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Engineering with Computers

, Volume 29, Issue 3, pp 251–272 | Cite as

Modelling and simulation of a new worm gear drive having point-like contact

  • László Dudás
Original Article

Abstract

The modelling and simulation of a worm gearing generated by an intermediary generating helicoid with a circle profile in the axle plane is considered. The goals of the paper are to verify the viability of such a worm gearing and to determine the proper parameters of a machinable construction. For the modelling and for the analysis of contact patterns the software Surface Constructor was applied. First the original theory behind the software is introduced and then its application to modelling gear types with point-like connection is explained. After a short overview of the capabilities of the software, two sample tasks are introduced: the generation of the gear surfaces of a hypoid gearing and the detailed description of modelling the novel worm gear drive with point-like contact. The modelling of the worm gearing proves that such a type is viable and machinable. The main advantage of the construction in comparison to other modified worm gearing types is the lack of transmission error, owing to the point-like connection. Due to this feature, the gearing tolerates misalignment well.

Keywords

Worm gear Mesh theory Generating helicoid Point-like contact Surface Constructor 

Abbreviations

e, f

p1, p2 parameter line indices

g, h

Indices of last p1, p2 parameter lines

is

Counter of intersected quadrangle edges

k1, k2

Contact lines

p1, p2

F1 surface parameters

p1e, p2f

F1 surface parameter values in an F1 grid point, \( e = 0, \ldots,g - 1;\;f = 0, \ldots,h - 1 \)

(x0k,l, y0k,l, z0k,l)

P k point in the K0 frame, determined at (T k , Z l ) grid point of F2

(x2k,l, y2k,l, z2k,l)

P k point in the K2 frame, determined at (T k , Z l ) grid point of F2

A, B

Points on F1

F1, F11, F12

Generating surface, 1st and 2nd intersection

F1(p1, p2)

Generating surface given by surface parameters p1 and p2

F2

Generated surface

F21, F22

Surfaces generated by the same F1 surface

Fi1, Fi2

Φ variables of different surface generations

Ki(xi, yi, zi)

Coordinate system (i = 0, 1, 2, 70, 100, 101, 130, 160, 201, 230, 231, 260)

M, Mi

F1 surface points

Mt

F1 surface point corresponding to P k

Pk

Contact point, \( k = 0, \ldots,r \)

\( P^{\prime}_{k} \)

Generated point of F2, \( k = 0, \ldots,r \)

Pn

Point of singularity

R

Space coordinate of κ coordinate system and reaching direction (mm)

R(Φ)

Reaching-coordinate function at T k division in κ l coordinate system associated with Z l (mm)

Ri

R value at index i (mm)

Ru

R value of P k point (mm)

Rt

R value of M t point (mm)

Rk,l

Minimum R value determined at (T k , Z l ) grid point of F2 (mm)

Rprev

R minimum determined in previous iteration cycle for same TZ grid point (mm)

R(x0, z0)

Changeover from the (x0, z0) Descartes coordinate system to the curved RT coordinate system

ΔR

Difference between the results of the last two cycles

Rho1, Rho2

R variables of different surface generations

T, Tk

Division coordinate, \( k = 0, \ldots,r \) (mm or grad)

TA, TB

Division coordinate values at points A and B

Tau1, Tau2

T variables of different surface generations

T(x0, z0)

Changeover from the (x0, z0) Descartes coordinate system to the curved RT coordinate system

T1, T11, T12

Body of the generating object, 1st and 2nd intersections

T2

Body of the generated object

v, w

Degree of partial derivation

Z, Zl

F2 grid-parameter value and identifying parameter of κ coordinate system at the same time, \( l = 0, \ldots,s \) (mm or grad)

Zeta1, Zeta2

Z variables of different surface generations (mm or grad)

n

Surface normal vector

r0

Vector form of F1 in K0 coordinate system, r 0 = r 0(p1, p2)

r0(h)

Vector form of F1 in K0 coordinate system given by symbolic algebraic expressions

r1(h)

Vector form of F1 in K1 coordinate system given by symbolic algebraic expressions

rAB[ ]

Array of crossing points of a p1 or p2 parameter-line and the F1 intersection curve given in K0 coordinate system

v1,2

Relative speed vector

Mi,j

Homogenous transformation matrix from Kj to Ki coordinate system given automatically by symbolic algebraic expressions

β, βA, βB

Angle between the path of motion and coordinate direction R, in points A and B (grad)

κ(Φ, R, T)

Non-Descartes coordinate system

κl

κ coordinate system, associated with Z l F2 grid-parameter value

Φ, Φi

Motion parameter and space coordinate of κ coordinate system, \( i = 0, \ldots,p \) (mm or grad)

Φt

Φ Value corresponding to P k contact point (mm or grad)

C

Center distance of worm gearing (mm)

BETA

Slicing plane orientation angle in worm-wheel generation (grad)

DELTAX

Intermediary generating surface shift in X direction (mm)

DELTAY

Intermediary generating surface shift in Y direction (mm)

FI1

Motion parameter in generation of the worm (grad)

FI2

Motion parameter in generation of the worm-wheel (grad)

I21

Reciprocal gear ratio of modelled worm gearing

P1

F1 surface parameter (grad)

P2

F1 surface parameter (grad)

PAX

Axial parameter of helicoid (mm/rad)

RHO1

Reaching direction coordinate in worm generation (mm)

RHO2

Reaching direction coordinate in worm-wheel generation (mm)

RO

Circle tooth profile radius in axial plane (mm)

Notes

Acknowledgments

This study was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project in the framework of the New Hungarian Development Plan. The realisation of this project is supported by the European Union, co-financed by the European Social Fund.

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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of Information EngineeringUniversity of MiskolcMiskolcHungary

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