a practical approach to the optimization of gear trains with spur gears

A practical approach to the optimization of gear trains with spur gears

Nenad Marjanovic a,, Biserka Isailovic b, Vesna Marjanovic a, Zoran Milojevic c,Mirko Blagojevic a, Milorad Bojic a

a Faculty of Mechanical Engineering in Kragujevac, 34000 Kragujevac, S. Janjic 6, Serbiab Zastava automobili, Thermo shaping plant, 34000 Kragujevac, 4 Kosovska str., Serbiac Faculty of Technical Sciences, 21000 Novi Sad, Bulevar D. Obradovica 6, Serbia

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 November 2010Received in revised form 3 November 2011Accepted 13 February 2012Available online 22 March 2012

Optimization of gear trains is a complex task, due to the characteristics of mathematical modelthat describes its behavior. This paper presents the characteristics and problems of optimiza-tion of gear trains with spur gears. It provides a description for selection of the optimalconcept, based on selection matrix, selection of optimal materials, optimal gear ratio andoptimal positions of shaft axes. The paper will further present the definition of mathematicalmodel, with an example of optimization of gear trains with spur gears, using original software.Using an approach like this for the optimization of gear trains with spur gears gives results thatcan be applied in practice.

2012 Elsevier Ltd. All rights reserved.

Keywords:OptimizationGear train with spur gearsMinimum volumeOptimal conceptMathematical modelSelection matrix

1. Introduction

Gear trains are complex technical systems. Numerous complex equations, depending on a large number of design variables,are used for their mathematical formulation and many influence factors have to be taken into consideration as well. Thepossibility of reducing the number of factors influencing a system is limited and it depends on good knowledge about the natureof the system and the ability of the designer to assess the importance of each influence factor in advance.

The designing of gear trains is very complex and it often requires the use of nonlinear functions, as well as discrete designvariables. In almost all structures, it is extremely important to design elements in such a manner that the whole constructionweight is minimal. Savsani et al. [1] described the design of minimum weight gear trains using particle swarm optimizationand simulated annealing algorithms. Yokota et al. [2] described genetic algorithm for the optimization of gear weight. Gologluand Zeyveli [3] presented an automated preliminary design of gear drives by minimizing volume of gear trains using a geneticalgorithm. Mendi et al. [4] presented a dimensional optimization using a genetic algorithm.

Thompson et al. [5] presented their work on the optimization of multi-stage spur gear reduction units taking into accountminimum volume and surface fatigue life, as objective functions, employing quasi-Newton method. Chong et al. [6] described amethod for reduction of geometrical volume and meshing vibration of cylindrical gear pairs while satisfying strength andgeometric constraints using a goal programming formulation.

Abuid and Ameen [7] have done the optimization based on minmax method combined with a direct search technique. Theypresented a problem containing seven objective functions gear volume, center distance and five dynamic factors of shafts andgears.

Mechanism and Machine Theory 53 (2012) 116

. Corresponding author. Tel.: +381 34 335990; fax: +381 34 333192.

E-mail address: [email protected] (N. Marjanovic).

0094-114X/$ see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2012.02.004

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It is possible to optimize gears using various criteria. Ciavarella and Demelio [8] developed a software package for gearoptimization that includes: kinematic optimization (minimisation and balancing of specific sliding), static stress analysis (tominimize stress concentrations) and crack propagation studies (to estimate fatigue life under a pre-existing defect). Mao [9]described optimization of gear surface micro-geometry for the fatigue wear reduction. Optimization of gears for the reductionof noise is shown in paper [10]. The rotational movement of gears is treated as the input of the gear system and the acousticalnoise signal as the output. Bonori et al. [11] performed the optimization of gear pairs aiming to reduce vibration and noise byusing a genetic algorithm.

Literature review shows that various authors apply various techniques for the optimization of gear trains. Nevertheless,this paper shows the review of specific problems that occur during the optimization of gear trains with spur gears. Unlikequoted literature, this paper offers a comprehensive original approach to optimization of gear trains with spur gears. Itprovides a description for selection of optimal concept, based on selection matrix, selection of optimal materials, optimalgear ratio and optimal positions of shaft axes. This paper presents the defining of a mathematical model for gear trainswith spur gears, with an example of their optimization using original software GTO (Gear Train Optimization) developed byMarjanovic [12].

The motivation for this paper comes from the idea to provide a mechanical engineer with a powerful tool that would offer thesolution to the majority of problems that arise during the design of gear trains with spur gears.

2. Characteristics and problems of optimization of gear trains with spur gears

Optimization of gear trains with spur gears is a complex task, due to the characteristics of mathematical model that describestheir behavior. The increase of complexity of the mathematical model gives a more accurate solution, but it aggravates and slowsdown the process of reaching the solution. It is obvious that it is necessary to make a compromise between complexity andcompatibility of the mathematical model.

When formulating a mathematical model, the following facts should be taken into consideration:

Mathematical model is just one of many possible approximations of behavior of the given system. The task of the model is to assist, and not replace the researcher, nor free him from making decisions. Themodel does not provide completely new information about the system, but it enables a better understanding of its behavior.

There are numerous factors that make the optimization of a gear train with spur gears a complex procedure: selection andcomplexity of objective function, complexity of mathematical model, number of optimization variables, complexity and numberof constraints, selection of method for solving, to name but a few.

3. Selection of optimal concept of gear trains with spur gears

Gear trains transmit energy from driving machine to working machine and in that, they adjust torque and number of revolu-tions of driving machine shaft to the torque and number of revolutions that the working machine needs. Partial functions of thisbasic function are realized through gear pairs, shafts, housings, bearings and other parts and assemblies. Various combinations ofthese elements give various concepts of gear trains. Limited number of design variables are combined when versions of conceptsare formed, but due to a large number of parameters and characteristics (of gear pairs primarily), the number of concept versionsbecomes quite large.

Conceptual design of gear trains with spur gears is conducted in two phases. The first phase includes the development ofversions and the second includes the selection of optimal concept and its optimal parameters.

3.1. Concepts selection matrix

Optimal concept of gear train with spur gears can be selected by selection matrix. Selection matrix is made by combination ofvarious gear pairs in certain stages.

Those combinations providing gear trains that cannot function or gear trains that will surely be worse by all selection criteriaare eliminated at the beginning.

The designation of gear train concept provides the information about: number of stages, type of gear pair (S spur gear,B bevel gear or W worm gear) in each stage, position of axes of input and output shafts (P parallel, I intersecting,A skew or C coincident), direction of rotation (+, or +/) as well as the position of intermediate shafts that isdefined by the number showing the number of planes in which the axes of all shafts lie. Selection matrix can be summarizedas in Fig. 1. In this manner, apart from ordinal number, name, sketch and designation of the gear train, the following can beadded to the table: positions of shaft axes, number of stages, gear ratio (u) that can be achieved, approximate efficiency ()and direction of rotation. Information about other concepts of gear trains can be seen in the Appendix A, but the focus of thispaper is turned to gear trains with spur gears.

2 N. Marjanovic et al. / Mechanism and Machine Theory 53 (2012) 116

A selection matrix with twenty-eight concept versions is developed as a part of this research, and the methodology of formingof these versions enables its expansion.

3.2. Selection of acceptable concepts of gear trains with spur gears

Selection matrix can be used for the selection of acceptable concepts of gear trains with spur gears, and factors that influencethe selection of concepts are used as constraints. The procedure includes the selection of concepts that satisfy the first adoptedconstraint and then the concepts satisfying the second constraint are chosen from them and so on. When all constraints areconsidered, a set of acceptable gear trains with spur gears is created which can be smaller or larger, depending on adoptedconstraints. Software following previously defined procedure is developed for the selection of acceptable concepts. During thedevelopment of the software, the following constraints were considered: position of input and output shaft axes, position ofintermediate shaft axes, maximum number of stages, gear ratio and direction of rotation. Designer manually enters theconstraints, while the software offers acceptable concepts after each entered constraint. Finally, we get the concepts that satisfyall set constraints.

3.3. Selection of optimal concept of a gear train with spur gears

If the set of acceptable concepts contains several members, this means that there are several concepts of gear trains with spurgears that satisfy all given constraints. To select the optimal concept of a gear train with spur gears it is necessary to do theoptimization of each acceptable concept and select the one that gives the best value of objective function for the chosen criterion,or for several criteria in case of multi-objective optimization [13,14].

To speed up the process of selection of optimal concepts, a continuous optimum for the chosen criterion can be adopted as ameasure of comparison of qualities of individual concepts. This method does not include the solution of problems with mixedvariables (i.e., continuous, integer and discrete), which can significantly speed up the process. Achieved continuous optima areapproximate, but they can be used for comparison of individual concepts.

Special software for the selection of optimal concept was developed. At this level, it is convenient to give the designer(decision maker) a possibility to change the solution suggested by the computer. In this way and with the possibility of changingthe set of acceptable concepts, the designer is actively involved in the process of decision making and is the key factor of the

Fig. 1. Selection matrix of gear train concepts.

3N. Marjanovic et al. / Mechanism and Machine Theory 53 (2012) 116

process. The information given by the computer is used as an objective estimation of certain factors that influence the selection ofgear train concept.

4. Selection of optimal parameters of a gear train with spur gears

4.1. Selection of optimal materials

In the process of optimization, materials that will provide maximum load carrying capacity of flanks and load carrying capacityof roots should be chosen for spur gears, which would enable reaching the optima by various criteria. For the case of pinion, thiscan be achieved by maximization of expression:

minS p HSHmin

;S p FSFmin

( )1

and for the case of gear:

minS g HSHmin

;S g FSFmin

( )2

where: SH and SF are factor of safety from pitting and factor of safety from tooth breaking, respectively, of pinion and gear, andSH min and SF min are their minimum recommended values.

SH and SF are determined according to standard calculations [15]. Maximization of expressions (1) and (2) ensures maximumload carrying capacity of teeth of both spur gears by criterion of tooth surface strength and tooth bending strength. Depending onthe purpose of the gear train with spur gears and conditions under which it functions, one can aim to achieve greater toothsurface strength or tooth bending strength. This is realized by the selection of values for SH min and SF min [16]. By introducingSH min and SF min into decision making of the material selection, wanted ratio of tooth surface strength and tooth bending strengthis maintained in this phase of optimization, as well. Suggested approach enables the inclusion of costs in the process ofoptimization as the only criterion or as one of the criteria. To obtain correct results according to this criterion, it is necessary touse reliable and current data about costs, which do not depend only on technical conditions. Optimization criteria used in thispaper generally are not opposed to costs criterion.

4.2. Selection of optimal gear ratios for gear trains with spur gears

In optimization of multi-stage gear trains it is important to select the number of stages and to properly distribute gear ratio toindividual stages [17]. For the criterion of minimum weight of spur gear pairs, it is possible to determine optimal gear ratios withLagrange Multipliers method [12].

The volume of spur gear pair (Fig. 2) can be determined as follows:

V b4 k

p ol d

p 2 k g ol d g 2

3

where: d(p) and d(g) are pitch diameters, and kol(p)

and kol(g)

are mass reduction factors of pinion (p) and gear (g) and b is gear width.Mass reduction factor is the ratio of approximate volume of spur gear and theoretical volume of the gear, i.e. the volume ofcylinder encompassing the gear.

As shown in detail in the literature [12], applying Lagrange Multipliers method gives the following equation:

2kru3k kru2k1u2k

kru3k1 kru2k12

u2k1; 4

where kr is the ratio kol(p)/kol

(g)that can be defined as a relative reduction factor of gear in relation to pinion. Based on thorough

analysis, it can be said that the value of reduction factor is between 0.3 and 0.9 [12]. Reduction of gear is greater than thereduction of pinion, so approximate value kr=0.7 can be adopted as a relative reduction factor. System of nonlinear Eqs. (4) issolved numerically.

Optimal number of stages is determined by varying index k in Eq. (4) and creating various systems of equations. By solving thesystem of equations and comparing the objective function values (relative volumes), the number of stages that provides itsminimum value is adopted. Fig. 3 shows the distribution of gear ratio to individual stages. This figure was created by varyingtotal gear ratio, with determined optimal number of stages and optimal gear ratio in each stage.


Fig. 3. Distribution of gear ratio of multi-stage gear train.

Fig. 2. Spur gear pair.


4.3. Selection of optimal position of shaft axes of gear trains with spur gears

Gear trains are usually made with shaft axes in the same (most commonly horizontal) plane. When the position of shaft axes ischanged, the space that gear pairs occupy and the dimensions of gear train are changed as well. Optimization of gear pairs fromthe point of view of minimization of center distance has been presented in papers [18,19].

To define mathematical model, multi-stage gear train with spur gears (Fig. 4) is observed. Origin is placed in the input shaftaxis and the position of other axes of the shaft is defined by angles i in relation to x axis.

Dimensions of space occupied by gear train (L and H) can be defined as follows:

L XRXLH YTYB 5

where:

XR max XRif g;XL min XLif gYT max YTif g;YB min YBif gi 1;2;;n:

6

Coordinates of points xi and yi are determined on the basis of geometry shown in Fig. 4.In this way, L and H can be defined and, using them, the objective functions describing the criteria of length, height or volume

of a gear train with spur gears are defined as well. Changing the angle i, changes the position of the spur gear in space, therefore itis necessary to introduce additional constraint that enables the assembly of the spur gear i.e. that prevents the gear from catchingon non-neighboring shaft.

In this paper an example is presented that shows the importance of optimization done in the previously defined manner.Objective function that is the product of length, width and the height of a gear train with spur gears is applied in thisexample.

Angles defining the position of shaft axes (1 and2) are used as optimization variables, while the dimensions of spur gears aretreated as parameters that do not change during the process of optimization. Complex BOX method is used for solving of thisoptimization task. Fig. 5a shows starting concept of a gear train with spur gears and Fig. 5b shows the concept with optimal

Fig. 4. Selection of optimal position of shaft axes of gear train with spur gears.


position of shaft axes from the perspective of minimization of total volume of the space occupied by the gear train with spur gears.This resulted in the following optimal values of optimization variables:

1 53:874 and 2 7:448:

With these values of angles 1 and 2 and with other measurements of the spur gears unchanged, the volume of the spaceoccupied by the gear train with spur gears is decreased by 22.5%, which is not insignificant.

5. Optimization of gear train elements

Optimization of gear pairs provides greatest possibilities in the optimization of gear trains. There are numerous reasons forthat and the most important are:

- Gears in gear trains carry the biggest load; they are the most complex and the most expensive elements of the whole geartrain.

- Gear pairs are defined by a large number of parameters, such as the type of gearing, number of teeth, module, gear width, helixangle, etc.

- Optimization of gear pair changes the conditions under which other elements of gear train function.- Objectives (criteria) of gear pairs optimization are not inconsistent, which means that the achievement of an objective in gearpair optimization does not (drastically) deteriorate that same objective in the optimization of the whole gear train.

Other gear train elements can be optimized as well, such as: shafts (that should have minimum weight in any case), housings(the dimensions of which depend on the dimensions of other elements), bearings (that are standard elements), as well as otherstandard elements such as: pistons, screws etc., but the possibilities of saving are significantly less than in gear train optimization.

5.1. Mathematical model of optimization of a gear train with spur gears

To define a mathematical model, it is necessary to mathematically define optimization criterion by objective function and todefine constraints by constraint function. Set of optimization variables, objective functions and the sets of constraints depend ongear train concept. Due to that, it is convenient to use matrices of variables, objective functions and constraints for the definitionof mathematical model.

Set of optimization variables is defined by vectors Xi that depend on the given concept of a gear train with spur gears i.e.: Xi={x1,x2,,xni}, where ni is the number of optimization variables for i-th concept.

5.1.1. Objective function matrixIt is convenient to form objective function matrix F from the sets of variables for each gear train concept. The elements of this

matrix are functions: fkj,k=1,2,,nkp,j=1,2,,nko, with: k index of gear train concept (nkp number of concepts taken intoconsideration), and j index of optimization criterion (nko number of optimization criteria taken into consideration).

5.1.2. Constraint matrixSimilar to objective function, it is convenient to form a constraint matrix G with constraint functions as its members: gkl,

k=1,2,,nkp,l=1,2,,nog, with: l constraint index (nog number of constraints taken into consideration). It is alsoconvenient to divide constraint function matrix into three parts, with the following equation as the result: G=Go+Gg+Ga,with: Go obligatory constraint matrix, Gg constraint matrix for given concept, Ga additional constraints matrix.

Fig. 5. Concept of a gear train with spur gears, before and after optimization.


Adaptive mathematical model of gear train optimization can be summarized in a table. For the gear train concepts defined inselection matrix it is possible to define objective functions, sets of variables, sets of constraints (sets of explicit constraints, setsof obligatory constraints and sets of constraints for given concept). Adaptive mathematical model is used for the developmentof software for gear train optimization. Within the research, method Complex BOX was used to solve the problem of gear trainoptimization. The problem of objective function minimization was solved with this method, and, in that, the optimizationvariables can be submitted to explicit and implicit constraints. Lower and upper limits of variables can be constant or arbitraryfunctions of optimization variables. If, in a specific problem, some variables are theoretically unlimited, so called safety limitsare adopted, and this makes the application of this method possible. Safety limits are of values that the specific variable cannotreach during the process of optimization.

Previously defined mathematical model can also be used for multi-objective gear train optimization. In case of multi-objectiveoptimization, it is necessary to select several objective functions from the set of objective functions and those functions have tomeet chosen criteria. In this case, the set of constraints is determined in the same manner as in single-objective optimization.To completely define mathematical model for multi-objective optimization, it is necessary, apart from objective function andset of constraints, to define the importance (preference) of individual optimization criteria in a certain manner i.e. it is necessaryto define weight coefficients. Numerous methods and techniques are developed for solving the problem of multi-objectiveoptimization. Marler and Arora [20] analyzed the problem of multi-stage optimization and the possibility of application oflarge number of methods in engineering.

5.2. Software

Optimization of gear trains is an optimization task that cannot be performed without a computer. This paper describes acomprehensive approach to this problem, which is the base of software GTO. The source code currently contains approximately7000 lines. General flowchart for this software is shown in Fig. 6.

Computer programs are developed in programming language C (C++). Due to the length of the source code, the program issplit to several smaller programs connected into one project.

Each program contains several functions that perform similar tasks. In this way, the program is developed faster and morecomprehensible source code is created. Apart from that, individual functions (or whole programs) can be used for solving ofother optimization tasks as well.

Optimization of gear trains is divided into two levels. Optimal concept is selected at the first level and optimal parameters ofselected concept are selected at the second level.

Fig. 7 shows a view of software GTO output screen.

5.3. An example of selection of optimal concept and the optimization of a gear train with spur gears

For the optimization of gear trains with spur gears, it is necessary to select acceptable concepts initially and then the optimalconcept is selected. Adopted constraints, which are the basis of selection of acceptable concepts, are shown in Table 1. There are11 acceptable concepts for the first criterion; this number is then reduced to only three concepts that satisfy all given criteria(Table 1).

Fig. 6. General flowchart for GTO software.


To select the optimal concept of a gear train with spur gears, it is necessary to define optimization criterion as well. In thiscase, the criterion is the length of the gear train with spur gears. This means that the objective function is expressed throughoptimization variables, such as pinion and gear diameters, as well as recommended values for the distance from the spur gearto the housing. The shape of the objective function is different for different concepts. Objective function for concept 4 is:

f 4 a1 a2 d p 12

dg 22

2c2, and for concepts 15 and 16: f 15;16 a1 a2 a3 d p 12

dg 32

2c2, where a is center distance,d(p), d(g) pitch diameters of pinion and gear, c2 minimum distance between gears and housing. Constraints for concept 4are:

- Explicit constraints: standard module 1 mmmn20 mm, number of teeth of pinion 13z(p)50, number of teeth of gear13z(g)500, gear width 0.4d(p)b1.6d(p), helix angle 030, profile shift coefficient 0.5x1.0, angle definingthe position of shaft axis 0360.

- Obligatory constraints: safety factor for tooth breakage of pinion (SFi(p)) and gear (SFi

(g)) in relation to minimal value (SF min)

SFi(p)SF min and SFi

(g)SF min, i=1,,nns, constraint safety factor for pitting of pinion (SHi(p)) and gear (SHi

(g)) in relation to

minimal value (SH min) SHi(p)SH min and SHi(g)SH min, i=1,,nns, where nns is the number of stages.

- Additional constraint:d g a12a2c2dsh3.

Explicit and obligatory constraints for concept 15 are the same as for concept 4, and there is additional constraint that is

introduced so that the gear of the second spur gear pair does not come into contact with the third shaft:d g a22a3

dsh42c2,

where dsh4 is the fourth shaft diameter.For concept 16, apart from explicit and obligatory constraints (as in concept 4), an additional constraint is introduced, so that

the gear of the first spur gear pair does not come into contact with the pinion of the third spur gear paird g 12a2

d p 32c2.

Table 1Selection of the set of acceptable concepts of a gear train.

Selection criterion Acceptable concepts No. of accept. conc.

Position of axes of input and output shafts Parallel 1, 4, 5, 13, 14, 15, 16, 18, 19, 26, 27 11Position of axes of intermediate shaft In one plane 4, 15, 16, 26, 27 5Maximum number of stages 3 4, 15, 16 3Gear ratio 40 4, 15, 16 3Direction of rotation Arbitrary 4, 15, 16 3

Fig. 7. Output screen of gear train optimization software GTO.


Optimization according to this criterion is done for each acceptable concept and in that, only continuous optima aredetermined. The values of these optima are shown in Table 2. It is obvious that, according to this criterion, the best value ofobjective function is realized by concept number 15, so it is adopted as the optimal concept. Mixed variables problem is solvedby the use of Branch and Bound method. Branching for discrete values of the module for all three stages is done for this conceptand Table 2 shows these values. Real values of optimization variables are adopted based on discrete values. Table 2 shows thevalues of optimization variables (mn module, z1 number of pinion teeth, z2 number of gear teeth, b gear width) for allthree stages (I, II, III) of gear trains with spur gears.

Fig. 8 shows the sketches of optimal designs of gear trains with spur gears for all acceptable concepts of the gear train. Sketchesfor concept numbers 4 (Fig. 8a) and 16 (Fig. 8b) are given on the basis of continuous optima, and for concept number 15 (Fig. 8c),which is adopted as the optimal one, the sketch is given based on real values of optimization variables [12].

6. Conclusion

Gear pairs are the most important elements of gear trains. This paper presents one practical approach to optimization of geartrains with spur gears, which was the basis for the development of GTO software. Two examples of gear train with spur gearsoptimization are shown using this software. First example shows the optimization of position of shaft axes for the purpose ofreducing the volume occupied by the gear train with spur gears. The results show that the volume of the gear train with spurgears is reduced by 22.5%. Software GTO accomplishes needed results in a very short time.

The process of designing of gear trains with spur gears is performed by using two operations. Versions of concepts ofgear trains with spur gears are developed within the first operation and the selection of optimal concept and their optimalparameters are selected within the second operation. The second example shows the complete procedure run during the

Fig. 8. Acceptable concepts of gear trains with spur gears.

Table 2Example of selection of optimal concept and the optimization of gear train with spur gears.

Gear train concept no.: 4, 15, 16 Optimization criterion: length of gear train with spur (no. 3) [mm]Input data:Power: P=40 kW, Total gear ratio: utot=40 Number of variables: 8/12

Number of constraints: 28/40Input speed (RPM): n1=1450 rpm Available materials: 14CrNi6Number of points in complex: 16/24

Set of variables Optimal valuesContinuous optima For optimal concept (no. 15)Conc. no. 4 Conc. no. 15 Conc. no. 16 Discrete Real

I mn 2.07 1.669 2.492 2 2z1 22.69 28.868 20.087 23.86 24z2 185.476 123.039 71.794 110.163 110b 65.0 76.436 76.455 76.347 76.4II mn 4.667 3.524 3.251 3.0 3z1 18.316 20.025 22.334 24.142 24z2 86.036 70.104 85.923 84.184 84b 107.059 100.00 109.483 115.881 116III mn 4.107 3.702 4.5 4.5z1 24.108 27.505 23.471 23z2 62.095 76.934 56.217 56b 157.472 146.192 139.183 140Objective function value 714.169 644.241 681.24 656.164


optimization of a gear train with spur gears, from the selection of acceptable concepts, through the optimization of eachacceptable concept to the selection of optimal concept with software GTO. Depending on set constraints, it is possible toselect the set of acceptable concepts using GTO, which is later used as a basis for the selection of optimal concept. Thiskind of approach can also be applied to gear trains with bevel and worm gear pairs, but this is not shown here, as it doesnot fall within the subject of this paper. The optimization of parameters of gear trains with spur gears is done in this mannerand the results can be applied in practice. GTO can be used for problems in single-objective optimization, as well as for thosein multiple-objective optimization.

Nomenclaturea center distanceb gear widthc2 minimum distance between gears and housingd(p),d(g) pitch diameters of pinion, gearda(p),da(g) diameter of the addendum circle of pinion and gear

dsh shaft diameterF objective function matrixfkj elements of objective function matrixG constraint matrixGa additional constraints matrixGg constraint matrix for given conceptgkl constraint functionsGo obligatory constraint matrixH height of space occupied by gear trainj index of optimization criterionk index of gear train conceptkol(g)

mass reduction factor of gearkol(p) mass reduction factor of pinion

kr relative reduction factorl constraint indexL length of space occupied by gear trainm moduleni number of optimization variables for i-th conceptnkp number of concepts taken into considerationnko number of optimization criteria taken into considerationnns number of stagesnog number of constraints taken into considerationP power to be transmitted (kW)SH(p),SH

(g) factor of safety from pitting of pinion, gearSH min minimum prescribed value of factor of safety from pittingSF(p),SF

(g) factor of safety from tooth breaking of pinion, gearSF min minimum prescribed value of factor of safety from tooth breakingu gear ratioutot total gear ratioV volume of gear pairx profile shift coefficientXi set of optimization variablesz(p),z(g) number of teeth on pinion, gear helix angle efficiencyi position of shafts defined in relation to origin and x axis

AbbreviationsA skew shaft axesB bevel gearC coincident shaft axesGTO Gear Train OptimizationI intersecting shaft axesP parallel shaft axesS spur gearW worm gear


Appendix A. Selection matrix of gear train concept (Complete)


Appendix A (continued)


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A practical approach to the optimization of gear trains with spur gears1. Introduction2. Characteristics and problems of optimization of gear trains with spur gears3. Selection of optimal concept of gear trains with spur gears3.1. Concepts selection matrix3.2. Selection of acceptable concepts of gear trains with spur gears3.3. Selection of optimal concept of a gear train with spur gears

4. Selection of optimal parameters of a gear train with spur gears4.1. Selection of optimal materials4.2. Selection of optimal gear ratios for gear trains with spur gears4.3. Selection of optimal position of shaft axes of gear trains with spur gears

5. Optimization of gear train elements5.1. Mathematical model of optimization of a gear train with spur gears5.1.1. Objective function matrix5.1.2. Constraint matrix

5.2. Software5.3. An example of selection of optimal concept and the optimization of a gear train with spur gears

6. ConclusionNomenclatureAbbreviationsReferences

a practical approach to the optimization of gear trains with spur gears

Documents

optimization of gear

volume of gear trains

designing of gear trains

optimization of gear

introductiongear trains

example of optimization

dimensional optimization

spur gearsnenad marjanovic