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1.College of Engineering，China Agricultural University，Beijing 100083，P.R.China；2.Key Laboratory of Optimal Design of Modern Agricultural Equipment in Beijing，China Agricultural University，Beijing 100083，P.R.China；3.Beijing Kuiente Technology Co.，Ltd.，Beijing 100043，P.R.China

（Received 27 July 2020；revised 20 October 2020；accepted 20 November 2020）

Abstract:The top relief surfaces of an hourglass worm gear hob are ground manually in the traditional manufacturing process，which cannot ensure the width of the land surfaces of the hob.Moreover，each geometric feature of the hob has been produced through different manufacturing techniques and machine tools，which results in low efficiency.To solve this problem，we propose a semi-automatic computer aided design（CAD）method for hobs.The point clouds of each feature surface of a hob are calculated by combing mathematical equations of the top relief surfaces built by the proposed method with other existing equations of hob surfaces.According to the point clouds，the method can achieve the automatic modeling for the hob in three-dimensional（3D）software by classifying and extracting the parameter information of the feature-hierarchical knowledge of the hob.Based on the generated 3D model，the entire surfaces of the hob can be manufactured on a four-axis computer numerical control（CNC）milling machine through only twice clamping.Verification of the width of the land surface of the hob manufactured by semi-automatic CAD method on a measuring projector proved the precision of the designed width can be ensured.The edge of the contact area on the worm wheel in a meshing experiment is clear and distinct，which means the worm gear drive is meshed well and the hob manufactured by the proposed method has improved machinability.The method simplifies the processing technique，and improves the design efficiency and production accuracy.

Key words：hourglass worm gear hob；top relief surface；feature-hierarchical knowledge；semi-automatic；computer aided design（CAD）

0 Introduction

Hourglass worm drives are extensively used in heavy-loaded and highly-efficiency fields，especially in manufactoring involving iron，steelmaking and hoisting ones［1-2］.Compared with that of cylinder worm gear sets，the lubrication and transmission efficiency of the hourglass worm gear sets are much better［3-5］，and they can carry the same load with a more compact volume because of their characteristics， like multi-tooth and double-line contact［6-10］.Therefore，the hourglass worm gear sets can ensure a long service life and a high reliability.However，the mesh theory of the hourglass worm drives is complicated.A precise manufacturing method and a high efficiency manufacturing technique are required because the performances of the drives are sensitive to the manufacturing errors and the assembly errors，and a little error can easily lead to an edge contact［11-12］.The hourglass worm wheel is generally hobbed by a specialized hob that is in accordance with the worm.But the hob is difficult to design and manufacture because of its complex geometry features［13］.

The traditional production procedures of the hourglass worm gear hob are as follows.First，the original spiral surfaces of the hob are turned or milled from a work blank［14-20］.Second，the spiral surfaces are roughly ground in an hourglass wormgrinding machine.Third，the gashes of the hob are milled and ground and the cutting faces of each cutting tooth can be obtained［21-22］.Fourth，the hob which can be called a semi-finished hob now is quenched to enhance its rigidity［23］.Fifth，each relief surface of the teeth is ground by a grinding wheel manually after the heat-treatment［24］.This process requires different machine tools with different techniques，including a turning or milling machine and a grinding one for rough generating spiral surfaces，a milling machine for rough and fine milling gashes，a grinding machine for fine finishing spiral surfaces［25］，a heat-treatment equipment for quenching and a grinding wheel for grinding manual.This leads to a low extent of standardization and a low efficiency in manufacturing［26-27］.Moreover，the manual works and the multiple times of clamping lead to a low precision and consequently a low cutting performance of the hob.Thus，a new production method for hobs is required.

Dong et al.put forward a new design for producing the flank relief surfaces of the hob based on a turning method，in which the flank relief surfaces were turned on a four-axis computer numerical control（CNC）worm turning machine［28-29］.3D models of some key surfaces of a hob including the flank land surfaces and the flank relief surfaces were established.Moreover，a method on direct-forming of the hob on the four-axis CNC milling machine was preliminary discussed.However，the hob produced by this method still cannot be applied in practice because of the lack of models for some key surfaces of the hob.Liu et al.put forward a flank relief grinding method of a planar double-enveloping worm gear hob by using a four-axis CNC hourglass worm grinding machine［30-31］.In this method，the flank relief surfaces could be ground automatically with a designed land width and an exact relief angle.However，similar to Dong et al.’s method and other existing studies，this method neither provided a model for the top relief surface nor produced a completed physical model of the worm gear hob.

This paper proposes a semi-automatic computer aided design（SA-CAD）method for manufacturing an hourglass worm gear hob.This method can automatically classify and extract the parameter information of the feature-hierarchical knowledge of the hob according to the point clouds of each feature surface，and thus automatically models the hob in 3D software.The point clouds were calculated by the given hob mathematical model.The model is established by combing apresented mathematical equation of the top relief surfaces with other existing equations of hob surfaces.Based on computer aided manufacturing（CAM）technology with the generated 3D model，a milling program for hob milling can be generated.Thus the entire surfaces of the hob can be manufactured on a four-axis CNC milling machine through only twice clamping.The width of the land surfaces can be measured for precision validation.An hourglass worm wheel is hobbed by the hob.Furthermore，a meshing experiment for contact area observation was conducted by installing the worm wheel and the corresponding worm in a gearbox.

1 Features of an Hourglass Worm Gear Hob

The spiral surfaces，the top and root torus of the hob are identical to the parts of the corresponding worm，as shown in Figs.1（a）and（b）.For the sake of draining chips，a series of gashes are cut by the cutting faces and the flank faces.In order to hob the worm wheel，a land surface with a certain width that is identical to the spiral surface and top torus must be kept.So the relief surface on every tooth of the hob can prevent interference when hobbing the worm wheel.

The land surface of the teeth can keep the cutting edge of the teeth be used persistently in hob-bing.The more accurate the width of the land surfaces of the hob，the smoother the hobbing process.A proper width can lessen the vibration of the cutting process and provide a burnishing effect.Fig.1（c）shows one of the teeth of the hob.For simplicity，only the right side of the tooth（when viewing from the cutting face side）is discussed.After grinding the flank relief surface and the top relief surface，the spiral surface and the top torus are remained as flank land surface and top land surface，respectively，with a designed width.However，when the grinding depends on manual work，the width cannot be ensured.

Fig.1 Schematic of an hourglass worm gear hob with straight gashes

2 The Proposed Semi-automatic CAD Method

This section introduces the proposed semi-automatic CAD method for the hourglass worm gear hob.This method deploys the feature hierarchical knowledge classification of the hourglass worm gear hob，the mathematical model of each feature of the hob，the information extraction and automatic 3D modeling algorithm based on the feature-hierarchical knowledge of the hob and the machining process on a four-axis CNC milling machine that only contains two clamping times.Fig.2 shows the flowchart of the proposed semi-automatic CAD method.

For the purpose of simplifying the whole manufacturing process and improving the accuracy of the features，the first step is to classify the features of the hourglass worm gear hob based on the machining methods and mechanical principles into surfaces based on the structure of the hob.As the surfaces of the hob structure are classified，the mathematical models with key parameters of each surface can be established according to the mechanical principles of the hob machining.In addition，the mathematical model of the top relief surface is built for the first time.The second step is to extract the point data from the mathematical model layer by layer based on the feature-hierarchical knowledge of the hob.In our approach，by using the API functions，an automated modeling algorithm is developed based on the feature-hierarchical knowledge of the hob and pointto-curve-to-surface reverse engineering method.The algorithm can extract the data points from the mathematical model of the hob and automatically deploy the CAD modeling command to model every surface of the hob layer by layer.A milling program is wrote based on the completed 3D model，and the hob is produced on a four-axis milling machine with one rough-milling time and one finish-milling time，only twice.

3 Construction of the CAD Method

3.1 Mathematical model of the hourglass worm gear hob

3.1.1 Coordinate systems

The coordinate systems in Fig.3（a） are all right-handed Cartesian coordinate systems.Fig.3（a） illustrates the positional relationship between the hob and the grinding machine in the hob manufacturing process.Subscript“h”represents the hob.OriginOhis the centre of the hob，stationary coordinate systemσoh（Ooh；ioh，joh，koh） is connected with the hob.The hob analysed in this paper is righthanded and is rotated around the axiskohwith a rotation angleφhwhen generated.Thus，the moving coordinate systemσh（Oh；ih，jh，kh） is fixed with the hob andkh=koh.The rotation angleφh=0 andσhcoincides with σohat the start of motion.Subscript“d”represents the grinding machine.OriginOdis the centre of the grinding machine and is also the centre of the base circle（It is the centre of the worm wheel whenOhis the centre of the worm）.Stationary coordinate system σod（Ood；iod，jod，kod） is connected with the grinding machine.The grinding machine rotates around the axiskodwith a rotation angleφdwhen generating the hob.So the moving coordinate systemσd（Od；id，jd，kd）is fixed with the grinding machine andkd=kod.Axesjodandiodare parallel with axeskohandioh，respectively.a=|OhOd|is the centre distance which represents the vertical distance between the hob axiskohand the grinding machine axiskod.

Fig.2 Flowchart of the proposed manufacturing method

Fig.3 Coordinate systems of the hob and the flat grinding wheel

3.1.2 Mathematical model of the spiral surfaces

The spiral surface of the hourglass worm gear hob，marked asΣh，is generated by a plane called generating plane，marked asΣd.The generating plane can be a plane，a single cone or a double cone when rotating aroundkodat a defined transmission ratio with the hob rotating aroundkohat the same time.

Fig.3（b）shows the process that a flat grinding wheel generating the hob.The generating plane is tangent to the main circle of the worm gear with radiusrb.βrepresents the angle between the rotation axis of the grinding machine and the generating plane.As the parameters of the generating plane，vanduare perpendicular to each other anduis parallel to axisid.The vector equation of this generating plane in σdcan be written as

The generated spiral surfaceΣhcan be replaced by the generating planeΣdusing coordinate transformation from coordinate systemσdtoσod，and then from coordinate systemσodtoσoh，and finally from coordinate systemσohtoσh.PointQin Fig.3 represents the contact point betweenΣdandΣh.If（rh）hstands for the position vector equation ofΣhin the moving coordinate system σh.The results can be written as

where

According to the gearing theory and the generating theory，the spiral surfaceΣhinσhcan be formulated as

where the first line of Eq.（6）means that the vector of the arbitrary point at the spiral surfaceΣhin coordinate systemσh；the second line of Eq.（6） is the conjugate condition function.A value ofudcan be calculated from the second line of Eq.（6） when a pair of parametersφdandvdofΣdare given；then a point with coordinate（xh，yh，zh）on the spiral surfaceΣhcan be found with the first line of Eq.（6）with the value ofud.An instantaneous contact line can be derived when a value ofφdand a series values ofvdare given.The spiral surfaceΣhcan be calculated as the value ofφdchanges gradually.

3.1.3 Mathematical model of the cutting face and the flank face

The cutting face and the land surface play important roles during a hobbing process as they are impacted by the wheel blank directly.For simplicity，straight gash is selected in this paper.The cutting faces of straight gashes are a series of surfaces passing through the hob rotation axiskoh，which is highlighted in red in Fig.4（a）.One of these surfaces must go through a point called measuring point of the tooth thicknessrSR，which can be solved by the

wheredFmeans the diameter of the reference torus.The first and the second eqution in Eq.（7）indicate that the measuring pointrSRis on the spiral surface.

Fig.4 Diagram of the cutting face,the flank face,the land surface,and the land edge

The third and the fourth eqution of Eq.（7）represent the geometric relationship ofrSRwith the hob model.Specifically，rSRis on the reference torus and it is at a distance between the middle point of the axial tooth thicknessSoto the throat plane.

According to the differential geometry，the expression of the cutting face which passes through the measuring point can be written as

where subscript“CF”represents the cutting face；pCFstands for the vertical distance between a point on the cutting face to the hob axis；pCF=0 when the point is on the hob axis；qCFis perpendicular topCF.

The other cutting faces can be calculated by rotating Eq.（8） with the hob axis in a designed rotation angle.The cutting edges highlighted in red in Fig.4（a）is defined as the intersection of the cutting face between the spiral surface and the top torus.nCFis a unit normal vector of an arbitrary point on the cutting face.lFAis the width of the tooth.The cutting face can turn to be the flank face by moving along the vectornCFby a distancelFA.Therefore，the equation of the flank face derived from Eq.（8）can be written as

where subscript“FA”represents the flank face.

3.1.4 Mathematical model of the land surface

Similar to the establishment of the flank face from the cutting face，an offset surface is built from parallelly moving the cutting face by a distancelE，which represents the width of the land surface and is highlighted in red in Fig.4（b）.Thus，the equation of the offset surface can be formulated as

where subscript“CFF”refers to the offset surface.The top torus between the offset surface and the cutting face is then defined as the top land surface；the spiral surface between the offset surface and the cutting face is defined as the flank land surface.The land edges illustrated in red in Fig.4（b）can be defined by the intersection of the cutting face with the spiral surface and the top torus，thus the expression of the flank land edge can be written as

and the top land edge can be formulated as

wheredwis the diameter of the top torus.

3.1.5 Mathematical model of the relief surfaces

As a spatial curved surface，the flank relief surface situated between the flank land edge and the flank face is generated by a type of configuration rules.Dong et al.［29］proposed a generating method where the flank relief surface is generated by a constantly changing transmission ratio modified from the worm meshing ratio during the tool tip turning.

The equation of the flank land edge is evaluated by a given thickness of the flank land surface and a given position of the cutting face.IfE（xE，yE，zE）means an arbitrary point on the flank land edge，γEmeans the lead angle of the spiral surface at pointE.Based on the given angle of the flank relief surface，the lead angleγHof arbitrary pointEon the flank land edge can be calculated，and the generating ratioof the flank relief surface can then be derived.When pointErotates around the grinding machine axis at an angleφ′d，with the coordinate transformation fromσdtoσod，then toσoh，and finally toσhat pointE，the equation of the flank relief surface can be formulated in coordinate systemσhas

where subscript“CH”stands for the flank relief surface；（rE）dthe vectorin the coordinate systemσd（Od；id，jd，kd）；xH，yHandzHmean the coordinate values of a point on the flank relief surface.

3.1.6 Mathematical model of the top relief surfaces

In previous studies in the field，little work has been devoted to the automated method for machining the top relief surface.As the hob is designed for a CNC milling machine，this paper proposes a geometry method instead of a grinding method like the ones proposed by Dong［28］or Liu［30］to build a mathematical model of the top relief surface.

Fig.5（a）is a diagram of the building process.PointsAandBare two endpoints of the top land edge，as well as the intersection points of the offset surface，two flank relief surfaces and the top torus.If the coordinates of pointsAandBare denoted asA（xA，yA，zA）andB（xB，yB，zB），they can be calculated by solving system of Eqs.（14），（15）.

In order to gain the top relief surface，an auxiliary planeMois built.Mopasses along the middle pointMof pointsAandBand is perpendicular to the line.It can be formulated as

Fig.5 Diagram of top relief surface and relationship between σh and σP

For simplicity，an arbitrary pointM（xM，yM，zM）on the planeMois used，and the equation of planeMoin the coordinate systemσhcan be rewritten as

As the intersection curve of planeMoand the top torus，curveD1can be calculated as

PointPis the intersection point of planeMoand the top land edgeAccording to the geometry theory，pointPis also the mid-point o fIt is formulated as

In order to provide a certain angle to the top relief surface，curveD1must rotate by a designed top relief angleθDaround pointPinto curveD2on planeMo.In another words，τD1represents the unit tangent vector of curveD1at pointP，which is also the unit director normal vector of the top land edgeτD1must rotate a designed top relief angleθDaround vectorτPto become vectorτD2，whereτPis the unit tangent vector of the top land edgeandτD2is the unit tangent vector of curveD2.The process is shown in Fig.5（a）.τD1can be formulated as

where（nP）his the unit normal vector of top torus at pointP.According to geometry theory，（nP）his also the unit deputy normal vector of the top land edgewhich can be formulated as

where

As one of the basic trigonal elements of space curve，τPcan be calculated as

where（rAB）his the equation of the top land edgeinσh，which has been discussed in 3.1.4.

According to the differential geometry，the three vectorsnP，τPandτD1can be composed to a coordinate system denoted asσP.Fig.5（b） describes the relationship between the coordinate systemσhandσP.As the value of these two coordinate systems and pointPis known，the angleβIDbetween axisihand vectorτD1as well as the angleβJNbetween axisjhand vectornPcan be calculated.According to coordinate transformation principle，the equation of curveD2in coordinate systemσhcan be formulated as

where（rD1）hpresents the equation of curveD1in coordinate systemσh，as described in Eq.（18）；R［τP，θD］the rotation transformation matrices when vectorτD1rotates around vectorτPin angleθD，it can be written as

where

when the vectorτPis written inτP（xτ，yτ，zτ）.

R［jh，βID］andR［ih，βJN］are the rotation transformation matrices from coordinate systemσhtoσP，which is changed with the location of each tooth.

As the equation of curveD2is known，denoteD（xD，yD，zD）as an arbitrary point on the curveD2.The top relief surface which is built by the top land edgemoving parallelly along curveD2to form a curved surface（fDP）h.It is represented by

where

3.2 3D automated modelling of the hourglass worm gear hob

3.2.1 Hob information extraction algorithms

The mathematical model of the hob is built in MATLAB according to section 3.1.The coordinates of points are evenly scattered on each surfaces of the hob and can be calculated.An algorithm is proposed to classify and extract the parameter information of the hob based on the feature-hierarchical knowledge method.The process is shown in Fig.6.

Firstly，the surfaces are classified into two groups according to the operability of modelling，the regular surface（root surface，gashes tours，end surfaces and top land surfaces）and irregular surfaces（top relief surfaces，flank relief surfaces，flank land surfaces，cutting faces and flank faces）.Secondly，the structure-related parametersPk（denoted asPk1，Pk2，… ，Pkn）and the centre distancea，are extracted as values for regular surfaces forming.Since the number of the hob teeth is updated continuously depending on some customer requirements，like the length of the hob，the algorithm has to extract this value for surfaces building subsequently.The irregular surfaces are built based on point-tocurve-to-surface process，so the users have to define the precision of the hob surface modelling，and the precision transfers into the number of points and curves of the surface.Otherwise，the algorithm can use the empirical rules to define them，usually preference as

wheredppmeans the distance between each points.

There are nine types of the irregular surfaces and each type is defined differently.The number of pointsAiand curvesBiare defined into the set

And then the sets are distributed to the dynamic arrayCTm+2which can be denoted as an ID for each surface.

wheremmeans the number of the teeth surfaces except the top relief surface that equals to 7.

3.2.2 3D automatic modeling algorithms for hob

As the dynamic arrays are defined，the APIs functions based on the reverse engineering method（points-to-curve-to-surface） are programmed to form every surface of the hob automatically.Fig.7presents the subsection of the automatic modelling of a tooth surface.

Fig.6 The proposed classification-extraction-modeling algorithm

3.3 Manufacturing processofthehourglass worm gear hob

Due to the teeth ranging a round the surface of the hob，there needs co-movements including the rotary movement around the hob axis，the rectilinear translation along the hob axis and the radial movement to the hob during the milling process.Therefore，a four-or above-four-axis universal CNC milling machine is recommended.Compared with tradi-tional manufacturing，the new method experiences less processing and clamping.

This paper suggests the manufacturing on a four-axis universal CNC milling machine.The three processing steps are as follows.

（1）NC rough-milling

Data of the 3D hob model are imported into the CNC system.The hob is clamped into the milling machine by aligning the centre axis，the centre rotation angle and the centre vertical axis of the hob.This is the first time the hob to be clamped.Rough milling the hob is roughly milled and only an allowance is left in the cutting faces and the land surfaces of the teeth for fine finish.In Fig.8，the red area stands for the remained part for fine finish after rough milling.

（2）Heat-treatment

In order to improve the solidity，strength and toughness，a heat-treatment，including quenching for the hob after rough milling，is conducted.

（3）NC finish-milling

The hob needs to be fine finish milling as there will be a little surface deformation after the heattreatment.The allowance on the cutting faces and the land surfaces of the teeth remains as before.This is the second time for the hob to be clamped.The centre axis and the centre rotation axis of the hob are located，as the rough milling process is completed.

4 Experiment

4.1 Design of the hob

This method can be applied to the production of different types of worm gear hobs to achieve higher convenience and accuracy.An hourglass worm gear hob is designed to illustrate this new method.As mentioned in Section 1，it is the cutting edge of the teeth actually plays the role of hobbing the worm wheel.The relief surface of the teeth is used to avoid interference or touching the surface of worm wheel that have just been hobbed.Thus，in the initial design，there is no need to keep the land surface between cutting edge and relief surface，and the existing land surface in the later design is to keep the cutting edge of the teeth be used persistently in hobbing worm wheel.Therefore，the tooth of the hourglass worm gear hob has a characteristic of non-resharpening.In the most design of small size hourglass worm gear hob，the width of the land surface is recommended as 0.1 to 0.2 mm.For the purpose of a better visual effect，the width of the land surface was designed as 1.00 mm in this paper；the angle of the relief surface was designed as 5°.Straight gashes was adopted instead of spiral flutes，as the number of spirals was only 1 and the reference lead angle at gorge of worm was small in this design.The parameters of the hourglass worm gear hob are given in Table 1.

The parameters were substituted into the mathematical model of the hob as represented by Eqs.（1）—（24）.The parameter information was extracted and the features of every surface of the hob were built in UG Modelling software automatically.Fig.9 illustrates the building program and the 3D model of the hob built in UG software.The hourglass worm gear hob was designed to have six straight gashes and 31 teeth.

Table 1 Main parameters of the worm gearing

Fig.9 Automatic program and 3D model of the hob in UG software

4.2 Hob manufacturing

The hob was manufactured on the model-JDVT700four-axisuniversalCNC machine.The speed of main shaft ranged from 4 000 rad/min to 14 000 rad/min in the rough milling process.The finishing allowance for the cutting faces and the land faces were remained at 0.5 mm thick.The speed of the main shaft remained at 1 000 rad/min in the finish milling step.Fig.10 is a photo of a manufactured hourglass worm gear hob.The land surfaces of the teeth can be seen clearly.

Fig.10 The produced hourglass worm gear hob

4.3 Verification of the width of the land surface

4.3.1 Virtual measurement in software

The width of the land surface of the 3D model of the hob was measured in UG.Fig.11（a）is a width measurement result of one of the land surfaces，including the top land surface and two flank land surfaces of a tooth.It is seen that the data is very close to the designed width，which means that the 3D hob model has a precise structure corresponding to the mathematical model.The measurement result of the other five arbitrary teeth is shown in Table 2.

4.3.2 Measurement of the finished product

Measurement of the finished product was conducted in a PJ-A3000 Series Measuring Project.Since the hob was designed to be straight gashes，it was placed on the objective table so that the hob axis was vertical to the glass stage.Fig.11（b）shows the placement method and measurement process.As the structure of the hob is special，only one tooth was measured from the top view.The measurement was carried out three times and the result is shown in Table 3.

Fig.11 Measurements of land surface’s width UG and measuring projector

4.4 Worm gear hobbing

The hob developed by the new method was used to hob the worm wheel on a hobbing machine.Compared with the conventionally-produced one，the hob produced by the new method has advantag-es of more stable cutting and much less vibration and noise.Fig.12（a）is a detailed view of teeth surface of the worm wheel hobbed by the hob.The surface roughness of the teeth of the worm wheel measured by engineers in the factory was smaller than the conventionally-produced one.

Table 2 Width measurement of land surface of 31 teeth mm

Table 3 Width measurement of land surface in the measuring projector

Fig.12 Detailed view of teeth surface of worm wheel and the contact area on the worm wheel teeth after the meshing experiment

4.5 Contact meshing

The worm wheel hobbed by the hob was installed in the gearbox with a corresponding worm.The surfaces of the worm and worm wheel were covered with red lead.Parts of the red lead on the teeth surfaces of the worm wheel were erased and those represent the contact area when the worm pair was meshing.Fig.12（b）shows the contact area on the worm wheel teeth.The edge of the area is clear and distinct，which means the worm gear drive was meshed well and machinability of the hob manufactured by the proposed method has been improved.

5 Conclusions

In order to simplify the processing technique，reduce the design time and improve the production accuracy，we propose a semi-automatic-CAD method for hourglass worm gear hob.The method can achieve the automatic modeling for the hob in a 3D software by classifying and extracting the parameter information of the feature-hierarchical knowledge of the hob，according to the point clouds of each feature surfaces of the hob.These point clouds are calculated by combing a presented mathematical equation of the top relief surfaces with other existing equations of hob surfaces.The hob is then manufactured on a four-axis CNC milling machine using a milling program generated by the 3D model only through twice clamping.The method simplifies the manufacturing technique of the hob.

In addition，the accuracy and efficiency of the manufacturing of the hob is improved.The milling of the land surface can ensure the design width，because the top relief surfaces are modeled by the presented mathematical equations in a 3D software.The edge of the contact area on the worm wheel in meshing experiment is clear and distinct，which demonstrated the worm gear drive is meshed well and the machinability of the hob manufactured by the proposed method has been improved.