1、91 FTM 12The Combined Mesh Stiffness Characteristicsof Straight and Spiral Bevel Gearsby: K. Yoon, J.W. David and M. ChoiNorth Carolina State UniversityAmerican Gear Manufacturers AssociationTECHNICAL PAPERThe Combined Mesh Stiffness Characteristics of Straight andSpiralBevelGearsK. Yoon, J. W. Davi
2、d and M. ChoiNorth Carolina State UniversityTheStatements andopinionscontainedhereinare thoseof the authorand shouldnotbe construed asan official action oropinion of the American Gear ManufacturersAssociation.ABSTRACT:The combined mesh stiffness of spiralbevel gearsis oneimportant factorfor dynamic
3、analysis. The totaldeflection onthe contact line of a tooth pair is composed of bending, shear and tooth contact deflections. The bending and sheardeflections on the contact line of the gear tooth are evaluated by the finite element method with isoparametric shellelements, and the tooth contact defl
4、ectionsare evaluatedusing Hertzian contacttheory. Based on these deflections, thestiffness is obtained using the so called flexibility method, and then the combined mesh stiffness is obtained by applyingthe contact and load sharing ratios.Copyright 1991American Gear Manufacturers Association1500 Kin
5、g Slreet, Suite 201Alexandria, Virginia, 22314October, 1991ISBN: 1-55589-609-XAvTHE COMBINED MESH STIFFNESS CHARACTERISTICS OFSTRAIGHT AND SPIRAL BEVEL GEARSK. Yoon, Graduate StudentJ. W. David, Associate ProfessorM. Choi, Graduate studentDepartment of Mechanical and Areospace EngineeringNorth Carol
6、ina State University, Raleigh, North Carolina, U.S.A.INTRODUCTON the reasons why spiral bevel gears are superseding straightbevel gears in many applications2,3.As a gear pair operates, the combined stiffness of the There have been some attempts to evaluate themeshing teeth changes, and this is known
7、 to have strong combined mesh stiffness for spur andhelical gearsI,4,5,6,7.influence on the dynamic behavior of geared systems. The Chakraborty and Hunashikatti1 evaluated the combined meshstiffness charateristics for one engagement cycle of a spur gearexact meshing action of a pair of spiral gears
8、is not easilyunderstood because of the complex, three dimensional pair. Umezawa6 evaluated the mesh compliance of a helicalgeometry, but the relationship between a spiral bevel gear and gear pair, Choi7 evaluated the combined mesh stiffnessa straight-toothed bevel gear is substantialy the same as th
9、at characteristics for spur and helical gears. In order to evaluatebetween a helical gear and a spur gearl, the combined mesh stiffness, the deflections on the contact lineDuring operation of a gear pair, tooth pairs are in which consist of bending, shear and tooth contact deflectionscontact and at
10、any contact position of the meshing teeth the must be found. Most attempts for calculating tooth deflectionsequivalent tooth stiffness can be evaluated by considering are based on plate theory or beam theory. Krenzer8 used abending, shearing, and local contact deflections. Spiral bevel cantilever be
11、am formulation as the basis for the calculation ofgears normally have two or more tooth pairs in contact at all tooth deflection per unit load for spiral bevel and hypoid gears.times, and the curved oblique teeth come into contact gradually Holl5 used the method of finite differences to obtainand sm
12、oothly from one end to the other. This gives a approximate solutions for the deflections of a thin cantileversmoother engagement and more evenly distributes the tooth plate of finite width. In this study, the finite element method isloads. The overlapping tooth action transmits motion much used to d
13、etermine the bending and shear deflections, and themore smoothly and quietly than straight bevel gears, which are deflections due to contact between the mating teeth under load1are evaluated by applying Hertzian contact theory and using O _ C_ _ _(1) (2) (3)Weber and Benascheks equation9. Then, addi
14、ng up thesedeflections, the total deflections on the contact line of a tooth P_pair are evaluated. Finally, by using the influence function (3, O C_al t all aat Vmethod (flexibility method), the mesh stiffness of a spiralbevel gear pair for one engagement cycle is evaluated, and then P2the combined
15、mesh stiffness of a bevel gear pair is evaluatedby applying the contact and load sharing ratios, a,2 a_ %2P3METHOD OF ANALYSISa13 a,23 a33Fig. 1 The Flexibility Influence Coefficients of the UniformExact analysis of the tooth deflections of a meshing Cantilever Beam for the Points 1, 2, and 3spiral
16、bevel gear pair in various positions is not easy becauseat _, and the subscript p is the pinion tooth and g is the gearof the complicated relationship between load and deflections.tooth. From the above equations, the cumulative defiection(W)In general, the relationship between any displacements (Wi)
17、 of both teeth together is obtained,and any forces (Pi) is expressed as Wi = aij Pj by means of anW(x) = Wp(X) + Wg(x) - SLA(X,_) P(_)d_ (3)influence function aij where i, j=l, 2, 3 The flexibilityinfluence coefficient aijis defined as the displacement at i due A(x,_) = Ap(x,_) + Ag(x,_) = ap(x,_) +
18、 ag(x,_)to a unit force applied atj. As an example shown in figurel, + ae(x,_) (4)the displacement in terms of the flexibility influencewhere av and ag are influence fuctions due to bending andcoefficients are expressed as following:shear of the pinion and gear, and ac is an influence functionW 1= a
19、llP 1+ a12P2 + a13P3W2 = a21P1+ a22P2+ a23P3 due to tooth contact. The function, ae(x,_), is linearized underW3 = attP t + a32P2 + a33p3 the assumption that the half width of the contact surface isconstant along the contact line at any instant. Under theFor a gear tooth pair, the general situation i
20、s shown in figure 2assumption of mathematically exact geometry, the teeth are inin case of the distributed load,p(x), along the entire contactperfect contact even when unloaded. Therefore, the totalline, and the deflections(W) of the pinion tooth and gear aredeflection under the load will be constan
21、t along the contact linecalculated based on reference 10 as following:as shown in figure 2(c) and expressed by the equation below.Wp(X)= SLAp(x,%)p(%)d% (1) W(x)= Wp(x)+ Wg(X)=constant (5)Wg(x) = SLAg(x,_)p(_)d_ (2) The above equation is the global condition, and the equation(3) can be solved, and c
22、an be rewritten as following:where L is the length of contact line, A(x,_) is the influencefunction which is the deflection at x when a unit load is applied W(xi) = E A (xi,xj) p(xj) _n for i=l, 2. n (6)2W i = _ Aij Pj = constant for i= 1,2. n (7) _ xwhere _n = L/n ( L is the length of contact and n
23、 is the number wpof points on the contact line ). Therefore, the equation(7) caneasily be solved if the influence matrix Aij and the total wgcx)Iransmitted load are known, and also the stiffness can easily be , _evaluatedwith the valueof the total deflectionandapplied w_ xloads on the contact line.
24、The deflection of the equivalentstiffiness is Keq = Ptotal/V_ where Ptotal is a transmitted load.Accordingto theEq. (6)and(7),theequivalentstiffnessis _ . xW=Keq = p(x 1) _n/W + p(xz) 8n/W + . + p(xa) _n/W= Pl/W + P2/W + . + Pnr,v= ( pl+p2+ .+ pn)/N_r = Ptota!xN (8) Fig. 2 Deflection of a Tooth Due
25、to Load Distributionon the Contact LineThen we can rewrite Eq. (6) and (7) in malrix form,following 6W = A p 5n = A P (9)1 = A PW (10) d(yl)f(yl) = d(yz)f(y2) . d(yn)f(yn) (11)P/W can be obtained after solving the above equation. Since successive positions of contact are spaced at one baseAccording
26、to the Eq. (8), the stiffness, K_a, is the sum of all pitch (Pb) approximately on the line of action ponents of p/w for the discretized system. Then the l yi+l- YiI= Pb (12)equivalent compliance, Ceq, can be expressed as Ceq = 1/Keq. Since the sum of the load sharing ratio must be one,As we have see
27、n so far, the equivalent stiffness, Keq, can be f(Yl) + f(Y2)+ .-. + f(Yn)= 1 (13)evaluated with respect to a contact line, and by repeating the From the Eq.(10) we have n unknowns f(Yl), f(Y2). f(YQprocedure for various positions of the contact line during the and n equations, we can easilyengageme
28、nt cycle, we obtain the mesh stiffness during the calculate the values of f(Yl), f(Y2). ffYn)-With these f(Yi)entire engagement cycle2,6. Then the combined mesh and d(yi), for i=1,2 . n, the combined mesh stiffness,stiffness can be obtained with these mesh stiffnesses during CMS, is calculated asthe
29、 one engagement cycle by using the contact ratio and loadCMS (Yi)= 1/(d(yi) f(Yi) ) for i=1,2 . n (14)sharing for the whole engagement period. Let Yl, Y2. Ynbe the positions on the line of action in the plane of rotation EVALUATION OF THE TOTAL DEFLECTIONwhere n is the number of tooth pairs in conta
30、ct, f(Yl) be the AND INFLUENCE COEFFICIENTSload sharing ratio at Yiand d(Yi)be the flexibility at Yi. ThenBending, shear and contact deflections must beunder the assumption that errors in the gear and pinion areevaluated to obtain the influence coefficients, Aijs, which arezero, we can represent the
31、 behavior of the driven gear as the3composed of bending, shear influence coefficients of the the plane Mt is inclined at angle co to the projected pitch line,pinion and gear and the contact influence coefficient. AB, in Fig. 5(b). Then, using the finite element method, theExact anaysis of the deflec
32、tion is not easy because finite element meshes are generated along the contact line andspiral bevel geargeometryis very complexandthe relationship define the nodes to xt and Ytcoordinate on the plane M t asbetween load and deflection is complicated. Thus, the finite shown in Fig. 5(b). Next, the def
33、ined nodal coordinates, xtelement method is adopted for evaluating the bending andand Yt, are transfered to the normal direction which is theshear deflection on a contact line by using isoparametric shellelements which have 5 degree of freedom, three displacements coordinate xn and Yn in Fig. 5(a).
34、Finally, bending and shearand two rotations as shown in figure 3. deflections are evaluated on the contact Linewhen unit forcesare applied on each node on the contact line.i_+/ J(a) (b) (c)C,y,vi _b_2- - _ _, Fig. 4 (a) The Center-Surface of Spiral Bevel Gear Tooth(b) The Planes on a Spiral Bevel Ge
35、ar Tooth Surfacex,uj at its Mean Point P (* source of figure : Ref. 12)(d) re)Fig.3 (a)TypicalSolidIsoparametricShellElement.(b)AfterSpecializationtoShellElement.(c) Typical Node i, with Thickness Vector V3i I(d)Translationald.o.f,at Nodei in GlobalDirectionxy=z y(e) Rotational d.o.f, oci and _i at
36、Node i. _ z_The bending and shear deflections are evaluated on the _,_ _ “_=contactlineasfollowingunderthe assumptionthatthecontact x,occurs over the entire face width, not effective tooth surface. / / _Referring Fig. 4 and 5, the plane M is defined as a /(./,/ ,a)lcenter-surface of the gear tooth a
37、s shown in Fig. 4(a), and the _ _ Fin Fig.5(b) is defined by the projection of the plane, / I c_plane MtM, to the transverse direction, Zt, in Fig. 5. The plane Mti s / / A I_ _:also the projection of the plane To to the direction Zt. The cplane TQ is the tangent to the tooth surface and which is _
38、_x_(b)containing the projected contact line kk, and this contact line isinclined at angle coto the projected pitch line tangent tt on this Fig. 5 The Coordinate of a Spiral Bevel Gearplane. Therefore, it has been assumed that the contact line I on4Thedeflectiondue to contactis evaluatedas the _ _o 5
39、analytical solution based on the scheme of Weber and x -or- A ft.D -41_B-_- CBanaschek9. When two cylinders are in contact, we have the _ r7 -a- Dfollowing equations according to Hertzian theory. _“ o -_- E._“0bh2= ( 4qr/x ) (I-1)12)1 El + (I-a)22)/ E.2 (15) I 47o 2 3with 1/r = 1/r1+ 1/1“2, Pmax = 2
40、q/xbh where bh is the half “xt“qwidth of the surface of contact. In reality, most bevel geartooth profiles have elliptical contact surface, but it is assumed o I -0.5 0 0.5to be constant along the contact line at any instant. Pmax is Xt(ia)maximum pressure, q is the load per unit length of surface o
41、f Fig. 6 Total Deflection on a Contact linecontact, and rI and r2 are the radii of the cylinders, E 1 and F_,2are Youngs moduli, and 9t and _2 are Poissons ratios. The the contact line of a tooth pair. The graph A shows the totaldeflection on the contact line by adding contact deflection tocontactin
42、g teeth are idealized as rollers, and r 1, r2 are equal tothe bending and shear deflection on the contact line when unitthe radii of curvature of the tooth profiles at the point ofload is applied at position 1, and the graph B shows when unitcontact. In this study, r1and r2 are taken as radii of cur
43、vature load is applied at position 2, and so on. Thus, all points onat the pitch point of the mating teeth of the equivalent spur the graph in Fig. 6 are the elements of a 5 by 5 influencegears. Next, the displacement, Wh(x) under the Hertzian coefflent matrix for only one contact line at any instan
44、t, and apressure is evaluated the following equation which is derived tooth flexibility can be evaluated at any instant of time orby Weber and Banaschek 9 under a force per unit length, q. position of a rotating tooth pair. Thus, the tooth flexibility forvarious positions of the contact line during
45、the engagementW h= 2q(1-u 2) / _E kl(4hlh 2 / bh)- x)/(1-u) (16)cycle can be evaluated by repeating the procedure. Fig. 7 iswhere h 1 and h2 ale the length on the lines of application from the tooth flexibility for one engagement cycle. Then thethe contact point to the center of teeth. Therefore, th
46、eFig. 7 ToothFlexibility(consideringonetooth)flexibility due to contact, ae, isi i i i i ,= Wh(X)/ (qLc) (17) _ _ o str=l_t bewlgeor(_=1.o in)ao(x)spiral bevel geor (FW=1.0 in)o splrol bevel geor (FW=l.Sin) Ttransmitted load / Lc. _The total deflections on the contact line of a tooth pairOare obtain
47、ed by summing bending, shear and contact =deflection. As a numerical example, standard 20 degree, ! i i i ipressure angle, 35 degree of spiral angle and, 25 teeth-25 teeth o -_ -o._ o o_ 1 gear pair is selected, and Fig. 6 shows the total deflections on distoncefrom the pitchpointon the line of oeti
48、on (in)5combined mesh stiffness is evaluated by appling the contact i i i i iand load sharing ratio.v RESULT AND DISCUSSION _ /Jo dAs an example, three gear pairs are given; 1.0“ face o _-width of the straight and the spiral bevel gear, and 1.5“ face -;., ; .2 0,width of the spiral bevel gear. In al
49、l case, both pinion anddistance from the pitch paint on the line of action (in)gear have 25 teeth, and tooth has a 20 degree pressure angle.Fig. 8 The Combined Mesh StiffnessThe 1.0 face width of the bevel gears have a 5.482 diametral of the 1.0“ Face Width Straight Bevel Gearpitch, and 1.5“ face width of the spiral bevel gear has a 5.088todiametralpitch. ,_ “Fig. 8 shows the combined stiffness of the straight K“I,ibevel gear pair which has 1.7 contact ratio. The stiffness doesno
