1、91 FTM 11Initial Design of Gears Using ArtificialNeural Netby: T. Jeong and T. P. Kicher, Case Western Reserve;and R. J. Zab, Joy TechnologyAmerican Gear Manufacturers AssociationA- TECHNICALPAPERInitialDesignof GearsUsingArtificialNeuralNetT. Jeong and T. P. Kicher, Case Western Reserve University;
2、 and R. J. Zab, Joy TechnologyThe Statementsandopinionscontainedhereinarethose of theauthorand shouldnot beconstrued as an officialactionoropinion of the American Gear Manufacturers Association.ABSTRACT:Most mechanical engineering design problems require both the computational and decision making as
3、pects. Thosedecision making taskscan be performed by an artificialneural net. The adaptabilityof theartificial neural net for initialgear design was demonstrated and the detailed application is explained throughout the paper.Copyright 1991American Gear Manufacturers Association1500 King Street, Suit
4、e 201Alexandria, Virginia, 22314October, 1991ISBN: 1-55589-608-1AwINITIAL DESIGN OF GEARS USING ARTIFICIAL NEURAL NETTaesik JeongCase Western Reserve University, Cleveland, OhioThomas P. Kicher, Armington Professor of EngineeringCase Western Reserve University, Cleveland, OhioRonald J. Zab, Engineer
5、ing ManagerJoy Technology Inc., Bedford Gear Division, Solon, OhioINTRODUCTION develop mechanical engineering CAD and expert systems1011. This simplified design model is adaptable toMany CAD (Computer Aided Design) systems most mechanical element designs including gear design.v have been developed a
6、nd implemented to produce a A specific model representative of gear design whichsuperior quality design and to increase the design corresponds to figure 1 is shown in figure2.productivity in the gear industry. In general, it is truethat a major portion of design task can be performed byCAD systems c
7、urrently available. However, they can IDesign Statement ionly address the computational aspects of gear design that /typically requires decision making as well. In mostindustrial gear design practices, the initial design is the I Initial Design critical task that significantly effects the final resu
8、lts. I/However, the decisions of estimating or changing gear _,developingSiZeparameters must be made by a gear designexpert.TotechniquesmVeonehaveStePbeenfOrward,investigated.twoneWoneSyStemisthe _1 Design E;aluation _!_ RosUIt_DesignReDesign I I Optional Design Iartificial neural net and the other
9、is the expert system /known as artificial intelligence. The former is well suitedfor estimating initial gear size while the latter is the choice II Final Output Ifor changing parameters. The adaptability of artificialneural net for the initial gear design is demonstrated inthis paper, which is a par
10、t of the Intelligent GearCAD Figure 1. Simplified Mechanical Design Stagessystem under developing that emulates the entire geardesign procedure including the decision making tasks.The first stage of designing a gear set isestimating the necessary gear size parameters based onA INITIAL GEAR DESIGN us
11、er specified requirements. Once these parameters areselected, gear and tool geometries will be calculated andIn figure i, a model of the mechanical design evaluated by the AGMA (American Gear Manufacturersprocedure is illustrated. Similar models have been used to Association) power rating standard 8
12、. If the power1rating result is unsatisfactory, the result will be analyzed Ne Pinion Teeth Numberand the necessary parameters will be changed. The NT Total Teeth Number _second and the third stages will be repeated in an iterativemanner until the AGMA power rating is satisfied. The The determinatio
13、n of one parameter in expressionfinal stage is designing a gear blank, which is customarily (1.a) is dependent on the two other parameters.done after a successful power rating is achieved. Therefore, at least two parameters must be estimated bythe engineer. There may be many combinations of I soluti
14、ons which satisfy the equation (1) for a singleUser Specificationf example. Finding a superior solution among a myriad ofIupon ability an engineer.possibilities depends the ofI Proper initial parameter estimations usually requires yearsInitial Parameter of experience as well as an organized knowledg
15、e of thefield. In most the accumulated datacases, design through the history of a company is also an essential factor. ThisGear Geometry i type of design task is known as decision making. Figure3 shows the factors involved in decisiona gear engineersAGMA power rating_ Result Design making.Knowledge
16、Standard Final Output Existing CompanyDesign DesignData HistoryFigure 2. Modeled Gear Design StagesPrevious IndustryExperience StandardIn practice, engineers go through the initialdesign stage only once during the entire design procedure. I IThe number of iterations carried out to complete the gear
17、I Design Estimation Idesign depends upon how well the gear size parametersare estimated in the initial design stage. Consequently, anefficient gear design can only be achieved by properly Figure 3. Engineers Decision Making Factorsestimating the initial gear size parameters.The parameters required t
18、o be estimated for the TWO STEPS OF INITIAL GEAR DESIGNinitial design stage consist of the center distance,diametral pitch, pinion teeth number and gear teeth The initial gear design stage consists of twonumber, or alternately the total number of teeth. These steps. First, an engineer refers to a st
19、andard productfour parameters are the essential parameters among the catalog to identify the proper model. The selection ismany parameters of gear design that are necessary to carry based on the users specifications which includeout the AGMA power rating procedures. Equation (1) horsepower, speed ra
20、tio, and input RPM. At this step,illustrates how these four parameters are related to each the center distance is obtained with the proper selection ofother while assuming the helix angle is zero. model size. Next, the number of pinion and gear teethDP - NT will be estimated by a trial and error met
21、hod. The ratio ofestimated number of pinion and gear teeth must not2 CD (1.a) exceed the pre-determined percentage of error over therequired speed ratio. The diametral pitch can then beNT = Np + NG (1.b) calculated using these estimated values. This procedure isonly one example of a number of initia
22、l gear designwhere, DP Diametral Pitch methods used in the industry. The method shown hereCD Center Distance was obtained from an engineer actively working in theNG Gear Teeth Number2gear industry, with many years of experience in both that the connecting weights can be learned anddesigningandmanufa
23、eturing, memorized. Once all the connecting weights areestablished, the net will produce the proper output whenthe same or similar input pattern is seen. Accordingly,ARTIFICIAL NEURAL NET the quality of the knowledge patterns used for traininginfluences the quality of the estimated outputs. The net
24、isThe artificial neural net is composed of highly said to be successfully trained if the estimated outputsinterconnected layers which attempt to achieve human match the target outputs within a certain level of error.neuron-like performance 3. It is designed to emulate the Because the training knowle
25、dge patterns may not behuman neural activities, exhibiting abilities such as perfect, there is always the chance that an errantlearning, generalization, and abstraction 4, using estimation may appear. In comparison, it can also be saidmathematical implementations. A typical model of the that the per
26、formance of the human engineer will beartificial neural net is illustrated in figure 4. The modeled inaccurate if incorrect knowledge was used in has three layers; input, hidden (or middle), and outputlayers. This model is extremely simple, compared to thehundred trillion connections of the human n
27、eural system ARTIFICIAL NEURAL NET ALGORITHMS2. The terms shown in parenthesis in figure 4 are theanatomic terms oft he human neural system. Many artificial neural net algorithms have beendeveloped and implemented. Although there are somestructural variations, the basic idea is equivalent in termsof
28、 implementing a human neural system. Each algorithmOutput has its own characteristics and applicable regime. Afterthe nature of initial gear design was investigated, twoalgorithms, namely LVQ (Learning Vector Quantization) and GDR (Generalized Delta Rule), were selected to_ emulate two steps of init
29、ial gear design.v LVQ is also known as the pattern recognition orclassification method, which classifies availableknowledge patterns in a pattern space 5. Each patternmust have its own class label (or class I.D.). LVQ formsclusters which include identically labeled patterns whileI Nodes remembering
30、their weight centers. When a new inputWeights pattern without a class label, not encountered previously,(Synapses) (Cell Bodies) is seen, LVQ locates the cluster weight center which isclosest to the new input pattern and sends the class labelof the selected cluster as the output. In other words, LVQ
31、Figure 4. Typical model of Artificial Neural Net simply tells where the new input pattern belongs.In figure 5, a single step of the LVQ isIn figure 4, each node in one layer receives illustrated. At any kth step, the distances between one ofmultiple signals from the nodes in the previous layer. The
32、training patterns Pi E R n, i = 1,2 . l, and the neuronsstrength of each signal is determined by the value of theconnecting weight between paired nodes. The signals (or reference vectors 3) Nj E Rn, j = 1,2 . m, areconveyed to the node are summed and averaged (or measured using Euclidean distance (E
33、L) metric in order tomathematically evaluated) to decide whether this node will find the nearest neuron Nc.activate or not. If the node activates, the signal generatednwill be transmitted to the nodes in the next layer. EDj = _, (pq - nqq-I (2)The artificial neural net is not functional withoutexist
34、ing knowledge, just as a human engineer can notA perform a task without pre-existing knowledge of the where, pq Elements of P ifield. The net must be trained with known knowledge nq Elements of2_I,- patterns that consist of input and the corresponding targetoutput. The knowledge patterns are fed thr
35、ough the net so)(2 GDR also requires knowledge patterns whichhave inputs and corresponding target outputs for training.The knowledge patterns are supplied to the net in aNj feedforward manner to find a connecting weight matrix and then those weights are adjusted by the back-propagation of error to r
36、educe the total net error. ThePi+I GDR net shown in figure 6 uses the typical artificial neural net construction introduced in figure 4. The_.,., outputs of the nodes in one layer are transmitted to nodesin the next layer through connections that amplify,Xl attenuate or inhibit such outputs through
37、connecting weights 1. The net may have a number of hiddenlayers. However, in practice, only one or two hiddenNI+I layers are sufficient for most applications 5.The output of a node in the input layer i isPattern Space q“ = li i = 1, 2, n (4)The net input to a node in layerj isr limiting radius of ea
38、ch clusternet j - Z l_ i Oi , j=l, 2, mFigure 5. Single Step of LVQ Algorithm i (5)The output of nodej isThe neurons, /_s, are initially located randomlyin the pattern space and the closest neuron, N e, becomes a Oj - 1candidate for one of the many cluster centers that will 1 + e-f (6)appear after a
39、ll steps are performed. If the closest neuronhas the identical class label as the pattern, this neuron is f -netj + Ojmoved toward the pattern as the reward for a correct (7)classification. Otherwise the neuron is moved away fromthe pattern as the punishment for an incorrect In expression (7), the p
40、arameter 0j serves as a threshold orclassification 3. Equation (3.a) is used to represent the bias. Similarly, input netk and output Ok can be found bymove toward the pattern and the equation (3.b) is used for substituting the subscript j to k in equations (5) throughthe move away. For all other neu
41、rons, the equation (3.c) (7).is k = _ WkjO j , k =1, 2, lN.ck+I = Nck + 6g( Pj - Nck ) (3.a) J (8)1N._ +t= N_ - Ot(Pj - Nck ) (3.b) Ok - I + e -f (9)N k+l = N k. forj;e cJ J (3.C) f = netk + Ok (10)where, a is a monotonically decreasing momentum rateand preferably less than 1.0 3. In practice, the
42、All knowledge patterns will be fed through the net by thedetermination of t_ is non-trivial. When the neuron Nc is feedforward procedures, equations (4) through (10).moving toward the pattern, it is known that the pattern Usually, outputs Opk generated by the net will not bebelongs to this neuron at
43、 _h iteration. The same method the same as the target or desired outputs Tp_. Thewill be applied to all available patterns and the step will square of the difference (or pattern error) between these -be repeated iteratively until all the clusters are formed, two values is41E p = _ _k ( Tpk - Opk )2
44、The 8s at an internal node can be evaluated in terms of- (11) the tSs at an upper layer. Thus, starting at the highestlayer (or output layer), t5k can be evaluated usingand theaverageneterroris expression (15) and the errors can be propagated1 backward to the lower layers. The connecting weightsErie
45、: - _ _, _, ( Tk _ Opk ) 2 now will be updated as follows,2Ppkp = 1, 2, P (12) W.n+l - W.n. -I- m W.n.jz j, .t, (18)where, P Number of Patterns where, A W.n.J, = /7 ( _jO/)+ txA W.n.JzIf Enet falls into the acceptable error range, the net issuccessfully trained. Otherwise, the following procedures T
46、he momentum rate xhas been added to the expressionsare necessary to minimize the error. The convergence (14) and (16) to reduce the risk of oscillations whiletoward improved values for the connecting weights and training the net in the iterative approach 1. The t_ alsothresholds can be achieved by t
47、aking incremental changes allows a larger value of 77, thereby speeding convergencyAWkj proportional to-BE/_Wkj 1. 4. Both 7/ and a influence the training results andshould be carefully selected by trial and error. TheO3E improved connecting weight matrix will be used at theA Wkj “- - 77 o3Wkj next
48、iteration and the procedure is repeated until the09E O3netk system error reaches the desired level.= - 17 o3netk O3Wkj (13) Hidden Layer- where, 7 Learning Rate Input Layer jv Therefore,A Wkj = - 77 (_kOj (14) Input _ / Output.o, / .o,where, t_k - o3E o3netk_i _o3etk %- O3WkjThe term _k , which is t
49、he error to be propagatedbackward for the/tzh node in the layer, can be rewritten as _ _et KO3E o3Ok Weights _ Weights_k = 03Ok o3netk Wji netj Wkj= (T k - 0k)f; (net k)Figure 6. Net Construction of GDR Algorithm= (T k - Ok)O k (1 - Ok) (15)By similar mathematical procedures (details can be found APPLICATIONSin Ref. 1), As mentioned earlier, two steps of the initial gear_j design are emulated using the artificial neural nets.A Wji = - /7 Oi (16) Although, it is
