ISSN 1805- 062X, 1 805-0638 (online), ETTN 072 -11-00002 -09-4 [620927]

GRANT journal
ISSN 1805- 062X, 1 805-0638 (online), ETTN 072 -11-00002 -09-4
EUROPEAN GRANT PROJECTS | RESULTS | RESEARCH & DEVELOPMENT | SCIENCE

Solution and modification in the profile the harmonic drive

Daniela Harachová1

1 Technická univerzita v Košiciach, Strojnícka Fakulta, KKDaL; Letná 9, 040 01 Košice; email: [anonimizat]
Grant:. VEGA 1/0515/13
Name of the Grant: Draft des ign layout and architecture of intelligent implants
Subject: JR – Other machinery industry

© GRANT Journal, MAGNANIMITAS Assn.

Abstra ct The h armonic gearing unquestionably include among a
prospective technology. A harmonic gear is basically a differen tial
gear with a train of spur gears where the mesh is achieved by the flexible deformation of one of the meshing wheels. The extent of the flexible wheel deformation is coherent to the character and the mesh quality. The difference lies in the fact that more cogs/teeth
participate in meshing and thus also in transmission at the same time. The existence of the flexible wheel within the harmonic gear which undergoes deformation during the process of usage requires a specific approach in the mesh examination of this gear.
During
recent years the issue of the tooth deformation has been solved with
modern methods of calculation including also one of the widely used numeric ones, the finite element method (FEM).

Key words Harmonic drive, elastic gear, finite element methods
(FEM), model, deformation analysis.

1. DESCRIPTION HARMONIC DRIVE

Fig.1 shows the complete set of harmonic gear, ie flexible toothed
wheel, solid sprocket and wave generator .
1- wave generator
2 – solid sprocket
3 – flexible wheel

Fig. 1 Harmonic drive
The meshing of a harmonic gear is achieved with the deformation of
a flexible wheel (w) under the application of a wave generator (g). There is a very small relative movement between the teeth in the
toothed mesh. In reality this relative movement of meshing teeth
happens in zones where their loading capacity is small, i.e. on their entrance into the mesh and on their leaving it. This deformation
influences the shape of the active walls of the teeth of the flexible
wheel. And as a result they do not mesh correctly. When properly
selected parameters of gearing between the teeth of the flexible
wheel – (a) a rigid wheel – (k), there is a relative movement along
such a path, which provides a small slip of tee th – ( sc ) (Fig. 2).

Fig. 2 Shot tooth flexible wheel and solid wheel

This deformation influences the shape of the active walls of the
teeth of the flexible wheel. And as a result they do not mesh
correctly. As a consequence of the meshing of the flexible wheel with the hard
wheel the impact and interference (and also contact ratio) are created. These occurrences result in quick wear and the increase of the general damage which consequently decrease the longevity of the harmonic ge ars.

2. THE ELASTIC WHEEL DEFORMATION
ESTIMATE USING FEM
Within the harmonic gear, the existence of the flexible wheel being deformed while being used requires an individual approach to the mesh examination for this gear. Primarily it is important to defin e
the effect of the flexible wheel deformation on the tooth shape. The
problem of the tooth deformation has been researched by many
authors. The older works emerged from the classic theory of elasticity and treated a tooth as a fixed beam. So the flexible wheel
is the limiting part of the harmonic gear’s bearing power in direct
coherence of an adverse wear. In the experiment conditions the
tooth deformation is mostly determined by a static measurement of the tooth deformation loaded with a constant power or it is
determined with the measurement of the divergence during a slow
rotation. Currently the finite element method (FEM) is one of the
1
2
3

73

GRANT journal
ISSN 1805- 062X, 1 805-0638 (online), ETTN 072 -11-00002 -09-4
EUROPEAN GRANT PROJECTS | RESULTS | RESEARCH & DEVELOPMENT | SCIENCE

most prevalent numeric methods. Modern program systems FEM
utilising the ever -growing facilities of the computer equipm ent
enables to solve even very complicated tasks. The user of the
program is able to work very effectively as the data feeding,
calculation itself and the result analysis are fully automated.
3. CREATING A TREE -DIMENSIONAL GEOMETRIC
MODEL

In standard cases a 3 -dimentional geometric model can be created
via volume calculations. In more complicated cases it is necessary to model it with the help of more complex geometric forms in the
program Cosmos/M. In this case we use the command for the
volume operations VLEXTR which creates the volume by shifting
the generating plane (or a group of planes) in the direction of a given axis of the active coordinate system. After the design of the 3 –
dimentional geometric shape we frame its grid. This was done by the command for the parametric griding M_VL which grids volume
to create a 3 -dimentional model of the object.

Fig. 3 Element of Solid type

Next step is to find out 3 -dimensional geometric model.

Enterin g the material constants
Limit conditions
geometric limit conditions
force limit conditions

In case solutions deformation tooth the elastic wheel the harmonic
gear using FEM, we select all the displacement and rotation in
place solid and the inflexiblelink zero bond. Saving bond is shown in (Fig. 3 ). As regards the prescription of surface forces, we
proposed the of force in the place where the action arises from the
wave generator (Fig.4 ).

Fig. 4 The final shape of computational geometric model
a) front view, b) view 3D

4. TREATMENT OF RESULTS
Processing of computed results is important, the final part of the
calculation by finite element method. The tasks of t he mechanics of
deformed bodies are generally the most important results of the
nodal displacements, stress and deformation.

To detect displacement, stress and deformation will the use animation. Animation is an easy way representation of the deformed
shape of the structure. Animation is a useful tool to better
understand the behavior of the proposed model. For animation mostly just when a command ANIMATE confirm the proposed parameters (Fig.5) .

Fig. 5 D eformed shape of the flexible wheel to move the field

Fig. 6 Comparison of shape

Fig.6 is a comparison of the shape of the elastic wheel undeformed
and deformed shape of the flexible wheel
In (Fig. 7) are illustrated teeth and nodes in which indicated values
of displacement along the axis X, Y, Z.

b) a)
deformed shape of
the flexible wheel
un-deformed shape
of the flexible wheel
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GRANT journal
ISSN 1805- 062X, 1 805-0638 (online), ETTN 072 -11-00002 -09-4
EUROPEAN GRANT PROJECTS | RESULTS | RESEARCH & DEVELOPMENT | SCIENCE

Fig. 7 Teeth with marked nodes
Measured figures are used in ascertaining whether the movement is
greater if the force in respective directions is on the middle of the
teeth.

Arithmetic mean of the displacement in the X -∆ X

i – the number of nodes in which they were subtract displacement

mmiUX
X43
10 . 400489 , 41210 . 280587 , 5 −−
= = = ∆∑
Arithmetic mean of the displacement in the Y – ∆Y

mmiUY
Y32
10 . 135833 , 31210 . 763 , 3 −−
= = = ∆∑
Arithmetic mean of th e displacement in the Z – ∆Z

mmiUZ
Z87
10 . 60233 , 31210 . 3228 , 4 −−
= = = ∆∑

5. CONSTRUCTING PROFILE AS ENVELOPES
CIRCLES – PONCELET’S METHOD
As mentioned objective is to determine size the deformatio n of a
flexible wheel harmonic transfer and subsequent toot h shape after
deform ation using FEM. After determining the shape of the
deformed tooth it is necessary to design an appropriate shape of the opposite profile so when meshing the flexible wheel with the rigid wheel of the harmonic gear it would not cause interference. Tooth flanks solid wheel must be enveloping curves of the tooth flanks of
the flexible wheel.
The internal gear is when the outer and the inner teeth mesh together. The harmonic gear is such a case where the outer teeth are
provided by the flexible wheel and the inner by the rigid wheel.
During meshing the associated teeth profiles are in point contact at
all times. The most used direct construction is to design a profile like envelope circles (Poncelet’s method).
Poncelet’s method is very graphic and is based o n the envelope
principle: following the movement of both profiles in the axis
system connected to one wheel (example 2 ) (Fig. 8), the sought p
1 profile is the envelope of the p 1 profile connected to the wheel 1
which spins off the rigid wheel 2 during this relative movement.

Fig. 8 Principle Poncelet’s method

The mentioned method is used in discovering the opposite profile to
the deformed tooth profile of the flexible wheel. The program “AUTOCAD” will be used for the design of the opposite profile. On
(Fig 9) is the active panel of the deformed tooth of the flexible
wheel and the opposite profile designed to it.

Fig. 9 Create opposite profile wheel using Poncelet’ s method

To design the opposite profile by the Poncelet’s method is time
consuming as the sides of the flexible wheel teeth must create the
pitch circle of the rigid wheel teeth. It would be necessary to design the envelope of the active panel for each tooth separately because the teeth have individual shape as a consequence of the flexible
wheel deformation. Also method is not very accurate, and especially
in this case where the elastic wheel dimensions are very small.
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GRANT journal
ISSN 1805- 062X, 1 805-0638 (online), ETTN 072 -11-00002 -09-4
EUROPEAN GRANT PROJECTS | RESULTS | RESEARCH & DEVELOPMENT | SCIENCE

For this reason, it is preferable to set up a calculation of the
math ematical model to determine the contra profiles to the flexible
wheel, but that is the aim of further work.

6. CONCLUSION

The harmonic toothed gear transmissions undoubtedly are a
prospective technology. Its uniqueness is in using a higher number
of teeth i n mesh and consequently also in conveyance. Within the
harmonic gear, the existence of the flexible wheel being deformed
while being used requires an individual approach to the mesh examination for this gear. Primarily it is important to define the
effect of the flexible wheel deformation on the tooth shape. The
shape of the active side of the tooth elastic deformation of the
wheel, we found using the finite element method. Processing of computed results is an important part of the final calculation by
finite element method. To detect displacement, voltage and
deformation to used the animation. As mentioned objective is to
determine size the deformation of a flexible wheel harmonic transfer
and subsequent tooth shape after deformation using FEM. After
determ ining the shape of the deformed tooth it is necessary to design
an appropriate shape of the opposite profile so when meshing the
flexible wheel with the rigid wheel of the harmonic gear it would not cause interference. Tooth flanks solid wheel must be env eloping
curves of the tooth flanks of the flexible wheel. The internal gear is
when the outer and the inner teeth mesh together. The harmonic gear is such a case where the outer teeth are provided by the flexible wheel and the inner by the rigid wheel. During meshing the
associated teeth profiles are in point contact at all times. The most
used direct construction is to design a profile like envelope circles (Poncelet’s method). To design the opposite profile by the
Poncelet’s method is time consuming as the sides of the flexible wheel teeth must create the pitch circle of the rigid wheel teeth. It
would be necessary to design the envelope of the active panel for
each tooth separately because the teeth have individual shape as a consequence of the flexible w heel deformation. Also method is not
very accurate, and especially in this case where the elastic wheel dimensions are very small.
For this reason, it is preferable to set up a calculation of the mathematical model to determine the contra profiles to the flexible
wheel, but that is the aim of further work.

Reference

1. IVANČOI, V., KUBÍN, K., KOTOLNÝ, K. (2000): Program
COSMOS/M. Elfa, Košice.
2. IVANČO, V., KUBÍN, K., KOSTOLNÝ, K.: Program
Cosmos/M.Elfa, Košice 2000.
3. HAĽKO, J., VOJTKO, I., (2008): Diferenciálny Harmonický
prevod a jeho simulácia, In: Mechanica S lovaca. Roč. 12, č. 3 –
C, s. 165 -172. – 1SSN 1335 – 2393.
4.
HAĽKO, J., SEDLÁKOVÁ, J. (2009): Integrovaný harmnicko
diferenčný prevod s možnosťou obojstranného vstupu. IN: 50.
Medzinárodná vedecká konferencia katedier časti
a mechanizmov strojov – Žilina: ŽU – S. 1 -7. –ISDN
9788055400815.
5. Harachová, D., Tóth T. (2013): Deformation analysis and
modification in the profile the harmonic drive In: Technológ. Roč. 5, č. 4 (2013), s. 63 -66. – ISSN 1337- 8996

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