1、14FTM13 AGMA Technical Paper A Practical Approach for Modeling a Bevel Gear By B. Bijonowski, Arrow Gear Company 2 14FTM13 A Practical Approach for Modeling a Bevel Gear Brendan Bijonowski, Arrow Gear Company The statements and opinions contained herein are those of the author and should not be cons
2、trued as an official action or opinion of the American Gear Manufacturers Association. Abstract The modern bevel gear design engineer is often faced with knowing the basic appearance of the bevel gear tooth that he is designing. The geometry of the bevel gear is quite complicated to describe mathema
3、tically, and much of the overall surface topology of the tooth flank is dependent on machine settings and the cutting method employed. AGMA 929-A06, Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius, lays out a practical approach for predicting the approximate top land thicknesse
4、s at certain points of interest regardless of the exact machine settings that will generate the tooth form. The points of interest that AGMA 929-A06 is concerned with consists of toe, mean, heel, and the point of involute lengthwise curvature. The following method expands upon the concepts described
5、 in AGMA 929-A06 to allow the user to calculate not only the top land thickness, but the more general case, the normal tooth thickness anywhere along the face and profile of the bevel gear tooth. This method does not rely on any additional machine settings; only basic geometry of the cutter, blank,
6、and teeth are required to calculate fairly accurate tooth thicknesses. The tooth thicknesses are then transformed into a point cloud describing both the convex and concave flanks in a global Cartesian coordinate system. These points can be utilized in any modern computer aided design software packag
7、e to assist in the generation of a 3D solid model. All pertinent macro tooth geometry can be closely simulated using this technique. Furthermore, a case study will be presented evaluating the accuracy of the point cloud data to a physical part. Copyright 2014 American Gear Manufacturers Association
8、1001 N. Fairfax Street, Suite 500 Alexandria, Virginia 22314 October 2014 ISBN: 978-1-61481-105-3 3 14FTM13 A Practical Approach for Modeling a Bevel Gear Brendan Bijonowski, Arrow Gear Company Introduction The first question that comes to mind from any engineer presented by this paper is why would
9、anyone be interested in a close approximation to the tooth flanks of a straight or spiral bevel gear? That is a very valid question. A bevel gear designer is typically interested in the minute details of the tooth flanks surface topology to gain the optimized contact conditions under the loads that
10、the bevel gears will be operated. The method presented in this paper does not address any of these needs. Purpose One of the main purposes of this method is to create a working three dimensional model for visual interrogation of fits and proportions. The bevel designer is typically concerned with th
11、e proportions of a myriad of geometric values describing the basic layout of the tooth form. Many of the values like top land balance, slot width tapers, spiral angle adjustments, etc., are hard to visualize for the engineer, and their impact to the tooth form can be drastic. The gear engineers tool
12、 kit should include a method for generating a three dimensional model of a bevel gear, as the dependency of using CAD for the general engineer is rapidly increasing. Many application engineers are demanded to provide three dimensional models to their customers. These customers utilize their CAD syst
13、em to validate fits and clearances between components in their systems. Gears should not be left out of this analysis. Advancements in 5 and 6-axis machining of gears are rapidly approaching the precision and capability of dedicated gear generating machines. Most of these machining centers require a
14、 three dimensional model for programming. If the application engineer provides a fully developed gear model to their customer, the 5 or 6-axis machining techniques may become very attractive. For this case alone, many gear companies opt out of providing models of their designs. Background The method
15、 outlined in AGMA 929-A06, Calculation of Bevel Gear Top Land and Guidance on Cutter Edge Radius, describes how to calculate the top land thickness for a bevel tooth at specific points of interest. These points of interest are at the toe, mean, heel, and point of involute lengthwise curvature. Reaso
16、ns regarding the purpose for these calculation points are beyond the scope of this document. The formulas inside AGMA 929 can be generalized so that a set of equations can be devised to calculate the normal circular tooth thickness anywhere along the profile and length of the tooth. Why is this meth
17、od approximate The geometry of a spiral bevel gear is mathematically complicated. The machine settings used to create a spiral bevel gear, whether using face milling or face hobbing, adjust the final flank form of the teeth. A strong understanding of the machine settings and motions are necessary to
18、 achieve an accurate tooth model. AGMA 929-A06 uses the technique of a virtual spur gear to approximate the profile of a bevel tooth in the normal plane without knowing the motions of the machine generating the final form. In doing so, the tooth thicknesses can be calculated quite simply using tradi
19、tional methods for spur gears. The virtual spur gear technique assumes that the tooth form will follow an involute in the profile direction in the normal plane. This is only a close approximation to the true form of a spiral bevel tooth. Most spiral bevels follow the octoid tooth form, which is simi
20、lar but not the same as the involute tooth form found on most cylindrical gears. What is missing in AGMA 929-A06 to complete a model A generalized set of equations can be produced from the content of AGMA 929-A06 to calculate the normal tooth thicknesses of a bevel tooth anywhere in the profile or l
21、engthwise direction. The purpose of this document is to fill in the gaps of AGMA 929-A06 so that a model can be generated. The majority of the content of this document is pertaining to how the normal tooth thicknesses are oriented in three dimensions, and resolving these thicknesses into an array of
22、 Cartesian coordinates. See Figure 1 for a 4 14FTM13 visual depiction of the calculation method described within this document. Additional information regarding the terminology described within the figure is in the subsequent sections. This methods shortcomings Enough discussion has taken place to i
23、lluminate the reader as to why this method creates only an approximate method, but there are additional shortcomings worth mentioning. This method does not currently take into account the root fillet. The coordinates calculated in this method are strictly points following the involute curve that des
24、cribes the approximate flank form. Coordinates for the root fillet is outside the scope of this document. The other major shortcoming is that all subsequent formulas are for spiral bevel gears without a hypoid offset. Additional provisions would need to be made to generalize the formulas to account
25、for hypoids. Coordinate system definition All bevel gears are designed using a reference right cone called the pitch cone. The pitch cone is used as a basis for describing all other geometric entities of the bevel gear. Since describing the motions of the generating process in three dimensions would
26、 be hard to comprehend, the general practice is to unwrap the surface of the pitch cone into a tangent plane, called the pitch plane. The point at the top of the pitch cone is called the pitch apex. The pitch apex is significant because the axes of both gear and pinion intersect at this point. Figur
27、e 2 displays the pitch cone, shows the pitch plane unwrapped, and also describes the global Cartesian coordinate system, CG. In addition, Figure 2 shows the pitch cone sectioned through the YZ plane. This describes the definition of the pitch angle, , and the face width, F, of a part. The global coo
28、rdinate system follows the right-hand rule and its origin is located at the pitch apex of the member to be modeled. Basic generation The majority of all generated spiral bevel gears are manufactured in one of two processes, face milling or face hobbing. When compared, Both manufacturing methods have
29、 advantages and disadvantages. For the purposes of this method, a brief understanding of the generating method utilized during these processes is necessary to realize the three dimensional model. Figure 1. Visual depiction of calculation method 5 14FTM13 Figure 2. Pitch cone Face milling The face mi
30、lling manufacturing method employs a circular cup shaped cutting tool moving in a timed relationship with the work piece to roll through the gear blank and generate an individual slot. The cutter is then withdrawn and the work is indexed to the location of the next slot and the process repeated. See
31、 Error! Reference source not found Figure 3 displays three instances of the cutter as it passes through the work piece; these points are at the toe, mean, and heel of the crown gear. The path of the cutter sweeps a circular arc in the lengthwise direction with the same radius as the radius of the cu
32、tter, nullnull. The axis of the crown gear, Cg, is known as the machine center. The local coordinate system for the pitch plane is located at the machine center. Xpp correlates to X and Ypp lies along the line describing Aoin Error! Reference source not found Figure 3. Generating triangle in pitch p
33、lane for face milling 6 14FTM13 The cutter axis, Cc. This location can be found by: coscmVR (1) sinmc mHA R (2) ,cCVH (3) The sign of the vertical term, V, may either be positive or negative depending on the hand of spiral to be modeled. Figure 3 shows a right hand member (use negative value for V w
34、hen calculating left hand members). The cutter sweep angle, r, is necessary to determine how much rotation is used during the cutting process. When calculating points to model the gear tooth, these will be the endpoints of the working portion of the cutter path. 22SVH (4) cos cos22222 222-1 -1co cir
35、ccSRA SRASR SR (5) cos cos22222 222-1 -1co cmroccSRA SRASR SR (6)ri r ro (7) Discrete points can be calculated along the cutter sweep on the pitch plane. Using the general parametric formulas for a circle, the solution for a point, p, along the cutter path is: cosxcpt VR t (8) sinycpt HR t (9) Where
36、 t is a parameter that has the following ranges: For left hand members, mri mrot (10) For right hand members, 22mri mrot (11) This cutter path can be broken into as many discrete sections as desired. Face hobbing The face hobbing manufacturing method is a continuously indexing process. The cutting t
37、ool has groups of staggered blades. The work piece moves in a timed relationship with the cutter so that a group of blades in the cutter passes through a slot of the work piece. The face hobbing method generates an extended epicycloidal shape in the lengthwise direction. See Error! Reference source
38、not found. for a detailed layout of the face hobbing generating triangle. Since the lengthwise shape of a face hobbed part creates an extended epicycloidal shape, a little trigonometry is necessary to calculate the discrete points along the path created by the cutter. The crown gear tooth count, sin
39、cNN (12) The lead angle of the cutter, sin cos-1 msmccANRN (13) 7 14FTM13 Figure 4. Generating triangle in pitch plane for face hobbing The first auxiliary angle, 2 (14) The center distance from crown gear center to cutter (Radial), 2cos221mcmcSARAR (15) The second auxiliary angle, coscos-1 mm1cs1cA
40、NNSN (16) The second roll angle, 21m (17) The auxiliary roll angle, 2rh 2 (18) The radius of the roll circle, sincosc2rhR (19) The radius of the primary circle, 112S (20) The first roll angle (assuming rolling without sliding), 2121(21) 8 14FTM13 Now that the dimensions of the cutting cycles epicycl
41、oid, the cutter path can be calculated, it would be very difficult to determine the angle of sweep that the cutter makes during the generating process because the cutter axis does not stay stationary during the cutting cycle. The face width of the bevel gear being modeled will be subdivided into dis
42、crete portions. The roll angles need to be recalculated for each discrete location individually. The second roll angle as a function of cone distance, cos22221c-1p21cSRAASR(22) Where, ioA AA (23) The first roll angle as a function of cone distance, 2p1 p21AA(24) The local Cartesian coordinates descr
43、ibing a point, null, along the cutter sweep path can be calculated. sin sinx c p1 p2 1 2 p1pR (25)cos coscp1p212p1ypR (26)Wrapping the pitch plane The pitch plane is a tangent plane to the lateral surface of a right cone. This cone, described earlier, is called the pitch cone. The diagrams take into
44、 account the setup of the machine, the cutter size, and the motion of the cutter. The rotation of the work piece also needs to be accounted for. This can be accomplished by wrapping the pitch plane around the pitch cone. This will transform the local coordinates calculated for the cutter sweep path
45、into a global coordinates. These points in the global coordinates will define the center of a tooth slot. The following formulas will transform a point, p, from the local XppYpp plane to the global Cartesian coordinate system, CG. The first step is to determine the location of the point in the globa
46、l Z axis direction, cos22pxyzpp (27) Figure 5. Normal view to pitch plane displaying wrap point 9 14FTM13 Calculate the rotation angle that the point will wrap around the cone, 1tansinxpypp(28) Calculate the radius of the cone at location, zp, tanppRz (29) Convert the cylindrical coordinates for the
47、 wrap point into global Cartesian coordinates. cospp pxR (30) sinpp pyR (31) Therefore, all the local cutter positions can be wrapped and transformed into the global Cartesian coordinate system. Calculating local cutter coordinate system Since AGMA 929 effectively calculates the normal circular toot
48、h thicknesses at a specific spot along the cutter path, the next step is to determine the correct orientation of the normal plane. A complete coordinate system will be oriented to have the xnynplane normal to the cutter path with the origin at each global coordinate of the cutter path calculated pre
49、viously. This coordinate system will be called Cn. Tangent axis, znThe tangent axis is defined by a vector that is tangent to the cutter sweep path at the location of the wrapped point. There are a couple of options for calculating this tangent vector. One could calculate the first derivative of the cutter sweep path formulas for bo
