Proceedings of the 11th World Congress in Mechanism and Machine Science August 18−21, 2003, Tianjin, China China Machinery Press, edited by Tian Huang

Optimization of a Multi−Stage Transmission Mechanism Jean−Christophe Fauroux LaRAMA, IFMA, Campus universitaire de Clermont−Ferrand / Les Cézeaux, BP 265, 63175 AUBIERE Cedex, FRANCE Tel : +33.(0)4.73.28.80.50 Fax : +33.(0)4.73.28.81.00 E−Mail : [email protected] Pascal Lafon LASMIS, Université de Technologie de Troyes 12 rue Marie Curie, BP 2060, 10010 TROYES Cedex, FRANCE Tel : +33.(0)3.25.71.56.55 Fax : +33.(0)3.25.71.56.75 E−Mail : [email protected]

Abstract: This work describes a method for optimizing the class of multi−stage transmission mechanisms. The problem formulation is a complex one and deals with continuous as well as discrete variables. First, the problem is expressed at a local level for one stage, including gear dimensioning with both contact stress and bending stress criteria. Then, the global problem is constructed and ensures global coherence of mechanism. Solving is achieved through a method using evolutionary strategies. An example of a three−stage speed reducer is provided and significantly optimized. This general method proved to be very robust and efficient for gears and is extendable to a wider category of transmission mechanisms. Keywords: Optimization, Evolutionary Strategy, Mixed Vari− ables, Discrete Variables, Gear, Multi−Stage, Speed Reducer.

1 Introduction This work describes the optimization of a class of real and very common mechanical systems, the class of transmission mechanisms. This class is interesting for two reasons. First, most of the transmission mechanisms can be split into more simple mechanisms called stages, each of them being well−known and defined by a design relation set [1]. Second, transmission mechanisms offer real−life optimization problems, with a great number of mixed variables (continuous and discrete ones), complex constraints and objective function. This often leads to poor convergence because of ill−conditioned problems. Moreover, such problems often exhibit multiple local optima [2, 3, 4]. In a first part, we present the optimization problem associated to one standard parallel gearing stage. Each stage contributes to the global problem with a set of bounded variables and a set of constraints. Secondly, the global problem has to be constructed by assembling the contributions of each stage. It is also enriched by adding several types of global constraints in order to ensure coherence of the mechanism. The global problem is an optimization problem with mixed variables and a great number of highly non linear and non differentiable constraints. Third, the problem has to be solved with efficient and well−suited methods that are able to deal with large scale problems, mixed variables, non linear constraints and non differentiable functions. We propose to use optimization algorithms based upon evolutionary strategies which are briefly presented [5, 6, 7]. A detailed

example of a three−stage speed reducer is provided showing the efficiency of the optimization algorithm. Then some further work is traced. 2 Optimizing Speed Reducers Designing a speed reducer could appear to be elementary. It is however a complex task because of the great number of variables, rules and know−how that are necessary. For this reason, many different formulations of speed reducer design have been presented for years and this rather common mechanism is a perfect occasion for demonstrating state−of−the−art optimization techni− ques. Datseris presented a work [8] on optimization of a single−stage speed reducer with decomposition techni− ques. The algorithm took into account the following variables : tooth number, tooth module, face width, shaft length and diameter. He already noticed that combina− torial heuristics were powerful for engineering design. Osman et al. [9] focused on combinatorial optimi− zation of gears in double composite gear trains to create a multiple−speed gear−box without multiple identical ratios. Problem variables were gear tooth number and relative geometric setting of gears on 3 coplanar and parallel shafts. Azarm et al. [10] performed a manual monotonicity analysis within a multi−level design optimization frame− work. They used the same speed−reducer as [8]. The complex decomposition process was interesting for improving resolution of problems too complex for global monotonicity analysis. However, most of real−life engineering problems are non monotonous ones. Caroll et al. [11] treated the dimensioning problem for one gear pair with dimensionless parameters. The optimization problem was also formulated with continuous variables and authors already noticed that the discrete optimal solution was not easy to obtain from the continuous problem. In [12], Savage et al. presented a global problem including one spur gear pair, two shafts and four bearings. Variables were continuous ones and a gradient method was used to optimize volume, weight and bearing life duration. Papalambros et al. [13, 14] demonstrated a very complex four−stage speed reducer example with 22 variables and 86 constraints. All variables were either integer or

discrete and represented number of teeth, gear face width and predetermined cartesian shaft location in the housing. Minimizing total volume required variables to be converted into 666 binary variables for solving, which is a tremendous work. More recently, progresses of optimization techniques such as genetic or evolutionary strategies proved useful for optimal designers. Azarm et al. [15, 16] applied a Pareto multi−objective genetic algorithm to a speed reducer, formulated as a two−level two−objective design problem. All these works demonstrate that real engineering tasks often lead to extremely complex optimization problem. Integrating discrete variables is fundamental as they appear everywhere in technology. Decomposing problems into sub−problems is an interesting but difficult path and complicates problem formulation. The recent maturity of optimization strategies based on evolution laws and genetics gives a perfect occasion to try to solve a very complex test−problem.

di,s KBP, KBR ms NInput PMin ur Ui,s V Vmax Yi,s Zi,s αn, α’t

Diameter for pinion (i=1) or wheel (i=2) of stage s Service factor for contact / bending stress, supposed constant Tooth real module of stage s Input rotational speed Minimal required input power Global required reduction ratio of the speed reducer Geometry coefficient for wheel i. Non lin. function of ms, Zi,s Linear velocity on teeth Maximal allowed linear velocity (typically 20m/s) Geometry coefficient for wheel i. Non lin. function of ms, Zi,s Number of teeth for wheel i of stage s Normal and working transverse pressure angle β Reference helix angle εα,s Transverse contact ratio. Non linear function of ms, Z1,s, Z2,s

Table 1. Nomenclature of sub−problem

g 1 x =1B

C 1⋅C 2⋅C 3⋅C 4⋅C 5⋅C 6 K B P Min

T0

(2)

s

P

g 2 x =1B

C B1⋅C B2⋅C B3⋅C B4⋅C B5⋅C B6⋅C B7 K B P Min

T0 (3) s

R

g 3 x =1.3Bε α ,s T0

3 Problem Setting This section describes a three−stage speed reducer problem that is rather comparable to the one described by Papalambros et al. However, in our case, the problem was formulated as a fully 3D problem and even if this example is quasi planar (all shafts are parallel), it is easily extendable to spatial configurations, as demons− trated in [3, 4]. 3.1 Sub−problem : optimizing one stage A first formulation of this sub−problem was presented in [2, 4] , including only continuous variables (gear primitive radius and face width but no tooth module) and contact stress constraint. This method was fast and suitable for pre−dimensioning. A more complete optimization problem of a parallel gearing stage with helical teeth can be found in [7]. In this problem, the mass and relative variation of the specific slips of the wheel and the pinion have to be minimized under several constraints issued from strength and functional conditions. The determination of the tooth shape is a complex operation. There are however simpli− fied and standardized methods usually used in the engineering and design departments. This formulation of the problem is based on the simplified "C" method recommended by standard [17, 18]. We recall below in table 1 and equations 2−12 the main constraints of this problem. The vector x of chosen variables is : (1) x= m Z , Z , b T s,

s

index bs Ci C1 C2 C3 C4 C5 C6 CBi CB1 CB2 CB3 CB4 CB5 CB6 CB7

1,s

2,s

s

Stage number Gear face width of stage s Contact stress factors (see [17] p. 64) Speed and ratio coef. Non linear function of ms, Z1,s and Z2,s Tooth shape coef. Depends on αn and β −> Constant coef. Speed coefficient. Non linear function of ms, Z1,s , Z2,s Load distribution coef. Non linear function of ms, Z1,s , Z2,s, bs Material coefficient considered constant Tooth contact coefficient considered constant Bending stress factors (see [17] p. 73) Speed and ratio coef. Non linear function of ms, Z1,s and Z2,s Contact and overlap ratio coef. Considered constant Dynamic behaviour coef. Non linear function of ms, Z1,s , Z2,s Shape and constraint coef. Non linear function on Z1,s , Z2,s Load distribution coef. Depends on ms, Z1,s , Z2,s, bs Fatigue stress coef., depends on material. Considered constant Stress concentration coef. Depends mainly on ms

g4 x =

Z 1,s V

Z 22,s

100

Z 22,s AZ 21,s

g 5 x =π Y 1,s U 1,s B

Z 2,s 2 Z 1,s

B10T0

⋅tan α’t T0

⋅tan α’t T0 2 d g 7 x =Max d 1 Min,s , 2 Min,s Bd 1,sT0 ur g 6 x =π Y 2,s U 2,s B

g 8 x =d 1,s BMin d 1 Max,s ,

d 2 Max,s 60V Max T0 , ur π N Input

(4) (5)

(6) (7) (8) (9)

g 9 x =b s BMin d 1,s ,b Max,s T0

(10)

g 10 x =Max 0.1⋅d 2,s ,b Min,s T0

(11)

with everywhere

d i,s =m s⋅Z i,s

(12)

Equations 2 and 3 express contact and bending stress criteria. Equation 4 gives a condition on the minimum value of the transverse contact ratio εα. Equation 5 expresses a condition on the linear velocity of teeth. Equations 6 and 7 express conditions on meshing interference. Equations 8 and 9 are conditions on the minimal and maximum values of the diameter of the pinion. Equations 10 and 11 limit the minimal value of the face width compared to the diameters of the pinion and the wheel. 3.2 Global problem : optimizing multiple stages Ce,s G la,s Oi Oo ra,s ri,s ∆ur θMax ξs

Input torque at stage s Shaft shearing modulus Length of input shaft (0) or output shaft of stage s (1,2,3) Input shaft location on the housing Output shaft location on the housing Radius of input shaft (0) or output shaft of stage s (1,2,3) Radius of wheel i of stage s Tolerance on required global ratio Maximum torsion angle for shafts (typically 0.1 °/m) Angular self rotation of stage along its input shaft

Table 2. Nomenclature of global problem

The global problem may now be constructed by assem− bling the contributions of each of the three serially connected stages (fig. 1) . Each one is associated to a set of gear constraints (equations 5−12). Supplementary variables are needed for the input shaft and for output shaft attached to each stage (ra,s and la,s for shaft radius and length). An angle ξs is also necessary for expressing angular self−rotation of a stage along the output shaft of previous stage. Input s haft

Stage 1

Stage 2

Stage 3

ξ1

Figure 1. Multi−stage structure

The list of variables for one stage s becomes : (13) x= ξs, m s, Z 1,s , Z 2,s , b s, r a,s, l a,s T The global variable vector for the problem is expressed in table 3 where the 9 discrete variables are emphasized with a gray background. There is a total of 23 mixted variables. Input shaft

ra,0

la,0

Stage 1

ξ1

m1

Z1,1

Z2,1

b1

ra,1

la,1

Stage 2

ξ2

m2

Z1,2

Z2,2

b2

ra,2

la,2

Stage 3

ξ3

m3

Z1,3

Z2,3

b3

ra,3

la,3

Table 3. Global problem variable vector

The objective function is the overall volume : f Obj x =V Shafts AV Gears

(14)

s=3

with

V Shafts =π ∑ r 2a,s⋅l a,s s=0 s=3

and

V Gears =π ∑ m s⋅Z 1,s 2 A m s⋅Z 2,s 2 ⋅b s

Finally the expression of this problem, requires 115 non−linear constraint functions, and 23 mixed variables including 14 continuous and 9 discrete variables. Then, the problem must be solved with an efficient and well− suited method capable to deal with large scale problems, mixed variables, non linear constraints and non differentiable functions. 3.3 A well−suited algorithm : Evolutionary Strategies Evolutionary algorithms are based on the principle of evolution, i.e. survival of the fittest. Unlike classical methods, they do not use a single search point but a population of points called individuals. Each individual represents a potential solution to the problem. In these algorithms, the population evolves toward increasingly better regions of the search space by undergoing statis− tical transformations called recombination and mutation. In this algorithm P(t) is a population of µ individuals. Each individual a is represented by four vectors : a= x , x , σ , p T where (20) C D •

x C ∈þ

nC

•

x D ∈þ

nD

•

σ ∈þAC contains standard deviations

is a vector of nC continuous variables is a vector of nD discrete variables

n

n

•

p∈ 0,1 D contains the mutation probabilities σ and p are strategy parameters which control the application of mutation to the continuous and discrete variables. The recombination operator generates a new population P’(t) of λ individuals (λ > µ). This mechanism allows a mixing of parental information and passes this information to their descendants (fig. 2). In the selection step, the best µ individuals are chosen among the overall population of (λ + µ) individuals. For evaluation of each individual, we apply a dynamic penalty method in which the penalty coefficient increases during the evolution. Several recombination and mutation operators exist. We have shown in [5] the most efficient operator for this type of problem and also more details on this algorithm are given in [5, 6].

s=1

•

• •

•

Closure conditions request that output shaft ends at specified location Oo (equations 17−19 on last page). Non−interference with housing : as all parts can be considered as cylinders (even gears are defined by their tip cylinder), these criteria are easy to formulate for a housing with a box shape. As there are 10 cylinders and 6 planes to test, it gives 60 constraints. Non−interference between parts (cylinders) was al− ready expressed in [2]. It gives 16 supplementary cylinder−cylinder constraints.

Penalty loop

Generation loop

Several other types of global functions have to be added to stage design constraints : Required reduction ratio should be satisfied with a tolerance : g Ratio x = u r B∏ s=3 Z ⁄Z 1,s B∆u r =0 (16) s=1 2,s

Best individual Statistics loop

For each stage, torsional shaft resistance must also be taken into account (equation 15). 2⋅10 3⋅C e,s⋅ Z 1,s ⁄Z 2,s (15) g 11 x = B1T0 G⋅ϑ Max π r a,s4

P’(t) = Recombination of P(t) P’’(t) = Mutation of P’(t) Evaluation of P’’(t)

Generation of a new population

Initialize population P(t)

P(t+1) = Selection of (P’’(t), P(t)) Initial evaluation of P(t)

Figure 2. Flowchart of the optimization algorithm

4 Example of a three−stage speed reducer 4.1 Algorithm settings There are three distinct loop levels in the algorithm : • Generation loop : at each iteration, it passes from generation P(t) to P(t+1). The stop test is : NG > NGMax = 40 OR “ fObj stays unchanged 10 times ”. • Penalty loop : its purpose is to increase progressively the penalty factor r. For each individual : fEvaluation = fObj + r * (Sum of >0 violated constraints) Stop test : NP > NPMax = 8

•

Statistics loop repeats all the previous operation till Ns < NsMax = 5. Evolutionary strategies are stochastic methods. Consequently, to obtain significant results, they require a great number of objective and constraint function evaluations. Calculations took a computing time of 45 min on a HP PA−RISC 8600 750 MHz. As for every genetic algorithm, we have no convergence criterion. Users pre− define a computing time before starting. The advantage is the possibility to stop at every moment and and to always obtain a better design than the initial one. Of course, we have no guarantee for global optimum, but evolutionary strategies are renown for their enhanced ability to locate global optimum compared to gradient methods, which only follow the steepest path from the initial conditions [5]. 4.2 Results Starting from initial design (fig. 3) and after 2 110 800 individual evaluations, a better solution is found (fig. 4). In spite of its impressive value, this number of evaluations is extremely small compared to the space to explore. Just considering discrete variables, we have 413 module combinations and 1356 tooth number combi− nations, which reach the outstanding value of 42.1016, even when neglecting continuous variables ! Observing 3D model of solution, we notice that modules are significantly smaller than before. In general, the optimization process led to smaller gear radius and larger face width. It is rather logical because radii are included in the objective function at the power of two whereas face width are only at the power one. When examining constraint values, one could also check that the initial design with its big tooth gears was highly over−dimensioned according to bending stress criterion, although perfectly correct according to contact stress. The new tooth geometry provides a far better balanced compromise. One could also notice on the top views that initial design puts stages in three layers whereas optimized design uses only two and has more compact axial layout. Finally, we can say that these results and the corresponding method are significantly improved (−14%) with respect to the initial mechanism. Of course, a better solution could probably be found with longer computing. 5 Conclusion We proposed the formulation of the optimization pro− blem of multi−stage speed reducers with an evolutionary algorithm. The algorithm allowed us to improve significantly an industrial design of a three−stage speed reducer. This problem was formulated with 14 conti− nuous variables, 9 discrete variables and 115 non linear constraint functions. Moreover, problem construction did not require any complex reformulation such as in [13, 14]. As a consequence, the provided method seems rather interesting for solving many real−life complex engineering problems. In further work we would like to enhance the formulation of this optimization problem to take into account shaft bearings [12], which may strongly modify the optimal solution currently proposed. Bearing dimensioning relies intensively on discrete variables, which is no problem for our method. It was also designed for a possible extension to other types of gears.

References 1. Fauroux, J.C., Towards Natural Optimization into CAD Software, In : Proc. of IDMME’2000, Montreal (Québec), May 16th−19th, 2000, 8 p., Proceeding CD−ROM, ISBN 2− 553−00803−1, 140.pdf file 2. Fauroux, J.C., Sartor, M., Conception optimale de structures cinématiques 3D. Application aux mécanismes de transmission en rotation, 6ème Colloque national sur la conception mécanique intégrée PRIMECA’99, La Plagne, France, April 7th−9th ,1999. 3. Fauroux, J.C., Sartor, M., Conception optimale de mécanismes. Application aux réducteurs à engrenages, In : Proc. of 4th Worldwide Conference on Gears and Power Transmissions, CNIT Paris, 16th−18th March 1999 4. Fauroux, J.C., Conception optimale de structures cinématiques tridimensionnelles. Application aux mécanismes de transmission en rotation , PHD thesis, 19th jan 1999, INSA de Toulouse, Laboratoire de Génie Mécanique de Toulouse, 228 p., http://www.ifma.fr/ recherche/larama/membres_larama/gb_fauroux.htm 5. Giraud L., Lafon P., A comparisation of evolutionary strategies for mechanical design, Journal of Engineering Optimization, 34(5):307−322, 2002 6. Moreau−Giraud L., Lafon P., Evolution strategies for optimal design of mechanical systems, In : Proc. of Third World Congress of Structural and Multidisciplinary Optimization (WCSMO−3), BUFFALO, USA, 17th−21st May 1999, p 93−95 7. Lafon P., Moreau−Giraud L., Coupling of evolution strategy and lagrangian augmented algorithm for optimal design of mechanical system, In : Proc. of IDMME’2000, May 16th−19th, 2000, Montreal (Québec), Proceeding CD− ROM, ISBN 2−553−00803−1, 193.pdf file 8. Datseris, P., Weight minimization of a speed reducer by heuristic and decomposition techniques, Mechanism & Machine Theory, 17(4):255−262, 1982 9. Osman, M., Dukkipati, R.V., Prasad, V.S., An efficient computational iterative for design synthesis of 4x3 double composite gear trains, Mechanism & Machine Theory, 22(1):21−26, 1987 10. Azarm, S., Li, W.−C., Multi−level design optimization using global monotonicity analysis, Journal of Mechanisms, Transmissions and Automation in Design, 111(6):259−263, 1989 11. Caroll, R.K., Johnson, G.E., Dimensionless solution to the optimal design of spur gear sets, Journal of Mechanical Design, 111(6):290−296, 1989 12. Savage, M., Lattime, S.B., Kimmel, J.A., Coe, H.H., Optimal design of compact spur gear reductions, Journal of Mechanical Design, 116(9):690−696, 1994 13. Pomrehn, L.P., Papalambros, P.Y., Discrete optimal design formulations with application to gear train design, Journal of Mechanical Design, 117(9):419−424, 1995 14. Pomrehn, L.P., Papalambros, P.Y., Infeasibility and non− optimality tests for solution space reduction in discrete optimal design, Journal of Mechanical Design, 117(9):425−432, 1995 15. Kurpati, A., Azarm, S., Immune network simulation with multiobjective genetic algorithms for multidisciplinary design optimization, Engineering Optimization, 33(2):245− 260, 2000 16. Kurpati, A., Azarm, S., Wu, J., Constraint handling improvements for multiobjective genetic algorithms, Structural and Multidisciplinary Optimization, 23(3):204− 213, 2002 17. AFNOR (Association Française de NORmalisation), Détermination de la capacité de charge des engrenages cylindriques extérieurs de mécanique générale, Experimental norm AFNOR E 23−015, AFNOR, Paris, 1982, 78 p. 18. Henriot G., Traité théorique et pratique des engrenages, Dunod, 1979, 6th edition, Volume 1/2, 670 p., ISBN 2−04− 010934−X

g Closure X x = X O Al a,0 Ab 1 Al a,1 Bl a,2 Al a,3 ⁄X O B1=0 i

(17)

o

g Closure Y x = Y O A r 1,1 Ar 2,1 ⋅cos ξ1 A r 1,2 Ar 2,2 ⋅cos ξ1 Aξ 2 A r 1,3 Ar 2,3 ⋅cos ξ1 Aξ 2 Bξ3 ⁄Y O B1=0

(18)

g Closure Z x = Z O A r 1,1 Ar 2,1 ⋅sin ξ1 A r 1,2 Ar 2,2 ⋅sin ξ1 Aξ2 A r 1,3 Ar 2,3 ⋅sin ξ1 Aξ2 Bξ3 ⁄Z O B1=0

(19)

i

o

i

Var

Value 30.5 mm

ra,0 la,0 130 mm ξ1 −17.9°

o

3D view

m1 5 mm Z1,1 15 Z2,1 50 b1 46 mm ra,1 39 mm la,1 12 mm

Stage 3 ξ3

ξ2 98.5° m2 6 mm Z1,2 15 Z2,2 54 b2 66 mm ra,2 60 mm la,2 84 mm ξ3 98.7° m3 8 mm Z1,3 18 Z2,3 66 b3 104 mm ra,3 83 mm la,3 150 mm

Oi

Oo ξ1

ξ2

Z

Face view Stage 2

Input shaft Z Y

Y

X Stage 1

X

Housing

Top view

Figure 3. 3D re pre s e ntation of initial de s ign Volume = 3.74.107 mm3 Var

Value

ra,0 26.7 mm la,0 106.8mm ξ1 m1 Z1,1 Z2,1 b1 ra,1 la,1 ξ2 m2 Z1,2 Z2,2 b2 ra,2 la,2 ξ3

19.4° 2.25 mm 28 92 47.3 mm 36.4 mm 10 mm 31.1° 2.5 mm

Small tooth modules

32 124 91.3 mm 51.1 mm 10 mm

Free axial space

83.8°

m3 5 mm Z1,3 27 Z2,3 93 b3 115.8mm ra,3 69.6 mm la,3 101.3mm

Layout in 2 layers

Figure 4. 3D re pre s e ntation of final optimize d de s ign Volume = 3.23.107 mm3