CN114019908A - Method for controlling tooth profile cambered surface chamfering of spiral bevel gear - Google Patents

Method for controlling tooth profile cambered surface chamfering of spiral bevel gear Download PDF

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CN114019908A
CN114019908A CN202111254043.9A CN202111254043A CN114019908A CN 114019908 A CN114019908 A CN 114019908A CN 202111254043 A CN202111254043 A CN 202111254043A CN 114019908 A CN114019908 A CN 114019908A
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tooth profile
gear
bevel gear
spiral bevel
chamfering
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CN114019908B (en
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洪荣晶
胡敏
李帅康
薛凤
张建坤
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Nanjing Tech University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/408—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by data handling or data format, e.g. reading, buffering or conversion of data
    • G05B19/4083—Adapting programme, configuration
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
    • B23F—MAKING GEARS OR TOOTHED RACKS
    • B23F19/00—Finishing gear teeth by other tools than those used for manufacturing gear teeth
    • B23F19/10—Chamfering the end edges of gear teeth
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B2219/00—Program-control systems
    • G05B2219/30—Nc systems
    • G05B2219/35—Nc in input of data, input till input file format
    • G05B2219/35356—Data handling

Abstract

The invention provides a method for controlling tooth profile cambered surface chamfering of a spiral bevel gear, which comprises the following steps of: the method comprises the steps of firstly calculating tooth profile line equations of the large end and the small end of a small wheel of the spiral bevel gear path, accurately controlling the track of the cambered surface chamfering of a chamfering cutter, then completely integrating the tooth profile line equations into software developed on the basis of C #, automatically controlling an open numerical control machine tool to perform chamfering through a fixed height motion control card, and similarly processing the tooth profile cambered surface chamfering of the two ends of the large wheel of the spiral bevel gear. The invention can really eliminate the problem of the edge of the gear, and reduce the stress concentration during the heat treatment of the gear to the maximum extent, thereby improving the working effect of the gear; the method provides man-machine interaction software, can automatically acquire the machining program of the tooth profile cambered surface chamfering of the spiral bevel gear, greatly reduces the machining difficulty, and enables an operator to work more easily.

Description

Method for controlling tooth profile cambered surface chamfering of spiral bevel gear
Technical Field
The invention relates to the technical field of gear machining, in particular to a method for controlling tooth profile cambered surface chamfering of a spiral bevel gear.
Background
The rapid development of modern technology for spiral bevel gear wide application in each field, but because can leave the incomplete thorn in the edge department of the teeth of a cogwheel during gear cutting, produce in transport or assembly and collide with and draw hair etc. for spiral bevel gear produces certain defect, for example can produce meshing noise, reduce gear drive's precision, shorten the life-span of gear use etc.. At present, few technologies are used for chamfering the cambered surfaces of the tooth profiles of the spiral bevel gears, the programming process is complex, mistakes are easy to make, and certain difficulty exists in operation of technical personnel. The chamfering technology of the spiral bevel gear is not mature at present, programming of chamfering is complex, requirements on operators are high, automatic programming software needs to be developed urgently, and technicians can chamfer the spiral bevel gear only through simple operation.
Disclosure of Invention
The invention aims to provide a method for controlling tooth profile cambered surface chamfering of a spiral bevel gear, which aims to solve the problems in the background technology.
The method combines the characteristics of tooth profile chamfering instantaneity, convenience and simplicity in machining process and the like, and selects an improved NURBS curve interpolation algorithm and a GTS-VB series fixed-height motion control card. Based on an open numerical control system platform, an operator operates software developed on a computer on the basis of C #, including calculating a curve equation of tooth profiles at two ends of a small tooth profile of a spiral bevel gear wheel and simulating a chamfering processing track of a cutter, finally inputting parameters of a machine tool, a workpiece, the cutter and the like, calling a dynamic link library in a fixed-height motion control card, sending a high-speed pulse instruction to the machine tool, driving a chamfering cutter on the machine tool to complete chamfering of arc surfaces at two ends of the small tooth profile, and enabling the tooth profiles at the two ends of the large tooth profile of the spiral bevel gear wheel to be identical in chamfering.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for controlling tooth profile cambered surface chamfering of a spiral bevel gear is characterized by comprising the following steps:
step 1: according to the spiral bevel gear structure and the gear cutting method, calculating and obtaining tooth profile curve equations of the large end and the small end of the small gear: the tooth profile curve equations of the large end and the small end of the small gear are obtained, firstly, a tooth crest line equation of the gear is required to be calculated, and then the tooth profile curve equations of the large end and the small end of the gear are calculated according to the structure of the spiral bevel gear and the coordinate relation of a machine tool;
step 2: simulating and extracting data points: simulating a three-dimensional curve graph in Matlab according to the tooth profile curve equation obtained in the step 1, and extracting a plurality of data points;
and step 3: in combination with a modified NURBS curve interpolation algorithm:
in the interpolation process, any node parameter u epsilon [ u ] of the NURBS curve is giveni,ui+1]The curves are represented in matrix form and the ith NURBS curve is calculated as:
Figure BDA0003323287160000021
wherein:
Figure BDA0003323287160000022
Figure BDA0003323287160000023
all coefficients in the formula (1) are determined by initially given weight factors, control vertexes and node vectors, so that the coefficients are selected only under the condition that the position of a current interpolation point is known, and the interpolation efficiency is improved;
carrying out parameter point densification processing on the feeding step length in the track space to ensure that the node vector increment delta u converted into the one-dimensional parameter spaceiFrom the formula ui+1＝ui+ΔuiThe parameter coordinate of the next interpolation point can be obtained;
according to the fourth-order Runge-Kutta algorithm, the node vector increment is calculated as follows:
Figure BDA0003323287160000031
will calculate the ui+1Substituting the position parameter into NURBS curve interpolation equation (1) to obtain the position parameter of the next interpolation point: p is a radical ofi+1＝p(ui+1)；
The self-adaptive control of NURBS curve interpolation reduces the interpolation step length by adjusting the interpolation speed, thereby reducing the chord height error; in each interpolation period, the chord height error is:
Figure BDA0003323287160000032
wherein: rhoiRadius of curvature,. DELTA.liInterpolation of the feed step length, Δ li＝viT is an interpolation period;
the feed speed v can thus be approximatediSum chord height error deltaiThe relationship is as follows:
Figure BDA0003323287160000033
by maximum chord height error delta of machine toolmaxSelecting a maximum allowable chordal height error value
Figure BDA0003323287160000034
Figure BDA0003323287160000035
The resulting acceleration error is:
Figure BDA0003323287160000036
according to the formula of the chord height error and the acceleration error, the chamfering of the cutter is more stable and accurate;
finally, combining the obtained data points on the tooth profile of the spiral bevel gear and an improved NURBS curve interpolation algorithm, so that the chamfering tool obtains an accurate and complete tooth profile arc surface chamfering track;
and 4, step 4: the whole control flow is integrated into software based on C #:
opening a Windows window application program under a Visual C # template in VS2015 software, compiling a control program, inputting a login name and a login password by a user on a login interface, and entering a functional interface, wherein the functional interface comprises functions of machine tool each-axis state detection, machine tool management, workpiece management, cutter management, database management, interpolation motion, NURBS curves and Matlab drawing calling;
and 5: the fixed height operation control card controls the machine tool to complete the cambered surface chamfering of the gear tooth profile:
after the operator inputs required parameters into the software on the computer, the operator calls the dynamic link library in the fixed-height motion control card and sends a high-speed pulse instruction to the machine tool so as to control the open type numerical control machine tool to finish the chamfering of the tooth profile cambered surface of the gear.
The tooth crest line equation calculation process of the gear in the step 1 is as follows:
the tooth cutting principle of the spiral bevel gear, and technological parameters and cutter parameters in the actual machining process are comprehensively analyzed to obtain that the parameters are related to the pitch cone surface of the spiral bevel gear, an analytic equation of an imaginary tooth line on the pitch cone surface at the center of a tooth groove of the spiral bevel gear is solved, and then an tooth crest line equation is derived according to the relationship between the obtained imaginary tooth line equation and the tooth crest line under an equivalent gear.
In the step 2, a three-dimensional curve graph of the tooth profile curve is simulated in Matlab software, and the steps of extracting a plurality of data points are as follows:
setting the configuration environments of C # and Matlab, calling Figure in Matlab by C # and embedding the Figure into a Winform window, drawing a three-dimensional graph according to the tooth profile curve equation calculated in the front, extracting corresponding data points to a TextBox control in C # and storing the data points to a txt file.
The chamfering tool in the step 3 is a fillet tool: the traditional tooth profile chamfering process is to process an edge into an inclined plane of about 45 degrees, but a new small edge can be generated at the same time, so a chamfering cutter is selected, a cambered surface chamfering is adopted for the tooth profile edge, and the circular surface is smoothly connected with two sides of a tooth profile edge line, so that the edge of a gear can be completely eliminated.
The machine tool in the step 5 finishes chamfering the arc surface of the gear tooth profile: the spiral bevel gear is a spiral bevel gear, the axes of the two gears are intersected, and the intersected angle is 90 degrees; the axis of the small spiral bevel gear is superposed with the Z axis of the machine tool, and the C axis of the machine tool drives the gear to rotate; firstly, a cutter executes a curved tooth chamfering instruction at the small end of a gear, after a complete tooth profile is finished, the cutter is lifted to a certain height, at the moment, a shaft C of a machine tool rotates by a certain angle, so that the gear to be machined rotates to the next complete tooth profile, the cutter starts to return to the initial position of the chamfer and starts chamfering of the next tooth profile, and the operation is repeated continuously until the complete machining of the gear teeth at the small end of the gear is finished, so that the tooth profile arc surface chamfer at the large end of the small spiral bevel gear and the tooth profile arc surface chamfer at the large end of the large spiral bevel gear are the same.
The invention has the following advantages and benefits:
(1) by adopting the chamfering tool, the problem that a new smaller edge can be left after chamfering of a common chamfering tool and the edge of the gear cannot be really eliminated can be avoided, and stress concentration during heat treatment of the gear is reduced to the maximum extent, so that the working effect of the gear is improved.
(2) By adopting NURBS curve interpolation based on a fourth-order Runge-Kutta algorithm, the tooth profile cambered surface chamfering of the chamfering tool can be more accurate.
(3) The C # based human-computer interaction software is provided, and a machining program of the tooth profile cambered surface chamfering of the spiral bevel gear can be automatically obtained by inputting the model parameters of a machine tool, the offset parameters of each shaft, the parameters of the spiral bevel gear and the parameters of a cutter, so that the machining difficulty is greatly reduced, and an operator can work more easily.
Drawings
FIG. 1 is a general flow chart of the method for controlling the cambered surface chamfering of the tooth profile of a spiral bevel gear according to the present invention;
FIG. 2 is a NURBS curve interpolation flow chart of the spiral bevel gear tooth profile arc surface chamfering control method of the invention;
FIG. 3 is a diagram of the tooth profile arc surface chamfering of the chamfering tool according to the tooth profile arc surface chamfering control method of the spiral bevel gear;
FIG. 4 is a software design drawing of the spiral bevel gear tooth profile arc chamfering control method of the present invention.
Detailed Description
The technical solutions of the present invention will be described clearly and completely with reference to the accompanying drawings, and it is to be understood that the described embodiments are only a part of the embodiments of the present invention, and not all embodiments, and all other embodiments obtained by those skilled in the art without any inventive work based on the embodiments of the present invention belong to the scope of the present invention.
As shown in fig. 1 to 4, in this embodiment, the tooth profile line equation of the large end and the small end of the spiral bevel gear small wheel is calculated, the trajectory of the chamfer of the arc surface of the chamfering tool is accurately controlled, and then the trajectory is integrated into software developed on the basis of C #, the open numerical control machine is automatically controlled by the fixed height motion control card to perform chamfering, and the tooth profile arc surface chamfering at the two ends of the spiral bevel gear large wheel is the same.
A method for controlling tooth profile cambered surface chamfering of a spiral bevel gear comprises the following steps:
step 1: according to the spiral bevel gear structure and the gear cutting method, tooth profile curve equations of the large end and the small end of the small gear are obtained through calculation:
the tooth profile curve equations of the large end and the small end of the small gear are obtained, firstly, a tooth crest line equation of the gear is required to be calculated, and then the tooth profile curve equations of the large end and the small end of the gear are calculated according to the structure of the spiral bevel gear and the coordinate relation of a machine tool;
step 2: simulating and extracting data points:
simulating a three-dimensional curve graph in Matlab according to the tooth profile curve equation obtained in the step 1, and extracting a plurality of data points;
and step 3: in combination with a modified NURBS curve interpolation algorithm:
in the interpolation process, any node parameter u epsilon [ u ] of the NURBS curve is giveni,ui+1]The curves are represented in matrix form and the ith NURBS curve is calculated as:
Figure BDA0003323287160000061
wherein:
Figure BDA0003323287160000071
Figure BDA0003323287160000072
all coefficients in the formula (1) are determined by initially given weight factors, control vertexes and node vectors, so that the coefficients are selected only under the condition that the position of the current interpolation point is known, and the interpolation efficiency is improved.
Carrying out parameter point densification processing on the feeding step length in the track space to ensure that the node vector increment delta u converted into the one-dimensional parameter spaceiFrom the formula ui+1＝ui+ΔuiThe parameter coordinates of the next interpolation point can be found.
According to the fourth-order Runge-Kutta algorithm, the node vector increment is calculated as follows:
Figure BDA0003323287160000073
will calculate the ui+1Substituting the position parameter into NURBS curve interpolation equation (1) to obtain the position parameter of the next interpolation point: p is a radical ofi+1＝p(ui+1)
The self-adaptive control of NURBS curve interpolation reduces the interpolation step length by adjusting the interpolation speed, thereby reducing the chord height error. In each interpolation period, the chord height error is:
Figure BDA0003323287160000074
wherein: rhoiRadius of curvature,. DELTA.liInterpolation of the feed step length, Δ li＝viT, and T is an interpolation period
The feed speed v can thus be approximatediSum chord height error deltaiThe relationship is as follows:
Figure BDA0003323287160000081
by maximum chord height error delta of machine toolmaxSelecting a maximum allowable chordal height error value
Figure BDA0003323287160000082
Figure BDA0003323287160000083
The resulting acceleration error is:
Figure BDA0003323287160000084
according to the formula of the chord height error and the acceleration error, the chamfering of the cutter is more stable and accurate;
and finally, combining the obtained data points on the tooth profile of the spiral bevel gear and an improved NURBS curve interpolation algorithm to ensure that the chamfering tool obtains an accurate and complete tooth profile arc surface chamfering track.
And 4, step 4: the whole control flow is integrated into software based on C #:
opening a Windows window application program under a Visual C # template in VS2015 software, compiling a control program, inputting a login name and a login password by a user on a login interface, and entering a functional interface, wherein the functional interface comprises functions of machine tool each-axis state detection, machine tool management, workpiece management, cutter management, database management, interpolation motion, NURBS curves and Matlab drawing calling.
And 5: the fixed height operation control card controls the machine tool to complete the cambered surface chamfering of the gear tooth profile:
after the operator inputs required parameters into the software on the computer, the operator calls the dynamic link library in the fixed-height motion control card and sends a high-speed pulse instruction to the machine tool so as to control the open type numerical control machine tool to finish the chamfering of the tooth profile cambered surface of the gear.
The tooth crest line equation calculation process of the gear in the step 1 is as follows:
the tooth cutting principle of the spiral bevel gear, and technological parameters and cutter parameters in the actual machining process are comprehensively analyzed to obtain that the parameters are related to the pitch cone surface of the spiral bevel gear, an analytic equation of an imaginary tooth line on the pitch cone surface at the center of a tooth groove of the spiral bevel gear is solved, and then an tooth crest line equation is derived according to the relationship between the obtained imaginary tooth line equation and the tooth crest line under an equivalent gear.
In the step 2, a three-dimensional curve graph of the tooth profile curve is simulated in Matlab software, and the steps of extracting a plurality of data points are as follows:
setting the configuration environments of C # and Matlab, calling Figure in Matlab by C # and embedding the Figure into a Winform window, drawing a three-dimensional graph according to the tooth profile curve equation calculated in the front, extracting corresponding data points to a TextBox control in C # and storing the data points to a txt file.
The chamfering tool in the step 3 is a fillet tool:
the traditional tooth profile chamfering process is to process an edge into an inclined plane of about 45 degrees, but a new small edge can be generated at the same time, so a chamfering cutter is selected, a cambered surface chamfering is adopted for the tooth profile edge, and the circular surface is smoothly connected with two sides of a tooth profile edge line, so that the edge of a gear can be completely eliminated.
The machine tool in the step 5 finishes chamfering the arc surface of the gear tooth profile:
the spiral bevel gear in the embodiment is a spiral bevel gear, the axes of the two gears are intersected, and the intersected angle is 90 degrees. The axis of the small spiral bevel gear coincides with the Z axis of the machine tool, the C axis of the machine tool drives the gear to rotate, firstly, the cutter executes a curved tooth chamfering instruction at the small end of the gear, after a complete tooth profile is finished, the cutter is lifted by a certain height, the C axis of the machine tool rotates by a certain angle, so that the gear to be processed rotates to the next complete tooth profile, the cutter starts to return to the initial position of the chamfer and starts to chamfer the next tooth profile, the operation is repeated continuously until the complete processing of the teeth of the small end of the gear is finished, and the tooth profile curved surface chamfer at the large end of the small spiral bevel gear and the tooth profile curved surface chamfer at the large end of the large spiral bevel gear are the same.
The foregoing shows and describes the general principles, essential features, and advantages of the invention. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that the preferred embodiments of the invention and described in the specification are only preferred embodiments of the invention and are not intended to limit the invention, and that various changes and modifications may be made without departing from the novel spirit and scope of the invention, which fall within the scope of the invention as claimed. The scope of the invention is defined by the appended claims and equivalents thereof.

Claims (5)

1. A method for controlling tooth profile cambered surface chamfering of a spiral bevel gear is characterized by comprising the following steps:
step 1: according to the spiral bevel gear structure and the gear cutting method, calculating and obtaining tooth profile curve equations of the large end and the small end of the small gear: the tooth profile curve equations of the large end and the small end of the small gear are obtained, firstly, a tooth crest line equation of the gear is required to be calculated, and then the tooth profile curve equations of the large end and the small end of the gear are calculated according to the structure of the spiral bevel gear and the coordinate relation of a machine tool;
step 2: simulating and extracting data points: simulating a three-dimensional curve graph in Matlab according to the tooth profile curve equation obtained in the step 1, and extracting a plurality of data points;
and step 3: in combination with a modified NURBS curve interpolation algorithm:
in the interpolation process, any node parameter u epsilon [ u ] of the NURBS curve is giveni,ui+1]The curves are represented in matrix form and the ith NURBS curve is calculated as:
Figure FDA0003323287150000011
wherein:
Figure FDA0003323287150000012
Figure FDA0003323287150000021
all coefficients in the formula (1) are determined by initially given weight factors, control vertexes and node vectors, so that the coefficients are selected only under the condition that the position of a current interpolation point is known, and the interpolation efficiency is improved;
carrying out parameter point densification processing on the feeding step length in the track space to ensure that the node vector increment delta u converted into the one-dimensional parameter spaceiFrom the formula ui+1＝ui+ΔuiThe parameter coordinate of the next interpolation point can be obtained;
according to the fourth-order Runge-Kutta algorithm, the node vector increment is calculated as follows:
Figure FDA0003323287150000022
will calculate the ui+1Substituting the position parameter into NURBS curve interpolation equation (1) to obtain the position parameter of the next interpolation point: p is a radical ofi+1＝p(ui+1)；
The self-adaptive control of NURBS curve interpolation reduces the interpolation step length by adjusting the interpolation speed, thereby reducing the chord height error; in each interpolation period, the chord height error is:
Figure FDA0003323287150000023
wherein: rhoiRadius of curvature,. DELTA.liInterpolation of the feed step length, Δ li＝viT, and T is an interpolation period, viIs the velocity, δiIs the chord height error;
the feed speed v can thus be approximatediSum chord height error deltaiThe relationship is as follows:
Figure FDA0003323287150000024
by maximum chord height error delta of machine toolmaxSelecting a maximum allowable chordal height error value
Figure FDA0003323287150000025
Figure FDA0003323287150000031
The resulting acceleration error is:
Figure FDA0003323287150000032
according to the formula of the chord height error and the acceleration error, the chamfering of the cutter is more stable and accurate;
finally, combining the obtained data points on the tooth profile of the spiral bevel gear and an improved NURBS curve interpolation algorithm, so that the chamfering tool obtains an accurate and complete tooth profile arc surface chamfering track;
and 4, step 4: the whole control flow is integrated into software based on C #:
opening a Windows window application program under a Visual C # template in VS2015 software, compiling a control program, inputting a login name and a login password by a user on a login interface, and entering a functional interface, wherein the functional interface comprises functions of machine tool each-axis state detection, machine tool management, workpiece management, cutter management, database management, interpolation motion, NURBS curves and Matlab drawing calling;
and 5: the fixed height operation control card controls the machine tool to complete the cambered surface chamfering of the gear tooth profile:
after the operator inputs required parameters into the software on the computer, the operator calls the dynamic link library in the fixed-height motion control card and sends a high-speed pulse instruction to the machine tool so as to control the open type numerical control machine tool to finish the chamfering of the tooth profile cambered surface of the gear.
2. The method for controlling tooth profile curved chamfer angle of a spiral bevel gear according to claim 1, wherein: the tooth crest line equation calculation process of the gear in the step 1 is as follows: firstly, solving an analytic equation of an imaginary tooth line on a pitch cone surface at the center of a spiral cone gear tooth groove, and then deducing an tooth crest line equation according to the relation between the obtained imaginary tooth line equation and the tooth crest line under an equivalent gear.
3. The method for controlling tooth profile curved chamfer angle of a spiral bevel gear according to claim 1, wherein: in the step 2, a three-dimensional curve graph of the tooth profile curve is simulated in Matlab software, and the steps of extracting a plurality of data points are as follows:
setting the configuration environments of C # and Matlab, calling Figure in Matlab by C # and embedding the Figure into a Winform window, drawing a three-dimensional graph according to the tooth profile curve equation calculated in the front, extracting corresponding data points to a TextBox control in C # and storing the data points to a txt file.
4. The method for controlling tooth profile curved chamfer angle of a spiral bevel gear according to claim 1, wherein: the chamfering tool in the step 3 is a fillet tool: the tooth profile edges adopt cambered surface chamfering, and the two sides of the tooth profile edge line are smoothly connected by using a circular surface, so that the edges of the gear can be completely eliminated.
5. The method for controlling tooth profile curved chamfer angle of a spiral bevel gear according to claim 1, wherein: the machine tool in the step 5 finishes chamfering the arc surface of the gear tooth profile: the spiral bevel gear is a spiral bevel gear, the axes of the two gears are intersected, and the intersected angle is 90 degrees; the axis of the small spiral bevel gear is superposed with the Z axis of the machine tool, and the C axis of the machine tool drives the gear to rotate; firstly, a cutter executes a curved tooth chamfering instruction at the small end of the gear, after a complete tooth profile is finished, the cutter is lifted, the C shaft of the machine tool rotates by an angle, so that the gear to be machined rotates to the next complete tooth profile, the cutter starts to return to the initial position of the chamfer and starts chamfering of the next tooth profile, and the operation is repeated continuously until the complete machining of the gear teeth at the small end of the gear is finished, so that the tooth profile arc surface chamfer at the large end of the small spiral bevel gear and the tooth profile arc surface chamfer at the large end of the large spiral bevel gear are the same.
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