COMPUTER-AIDED DESIGN OF MULTISTAGE INSTRUMENTAL TOOTHED MECHANISMS

OPTIMIZED BY CRITERION OF VOLUME MINIMIZATION

АВТОМАТИЗИРОВАННОЕ ПРОЕКТИРОВАНИЕ МНОГОСТУПЕНЧАТЫХ ПРИБОРНЫХ ЗУБЧАТЫХ МЕХАНИЗМОВ, ОПТИМИЗИРОВАННЫХ ПО КРИТЕРИЮ МИНИМИЗАЦИИ ОБЪЕМА РЕДУКТОРА

Prof. Dr. Eng. Shalobaev E.V. 1, Prof. Dr. Eng. Starzhinsky V.E. 2,

Prof. Dr. Eng. Basiniuk U.L. 3, Dr. Eng. Mardasevich A.I. 3 Saint-Petersburg National Research University of Information Technologies, Mechanics and Optics1, Saint-Petersburg, Russia

V.A. Belyi Metal-Polymer Research Institute of National Academy of Sciences of Belarus2, Gomel, Belarus Joint Institute of Mechanical Engineering of National Academy of Sciences of Belarus3, Minsk, Belarus

Abstract: On the base of studying multistage instrumental toothed mechanisms provided earlier computer-aided design of multistage gear drives, optimized by criterion of minimum volume is considered. The sowfware’s CAD-systems including computer programs for multistage gear drives of ledge wise, coaxial and orbital arrangements are developed. Different schemes of arrangement are analyzed and results of optimization and automation design are described. Developed CAD-systems provide design of toothed mechanisms with optimized kinematic parameters on the criterion of minimum volume. KEYWORDS: MULTISTAGE INSTRUMENTAL TOOTHED MECHANISM, OPTIMIZATION, PC-AIDED DESIGN, LEDGE-WISE AR-RANGEMENT, COAXIAL ARRANGEMENT, ORBITAL ARRANGEMENT

1. Introduction To optimize a toothed mechanism, one of the following pa-

rameters (or a combination of them) can be taken as a criterion: efficiency of the mechanism; volume of the mechanism (or the sum of distances between axes); weight of gears; reduced inertia mo-ment; cumulative kinematic error and some other.

A toothed gear pair used in instrumental drives differs from others by large values of the total gear ratio ur – from a few hun-dreds to a few tens of thousands. When such drives design the mul-tistage fine-pitch cylindrical gear trains are used. The problems of arrangement and optimization of the instrumental drive kinematic parameters are systematically discussed in technical papers (see [1-3]), the ledgewise arrangements on the main. Along with the analy-sis of optimum versions of the ledgewise arrangements, the recom-mendations on optimization of the dimensional and efficiency crite-ria for the orbital arranged mechanisms are given in [4] as well as the kinematic parameters for toothed mechanisms of worm, three-stage, two-route types.

In our previous papers [5-7] the methods of optimization of the kinematic parameters of ledgewise (in-line diagram) and orbital arrangements are analyzed. In addition to ledgewise arrangement placed in-line, a specific ledgewise arrangement is given in [8-10], where the stages are arranged at certain angle φ to the line connect-ing the centers of the input and output shafts, and a coaxial ar-rangement is analyzed too. The advantages of the coaxial arrange-ment are shown, especially of the trains equipped with the plastic gear units rotating about stationary axles of real reducers and its optimization is described.

The results of optimization of the multistage toothed mecha-nisms are reviewed, analyzed and summarized in the paper [11]. For a ledgewise mechanism with φ-angle the dependencies are received of the minimal reducer volume versus angle φ with account of the stage gear ratio. In the considered arrangements the diagram with φ-angle is a summarized one, which at φ = 0o transforms into in-line ledgewise arrangement and at φ = 90o with minimal alterations transforms into a reducer of coaxial arrangement. The coefficients determining relation of the stage number n – total gear ratio ur for reducers of different arrangements have been determined (Table 1), total reducer gear ratio Ur as well as the optimal stage number has been recommended for studied kinematic arrangements (Fig. 1).

Table 1: Generalized data by the optimal stage ratio uj and stage number n for different arrangements

Arrangement diagram

Stage gear ratio uj

Optimal stage num-ber nopt

Formulae number

In-line φ = 0o

1.668 nopt = 4.55 lg ur (1)

30o 1.698 nopt = 4.35 lg ur (2) 60o 1.632 nopt = 4.70 lg ur (3) Le

dgew

ise

arra

ngem

ent

80o 1.468 nopt = 6.0 lg ur (4) Coaxial arrange-ment φ = 90o

1.931 nopt = 3.6 lg ur (5)

Orbital arrange-ment

1.655 (un =

1.672)

nopt = 1.95 + 2.60 lg ur

(6)

Fig. 1 Zones of optimal stage number versus total gear ratio for next ar-rangements: 1 – ledgewise with in-line stages; 2 – ledgewise with φ-angle arrangement; 3 – coaxial; 4 – orbital arrangement

It has been established too that the choice of a reducer arrange-

ment with a minimal volume depends on the total gear ratio ur. With high ur values the orbital arrangement is recommended, whereas with low ur values smaller total dimensions are typical of the ledgewise with ϕ-angle arrangement.

2. Analytical formulae and diagram arrangements Considered diagrams of reducer arrangements are shown in Fig.

2.

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a b

c d

Fig. 2 Diagrams and schematic 3D-representation of reducers with different arrangements: a – ledgewise in-line placed stages;

b – ledgewise arrangement with ϕ-angle placed stages;c – coaxial arrangement: d – orbital arrangement.

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2.1. Ledgewise and coaxial arrangements Specific volume of multistage reducer is determined by follow-

ing formula common for ledgewise (two types) and coaxial reducer arrangements:

,0000 HBAV = (7) where structure of components A0, B0, H0 are devoted in the table 2 [11].

Table 2: Structure of formula (7) for determination of specific volume of multistage drives of different arrangements

Arrangement Parameter nomina-tion and designation In-line

stage ϕ = 0

ϕ-angle stage ϕ ≠ 0

Coaxial ϕ = π/2

Specific length, A0 (1+uj)(1+n) (1+uj)(1+ncosϕ) 1+3uj Specific height, H0 uj 2u+(1+uj)sinϕ 2uj Specific width, B0 1 1 n+(n+1)k1 Specific volume, V0 2uj

(1+n)(1+uj) (1+uj)(1+ncosϕ)* [2uj+(1+uj)sinϕ]

2u(1+3uj)[ n+(n+1)k1]

Width ratio, K N+(N+1)k1 N+(N+1)k1 1 2.2. Orbital arrangement Volume V0

occupied by the reducer with orbital arrangement, is [11]:

V R Bk02= π , (8)

where B – the reducer width determined from expression (9): ,KmB bmψ= (9)

where ;2,0...1,0 ];)1([ 11 =++= kkNNK (10)

Rk – radius of the circle tangent to the pitch circle of the gears on the orbit will be:

Rk = mz (1+uj + un)/2. (11) Then specific reducer volume

00 γV will be [11]: 2

nj00 )uu1(/V ++=γ , (12)

where γ0 = (π/4)m3z2ψb1k0, K0 = 3,4…3,8 (13)

The factor of orbit filling α should be not exceeding a unit ( 1≤α ), and it is determined by expression (12):

).1(2/)1)(1( nj unu +−+= πα (14)

For the optimal stage number nopt determination the formula (6) is used.

The present paper is devoted to new results obtained during fur-ther subject investigations in this direction, e.g. the software-programs of CAD-systems for designing above mentioned ar-rangements are developed.

3. Procedure of automated designing multistage

reducers with ledgewise and coaxial arrangements Below the computation algorithm with numerical examples is

considered (Table 3).

Table 3: Initial data and procedure of computation for ledgewise and coaxial arrangement

Parameter nomination Parameter designation

Numerical value

Initial data 1 Total reducer gear ratio

ru 100 2 Module, mm m 1,0 3 Tooth numbers at reducer driving shafts

jz 17

4 Range member N 3 5 Facewidth ratio mbwbm /=ψ bmψ 6

Procedure of computation a) Ledgewise of in-line stage arrangement ( 0=ϕ )

1 Optimal stage number – according the formula (1) optn 9 2 Stage gear ratio optn

rj uu /1)(= (15) ju 1,668

3 Further computation is carried out for versions:

optn = 1n , 2n = ( 1+optn ), 3n = ( 1−optn ) 1n

n2

3n

9 10 8

4 Reducer dimensions, mm 2/)1)(1(1 ++= junmzA ,

umzH 1= ,

KmB bmψ=

A H B K

226,78 28,36 21,6 3,6

5 Reducer volume, mm3

00γVAÂÍV ==

V

138920

6 Specific reducer volume

)1)(1(2/ 00 ++== unuVV γ

4/230 Kzm bmψγ = (16)

0V

0γ 89,0

1560,6

7 Compare V at n1, n2 and n3 and select minV

minV

b) Ledgewise 0≠ϕ angle stage arrangement 1 Optimal stage number – according the formula (4)

optn 12

2 Stage gear ratio – according the formula (15) ju 1,468

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Тable 3: Continuation

Parameter nomination Parameter designation

Numerical value

3 Further computation is carried out for versions:

1n = optn , 2n = 1+optn , 3n = 1−optn

n1 n2 n3

12 13 11

4 Maximum angle maxϕ

)1(2/1,1cos max ju+=ϕ

maxϕ

77,23

5 Reducer dimensions, mm 2/)1)(cos1(1 junmzA ++= ϕ

2/]sin)1(2[1 ϕjj uumzH ++=

B - according the formula (9) K- according the formula (10)

1A

1H

1B K

77,079 45,406

21,6 3,6

6 Reducer volume, mm3

00γVABHV ==

V

75597,4

7 Specific reducer volume )1](sin)1(2)[cos1(/ 001

uuunVV ++++== ϕϕγ ,

0γ - according the formula (16)

10V

0γ

48,442

1560,6

8 To compare 321 ,, VVV and select minV

c) Coaxial arrangement ( 2/π=ϕ ) 1 Optimal stage number – according the formula (5) optn 7 2 Stage gear ratio

ju - according the formula (15)

3 Further computation is carried out for versions: n1 =

optn , 2n = 1+optn , n3 = 1−optn

ju

n1

n2 n3

1,931 7 8 6

4 Reducer dimensions, mm

jumzA 1=

KmB bmψ=

2/)31(1 jumzH +=

[ ]knnK )1( ++= , 15,0=k

A B H K

32,827

49,2 57,7 8,2

5 Reducer volume

00γVABHV ==

V

93256,8

6 Specific reducer volume )31(2/ 00 jj uuVV +== γ

0γ - according the formula (16)

0V

0γ

26,235

433,5,7

7 Compare V at n1, n2 and n3 and select minV minV

Preliminary check has been carried out, results of which is shown in the Tables 4-6. Table 4: To analysis of V0min for ledgewise in-line stage arrangement (ver-sion a)

Numerical parameter value for version Parameter 1 2 3 4

n 7 8 9 10 ju 1,931 1,778 1,668 1,585

0V 90,5 88,9 89,0 90,1 Table 5: To analysis of V0min for ledgewise of ϕ-angle stage arrangement (version b)

Numerical parameter value for version Parameter 1 2 3 4

n 10 11 12 13

ju

1,585 1,520 1,468 1,425

0V 47,8 47,1 48,4 49,9

Table 6: To analysis of V0min for ledgewise of coaxial arrangement (version c) Numerical parameter value for version

Parameter 1 2 3 n 6 7 8

ju 2,154 1,931 1,778

0V 226,3 215,1 233,1 K 7,05 8,2 10,35

The analysis of draft calculations presented in Tables 4-6 shows

that assembling of the drive following variant b) and ur = 100 gives a minimal volume of the reducer rather at n = n0pt-1 number of stages than at n = n0pt, which proves the adequacy of accepted cal-culation variants V0min at n0pt, n0pz-1 and n0pz+1. For the rest variants of assembly V0min corresponds to n0pt.

4. Procedure of automated designing multistage

reducers with orbital arrangement Below the computation algorithm of orbital arrangemented re-

ducer, with numerical example is considered.

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Table 7: Initial data and procedure of computation for orbital arrangement Parameter nomination Parameter

designation Numerical

value Initial data

Total reducer gear ratio ru 10000

Module, mm m 1,0 Tooth numbers at reducer driving shafts

jz 17

Range numbers N 3 Facewidth ratio m/bwbm =ψ bmψ 6

Factor of orbit filling α 0,95 Procedure of computation

1 Optimal number of reducer stages – according to formula (6) nopt 12 2 Gear ratio of last stage

1/ −= njrn uuu (17)

3 Unknown value ju has been found by decision of equation (18)

)1)(1()/1(2 1 −+=+ − nuuu jnjrπα (18)

Then actual value α have been found by the formula (14)

ju

nu

α

2,012 4,572 0,946

4 Reducer casing radius, mm

)1(2 njk uumzR ++=

kR

64,463

5 Specific reducer volume 2

00 )1(/ nj uuVV ++== γ , where

0γ - according to formula (13)

0V

0γ

K

57,575 17432,1

3,6 6 Reducer volume, mm3

002 γπ ⋅== VBRV K

KmB bmψ=

V B

1002602,0

21,6

7 Further computation is carried out for versions: nopt, nopt + 1 and nopt - 1 nopt nopt + 1 nopt - 1

12 13 11

Table 8: Comparison of approximation errors for computation results of [11] version and PC-program one with approaches by equations (17) and (20)

Numerical value of parameter at total gear ratio ur Parameter values 102 103 104 105

optn 7,272 9,584 12,230 15,061

ju 1,878 1,960 1,989 2,001

nu 1,917 3,105 4,424 5,818

Theoretical [11]

TV min0 23,0 36,8 55,0 77,8

optn 7,15 9,75 12,35 14,95

ju 1,802 1,935 1,974 2,012

nu 1,917 3,105 4,424 5,818

At optn approximated

by equation (6)

aV min0 22,3 36,5 54,7 78,0

Approximation error %100

min0

min0min0 ⋅−c

ac

VVV 3,0 0,8 0,6 0,3

optn 7 9 12 13

ju 1,928 2,058 2,012 2,263

nu 1,947 3,408 4,572 5,558

Computed by PC-program

cV min0 23,8 38,0 57,5 77,8

optn 7 9 12 13 Approximated by equation (20)

bV min0 25,0 35,2 62,4 74,7

Approximation error %100

min0

min0min0 ⋅−c

bc

VVV 5,0 7,4 8,5 4,0

Difference between V0 values calculated by PC-program and

equation (6)

%100min0

min0min0 ⋅−c

ac

VVV 6,3 3,9 4,9 0,03

The analysis of draft calculations presented in Tables 4 - 6 shows that assembling of the drive following variant c) and ur = 100 gives a minimal volume of the reducer rather at n = n0 pt – 1 number of stages than at n = n0 pt, which proves the adequacy of accepted

calculation variants V min at n0 pt , n0 pz – 1 and n0 pz + 1. For the rest variants of assembly V0 min corresponds to n0 pt.

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The calculation results using a PC program were approximated by a quadratic equation of the type ax2 + bx + c with the correlation factor R2 = 0.973:

V0 min = 0.798 n20 pz + 7.678 n0 pt + 39.668 (20)

The approximation results are shown in a diagram (Fig. 3) whe-re a curve is drawn corresponding to V0 min calculated by n0 pt using formula (17).

Fig. 3 Dependence of specific volume V0 versus stage number nopt for re-ducer orbital arrangement: 1 – in accordance with PC-program; 2 – ap-proximated by equation (20); 3 – in accordance with nopt approximated by equation (17).

The analysis of data presented in Table 8 and Fig. 3 has proved,

firstly, that calculation of V0 min when determining n0 pt by Eq. (17) allows for the accurate enough compliance with the accepted crute-rion at ur ≤ 104, while at ur > 104 the value of n0 pt is elevated (n0 pt = 15 when Eq. (17) is used and n0 pt = 13, when it is determined by a PC program) where the error in calculations by Eq. (17) is rather large in contrast to that by a PC program (6.3%), and secondly, approximation of calculation results by Eq. (20) use in a PC pro-gram gives a larger error (up to 8.5%) making thereby dependence V0 оси n0 pt inadequate.

The display windows with the initial data and calculation results of multistage gear mechanisms using CAD software is illustrated in Fig. 4.

a

b

c

d

Fig. 4 Windows with initial data and computation results for arrangements: co-locating ledgewise (a, b), coaxial (c) and orbital (d) versions: a, b – ur = 1000; c – ur = 1000; d – ur = 100000.

5. Conclusion Based on the developed CAD algorithms for the multistage

gearing the programs for their computer-aided design were created with optimized kinematic parameters that ensure minimization of the reducer volume. The software program makes possible to calcu-late the minimal volume of the reducer at optimal correlation of uj for the next gearing design schemes: unfolded, with angular stage location towards the line connecting the input and output shaft axes, coaxial and orbital. As a result a designer is able to choose most adequate in size and geometry design scheme of the reducer.

6. References

[1] OLEKSIUK, W. To the Problem of Assortment of Optimum Gear Ratios in the Toothed Reducers, Measurements, Automation, Control,1964 No 12, pp 20-23 (Polish) [2] DMITRIEV, F.S. Design of reducers for precise instru-ments, Mashinostroenie, 1971 (Rus) [3] Elements of Instruments: Design Works for Students Edited by O.F. Tishchenko, Vol. 1, Calculations, Vysshaya Shkola, 1978 (Rus.) [4] ISTOMIN, S.N., Designing of Gearing for Instruments Us-ing Computers, Mashinostroenie, 1985 (Rus.) [5] STARZHINSKY, V.E., SHALOBAEV, E.V., OS-SIPENKO, S.A., BABTCHENKO, A.A. (1997) Optimization of Multistage Orbital Arrangement of Instrument Gear Reducers, Gearing and Transmissions, No 2, pp 16-24 [6] Plastic gears in instrumental drive mechanisms, Edited by V.E. Starzhinsky and E.V. Shalobaev, MPRI NASB, 1998 (Rus) [7] STARZHINSKY, V.E., OSSIPENKO, S.A., SHALOBAEV, E.V., Optimization of Multistage Toothed Mechanisms, Proceedings of the International Conference “Mechanics in Design-98”, Not-tingham Trent University, 1998, pp 111-119 [8] STARZHINSKY, V.E., OSSIPENKO, S.A., SHALOBAEV, E.V., Assortment of Kinematic Parameters of Multistage Toothed

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Mechanisms, Bulletin of Kharkov State Politechnical University No. 109, Technologies in Machine Engineering, KSPU, 2000, pp 173-180 (Rus) [9] SHALOBAEV, E.V., STARZHINSKY, V.E., OS-SIPENKO, S.A., Arrangement schemes and optimization of kine-matic parameters of instrumental reducers, Proceedings of scien-tific and practical conference “Russian Reducer-making: Statement, problems, perspectives”, Center of numerical printing “Svetoch”, 2003, pp 56-60 (Rus) [10] SHALOBAEV, E.V., MONAHOV, YU., S., STARZHIN-SKY, V.E., Statement and perspectives of development of coaxial multistage reducers of new generation, Proceedings of scientific and practical conference “Russian Reducer-making: Statement, problems, perspectives”, Center of numerical printing “Svetoch”, 2003, pp 55 (Rus)

[11] STARZHINSKY, V.E., OSSIPENKO, S.A., SHALOBAEV, E.V., MONAHOV, YU., Optimization of Multistage Instrumental Toothed Redusers by Volume Minimization Criterion, Proceedings of the 2nd International Conference “Power Transmission-2006”, Novi Sad, 2006, pp 95-102

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