# Optimising involutes asymmetrical teeth gears software.

1. Introduction

1.1 Nomenclature

A few years ago the aim of design engineers in gear transmission was to obtain large quantities of products at low costs. Now it is more important to obtain better performances in function by means of optimal design. The tendencies in present toothed gear research are aimed to obtain best performances of the transmissions by using special gearing with non-standard parameters.

One of the methods to add supplementary advantages to those already known of the involutes gears, by improving the load capacity and efficiency (Kapelevich, 2008) and reducing the transmission error and vibrations (Karpat et al., 2006), is to use the asymmetric gears (Novikov et al., 2008).

These special gears are characterized by asymmetric involutes profiles of the tooth, having different diameters for the base circles of the opposite profiles (Litvin et al., 2000).

By using the computational method of design, taking into consideration the great numbers of geometrical parameters in relation with the classical gears, the difficulties in manufacturing asymmetric gears can be solved without significant supplementary costs.

There are authors who study and emphasize the advantage of gears with the coefficient of asymmetry bigger than the unit and others who emphasize the advantage of using gears having the coefficient of asymmetry smaller than the unit.

In this paper, the authors consider that an asymmetric gear transmission, formed by two gears with asymmetric teeth profiles, with geometric parameters established, represent two gear transmissions with different performances, depending on the profile chosen as active profile (Chira & Banica, 2007).

[FIGURE 1 OMITTED]

Two circular arcs, on the addendum circle and on the dedendum circle, limit an involute tooth with asymmetrical profiles and lateral by two asymmetrical involutes arcs and two fillet curves of the involutes arcs with the root circle.

The profile called "direct profile" belongs to the positive side of an involute with the base circle diameter dbd--the direct profile base circle diameter.

The profile called "inverted profile" belongs to the negative side of an involute with the base circle diameter dbi--the inverted profile base circle diameter.

In what follows, the direct profile elements are attributed the index "d" and the inverted profile elements are attributed the index "i" (Fig. 1).

If the direct profile is the active one, the gear is called "direct asymmetric gear", having k>1. If the inverted profile is the active one the gear is named "inverted asymmetric gear", having k<1. With "k" has been noted the coefficient of asymmetry, defined as the ratio between the base circle diameter of the inactive flank and the base circle diameter of the active flank (Litvin et al., 2000).

1.2 Stages of the research

The authors' research developed on the asymmetric gears has five important stages what will be shortly presented as follows:

1) The determination of the algorithms and realisation of the routines for calculating the geometrical parameters of the asymmetric gears.

2) The realisation of the programs necessary to representing the asymmetric gear, in order to obtain the 2D and 3D models of the pinion and of the gear.

3) The determination of the algorithms and setting up of the routines for calculating and determining the variation during the meshing cycle of some important functional parameters. Some of the functional parameters that can be determined with the developed applications are: the elasticity, implicitly the rigidity, of the asymmetric tooth and of the pairs of teeth in contact; the normal force on the active profile; the transmission error; the deformations of the teeth; the relative sliding speed and the specific sliding between flanks; the instantaneous efficiency and then the medium efficiency for a meshing period; the bending stress at the bottom of the tooth for pinion and for the gear and the contact stress. All the parameters are determined for the "j" point of contact, whose number, during the meshing cycle, can be established by the designing engineer. With the values obtained for all the points of contact are represented the variation diagrams during the meshing cycle. The significant values from these diagrams, the maximum values or the minimum values have been used as quality indicators of the asymmetric gears, in order to compare different possible solutions for the given initial data.

4) The realisation of multiple routines for studying the variation of the asymmetric gear performances in relation with different designing variables (Chira et al., 2008).

5) The setting up of an optimising programme, based on genetic algorithms, with one or multiple objective functions, for optimal design of the asymmetric gears. In this part, the functional parameters, determined in the mentioned stage, in anterior routines of the computer application, are used single as evaluation function or in combination to obtain the multiple objective evaluation functions.

2. The geometrical parameters

2.1 Determination of the geometrical parameters

The method developed to determine the geometrical parameters of the asymmetrical gears, uses the "direct design" of the spur gears. This method offers the possibility of determining first the parameters of the gears, followed by the determination of the asymmetric gear rack's parameters, by using those of the gears (Kapelevich, 2008).

Based on a large study on fundamental theoretical aspects about asymmetric gears, a MATLAB application has been written to determine all the geometrical parameters of the pinion and gear that compose an asymmetric spur gear transmission. For a given centre distance "a", number of teeth "[z.sub.1]", "[z.sub.2]" as initial data, the design engineer chooses, as designing variables, the pressure angle for the direct profile " [[alpha].sub.wd]" and the pressure angle for the inverted profile "[alpha].sub.wi]. Implicitly the coefficient of asymmetry of the tooth is also chosen. Another designing variable is the "f" coefficient, which is called coefficient of modification of the gear rack angle. The value of the "f" coefficient shows how much the angle of the direct gear rack profile "[alpha].sub.dc] is different from the pressure angle of the direct profile of the " a wd "gear. The shift rack of the pinion and the shift rack of the gear is determined by the "[alpha].sub.dc] angle of the rack. So, we can also establish the ratio of the pinion tooth thickness on the pitch circle and gear tooth thickness on the pitch circle [S.sub.w1] / [S.sub.w2].

2.2 The asymmetric profile involutes and filet curve parametrical equations

With the aim of representing the asymmetric gear transmission, we have written the equations of the direct profile curves [E.sub.vd], [R.sub.d] in relation with the reference system xOy, and of the inverted profile curves [E.sub.vi], [R.sub.i] in relation with the reference system x'Oy' (Fig. 2).

[FIGURE 2 OMITTED]

Both reference systems have as y respectively y' axis the radius that goes through the tip of the tooth, the only common point of the two involutes profiles. Having the parametrical equations of the involutes and filet curves of the direct profile, also of the inverted profile, and developing a MATLAB application on this basis, one can obtain the graphical representation of the four curves for the pinion and also for the gear (Fig. 3).

[FIGURE 3 OMITTED]

2.3 The 3D model of the asymmetric gear

To carry out the representation of the gearing in AutoCAD Mechanical (Fig. 4), two programmes have been developed, as applications in Auto LISP, for the exact representation of the pinion and of the gear. In the gearing representation, the model of the pinion and the model of the gear have the teeth in contact in the pitch point, on the direct profile. In these programmes, the representation of the curves that limit the tooth body is performed on the basis of the parametrical equations of these curves.

[FIGURE 4 OMITTED]

The geometric parameters which are used as input data in the Auto LISP applications, one can obtain by means of the developed MATLAB application: the numbers of the teeth; the centre distance; the meshing angles on the direct and inverted profiles; the profile angles of the asymmetric generation gear rack; the profile angles on the addendum circles, on the direct teeth profiles, for the pinion and for the gear; the tip radius of the gear rack, or racks in the case of generation pinion and gear with two different racks; the pinion shift rack value.

The models of the gear can be used in 3D Max program for the simulation of the meshing. Method analysis of the stress and displacements is also necessary for the finite elements (Fig. 5, Fig. 6).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The model of the gear obtained as drawing file can be transformed in initial graphics exchange file. So in the soft of coordinate manufacturing machine, the obtained data will be transformed in machine language and thus all the data necessary for manufacturing are provided.

3. The functional parameters

One can rapidly evaluate the behaviour of the designed asymmetric gear under the load by algorithmic calculus and the MATLAB applications that solve step by step the following aspects:

(1) Mathematical modelling for the meshing of the asymmetric gears by determining the profile angles for the pinion tooth and for the gear tooth for an established number of contact points for a meshing period.

(2) The elasticity, implicitly the rigidity, of the asymmetric tooth and of the pairs of teeth and the variation of these parameters during the meshing cycle (Fig. 7).

(3) On the basis of tooth elasticity the statically unknown problem of load distribution between the two pairs of teeth in meshing could be solved and so the diagram of variation of the normal force and then the variation of the transmission error were determined (Fig. 8).

(4) The relative sliding speed and the specific sliding variation during the meshing period (Fig. 9).

(5) The instantaneous power loses and the variation of the instantaneous efficiency (Fig. 9), then the medium efficiency for a meshing period.

(6) The bending stress at the bottom of the tooth in relation with the number of the contact points. And thus has resulted the variation during the meshing cycle of the bending stress (Fig. 10).

(7) Also, the contact stress has been determined for every contact point and the diagram of variation for a meshing cycle has been represented (Fig. 10).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

In the figures 7-10 are presented the variations of the functional parameters during the meshing cycle, or for the instantaneous efficiency only for the meshing period, for an asymmetric gear designed with the following initial data: number of teeth, [z.sub.1] = 15, [z.sub.2] = 60, centre distance a = 120mm, direct profile pressure angle

[[alpha].sub.wd] = 40[degrees], inverted profile pressure angle [[alpha].sub.wi] = 20[degrees], transmitted power P = 18kW, rotation speed n = 1000rot/min .

On the horizontal axis of these diagrams are given the numbers "j" of the contact points. The first point of contact is the point A on the line of action, at the bottom of the pinion tooth active profile. The last point of contact is the point E on the line of action, at the bottom of the gear tooth active profile. For the distance on the line of action equal with the pitch on the base circle the number of point of contact are 200. For each number "j", we can find out the corresponding values of the profile angles for the pinion tooth and for the gear tooth.

All the presented diagrams have been obtained with the application developed for the direct asymmetric gears, having the direct profile as the active one. The variation of the same functional parameters can be obtained for the inverted asymmetric gear, with the MATLAB application developed for this case, resulting different values.

Considering as significant values that can be used as transmission quality indicators the maximum values of transmission error, specific sliding, bending stress, contact stress or the value of the medium efficiency, we can compare gears that have been designed as different solutions for the same problem having the possibility to emphasize which the best one in the relation of the beneficiary requests is.

4. Applications for analysing and comparing the asymmetric gears performances

4.1 The variation of gear parameters in relation with the asymmetric generation gear rack's angles

For a given centre distance, number of teeth and pressure angles, the sum of the shift rack of the pinion and the shift rack of the gear is determined by the "[[alpha].sub.dc]" angle of the rack. The maximum value for the "[[alpha].sub.dc]" profile angle of the rack is equal (Kapelevich, 2008) to the pressure angle for the direct profile "[[alpha].sub.wc]"; the minimum value is determined by the coefficient of asymmetry [[alpha].sub.dcmin] = arccos(1 / k).

The shifts are established in the computer program, finding the values that satisfy the conditions to avoid the interference, to avoid the undercutting, to ensure the contact ratio greater than one. Because the rack's angle for the active profile is considered as a designing variable that one can choose between minimum and maximum values by introducing the value of "f" coefficient in the program, it is necessary to study how this angle influences the geometrical and functional parameters of the gears. The developed applications also make possible the study of the influence of generating the pinion and the gear with different racks, or with the same rack.

In figures 11 and 12 is given information that shows the variation of the bending stress for different values of the gear rack direct profile angle, for one or two generating gear racks. For the presented examples have been used the same initial data mentioned before [z.sub.1] = 15, [z.sub.2] = 60, a = 120 mm, n = 1000rot/min , P = 18kW ,

[[alpha].sub.wd] = 40[degrees], [[alpha].sub.wi] = 20[degrees]. The different values for the "[alpha].sub.dc] angle have been obtained for the following values chosen for the coefficient of modification of the gear rack angle f = [1, 5, 9, 13, 17].

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

We can see that the maximum bending stress to the pinion increases, in the same time to the gear decreases, having more close values by choosing bigger values for the gear rack angle. Using different generation gear racks is advantageous.

Also, the influence of the generation gear rack profile angle on the functional parameters of the gears for different coefficients of asymmetry can be studied. In figures 13 and 14 are presented examples of results obtained for [z.sub.1] = 15, [z.sub.2] = 60, a = 120 mm, n = 1000rot/min, P = 18 kW, awd = [30[degrees], 32[degrees], 34[degrees], 36[degrees], 38[degrees], [[ALPHA].SUB.WI] = 20[degrees], f = [2, 4, 6, 18, 10].

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

4.2 The variation of gear parameters in relation with the pressure angles of the asymmetric profiles

In order to make a rapid comparison between the functional parameters of the direct asymmetric gear, inverted asymmetric gear or symmetric classical gear, to choose which the best solution for one particular request is, it is possible to use a developed application that allows us to carry out on the same diagram the variation of some parameter for all three cases of transmissions designed for the same initial data.

In figure 15 is given the above-mentioned comparison, having as evaluation function or quality indicator the medium efficiency, for gears designed for [z.sub.1] =13, [z.sub.2] = 60, a = 100 mm, n = 1000rot/min, awd = [34[degrees], 35[degrees], 36[degrees], 37[degrees], 39[degrees], P = 18kW, k= 1,15. The horiZontal line corresponds to the value of efficiency of the classical symmetric gear generated with the gear rack with a 20-degree profile angle.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

There is also the possibility of evaluating, in the same time, the influence on the performances of transmission of the coefficient of asymmetry, for both usage cases of the asymmetric gear, with k>1 or k<1. In the programme, the coefficient of asymmetry is established by choosing the different pressure angles of the asymmetric profiles. In figure 16 is presented such a diagram obtained for gears designed with the following initial data: [z.sub.i] = 16 , [z.sub.2] = 57 , a = 120 mm, n = 1000 rot/min, P = 18kW ,

[[alpha].sub.wi] = 20[degrees], [[alpha].sub.wd] = [20[degrees], 25[degrees], 30[degrees], 35[degrees], 40[degrees].

The intersection point of the lines corresponds to the medium efficiency of the symmetrical gear with the pressure angles equal to 20 degrees. The comparison can be made having as quality indicator those that are important for the beneficiary.

In figures 17 and 18 are presented examples of results obtained with an application developed for analysing the functional parameter variation by modifying both pressure angles. These have been obtained by introducing the initial data in the programme: [z.sub.1] = 15 , [z.sub.2] = 60 , a = 120 mm , P = 18kW , n = 1000rot/min and [[alpha].sub.wd] = [30[degrees], 32[degrees], 34[degrees], 36[degrees], 38[degrees], [[alpha].sub.wi] = 22[degrees], 24[degrees], 26[degrees], 28[degrees], 30[degrees] as designing variables.

The aim of the examples presented is to show how many data about the influence of designing variable on the transmission performances can be obtained very easily and in a short time by using specialiZed applications. It is necessary to establish those variables, in relation with the designed transmission beneficiary requirements such as the increase of efficiency, reduction of the bending stress, reduction of the contact stress, minimiZing the transmission error.

5. Conclusions

The packet of original software for asymmetric gears design and study, developed by the authors of this paper, and the few examples from several other results, that any design engineer can carry out as a user, emphasiZe the advantages of the computational method for designing, modelling, evaluating and establishing the best design solutions. By using the above-described method of design, the difficulties in manufacturing asymmetric gears can be solved without significant supplementary costs. The domain of possible solutions being very wide, a routine for optimal design based on genetic algorithms has also been developed in order to establish which the best is. The future research will be focused on the asymmetric gears dynamics.

DOI: 10.2507/daaam.scibook.2009.38

6. References

Chira, F. & Banica, M. (2007). Studies on the gears with asymmetric teeth, Risoprint

Cluj-Napoca, ISBN 978-973-751-495-0, Romania Kapelevich, A.L. (2008). Direct Design Approach for High Performance Gear Transmissions, Gear Solution, January 2008, 22-31. Available from: http://www.akgears.com Accessed: 2009-07-05

Karpat, F., Cavdar, K. & Babalic, C.F. (2006). An investigation on analysis of involutes spur gears with asymmetric teeth: Dynamic load and Transmission errors, Proceedings of the 2th International Conference "Power Transmissions 2006", pp. 69-74, ISBN 86-85211-78-6, Novi-Sad, April 2006, Serbia and Montenegro

Litvin, F.L., Lian, Q. & Kapelevich, A.L. (2000). Asymmetric Modified Gear Drives: Reduction of Noise, LocaliZation of Contact, Simulation of Meshing and Stress Analysis. Computer Methods in Applied Mechanics and Engineering, No.188, 363-390, ISSN 0045-7825

Novikov, A.S., Paikin, A.G., Dorofeyev, V.L., Ananiev, V.M., & Kapelevich, A.L. (2008). Application of Gears with Asymmetric Teeth in Turboprop Engine Gearbox, Gear Technology, January/February 2008, 60-65

This Publication has to be referred to as: Chira, F[lavia]; Banica, M[ihai] & Lobontiu, M[ircea] (2009). Optimising Involutes Asymmetrical Teeth Gears Software, Chapter 38 in DAAAM International Scientific Book 2009, pp. 363-376, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

Authors' data: Univ. Lecturer Dipl.-Ing. Dr. Chira, F[lavia]; Univ. Assoc. Prof. Dipl.-Ing. Dr. Banica, M[ihai]; Univ. Prof. Dipl.-Ing. Dr. Lobontiu, M[ircea], North University of Baia Mare, Dr. Victor Babes 62A, 430083, Baia Mare, Romania, Flavia.Chira@ubm.ro, Mihai.Banica@ubm.ro, Mircea.Lobontiu@ubm.ro

1.1 Nomenclature

A few years ago the aim of design engineers in gear transmission was to obtain large quantities of products at low costs. Now it is more important to obtain better performances in function by means of optimal design. The tendencies in present toothed gear research are aimed to obtain best performances of the transmissions by using special gearing with non-standard parameters.

One of the methods to add supplementary advantages to those already known of the involutes gears, by improving the load capacity and efficiency (Kapelevich, 2008) and reducing the transmission error and vibrations (Karpat et al., 2006), is to use the asymmetric gears (Novikov et al., 2008).

These special gears are characterized by asymmetric involutes profiles of the tooth, having different diameters for the base circles of the opposite profiles (Litvin et al., 2000).

By using the computational method of design, taking into consideration the great numbers of geometrical parameters in relation with the classical gears, the difficulties in manufacturing asymmetric gears can be solved without significant supplementary costs.

There are authors who study and emphasize the advantage of gears with the coefficient of asymmetry bigger than the unit and others who emphasize the advantage of using gears having the coefficient of asymmetry smaller than the unit.

In this paper, the authors consider that an asymmetric gear transmission, formed by two gears with asymmetric teeth profiles, with geometric parameters established, represent two gear transmissions with different performances, depending on the profile chosen as active profile (Chira & Banica, 2007).

[FIGURE 1 OMITTED]

Two circular arcs, on the addendum circle and on the dedendum circle, limit an involute tooth with asymmetrical profiles and lateral by two asymmetrical involutes arcs and two fillet curves of the involutes arcs with the root circle.

The profile called "direct profile" belongs to the positive side of an involute with the base circle diameter dbd--the direct profile base circle diameter.

The profile called "inverted profile" belongs to the negative side of an involute with the base circle diameter dbi--the inverted profile base circle diameter.

In what follows, the direct profile elements are attributed the index "d" and the inverted profile elements are attributed the index "i" (Fig. 1).

If the direct profile is the active one, the gear is called "direct asymmetric gear", having k>1. If the inverted profile is the active one the gear is named "inverted asymmetric gear", having k<1. With "k" has been noted the coefficient of asymmetry, defined as the ratio between the base circle diameter of the inactive flank and the base circle diameter of the active flank (Litvin et al., 2000).

1.2 Stages of the research

The authors' research developed on the asymmetric gears has five important stages what will be shortly presented as follows:

1) The determination of the algorithms and realisation of the routines for calculating the geometrical parameters of the asymmetric gears.

2) The realisation of the programs necessary to representing the asymmetric gear, in order to obtain the 2D and 3D models of the pinion and of the gear.

3) The determination of the algorithms and setting up of the routines for calculating and determining the variation during the meshing cycle of some important functional parameters. Some of the functional parameters that can be determined with the developed applications are: the elasticity, implicitly the rigidity, of the asymmetric tooth and of the pairs of teeth in contact; the normal force on the active profile; the transmission error; the deformations of the teeth; the relative sliding speed and the specific sliding between flanks; the instantaneous efficiency and then the medium efficiency for a meshing period; the bending stress at the bottom of the tooth for pinion and for the gear and the contact stress. All the parameters are determined for the "j" point of contact, whose number, during the meshing cycle, can be established by the designing engineer. With the values obtained for all the points of contact are represented the variation diagrams during the meshing cycle. The significant values from these diagrams, the maximum values or the minimum values have been used as quality indicators of the asymmetric gears, in order to compare different possible solutions for the given initial data.

4) The realisation of multiple routines for studying the variation of the asymmetric gear performances in relation with different designing variables (Chira et al., 2008).

5) The setting up of an optimising programme, based on genetic algorithms, with one or multiple objective functions, for optimal design of the asymmetric gears. In this part, the functional parameters, determined in the mentioned stage, in anterior routines of the computer application, are used single as evaluation function or in combination to obtain the multiple objective evaluation functions.

2. The geometrical parameters

2.1 Determination of the geometrical parameters

The method developed to determine the geometrical parameters of the asymmetrical gears, uses the "direct design" of the spur gears. This method offers the possibility of determining first the parameters of the gears, followed by the determination of the asymmetric gear rack's parameters, by using those of the gears (Kapelevich, 2008).

Based on a large study on fundamental theoretical aspects about asymmetric gears, a MATLAB application has been written to determine all the geometrical parameters of the pinion and gear that compose an asymmetric spur gear transmission. For a given centre distance "a", number of teeth "[z.sub.1]", "[z.sub.2]" as initial data, the design engineer chooses, as designing variables, the pressure angle for the direct profile " [[alpha].sub.wd]" and the pressure angle for the inverted profile "[alpha].sub.wi]. Implicitly the coefficient of asymmetry of the tooth is also chosen. Another designing variable is the "f" coefficient, which is called coefficient of modification of the gear rack angle. The value of the "f" coefficient shows how much the angle of the direct gear rack profile "[alpha].sub.dc] is different from the pressure angle of the direct profile of the " a wd "gear. The shift rack of the pinion and the shift rack of the gear is determined by the "[alpha].sub.dc] angle of the rack. So, we can also establish the ratio of the pinion tooth thickness on the pitch circle and gear tooth thickness on the pitch circle [S.sub.w1] / [S.sub.w2].

2.2 The asymmetric profile involutes and filet curve parametrical equations

With the aim of representing the asymmetric gear transmission, we have written the equations of the direct profile curves [E.sub.vd], [R.sub.d] in relation with the reference system xOy, and of the inverted profile curves [E.sub.vi], [R.sub.i] in relation with the reference system x'Oy' (Fig. 2).

[FIGURE 2 OMITTED]

Both reference systems have as y respectively y' axis the radius that goes through the tip of the tooth, the only common point of the two involutes profiles. Having the parametrical equations of the involutes and filet curves of the direct profile, also of the inverted profile, and developing a MATLAB application on this basis, one can obtain the graphical representation of the four curves for the pinion and also for the gear (Fig. 3).

[FIGURE 3 OMITTED]

2.3 The 3D model of the asymmetric gear

To carry out the representation of the gearing in AutoCAD Mechanical (Fig. 4), two programmes have been developed, as applications in Auto LISP, for the exact representation of the pinion and of the gear. In the gearing representation, the model of the pinion and the model of the gear have the teeth in contact in the pitch point, on the direct profile. In these programmes, the representation of the curves that limit the tooth body is performed on the basis of the parametrical equations of these curves.

[FIGURE 4 OMITTED]

The geometric parameters which are used as input data in the Auto LISP applications, one can obtain by means of the developed MATLAB application: the numbers of the teeth; the centre distance; the meshing angles on the direct and inverted profiles; the profile angles of the asymmetric generation gear rack; the profile angles on the addendum circles, on the direct teeth profiles, for the pinion and for the gear; the tip radius of the gear rack, or racks in the case of generation pinion and gear with two different racks; the pinion shift rack value.

The models of the gear can be used in 3D Max program for the simulation of the meshing. Method analysis of the stress and displacements is also necessary for the finite elements (Fig. 5, Fig. 6).

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

The model of the gear obtained as drawing file can be transformed in initial graphics exchange file. So in the soft of coordinate manufacturing machine, the obtained data will be transformed in machine language and thus all the data necessary for manufacturing are provided.

3. The functional parameters

One can rapidly evaluate the behaviour of the designed asymmetric gear under the load by algorithmic calculus and the MATLAB applications that solve step by step the following aspects:

(1) Mathematical modelling for the meshing of the asymmetric gears by determining the profile angles for the pinion tooth and for the gear tooth for an established number of contact points for a meshing period.

(2) The elasticity, implicitly the rigidity, of the asymmetric tooth and of the pairs of teeth and the variation of these parameters during the meshing cycle (Fig. 7).

(3) On the basis of tooth elasticity the statically unknown problem of load distribution between the two pairs of teeth in meshing could be solved and so the diagram of variation of the normal force and then the variation of the transmission error were determined (Fig. 8).

(4) The relative sliding speed and the specific sliding variation during the meshing period (Fig. 9).

(5) The instantaneous power loses and the variation of the instantaneous efficiency (Fig. 9), then the medium efficiency for a meshing period.

(6) The bending stress at the bottom of the tooth in relation with the number of the contact points. And thus has resulted the variation during the meshing cycle of the bending stress (Fig. 10).

(7) Also, the contact stress has been determined for every contact point and the diagram of variation for a meshing cycle has been represented (Fig. 10).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

In the figures 7-10 are presented the variations of the functional parameters during the meshing cycle, or for the instantaneous efficiency only for the meshing period, for an asymmetric gear designed with the following initial data: number of teeth, [z.sub.1] = 15, [z.sub.2] = 60, centre distance a = 120mm, direct profile pressure angle

[[alpha].sub.wd] = 40[degrees], inverted profile pressure angle [[alpha].sub.wi] = 20[degrees], transmitted power P = 18kW, rotation speed n = 1000rot/min .

On the horizontal axis of these diagrams are given the numbers "j" of the contact points. The first point of contact is the point A on the line of action, at the bottom of the pinion tooth active profile. The last point of contact is the point E on the line of action, at the bottom of the gear tooth active profile. For the distance on the line of action equal with the pitch on the base circle the number of point of contact are 200. For each number "j", we can find out the corresponding values of the profile angles for the pinion tooth and for the gear tooth.

All the presented diagrams have been obtained with the application developed for the direct asymmetric gears, having the direct profile as the active one. The variation of the same functional parameters can be obtained for the inverted asymmetric gear, with the MATLAB application developed for this case, resulting different values.

Considering as significant values that can be used as transmission quality indicators the maximum values of transmission error, specific sliding, bending stress, contact stress or the value of the medium efficiency, we can compare gears that have been designed as different solutions for the same problem having the possibility to emphasize which the best one in the relation of the beneficiary requests is.

4. Applications for analysing and comparing the asymmetric gears performances

4.1 The variation of gear parameters in relation with the asymmetric generation gear rack's angles

For a given centre distance, number of teeth and pressure angles, the sum of the shift rack of the pinion and the shift rack of the gear is determined by the "[[alpha].sub.dc]" angle of the rack. The maximum value for the "[[alpha].sub.dc]" profile angle of the rack is equal (Kapelevich, 2008) to the pressure angle for the direct profile "[[alpha].sub.wc]"; the minimum value is determined by the coefficient of asymmetry [[alpha].sub.dcmin] = arccos(1 / k).

The shifts are established in the computer program, finding the values that satisfy the conditions to avoid the interference, to avoid the undercutting, to ensure the contact ratio greater than one. Because the rack's angle for the active profile is considered as a designing variable that one can choose between minimum and maximum values by introducing the value of "f" coefficient in the program, it is necessary to study how this angle influences the geometrical and functional parameters of the gears. The developed applications also make possible the study of the influence of generating the pinion and the gear with different racks, or with the same rack.

In figures 11 and 12 is given information that shows the variation of the bending stress for different values of the gear rack direct profile angle, for one or two generating gear racks. For the presented examples have been used the same initial data mentioned before [z.sub.1] = 15, [z.sub.2] = 60, a = 120 mm, n = 1000rot/min , P = 18kW ,

[[alpha].sub.wd] = 40[degrees], [[alpha].sub.wi] = 20[degrees]. The different values for the "[alpha].sub.dc] angle have been obtained for the following values chosen for the coefficient of modification of the gear rack angle f = [1, 5, 9, 13, 17].

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

We can see that the maximum bending stress to the pinion increases, in the same time to the gear decreases, having more close values by choosing bigger values for the gear rack angle. Using different generation gear racks is advantageous.

Also, the influence of the generation gear rack profile angle on the functional parameters of the gears for different coefficients of asymmetry can be studied. In figures 13 and 14 are presented examples of results obtained for [z.sub.1] = 15, [z.sub.2] = 60, a = 120 mm, n = 1000rot/min, P = 18 kW, awd = [30[degrees], 32[degrees], 34[degrees], 36[degrees], 38[degrees], [[ALPHA].SUB.WI] = 20[degrees], f = [2, 4, 6, 18, 10].

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

4.2 The variation of gear parameters in relation with the pressure angles of the asymmetric profiles

In order to make a rapid comparison between the functional parameters of the direct asymmetric gear, inverted asymmetric gear or symmetric classical gear, to choose which the best solution for one particular request is, it is possible to use a developed application that allows us to carry out on the same diagram the variation of some parameter for all three cases of transmissions designed for the same initial data.

In figure 15 is given the above-mentioned comparison, having as evaluation function or quality indicator the medium efficiency, for gears designed for [z.sub.1] =13, [z.sub.2] = 60, a = 100 mm, n = 1000rot/min, awd = [34[degrees], 35[degrees], 36[degrees], 37[degrees], 39[degrees], P = 18kW, k= 1,15. The horiZontal line corresponds to the value of efficiency of the classical symmetric gear generated with the gear rack with a 20-degree profile angle.

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

[FIGURE 18 OMITTED]

There is also the possibility of evaluating, in the same time, the influence on the performances of transmission of the coefficient of asymmetry, for both usage cases of the asymmetric gear, with k>1 or k<1. In the programme, the coefficient of asymmetry is established by choosing the different pressure angles of the asymmetric profiles. In figure 16 is presented such a diagram obtained for gears designed with the following initial data: [z.sub.i] = 16 , [z.sub.2] = 57 , a = 120 mm, n = 1000 rot/min, P = 18kW ,

[[alpha].sub.wi] = 20[degrees], [[alpha].sub.wd] = [20[degrees], 25[degrees], 30[degrees], 35[degrees], 40[degrees].

The intersection point of the lines corresponds to the medium efficiency of the symmetrical gear with the pressure angles equal to 20 degrees. The comparison can be made having as quality indicator those that are important for the beneficiary.

In figures 17 and 18 are presented examples of results obtained with an application developed for analysing the functional parameter variation by modifying both pressure angles. These have been obtained by introducing the initial data in the programme: [z.sub.1] = 15 , [z.sub.2] = 60 , a = 120 mm , P = 18kW , n = 1000rot/min and [[alpha].sub.wd] = [30[degrees], 32[degrees], 34[degrees], 36[degrees], 38[degrees], [[alpha].sub.wi] = 22[degrees], 24[degrees], 26[degrees], 28[degrees], 30[degrees] as designing variables.

The aim of the examples presented is to show how many data about the influence of designing variable on the transmission performances can be obtained very easily and in a short time by using specialiZed applications. It is necessary to establish those variables, in relation with the designed transmission beneficiary requirements such as the increase of efficiency, reduction of the bending stress, reduction of the contact stress, minimiZing the transmission error.

5. Conclusions

The packet of original software for asymmetric gears design and study, developed by the authors of this paper, and the few examples from several other results, that any design engineer can carry out as a user, emphasiZe the advantages of the computational method for designing, modelling, evaluating and establishing the best design solutions. By using the above-described method of design, the difficulties in manufacturing asymmetric gears can be solved without significant supplementary costs. The domain of possible solutions being very wide, a routine for optimal design based on genetic algorithms has also been developed in order to establish which the best is. The future research will be focused on the asymmetric gears dynamics.

DOI: 10.2507/daaam.scibook.2009.38

6. References

Chira, F. & Banica, M. (2007). Studies on the gears with asymmetric teeth, Risoprint

Cluj-Napoca, ISBN 978-973-751-495-0, Romania Kapelevich, A.L. (2008). Direct Design Approach for High Performance Gear Transmissions, Gear Solution, January 2008, 22-31. Available from: http://www.akgears.com Accessed: 2009-07-05

Karpat, F., Cavdar, K. & Babalic, C.F. (2006). An investigation on analysis of involutes spur gears with asymmetric teeth: Dynamic load and Transmission errors, Proceedings of the 2th International Conference "Power Transmissions 2006", pp. 69-74, ISBN 86-85211-78-6, Novi-Sad, April 2006, Serbia and Montenegro

Litvin, F.L., Lian, Q. & Kapelevich, A.L. (2000). Asymmetric Modified Gear Drives: Reduction of Noise, LocaliZation of Contact, Simulation of Meshing and Stress Analysis. Computer Methods in Applied Mechanics and Engineering, No.188, 363-390, ISSN 0045-7825

Novikov, A.S., Paikin, A.G., Dorofeyev, V.L., Ananiev, V.M., & Kapelevich, A.L. (2008). Application of Gears with Asymmetric Teeth in Turboprop Engine Gearbox, Gear Technology, January/February 2008, 60-65

This Publication has to be referred to as: Chira, F[lavia]; Banica, M[ihai] & Lobontiu, M[ircea] (2009). Optimising Involutes Asymmetrical Teeth Gears Software, Chapter 38 in DAAAM International Scientific Book 2009, pp. 363-376, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

Authors' data: Univ. Lecturer Dipl.-Ing. Dr. Chira, F[lavia]; Univ. Assoc. Prof. Dipl.-Ing. Dr. Banica, M[ihai]; Univ. Prof. Dipl.-Ing. Dr. Lobontiu, M[ircea], North University of Baia Mare, Dr. Victor Babes 62A, 430083, Baia Mare, Romania, Flavia.Chira@ubm.ro, Mihai.Banica@ubm.ro, Mircea.Lobontiu@ubm.ro

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Title Annotation: | Chapter 38 |
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Author: | Chira, F.; Banica, M.; Lobontiu, M. |

Publication: | DAAAM International Scientific Book |

Article Type: | Report |

Geographic Code: | 4EXRO |

Date: | Jan 1, 2009 |

Words: | 3329 |

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