machine design, Vol.3(2011) No.2, ISSN 1821-1259 pp. 75-78

*Correspondence Author’s Address: University of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva c. 6, SI-1000 Ljubljana, Slovenia, gorazd.hlebanja@fs.uni-lj.si

Original scientific paper

PARABOLIC WORM-GEARING CONTACT CIRCUMSTANCES

Gorazd HLEBANJA1, * 1 University of Ljubljana, Faculty of Mechanical Engineering, Ljubljana, Slovenia

Received (20.03.2011); Revised (05.05.2011); Accepted (27.06.2011) Abstract: Recently proposed parabolic type worm-gearing is distinguished by a convex-concave contact which in general implies higher load capability, improved reduced radii of curvature along the path of contact, and thus better sliding circumstances, which means lower friction and power losses. The worm tooth shape is defined in the axial section by a generalized parabola, which enables progressive curvatures of profiles and implies a convex-concave contact through the entire contact area. The corresponding shape profile implies a progressively curved path of contact and thus a closer fit and correspondingly thicker oil film and better lubrication already in the contact start area. From the basic worm profile in the axial plane, the worm flank surface and the mating worm gear profile and flank surface are derived. Kinematic circumstances are discussed, disclosing the emergence of contact lines (surfaces). The essential characteristics of the proposed worm-gearing are that its entire teeth flank surfaces in contact are involved in power transmission and that concave-convex contacts exist anywhere on flank surfaces. Recent study, conducted at the FZG of the TU München, supports the theory. Key words: power transmission, worm-gear drives, concave-convex contact; parabolic profile 1. INTRODUCTION Archimedes is considered as the inventor of the worm drives [1], which have become inadmissible in various technical applications since then. They can be found in mining industry, manufacturing machines in transporting conveyors, etc., due to a high reduction of the rotational speed in a small volume [2]. A continuous development has made worm-gearings feasible even in high precision applications like mechatronic machines, e.g. robots and automotive industry. Thus, worm-gearings are still capable of improvements. Many different wormgearing types, e.g. ZA, ZK, ZN, ZI, were developed during decades with the prevailingly convex-convex contact, which is due to involute contact pairs. On the other hand, the Holroyd worm tooth form derives from modifications and optimization of the worm thread, and to recommendations in BS 721. This tooth form is characterized by the particularly shaped contact surface enabling good lubrication and low contact pressure [3]. This kind of gear transmission can reach the efficiency up to 98% and a very long life cycle. Thus, the worm-gearings with 873 kW, the module 21 mm, z1=5, z2=62, centre distance 762 mm, and 15000 hours life cycle are available. A high quality fabrication is assumed. Yet another example of the high quality worm-gearings are Flender’s, with the trademark name Cavex [4], having the concave-convex contact. Worm gear sets ranges with the centre distance up to 1400 mm, the module up to 40 mm, the number of threads up to 12, the worm diameter up to 400 mm, and power transmission up to 200 kW. A high quality assures a good alignment of the teeth flanks. Other solutions are also available in the precision technique [5, 6].

2. PARABOLIC WORM-GEARING GEOMETRY

The parabolically profiled worm-gearings derive from the experience with gears, formed with a curved (S-shaped) path of contact [7] and from analytical considerations of Holroyd’s [3] and Flender’s [4] worm drives. A worm tooth shape is defined in the axial plane by:

( )[ ]nppp bzbay −−⋅⋅= 11 (1)

where y and z are coordinates, aP is a height factor, bP is a width factor and n is a power. Actually Eq. (1) represents a generalized higher order parabola with its origin in the pitch point C. Fig. 1 illustrates the worm flank definition and the path of contact emergence, whereas Fig. 2 reveals the mating worm drive. The path of contact is generated by a trace of the contact point U0i of the driving worm tooth flank and the driven worm gear tooth flank surfaces. The contact point U0i moves from the starting point AU to the end point which is in the pitch point C, as illustrated in Fig. 2. In order to define the worm and worm gear teeth profiles in planes parallel to the y-z plane, i.e. the worm axis plane in which the basic worm tooth profile is defined by Eq. (1), and the corresponding worm gear tooth profile. Parallel planes to the worm axis plane should be defined whereas in each of them a worm tooth profile can be defined. Helices can be drawn on the worm tooth flank surface. These are defined by their circular radii ρi and helical angles γi. A point on an arbitrary worm profile is Pki, i – index in the radial direction and k – index in direction normal to the worm axis plane. Thus, the coordinates of an arbitrary point Pki are

Gorazd Hlebanja: Parabolic Worm-Gearing Contact Circumstances; Machine Design, Vol.3(2011) No.2, ISSN 1821-1259; pp. 75-78

76

iiiiPPki

iiiPPki

-=zz

--=yy

γωρ

ωρ

tan

and)cos1(

0

0 (2)

It is also true sinϕi=ak/ρi. In this way the worm tooth flank surface is defined. Similar to the estimation procedure for the path of contact and the teeth profiles in the worm axial plane, illustrated in Fig. 1, is estimation of profiles in any parallel plane k. Thus, the point Pki is translated in the axial direction for the distance zPki, which is modified due to a changed radius of rotation, to the point Uki, and rotated from there around the worm gear axis O2 to the point Gki on the worm gear profile gk. Thus paths of contact in any parallel plane and counterpart worm gear profiles can be defined.

path of cont tacP

C

A

worm datum line

wor

m to

oth

f lan

k

worm

gea

r tooth

fl na k

worm gear reference circle

worm

worm gear, z=24

Fig.2. Worm and worm gear in the axial section 3. WORM-GEARING CONTACT LINES Manufacturing process of worm gear tooth profiles is based on the fact that a gear cutting edge point can be defined for each point of the path of contact. The cutting edge point forms a helical line on a worm tooth and consequently it forms a worm tooth surface when moved in the axial direction and at the same time a worm is rotating in synchronized manner. This is the easiest way to define the contact lines on a worm gear tooth, which are defined in detail, based on velocities distribution, in [8]. The contact lines, which have been calculated in Matlab and additionally verified by an ACIS based modeller, Fig. 3, are illustrated in Fig. 4.

Fig.3. Computer modeled contact surfaces of a worm gear

contact linevgivgi vgi

vgi

tooth flank top

tooth flank bottom

wor

m a

xial

pla

ne

Fig.4. Worm gear tooth flank contact lines The contact lines, Fig. 4, clearly distinguish two cases: a) a helical contact part from the contact start to the contact line deflection (and similar part towards contact line end) enabling thread-like movement of the contact spot, and b) sliding motion contact in the worm gear flank central area, which is an important feature of lubrication conditions. Influence of the reduced radius of curvature ρred on the oil-film thickness is of considerable importance. Keeping other factors (viscosity, load, summary velocity, elasticity module) in Dowson-Higginson’s equation [9] constant, it can be expressed by: h=κ⋅ρred

0,43. The expression κ for known parameters can be used as a normalization factor when representing functional dependence of oil film thickness on ρred, as discovered from Fig. 5, whereas Fig. 6 presents the reduced radii of curvature along the path of contact in the axial plane.

V

t

n

E

C V’

AP

z

yu

b = 2 mp ·

A

C

Z0i

zu

y

90-A

worm flank zonepath of contact zone

90-

E,

pa ht of contactP0i

worm datum line

active worm tooth flank

theoretical worm tooth flank curve

reference c ri cle

h

G0iU0i

ab

pp

·

Cp

Cg

p0

worm gear tip diameterAU

Fig.1. Worm tooth basic profile and the corresponding path of contact in the axial section

Gorazd Hlebanja: Parabolic Worm-Gearing Contact Circumstances; Machine Design, Vol.3(2011) No.2, ISSN 1821-1259; pp. 75-78

77

20 40 60 80 10010

2

4

6

8

red(mm)

h/κ

Fig.5. Functional dependence of oil film thickness on ρred The proposed worm drives have been manufactured and assembled in a standard housing of a Hydro-Mec s.p.a. to experimentally verify such an arrangement in proper working conditions. Thus, such a worm-gearing with module m=3 mm and axial distance a=50 mm consisting of a single threaded worm and a worm gear with z2 = 21, Fig. 7, was successfully tested under load and teeth flanks did not suffer any considerable damage.

C

path of contact of teeth flanks

datum line of worm teethC

red

16

23

34

56 mm

10

tooth f alnk

worm g

ear

U1

U2

U4

U3

U5

1,2 p.reference circle

P 2

P 3

P 4

P5

P 1

G 2

G 3

G 4

G5

A

worm gear

worm gear top circle

worm

toot

h fla

nk

Fig.6. Radii of curvature along the path of contact

in the axial section

Fig.7. Parabolic worm drive mounted in the industrial worm drive housing

The profile geometry leads to a conclusion that the contact area expands over the entire contact area of both teeth flanks, which consequently implies better conditions for power transmission. The basic feature of the proposed worm-gearing is that the teeth flanks assure progressive curvature and continual concave-convex contact. Worm and worm gear meshing in such an arrangement generates a better lubricating oil film, resulting in better EHD lubrication conditions; therefore, reduced energy losses and lower wear damages are anticipated. Tests with the worm-gearing inserted in the industrial case were also promising. So we proceeded with a programmatic simulation.

4. SIMULATION WITH SINETRA PROGRAM [10]

The Program SNETRA is specialized in worm-gear calculations. The program can be used to calculate contact lines, figure of carrying capacity and line load distribution, provided that the flank form is modeled in Sinetra. A standard worm-gearing with axial distance 100 mm, module 4 mm, worm gear width 30 mm and gear ratio 41/2 was used in this context. The chosen parabola parameters were n=2,66, bp=8 and ap=1,4. Such an arrangement can be compared to other worm-gearing types, e.g. involute type (ZI). Synthetic lubricating oil (S2/220) was used. Material of worm was steel and that of worm gear bronze alloy CuSn12Ni. Rotational speed of the worm was 500 rpm and the worm gear shaft torque 380 Nm. The simulated contact lines course, shown in Fig. 8, resembles course of Cavex ZC-worm-gearings.

Fig.8. Computer simulated contact line course for the parabolic worm drive (worm gear – above, worm – below)

Fig.9. Reduced radii of curvature along the contact lines

Fig. 9 shows reduced radii of curvature along the contact lines as computed by Snetra program. Fig. 6 indicates the shape of the reduced radii of curvature curve along the path of contact in the axial direction where the curvature

Gorazd Hlebanja: Parabolic Worm-Gearing Contact Circumstances; Machine Design, Vol.3(2011) No.2, ISSN 1821-1259; pp. 75-78

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progressively grows from the wheel root to its tip. This conforms to the curvatures in Fig. 9.

Fig.10. Local Hertzian pressures

Fig.11. Distribution of line load Furthermore, Fig. 10 represents local Hertzian pressures along the contact lines. According to the concentration of the contact lines in vicinity of the worm gear root area, rather high pressures in this area can be expected. The line load distribution, Fig. 11, appears rather uniform with two pick areas toward worm gear sides. 5. CONCLUDING REMARKS The basic feature of the proposed worm-gearing is that the teeth flanks assure progressive curvature and continual concave-convex contact. Worm and worm gear meshing in such an arrangement generates a better lubricating oil film, resulting in better EHD lubrication conditions; therefore, reduced energy losses and lower wear damages are anticipated. An experimental worm-gearing loaded under working conditions verified theoretical considerations. Computer simulation also confirmed the contact theory of the proposed gearing. The computer simulation discovered that the contact line flow appears similar to the theoretical one. However, the theoretical contact has two distinct modes, the threaded movement near the worm gear sides and sliding area in the central part. The former is missing in Fig. 8. One can also observe contact line concentration near the worm gear root part, which can be also observed with the geometrical modeling, where contact lines emerge up to the worm gear root contact area. “Irregularities” can be attributed to various factors, among other numerical resolution. Fig. 12 photo represents the worm gear wheel of the worm-gear assembly from Fig. 7. One can observe clearly visible traces of the worm on the worm gear flanks. These traces flow over the entire gear width, yet contact lines

course appears multi directional, which is similar to the theoretical one.

Fig.12. Worm gear wheel

Computed reduced radii of curvature indicate good lubrication conditions particularly in the worm gear tip area, where these radii are the longest. These first computations with Snetra program in general confirm theoretical considerations. However, additional simulations with changes in geometry and in working parameters are still necessary in order to get clear picture. ACKNOWLEDGEMENT The work has been partly conducted in cooperation with the Institute for machine elements (FZG) of the Technical University Munich. Mr. Sigmund provided figures made by Snetra program. REFERENCES [1] SEHERR-THOSS, H.Chr. Graf v. (1965) Die

Entwicklung der Zahnrad-Technik. Springer Verlag, Berlin, Heidelberg, New York, p. 48.

[2] NIEMANN., G., WINTER, H. (1983) Maschinen Elemente, Band III, Springer Verlag.

[3] JÜNGEL, H. (1992) Holroyd-Zahnform für Schneckenradsätze. Antriebstechnik (Mainz, 1982), Vol. 31, No.8, p. 58-60.

[4] Flender Worm Gear Units. A.Friedr. Flender AG, http://www.flender.com/.

[5] Renold Worm Gears. Renold plc, http://www.renold.com/.

[6] PREDKI, W. (1995) Status of worm gear drive development, Gear Transmissions’95, Sofia, Bulgaria.

[7] HLEBANJA, J., HLEBANJA, G. (2005) Anwendbarkeit der S-Verzahnung im Getriebebau : Nichtevolventische Verzahnungen weiterentwickelt. Antriebstechnik (Mainz, 1982), Vol. 44, No. 2, p. 34-38.

[8] HLEBANJA, G., HLEBANJA, J., ČARMAN, M. (2009) Cylindrical Wormgearings with Progressively Curved Shape of Teeth Flanks. Journ. of Mech. Eng. 55(2009)1, p. 5-14.

[9] DOWSON, D., HIGGINSON, G.R. (1977) Elasto-Hydrodynamic Lubrication. SI Edition. Pergamon Press.

[10] SIGMUND, W. (2011) Schneckenräder mit gekrümmter Eingriffslinie. Bericht über die Simulation mit SINETRA Programm, FZG - Forschungsstelle für Zahnräder und Getriebebau, TU München.