渐开线齿轮滚刀设计

A NOVEL HOB DESIGN FOR PRECISION INVOLUTE GEARS: PART II38By Stephen P. Radzevich, Ph.D.AbstractThis pa per is a imed a t the development of a novel design of precision gea r hob for the ma chining of involute gea rs on a conventiona l gea r-hobbing ma chine. The reported resea rch is ba sed on the use of funda menta l results obta ined in a na lytica l mecha nics of gea ring. For solving the problem, both the descriptive-geometry-ba sed methods (further DGB-methods) together with pure a na lytica l methods ha l problems, which consequently ha ve a n a na lytica l solution. These a na lytica l methods provide a n exa mple of the a pplica tion of the DG/K-method of surfa ce genera tion ea rlier developed by the a uthor. For interpreta tion of the results of resea rch, severa l computer codes in the commercia l softwa re Ma thCAD/Scientificstra ight-line la tera l cutting edges of the hob with the stra ight-line cha ra cteristics of its genera ting surfa ce elimina tes the ma jor source of devia tions of the hobbed involute gea rs. The rela tionship between ma jorI can be downloaded at []. • MAY 2007 • GEAR SOLUTIONS 39W R= pitch plane of the auxiliary phantom rack Rλψg N = number of gear teethh N = number of starts of the involute hobg O = gear axis of rotation h O = hob axis of rotation.x h P = axial pitch of the involute hobR= auxiliary phantom rack of the involute hobh S = involute hob feed-rateT = the generating surface of the involute hobU = idle distance in gear hobbing operationg a = gear tooth addenduma R= the auxiliary rack tooth addendumg b = gear tooth dedendumb R= the auxiliary rack tooth dedendum.b g d = base diameter of a gear.b h d = base diameter of an involute hob.f g d = gear root diameterh d = gear hob pitch diameter.o h d = outside diameter of the involute hob.t g h = gear tooth whole depthm = gear modulus.b h p = base pitch of the involute hobc t = normal tooth thickness/g h C = center distanceg D = pitch diameter of the gear .o g D = outside diameter of the gearG = gear tooth surface being machiningΣ= cross-axis angleh ζ= hob-setting angle.b h λ = involute hob base lead angle (=90deg −)n φ= normal pressure angle .b h ψ= involute hob base helix angle g ψ= gear pitch helix angleh ψ= involute hob pitch helix angle ψR= auxiliary rack pitch helix angleg ω= gear rotation h ω= involute hob rotationg = gear to be machined h = involute hob to be appliedb.h g.h NomenclatureGreek SymbolsSubscriptsThe pitch diameter neither of the new hob, nor of the completely worn hob, could be used for the computation of parameters of design of the gear hob. For accurate computations it is rec-ommended to use pitch diameter of the partly worn gear hob that correlates with outside diameter of the cutting tool. Outside diameter of the new gear hob is equal to d o.h (Fig. 10), while outside diameter of the completely worn gear hob could be computed from the equation (d o.h − Δd o.h ). For computation of the outside diameter reduction Δ d o.h , the following approxi-mate equation.tan 2tan 5.585rho h rh hd L n ααΔ≅⋅⋅=⋅is derived. H ere it is designated that: L is adistance between two neighboring hob teeth that is measured along the helix on the outside cylinder of the hob (Fig. 10), αrh is clearance angle at the top cutting edge of the hob tooth, and n h is effective number of the hob teeth.For involute hobs with straight slots, n h is always an integer number, and it is always equal to the actual hob teeth number n (a)hwhich is usually in the range of n (a)h = 8~16 [because of this, the distance L can be computed from equation ∪L =(π⋅d o.h)/n (a)h]. For gear hobs with helical slots, the effec-tive hob teeth number n h is always a number with fractions. Moreover, the actual value of n h depends upon the hand of helix of slots. This is due to that in the last case the distance L is computed from the equation ∪L =(π⋅d o.h )/n (a)h ]+P x.h ⋅N h ⋅ cos λrf ⋅ sin λrf [1], [13], [14] and oth-ers. Here is designated: d o.h is outside diameter of the hob, P x.h is axial pitch of the hob, N h is the hob starts number, λrf is lead angle of the hob rake face (λrf is the signed value).Equation (25) is a simple one. It is an approximation, which returns reasonably accu-rate results of computation.The performed analysis of the gear hob design reveals that decrease of number of starts N h of the hob (Fig. 11) (a), increase of normal pressure angle φn (Fig. 12) (b), increase of the hob-setting angle ζh (Fig. 13) (c), and increase of the hob pitch diameter d h (Fig. 11 through Fig. 13) (d) result in reduction of the angle ξ of the rake surface orientation.The plots (Fig. 11 through Fig. 13) are cre-ated using commercial software MathCAD/Scientific . Unfortunately, the lack of capabilities of MathCAD/Scientific imposed restrictions on graphical interpretation of the functions ξ = ξ (N h ) , ξ =ξ(φn ), ξ =ξ(ζ h ), and ξ = ξ (d h ). The lack of capabilities is the sole reason that the listed functions are interpreted as a function ξ = ξ (d h ) under various values of the gear hob number starts N h (Fig. 11), normal pressure angle φn (Fig. 12), and the gear hob-setting angle (Fig. 13). However, Fig. 11 through Fig. 13 provide clear understanding of the impact of the above mentioned parameters of the gear hob design onto the rake face inclination (ξ).BEEN REGROUND.h OFTHE INVOLUTE HOB ONTO THE ACTUAL ORIENTA-TION OF THE RAKE PLANE DETERMINED BY THE ANGLE ξ ( M = 10 MM, φn = 20 DEG , ζh = 3 DEG , n h = 10 , αt = 12 DEG ).n h FIG. 13. IMPACT OF THE HOB-SETTING ANGLEζ h ONTO THE ACTUAL ORIENTATION OF THE RAKE PLANE DETERMINED BY THE ANGLE ξ ( M = 10 MM, 20 DEG φn = 20 DEG, N g = 1, n h =10, αt = 12 DEG ).40 GEAR SOLUTIONS • MAY 2007 • It is important to single out here that the hob pitch diameter could be significantly increased due to the application of hobs with internal location of teeth, the use of which allows hob-bing of numerous gears either in one set-up or simultaneously.Internal hobbing could be performed with the application of a gear hobbing machine ofspecial design [13].One could suppose that in the ideal case, the equality ξ = −ψh has to be observed. Actually, this equality is not of importance for the design of finishing, as well as of semi-finishing preci-sion involute hob. Finishing and semi-finishing gear hobs cut thin chips, the thickness of which is comparable with the hob cutting edge round-ness ρh . Therefore, not the rake angle but the cutting edge roundness directly affects the chip removal process in hobbing of precision involute gears.For precision gear finishing hobs of big modulus m , for example for semi-finishing and finishing skiving hobs, the top cutting edges are out of contact since the gear bottom land is completely machined on a gear roughing operation [8], [18].Therefore, the geometry of the top cutting edge is out of importance for the finishing and semi-finishing precision gear hobs of the developed design.The geometry of the active part of the cutting teeth of the involute hob is a subject of another paper to be submitted. Investigation of this problem is of importance, firstly because the rake face is not orthogonal to the generating surface T of an involute hob.An approximation of the rake surface of the gear hob could be feasible. In the event of approximation of the rake surface, the rake surface could be shaped in the form of a screw surface of that same hand as the hand of the screw involute surface of the generating sur-face T of the gear hob. Helix angle of the screwrake surface is equal to ψrf = 90˚−ξ .Either the rake surfaces or the clearance sur-faces of the worn gear hob could be reground. A novel technology of the hob regrinding opera-tion has been developed. A comprehensive analysis of the gear hob regrinding operation is a topic to be reported in another paper.4. Hob Design ExtensionHere we consider a precision hob for machining of a modified involute gear as an extension of the original design of a gear hob. The reported results of analysis of inclination of the rake sur-face of the involute hob teeth (see Section 3)FIG. 14. MODIFICATION OF THE GEAR HOB TOOTH PROFILE. • MAY 2007 • GEAR SOLUTIONS 41uncovered an opportunity of hobbing of modified involute gear (Fig. 14).The straight-line lateral cutting edges of the gear hob align with the straight-line characteristic E h of the hob. Searching for an opportunity of reduction of the rake face inclination (i.e. reduction of ξ ), one could turn his/her attention to a possibility of turning of the characteristic E h (and the gear hob cutting edge as well) through a certain angle ϕ about a point K on the pitch line of the auxiliary rack R of the gear hob. The rotation of E h definitely reduces the rake-face inclination ξ . However, at that same time the rotation of E h results in curved lateral profile of the auxiliary rack R m tooth 5. The last gives a possibility of hobbing of modified involute gear. For this purposes a gear hob of novel design is developed [17].The required angle ϕ can be computed from the Euler’s formulaHere, the principal radii of curvature R 1.T and R 2.T of the modified auxil-iary rack surface R mare equal to [1] [19]ϕ .A possibility of modification of the gear tooth profile could easily be illustrated by the characteristic curves of novel kind recently developed by the author [20], say by the An R (T ) -indicatrix of the first kind, and the An k(T ) -indicatrix of the second kind.The An R (T ) -indicatrix of the first kind could be given in matrix represen-tation This characteristic curve illustrates the distribution of normal radii of curvature R T (ϕ) of the surface T in differential vicinity of K (Fig. 15).The An k (T )-indicatrix of the second kind could also be given in matrixrepresentation This characteristic curve illustrates the distribution of normal curvature k T (ϕ) of the surface T in differential vicinity of K (Fig. 15).Figure 15 describes that the gear hob of the proposed design [17] enables any desirable value of the involute gear tooth modification (R T ).Both the characteristic curves An R (T ) and An k (T ) are derived using a generalized equation for the Plücker’s conoid [21], [22].The possibility of the involute gear tooth profile modification can also be proven in another way. For this purpose it is convenient to represent the well known equation for R T =Φ 1.ΤΦ 2.T in exploded form. Then, the expres-sions for R T (ϕ) and for k T (ϕ) could be replaced with the similar expressions R T (υ) and k T (υ) in terms ofυHere is designated υ = dV T /dU T .The extreme values R 1.T and R 2.T , as well as k 1.T and k 2.T occur at roots υ1 and υ2 ofHere, E h , F h and G h designate Gaussian coefficients of the first funda-mental form Φ1.T of the machining surface T of the involute hob. They arefunctions of the U h − and V h − parameters [see Eq. (12)], i.e. E h =E h (U h ,V h ), F h =F h (U h ,V h ) and G h =G h (U h ,V h ). The coefficients E h , F h and G h are derivedfrom Eq. (12) using for this purpose equations:Gaussian coefficients of the second fundamental form Φ 2.T of the machining surface T of the involute hob are designated as L h , M h and N h . They also are functions of the U h − and V h − parameters [see Eq. (12)], i.e. L h =L h (U h ,V h ), M h =M h (U h ,V h ) and N h =E h (U h ,V h ). The coefficients L h , M h and N hare derived from Eq. (12) using for this purpose equations:Both the characteristic curves R T =R T (υ) and k T =k T (υ) (Fig. 16) perfectly correlate with the An R (T ) - indicatrix of the first kind and the An k (T ) -indicatrix of the second kind of the surface T [1].Computation of parameters of design of the gear hob with modified tooth profile is almost identical to computation of parameters of designof the gear hob with non-modified tooth profile. The difference is just incomputation of the parameter R m and the distance d m . The parameter R m differs from the parameter R [see Eq. (13)], and the distance d m is not equal to the gear hob base diameter d b.h .Actual value of R is required to be expressed in terms of the gear hob tooth modification R T . For this purpose, it is convenient to solve an ele-mentary geometrical problem, say to determine coordinates of a certain point S of intersection of the circular arc of the radius R T (Fig. 14) with the centerline of the modified tooth profile. The point S is not shown in Fig. 14. Then, the parameter R m can be determined as a distance of the point S from the gear hob axis of rotation O h . Following the described routine,one could come with the equation for RmThe corresponding equation for the distance d m could be representedin the formwhereFIG. 15. DISTRIBUTION OF NORMAL CURV ATURE IN THE DIFFERENTIALVICINITY OF A POINT ON THE SURFACE T OF THE FETTE GEAR HOB(DIN 8002A, CAT.-NO 2022, IDENT. NO 1202055).STOR-LOC MODULAR DRAWER SYSTEM880 N. Washington Ave. Kankakee, IL 60901Toll Free: 1.800.786.7562 • Fax: 1.800.315.8769email: sales@42 GEAR SOLUTIONS • MAY 2007 • • MAY 2007 • GEAR SOLUTIONS 43The first principal radius of curvature R 1.T of the generating surface Tof the hob [Eq. (27)]The required value of angle ϕ[Eq. (28)]The angle ν * (Fig. A2) that projection of the lateral cutting edge onto X h Y h coordinate plane makes with the centerline is of the value of ν* = 3.835deg .The above computed design parameters of the precision involute hobyield computation of R * = 123.175mm and d *b.h= 16.478mm . These val-ues are obtained on solution of triangles (Fig. A2).Finally, Eq. (23) yields for ξFor computation of φ*rCONTINUED ON PAGE 50 >50 GEAR SOLUTIONS • MAY 2007 • 。