PRINCIPLE OF GENETIC ALGORITHM

Transcription

1 22 Optimization of Internal Spur Gear Design Using Genetic Algorithm Ashtosh Kumar Singh M.Tech. Student Mechanical Engineering G.B.Pant University of Agriculture & Technology Pantnagar, Uttarakhand Rakesh Saxena Professor Mechanical Engineering G.B. Pant University of Agriculture & Technology Pantnagar, Uttarakhand Hitendra Pal Gangwar M.Tech. Student Mechanical Engineering G.B.Pant University of Agriculture & Technology Pantnagar, Uttarakhand Anadi Misra Professor Mechanical Engineering G.B.Pant University of Agriculture & Technology Pantnagar, Uttarakhand ABSTRACT The optimization of internal spur gear is treated with the objective of minimizing the centre distance of gear for a given specification. The designs obtained satisfy all the necessary conditions for kinematic limits on contact ratio, interference and gear ratio tolerance and meet strength and wear criteria according to the design standard selected. A set of design variables are defined in terms of the number of pinion teeth, annulus teeth and the module. The objective functions of minimum centre distance are expressed. Internal spur gear design is being optimized by the application of genetic algorithm technique. The reason of using genetic algorithm is that it gives global optima instead of local optima in which the conventional technique usually get stuck. So genetic algorithms are presented in order to solve the problems of a discrete number of variables and to reduce the calculation time. The effects of tolerances placed on the design specification and the influence of acceptable facewidth limits on the optimum results are described. Keywords: Optimization, Internal Spur Gear and Its Design Parameter, Genetic Algorithm. NOTATION C Centre distance m n Normal module T 2 Annulus tooth number T 1 Pinion tooth number â Helix angle on the pitch circle Annulus addendum coefficient R i Actual gear ratio? t Transverse pressure angle? Pressure angle X n1 Normal profile shift coefficient of pinion X n2 Normal profile shift coefficient of annulus R Nominal gear ratio R U Upper tolerance gear ratio R L Lower tolerance gear ratio Outside diameter of external gear d a1 d b1 Base circle diameter of external gear d a2 Outside diameter of internal gear d b2 Base circle diameter of internal gear? w Operating pressure angle b Facewidth W t Tangential load d p Diametral pitch S b Bending stress factor X b Speed factor (in bending) Y Strength factor S c Surface stress factor X c Speed factor (in wear) Z Zone factor D o1 Outside diameter of pinion D o2 Inside diameter of annulus D b1 Base circle diameter of pinion Base circle diameter of annulus D b2

2 23 T d P n V Torque Pitch circle dia. in mm Power in KW Speed in rpm Circumferential velocity in m/s INTRODUCTION The process of designing a gear aims towards obtaining a perfect combination of minimization of cost, manufacturing time, and volume of the material and other resources with maximization of the gear life. Gear optimization must take into consideration that the constraints are within the acceptable limits. The designers working constantly towards improving the design and many different approaches have come in the light in the previous literary works. The use of optimization techniques has been given a wide importance in the design process because of less time consumption and gives best result. So it is now a day s being quite extensively applied in mechanical design. There are different techniques available for the solution of optimization problems which satisfy the given design constraints and the aim is basically to minimize or maximize a design objective function. The constraints can be either inequalities or equalities. These techniques usually use the previous solution as a key to get the closer solution, but do not really incorporate the earlier solution into the search for the new closer solution. In the present work, an internal spur gear design is being optimized by the application of genetic algorithm technique. The reason of using genetic algorithm is that we obtain global optima instead of local optima in which the conventional technique usually get stuck. In general, the most desirable gear set is the one that is minimal in size, so the design of a pair of internal spur gear is treated with the objective of minimizing the centre distance for a given specification. Smaller gears are easier to manufacture, they run more smoothly due to smaller inertial loads and pitch line velocities, and are less expensive. Spur gears find their application in automobile sector, manufacturing companies, cnc machines and machines. Researchers presented different approaches for optimum spur gear design. Methods were developed for spur gear optimization and design analysis by Hughson [3] and investigation about computer aided design of spur gear by White and Henderson [2]. Tong and Walton [11] optimized the internal spur gear centre distance and volume with the help of pinion teeth, annulus teeth and module taken as a variable. The genetic algorithm technique which was originally introduced by Holland [7] has been extensively used in gear design in addition to other engineering applications and results are found to be better and their reliance has been repeatedly established by the various researchers. At present US scientists are using the genetic algorithm in the much sophisticated space craft equipment design. It itself shows the importance given to genetic algorithm technique in modern design. In continuation to research going on, an attempt has been made in the present work to optimize the centre distance of a internal spur gear set by genetic algorithm. A work on spur gear by an earlier researcher is taken as reference problem and genetic algorithm is applied on it. Centre distance of internal spur gear is chosen as the objective function which is to be minimized. A set of design variables are defined in terms of the number of pinion and annulus teeth and the module. A number of practical constraints are built-in to ensure that the solutions meet all the requirements for adequate contact ratio and avoidance of tip interference and undercutting. The design obtained satisfies all the necessary conditions for kinematic limits on contact ratio, interference, and gear ratio tolerance and meet strength and wear criteria according to the design standard selected. Checks are also made to keep the tooth numbers to reasonable limits. A complete program of Genetic Algorithm has been developed in C language to solve the problem. De Jong s [7] experiments indicate that the best population size is individuals, the single point crossover rate is 0.6 per pair of parents, and the best mutation rate is per bit. In the program population size is 100, single point crossover is used with a probability of mutation is.016 and stopping criteria is given as 50 number of generation. By running the program, value of pinion teeth, annulus teeth and module are found at which the centre distance is optimum and all the constraints are satisfied. Internal Spur Gears The internal involute gear can have either spur or helical teeth (double helical are also used) cut on the inside of the annular ring. It is also known as an annular gear since the rim on which the gear is produced is in the shape of an annulus. The internal gears have many applications, the most frequent being the planetary gear systems. The tooth space on an internal gear more or less corresponds to the tooth of the external gear with which it mates, and the tooth of an internal gear corresponds to the tooth gap of its mating external gear. The tooth and the tooth-space of an internal gear can be proportioned like a standard gear with the addendum and the dedendum in reversed positions, but this is not generally done in order to alleviate the interference effects and also to improve tooth action. Normally, both the internal diameters of the internal gear and the outside diameter of the mating pinion are made slightly larger than the size calculated according to the conventional tooth proportions. Figure 1 shows the internal gear parameters. Usually, the pinion is the driver and the internal gear is the driven one. Internal gears are most commonly used in planetary or epicyclic gear trains but they can also be used as a compact ordinary reduction unit. Thus internal gears are used particularly where space is limited and when the rotational direction of the input and output shafts must be maintained. For a given tooth size they are stronger in bending and the contact stresses are lower than the equivalent external gear, giving higher power ratings, less wear and greater pitting

3 24 resistance. In general, the contact ratio is larger for a given gear ratio than the equivalent external gears and this, together with the convex-concave contact, results in smoother and quieter running. A further advantage is that the shaft centre distance may be kept small, often allowing only a single pair of internal gears to be used where otherwise two sets of external gears are needed. A disadvantage of internal gears is that the annular gear is more expensive to manufacture than external wheels. Otherwise, internal gears are limited to much the same constraints as external gears in terms of permissible reduction ratios (about 10: 1) and pitch line velocities. Figure 1: Internal gear parameters A. Advantages 1.Geometry ideal for epicyclic gear design. 2.Allows compact design since the center distance is less than for external gears. 3.A high contact ratio is possible. 4.Good surface endurance due to a convex profile surface working against a concave surface. B. Disadvantages 1.Housing and bearing supports are more complicated, because the external gear nests within the internal gear. 2.Low ratios are unsuitable and in many cases impossible because of interferences. 3.Fabrication is limited to the shaper generating process, and usually special tooling is required. PRINCIPLE OF GENETIC ALGORITHM Genetic Algorithm (GA) is referred to as a search method of optimal solution to simulating Darwin s genetic selection and biological evolution process. Genetic algorithm is a series of random iterations and evolutionary computations simulating the process of selection, crossover and mutation occurred in natural selection and population genetic, in according to the survival of the fittest, through crossover and mutation, good quality gradually maintained and combined, while continually producing better individuals and out of bad individuals. Through the generational produce and optimizing the individual, the whole group evolves forward and constantly approaches to the optimal solution. Genetic algorithm, not requiring gradient information and continuous function, optimization results being global, applied to mechanical design optimization problems, can effectively avoid local optimal solutions, and get the global optimal solution. So genetic algorithm is selected for gear optimization, and achieved through the C Language, optimizing process simple and efficient. OBJECTIVE FUNCTION Along with the new age of the development computers came the growth of optimization theory applied to the design problem. Optimization theory would require formulation of objective function that precisely measure cost or profit, and the expression of all side condition as mathematical equation or inequalities. The combination of design variables, giving the best possible value of the objective, which is consistent with constraints, is then sought by certain optimization procedures. Gear can be designed either on the basis of a minimum centre distance, or minimum volume and hence weight. The program described goes a step beyond IGD [11] in that a search of tooth numbers of meet the required kinematic conditions is made while at the same time the face width is computed and a search made for the optimum combinations. The design objective of the work is to obtain the minimum centre distance satisfying design requirements that include tooth numbers of pinion and annulus, module, gear ratio and material parameters. The gear design must satisfy operational constraints of tooth numbers, face width, contact ratio and involute interference. The centre distance between the pinion and annulus, CD is chosen parameter to be optimized and is given by [9] following equation. The criteria on which the optimization is based, i.e. the objective function is, Minimum centre distance, mn ( T2 T1) CD f ( x) (1) min 2 Here T 1, T 2 and m n are the design variables.t 1 and T 2 are the pinion and annulus tooth numbers respectively and m n is the module. This is the required objective function equation for the optimization problem for the internal spur gear set. The various gear parameters and the design variables are given in the Table 1 as shown on next page.

4 25 Table 1: Basic gear design parameters and variables Gear Parameters Design Variables 1. Bending strength and wear 1. Pinion teeth number Strength limits factor 2. Tangantial load 2. Annulus teeth number 3. Face width 3. Normal module 4. Gear ratio 5. Contact ratio DESIGN CONSTRAINTS A number of practical constraints are built-in to ensure that the solutions meet all the requirements for adequate contact ratio and avoidance of tip interference and undercutting. Checks are also made to keep the tooth numbers to reasonable limits and to ensure that the gear ratios are within tolerances. The calculated facewidth are kept within the limits [11]. A. Involute Interference Involute interference is defined as a condition in which there is a blockage on the tooth surface that prevent proper tooth contact at contact between portions of tooth profiles that are not conjugate. Involute interference or undercutting will occur if the tooth flank of the annulus meshes with the pinion flank below the pinion base circle. An undercut tooth is weaker, less resistant to bending stress, and proves to premature tooth failure. Undercutting has the following disadvantages: (i) It reduces contact ratio, resulting in a noises and rough gear action. (ii) It weakens the base of the tooth. The minimum number of pinion teeth to avoid interference is given [5] by the following equation T 1 Ë 2 2 Û E hn2 Xn1 Xn1 Xn2 Ri Dt Ri D Í t Ý 2cos ( ) ( ) [( cos )( 1)] sin R [( 1)sin ] 2 i Ri Dt B. Gear Ratio Internal spur gear, gear ratio is a important constraints. Checks are also made to keep the tooth numbers to reasonable limits and to ensure that the gear ratios are within tolerance describe in [11]. In internal gear basically two type of tooth are used. First is pinion teeth (T 1 ) and second is annulus or gear teeth (T 2 ). To avoid excessive tooth numbers T 1min < T 1 < 120 (3) T 1min is the minimum number of pinion teeth which may be set by the user, otherwise T 1min = 11. And for the annulus number of teeth T2 R RL R RU T (4) 1 Here R is 4 and value of R L and R U are 0.10 C. Contact Ratio To assure continuous smooth tooth action, as one pair of teeth ceases action a succeeding pair of teeth must already have come into engagement. It is desirable to have as much overlap as is possible. A measure of this overlap action is the contact ratio [4]. This is a ratio of the length of the line-of-action to the base pitch. Figure 2 shows the geometry for a spur gear pair, which is the simplest case, and is representative of the concept for all gear types. The length-of-action is determined from the intersection of the line-of-action and the outside radii. Figure 2: Geometry of contact ratio The ratio of the length-of action to the base pitch is determined from; CR Èda1Ø Èdb1Ø Èda2 Ø Èdb2 Ø É Ù É Ù É Ù É Ù c sin D Ê Ú Ê Ú Ê Ú Ê Ú Z Sm cos D To ensure smooth and continuous operation, the contact ratio must be as high as possible, which the limiting factors permit [11]. So CR > [CR] (6) Where CR is the contact ratio which is also a function of the design variables. [CR] is the minimum permissible contact ratio as specified by the user. If this is not specified the default value is set at 1.4. D. Tip Interference A check has to be made to ensure that tip interference does not occur. If the circumferential tip clearance is designated by d x, then tip interference is deemed not to occur if d x >0.2mm. d x may be determined as follows: D02( d J2 T) dx 2 (\J )T1 Where d T2 and y and q are obtained from (9) and (10) CD01 (5) (7) (8) D D C cos \ (9)

5 26 D sin \ sin T (10) 01 D02 In the given (8) value of g 1 is given by (11) S 2X n1 tandt cose J invdt invd 01 t (11) T1 cos and value of g 2 is given by (13), D b1 D t1 (12) D01 S 2Xn2 tan Dt cos E J 2 invdt invd 0t2 (13) T2 Here value of a ot2 is given by (14) E. Facewidth Db 2 cos D 02 t (14) D The last constraint is the gear pair facewidth. This involves selecting the gear pair materials and stressing the gears according to national design standards. Nearly in all standards, for example ISO, BS and AGMA, this requires both the pinion and annulus to be stressed on the basis of bending strength and wear (ISO also considers scuffing). Wear calculations require the life of the gear train to be specified. If the gears run at a fixed speed the life can be simply stated, but should the operating power and speed cycle alter throughout the day, this requires an equivalent life to be calculated. The method used in the British Standards Institution formula [1] for determining the strength of gear teeth is essentially the same as the Lewis equation [8], but instead of taking the load to act at the extreme corner of the tooth as in the Lewis method, in the BSI analysis the load is taken to act at the position it occupies at the moment when the whole load comes on to one tooth (Figure 3). Face width for the bending strength of a tooth is given[1] by following (15) 02 Wd t p b (15) ( SbXbY) This introduces a strength factor, Y, which is dependent only on the tooth system used, and the number of teeth on the pinion and annulus. Figure 4 shows the strength factor given in the standard for external gears. For the internal wheel the strength factor should be the same as that of a rack gearing with a pinion having the same number of teeth as the actual pinion, multiplied È 3 Ø by 1, where T B is the number of teeth in the internal É Ê Ù T B Ú gear. In addition a speed factor, X b is introduced to account for fatigue loading, which depends on the gear running speed and time. Figure 5 shows the speed factor for bending. Figure 4: Strength factor (Y) chart for external spur gear Figure 5: Combined speed factor (X b ) chart (in bending) Figure 3: The point V where the tooth carries the full load The safe load on a tooth based on surface stresses employs Hertzian contact stress theory, which has been modified in the British Standards to include a speed factor, X, and a zone factor Z. So the Face width on the basis of surface stresses or wear is given by (16) WK t b (16) ( S X Z) C C

6 27 The zone factor Z, like the strength factor Y, depends on the numbers of teeth in both mating gears and is found from a chart Figure 6. For internal gears the zone factor should be equal to that or the same combination of external teeth 0.8 È multiplied by R 1Ø, where R is the gear ratio. The speed É Ù ÊR 1Ú factor, for surface stresses, depends on the speed of the gears and the equivalent running time, and is similar to the chart shown in Figure 7. Figure 6: Zone factor (Z) chart for spur gear Figure 7: Combined speed factor (X c ) chart (in wear) One of the fundamental parameters to be considered, analyzed and checked for designing a gear system is the loadtransmitting capability of gear teeth. For this the circumferential force effective on the tooth at the pitch circle of the gear when in mesh, must be known. Depending on the given data, this force W t, known as the tangential force or transmitted load, can be found from the following [6] standard equations W t 2000T P d d n 1000 P Wt (17) v Here v is the pitch line velocity and which is given by (18). dcm ( ) nrpm ( ) vms ( / ) (18) 1910 Put the value of v in (16) and find the value of tangential load. And on the basis of tangential load and other factors find the value of face width of internal spur gear. For finding the optimum value of centre distance face widths must satisfied the given limits [11] given by (19). b L < b < b U (19) and value of b L and value of b U are and 29.8mm respectively. RESULT AND DISCUSSION Based on the concepts of the genetic algorithm, a complete program in C language has been developed for optimization of centre distance of internal spur gear. By using this program the string combination from the search area with the maximum fitness and satisfying the given constraints is obtained. In this study, optimum parameters of the pinion are determined satisfying the constraints such as involute interference, contact ratio, tip interference, gear ratio and face width. The problem of internal spur gear was also solved by Tong and Walton [11]. In this chapter, results obtained by GA are discussed and compared with those of Tong and Walton [11] in tabular as well as in graphical form. Table 2: Basic design data specification of internal spur gear 1. Power kw 2. Gear ratio :1 3. Gear ratio tolerance ± Pinion speed 3000 rpm 5. Minimum pinion teeth Total life hours 7. Contact ratio Pinion material En 24 H&T 9. Annulus material En 8 H&T 5

7 28 Table 2 shows the basic gear parameters for a sample gear being studied. They were used in an internal spur gear design by Tong and Walton [11]. They worked on a internal spur gear and obtained the optimized centre distance as mm. The optimum gear designed using GA has reduced the centre distance by 12.76%. In other words, a better size and compact gear design is found. The ranges of the variables are described in the Table 3. Table 3: Range of variables Variable Parameter Range 1 Pinion teeth number Annulus teeth number module ( ) mm In the program, input design parameters are taken as shown in Table 4. A complete program of genetic algorithm has been developed in C language to solve the problem. De Jong s [7] experiments indicate that the best population size is individuals, the single point crossover rate is 0.6 per pair of parents, and the best mutation rate is per bit. So in this program the population size is 100, single point crossover is used with a probability of 0.6, probability of mutation is.016 and stopping criteria is given as 50 number of generation. By running the program, value of pinion teeth number, annulus teeth number and module was computed at which the centre distance was optimum and all the constraints were satisfied at the maximum fitness function value. A criterion for the optimization is fitness value; the point at which fitness value is maximum is the optimum points. After running the program fifty generations were developed, in each generation one optimum result was printed. The optimum result refers to string combination at which the value of fitness was maximum in that particular generation of strings after the application of the operators of crossover and mutation. Then new generation is produced and the best string in that generation is obtained. Table 4: Comparison of result obtained by GA with tong and walton for internal spur gear T 1 T 2 M n CD d A d B b Tong method GA method Here T 1 Pinion teeth, T 2 Annulus teeth M n Module (mm), CD Central Distance (mm) d A Pitch diameter of pinion (mm) d B Pitch diameter of annulus (mm) b Face width (mm) The maximum fitness is which is found after running the program and in fifty generation, at which the pinion teeth number is 14, annulus teeth number is 55 and module is 2.00 mm. This is the combination at which the global optima are achieved. In this case, the objective function is the minimization type which means that a global minimum has been reached. The value of the objective function is obtained at the combination of the string of the maximum fitness and that gives the global solution to the design problem in the given search space. The results obtained with GA are compared here in tabular form with the previous work of Tong and Walton [11]. In Table 4, a comparison has been done between the results that are obtained with GA to those of Tong and Walton [11]. The centre distance in case of Tong and Walton was mm which in the case of GA has reduced to mm. The diameter of pinion has been decreased from 32.0 mm as in case of Tang and Walton to 29.6 mm in case GA. The diameter of annulus has been decreased from mm to mm. The facewidth has been increased from mm as in case of Tong and Walton to mm as in case of GA. Figure 8 shows the variation of centre distance with number of pinion tooth. It is clear from the figure that the value of centre distance with respect to different number of pinion tooth given by genetic algorithm method is found to be lesser than the centre distance corresponding to Tong and Walton method. Figure 8: Variation of centre distance with number of pinion tooth The optimize centre distance in previous work of Tong and Walton method was mm.in case of GA the centre distance is reduced to mm. So the size of gear will be more compact comparison to Tong and Walton result. Figure 9 shows the variation of pitch diameter of pinion with centre distance. At each point the value of pitch diameter with respect to centre distance, given by Tong and Walton is higher than the value of pitch diameter found by GA. The value of pitch diameter of pinion at optimized value of centre distance given by Tong and Walton is 32.0 mm whereas pitch diameter of pinion at optimize value of centre distance found

8 29 by GA is 29.6 mm. All lower value of different pitch diameter at different centre distance found by GA. So centre distance is better optimized comparison to Tong and Walton method. distance, pinion tooth number is 14 and corresponding that value the facewidth is mm. Figure 9: Variation of pitch diameter of pinion with different centre distance Figure 10 shows the variation of centre distance with number of pinion tooth at different values of module. Following inferences, can be drawn from the figure. First, for different modules centre distance always decreases with decreases in number of pinion tooth. And second at 2.00 mm module, the graph shows the value of centre distance is mm which is minimum as compared to other value of centre distance at other modules. Hence it can be said that at mm module, the centre distance is mm and the number of pinion tooth is 14 are the optimized value as per the optimum design obtained by GA. Figure 11: Variation of face width with different number of pinion tooth Figure 12 shows the flow chart of internal spur gear optimization process.in which all the steps shows which are taken during programming. Figure 10: Variation of centre distance with different number of pinion tooth at different module Figure 11 shows the variation pinion tooth with face width as per optimum design obtained by GA method. It is clear from the figure that for increasing value of facewidth, pinion tooth number is decreasing and decreasing value of facewidth, the pinion tooth number is increasing. Facewidth increase for maintain the gear strong. At the optimum value of centre Figure 12: Flow chart of internal spur gear optimization process

9 30 CONCLUSION In the internal spur gear design optimization problem minimum centre distance of internal spur gear is prime consideration; it depends largely upon tangential load on tooth, facewidth, pinion teeth number, annulus teeth number and module. In the genetic algorithm program following parameters are used, population size is 100, single point crossover is used with a probability of 0.6, probability of mutation is.016 and stopping criteria is given as 50 number of generation. On running the program for centre distance, following conclusion can be drawn: 1.By using genetic algorithm, the minimum centre distance in gear design approaches optimum value when the pinion teeth number is 14, annulus teeth number is 55 and module is 2.00 mm, and optimum minimum centre distance is mm. 2.Previous researcher Tong and Walton [11] studied the sample internal spur gear and gave optimized centre distance mm. The optimum gear set using genetic algorithm has a centre distance mm that optimize the Tong and Walton method by 12.76%. In other words, a better size and more accurate gear design is found for same work output. 3.At the point where centre distance is obtained, all the constraints are satisfied. The value of pitch diameter of pinion is mm and pitch diameter of annulus is These values are lower than corresponding value of Tong and Walton design. It indicates that the internal spur gear design is more compact in size as compared to the previous gear design. 4.Compared to Tong and Walton the value of facewidth in case of GA is increased from mm to mm, which indicate that for sustaining previous stress and strain levels, the facewidth has to increase. REFERENCES [1] British Standard BS436 [1940], Specification for Machine Cut Gears. A. Helical and Straight Spur, British Standards Institution, London, (1973). [2] D.D. White and J.L. Henderson [1969], Computer-aided Spur Gear Design. National Farm Construction and Machinery Meeting, Milwaukee, Wisconsin. SAE Technical Paper No , September. [3] D. Hoghson [1980], GODA5 (Gear Optimization and Design Analyses 5,) International Highway Meeting and Exposition, Mecca, Milwaukee. SAE Technology, Paper No , September. [4] Dr. George Michalec, Handbook of Metric Drive Components, Catalog 785, SDP/SI, New Hyde Park, NY. [5] E. Buckingham [1963], Analytical Mechanics of Gears, Dover Publications, New York. [6] Gitin M. Maitra [1997], Handbook of Gear Design, Second Edition, Tata McGraw-Hill Publishing Company Limited, New Delhi. [7] J.H. Hollend [1975], Adaptation in Natural and Artificial Systems. University of Michigan Press. [8] R.G. Mitchiner and H.H. Mabie [1982], ASME J. Mech. Design, 104, [9] Savage, M., Coy, J.J., and Townsend, D.P. [1982], Optimum Design of Standard Spur Gear Sets, J. Mech. Design, 104, [10] Tong, B.S and Walton, D. [1986], A Computer Design Aid for Internal Spur and Helical Gears, Int. J. Mach. Tools Manufact., Vol. 27, No. 4, pp [11] Tong, B.S and Walton, D. [1987], The Optimization of Internal Gears, Int. J. Mach. Tools Manufact., Vol. 27, No.4, pp