balkan journal of mechanical transmissionsbostan, i., dulgheru, v., sochirean, a. research and...

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)



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



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.



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.



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.



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

[email protected]

Valeriu DULGHERU, Prof. pHd, Dr.Sc.



in Machine Building



[email protected]

Anatol SOCHIREAN, assoc. prof. Dr.



in Machine Building



[email protected]

a.

b. Fig. 9. Planetary precessional reducer: general view (a);

section view (b).

balkan journal of mechanical transmissionsbostan, i., dulgheru, v., sochirean, a. research and...

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