- Email: [email protected]
The ‘Constructive-Deductive’ Design Approach — Application to Power Transmissions
The 'Constructive-Deductive' Design Approach - Application to Power Transmissions Dr. J.-P. van Griethuysen; Swiss Federal institute of Technology, Lausanne/Switzerland by Prof. A. Wirtz (1) Received on January 14,1992
- Submitted
ABSTRACT : The construcrivedeductive approach is a "bortom-up" optimized design method. The mehod proposes a sequence of Iogcal consmctive rules up IO the product solution from the functional specifications. It is easy to automate in an intelligent CAD expen system as it is sequenaai. A description of this approach will precede an application to demonstrate the fibst step of this method : the design onencd analysis.
KEYWORDS :design, modelling, bottom-up design. CAD
I.
INTRODUCTION
Todays rapid pace of technological developments forces the design and engineering depanments of indusmal companies to adapt to shorter product life. They must be able to decrease their development time. while continuing to increase quality. Therefore the design process has attracted many researchers who have produced amount of literature on this topic. In the annals of CIRP many articles deal with this problem and readers C M refer to the keynote ones [ 1.8, 101 where the state of the an in design is presented. PETERS et al. [I] stated, that two main orientations exist in design. The first orientation develops universal design methods. the second aims to integrate the overall expertise in the design itself. The axiomatic theory presented by SUH [9]attempts to create fundmental principles and methodologies to put the design field on a fm scientific basis [8]. This approach clearly involves the first orientation. The second approach vies to incorporate current design practice and knowledge (possessed by "design expens") into algorithmic theory. The consrmctive-deductive approach, was first applied by PRUVOT in his theory of Synthedc Intelligence [5]. It is an algorithm oriented method which can be automated.The knowledge to design is extracted from the product itself through analysis. By using scientific analysis. this approach guarantees firm bases. Starting from the specifications, this method is bottom-up oriented. By using logical rules, this design method suggests all the possible solutions. Sorting criteria must then be used to select the best solution(s). The aim of this paper is to present the latter approach and the state of the work using the constructive-deductive design method applied to analysis of power transmissions. Therefore a practical example of an automotive transmission is proposed.
11.2
THE U S U A L
ANALYSIS
Using usual analysis methods, a finished product can be progressively transformed until its performances are found (Figure I).
I
FINISHED PRODUCT
1
SPECIFICATIONS
I
Figure 1 : Usual analysis of a product During this analysis. MODELS have to be used. As models are obtained by means of THEORIES and are simplified as formal representations of actual things; thus analysis requires knowledge in order to be performed and its final result is made up of the PERFORMANCES of the product. Therefore, analysis necessary yields a loss of information (all that is materialistic). "Performances" are purely absuact. From a design point of view. this "lost" information will be called the "missing" information.
11.3
EMPIRICALDESIGN
During the design phase, a traditional engineer tries to build up a solution starting from the specifications (Figure 2). Then the "performances" of the solution are compared to the specifications using an analysis method as described above.
r
FINISHED PRODUCT iteration
XI.
THE CONSTRUCTIVE-DEDUCTIVE APPROACH
11.1
INTRODUCTION
Design is defined as the creation of all the necessary information for manufacturing a product, starting from its functional requirements. The product specifications never include all the needed information. In a NEW design - different from PARAMETRIC design -,the specifications are very simple and should only concern its PERFORMANCES. Therefore a design method has to create information. The different methods for creating information will not be compared in the present paper. Where as Artificial Intelligence (AI) tries to extract the knowledge of experienced engineers (experts) and to store it into an database, this connibution will propose a new approach to generate this information.
USUAL ANALYTICAL PRACTICE
I knowledge, intuition and
B
SPECIFICATIONS Figure 2 : Empirical design process
Annals of the CIRP Vol. 41/1/1992
169
Many researchers deal with the creation by the designer of the missing information. for example YOSHIKAWA [lo, I I ] presented "transitions between thinking units" and more generally speaking "mapping between spaces". A traditional engineer has therefore to jump From precise requirements to a nebulous solution. In this fashion, the creative act is never accomplished in a rational way. The designer "jumps" to the conclusion that is a "finished solution (at least piecewise). As a matter of fact. using his experience, a designer can pick up pans of old solutions to imagine a new one.
11.4
ANOTHER POINT OF VIEW
In an authentic design method, these "jumps" a ~ unacceptable e since they ;tre unable to generate all the solutions and particularly the best one. A good design method should be able to propose all the solutions. then the best one should be sorted out according to some clearly defined criteria. SoKing criteria may be geomemcal or physical, while others are economic or manufacturing oriented (design for production). However, we shall continue to concentrate on the design mechanism which enables decision trees to be analyzed funher. The proposed approach mes to reverse the analysis process to improve the empirical method. It mes to generate all the products by adding missing information to the specifications. If the missing information cwld be structured and stored in a database during the analysis, then it could eventually k used during the conceptual phase. At this point, this conclusion must be considered as purely hypothetical. 11.5
DESIGN ORIENTED ANALYSIS
The first difficulty of the proposed approach is the non-revenibility of the usual analysis. Figure 1 shows that it is impossible since the analysis is "tree oriented". A "linear" analysis method would stand a better chance of being reversible, but its possibility has to be demonstrated. A reversible analysis will be called design oriented (Figure 3). "lost" knowledge
IIr
FINISHED PRODUCT
11.6
DESIGN
Assuming that a linear design oriented analysis has been created. it might be possible to reverse it in order to build up a design method. This is the basis of the consouctive-deductive design method. "Consrrucrive" is used here with its philosophical meaning of "creorion ex nihilo"; its means that knowledge has to be generated from a base that doesn't contain it. "Deducrive" means that formal logic is used throughout. The term "constructive-deductive" has been defined by the logician/philosopher E. GOBLOT (1858-1935). [f one considers an analysis linear. then a reversible method should be h e x as well. Therefore, during the design phase a stepwise construction is used including qualitattve and quantitative reasoning. After each step, the nonoptimal proposals will be canceled using design filtering rules. Assuming that design can be a stepwise construction. this implies that the method will be ranonal and will stand on the same firm scientific bases as analysis. This also means that without the jump, based on previously acquired knowledge, intuition and luck, any engineer may create the best products. This idealistic view has to be moderated : in fact. k i n g a reversed analysis, the design method will bear an image of the quality of the linear analysis. 11.7
REMARKS
* At this point, the proposed method is purely hypothetical. The possibility of a linear analysis has to be demonstrated, and as its reversibiiity. Nevertheless. the generality of this approach needs to be pointed out. The description above is independent of the product domain. This implies that if a reversible linear analysis can be build up for any kind of product, a consauctive-deductive design method may exist. This verifies that a good algorithmic approach must deal with fundamental concepts or principles from which a ,geater generalization can be made, as mentioned by SUH (81 . * As the proof of the existence of such an algorithmic method cannot be obtained, it has to be demonstrated by means of examples. The potential of the method will bc indicated if a demonsnation is possible using a real industrial complex product design. The next step will be to apply the new design method in other domains, in order to generalize it. 111.
AN APPLICATION
- A A U T O M O T I V E TRANSMISSION
111.1 I N T R O D C C T I ~ N
1
SPECIFICATIONS
I
I
Figure 3 : Design oriented analysis As design needs to create information. the aim of design oriented analysis is to extract the information hidden in the product and to store it. This certainly means that a development of intermediaterepresentations will be necessary as well as the development of new tools. Therefore, design oriented analysis has to include a generalization of knowledge smcture. By contrast. the A1 approach tries to find how the information has been introduced in the product through the analysis of the designer's behavior. Without any hypothesis on the information structure, A1 is unable to geneniize.
An industrial product of reasonable complexity may be selected to demonstrate the practicality of the approach. An automotive msmission is a suitable object as it is a highly sophisticated industrial product that can be treated by formal methods. It is therefore perfectly adapted to an engineers education. The goal is to design the transmission. control system. gearbox housing, and other related components. The method will be implemented after generalizanon on a intelligent CAD system which can assist the design engineer during the conceptual phase. The present paper will only deal with the gearbox itself. and will present how a set of logical rules can be devised that will create a product. 111.2 ANALYSISOF AN A C I U A L TRANShllSSlON
In order to apply the constructive-deductive method, a real transmission will be considered. The performances of the transmission will be extracted using logical deductions, from c o r n t i engtneenng knowledge. mathematical tools and axioms. To present the linearity of the analysis each development step (shown in Figure 3) will be described; however the complete mathematical demonsaxions will not be presented here.
-
1st step : From the real transmission t o the technieal drawing Starting from an automatic car transmission : its design is unknown, therefore it is necessary to disassemble i t in order to understand the mechanism. Then each pan will be drawn for cremng the assembly drawing.
170
Figure 4 shows the assembly drawing of the analyzed transmission (Renault).
-
.
" L
Figure J : automatic transmission *
2nd step : From the technical drawing to the schematic Since figure 4 results in a complicated representation that is difficult to understand. a new representation, simpler and more explicit, will be used to continue the analysis. The representation is called the "functional diagram". With this diagram it is easy to identify several subsystems. The torqueconvener, the power transmission to the wheels, the conuol system will be omitted as the present paper will only deal with the gearbox itself. The diagram of the gearbox is simplified by using formal symbols in order to obtain fig.5.
*
4th step : From the graph to the matrix PICHARD and BESSON [3] demonstrated that. using mamx relations, the transmission graph can be represented in two isomorphic onhogon31 spaces (speed and torque spaces). A consequence of isomorphism is that. to some extent, all the solutions don't have to be examined; some of them can be implicitly drawn from the others. The PICHARD and BESSON graph theory is also based on a fundamental theorem : 3.1 I + J = U + C where : I is the linear independence in speed, J is the linear independence in toque. LI is the number of inputlourpur shafis and C is the number of conrrol shafts. This means that the sum of the linear independences in the two spaces is equal to the number ofinpudourput and conuol shafts. With one input shaft from the engine and one output shaft to the wheels, a Car uansmission has two inpur/output shafts ( U = D The graph (Figure 7) shows that U+C=6; this indicares that the uansmission has four control shafts (C=4), which are virtual shafts. By analyzing the graph, the value of the independences (or degrees of freedom) (IJ) can be deduced : I =J = 3 With PICHARD and BESSON relations, the value of the degradation facror (k)and the vec:or base (Xi) can be found: and also :
i
m = p = M =
- (I+J) -
I J IJ
(J-1)
k
( 1- 1)
k k+l
3.2 3.3 3.4
where : m is the number of mechanical nodes. p is the number of plmtary nodes and M is the number of loops of the graph. k is the "degradation factor" which corresponds to the number of zeros in the mamx. If the degradation factor (k) is different from zero, the choice of the vector base (Xi) is not indifferent. With these parameters, the vector base can be Figure 5 : Functional diagram using formal symbols 3rd step : From the functional diagram to the graph Among the different graphic representations used for transmission design. PRUVOT [6] has chosen a representation from PICHARD and BESSON [2]. They have demonstrated that every transmission is composed of shafts and gears, and can be represented by a gaph of order 3 as shown in figure 6. *
FUNCTIONAL DIAGRAM
chosen and the matrix coefficients can be identified. In our case the mamx is
I 1 =I a
a11 313 a13 a21 0 a n 1 a32 '33
a: a3
1 Ei I X
3.5
with ai, Xi the speed of the shafts. real or virtual
GRAPH Using the laws of kinematics of mechanical and planetary trains and of brakes and clutches. the general mauix can be transformed into the following literal matrix :
-n
-1
a 0
n,nj
"4
+ n3n6
14
X
"1
1
-n4ni
* /
1
3.6
n3n6 n3n6 where ni corresponds to the number of teeth of the gears.
-
Figure 6 : Examples of functional diagrams and their graphs PRUVOT [6] has improved this representation by adding brakes and clutches. Figure 7 shows the graph of the gearbox using the complete representation LIRn
RMXTMV NOW I
5th step : From the matrix to the gear ratios The transmission has three degrees of freedom in speed (1=3) with 6 shafts. In order to define a ratio, the speeds of three (I) shafts have to bc defined. As the input speed from the motor is given, the definition of the complete system needs setting the speed of two (I- 1) control shafts. The number of gear ratios is then defined by the number of combinations
R' = cl-' C
X
Cluich I
%
- (I-I)!
C! 4! ( C - l + l ) ! = 2! ( 4 - 2 ) ! =
3'7
In the application, two combinations (degradation factor k=2) block the transnussion and can not be used. In fact the actual value is :
Brake I
n
Clutch 2
R
z
R'-k
=
4
3.8
Setting the speed (zero) of control shafts for each combination into the literal mamx. the expression of each gear ratio can be obtained. Substituting the actual number of teeth in 3.6, it is easy to determine the value of the different g e a ratios. T h e gear ratios a r e part of the performances of the automatic transmission. Figure 7 : Graph of the gearbox
171
6th step : Torque space As mentioned above. using the isomorphism, the general aansfer mamx in the torque space is easy to obtain from the former matrix.This torque matrix is transposed from the sped niamx and its sign is changed : 3. 9 where Y,(resp. PJ are the torque of the shafts of speed XI (resp. a,). *
7th step : Power transfer and stresses Using speeds and torques, the power circulation for each rat10 can be calculated Figure 8 shows this nansfer for a particula~gear ratio (gear ran0 = I : direct drive). P I = .0.s2 Yl la,
111.4 CONSTHUCTIVE-DEUUC1'IVE D E S I G N
Using the linear analysis results. a constructive-deductivedesign method is under development. In order to evaluate the quality of the design method, the specifications used during the design phase will be the performances of the analyzed transmission. As design is - following the new approwh - a reverse analysis, all the analysis results will be reused to generate the missing information needed during the design. To generate information the method will also be based on different notions introduced by PRUVOT [ 5 ] . among them: recurrence (complete inductionr and two axioms : the simpiiciy axium and the marhemarical beauty uibm. In engineering, mathematical beauty can be cransformed into iechnical and fechnoiogicuibeauty., which were defined by POLUCARE 141. A s this deveiopment is not achieved, the present results will not be presented in this paper. At this moment, the constructive-deductivedesign method can create, by formal. rational steps, graphs and even functional diagrams (sorted out by decreasing quality) verifying the specifications.
v.
7
b
a3
Figure 8 : Power circulation for direct drive With rhe usual engineering knowledge, the characteristicsof the mnsmission can be estimated : power dissipation, temperature, etc. Then the stress of all the components can be calculated. All this information belongs to the PERFORMANCES of the transmission. The technical analysis is completed. Other analysis can be conducted. like factors which influence economics and manufacturing. 111.3 CONCLUSIONS ON THE DESIGN ORIENTED ANALYSIS
In this paper, a complex application has been analyzed. In figure 9 all the steps of the analysis an shown. The main column represents the evolution of the product during the analysis. On the left side of the figure is the information "lost" during the analysis. On the right side the tools or the methods (knowledge) which are needed to accomplish the steps are indicated. In order to linearize the usual analysis, new tools have been developed (PICHARD and BESSON, PRUVOT). New data representations (m, p. I. J , M, k) have to be generated to characterize the gearbox.
Knowledge "lost" during analysis
Knowledge necessary for the analysis
.
PFR FORUANCEF SPECIRCATDNS (ABSTRACTION)
Figure 9 : The complete gearbox linear design oriented analysis
172
CONCLIjSIOK A new design approach has been proposed. Using an application, a linear analysis has been developed. To this purpose new tools. new data representations, new methods have k e n generated. The transmission applicanon has demonstrated the possibility of a linear analysis. The proposed design approach is independent of the application domain. It has been shown that the concepts are general and deal with formal. rxional. scientific bases. As an ambitious goal has been set, only the first steps are presented (the design oriented analysis). Taking into consideration the promising results of the design method. the development of [his approach will be continued and will be the subject of a next presentation.
In parallel. an automatic design oriented analysis software is under development that will generate and store the knowledge used during the design (CAD). REFERENCES Peters. J.. Leuven, K.U.. Krause. K.L. and Agerman, E.. 1990. Design : an Integrated Approach. CIRP Annals Manufacturing Technology, Vol. 39/2/1990 p. 599-607. Peters. J. and Van Campenhout. D.. 1988, Manufacturing Oriented and Functional Design. CIRP Annals Manufacturing Technology, Vol. 37/1/1988 p. 153-157. Pichard. J. and Besson B.. 1976, Proposition d'un modtle genera! reprtsentadf des uansmissions de puissance. Rappon interne, Societt "Le Moteur moderns". Poincark. H.. 1908, Science et ,Methode. Flammarion. Pruvot. F.C., LIntellipnce SynrhCtique. EPFL (to be published). Pruvot, F.C., Cinkmatique IinCaire. EPFL (to be published). Pruvot, F.C.. 1991, CAO du futur. futur de la CAO. Keynote paper. MSM March6 Suisse des Machines, No 2, 6 . 9. 14, 20, 22. 26, Fachpresse Goldach. Suh. N.P.. 1988, Basic Concepts in Design for Producibility. CIRP Annals Manufacturing lechnology, Vol. 37/2/1988 p. 559-569. Suh. N.P.. 1989, The Principlcs of Dcsien. Oxford University Press, New York. NY. 10 Yoshikawa. H.. 1989. Design Philosophy: The state of the Art. CIRP annals Manufacturing Technology. Vol. 38/2/1989. p. 579-587. I 1 Yoshikawa, H..1987. Intelligcnt CAD in Manufacluring. CIRP Annals Manufacturing Technology, Vol. 36/1/1987 p. 77-80.
- Submitted
ABSTRACT : The construcrivedeductive approach is a "bortom-up" optimized design method. The mehod proposes a sequence of Iogcal consmctive rules up IO the product solution from the functional specifications. It is easy to automate in an intelligent CAD expen system as it is sequenaai. A description of this approach will precede an application to demonstrate the fibst step of this method : the design onencd analysis.
KEYWORDS :design, modelling, bottom-up design. CAD
I.
INTRODUCTION
Todays rapid pace of technological developments forces the design and engineering depanments of indusmal companies to adapt to shorter product life. They must be able to decrease their development time. while continuing to increase quality. Therefore the design process has attracted many researchers who have produced amount of literature on this topic. In the annals of CIRP many articles deal with this problem and readers C M refer to the keynote ones [ 1.8, 101 where the state of the an in design is presented. PETERS et al. [I] stated, that two main orientations exist in design. The first orientation develops universal design methods. the second aims to integrate the overall expertise in the design itself. The axiomatic theory presented by SUH [9]attempts to create fundmental principles and methodologies to put the design field on a fm scientific basis [8]. This approach clearly involves the first orientation. The second approach vies to incorporate current design practice and knowledge (possessed by "design expens") into algorithmic theory. The consrmctive-deductive approach, was first applied by PRUVOT in his theory of Synthedc Intelligence [5]. It is an algorithm oriented method which can be automated.The knowledge to design is extracted from the product itself through analysis. By using scientific analysis. this approach guarantees firm bases. Starting from the specifications, this method is bottom-up oriented. By using logical rules, this design method suggests all the possible solutions. Sorting criteria must then be used to select the best solution(s). The aim of this paper is to present the latter approach and the state of the work using the constructive-deductive design method applied to analysis of power transmissions. Therefore a practical example of an automotive transmission is proposed.
11.2
THE U S U A L
ANALYSIS
Using usual analysis methods, a finished product can be progressively transformed until its performances are found (Figure I).
I
FINISHED PRODUCT
1
SPECIFICATIONS
I
Figure 1 : Usual analysis of a product During this analysis. MODELS have to be used. As models are obtained by means of THEORIES and are simplified as formal representations of actual things; thus analysis requires knowledge in order to be performed and its final result is made up of the PERFORMANCES of the product. Therefore, analysis necessary yields a loss of information (all that is materialistic). "Performances" are purely absuact. From a design point of view. this "lost" information will be called the "missing" information.
11.3
EMPIRICALDESIGN
During the design phase, a traditional engineer tries to build up a solution starting from the specifications (Figure 2). Then the "performances" of the solution are compared to the specifications using an analysis method as described above.
r
FINISHED PRODUCT iteration
XI.
THE CONSTRUCTIVE-DEDUCTIVE APPROACH
11.1
INTRODUCTION
Design is defined as the creation of all the necessary information for manufacturing a product, starting from its functional requirements. The product specifications never include all the needed information. In a NEW design - different from PARAMETRIC design -,the specifications are very simple and should only concern its PERFORMANCES. Therefore a design method has to create information. The different methods for creating information will not be compared in the present paper. Where as Artificial Intelligence (AI) tries to extract the knowledge of experienced engineers (experts) and to store it into an database, this connibution will propose a new approach to generate this information.
USUAL ANALYTICAL PRACTICE
I knowledge, intuition and
B
SPECIFICATIONS Figure 2 : Empirical design process
Annals of the CIRP Vol. 41/1/1992
169
Many researchers deal with the creation by the designer of the missing information. for example YOSHIKAWA [lo, I I ] presented "transitions between thinking units" and more generally speaking "mapping between spaces". A traditional engineer has therefore to jump From precise requirements to a nebulous solution. In this fashion, the creative act is never accomplished in a rational way. The designer "jumps" to the conclusion that is a "finished solution (at least piecewise). As a matter of fact. using his experience, a designer can pick up pans of old solutions to imagine a new one.
11.4
ANOTHER POINT OF VIEW
In an authentic design method, these "jumps" a ~ unacceptable e since they ;tre unable to generate all the solutions and particularly the best one. A good design method should be able to propose all the solutions. then the best one should be sorted out according to some clearly defined criteria. SoKing criteria may be geomemcal or physical, while others are economic or manufacturing oriented (design for production). However, we shall continue to concentrate on the design mechanism which enables decision trees to be analyzed funher. The proposed approach mes to reverse the analysis process to improve the empirical method. It mes to generate all the products by adding missing information to the specifications. If the missing information cwld be structured and stored in a database during the analysis, then it could eventually k used during the conceptual phase. At this point, this conclusion must be considered as purely hypothetical. 11.5
DESIGN ORIENTED ANALYSIS
The first difficulty of the proposed approach is the non-revenibility of the usual analysis. Figure 1 shows that it is impossible since the analysis is "tree oriented". A "linear" analysis method would stand a better chance of being reversible, but its possibility has to be demonstrated. A reversible analysis will be called design oriented (Figure 3). "lost" knowledge
IIr
FINISHED PRODUCT
11.6
DESIGN
Assuming that a linear design oriented analysis has been created. it might be possible to reverse it in order to build up a design method. This is the basis of the consouctive-deductive design method. "Consrrucrive" is used here with its philosophical meaning of "creorion ex nihilo"; its means that knowledge has to be generated from a base that doesn't contain it. "Deducrive" means that formal logic is used throughout. The term "constructive-deductive" has been defined by the logician/philosopher E. GOBLOT (1858-1935). [f one considers an analysis linear. then a reversible method should be h e x as well. Therefore, during the design phase a stepwise construction is used including qualitattve and quantitative reasoning. After each step, the nonoptimal proposals will be canceled using design filtering rules. Assuming that design can be a stepwise construction. this implies that the method will be ranonal and will stand on the same firm scientific bases as analysis. This also means that without the jump, based on previously acquired knowledge, intuition and luck, any engineer may create the best products. This idealistic view has to be moderated : in fact. k i n g a reversed analysis, the design method will bear an image of the quality of the linear analysis. 11.7
REMARKS
* At this point, the proposed method is purely hypothetical. The possibility of a linear analysis has to be demonstrated, and as its reversibiiity. Nevertheless. the generality of this approach needs to be pointed out. The description above is independent of the product domain. This implies that if a reversible linear analysis can be build up for any kind of product, a consauctive-deductive design method may exist. This verifies that a good algorithmic approach must deal with fundamental concepts or principles from which a ,geater generalization can be made, as mentioned by SUH (81 . * As the proof of the existence of such an algorithmic method cannot be obtained, it has to be demonstrated by means of examples. The potential of the method will bc indicated if a demonsnation is possible using a real industrial complex product design. The next step will be to apply the new design method in other domains, in order to generalize it. 111.
AN APPLICATION
- A A U T O M O T I V E TRANSMISSION
111.1 I N T R O D C C T I ~ N
1
SPECIFICATIONS
I
I
Figure 3 : Design oriented analysis As design needs to create information. the aim of design oriented analysis is to extract the information hidden in the product and to store it. This certainly means that a development of intermediaterepresentations will be necessary as well as the development of new tools. Therefore, design oriented analysis has to include a generalization of knowledge smcture. By contrast. the A1 approach tries to find how the information has been introduced in the product through the analysis of the designer's behavior. Without any hypothesis on the information structure, A1 is unable to geneniize.
An industrial product of reasonable complexity may be selected to demonstrate the practicality of the approach. An automotive msmission is a suitable object as it is a highly sophisticated industrial product that can be treated by formal methods. It is therefore perfectly adapted to an engineers education. The goal is to design the transmission. control system. gearbox housing, and other related components. The method will be implemented after generalizanon on a intelligent CAD system which can assist the design engineer during the conceptual phase. The present paper will only deal with the gearbox itself. and will present how a set of logical rules can be devised that will create a product. 111.2 ANALYSISOF AN A C I U A L TRANShllSSlON
In order to apply the constructive-deductive method, a real transmission will be considered. The performances of the transmission will be extracted using logical deductions, from c o r n t i engtneenng knowledge. mathematical tools and axioms. To present the linearity of the analysis each development step (shown in Figure 3) will be described; however the complete mathematical demonsaxions will not be presented here.
-
1st step : From the real transmission t o the technieal drawing Starting from an automatic car transmission : its design is unknown, therefore it is necessary to disassemble i t in order to understand the mechanism. Then each pan will be drawn for cremng the assembly drawing.
170
Figure 4 shows the assembly drawing of the analyzed transmission (Renault).
-
.
" L
Figure J : automatic transmission *
2nd step : From the technical drawing to the schematic Since figure 4 results in a complicated representation that is difficult to understand. a new representation, simpler and more explicit, will be used to continue the analysis. The representation is called the "functional diagram". With this diagram it is easy to identify several subsystems. The torqueconvener, the power transmission to the wheels, the conuol system will be omitted as the present paper will only deal with the gearbox itself. The diagram of the gearbox is simplified by using formal symbols in order to obtain fig.5.
*
4th step : From the graph to the matrix PICHARD and BESSON [3] demonstrated that. using mamx relations, the transmission graph can be represented in two isomorphic onhogon31 spaces (speed and torque spaces). A consequence of isomorphism is that. to some extent, all the solutions don't have to be examined; some of them can be implicitly drawn from the others. The PICHARD and BESSON graph theory is also based on a fundamental theorem : 3.1 I + J = U + C where : I is the linear independence in speed, J is the linear independence in toque. LI is the number of inputlourpur shafis and C is the number of conrrol shafts. This means that the sum of the linear independences in the two spaces is equal to the number ofinpudourput and conuol shafts. With one input shaft from the engine and one output shaft to the wheels, a Car uansmission has two inpur/output shafts ( U = D The graph (Figure 7) shows that U+C=6; this indicares that the uansmission has four control shafts (C=4), which are virtual shafts. By analyzing the graph, the value of the independences (or degrees of freedom) (IJ) can be deduced : I =J = 3 With PICHARD and BESSON relations, the value of the degradation facror (k)and the vec:or base (Xi) can be found: and also :
i
m = p = M =
- (I+J) -
I J IJ
(J-1)
k
( 1- 1)
k k+l
3.2 3.3 3.4
where : m is the number of mechanical nodes. p is the number of plmtary nodes and M is the number of loops of the graph. k is the "degradation factor" which corresponds to the number of zeros in the mamx. If the degradation factor (k) is different from zero, the choice of the vector base (Xi) is not indifferent. With these parameters, the vector base can be Figure 5 : Functional diagram using formal symbols 3rd step : From the functional diagram to the graph Among the different graphic representations used for transmission design. PRUVOT [6] has chosen a representation from PICHARD and BESSON [2]. They have demonstrated that every transmission is composed of shafts and gears, and can be represented by a gaph of order 3 as shown in figure 6. *
FUNCTIONAL DIAGRAM
chosen and the matrix coefficients can be identified. In our case the mamx is
I 1 =I a
a11 313 a13 a21 0 a n 1 a32 '33
a: a3
1 Ei I X
3.5
with ai, Xi the speed of the shafts. real or virtual
GRAPH Using the laws of kinematics of mechanical and planetary trains and of brakes and clutches. the general mauix can be transformed into the following literal matrix :
-n
-1
a 0
n,nj
"4
+ n3n6
14
X
"1
1
-n4ni
* /
1
3.6
n3n6 n3n6 where ni corresponds to the number of teeth of the gears.
-
Figure 6 : Examples of functional diagrams and their graphs PRUVOT [6] has improved this representation by adding brakes and clutches. Figure 7 shows the graph of the gearbox using the complete representation LIRn
RMXTMV NOW I
5th step : From the matrix to the gear ratios The transmission has three degrees of freedom in speed (1=3) with 6 shafts. In order to define a ratio, the speeds of three (I) shafts have to bc defined. As the input speed from the motor is given, the definition of the complete system needs setting the speed of two (I- 1) control shafts. The number of gear ratios is then defined by the number of combinations
R' = cl-' C
X
Cluich I
%
- (I-I)!
C! 4! ( C - l + l ) ! = 2! ( 4 - 2 ) ! =
3'7
In the application, two combinations (degradation factor k=2) block the transnussion and can not be used. In fact the actual value is :
Brake I
n
Clutch 2
R
z
R'-k
=
4
3.8
Setting the speed (zero) of control shafts for each combination into the literal mamx. the expression of each gear ratio can be obtained. Substituting the actual number of teeth in 3.6, it is easy to determine the value of the different g e a ratios. T h e gear ratios a r e part of the performances of the automatic transmission. Figure 7 : Graph of the gearbox
171
6th step : Torque space As mentioned above. using the isomorphism, the general aansfer mamx in the torque space is easy to obtain from the former matrix.This torque matrix is transposed from the sped niamx and its sign is changed : 3. 9 where Y,(resp. PJ are the torque of the shafts of speed XI (resp. a,). *
7th step : Power transfer and stresses Using speeds and torques, the power circulation for each rat10 can be calculated Figure 8 shows this nansfer for a particula~gear ratio (gear ran0 = I : direct drive). P I = .0.s2 Yl la,
111.4 CONSTHUCTIVE-DEUUC1'IVE D E S I G N
Using the linear analysis results. a constructive-deductivedesign method is under development. In order to evaluate the quality of the design method, the specifications used during the design phase will be the performances of the analyzed transmission. As design is - following the new approwh - a reverse analysis, all the analysis results will be reused to generate the missing information needed during the design. To generate information the method will also be based on different notions introduced by PRUVOT [ 5 ] . among them: recurrence (complete inductionr and two axioms : the simpiiciy axium and the marhemarical beauty uibm. In engineering, mathematical beauty can be cransformed into iechnical and fechnoiogicuibeauty., which were defined by POLUCARE 141. A s this deveiopment is not achieved, the present results will not be presented in this paper. At this moment, the constructive-deductivedesign method can create, by formal. rational steps, graphs and even functional diagrams (sorted out by decreasing quality) verifying the specifications.
v.
7
b
a3
Figure 8 : Power circulation for direct drive With rhe usual engineering knowledge, the characteristicsof the mnsmission can be estimated : power dissipation, temperature, etc. Then the stress of all the components can be calculated. All this information belongs to the PERFORMANCES of the transmission. The technical analysis is completed. Other analysis can be conducted. like factors which influence economics and manufacturing. 111.3 CONCLUSIONS ON THE DESIGN ORIENTED ANALYSIS
In this paper, a complex application has been analyzed. In figure 9 all the steps of the analysis an shown. The main column represents the evolution of the product during the analysis. On the left side of the figure is the information "lost" during the analysis. On the right side the tools or the methods (knowledge) which are needed to accomplish the steps are indicated. In order to linearize the usual analysis, new tools have been developed (PICHARD and BESSON, PRUVOT). New data representations (m, p. I. J , M, k) have to be generated to characterize the gearbox.
Knowledge "lost" during analysis
Knowledge necessary for the analysis
.
PFR FORUANCEF SPECIRCATDNS (ABSTRACTION)
Figure 9 : The complete gearbox linear design oriented analysis
172
CONCLIjSIOK A new design approach has been proposed. Using an application, a linear analysis has been developed. To this purpose new tools. new data representations, new methods have k e n generated. The transmission applicanon has demonstrated the possibility of a linear analysis. The proposed design approach is independent of the application domain. It has been shown that the concepts are general and deal with formal. rxional. scientific bases. As an ambitious goal has been set, only the first steps are presented (the design oriented analysis). Taking into consideration the promising results of the design method. the development of [his approach will be continued and will be the subject of a next presentation.
In parallel. an automatic design oriented analysis software is under development that will generate and store the knowledge used during the design (CAD). REFERENCES Peters. J.. Leuven, K.U.. Krause. K.L. and Agerman, E.. 1990. Design : an Integrated Approach. CIRP Annals Manufacturing Technology, Vol. 39/2/1990 p. 599-607. Peters. J. and Van Campenhout. D.. 1988, Manufacturing Oriented and Functional Design. CIRP Annals Manufacturing Technology, Vol. 37/1/1988 p. 153-157. Pichard. J. and Besson B.. 1976, Proposition d'un modtle genera! reprtsentadf des uansmissions de puissance. Rappon interne, Societt "Le Moteur moderns". Poincark. H.. 1908, Science et ,Methode. Flammarion. Pruvot. F.C., LIntellipnce SynrhCtique. EPFL (to be published). Pruvot, F.C., Cinkmatique IinCaire. EPFL (to be published). Pruvot, F.C.. 1991, CAO du futur. futur de la CAO. Keynote paper. MSM March6 Suisse des Machines, No 2, 6 . 9. 14, 20, 22. 26, Fachpresse Goldach. Suh. N.P.. 1988, Basic Concepts in Design for Producibility. CIRP Annals Manufacturing lechnology, Vol. 37/2/1988 p. 559-569. Suh. N.P.. 1989, The Principlcs of Dcsien. Oxford University Press, New York. NY. 10 Yoshikawa. H.. 1989. Design Philosophy: The state of the Art. CIRP annals Manufacturing Technology. Vol. 38/2/1989. p. 579-587. I 1 Yoshikawa, H..1987. Intelligcnt CAD in Manufacluring. CIRP Annals Manufacturing Technology, Vol. 36/1/1987 p. 77-80.