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Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
12
Balkan Journal of
Mechanical Transmissions
Volume 1 (2011), Issue 1, pp. 12-17
ISSN 2069–5497
RESEARCH AND DEVELOPMENT OF PLANETARY PRECESSIONAL TRANSMISSIONS
Ion BOSTAN, Valeriu DULGHERU, Anatol SOCHIREAN
ABSTRACT. Various technological and energetic systems need updated multipliers. Planetary precessional transmissions
function efficiently in the regime of reducer, differential and multiplier. This work presents some aspects concerning the
justification of the precessional gear geometrical parameters and of the joining mechanism for satellite block and inlet shaft.
On the basis of carried out research, during the last 25 years, at the Department of Theory of Mechanisms and Machine
Parts, Technical University of Moldova, a new type of mechanical transmission was elaborated – precessional planetary
transmission, protected by more than 100 patents. Precessional planetary transmission has a number of advantages:
increased bearing capacity, high mechanical efficiency, large kinematical possibilities. The mechanical efficiency of
precessional planetary reducer is η=(0,85÷0,95 (at transmission ratio i = 10…300) and the specific weight, reported to the
moment of torsion, is (0,02÷0,05) Nm/kg.
KEYWORDS. precessional transmission, transmission ratio, computerised model, tooth profile
NOMENCLATURE
Symbol Description
i Transmission ratio η Mechanical efficiency
δ Conical axoid angle θ Nutation angle β Conical angle ∆ψ3 Tool position error
1. INTRODUCTION
Diversity of requirements forwarded by the beneficiaries of
mechanical transmissions consists, in particular, in
increasing reliability, efficiency and lifting capacity, and in
reducing the mass and dimensions. It becomes more and
more difficult to satisfy the mentioned demands by partial
updating of traditional transmissions. The cylindrical wheel
planetary transmissions with their new variety correspond
to these requirements: transmissions of CYCLO type
(Lynwander, 1983 and Rudenko 1965), harmonic
transmissions and precessional transmissions which have
appeared not long ago. The target problem can be solved
with special effects by developing new types of reducers
and multipliers based on precessional planetary
transmissions with multiple gear, that were developed by
the authors. As mentioned in literature (Bostan, 1991,a,
Bostan, Dulgheru, Sochireanu, Babaian, 2011, b) absolute
multiplicity of precessional gear (up to 100% pairs of teeth
simultaneously involved in gearing, compared to 5%-7% -
in classical gearings) provides increased lifting capacity and
small mass and dimensions. To mention that until now
precessional planetary transmissions have been researched
and applied mainly in reducers. Therefore it was necessary
to carry out theoretical research to determine the
geometrical parameters of the precessional gear that
operates in multiplier mode. Also, it was necessary to
develop new conceptual diagrams of precessional
transmissions that function under multiplier regime.
Depending on the structural diagram, precessional
transmissions fall into two main types – K-H-V and 2K-H,
from which a wide range of constructive solutions with
wide kinematical and functional options that operate in
multiplier regime. The kinematical diagram of the
precessional transmission K-H-V (fig. 1,a), comprises five
basic elements: planet career H, satellite gear g, two central
wheels b with the same number of teeth, controlling
mechanism W and the body (frame). The roller rim of the
satellite gear g gears internally with the sun wheels b, and
their teeth generators cross in a point, so-called the centre
of precession. The satellite gear g is mounted on the planet
(wheel) career H, designed in the form of a sloped crank,
which axis forms some angle with the central wheel axis θ
Revolving, the sloped crank H transmits sphero-spatial
motion to the satellite wheel regarding the ball hinge
installed in the centre of precession. For the transmission
with the controlling mechanism designed as clutch coupling
(fig.1, a), the gear ratio (gear reduction rate) varies in the
limits:
g bg
HV
b
z cos zi ;
z
Θ −= − (1)
g bg
HV
b
z cos zi ,
z cos
Θ
Θ
−= − (2)
reaching the extreme values of 4 times for each revolution
of the crank H. If necessary this shortcoming can be
eliminated using as a controlling mechanism the constant
cardan joint (Hooke’s joint), the ball synchronous
couplings, etc.
For zg = zb+1, gHV
b
1i ,
z= − the driving and driven shafts
ROmanian
Association of
MEchanical
Transmissions
(ROAMET)
Balkan Association of
Power Transmissions
(BAPT)
-
Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
13
have opposite directions. For zg = zb-1, g
HV
b
1i ,
z= the shafts
revolve in the same direction.
This kinematical diagram of the precessional transmission
ensures a range of gear ratios i = 8...60, but in the
multiplication regime it operates efficiently only for the
range of gear ratios i = 8…25. As well, in the coupling
mechanism W, that operates with pitch angles of the semi
couplings up to 3o, power losses occur reducing the
efficiency of the multiplier on the whole.
The precessional transmission 2K-H has higher
performances, including the kinetostatic one as well
(fig. 1,b). The transmission comprises a satellite gear g with
two crown gears Zg1 and Zg2, that gears with the unshiftable
b and movable a central wheels.
1
2 1
g a
b g g a
z zi .
z z z z= −
− (3)
The analysis of this relation demonstrates that precessional
transmissions 2K-H provide the fulfillment of a large range
of transmission ratios i = ± (12...3599). But in multiplier regime the transmission operates efficiently only in the
limits i = ± (12...40). The process of self-braking (self-stopping) occurs at bigger transmission ratios. It is
necessary to point out the series of peculiarities of the
precessional transmissions 2K-H that ensure higher
performances compared to similar planetary transmissions
with cylindrical gears: precessional transmissions do not
demand conditions of distance equality between the axis.
This factor widens the area of their optimal design;
precessional transmission kinematics does not limit the
selection of the gear couples modules or of the rollers
placement pitch. This factor increases the possibilities of
shaping teeth pairs and of the transmission ratios interval;
the peculiarities of the designed precessional gears allow
increasing in the number of teeth that transmit the load
simultaneously and this fact reduces significantly the
dimensions and mass for the same loads compared to the
traditional involute gearings.
Based on the carried out analysis a constructive diagram of
the precessional planetary transmission was designed, taken
as the base of precessional multipliers design. The
precessional planetary transmission (Fig. 2)comprises the
crank shaft 1 on which the satellite block 2, and the fixed
and the movable wheels 3 (the movable wheel is connected
to the shaft 5) are installed. The satellite block 2 has two
crown gears (6 and 7) with the teeth executed as conical
rollers mounted on the axle with the possibility of revolving
around them. The transmission operates in the multiplier
mode, as follows: at the rotation of the input shaft 5 with
the gear 4, due to the difference in the number of geared
teeth (Z4 = Z7 - 1, Z3 = Z6 - 1), the satellite block 2 will
perform a spherical-spatial motion around the point – centre
of precession (the point of intersection of the crown gear
roller axes and of the crank shaft axes 1), producing a
complete precessional cycle at the rotation of the gear 4 at
an angle equal to the angular pitch. Due to its mounting on
the sloped side of the crank shaft 1, the precessional motion
of the satellite block 2 is transformed into rotational motion
of the crank shaft 1 that will produce a complete rotation
during a complete precessional cycle of the satellite block.
The computerised model of the planetary precessional gear
designed in Autodesk MotionInventor on the figure 3 is
presented.
2. ANALYTIC DESCRIPTION OF TEETH PROFILE
AND JUSTIFICATION OF PRECESSIONAL GEAR
PARAMETERS SELECTION
Teeth profiles have an important role in the efficient
transformation of motion in the precessional transmissions
W
O
V
H
g
bb
ZbZb
V
Zg
a.
V
a
Zg2
Zg1
Zb
H
gb
O
Za
b.
Fig. 1. Conceptual diagrams of precesional transmissions.
1 3 6 2 7 4 5
Fig. 2. Constructive diagram of the planetary precessional
transmission 2K-H.
Multiplier mode Reducer
mode
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Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
14
Fig. 3. Computerised model of the planetary precessional
gear.
that operate as multiplier. Multiple precessional gear theory,
previously developed, did not take into consideration the
influence of the diagram error of the linking mechanism in
the processing device for gear wheel on the teeth profile.
Functioning under the multiplication regime, these errors
have major influence, which can lead to instant blocking of
gear and to power losses. With this purpose, a thorough
analysis was conducted on the motion development
mechanism under multiplication, and on the teeth profile
error generating source. As mentioned in literature (Bostan,
Dulgheru, Sochireanu, Babaian, 2011, b) on the basis of
fundamental theory of multiple precessional gear,
previously developed, a new gear with modified teeth
profile and the technology for its industrial manufacturing
was proposed and patented (Bostan, Dulgheru, Ţopa,
Vaculenco, 2002, d) .
Kinematically, the link between the semi product and the
tool, in which one of them (the tool) makes spherical-
spatial motion being, at the same time, limited from rotating
around the axis of the main shaft of the teething machine
tool, is similar to the „satellite-driven shaft” link from the
precessional planetary transmission of the K-H-V type. The
kinematical link between the tool and the stationary part of
the device represents a Hooke articulation that generates the
variability of transfer function in the kinematical link „tool-
semi product”. This variation will influence the teeth
profile. Thus, the connection of tool with the housing
registers a certain diagram error ∆ψ3 (to understand the deviation of the semi product angle of rotation ψ3 from the angle of rotation of the semi product itself m
3ψψψψ at its uniform rotation):
( )
m m2 331 3 3 31
3
2
3
z zi ; i
z
zarctg(cos tg ) .
z
∆ψ ψ
ψ θ ψ
−= − = − =
= − ⋅
(4)
Fig. 4 shows the dependence of the tool position diagram
error ∆ψ3 at a revolution of the machine tool main shaft ψ. This error is transmitted to the tool that shapes the teeth
profile with the same error. To ensure continuity of the
transfer function and to improve the performances of
precessional transmission under multiplication it is
necessary to modify teeth profile with the diagram error
value ∆ψ3 by communicating supplementary motion to the
tool. In this case the momentary transmission ratio of the
manufactured gear will be constant. Usually, in theoretical
mechanics the position of the body making spherical-spatial
motion is described by Euler angles. The mobile coordinate
system OX1Y1Z1 is connected rigidly with the satellite
wheel, which origin coincides with the centre of precession
0 (Fig. 5) and performs spherical-spatial motion together
with the satellite wheel relative to the motionless coordinate
system OXYZ.
The elaboration of the mathematic model of the modified
teeth profile is based integrally on the mathematic model of
teeth profile, previously developed by the authors. With this
purpose it is necessary to present the detailed description of
teeth profile without modification and, then, to present of
the description of modified profile peculiarities.
Description of teeth profile designed on sphere. An
arbitrary point D of the tool axis describes a trajectory
relative to the fixed system according to the equations:
-0,050
-0,040
-0,030
-0,020
-0,010
0,000
0,010
0,020
0,030
0,040
0,050
0 30 60 90 120 150 180 210 240 270 300 330 36033 33
θ = 1
θ = 1.5
θ = 2
θ = 2.5
θ = 3
Fig. 4. Dependence of the error diagram of tool position
∆ψ3 at a revolution of the machine-tool main shaft ψ.
Fig. 5. Tooth profile in normal section.
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Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
15
( )
( )
m m m
D C C
m m m 2 2
D C C
m m 2 2 m
D C C
X sin sin Y sin Z 1 cos cos ;
Y Y cos Z sin cos cos sin ;
Z Y sin cos cos sin Z cos .
δ θ θ ψ
δ δ ψ θ ψ
δ ψ θ ψ δ
= − + −
= − + +
= − + −
(5)
Index m means „modified”. The motion of point Dm
compared to the movable system connected rigidly to the
semi product is described by formulas:
m m m1D D D
1 1
m m m
1D D D
1 1
m m
1D D
X X cos Y sin ;Z Z
Y X sin Y cos ;Z Z
Z Z .
ψ ψ
ψ ψ
= −
= +
=
(6)
The projections of point Dm
velocities OXYZ and OX1Y1Z1 is
expressed by formulas:
( )
( ) ( )
[ ]
m m m
D C C
m m m
C C C
m m m 2 2
D C C
m
C
X sin cos Y sin Z 1 cos cos
sin sin Y sin Z 1 cos cos Z 1 cos sin ;
Y Y cos Z sin cos cos sin
Z sin 2cos sin 2cos sin cos ;
δ ψ θ θ ψ ψ
δ ψ θ θ ψ θ ψ ψ
δ δ ψ θ ψ
δ ψ ψ θ ψ ψ ψ
• •
• • •
• • •
•
= − + − −
− + − − − ⋅
= − + + +
+ − +
m m m m m
1D D D D D
1 1 1 1 1 1
m m m m m
1D D D D D
1 1 1 1 1 1
X X cos X sin Y sin Y cos ;Z Z Z Z Z Z
Y X sin X cos Y cos Y sin .Z Z Z Z Z Z
ψ ψ ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψ
• •• • •
• •• • •
= − − −
= + + −
(7)
The coordinates of point Em on the sphere is calculated by
formulas:
m m m m
1E 2 1E 2
m m m m
1E 1 1E 1
m m m mm 1 1 2 21E m2 m2
1 2
m m m m 2 m2 m2 2 m2 m2
1 1 2 2 1 2 D 1 2
m2 m2
1 2
X k Z d ;
Y k Z d ;
( k d k d )Z
k k 1
(k d k d ) ( k k 1) ( R d d ),
k k 1
= +
= −
−= −
+ +
− + + + ⋅ − −−
+ +
(8)
where:
( )
( )
m m m m m m 2 m
1 D 1 D 1 D 1 D 1 D 1 D 1 D
m
1 m m m m m
1 D 1 D 1 D 1 D 1 D
m m m
1 1 D 1 Dm
2 m
1 D
2 m
m D 1 D1
m m m m
1 D 1 D 1 D 1 D
2 m m
D 1 1 Dm
2 m
1 D
X X X Y Y Z X
k ;
Z X Y Y X
k Y Zk ;
X
R cos Xd ;
X Y X Y
R cos d Yd .
X
β
β
• • •
• •
•
• •
+ +
=
−
+= −
=
−
+=
(8)
According to the obtained analytical relations a soft for the
calculation and generation of teeth was developed in
CATIA V5R7 modelling system that allowed obtaining the
modified trajectories of points Em
e and Em
i on the spherical
front surfaces, both exterior and interior ones, by which the
teeth surface was generated (Fig. 4).
Description of modified teeth profile projected on a
transversal surface. Projection of point Em on the tooth
transversal plane has the following coordinates:
m m m m m mE 1E E 1E
m m m
E 1E
X X , Y Y ,
Z Z ,
ε ε
ε
′′ ′′ ′′= ⋅ = ⋅
′′ ′′= ⋅ (8)
where mm m m
1E 1E 1E
D.
AX BY CZε = −
+ +
The modified teeth profile in plane is described by the
equations:
( )
( )
( )
m m m
E D E
1 1
m m m
E D E
1 1
m
D E
X cos R cos Y sin ;Z Z
X sin sin R cos Y sin cosZ Z
R sin Z cos .
π πξ δ θ β
π πζ γ δ θ β γ
δ θ β γ
′′ ′′ = + + + +
′′ ′′ = − + + + +
′′ + + + +
(8)
A wide range of modified teeth profiles with different
geometrical parameters were generated in MathCAD 2001
Professional software (Fig. 6 a,b). The analysis of the
a.
b.
Fig. 6. Sample of teeth profiles for precessional gearings.
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Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
16
obtained teeth profiles, based on the fundamental
conditions of gearing selection (high bearing capacity due
to gearing multiplicity, small dimensions and mass,
technology, etc.) has allowed the selection of the teeth
profiles parameters.
This continuing dramatical change in available
computational resources offers new options in gear design
for gear manufacturing processes. They include the
complete 3D model of the whole gear, including the gear
body and al gear flanks, obtained by using of modelling
system CATIA V5R7 (figure 7).
Some steps of the teeth generating surface is shown in Fig.
8 a-d. The solid model of a gear wheel is shown in Fig. 8
e. Based on the carried out research it was established that
from the point of view of decreasing energy losses in
gearing, in the multiplication mode of operation, the
gearing angle should be α > 450, and the nutation angle (the pitch angle of the crank shaft) should be– θ ≤ 2,50. This is dictated by the reverse principle of movement in the
multipliers compared to the reducers: the axial component
of the normal force in gear must be maximal to drive the
crank shaft in the rotation movement through the satellite
wheel.
3. DESIGN OF PRECESSIONAL REDUCER
STRUCTURE
On the basis of the undertaken study, diagram 2K-H was
selected for the development of precessional multiplier of
the micro hydropower plant. As a result of analysis of a
wide range of tooth profiles with different geometrical
parameters of gear by using the mathematical modelling
package MathCAD 2001 Professional, the optimum tooth
profiles were selected with account of their functioning in
conditions of multiplication. Also, in MathCAD 2001
Professional software the calculation of geometrical
parameters of precessional gear was done. The structures of
precessional reducers for were designed in SolidWorks
software. The precessional multiplier is connected by flange
with an electric generator, which allows obtaining a
compact module, coaxial with the micro hydropower plant
rotor. The structure from fig. 9 is proposed for motoreducer
with general destination. To simulate the reducer assembly
and functioning, the dynamic computerized model of the
precessional reducer was developed in AutoDesk.
a. b.
c.
d.
e.
Fig. 8. Teeth generating surface (a-d) and computerized
model of the wheel (e).
MotionInventor (fig. 10). The precessional reducer has
transmission ratio up to i = 3600 (based on one stage
diagram) with satisfactory mechanical efficiency.
Having a high lifting capacity and large range of kinematic
possibilities (ratio of transmission up to 3600 in one step
realized only by 4 basic elements), high kinematic
accuracy, reduced dimensions and mass, simplicity of
construction the precessional gearings can be widely
utilized in various fields.
4. CONCLUSIONS
HIGH EFFICIENCY, rating 96%, is due to the use of a
gear/rolling coupling with the convexo-concave teeth
profile and because there is no special mechanism
connecting the planet pinion with the driven shaft.
A WIDE RENGE OF TRANSMISSION RATIO is from
±8,5 to ±3599 in the reducer with the only planet unit. In a solid reducer with two planet pinions, one enclosed into the
other, it is possible to realize the ratio to 12960000.
Normal section of the teeth
O
E1 E2
E1 E2 ξ
ζ
Fig. 7. 3D model of the gear.
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Bostan, I., Dulgheru, V., Sochirean, A. Research and development of planetary precessional transmissions.
Balkan Journal of Mechanical Transmissions, Volume 1 (2011), Issue 1, pp. 12-17, ISSN 2069–5497.
17
Fig. 10. Dynamic computerized model of the precessional
reducer.
HIGH LIFTING CAPACITY is maintained by meshing
when about 100% teeth couples are simultaneously mated
resulting in load distribution among the teeth on the line of
action.
COMPACTNESS AND SMALL WEIGHT has become
possible because of the principle of operation and
multicouple gearing. Specific material capacity of the
reducers ranges from 0,022 to 0,05 kg/Nm.
HIGH KINEMATIC ACCURACY from 30 to 90 ang/sec.
has been achieved due to the application of multicouple
convexo-concave engagement with the ground profile. The
teeth form is shaped so as to compensate the irregularity of
the driven shaft rotation connected with the spherical
motion of the planet pinion. The meshing may be without
clearance that makes it possible to set some controllable
interference.
HING RATING LIFE is mainly provided by the good
quality of engagement with the ground teeth profile and the
possibility to mount the planet pinion on the bearing with
great load rating without fitting the wheel diameter.
LOW LEVEL OF NOISE AND VIBRATION from 50 to
60 dB has become possible due to the accuracy of the teeth
profile, spotlessness of the driven shaft rotation, resting on
shaping the teeth profile in accordance with the
peculiarities of the spherical motion of the planet pinion.
LOW MOMENT OF INERTIA accounts for the
peculiarities of the spherical motion of the planet pinion.
THE CONDITIONS TO OPERATE in the self-holding
regime but in special designs - in the multiplication and
differential systems.
Most of the advantages of precession redactors are reasoned
by the new type of toothed-rolling engagement with
convexo-concave teeth profile with all teeth simultaneously
mating. To produce bevel gears with the presented
engagement there has been developed a new method of
processing the teeth by grinding and milling.
REFERENCES
BOSTAN, I. (1991,a). Precessional transmission with
multicouple gear. Chişinău, Ştiinţa, 1991, 356p. ISBN 5-
376-01005-8.
BOSTAN, I., DULGHERU, V., GRIGORAŞ, S. (1997,c).
Planetary, precessional and harmonic transmissions:
[Atlas]. Bucharest: Tehnica Publ House; Chişinău: Tehnica.
ISBN 973-31-1069-8.
BOSTAN, I., DULGHERU, V., SOCHIREANU, A.,
BABAIAN, I. (2011, b). Anthology of Inventions: Vol. 1.
Planetary Precessional Transmissions, 593p. Chişinău: S.n.
Combinatul poligrafic. ISBN 978-9975-4100-9-0.
BOSTAN, I., DULGHERU, V., ŢOPA M., VACULENCO,
M. (2002,d). Precession gear and process for realization
thereof. Patent MD Nr. 1886. Publ. BOPI Nr. 3.
LYNWANDER, P. (1983). Gear Drive Systems: Design and
Application. Marcel Dekker, New York, 1983, 415p. ISBN
082 47 18 968
RUDENKO, V. N. (1965). Planetarnye i volnovye
peredachi. Mashinostroenie, Leningrad, 1965.
CORRESPONDENCE
Ion BOSTAN, Prof. PhD, Dr.Sc.
Technical University of Moldova
Faculty of Engineering and Management
in Machine Building
168 Ştefan cel Mare
2004, Chişinău, Republic of Moldova
Valeriu DULGHERU, Prof. pHd, Dr.Sc.
Technical University of Moldova
Faculty of Engineering and Management
in Machine Building
168 Ştefan cel Mare
2004, Chişinău, Republic of Moldova
Anatol SOCHIREAN, assoc. prof. Dr.
Technical University of Moldova
Faculty of Engineering and Management
in Machine Building
168 Ştefan cel Mare
2004, Chişinău, Republic of Moldova
a.
b. Fig. 9. Planetary precessional reducer: general view (a);
section view (b).