Geometrical Analysis of the Worm-Spiral Wheel Frontal Gear

SOFIA TOTOLICI, NICOLAE OANCEA, VIRGIL TEODOR, GABRIEL FRUMUSANU Manufacturing Science and Engineering Department,

“Dunarea de Jos” University of Galati, Domneasca str. 111, 800201,

Romania virgil.teodor@ugal.ro http://www.tcm.ugal.ro/personal/Site%20TV/CV_english.htm

Abstract: - The worm-spiral wheel frontal gear, known as Helicon gear is a constructive variant of the worm gearing. The advantage of these construction is a reduced overall size in the plane of spiral wheel, regarding the usually worm gearing. In this paper is presented the geometrical modeling of this gearing type in order to define the constructive elements of this. In the proposed modeling is accepted that the worm is an Archimede’s worm, due of technological considerations. Key-Words: - geometrical analysis, worm-spiral wheel gear, tooth profile, gearing surface. 1 Introduction The worm-spiral wheel frontal gears represent a constructive variant of the worm gearing. These gearing types present the advantage of a reduced overall size regarding the normal worm gearing. Still, technological difficulties make that this gearing type to not be very used, especially for large modulus. In this paper is presented the geometrical modeling of the worm-spiral wheel frontal gear, in order to define some of the constructive elements of this gearing type. In the proposed geometrical modeling is assumed that it is used an Archimede’s worm due of technological considerations. 2 Worm’s Helical Surface In figure 1, is presented the reference systems of the spiral wheel and worm which compose the worm gearing: xyz is the global reference system, with z axis overlapped to the spiral wheel axis; XYZ — mobile reference system, joined with the frontal spiral wheel; ξηζ — mobile reference system, joined with the generating worm. The generating worm profile, in the ξηζ reference system is

( )0;

sin ;cos ,e

b ur u

ξη αη α

=Δ = +

= − + (1)

where: u is variable parameter; b, α and re are constants. In figure 2, is presented the axial generatrix of the worm.

Fig. 1. Reference systems

Fig. 2. Worm’s axial generatrix

By the helical movement of the generatrix (1), around the worm axis

( )2

0 0sincos 0 ,

T

e

b u pr u

ξ ω θ α θα

= ⋅ + ⋅− +

(2)

are obtained the parametrical equations of the generating worm’s helical flank surface

NON-LINEAR SYSTEMS and WAVELET ANALYSIS

ISSN: 1790-2769 51 ISBN: 978-960-474-189-2

( )

( )

cos sin ;sin ;

cos cos .

e

r

e

r ub u p

r u

ξ α θη α θζ α θ

= − +

Σ = + +

= − +

(3)

In the equations (3), are defined the constants: re the generating worm external radius,

1.2 ;2e

mqr = + ⋅m (4)

q — diametrical coefficient; m — axial module; α — angle of the worm generatrix (usually ). 20α = o

Similarly, are determinate the equations of the worm’s left flank

( )

( )

cos sin ;sin ;

cos cos .

e

l

e

r ub u p

r u

ξ α θη α θζ α θ

= − +

Σ = − +

= − +

(5)

2.1. Generating Movements In the gearing process, the two bodies, spiral wheel and worm, have rotations around its own axis. Because in the rotation around its own axis the worm generate an “endless worm” result that only the apparent translation of worm constitute a kinematics factor in the determination of the spiral wheel flank’s form. In consequence are defined: - the rotating motion of the spiral wheel: ( )3 1 ;Tx Xω ϕ= (6) - the worm teeth helical flank’s translation, along its own axis:

2, ;

rpRx a a p

aξ

−= − = − ϕ (7)

with p helical parameter of the generating worm’s flank and 2ϕ parameter of revolution around η axis. From (6) and (7) may be determinate the relative motion ( )[3 1

T ]X aω ϕ ξ= + (8) and the reverse relative motion ( )3 1 X aξ ω ϕ= − . (9) The correlation between the 1ϕ and 2ϕ parameters is defined through the gearing ratio as:

2zk 1ϕ ϕ= . (10)

2.2. Spiral Wheel Teeth Flank’s Surface From (3), in the plane Hζ = − , is determined the generating worm’s profile:

tan ;

tan ,cos

rHe

HHb r p

ξ θ

η α θθ

= −Σ ⎛ ⎞= + − +⎜ ⎟

⎝ ⎠

(11)

for the right flank, and

tan ;

tan ,cos

lHe

HHb r p

ξ θ

η α θθ

= −Σ ⎛ ⎞= − − − +⎜ ⎟

⎝ ⎠

(12)

for the left flank. From (8) and (11) is determined the profile’s family

rHΣ in the reference system of the spiral wheel:

( )

( )

1

1

1

1

tan cos cos

tan sin ;cos

tan cos sin

tan cos .cos

r

e

r

e

x H R

Hb r p

y H R

Hb r p

θ β ϕ

α θ ϕθ

θ β ϕ

α θ ϕθ

= − − +

⎡ ⎤⎛ ⎞+ + − +⎜ ⎟⎢ ⎥⎝ ⎠⎣= − − +

⎡ ⎤⎛ ⎞+ + − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎦ (13)

Similarly, is determined the profile’s family lHΣ generated by the left flank, from :

( )

( )

1

1

1

1

tan cos cos

tan sin ;cos

tan cos sin

tan cos .cos

r

e

r

e

x H R

Hb r p

y H R

Hb r p

θ β ϕ

α θ ϕθ

θ β ϕ

α θ ϕθ

= − − +

⎡ ⎤⎛ ⎞+ − − − +⎜ ⎟⎢ ⎥⎝ ⎠⎣= − − +

⎡ ⎤⎛ ⎞+ − − − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎦ (14)

Associating to the surfaces family (13) and (14) the enwrapping condition, are determined, in the XYZ reference system, the parametrical equations of the spiral wheel teeth’s flank. 2.3. Enwrapping Condition The enwrapping condition

10N RϕΣ ⋅ =

r r, (15)

where: NΣ

r is the normal to the surface Σ (11),

respectively (12) and 1

Rϕ

r the vector with the direction of

velocity in the relative movement of the spiral wheel regarding the generating worm. There are defined: - the directix parameters of the normal to the profile ΣrH (11),

2

2

sin tan ;cos

,cos

r

HN pN

HN

ξ

η

θ αθ

θ

Σ

= −=

=

r (16)

and, similarly for ΣlH (12)

NON-LINEAR SYSTEMS and WAVELET ANALYSIS

ISSN: 1790-2769 52 ISBN: 978-960-474-189-2

2

2

sin tan ;cos

.cos

l

HN pN

HN

ξ

η

θ αθ

θ

Σ

= +=

=

r (17)

It is defined the matrix associated to the vector 1

Rϕ

r,

1

1

dRdϕξϕ

= , (18)

which, from (10) and (11), may be bring at the form:

]1 2

tancos

sintan cos

e

r

r

Hb r

R p R pH R p

ϕ

αθ

θ β ϕθ β

⎡ ⎛ ⎞− − − +⎜ ⎟⎢ ⎝ ⎠⎣= + + +

− − + z (19)

for the right flank and, similarly, from (10) and (12)

]1 2

tancossin

tan cos

e

r

r

Hb r

R p R pH R p

ϕ

αθ

θ β ϕθ β

⎡ ⎛ ⎞+ − +⎜ ⎟⎢ ⎝ ⎠⎣= + + +

− − + z (20)

for the left flank. In this way, for the right flank, from (16) and (19), result the enwrapping equation:

( )1 2

tan cos1cos sin tan

sin tancos

rp

rp e

H R pz Hpz p H

HR p b r

θ βϕ

θ θ α

β θθ

⎡ + −⎢= +

−⎢⎣⎤⎛ ⎞+ + + + −⎜ ⎟ ⎥⎝ ⎠ ⎦

α

(21)

and, for the left flank, from (17) and (20),

( )1 2

tan cos1cos sin tan

sin tan .cos

rp

rp e

H R pz Hpz p H

HR p b r

θ βϕ

θ θ α

β θθ

⎡ + −⎢=

−⎢⎣⎤⎛ ⎞+ + − − −⎜ ⎟ ⎥⎝ ⎠ ⎦

α

+

(22)

The assembly of equations (13), (21) and (14), (22) represent the spiral wheel teeth’s flank, as surfaces reciprocally enwrapping with the helical flanks of the generating worm, in plane Hζ = − . 2.4. Gearing surface The gearing surface between worm and spiral wheel is defined as the geometrical place of contact points between the two conjugated surfaces in the global reference system. In this way, the gearing lines in the plane Hζ = − is determined from the equations assembly

x aξ= + (23) where ξ is give by (11) or (12) and the enwrapping condition (21) or (22). So, the gearing line, for the right flank is give by equations:

2

tan cos ;

tan ,cos

rp

re

x H RLA Hy b r p p

θ β

α θ ϕθ

= − −

⎛ ⎞= + − + −⎜ ⎟⎝ ⎠

(24)

associated to condition (21), for the right flank, and:

2

tan cos ;

tan ,cos

rp

le

x H RLA Hy b r p p

θ β

α θ ϕθ

= − −

⎛ ⎞= − − − + −⎜ ⎟⎝ ⎠

(25)

associated to condition (22) for the left flank. 2.5. Contact Lines on Conjugated Surfaces The contact lines on the conjugated surfaces represent the geometrical place of common points for a given value of the movement parameter. Regarding the specifically kinematics of the worm gearing, see figure 1, as so as the movement parameter ϕ1 from the equations assembly (3), (21) for the right flank and (5), (22), for the left flank, for value 1 0ϕ = , (26) is determined, on the worm flanks, the contact lines between worm and in-plane spiral wheel. 3 Numerical Application In following is presented a numerical application regarding the spiral wheel profile’s form for a gearing worm with: re=37.2 mm; α=20°; Rrp=225 mm; m=6 mm; b=2.09 mm; β=45°. In figure 3 and table 1, (HS=z=12 mm), there are presented the form and the coordinates of the spiral wheel in planes perpendicularly on its axis.

Fig. 3. Theoretically profile of the in-plane spiral wheel

NON-LINEAR SYSTEMS and WAVELET ANALYSIS

ISSN: 1790-2769 53 ISBN: 978-960-474-189-2

Table 1 Right flank Left flank

x [mm] y [mm] x [mm] y [mm] -188.447 161.243 -187.976 157.060 -188.429 161.249 -187.958 157.058 -188.411 161.256 -187.939 157.055

M M M M -136.948 157.793 -138.529 154.046 -136.940 157.788 -138.519 154.049 -136.932 157.783 -138.509 154.052

In table 2 and figure 4, there are presented the gearing lines in plane ξ=-H=z=7.2 mm.

Fig. 4. Gearing lines

Table 2 Gearing line for

right flank Gearing line for

left flank x [mm] y [mm] x [mm] y [mm]

-162.492 181.693 -161.946 178.055 -162.487 181.681 -161.940 178.035 -162.481 181.669 -161.934 178.016

M M M M -167.561 128.886 -169.196 123.050 -167.575 128.846 -169.207 123.022 -167.589 128.806 -169.218 122.993

In figure 5 and table 3, there are presented the contact lines between the two conjugated profiles.

Fig. 5. Gearing lines

Table 3 ξ [mm] η [mm] ζ [mm] 0.246 6.488 25.200 0.253 5.946 26.688 0.000 5.376 28.176 M M M

0.000 3.210 34.128 0.292 2.693 35.616 0.298 2.150 37.104

The real flanks of the spiral wheel should be limited as active length due of the interference process, in the same planes, with the conjugated worm. From constructive considerations, the real profiles of the spiral wheel are limited between cylinders with radius: Ri=Rrp-0.2 mm; Re=Rrp+1 mm.

Fig. 6. Real flanks of the spiral wheel

4. Conclusions The presented method allows a rigorous definition of the flanks surface’s form in enveloping with the Helicon gearing worm elements. It is also possible to be determined the contact lines between the surfaces in gearing, as well as the gearing line’s form. The software elaborated for these allows to draw all the needed elements for the geometrical modeling of this gear. Acknowledgement The authors gratefully acknowledge the financial support of the Romanian Ministry of Education, Research and Innovation through grant PN_II_ID_791/2008. References: [1] X1. Author, Title of the Paper, International Journal

of Science and Technology, Vol.X, No.X, 19XX, pp. XX-XX.

[2] Guja, N., Amgrenaje cilindrice şi hipoide, Editura Tehnică Bucureşti, 1990;

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ISSN: 1790-2769 54 ISBN: 978-960-474-189-2

[3] Litvin, F.L., Theory of Gearing, NASA Reference Publications, Washington, DC, 1989;

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ISSN: 1790-2769 55 ISBN: 978-960-474-189-2