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A rapidly convergent algorithm for computerized synthesis of machine tool gear drives
0094-114X/85 $3.00 + .00 Perg~non Press Ltd.
Mechanism and Machine Theoo Vol. 20, No. 4, pp. 361-365. 1985 Primed in the U.S.A.
A RAPIDLY COMPUTERIZED
CONVERGENT
ALGORITHM
SYNTHESIS
OF
FOR
MACHINE
TOOL
GEAR
DRIVES V. S I V A P R A S A D ~ Department of Mechanical Engineering, Andhra University, Waltair, India R A O V. D U K K I P A T I ~ Division of Mechanical Engineering, National Research Council of Canada, Ottawa, Canada and M. O. M. O S M A N § Department of Mechanical Engineering, Concordia University, Montreal, Canada AbstractBA complete analysis of all the kinematic versions of a drive having output speeds larger than six and employing the algebraic equations method involves a great deal of manual manipulations. The general theory of composite gear trains could reduce the tedium to some extent, but it could not lead to a general computer algorithm for design synthesis. The present algorithm based on the unified theory provides a computerized method to find the rank of any version in the kinematic merit order. The input data is simplified to that of feeding index exponent values for the applicable versions. The algorithm obviates the tedium of the manual manipulations involved when the algebraic equations method is used instead. Further. the algorithm provides values at a new design optimization point having attractive kinematic features hitherto not obtained. 1. INTRODUCTION
2. KINEMATIC OPTIMUM LAYOUT BASED ON S
It is known that for a given set of spindle speeds, there is no unique solution, due to the availability of a number of alternative kinematic versions of a gear drive. A complete analysis of all kinematic versions of a drive with output speeds higher than six and employing algebraic-equations methods involves many manual algebraic manipulations and much time and is truly a considerable task. as acknowledged in Ref. [l]. Researchers have duly recognized the tedium of the problem and employed new techniques, such as computerized multiparameter optimization[2], to eliminate these manual manipulations. The general theory of composite gear trains[3] reduces the tedium to some extent in the process of obtaining open-form solutions for gear size expressions. But it could not lead to a single computer algorithm for any general composite gear train, due to the need for separately identifying expressions for large gears in a transmission group for all kinematic versions. It is known that the unflied theory[4] requires only four equations for any composite gear train, compared to nine in the general theory. Utilizing this advantageous feature, we have developed an algorithm for the synthesis of a composite gear train based on a new design-optimization point revealed in Ref. [5].
It is shown in Ref. [5] that attractive values of envelop parameters are obtained at a new designoptimization point based on S. It requires identification of the smallest gears in G~ of a gear train. The unified theory[4] provides equations for sizes of these gears:
[Zij]l; =
Z,~6qcb"- l)(l - Sd¢' - ~ ) 6~[6 ' -
1 -
$6"-~(+
"-
I)]6"-
+~" (I)
[Z2j]nn,
=
Z,8(6"
-
I)(6"-
{6~[6 ~+ d,"-
6"~)(I
I -56"'*(6
cb"}[cb ~ -
l -
-
$ 6 .....
''S~U-x(6"
I)] -
l)] (2)
Since the smallest gear on the input shaft occurs in the first transmission in G;, indicated by h = 1, the ym,, value should be substituted for y in (1). In the same way the smallest gear on the intermediate shaft occurs in the last transmission in Gj, indicated by h = H ; , and the Yma, value should be substituted for y in (2). Depending on the value of S, one of these gears will be the smallest gear of a gear train. The technique of kinematic synthesis based on the 't
Reader in Production Engineering.
;t Associate Research Officer and member ASME. § Professor of Engineering and member ASME.
optimization criterion S requires finding the value of S for which the sizes of gears given by eqns (1) and (2) will be the same. 361
V. Szv^ PRASA.Det al.
362
3. DEVELOPMENT OF A CO/VIPUTER ALGORITHM F o r the example of a nine-speed gear train presented in this work, index exponents from Table 6b of Ref. [6] are used for substitution in (1) and (2). The index exponent y has been used here in its unflied mode according to Ref. [4]. Limits on the value of S have been established in[3, 6]. These limits, which ensure that the gear sizes remain finite and positive, as applicable for eqns (1) and (2), are 0 < S 6 U < ( ~ ~ - l)/~x(~ ~ -
])
(3)
for the case of kinematic versions with open-struc-
ture, i.e. w > x, and
(~
-
I)/(I)~'((I) '
-
I ) < S(b" < ~c
for the case of kinematic versions with closed-structure, i.e. w < x. Although expressions (3) and (4) are applicable for any general layout, only those kinematic versions having open-structure which mainly give speed reduction are useful for machine tool applications as shown in Refs. [1, 2, 3]. It was found useful to adopt the initial value of S as half of Smax for use in (1) and (2) for the first iteration. In case the subsequent value of S at any
F
C
Read
Type number, u, w, x. Ymin' Ymax' ¢
½ L
Initialize ~ o r y sticks
1
Set S - SNX/2
]
½
3"
I I
L Yes
S • S÷(DISOR)
],°
Set -I S • Smax/3 I Initialize
DI'D
I
(4)
l Yes Find ZI1 arid Z21 fnm (1) and Set 0 - ZI1 -
(2)
Z21
-re
iS- S1, S1-S2 L
F )3- sz L'i~- s,. s,-sj_. No
C Print Type ntllber, u, w, x, Ymin' Ymax" Z13 ) Fig. 1. Flowcharl.
363
Synthesis of machine tool gear drives time is greater than S . . . . according to expression (3), during iterations, one-third of Sma~ is then substituted as the new initial value of S. Memory locations SI, $2 and $3 are used to keep track of the previous three iterated values of S. When the it.lit. eration value of S crosses the converging point S during stepping, previous values from SI, $2 and $3 are retrieved until iterations can be continued from a previous step with a reduced stepping space. The difference between sizes of the two gears given by eqns (1) and (2) is computed as D to obtain successive iteration values of S. Memory location DI stores the value of D at the previous iteration for checking zero crossing at S. Whenever the iteration value of S crosses the converging point S, the stepping space is reduced for rapid convergence by incrementing R. F o r our purpose here, Z~e is taken to be unity, as in Ref. [2]: however, sizes of gears thus obtained must be multiplied by the minimum number of teeth permissible on the pinion according to the mesh conditions in the final design. The value of D is converted to integer mode for zero checking with an accuracy of one digit after the decimal point. This approach is found to give an accuracy up to the third decimal place in the computed value of S. A flow chart of the algorithm is given in Fig. 1. The technique is illustrated for a nine-speed composite gear train employing 10 gears[l], whose transmission scheme is shown in Fig. 2. The results for the nine applicable kinematic versions tabulated in Ref. [6] are given in Table 1. After final iteration. )'ma~ is substituted in expression (1) to give the size of ZL~ for comparing various versions. 4. DESIGN SYNTHESIS
Z,13 Zll rL
S. No.
u
m
x
Ymm
)'max
~
Z13
1. 2. 3. 4. 5. 6. 7. 8. 9.
0 0 3 0 0 3 1 1 4
3 6 3 3 6 3 3 6 3
1 1 1 2 2 2 1 1 1
0 0 -3 0 0 -3 0 0 -3
6 6 3 6 6 3 6 6 3
0.062 0.018 0.031 0.240 0.041 0.091 0.057 0.017 0.029
2.512 3.365 3.024 1.999 2.858 2.197 2.544 3.399 3.113
based on S and its structural scheme are shown in Figs. 3 and 4, respectively. Sizes of various gears *
of this kinematic version at S may be obtained from unified theory[4] as Zll = 1.000.
ZI2 = 1.499.
Z13 = 1.999.
Z_,3 = 1.744,
Z.,4 = 2.000.
Z31 = 2.623.
Z.,, = 1.001.
Z.,: = 1.500.
Z3: = 2.881.
Z3.~ = 3.123.
The two connecting gears between Gj and G.~ are Z_,_, and Z_,4. Transmissions containing these connecting gears are shown by dashed lines in Fig. 4. This layout obtained with the present algorithm, defined by the exponents of 6 given in Fig. 4. tallies with the kinematic optimum gearing layout of the nine-speed composite gear train obtained by the algebraic-equations method[ 1]. Envelope Tn. proposed in Ref. [1]. may be calculated for this version from Fig. 3 [T~]z,:~ = Zi, + Z_-4 ÷ Z31
Table 1 shows that the smallest value of ZL~ occurs for the Type 4 kinematic version, and the corresponding kinematic optimum gearing layout
AI ~
Table 1. Computed values of ~ when cb = ~, 2
Z?.3
= 1.000 + 2.000 -,- 2.623 = 5.623. Assuming Zr~. = 20. we get T~ = 113 in terms of the number of teeth, which is better than the corresponding minimum value of 129 obtained in Ref. [I]. Selecting an appropriate module for gears and multiplying it by T,, we get values of this envelope dimension in millimeters. Similarly, it may be shown that values of other envelope parameters are ~t
- -
A2
X
Z21
Z24 A3
also optimal at this design point based on S.
X m
1
CJ m
Z3 3
'32
Fig. 2. Transmission scheme.
13
Aa
Fig. 3. Kinematic optimmum layout.
364
V. SIvA PRXSADet al.
/
~' %'%%%%,
yl "-0
%%
.
/
y3:6
Z~
w= Y2:3
2~
",, Z x Zl~,,
Z22
Z21 "~~.
A2
Zl=0 s ~l.--x , 2 ' A3
s
s~
Z22
s~
I
s~
Z
so'
s@
s®6
so'
Fig. 4. Structural scheme. The values of S for the nine applicable kinematic versions given in Table 1, which were hitherto not available in the literature, suggest a better optimization criterion for choosing a kinematic version judged by the largest value of S. Since the present algorithm is based on the unflied theory, it obviates the necessity of feeding specific gear size expressions for each kinematic version of a composite gear train. Thus, the input data for the computation has been simplified to that of feeding the values of index exponents. Since the algorithm is general in nature, it would be easy for a designer to extend the technique for the synthesis of other composite gear trains based on various kinematic optimum design points, depending on the specific application.
5. CONCLUSIONS The algorithm based on the unified theory is found to be useful for the design synthesis of any composite gear train employing three shafts for speed reduction applications in machine tools. The algorithm obviates the necessity of feeding specific gear size expressions for each kinematic version of a composite gear train. The algorithm provides computations at a new design point with better values of envelope parameters. A better criterion for choosing a kinematic op,ik
timum version judged by the largest value of S has been outlined. The input data has been simplified to that of feeding values of index exponents. The algorithm facilitates finding the rank of any llr
version in the kinematic merit order based on S. employing a generalized method of design synthesis.
NOMENCLATURE D Arithmetic difference between sizes of the smallest gears in Gj. D~ Memory location for storing the previous value of D. G~ First transmission group in the power transmission order, i.e. between the input and intermediate shafts. G: Second transmission group in the power transmission order, i.e. between the intermediate and output shafts. h Number identifying position of a transmission in G~, taking the lowest speed transmission as the first (h = 1, 2, 3. . . . . HD. H~ Total number of transmissions in kth group. i Number identifying the position of a shaft in the power transmission order, taking the input shaft as the first (i = 1, 2.3 . . . . . ). j Number identifying the size order of a gear on a shaft, taking the smallest gear as the first I,i = 1.2 . . . . ). k Number identifying the position of a gear transmission group in the power transmission order (k = 1, 2). R Arbitrary variable used for step increment. S Lowesl output/input speed ratio. S l, $2. $3 Memory locations for storing the previous value of S. Sma× Maximum limit of the value of S. Value of S when the smallesl gears in G~ have equal size[5]. TI Radial envelope proposed in[l]. U Integer exponent of d~where 4" defines ratio of the lowest speed from the double-composite section and S. w Integer exponent of ~, where ~b~ defines the ratio of output speeds obtained by transmissions of the double-composite section in G~. x Integer exponent of 6, where ,x defines the ratio of output speeds obtained by transmissions of the double-composite section in G,.. Y Transmission index of G~. Zo Number of teeth or diameter of a gear on ith shaft with jth size. Z u (driving gear if i = k or driven gear if i + I = k) participating in hth transmission in G~. Number of teeth or diameter of the reference gear. Geometric progression ratio of output speed s. cb>l.
Synthesis of machine tool gear drives
Acknowledgements--The support of the Natural Sciences and Engineering Research Council of Canada Grant No. A5181 and La Formation de Chercheures et d'Action Concertee of the Gouvernment of Quebec, Grant No. 042-II0, is gratefully acknowledged. REFERENCES 1. G. White and D. J. Sanger. Int. J. MTDR 8, 141 (1968). 2. M. O. M. Osman, S. Sankar and R. V. Dukkipati, Trans. ASME, J. Mech. Design 100, 303 (1978). 3. D. J. Sanger and G. White. A General Theou" of Corn-
365
posite Gear Trains. Proc. Inst. Mech. Engrs., Vol. 184, Part I, No. 58, 1969-70, p. 1063. 4. V. Siva Prasad, A UnO~ed Theory of Three-Shaft Composite Gear Trains. Proc. 23rd Congress on Theoretical and Applied Mechanics, p. 55. Warangal, Dec. (1978). 5. V. Siva Prasad, On the Kinematic Synthesis of Gear Trains Employing Minimum Number of Gears to Obtain Minimum Overall Size. Paper No. k2, p. 453, Proc. 7th All India Machine Tool Design Res. Conf., Coimbatore, June (1976). 6. D.J. Sanger and G. White, Synthesis of composite gear trains. J. Mechanisms 6, 143-165 (1971).
ALGORITHME A CONVERGENCE RAPIDE POUR LA CONCEPTION ASSISTEE PAR ORDINATEUR DE COMMANDES PAR ENGRENAGES DE MACHINES-OUTILS R~sum~--Une analyse compl6te de toutes les versions cin6matiques d'une commande par engrenages plus de six vitesses de sorties, bas6e sur les ~quations alg~briques, n6cessite un tr6s grand nombre de manipulations manuelles. La th~orie g~n~rale des trains d'engrenages compos~s pourrait r6duire ce travail fastidieux jusqu'fi un certain point, mais elle ne pourrait pas donner un aigorithme informatique g6n~ral utilisable pour la conception assist~e par ordinateur. L'algorithme present6 dans cet article est bas6 sur une th~orie unitize et il g~n6re des m6thodes informatis~es permettant de trouver le rang de toute version dans I'ordre du m~rite cin~matique. En outre, l'algorithme fournit des valeurs un nouveau point d'optimisation de la conception ayant des caract~ristiques cin6matiques int6ressantes, et qu'on n'aurait pas pu obtenir autrement.
Mechanism and Machine Theoo Vol. 20, No. 4, pp. 361-365. 1985 Primed in the U.S.A.
A RAPIDLY COMPUTERIZED
CONVERGENT
ALGORITHM
SYNTHESIS
OF
FOR
MACHINE
TOOL
GEAR
DRIVES V. S I V A P R A S A D ~ Department of Mechanical Engineering, Andhra University, Waltair, India R A O V. D U K K I P A T I ~ Division of Mechanical Engineering, National Research Council of Canada, Ottawa, Canada and M. O. M. O S M A N § Department of Mechanical Engineering, Concordia University, Montreal, Canada AbstractBA complete analysis of all the kinematic versions of a drive having output speeds larger than six and employing the algebraic equations method involves a great deal of manual manipulations. The general theory of composite gear trains could reduce the tedium to some extent, but it could not lead to a general computer algorithm for design synthesis. The present algorithm based on the unified theory provides a computerized method to find the rank of any version in the kinematic merit order. The input data is simplified to that of feeding index exponent values for the applicable versions. The algorithm obviates the tedium of the manual manipulations involved when the algebraic equations method is used instead. Further. the algorithm provides values at a new design optimization point having attractive kinematic features hitherto not obtained. 1. INTRODUCTION
2. KINEMATIC OPTIMUM LAYOUT BASED ON S
It is known that for a given set of spindle speeds, there is no unique solution, due to the availability of a number of alternative kinematic versions of a gear drive. A complete analysis of all kinematic versions of a drive with output speeds higher than six and employing algebraic-equations methods involves many manual algebraic manipulations and much time and is truly a considerable task. as acknowledged in Ref. [l]. Researchers have duly recognized the tedium of the problem and employed new techniques, such as computerized multiparameter optimization[2], to eliminate these manual manipulations. The general theory of composite gear trains[3] reduces the tedium to some extent in the process of obtaining open-form solutions for gear size expressions. But it could not lead to a single computer algorithm for any general composite gear train, due to the need for separately identifying expressions for large gears in a transmission group for all kinematic versions. It is known that the unflied theory[4] requires only four equations for any composite gear train, compared to nine in the general theory. Utilizing this advantageous feature, we have developed an algorithm for the synthesis of a composite gear train based on a new design-optimization point revealed in Ref. [5].
It is shown in Ref. [5] that attractive values of envelop parameters are obtained at a new designoptimization point based on S. It requires identification of the smallest gears in G~ of a gear train. The unified theory[4] provides equations for sizes of these gears:
[Zij]l; =
Z,~6qcb"- l)(l - Sd¢' - ~ ) 6~[6 ' -
1 -
$6"-~(+
"-
I)]6"-
+~" (I)
[Z2j]nn,
=
Z,8(6"
-
I)(6"-
{6~[6 ~+ d,"-
6"~)(I
I -56"'*(6
cb"}[cb ~ -
l -
-
$ 6 .....
''S~U-x(6"
I)] -
l)] (2)
Since the smallest gear on the input shaft occurs in the first transmission in G;, indicated by h = 1, the ym,, value should be substituted for y in (1). In the same way the smallest gear on the intermediate shaft occurs in the last transmission in Gj, indicated by h = H ; , and the Yma, value should be substituted for y in (2). Depending on the value of S, one of these gears will be the smallest gear of a gear train. The technique of kinematic synthesis based on the 't
Reader in Production Engineering.
;t Associate Research Officer and member ASME. § Professor of Engineering and member ASME.
optimization criterion S requires finding the value of S for which the sizes of gears given by eqns (1) and (2) will be the same. 361
V. Szv^ PRASA.Det al.
362
3. DEVELOPMENT OF A CO/VIPUTER ALGORITHM F o r the example of a nine-speed gear train presented in this work, index exponents from Table 6b of Ref. [6] are used for substitution in (1) and (2). The index exponent y has been used here in its unflied mode according to Ref. [4]. Limits on the value of S have been established in[3, 6]. These limits, which ensure that the gear sizes remain finite and positive, as applicable for eqns (1) and (2), are 0 < S 6 U < ( ~ ~ - l)/~x(~ ~ -
])
(3)
for the case of kinematic versions with open-struc-
ture, i.e. w > x, and
(~
-
I)/(I)~'((I) '
-
I ) < S(b" < ~c
for the case of kinematic versions with closed-structure, i.e. w < x. Although expressions (3) and (4) are applicable for any general layout, only those kinematic versions having open-structure which mainly give speed reduction are useful for machine tool applications as shown in Refs. [1, 2, 3]. It was found useful to adopt the initial value of S as half of Smax for use in (1) and (2) for the first iteration. In case the subsequent value of S at any
F
C
Read
Type number, u, w, x. Ymin' Ymax' ¢
½ L
Initialize ~ o r y sticks
1
Set S - SNX/2
]
½
3"
I I
L Yes
S • S÷(DISOR)
],°
Set -I S • Smax/3 I Initialize
DI'D
I
(4)
l Yes Find ZI1 arid Z21 fnm (1) and Set 0 - ZI1 -
(2)
Z21
-re
iS- S1, S1-S2 L
F )3- sz L'i~- s,. s,-sj_. No
C Print Type ntllber, u, w, x, Ymin' Ymax" Z13 ) Fig. 1. Flowcharl.
363
Synthesis of machine tool gear drives time is greater than S . . . . according to expression (3), during iterations, one-third of Sma~ is then substituted as the new initial value of S. Memory locations SI, $2 and $3 are used to keep track of the previous three iterated values of S. When the it.lit. eration value of S crosses the converging point S during stepping, previous values from SI, $2 and $3 are retrieved until iterations can be continued from a previous step with a reduced stepping space. The difference between sizes of the two gears given by eqns (1) and (2) is computed as D to obtain successive iteration values of S. Memory location DI stores the value of D at the previous iteration for checking zero crossing at S. Whenever the iteration value of S crosses the converging point S, the stepping space is reduced for rapid convergence by incrementing R. F o r our purpose here, Z~e is taken to be unity, as in Ref. [2]: however, sizes of gears thus obtained must be multiplied by the minimum number of teeth permissible on the pinion according to the mesh conditions in the final design. The value of D is converted to integer mode for zero checking with an accuracy of one digit after the decimal point. This approach is found to give an accuracy up to the third decimal place in the computed value of S. A flow chart of the algorithm is given in Fig. 1. The technique is illustrated for a nine-speed composite gear train employing 10 gears[l], whose transmission scheme is shown in Fig. 2. The results for the nine applicable kinematic versions tabulated in Ref. [6] are given in Table 1. After final iteration. )'ma~ is substituted in expression (1) to give the size of ZL~ for comparing various versions. 4. DESIGN SYNTHESIS
Z,13 Zll rL
S. No.
u
m
x
Ymm
)'max
~
Z13
1. 2. 3. 4. 5. 6. 7. 8. 9.
0 0 3 0 0 3 1 1 4
3 6 3 3 6 3 3 6 3
1 1 1 2 2 2 1 1 1
0 0 -3 0 0 -3 0 0 -3
6 6 3 6 6 3 6 6 3
0.062 0.018 0.031 0.240 0.041 0.091 0.057 0.017 0.029
2.512 3.365 3.024 1.999 2.858 2.197 2.544 3.399 3.113
based on S and its structural scheme are shown in Figs. 3 and 4, respectively. Sizes of various gears *
of this kinematic version at S may be obtained from unified theory[4] as Zll = 1.000.
ZI2 = 1.499.
Z13 = 1.999.
Z_,3 = 1.744,
Z.,4 = 2.000.
Z31 = 2.623.
Z.,, = 1.001.
Z.,: = 1.500.
Z3: = 2.881.
Z3.~ = 3.123.
The two connecting gears between Gj and G.~ are Z_,_, and Z_,4. Transmissions containing these connecting gears are shown by dashed lines in Fig. 4. This layout obtained with the present algorithm, defined by the exponents of 6 given in Fig. 4. tallies with the kinematic optimum gearing layout of the nine-speed composite gear train obtained by the algebraic-equations method[ 1]. Envelope Tn. proposed in Ref. [1]. may be calculated for this version from Fig. 3 [T~]z,:~ = Zi, + Z_-4 ÷ Z31
Table 1 shows that the smallest value of ZL~ occurs for the Type 4 kinematic version, and the corresponding kinematic optimum gearing layout
AI ~
Table 1. Computed values of ~ when cb = ~, 2
Z?.3
= 1.000 + 2.000 -,- 2.623 = 5.623. Assuming Zr~. = 20. we get T~ = 113 in terms of the number of teeth, which is better than the corresponding minimum value of 129 obtained in Ref. [I]. Selecting an appropriate module for gears and multiplying it by T,, we get values of this envelope dimension in millimeters. Similarly, it may be shown that values of other envelope parameters are ~t
- -
A2
X
Z21
Z24 A3
also optimal at this design point based on S.
X m
1
CJ m
Z3 3
'32
Fig. 2. Transmission scheme.
13
Aa
Fig. 3. Kinematic optimmum layout.
364
V. SIvA PRXSADet al.
/
~' %'%%%%,
yl "-0
%%
.
/
y3:6
Z~
w= Y2:3
2~
",, Z x Zl~,,
Z22
Z21 "~~.
A2
Zl=0 s ~l.--x , 2 ' A3
s
s~
Z22
s~
I
s~
Z
so'
s@
s®6
so'
Fig. 4. Structural scheme. The values of S for the nine applicable kinematic versions given in Table 1, which were hitherto not available in the literature, suggest a better optimization criterion for choosing a kinematic version judged by the largest value of S. Since the present algorithm is based on the unflied theory, it obviates the necessity of feeding specific gear size expressions for each kinematic version of a composite gear train. Thus, the input data for the computation has been simplified to that of feeding the values of index exponents. Since the algorithm is general in nature, it would be easy for a designer to extend the technique for the synthesis of other composite gear trains based on various kinematic optimum design points, depending on the specific application.
5. CONCLUSIONS The algorithm based on the unified theory is found to be useful for the design synthesis of any composite gear train employing three shafts for speed reduction applications in machine tools. The algorithm obviates the necessity of feeding specific gear size expressions for each kinematic version of a composite gear train. The algorithm provides computations at a new design point with better values of envelope parameters. A better criterion for choosing a kinematic op,ik
timum version judged by the largest value of S has been outlined. The input data has been simplified to that of feeding values of index exponents. The algorithm facilitates finding the rank of any llr
version in the kinematic merit order based on S. employing a generalized method of design synthesis.
NOMENCLATURE D Arithmetic difference between sizes of the smallest gears in Gj. D~ Memory location for storing the previous value of D. G~ First transmission group in the power transmission order, i.e. between the input and intermediate shafts. G: Second transmission group in the power transmission order, i.e. between the intermediate and output shafts. h Number identifying position of a transmission in G~, taking the lowest speed transmission as the first (h = 1, 2, 3. . . . . HD. H~ Total number of transmissions in kth group. i Number identifying the position of a shaft in the power transmission order, taking the input shaft as the first (i = 1, 2.3 . . . . . ). j Number identifying the size order of a gear on a shaft, taking the smallest gear as the first I,i = 1.2 . . . . ). k Number identifying the position of a gear transmission group in the power transmission order (k = 1, 2). R Arbitrary variable used for step increment. S Lowesl output/input speed ratio. S l, $2. $3 Memory locations for storing the previous value of S. Sma× Maximum limit of the value of S. Value of S when the smallesl gears in G~ have equal size[5]. TI Radial envelope proposed in[l]. U Integer exponent of d~where 4" defines ratio of the lowest speed from the double-composite section and S. w Integer exponent of ~, where ~b~ defines the ratio of output speeds obtained by transmissions of the double-composite section in G~. x Integer exponent of 6, where ,x defines the ratio of output speeds obtained by transmissions of the double-composite section in G,.. Y Transmission index of G~. Zo Number of teeth or diameter of a gear on ith shaft with jth size. Z u (driving gear if i = k or driven gear if i + I = k) participating in hth transmission in G~. Number of teeth or diameter of the reference gear. Geometric progression ratio of output speed s. cb>l.
Synthesis of machine tool gear drives
Acknowledgements--The support of the Natural Sciences and Engineering Research Council of Canada Grant No. A5181 and La Formation de Chercheures et d'Action Concertee of the Gouvernment of Quebec, Grant No. 042-II0, is gratefully acknowledged. REFERENCES 1. G. White and D. J. Sanger. Int. J. MTDR 8, 141 (1968). 2. M. O. M. Osman, S. Sankar and R. V. Dukkipati, Trans. ASME, J. Mech. Design 100, 303 (1978). 3. D. J. Sanger and G. White. A General Theou" of Corn-
365
posite Gear Trains. Proc. Inst. Mech. Engrs., Vol. 184, Part I, No. 58, 1969-70, p. 1063. 4. V. Siva Prasad, A UnO~ed Theory of Three-Shaft Composite Gear Trains. Proc. 23rd Congress on Theoretical and Applied Mechanics, p. 55. Warangal, Dec. (1978). 5. V. Siva Prasad, On the Kinematic Synthesis of Gear Trains Employing Minimum Number of Gears to Obtain Minimum Overall Size. Paper No. k2, p. 453, Proc. 7th All India Machine Tool Design Res. Conf., Coimbatore, June (1976). 6. D.J. Sanger and G. White, Synthesis of composite gear trains. J. Mechanisms 6, 143-165 (1971).
ALGORITHME A CONVERGENCE RAPIDE POUR LA CONCEPTION ASSISTEE PAR ORDINATEUR DE COMMANDES PAR ENGRENAGES DE MACHINES-OUTILS R~sum~--Une analyse compl6te de toutes les versions cin6matiques d'une commande par engrenages plus de six vitesses de sorties, bas6e sur les ~quations alg~briques, n6cessite un tr6s grand nombre de manipulations manuelles. La th~orie g~n~rale des trains d'engrenages compos~s pourrait r6duire ce travail fastidieux jusqu'fi un certain point, mais elle ne pourrait pas donner un aigorithme informatique g6n~ral utilisable pour la conception assist~e par ordinateur. L'algorithme present6 dans cet article est bas6 sur une th~orie unitize et il g~n6re des m6thodes informatis~es permettant de trouver le rang de toute version dans I'ordre du m~rite cin~matique. En outre, l'algorithme fournit des valeurs un nouveau point d'optimisation de la conception ayant des caract~ristiques cin6matiques int6ressantes, et qu'on n'aurait pas pu obtenir autrement.