles and conversion matrix , the tooth profile of target gear can be proposed as: Eq. ( 3) illustrates the mathematical equation for target gear,   and   are the translation distance of movable coordinate  2 ′ - 2 ′  2 ′  2 ′ with meshing from  (rotating angle of cylindrical gear is  1 ) to ′ (rotating angle of cylindrical gear is  1 + Δ 1 ).As described in Fig. 2(b), gear hob always keeps the distance  away from  2 ( 2 ′ ), so   and   can be expressed by the component of arc length  3 in the  or  directions.Compared with Eq. ( 2), Eq. ( 3) has the same form for tooth profile, the target gear is cylindrical gear with the constant radius  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.
Gear hobbing of non-circular gear

. Simulated hobbing of non-circular gear with conventional method
As shown in Fig. 2(b), the generating profile of gear hob is overlapped with the tooth profile of hypothetical external gear at point .During the generating process of non-circular gear, the radius   (1′) (  ) of non-circular gear is angle-varying, related parameters ,   ,  can be derived by external gear rolling on the pitch curve of non-circular gear as: The coordinate transformation matrix  12′ from  2 ′ - 2 ′  2 ′  2 ′ to  1 - 1  1  1 can be established as: Matrix  11′ is the coordinate transformation from 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   (1′) (  ,   ,  1 ) =  1′2′ ⋅   (2′) (  ,   ), and the responded rotating angle  ℎ and translation displacements  ℎ of gear hob can be established as: Taken the parameters of gear hob in Table 1 into Eq.( 7), the responded displacement  ℎ and rotating angle  ℎ of equivalent external gear with rotating angle   or Δ  of non-circular gear are calculated by MATLAB, illustrated as Fig. 3.In Fig. 3(a) and (b), the parameters of gear hob and step value are constant, while the value of translation parameters  ℎ| ,  ℎ| ,  ℎ and step increment Δ ℎ , Δ ℎ| , Δ ℎ| are variable.In addition, with hobbing goes from short axis to long axis of non-circular gear, the change trend of  ℎ| ,  ℎ| ,  ℎ and Δ ℎ , Δ ℎ| , Δ ℎ| 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  ℎ| ,  ℎ| ,  ℎ will follow the regular pattern of axis.Additionally, Δ ℎ| , Δ ℎ| are range from -0.03 to 0.03 mm and the value of Δ ℎ is about 0.018 rad with Δ  = 0.001 rad.Fig. 3(c) and (d) are values of step parameters with Δ  = 0.01 rad, demonstrate  ℎ| ,  ℎ| ,  ℎ and Δ ℎ , Δ ℎ| , Δ ℎ| are also varying periodically, the lager step value Δ  , the lager step parameters  ℎ| ,  ℎ| ,  ℎ and Δ ℎ| , Δ ℎ| , Δ ℎ .

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   , even the step angle Δ  is controlled constant or step increment Δ ℎ , Δ ℎ| and Δ ℎ| 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 Δ ℎ| , Δ ℎ| and amplitude of ultrasonic-assisted processing method into consideration, ultrasonic-assisted processing will improve the step increment Δ ℎ| , Δ ℎ| to a certain degree.The manufacturing model with elliptical ultrasonic-assisted method is set up, shown as Fig. 4. Fig. 3 shows the step distance √ (Δ ℎ| ) 2 + (Δ ℎ| ) 2 of any adjacent two hobbing points on tooth surface ranges from 0.0315 mm to 0.0355 mm with different   , 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   = sin(  ) at the  direction and   = cos(  ) at the  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, JOURNAL OF VIBROENGINEERING.FEBRUARY 2023, VOLUME 25, ISSUE 1 the trajectory of cutting point  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  ℎ ′ and translation displacements  ℎ ′ are obtained as: where  = 0.005 mm,  = 0.004 mm,   ′ is the rotating angle of non-circular gear with ultrasonic-assisted method, Δ  ′ is the step angle of non-circular gear and  =   ′ /  ′ ,   ′ is the angular velocity of non-circular gear.

Micro peak and groove on shaped surface
Compared Eq. ( 7) and Eq. ( 8), the responded rotating angle  ℎ ′ , translation displacements  ℎ ′ |  and  ℎ ′ |  are affected by added ultrasonic motion.With the expansion and simplification of Eq. ( 7), the ultrasonic-assisted values   ′ and  ℎ ′ 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. 5. Micro surface of shaped tooth profile
As shown in Fig. 5, the unsmooth surface is caused by the retained material between two adjacent cutting steps  th and ( + 1) th on non-circular gear blank, making undulating continuous tooth surface as  ̂ or ′′′′ ̂ in Fig. 5(b) and (c).Based on the translation displacements Eq. ( 8), the upper cutting point , which is shown as micro peak in Fig. 5(c), at th cutting position can be derived as: +|  (2) ( ℎ ,  1 ) The mathematical expression for the other three points (, , ) can be derived by   (2′) ( ℎ ,  2 ), and the area  ̂ depits the material removed at the initial position with angle   .
With the same method, the cutting area ′′′′ ̂ with rotating angle   + Δ  at ( + 1)th cutting point can be built, so the remained material ′ ̂ between  ̂ and ′′′′ ̂ is the unsmooth tooth surface for hobbing step increases from th to ( + 1)th.The value of micro peak   and spacing   of micro remained material can be derived as follows: With the solution of Eq. ( 11), the phase angle   of  on hypothetical equivalent external gear can be established by  ℎ ,   ,  ℎ and Δ  .Similarly, the micro groove (′) can be built by the tooth tip point (′) of gear hob.

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  ℎ ,   of gear hob, starting angle  ℎ of tooth profile, step length Δ  , radius   (1′) (  ) 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 62 JOURNAL OF VIBROENGINEERING.FEBRUARY 2023, VOLUME 25, ISSUE 1 regular, the height   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   and spacing   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   and spacing   in this paper.According to the conjugate generating process in Eq. (11) and Fig. 5, spacing   and height   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 Δ ℎ and depth | ℎ | of equivalent generating gear will engender variable Δ ℎ| and Δ ℎ| , causing changeable spacing   and micro peak   .
As shown in Eq. ( 2), the step distance Δ of two adjacent cutting positions on gear hob can be established as Δ =  ℎ ⋅ Δ ℎ , showing the same trend with radius  ℎ , while the modulus   and tooth number of gear is series values not arbitrary, so the height and thickness of tooth will change with different  ℎ .Taken these decrease parameters into Eq.( 12), the height   and spacing   decrease, too.In addition,   is not only influenced by the step distance Δ, but also the cutting depth of gear hob and non-circular gear, the thickness of tooth profile, the spacing   is merely affected by the step distance, so the change of   is more pronounced than it of height   with different radius  ℎ .Based on Eq. ( 4), with angle   ranges from 0 to 2, radius increases first and then decreases, so the height   on the tooth surface of non-circular gear should be added by the additional values of variable radius Δ  = ||  (1′) (  )| − |  (1′) (  + Δ  )|| with much obvious range than spacing   at the direction of rotating angle   in Fig. 7(a).The changeable height   and spacing   create different micro topography on generated tooth surface.  mainly influence the height of peaks and spacing   influence both the spacing and smoothness of peaks, with radius  ℎ decrease from 28 mm to 24 mm, the range   changes from -3.04 um-4.96um to -1.81 um-2.71um 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   and width   with different parameters: step length Δ  , radius parameter  and eccentricity  of non-circular gear, getting the similar trend as Fig. 7(a).With the decrease of step length Δ  , eccentricityand parameter , the range of height   and spacing   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   of adjacent cutting tracks and height   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 | ℎ |, step angle Δ  , eccentricity  and parameter  , the height   and spacing   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.

Experiment
For a better understanding and verification of theoretical micro tooth surface of non-circular gear, the experimental hobbing of non-circular gear is operated with cutting parameters in Table 2 and control parameters of non-circular gear in Table 3 with 17Cr2Ni2Mo.As alloy steel with high property parameters, 17Cr2Ni2Mo is often used on gear transmission parts.So this paper focused on the micro surface topography of non-circular gear with two Fig. 8 shows the micro surface of non-circular gear, which were photographed by the Super depth of field 3D microscopy system (VHX-1000C/VW-6000).Comparing the two figures, it can be seen that the machining traces and boundaries of every step during ultrasonic-assisted hobbing are much less than that of conventional fabricating, resulting in lower height   and width   .
Specifically, the width   of adjacent cutting steps with conventional method is obvious, getting a value about 14 um, which can be considered as the same with theoretical step distance as the parameters in tables. 1 to 3. In addition, the axial feed rate will affect the formed surface periodically, so the micro surface is affected by both cutting steps of gear hob and feed speed of conjugate surfaces, dividing the tooth surface into series of independent areas, and the spacing of every area is about mm, which is approximately equal to the feed rate and offset translation distance of gear hob in one rotation.Although the change process among these independent areas is uniform and smooth, the difference of height   between micro peak and grooves is large, ranging from 0.3 um to 6.5 um, getting the maximum Δ  =   + −   − = 6.2 um (  + is the height of micro peak, and   − is the height of micro grove); Fig. 8(b) shows the micro surface with ultrasonic vibration, the added motion can perform secondary micro-processing on the boundary surface, cause curling and burrs, make the peaks and boundaries of cutting trajectory be more obvious.But it will break the peaks with large micro cutting, reducing the height of peak to -0.3-2.7 um.
Although the peaks and grooves on micro surface are more compact with more burrs in ultrasonic-assisted hobbing process than that in normal hobbing methods, the roughness and topographic error is smaller with the same after cutting condition and stabilization, which can be consistent as the same with the conclusions in REFs.[18] and [19].

Conclusions
According to the friction theory of conjugate surfaces, the transmission stability, lubrication, vibration will be affected by the micro topography of tooth surface.Based on the variable radius   (1′) (  ) of non-circular gear, the step translation  ℎ ′ |  ,  ℎ ′ |  and rotating angle Δ ℎ of gear hob

3 Z 3 X
Gear hobbing of non-circular gearGear hobVariable tooth surface Constant hob surface 3

3 .
Incremental displacement and angle a) Incremental displacement and angle Rotating angle of non-circular gear (rad) n Response displacement and angle b) Responsible displacement and angle Step angle of non-circular gear (rad) displacement and angle Rotating angle of non-circular gear (rad) displacement and angle Fig. Displacement  ℎ and rotating angle  ℎ of equivalent external gear

Fig. 7 ( 6 .Sp 7 .
a)-(d) illustrate the trend of micro peak   and spacing   with different parameters.surface involved with ultrasonic-assisted motion Fig. Micro tooth surface with different methods Ro tat in g an gl e Ra di us of ge ar ho b and spacing with different radius  ℎ Ro tat ing ang le St ep len gt h and spacing with different eccentricity  Fig. Micro surface with different control parameters 1  1  1 ,  2 - 2  2  2 and  3 −  3  3  3 are fixed coordinates on cylindrical gear, hypothetical generating gear and gear hob respectively.
* +  * )  ,   is equivalent modulus of non-circular gear,   (1′) (  ) is obtained as an equation for a tooth profile  ̂ at point  on non-circular gear.As a common point between arcs ′′ ̂ and  ̂, the two mathematical expressions for cutting point  should be consistent with each other:

Table 2 .
Ultrasonic vibration parameters of gear cutter