Scientific English writing The paper

Scientific English writingThe paperMesh Analysis and Test Verification on Two Degrees of Freedom LinearCutting Tools Enveloping Worm GearsLI Haitao1, WEI Wenjun2, DONG Xuezhu31College of Engineering,China Agricultural University, Beijing, 100083, China2 College of Engineering,China Agricultural University, Beijing, 100083, China3College of Engineering,China Agricultural University, Beijing, 100083, Chinalihaitao0402@AbstractIn order to improve the contact quality of Hindley worm gears, a dependent linear motion along the axis of the basic member is added to the linear cutting tools besides its rotation when the Hindley worm is cut, i.e. the single degree of freedom motion of the linear cutting tools is changed into two degrees of freedom one. So a new method of manufacturing enveloping worm and a new type of enveloping worm gears, which is called two degrees of freedom linear cutting tools enveloping worm gears, are obtained. The mathematical model of meshing analysis of the worm gears is built and the formula of calculating thehelical parameter of the linear cutting tools motion is given. The affecting regularities of the number of the thread of the worm and the helical parameter of the linear cutting tools motion on the undercutting line, the addendum thickness of the side tooth, the contact pattern, the length of the double line contact pattern on the worm and the micro contact quality on the worm gear are studied. A pair of worm gears is manufactured on a normal hob machine. The contact pattern test of the worm gears is carried out to verify the theoretical calculation results. The worm gears can be used with up to six threads of the worm; better contact quality and transmission performance than those of Hindley worm gears can be obtained by selecting suitable helical parameter of the linear cutting tools motion, and which has practical value.Keywords：Degrees of freedom; Tooth contact; analysis Worm gears Verification; Introduction1. IntroductionHindley worm gears are widely used and the leading brand is Cone Drive, America, which had a history of more than 80 years. The primal type Hindley worm gears are double-line contact, however, the intersecting line of the worm gear median plane and the tooth surface is a constant contact straight line, and the remaining contact lines distributing in the half tooth surface of the worm gear. It may result in not only the failing to make full use of the tooth surface, but also poor working conditions of the constant straight line which is prone to pitting corrosion.In order to improve the contact quality and load capacity of the Hindley worm gears, a number of authors have explored in different ways. ZAK [1] originally proposed the parabolic modification method. Qin Lin et al. [2] described several conventional modification methods, presented the formulas to do mesh analysis of tooth surface and made evaluations on microcosmic mesh quality of Hindley worm gears. Luo Jiashun [3]-[4] ameliorated the parabolic method to expand the contact area by adjusting the position of the two boundary lines. Wen Qingming [5] introduced the equal pressure angle profile modification and equal base circle profile modification of Hindley worm gears. Dong Xuezhu [6]-[7] et al. showed the microcosmic and macroscopical mesh quality of Hindley worm gears, put the modification methods into categories and recommended the preferred transmission type and the range of modification parameters. Zhou Liangyong [8]-[9] proposed a curvature modification method which eliminated the constant contact straight line. So far, profile modification removes the badly contact quality constant straight line, and improved the contact quality to some extent, however, single tooth contact, the contact line focusing on the left half of worm tooth surface, fails to make full use of the whole tooth surface.In the research described here, we proposed a new method of generating enveloping worm and a new type of enveloping worm gears, which was called two degrees of freedom linear cutting tools enveloping worm gears. The distinction was that a dependent linear motion along the axis of the basic member s was added to the linear cutting tools besides its rotation angleϕ, when the Hindley worm was cut. The equation of the helical worm surface 1∑, curvature parameters and cutting mesh equations were obtained in paper [10]. The worm processing and measurement may verify the theoretical results.2. Principle of two degrees of freedom linear cutting tools enveloping worm gearing2.1 The selection of coordinate systemThe hobbing cutter generating surface is the same with the helical worm surface and the center distance between the hobbing cutter and gear blank is a, gear crossed angle is 90 and the gear ratio are also the same with the worm gears. So it can make meshing analysis in the same sets of formulas.The processing coordinate systems are shown in Fig.1. The dynamic coordinate system(,,,)i i i j k O i j k σ (i=1,2, when i=1, it represents theworm and when i=2, it represents the worm wheel) is connected with the worm 1 and the worm wheel 2, while the initial position coincide with the static coordinate system00000(,,,)i i i j k O i j k σ whichisconnected with the machine rack.i O represents the public two-axis perpendicular pedal; the direction can be described1O points to 2O when 0102i i =; thecenter distancea equals to 12OO ; the gear crossedangle ∑equals to 90; the worm rotation angle around011k k = is 1ϕ and the worm wheel rotationangle around022k k = is 211i ϕ; the steering is determined by right-hand rule by the angular velocity vector 1ω and 2ω.Fig.1 Worm and worm wheel position and coordinatesystem2.2 Helicoid of two degrees of freedom linear cutting tools enveloping worm gearsThe helicoid equations surface1∑ of two degreesof freedom linear cutting tools enveloping worm gears are composed of two equations in paper [10], which are111101()(,)()r R k r ϕ=- (1)and21(,,)[cos d d d u u Dδϕϕ-Φ=Φ=+ 21(sin )sin ]0b d d d d d d r i p a u p ϕδϕ+--=. (2)The other equations can be denoted as,01010010()(,90)[()d e e d r a i R i ue ϕ=+-+00()]b e d d e r g p k ϕδ+, (3)D =, (4)1d d i ϕϕδ=+ , (5)1121d i i i=+∆ , (6)d a a a =+∆ ,(7)whereurepresents the parameters of the bladestraight line andb r represents the forming radius.In this paper, we discussed standard gears, so thegear ratio correction value i ∆ equals to zero and thecenter distance correction value a ∆ equals to zero,i.e.121,d d i i a a ==. The definition of the rotationmatrix R and circle vector function is described in paper [7]. 2.3 Mesh analysis on worm gearsIn paper [10], the helicoid of two degrees of freedom linear cutting tools enveloping worm gears 1∑ is considered to be generated by the virtual generating surfaced ∑ and its process is viewed as adouble-enveloping method, so we can make the meshing analysis in the general formulas shown in paper [7].(1) The equations of the family of surfaces{}101∑can be obtained after the rotation of the worm gear1ϕ, which are*10101111()(,)()r R k r ϕ=={}***011101111(,)(),,R k r x y z ϕϕ-= (8)*101011010()(,)(,90)()e d R k R i e ξαϕϕϕ=-- (9)*101011011()(,)(,90)R k R i Dηαϕϕ=-⨯00[()]e d d e ug p k ϕ-+ (10)()*101101011(,)(,90)n R k R i Dϕϕ=-⨯ []00()d e d e p g uk ϕ+. (11)(2) Equations of the relative velocity and relative speed of the worm gears()1221010101i j k ω=-+ (12) ()()***1221110110101i z y i x j υ=-+-*21101()i x a k - (13)(3) The equations of the worm wheel surface2∑ in2σ:()()()*2221102102201,(,90)r R k i R i r ai ϕ⎡⎤=---⎣⎦(14)120,0d Φ=Φ=ΦΦ= (15)112sin2ϕϕ-Φ= (16)21111sin ()cos ()22A B φφφφ⎡⎤⎡⎤Φ=-+-⎢⎥⎢⎥⎣⎦⎣⎦(17)221[cos ()sin d b d d A u ur p Dϕδϕ=+-+ 21(cos()]d d a i p u ϕ- (18)2221sin ()cos d b d d d i B u ur p Dϕϕδϕ-⎡⎤=--=-Φ⎣⎦ (19)There are two lines of contact on surface2∑ whentaken the same value: one of them is an original contact pattern2A ∑ with 20Φ=, the other is anew contact pattern2B ∑ in the worm wheel with20Φ=.(4) The meshing line of helicoid worm surface111sin()cos()A B ϕϕϕϕϕΦ=-+- (20)(5) Equations of the worm undercut line in the worm wheel surface are:()()1210101N ϕυψ=+Φ (21)()**10110101()()N N N ξξηηαα=+ (22)()()**11210111210101()()N ξξξξηκυατυα=++()*1210101()ηωα (23)()()**11210111210101()()N ηξξξητυακυα=+-()*1210101()ξωα. (24)The undercut line of2∑ could be obtained by (2),(14), (15) with 0ψ=.(6) The induced curvature function, induce normal curvature and induced geodestic torsion are212N ξξκ=ψ (25)212N ηηκ=ψ(26)12N N ξηξτ=ψ(27)(7) Root fillet radius of induced normal curvature andinduced curvature along the contact straight line are calculated by these two equations:121212N ξηκκκ=+ (28)121Nk aK ρ=. (29)(8) The transfer angle is()()1120101arcsinvt N ϕθυΦ-ψ= (30)3. Formulas of blade spiral parameters movementThe blade spiral movement parameterd P affectsthe two-lane distribution and double contact area length, so it is necessary to be correctly selected. Fig.2 shows the relative position of the worm wheel and the worm of two degrees of freedom linear cutting tools enveloping worm gears. The worm axis plane T coincides with the worm wheel gear median plane. If we turn the meshing line1ϕΦ into the worm axis T,1ϕΦ will divided the worm spiral surface into twoparts which are double-line contact area pointing to the top of the worm gear and single-line contact area pointing to the worm axis. The meshing line1ϕΦand worm gears the public perpendicular axes intersect at point P close to the worm axis2O , which results inreducing the length of the double-line area. So it is recommended to take212P P f d r C h =+, (31) where 2d represents the worm gear pitch diameter, 1f h represents the worm dedendum of thetooth andP C represents the'd P s spiral regulationfactor.Fig.2 The meshing line1ϕΦ two degrees of freedomlinear cutting tools enveloping worm gearsUltimately, the blade spiral movement parameterd P is obtained by the formula121211()(/2)d P P f P i a r i d C h =-=-, (9)where represents the worm throat section reference circle. By adjusting coefficient valueP C , which isdetermined by meshing analysis, the blade spiral movement parameter d P can be obtained simply andintuitively.4. Influence on worm gears of numbers of threads and spiral regulation factorThe series of corresponding transmission can be obtained by the number of threads from 1 to 6 on basic parameters of Hindley worm gears: the center distance250a mm =, the right-hand worm referencediameter190d mm =, the right-hand worm wheelreference diameter 2410d mm =, the formingradius80r mm =b . The other parameters can becalculated and the results are shown in Tab.1. From Tab.1, it can be observed thatP C varies with 1Zunder the situation of gear undercut. In Tab.1 M represents the minimum spiral regulation factorP Cwithin the tooth undercut and N represents the maximum spiral regulation factor P C without thecrest width tracking taper.Tab.1 Influence on worm gears undercut of number of threads1Z and the spiral regulation factor P CParameter svaluesNumber of threads 123456Number of worm teeth 40 40 48 40 40 48transmissi on ratio 40 20 16 10 8 8 M __-0.40.10.40.6 0.7 N 2.38 1.22 0.55 0.14-0.12-0.29The curvature undercut limit can obtained from helical worm surface1∑ by adjusting the parameterP C from -1.5 to 3.5 at a rate of 0.5. The position ofthe gear undercutd ψ is shown in Fig.3, where thenumber of threads is 1, 3 and 5 corresponding to Fig.3a, 3b and 3c. From Fig.3, it could be observed that with the decrease ofP C , the gear worm undercutline removes from the screw worm tooth surface to helical worm surface; the undercut line spreads more intensively in the out-meshing ends than that in the in-meshing ends and the greater1Z , the fewerundercut in out-meshing ends; if the dropping worm dedendum of the tooth method is applied, the crest width tracking taper may be avoided.Fig.3 Position of undercut line corresponding to thenumber of threads1. 1.5P C =-2. 1.0P C =-3.0.5P C =-4.0P C =5.0.5P C =6. 1.0P C =7. 1.5P C = 8. 2.0P C = 9. 2.5P C =10. 3.0P C = 11. 3.5P C =5. Influence on the edge tooth crest thick of numbers of threads and spiral regulation factorThe maximum of p C can be calculated by using theformula1b s t S k m ≥ of the edge tooth crest thick inpaper [7]. The results are shown in Tab.1, where the edge tooth crest thick parameter0.35s k =, theworm gear transverse module22/t m d Z =. It canbe seen from Tab.1, the greater 1Z , the smallermaximum of parameter p C . It is also easy observedthat for1Z =4, 5, 6, p C can ’t satisfy the requirement of worm undercut and the crest width tracking taper. However, if the dropping worm dedendum of the tooth method is applied, the crest width tracking may be avoided. For example, for the ensurence0.35s k =, under the condition 14,0.4p Z C ==, we can adjust the tooth crest arceccentricity of the edge worm dedendum of the tooth21.74e = and change the dedendum of the toothfrom 17.4mm to 16.53mm. So,it is suitable for the worm gear 1Z =4, 5, 6.length of double contact pattern of numbers of threads and spiral regulation factorFrom the analysis above, on the basis of Tab.1, we calculated the contact pattern and the length of double contact pattern which is shown as Fig.4. And also, the length of double contact straight line s L and thelength parameter L k in shown in Tab.2 along 1Zandp C . On the basis of Tab.2, the relation of 1Z ,p Cand L k can be drowned into a figure, which isFig. 4.Tab.2 The relationship between1Z ,p C s L and L kFig.4 The relationship of1Z , p C and L kThe mesh analysis of number of threads 1Z , thespiral regulation factorp Cmake influences on theworm undercut, the crest width tracking taper, the length of double contact straight line parameter and the contact pattern of the worm surface. From their analysis we can obtain the suitable range of the suitable range of the regulation factorp C of 1Zfor the two degrees of freedom linear cutting tools enveloping worm gears, as shown in Tab.3Tab.3 The relationship of1Z and p C1Zp C1 0~3.02 0~2.0 30.5~1.4 4 0.6~1.1 5 0.6~1.0 60.7~1.07. Test verification on contact pattern of the worm gearsIn paper [10], a two degrees of freedom linear cutting tools enveloping worm gears was processed by a common hobbing machine on the basic parameters: center distance125a mm =; the number ofright-hand worm threads 11Z =; the number ofworm teeth240Z =; the gear transmission ratio1240i =; the worm gear transverse module5t m mm =; the throat reference circle diameter 150d mm =; the forming radius 40r mm =b ; theworm wheel reference diameter 2200d mm =; thebladespiralmovementparameter563/d P mm rad =. The distribution of the contactpattern and the instant straight lines are shown in Fig.5.Fig.5 The distribution of the contact pattern and theinstant straight lines in the experimentThen, the worm gear blank with the same spiral surface could be processed in the method as shown above. After processing the cutter grooves and teeth grinding angle, the worm wheel could be obtained. The contact pattern would be gained after the last procedure of rolling the worm and the worm wheel together which is shown in Fig.6.Fig.6 Photo of the contact pattern in worm gearsIt can be observed that the contact pattern spreads to all of the tooth surface and it meets the theoretical results.8. ConclusionsBased on the analytical and experimental investigations presented in the research, the following conclusions may be drawn:1. A dependent linear degree of freedom adding in the process can improve the contact quality of Hindley worm gears. The worm gear surface covered withdouble contact straight lines which was totally different with the modification of Hindley worm gears, then a new type of enveloping worm gears could be obtained.2. Calculating formulas for linear cutting tools could be obtained by adjusting the spiral regulation factorP C in processing two degrees of freedom linearcutting tools enveloping worm gears.3. The two degrees of freedom linear cutting tools enveloping worm gears could apply to the number of threads up to 6, if the spiral regulation factorP C wasadopted and the crest dedendum modification method was added.4. In this paper, the best range ofP C was providedfor the two degrees of freedom linear cutting toolsenveloping worm gears.5. The two degrees of freedom linear cutting tools enveloping worm gears could be manufactured in the common hobbing machine and the reasonable agreement of the experimental values and the theoretical calculation shows that the proposed mesh analysis method is feasible.9. References[1]ZAK P S. Globoidnaya transmission [M].Moscow :Machinery,1962.[2]QI Lin，ZHANG Yaxiong，LI Shangwei，et al.Design of worm，V olume II. [M]. Beijing：ChinaMachine Press，1987. (in Chinese)[3]LUO Jiashun. On straight line tooth worm drivewith arcsurface of parabolic modification form[J]. Chinese Journal of Mechanical Engineering，1987，23(4)：28-42. (in Chinese)[4]LUO Jiashun. On straight line tooth worm drivewith arcsurface of parabolic modification form[J].Chinese Journal of Mechanical Engineering，1988，1(1)：75-89.[5]WEN Qingming. Research on invariable basiccircle modification of hindly worm[J].Mechanical Design，2001 (8)：45-48. (in Chinese) [6]DONG Xuezhu，BI Cheng. Tooth modification ofdouble enveloping worm gearing with straightside profile[J]. Journal of China AgriculturalUniversity，2001，6(4)：51-55. (in Chinese) [7]DONG Xuezhu. Design and modification oftoroidal worm gearing[M]. Beijing：ChinaMachine Press，2004. (in Chinese)[8]ZHOU Liangyong. Curvity modification oftoroidal worm[J]. Chinese Journal of MechanicalEngineering，2002，38(2)：112-115. (in Chinese) [9]ZHOU Liangyong. Modification theory andmanufacturing technology of toroidal worm[M].Changsha：National University of DefenseTechnology Press，2005. (in Chinese)[10]LI Haitao, WEI Wenjun, DONG Xuezhu. Theoryand application of surface generated bycurvilinear cutter edge with two degrees offreedom [J]. Journal of Mechanical Engineering，2010，46(9)：52-58. (in Chinese)[11]LI Haitao, WEI Wenjun, DONG Xuezhu. Twodegrees of freedom straight line-generatedenveloping worm gearing and manufacturingmethod: China , 200710063830.9[P]. 2007-02-12.(in Chinese)。