The engineering complex study of the triad “gear-technology-transmission” has permitted to elaborate a new type of precessional transmissions with multicouple gear. In this paper, the authors present the mathematic model of the multicouple gear. A computer program for doing this it is also elaborated. It is shown a blockscheme of the algorithm of the program modules including the calculus modalities exposed in the paper

Ion BOSTAN

Valeriu DULGHERU

The Annals of “Dunarea de Jos” University of Galati Fascicle IX Metallurgy and Materials Science

Directory of Open Access Journals

 
THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI 
0FASCICLE IX METALLURGY AND MATERIALS SCIENCE, ISSN 1453 – 083X. N . 2 – 2006 
 
 
 
 
SOME ASPECTS REGARDING THE ELABORATION OF  
MULTIPLE PRECESSIONAL GEAR THEORY AND  
MODERN MANUFACTURING TECHNOLOGY 
 
Ion Bostan, Valeriu Dulgheru 
“Technical University of Moldavia” Kishinew, Moldova Republic 
e-mail: dulgheru@mail.utm.md
 
ABSTRACT 
 
The engineering complex study of the triad “gear-technology-transmission” has 
permitted to elaborate a new type of precessional transmissions with multicouple 
gear. In this paper, the authors present the mathematic model of the multicouple 
gear. A computer program for doing this it is also elaborated. It is shown a block-
scheme of the algorithm of the program modules including the calculus modalities 
exposed in the paper. 
 
KEYWORDS: multicouple gear, technology, precessional,  transmission,  
 
 
1. Introduction 
 
The engineering complex study of the triad ''gear-
technology-transmission'' has permitted to elaborate a 
new type of precessional transmissions with 
multicouple gear, which, from the technological point 
of, view can be manufactured via a new method of 
conical teeth with convex-concave profile processing. 
The specific character of sphere-spatial 
(precessional motions of the precessional 
transmissions pinion makes impossible the utilisation 
of teeth classical involute profiles. This fact requires 
the elaboration of new profiles adequate to the 
sphere-spatial motion of pinion which would ensure 
high performances to the precessional transmission. 
Carrying out on the principle of the transfer function 
continuity and gear based on the principles of the 
transfer function continuity and gear multiplicity 
which aims to:  
 the elaboration of the gear mathematics model 
with account of the peculiarities; 
 the analytical description of teeth profiles by a 
system of parametric equations on spherical 
surface and normal teeth section for inner and 
plane gear;  
 the determination by CAD of geometrical and 
cinematic parameters influence of the gear upon 
the teeth profiles shape and the justification of 
their rational limits of variation; 
 the elaboration of the theoretical basis 
evaluation of teeth gear multiplicity in 
precessional transmissions; 
 the definition of area of gear multiplicity 
existence by 100% teeth couples. 
The production of non-standard teeth profiles 
requires a new manufacturing technology. In the 
complexity of problem “gear-synthesis-profile study- 
manufacturing “ an important role plays the 
elaboration of efficient methods of teeth 
manufacturing which ensures a maximum 
productivity and reduced cost while satisfying the 
requirements related to the gear with precessional 
motion. To solve this problem the following has been 
done: 
 we elaborated the mathematics model of teeth 
generation which shows the interaction of teeth 
in precessional gear; 
 we investigated  the kinematics of the 
mechanism of method realisation for teeth 
generating; 
 we determined the trajectory of the tool motion 
and the wrapping of the generating surface 
family of it by using the computer; 
 we elaborated and manufactured from metal 
milling and tooth grinding tools, inclusively 
their longitudinal modification. 
Constructions peculiarities and high multiplicity 
of gear create favourable premises for the 
improvement of precessional transmissions 
kinematics accuracy. Within these activities we 
elaborated: 
 theoretical basis for the identification of 
kinematics error generated by various primary 
error (frontal and radial knocking), on the basis 
of error independent action principle by 
fulfilling computer assisted mathematics 
experiment; 
 compensation method for manufacturing and 
assembling errors; 
- 12 -
 
THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI 
0FASCICLE IX METALLURGY AND MATERIALS SCIENCE, ISSN 1453 – 083X. N . 2 – 2006 
 
 
 method of determination of probable limit error 
for precessional reducers with account of the 
stochastic character of manufacturing and 
assembling errors. 
Special attention was paid to precessional 
reducer’s experimental research. For this purpose two 
laboratories were set up: 1) for mechanical tests and; 
2) working technology for gear wheels. The 
laboratories are equipped with stands for testing and 
with control and modern measuring devices. 
Know-how in the elaboration of the multicouple 
precessional gear, manufacturing technology and 
control methods, and a range of precessional 
transmission diagrams belong to research team from 
the Technical University of Moldova. During the last 
20 years the team patented about 127 inventions. 
 
2. Analytical description of the teeth 
profiles 
 
The engineering complex study of the triad 
''gear-technology-transmission'' has permitted to 
elaborate a new type of precessional transmissions 
with multicouple gear, which, from the technological 
point of, view can be manufactured via a new method 
of conical teeth with convex-concave profile 
processing. 
R
O
E2E1 ξ
ψ
ω1
ω2
ΘZ1
Z
Y
Y1
X
X1
E
D
ζ
E2E1 ξ
ζ
EN
 
 
Fig. 1. Profile teeth’s determination 
 
In precessional transmission the gear wheel 
produces sphere-spatial motion round a fixed point. 
As mentioned the literature, that the body, which 
produces spherical motion, has three degrees of 
freedom. As a rule, in theoretical mechanics, the 
position of the body, which produces precessional 
motion, is determined by Euler angles. In this case, 
the mobile system of co-ordinates OX1Y1Z1 is bound 
rigidly with the gear wheel, as origins of the system 
of co-ordinates the immobile point O (centre of 
precession) being chosen (Figure 1). This system of 
co-ordinates produces jointly with the gear wheel 
spherical motion related to the immobile system of 
coordinates OXYZ. Analytical relationship between 
the co-ordinates of gear wheel points, shown in the 
mobile OX1Y1Z1 and immobile OXYZ system of co-
ordinates, was obtained analysis of the two pinion 
positions - initial one, when the system of coordinates 
axis coincide, and final one, when the axis al removed 
(displaced). 
By expressing vectors '1'1'1 k,j,i  via basic vectors 
''
1
''
1
''
1 k,j,i  of the immobile system of coordinates 
OXYZ we obtain: 
 
 
kk  ,cosjsinj
,sinjcosii
'
1
'
1
'
1
=+−=
+=
ψψ
ψψ
 (1) 
 
The second rotation is produced at angle θ 
(O≤Θ≤π) round joint lines, after which the unit vector 
'
1
'
1
'
1 k,j,i  will move into directions ''1''1''1 k,j,i  
(respectively, they coincide with the directions of axis 
OX”, OY”, OZ”), at the same time vector ''1k , which 
coincide with vector 1k , defines the position of axis 
OZ1 in final position. By expressing vectors ''1''1''1 k,j,i  
via '1'1'1 k,j,i  we obtain: 
 
.cosksinjk
  ;sinkcosjj
;ii
'
1
'
1
"
1
'
1
'
1
"
1
'
1
"
1
Θθ
θθ
+−=
+−=
=
 (2) 
By operating matrix theory, transition from gear 
wheel point coordinates is produced (for example, of 
the roller centre D), given in the mobile system of co-
ordinates OX1Y1Z1 to the coordinates of the same 
points in the immobile system OXYZ. After some 
modifications we obtain:  
 
( ) ( )[ ]
;sinsinsinR
ZZcossinZZsincoscosRX 2121D
θψδ
ψψψδ
−
−+−=
( ) ( )[ ]
;sincossinR
ZZcoscosZZsinsincosRY 2121D
θψδ
ψψψδ
−
−+−=
( ) θδθψδ cossinRsinZZcoscosRZ 21D −−=  
(3) 
Point D moves at the spherical surface by radius R 
with its centre in the precessional centre O (figure 1). 
Being familiar with the trajectory of roller centre 
motion, the position of the contact point by central 
wheel tooth is determined, which family in a 
- 13 -
 
THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI 
0FASCICLE IX METALLURGY AND MATERIALS SCIENCE, ISSN 1453 – 083X. N . 2 – 2006 
 
 
precessional cycle represent the shape of the wheel 
tooth (Bostan, 1992).  
After some modifications we obtain:  
 ( )[ ]
( )[
( )[ ] .cosZsinR
Z
cosY
cosR
Z
sinsinX
;
Z
sinYcosR
Z
cosX
ED
1
E
D
1
E
1
ED
1
E
γβδπ
βδπγζ
πβδπξ
′′+++′′+
++−′′=
′′+++′′=
 (4) 
Were: 
 
( ) ( )
( ) .tg
Z
cos
Z
coscos
;tg
Z
costgsin
21
1
2
1
21
1
2
⎥
⎦
⎤
⎢
⎣
⎡
++=
⎥
⎦
⎤
⎢
⎣
⎡
+++=
βδππγ
βδπβδγ
  
1
2
 
 
Fig. 2. The diagrams of teeth with  
convex-concave profile. 
 
   In Figure 2 the diagrams of teeth profile, obtained 
for the various geometrical parameters of gear are 
shown.  The analyses of diagrams demonstrate the 
lever and direction of influence upon the angle of 
conical axoid δ , medium radius of wheel Rm, rollers 
radius R, conical angle of rollers β and correlation 
between  number of teeth Z1 and Z2. 
 
3. Gear manufacturing technology 
3.1. Kinematics of the realizing mechanism for 
the teeth generating method 
 
To realise the method of teeth processing we have 
elaborated the mechanism (Figure 3, Figure 4). In the 
elaborated mechanism the node, which involves the 
tool into precessional motion, is fixed not to rotate 
round the common axis of the principal shaft - 
semiproduct shaft with a binding mechanism.  
 
 
Fig. 3.  Spatial scheme of the realizing for the 
processing method by rolling using  
precessional tool. 
 
 
Fig. 4.  The device of the realizing for the teeth 
processing method by rolling using  
precessional tool. 
 
The constructive execution of the tool binding 
mechanism with the shaft ensures the continuity of 
the transmission function ω1/ω3 = const. and is 
determined by the motion trajectory of point C which 
belongs to the movable part. Setting up the position 
- 14 -
 
THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI 
0FASCICLE IX METALLURGY AND MATERIALS SCIENCE, ISSN 1453 – 083X. N . 2 – 2006 
 
 
function of the binding mechanism and the motion 
equation of the generating wheel and using the matrix  
device for transferring from the movable system of 
coordinates X1Y1Z1 to the immovable one OXYZ, we 
determined the coordinates of point C:  
 
 ( )
( )
.sinsinRZ
;coscossinRY
;sincoscos1RX
cc
22
cc
cc
ψθ
ψθψ
ψψθ
=
+=
−=
 (5) 
 
       These equations (5) represent the parametric 
equations of the supporting surface of the binding 
mechanisms, inserted in the device. The shape of this 
device ensures the continuity of the transmission ratio 
of the cinematic chain to the spindle axis-semiproduct 
shaft. 
 
 
3.2. Determination of the tool motion trajectory 
 
For the angle of the conical axoid of the teeth 
wheel 0=δ  the equation of tool motion are identical 
to the equation (4), having only opposite values.  
In the case of toothed wheels processing with an 
angle of the conical axoid 0>δ  the centre of the tool 
will have the co-ordinates in the movable system 
111 ΖΥΟΧ : 
 
 .sinR;cosR;0 D1D1D1 δΖδΥΧ −=−==
 
 
 
Then the equation of tool motion in the immovable 
system of coordinates ΟΧΥΖ  will be: 
 
 ( )
( )
.sinRcoscosR
;cossinRcoscossincosR
;sinsinRsincoscos1cosR
UUD
U
22
UD
UUD
δψδΖ
ψδψθψδΥ
ψδψψθΧ
−−=
+−−=
−−−=
 
(6) 
The motion trajectory of point D (curve 1, Figure 
2) for 0=δ  presents a symmetrically closed curve 
related to the big axis whose shape changes according 
to the angle value of conical axoid δ . Curve 2 
(Figure 2) presents the motion trajectory of tool’s 
centre in the movable system of coordinates 
.111 ΖΥΟΧ  
 
 
3.3 Determination of the family wrapping of 
tool surfaces 
 
Tooth profile of the processed wheel represents 
the family wrapping of the generating contour 
profiles of the tool.  
The wrapping is determined by the equations of 
the working surface of the generating tool and by the 
relative motion parameters while wrapping. 
To make easier the determination of wrapping we 
pass to the tool centre co-ordinates in the movable 
system of co-ordinates, bound to the semiproduct: 
 
 
.D
3D3D
;3D3D
;cossin
sincos
ΖΖ
ψΥψΧΥ
ψΥψΧΧ
=
+−=
+=
 (7) 
 
Where: DDD ,, ΖΥΧ  are the coordinates of the tool 
centre in the movable system of co-ordinates; 
i3 ψψ =  is the rotation angle of the semiproduct; i - 
is the transmission ratio of the cinematic chain 
“principal shaft - semiproduct“. The equations (7) 
define the motion trajectory of tool centre, evaluated 
on the sphere. Then we determined the equation of 
wrapping on the sphere (Figure 5).  
 
 
Fig. 5.  The family wrapping of tool surfaces. 
 
The analyses of diagrams demonstrate the lever 
and direction of influence upon the angle of tool 
insertion δ  by the rotation axis of the semiproduct, 
tool radius R and transmission ratio i of the cinematic 
chain “principal shaft - semiproduct ” on processed 
tooth profile. 
 
4. The elaboration of precessional reducers 
 
The elaboration of working machines driving 
mechanisms is based on the diagram of precessional 
transmissions, presented in Figure 6.  
 
Fig. 6.  Precession reducer. 
 
3
1
5
6 
4 
7 
2 
 
- 15 -
 
THE ANNALS OF “DUNAREA DE JOS” UNIVERSITY OF GALATI 
0FASCICLE IX METALLURGY AND MATERIALS SCIENCE, ISSN 1453 – 083X. N . 2 – 2006 
 
 
The rotating motion of the crank shaft 1 is 
transformed into sphere-spatial motion of the block 
pinion 2 with two teethed crowns 3 and 4, which are 
rolling without sliding on the immovable and driven 
toothed wheel teeth 5 and 6. Due to the minimum 
difference between the numbers of teeth 
,  the 
transmission ratio is:  
1,1 4635 −=−= ΖΖΖΖ ...3,2,143 += ΖΖ
 
 ;i
6345
63
ΖΖΖΖ
ΖΖ
−
±=  (8) 
 
The teeth of crowns 3 and 4 are manufactured in 
the shape of conical rollers installed on axis having 
the possibility to rotate round them, and the teeth of 
central wheel 5 and 6 have non-standard convex-
concave profile (Figure 2). 
 
References 
 
[1]. Bostan I. Precessional transmissions with multi-pair gearing. 
Stiinta Publishing House, 1991, 356p.  
[2]. Bostan, I., Babaian I., Precessional gear-engagement / 
Patent RU nr. 1455094 (patent MD 560), 1990 
[3]. Bostan, I., Dulgheru V., Vaculenco M. Precessional gear-
engagement  and method of its realisation/ Patent  MD 1886, 2001. 
[4]. Bostan, I., Dulgheru, V., Grigoraş, Ş., Planetary, 
precessional and harmonic transmissions, Bucureşti - Chişinău, 
1997.  
 
- 16 -


Some Aspects Regarding the Elaboration of Multiple Precessional Gear Theory and Modern Manufacturing Technology

https://www.gup.ugal.ro/ugaljournals/index.php/mms/article/download/3196/2876

Some Aspects Regarding the Elaboration of Multiple Precessional Gear Theory and Modern Manufacturing Technology

Abstract

Similar works

Full text

Available Versions

Directory of Open Access Journals