SYNTHESIS AND OPTIMIZATION OF EPICYCLIC-TYPE AUTOMATIC TRANSMISSIONS BASED ON NOMOGRAPHS

Transcription

1 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 SYTHESIS AD OPTIMIZATIO OF EPICYCLIC-TYPE AUTOMATIC TASMISSIOS BASED O OMOGAPHS E. L. Esmail K. H. Salih Leturer Assistant Leturer Deartment of Mehanial Engineering Mahinery and equiment deartment College of Engineering Institute of Tehnology University of Qadisiya Foundation of Tehnial Eduation Dewaniya, Iraq. Baghdad, Iraq ABSTACT A new methodology is develoed to design and otimize eiyli-tye automati transmission gear trains using kinemati nomograhs. From suh nomograhs, the kinemati harateristis of an eiyli gear mehanism an be exressed in terms of the gear ratios of its gear airs. The main roerties of this methodology are that; from a single nomograh, the angular veloities for all of the oaxial links an be estimated and omared diretly without seifying the exat size of eah gear, the angular veloities an be arranged in a desending sequene without using omliated tehniques, and all of the feasible luthing sequenes an be enumerated diretly. The otimization roedure to find the otimum gear ratios is aliable to any transmission mehanism omosed of two or more fundamental gear entities (FGEs) deending on kinemati and geometri onstraints. The reliability of the methodology is established by alying it to the onventionally available three-veloity Simson gear train for whih otimal gear ratios are fully available. The theoretial results are in omlete agreement with the ratial aliations. Keywords: Automati transmission, eiyli gear train, gear ratio, nomograh, and otimization / /. / :. یه. سیه..

2 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 صناعیا ه.. ITODUCTIO Most automati transmission mehanisms emloy eiyli gear trains (EGTs) to ahieve a set of desired veloity ratios. Figure shows an EGT emloying the Simson gear set as the ratio-hange gear train. This gear train with the above luthing sequene is the most oular gear train. It has been develoed by nearly all automotive manufaturers as three-veloity automati transmissions [Tasi 00]. It an be found in Ford C3, Ford C5, Meredes Benz, Toyota A40 and issan, to name a few. In an EGT, the veloity ratio is defined as the ratio of the inut shaft veloity to the outut shaft veloity. Various veloity ratios are obtained by using luthes to onnet various links to the inut ower soure and to the asing of a transmission gearbox, resetively. Tyially, a rotating luth is used for onneting two rotating links and a band luth is used to fix a link to the asing. In Figure rotating and band luthes are denoted by C and B, resetively. Also it is always ossible to ahieve a diret drive by loking all the links in the EGT together suh that they rotate as a single link. The veloity ratios seleted for a transmission are tailored for vehile erformane and fuel eonomy. Tyially, they inlude a first gear for starting, a seond and/or third gear for assing, an overdrive for fuel eonomy at road seeds and a reverse. A table deiting a set of veloity ratios and their luthing onditions is alled a luthing sequene. Table shows the luthing sequene of the transmission shown in Figure, where an Xi indiates that the orresonding luth is ativated on the ith link for that gear. For examle, when the mehanism is in the first gear, the rotating luth C and the band luth B are ativated. Hene, link 4 is onneted to the inut ower and link is fixed to the asing. In arranging a luthing sequene, it is highly desirable to ahieve a single-shift transition, i.e., only one luth is turned on while another is simultaneously turned off between two suessive veloity ratios. The gear ratios for the Ford C4 transmission as an examle of a three-veloity automati transmission that uses a Simson gear train, are:.46: st gear,.46: nd gear,.00: 3rd gear, and.0: reverse []. Ford Motor Comany has five basi transmissions that use the Simson gear train and are quite similar to eah other: the C-3, C-4, C-5, C-6, and A4LD. The A4LD is a 4- seed, whereas the others are 3-seeds. Most General Motors' transmission and transaxles are based on the Simson gear train. The turbo hydramati (THM) 375, 400, and 475 transmissions are heavy-duty transmissions generally used in full-sized rare wheel drive (WD) ars and truks. The 45 is based on the same design as the others but had been modified for use in forward wheel drive (FWD) vehiles with a longitudinally laed engine. The 3L80 (THM 400) was introdued in 964 and has been used by all divisions of General Motors, as well as by many different manufaturers, suh as Jaguar and olls-oye. The THM 50, 50C, 350, 350C, and 375B transmissions are 3-seed units. The 50 and 50C are light-duty, the 350 and 350C are medium-duty, and the 375B is a heavy-duty version of the 350 []. [Mogalaalli 993] develoed an interative design system for the design of automati automotive transmission gear trains. Mogalaalli's system an otimize the gear ratios using the Augmented Lagrangian Multilier Method. In what follows an overview to the otimization theory is given [Arora 004].. OPTIMIZATIO THEOY OVEVIEW Otimization tehniques are used to find a set of design arameters, x { x, x,..., x n }, that an in some way be defined as otimal [Mogalaalli 993]. In a simle ase this might be the minimization or maximization of some system harateristi that is deendent on x. In a more advaned formulation the objetive funtion, f(x), to be minimized or maximized, might be subjet

3 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 to onstraints in the form of equality onstraints, G i ( x ) 0 ( i,..., m e ) ; inequality onstraints, G i ( x ) 0 ( i m e,..., m ) ; and/or arameter bounds, x l, x u. A General Problem (GP) desrition is stated as min f ( x ), subjet to G i ( x ) 0 ( i,..., m e ) G i ( x ) 0 ( i m e,..., m ) where x is the vetor of length n design arameters, f(x) is the objetive funtion, whih returns a salar value, and the vetor funtion G(x) returns a vetor of length m ontaining the values of the equality and inequality onstraints evaluated at x. An effiient and aurate solution to this roblem deends not only on the size of the roblem in terms of the number of onstraints and design variables but also on harateristis of the objetive funtion and onstraints. When both the objetive funtion and the onstraints are linear funtions of the design variable, the roblem is known as a Linear Programming (LP) roblem. Quadrati Programming (QP) onerns the minimization or maximization of a quadrati objetive funtion that is linearly onstrained. For both the LP and QP roblems, reliable solution roedures are readily available. More diffiult to solve is the onlinear Programming (P) roblem in whih the objetive funtion and onstraints an be nonlinear funtions of the design variables. A solution of the P roblem generally requires an iterative roedure to establish a diretion of searh at eah major iteration. This is usually ahieved by the solution of an LP, a QP, or an unonstrained sub-roblem. In transmission design, otimization based on desired veloity ratios and mehanism kinemati and geometri onstraints generates the best ossible gear ratios. Otimization uses an objetive funtion, based on a set of design variables and onstraints, assigns a numerial value to the desired or required veloity ratios. The design variables stem from the hysial onstraints. Along with the numerial algorithm, the objetive funtion lays a big role in the effetiveness of the otimization roblem [Arora 004]. 3. OMOGAPHS omograh is defined as three or more axes, or sales, arranged suh that roblems of three or more variables an be solved using a straightedge. In the artiular ase of EGTs, a nomograh an be onstruted using three or more vertial arallel axes [Esmail 007, 008]. A basi EGT onsists of a sun gear, a ring gear, a lanet, and a arrier as shown in Fig.. Figure 3 shows the basi form of the grah to be reated for a basi EGT. The term "gear ratio" is used in this aer to denote the ratio of a meshing gear air. It is defined by a lanet gear with reset to a sun or ring gear x x Z Z, x x () Where Z and Zx denote the numbers of teeth on the lanet and the sun or ring gear, resetively, and the ositive or negative sign deends on whether x is a ring or sun gear. Considering the kinematis of a fundamental iruit, the fundamental iruit equation an be written as [Buhsbaum and Freudenstein 996]: ( ) ( ) () x, x Equation () an be re-written for the links of the basi EGT as follows

4 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 ( ) ( ) (3) r, r ( ) ( ) (4) s, s and ( ) ( ), (5) ( ) ( ), 0 (6) These values have been used to lae the axes of the nomograh shown in Fig. 3. The axis asses at the origin, and the axis is one unit aart from it. z Let the symbol x, y denote the veloity ratio between links x and y with referene to link z, where x, y and z are any three links in the gear train. Alying Eq. () twie for links y and z and simlifying it yields: z x z, x, z x, y y z, y, z (7) From the nomograh shown in Fig. 3, we an find diretly that s s, r r, s, r (8) and s, r, s r, s s, (9), r r, s r, s, r s, s, r, r, r (0) s r, r, s ( () s, r ) s, r s, r 3. Cluthing Sequene omograh Due to the fat that an EGT an be deomosed into several FGTs [Chatterjee and Tsai, 995, 996] the kinematis of an EGT is losely related to the kinematis of eah individual FGE. In order to derive a luthing sequene, it is imortant to unify the nomograhs of the FGEs in one nomograh alled the system nomograh and develo a methodology for arranging the outut veloities in a sequene. We first unify FGEs that are onneted to eah other by two ommon links i.e. having two links with the same labeling, as shown in Fig. 4. Let b and b be the ommon links, P ( ) be the first lanet of the first FGE or subsystem to be unified, y be any link from the seond FGE (not the basi one) other than b and b, then by using link b as a bridge, we exress the veloity ratio b ( ) y, of link y with reset to the basi FGE in terms of two of the veloity ratios of the FGEs or subsystems as

5 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 b y, ( ) b b y, b. () b, ( ) Where b b, ( ) is assoiated with one of the FGEs or subsystems (basi one) and b y, b is assoiated with the other FGE or subsystem. ow, we shall deal with link y as if it were a link that belongs to the basi FGE. From the unified nomograh, and making use of Eq. (7), we an write b y, ( ) ( ), y ( ), b (3) ( ), ( ) ( ), b where ( ), ( ) is equal to one and ( ), y is the virtual gear ratio of link y in terms of the lanet of the basi FGE or the distane between the first FGE arrier and link y axes on the unified nomograh. For a artiular EGT, two nomograhs are drawn in Fig. 5. Sine b is the arrier in this train, then making use of Eq. (6), ( ), b is zero. Alying Eq. (3) yields b ( ) y (4) Combining equations () and (4) yields ( ) y,, b b ( ), y y, b. (5) b, ( ) For a system having more than two FGEs, the unifiation roess ontinues between FGEs and/or unified subsystems until the highest-level system beomes the required mehanism. This way, a system nomograh an be obtained in terms of the gear ratios of its gear airs. The Simson gear train whih is omosed of two single-lanet FGEs will be used as a design examle to illustrate the resent methodology. Figure 6 shows the unifiation roess and the system nomograh for the Simson gear train. The gear ratios for this train are Z Z (6) 5, 5 Z Z (7) 5 3 and It an be shown that (8) 6, 3 6, 4 Z 6 Z 4 Z Z (9) 0 5, 6 3 (0) 0 ()

6 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 and From Eq. () From the first FGE nomograh And from the seond FGE nomograh 0 6, 4 () 0 (3), 3 6, 3, 3 3, 6 (4), 6. (5) 3, 6 6, 3 5, (6) 5, Substituting Eqs (5) and (6) into Eq. (4), yields But from the unified nomograh Therefore. 5,, 6 6, 6, 3 (7), 6 6, 5, (8). 5, 6, 3 (9) 5, Figure 7 shows the basi form of the unified system nomograh for the Simson gear train without the lanet axis. Here it is alled the basi form of luthing sequene nomograh. As shown in Fig.7, the ranges of outut veloities an be lassified into three kinds: drive (D), overdrive (OD), and reverse () aording to whether the veloity is between zero and the inut veloity, greater than the inut veloity, or less than zero. ote that a ositive veloity is divided into D and OD beause there is usually a "diret drive" (DD) between them [Hsieh and Tsai, 996]. If the outut veloity ranges of an EGT an be identified without seifying the gear sizes, a desending sequene of luthing onditions an be roughly onfigured. This will effetively failitate the enumeration of all the feasible luthing sequenes of an EGT. 3. Proedure for Enumeration of Cluthing Sequenes An EGT an rovide several veloities deending on the assignment of the inut, outut, and fixed links. These various veloities need to be estimated and arranged in a desending order to arrive at a roer luthing sequene. In a transmission mehanism, usually the outut link is ermanently attahed to a final redution unit, and the hange of veloity is aomlished by swithing either the fixed link or the inut link [Hsieh,996]. To a redetermined outut link, o, hoose two oaxial links (x, y), from the EGT at a time to onstrut the luthing sequene nomograh. For an EGM with m oaxial links, there is (m- )! / ossible sets of two oaxial links. By taking one of the two oaxial links as the ommon inut link the other as the fixed link and by drawing a line through them, two outut veloities are obtained at a time: a total of (m-)! veloities. The ends of eah line are labeled with B

7 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 and C to denote a band luth on the line of zeros and a rotating luth on the line of the inut veloity. Perform this oeration as many times as ossible until all ossible arrangements of (x, y, o) are exhausted. In the resent methodology the veloities are arranged automatially in a desending order into three kinds: D, OD and. eturning to our design examle, the Simson gear train, beause there are four oaxial links in this mehanism and link two is re-assigned as the outut link, this mehanism an rovide six luthing onditions. Figure 8 shows the luthing sequene nomograh for this mehanism. In arranging a luthing sequene, it is highly desirable to ahieve a single-shift transition [Hsieh, and Tsai, 998]. In order to ahieve single-shift transitions, the D-veloities an be further lassified into two sets as shown in Fig. 9. A diret drive is obtained by simultaneously luthing two oaxial links of an EGT to the inut ower soure. A reverse drive an be obtained by alying one or two of the luthes designed for the forward drives for the reverse drive. As a result, we obtain two desending sequenes of veloities as shown in Figures 9(a) and (b), whih result in two three-veloity and two four-veloity luthing sequenes as shown in Tables (a) through (d). Table () shows one feasible luthing sequene with rotating luthes attahed to links 3 and 4, and band luthes attahed to links and 3. This luthing sequene has been alied in most threeveloity automati transmission. Figure shows the funtional reresentation of this three-veloity luthing sequene for the seond redution set. From Fig. 9 (b), the veloity ratios an be written for three- or four-veloity automati transmission as D 3 5, 5, 6, 4, (30) 5, 6, 4 ( 6, 3 ) 3 D 4 6, 4, (3) 6, 3 3 5, (3) 3, 5, OD, 3 (33) 5, Under ertain luth ombinations the gear train may at as a rigid body, in whih ase the veloity ratio will equal to one ( DD ), and the designer will not affet it by his seletion of gear ratios. The other luthing sequenes obtained are in agreement with those reorted in the literature [Chatterjee and Tsai,995,996], exet that there is no need to any information ontaining the aroximate gear sizes arranged in a desending order that is required in other methods.

8 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 4. OPTIMIZATIO TECHIQUE AD POBLEM FOMATIO In the systemati reation of mehanisms the funtional evaluation stage deals with the seletion of aetable mehanisms from the enumerated list of mehanisms, in order to satisfy the funtional requirements. The next stage in the reation of mehanisms is its design and analysis. In this stage dimensional synthesis tehniques are used to determine the mehanism's roortions. The designer has to hoose a gear train with a set of luthes that are to be oerated in a hosen sequene, and a set of gear ratios that will rovide a set of veloity ratios. The lassial aroah of finding roer gear ratios has been to hoose a gear train and a orresonding luthing sequene, and then to vary the gear ratios by trial and error until the best ossible veloity ratios are obtained. To eliminate trail and error, an objetive funtion F is defined in terms of the veloity ratios k for the gear train, the minimization of whih ensures that the best ossible gear ratios are ahieved, subjeted to ertain onstraints. The objetive funtion an be written as k i F ( ) ( ) (34) i di th Where i ; is the i veloity ratio arameter obtained by the minimization roess and di is the th i desired veloity ratio. The veloity ratios,... k deend on the luth seletion and the toology of the gear train. It should be realized that for an arbitrary ombination of di there may not be a set of solutions for the gear ratios j, i that satisfy the veloity ratio equations. In other words, the ideal minimum value of the objetive funtion F() 0 may or may not be ahievable. The onstraints under whih the funtion F() is minimized are now determined. 4.. Geometri Constraints Let d be the diameter of a gear element i. If the diametral ith P of all the gears is the same, then d P. Z d i and we an get the geometri onstraints for the Simson gear train from Fig. as i i 4 d d 3 6 or Z 4 Z 3 Z 6 and therefore, Z Z Z Z thus And in a similar manner ote that both and meshes. 6, 3 6, 4 6, 3 (35) 5, (36) are negative numbers sine they orresond to external gear 4.. Kinemati Constraints The kinemati onstraints are determined from the system nomograh shown in Fig.9. They are rewritten here as

9 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 0 5, (37) 0 (38) 0 6, 4 (39) 0 (40) 6, 3 5. COSTAIED OPTIMIZATIO USIG MATLAB'S fminon Matlab rovides the ommand (fminon) for onstrained minimization of an objetive funtion f(x). Fminon finds a vetor x that is a loal minimum to a salar funtion f(x) of several variables subjet to onstraints on the allowable x starting at an initial estimate[brian, 007]. This is generally referred to as onstrained nonlinear otimization. It finds a minimum of a onstrained nonlinear multivariable funtion min f ( x ), subjeted to one or more of the following onstraints: (x) 0, eq(x) = 0, A x b, Aeq x = beq, l x u. where x, b, beq, lb, and ub are vetors, A and Aeq are matries, (x) and eq(x) are funtions that return vetors, and f(x) is a funtion that returns a salar. f(x), (x), and eq(x) an be nonlinear funtions. The objetive funtion must be oded in a funtion file. In this work this file is alled Simson and saved as Simson.m in the working diretory. Without any extra otions, fminon with linear inequality and equality onstraints, lower and uer bounds, and nonlinear inequality and equality onstraints is alled as follows: [x,fval]=fminon(@simson,x0,a,b,aeq,beq,lb,ub,'onstraint') The last inut argument in this all is the name of a funtion file (denoted onstraint in this work and saved as Simson_onfun.m in the working diretory), in whih the nonlinear onstraints are oded. 5.. Constraint funtion file: Simson_onfun.m is a funtion file (any name an be hosen) in whih both the inequality funtions (x) and the equality onstraints eq(x) are oded and rovided in the form of olumn vetors. The funtion all [,eq]=onstraint(x) must retrieve (x) and eq(x) for given inut vetor x. If only equality onstraints are given, define = [ ]. 5.. Interretation: The retrieved eq (x) is interreted by fminon as equality onstraint eq(x) = 0. The inequalities assoiated with (x) are interreted as (x) 0. Thus, if a onstraint of the form (x) 0 is given, rewrite this as -(x) 0 and ode -(x) in the onstraint funtion file Plaeholders: As shown above, the onstraints have to be assed to fminon in the following order:. linear inequality onstraints,. linear equality onstraints, 3. Lower bounds, 4. Uer bounds, and 5. onlinear onstraints 6. If a ertain onstraint is required, all other onstraints aearing before it have to be inutted as well, even if they are not required in the roblem. If this is the ase, their inut argument is relaed by the laeholder [ ] (emty inut). x

10 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 6. OPTIMIZATIO ESULTS The designer initially seifies the gear train, enumerates the luthing sequene and then seifies the redution ratios to be ahieved. The otimization roblem starts at an arbitrary design oint and at the end of its exeution dislays the otimized gear ratios. esults of samle inuts for the threeveloity Simson gear train are shown in Table 3. The trial and error aroah of finding the gear ratios for an automotive gear train to obtain a set of veloity ratios has been eliminated by formulating the task as a onstrained nonlinear otimization roblem. The otimization roedure was suessfully alied to find those gear ratios for whih the veloity ratios are losest to the original seifiation. The theoretial results are in omlete agreement with the ratial aliations as shown in olumns and 3 in Table 3. The Otimized gear ratios shown in the last olumn of Table 3 satisfy the geometri and kinemati onstraints given by Eqs. (35) Through (40). Figure 0 shows a MATLAB lot to the otimization results given in Aendix B for Ford C4, C5 and C6 Simson gear train given in the first row of Table 3. EFEECES []Arora, S. J., 004, "Introdution to otimum Design". Elsevier/Aademi Press, Amsterdam. []Buhsbaum, F., and Freudenstein, F., 970, "Synthesis of Kinemati Struture of Geared Kinemati Chains and Other Mehanisms," Journal of Mehanisms and Mahine Theory, 5, [3]Brian D. Hahn and Daniel T. Valentine, Third edition 007, "Essential MATLAB for Engineers and Sientists", Elsevier, Amsterdam. [4]Chatterjee, G., and Tsai, L. W., 995, " Enumeration of Eiyli-Tye Transmission Gear trains, " Transations of SAE, Vol. 03, Se. 6, Paer o. 940, [5]Chatterjee, G., and Tsai, L. W., 996, " Comuter-Aided Skething of Eiyli-Tye Automati Transmission Gear trains, " Transations of the ASME, Journal of Mehanial Design,, Vol. 08, o. 3, , Setember. [6]Esmail, E.L., 007, "Kinemati omograhs of Eiyli-Tye Transmission Mehanisms, Emirates Journal for Engineering esearh, volume, issue 3, [7]Esmail, E. L., 008 "omograhs for enumeration of luthing sequenes assoiated with eiyli tye automati transmission mehanisms", ASME International Mehanial Engineering Congress and Exosition (IMECE08), Otober 3- ovember 6, 008 Boston, Massahusetts, USA, aer o [8]Hsieh,H.I., and Tsai, L. W., 996, "Kinemati Analysis of Eiyli-Tye Transmission Mehanisms Using the Conet of Fundamental Geared Entities", Transation of the ASME, 8, [9]Hsieh, H. I.,996, "Enumeration and Seletion of Cluthing Sequenes Assoiated with Eiyli- Tye Transmission Mehanisms", Ph.D. Thesis, Deartment of Mehanial Engineering, University of Maryland at College Park, Maryland, USA. [0]Hsieh, H. L., and Tsai, L. W., 998, "The Seletion of a Most Effiient Cluthing Sequene Assoiated with Automati Transmission," ASME J. of Mehanial Design, 0, [] htt://transmissions0.om/automati-transmissions.h

11 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 [] htt:// [3]Mogalaalli, S.., Magrab, E.B. And Tsai, L-W, 993, "A CAD System for the Otimization of Gear atios for Automotive Automati Transmissions", SAE aer, no (htt://aers.sae.org/930675) [4]Tsai, L. W., 00, "Mehanism Design: Enumeration of Kinemati Strutures Aording to Funtion", CC Press, Boa aton, FL. Table Cluthing sequene of the mehanism shown in Figure. Ativated luthes ange C C B B First X4 X Seond X4 X3 Third X4 X3 everse X3 X Table (a) Three-veloity luthing sequene for the first redution set. Ativated luthes ange C C B B First X3 X4 Seond X X4 Third X3 X everse X3 X Table (b) Four-veloity luthing sequene for the first redution set. Ativated luthes ange C C B B B3 First X3 X4 Seond X X4 Third X3 X Fourth X X3 everse X3 X Table () Three-veloity luthing sequene for the seond redution set. Ativated luthes ange C C B B First X4 X Seond X4 X3 Third X4 X3 everse X3 X Table (d) Four-veloity luthing sequene for the seond redution set. Ativated luthes ange C C C3 B B First X4 X Seond X4 X3 Third X4 X Fourth X X3 everse X3 X

12 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Table 3 Table of otimized gear ratios for samle inuts for Simson gear train.

13 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Figure : A Tyial transmission mehanism. Figure : A basi eiyli gear train. Figure 3: omograh for the basi eiyli gear train. Figure 4: Unifiation of FGEs and/or subsystems.

14 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Figure 5: Unifiation of two FGE nomograhs into one nomograh. (a) omograh unifiation of the two FGEs of the Simson gear train shown in Fig.. (b) Unified system nomograh Figure 6: System nomograh of the Simson gear train shown in Fig..

15 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Figure 7: Basi form of the luthing sequene nomograh for the Simson gear train. Figure 8: Cluthing sequene nomograh for the Simson gear train. (a) First redution set. (b) Seond redution set. Figure 9: Cluthing sequene nomograhs for the two redution sets of Simson gear train.

16 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Figure 0: Examle of MATLAB lot to the otimization results given in Aendix B.

17 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 APPEDIX A: () MATLAB objetive funtion m.file (Simson.m) funtion f = Simson(x) =(x()*x()+(x(3)-x())*x(4))/(x()*x());=-(x(4)/x());3 = ;4=(x(3)/x()); k =.840; k =.6; k3 =.0; k4 = ; % Ford C4, C5, And C6 f=((/k)-)^+((/k)-)^+((3/k3)-)^+((4/k4)-)^; () MATLAB onlinear onstraints m.file (Simson_onfun.m) funtion [, eq] = Simson_onfun(x) =[]; eq = [(/x())+(/x(3))-; (/x())+(/x(4))-]; (3) MATLAB otimization m.file (simson_main_file.m) funtion [x,fval,exitflag,outut,lambda,grad,hessian] = simson_main_file(x0,lb,ub) x0 = [0,0,0,0] lb = [-Inf,0,-Inf,0] ub = [0,,0,] k =.840; k =.6; k3 =.0; k4 = ; % Ford C4, C5, And C6 otions = otimset; otions = otimset(otions,'dislay','iter'); otions }); otions = otimset(otions,'algorithm','interior-oint'); otions = otimset(otions,'diagnostis','on'); [x,fval,exitflag,outut,lambda,grad,hessian] =... fminon(@simson,x0,[],[],[],[],lb,ub,@simson_onfun,otions); =(x()*x()+(x(3)-x())*x(4))/(x()*x());=-(x(4)/x());3 = ;4=(x(3)/x()); Desired_Veloity_atios_k=[k k k3 k4] Otimal_Veloity_atios_=[ 3 4] Gear_atios_r=[x() x() x(3) x(4)] APPEDIX B: MATLAB otimization results samle x0 = lb = -Inf 0 -Inf 0 ub = 0 0 Diagnosti Information umber of variables: 4 Funtions Objetive Simson Gradient: finite-differening Hessian: finite-differening (or Quasi-ewton) onlinear onstraints: Simson_onfun Gradient of nonlinear onstraints: finite-differening Constraints umber of nonlinear inequality onstraints: 0 umber of nonlinear equality onstraints: umber of linear inequality onstraints: 0 umber of linear equality onstraints: 0 umber of lower bound onstraints: umber of uer bound onstraints: 4

18 Al-Qadisiya Journal For Engineering Sienes Vol. 4 o. 3 Year 0 Algorithm seleted interior-oint End diagnosti information First-order orm of Iter F-ount f(x) Feasibility otimality ste e e e e e e+000.4e e e e e e-00.80e e e e e e e e e e-00.84e e-00.4e e e e e e-00.86e e e e-00.e e e e e e e e-00.93e e-005.6e e e e e e-00.93e e e e e e e e e e e e e e e e e e e e e e-009.5e e e e e e e e e e e e-00.04e-0.600e e e e e e-006 Otimization terminated: first-order otimality relative error less than otions. TolFun, and relative onstraint violation less than otions.tolcon. Desired_Veloity_atios_k = Otimal_Veloity_atios_ = Gear_atios_r =