Abstract:

Based on a new algorithm variant of the conceptual design modeling,
proposed by the authors, an application of this algorithm is analyzed on
a product of motor--planetary reducer type. According to the new
variant, the paper exemplifies the generation of the reducer solving
structural variants and the reducer concept establishment.

Key words: conceptual design, product concept, generalized algorithm, planetary reducer concept

Key words: conceptual design, product concept, generalized algorithm, planetary reducer concept

Article Type:

Report

Subject:

Algorithms
(Usage)

Product development (Methods)

Gearing, Reduction (Design and construction)

Speed reducers (Design and construction)

Product development (Methods)

Gearing, Reduction (Design and construction)

Speed reducers (Design and construction)

Authors:

Neagoe, M.

Diaconescu, D.

Jaliu, C.

Saulescu, R.

Diaconescu, D.

Jaliu, C.

Saulescu, R.

Pub Date:

01/01/2009

Publication:

Name: DAAAM International Scientific Book Publisher: DAAAM International Vienna Audience: Academic Format: Magazine/Journal Subject: Engineering and manufacturing industries Copyright: COPYRIGHT 2009 DAAAM International Vienna ISSN: 1726-9687

Issue:

Date: Annual, 2009

Topic:

Computer Subject: Algorithm; Time to market

Product:

Product Code: 9914420 Product Development; 9914400 Market Research & Product Development; 3566020 Speed Reducers; 3566050 Parts & Components for Speed Reducers NAICS Code: 333612 Speed Changer, Industrial High-Speed Drive, and Gear
Manufacturing SIC Code: 3566 Speed changers, drives, and gears

Geographic:

Geographic Scope: Romania Geographic Code: 4EXRO Romania

Accession Number:

224335571

Full Text:

1. Introduction

Based on the analysis of the literature (Cross, 1994; Ulrich & Eppinger, 2008; Dieter & Schmidt, 2009; Pahl et al., 2007; VDI, 1997), regarding product conceptual design, a generalized modeling algorithm is proposed by authors in a recent published issue (Diaconescu et al., 2008).

This paper exemplifies an application of this algorithm on a product of motor-reducer type, which consists of a planetary speed reducer and an electric motor. This motor-reducer is used in a rotational platform on which a parabolic antenna is installed; the motor-reducer must drive the platform in different angular positions, under imposed conditions of accuracy and stability. The first steps of the algorithm up to the generation of solving structural variants inclusive are approached in this paper.

2. On the requirements list

The following requirements (main objectives) regarding the rotational platform and the motor-reducer come out from the requirements list of the global product formed by the parabolic antenna assembly:

1) The maximum diameter of the crown gear that is fixed to the platform: [approximately equal to] 5 m.

2) The maximum diameter of the pinion driving the crown gear: [approximately equal to] 100 mm.

3) The platform maximum total weight: [approximately equal to] 31.2 tones.

4) The platform maximum inertia moment vs. the fixed revolute axis: [approximately equal to] 9.75 * [10.sup.4] [kgm.sup.2].

5) The platform axial and radial support: through some peripheral rollers with fixed axes.

6) The platform maximum speed during the angular displacement: 0.6 rot/min.

7) The positioning accuracy: [+ or -] 5[degrees].

8) The stable maintenance of the platform angular position: the motor stopping must simultaneously induce the platform blocking.

9) The energy source: electrical.

10) The reducer must be a simple planetary gear with a kinematical transmission ratio: [absolute value of i] = 100 [+ or -] 1.5% (therefore, the reducer must reduce the motor speed 100 times).

11) The reducer admissible minimum efficiency: [[eta].sub.min] = 0.4 (thus, the reducer must amplify the motor moment at least [[eta].sub.min] * [absolute value of i] = 40 times).

The following four optimization objectives that are enumerated in their relative importance order are associated to these requirements:

A. Minimization of the production costs,

B. Reducing the friction losses,

C. Minimization of the radial size and

D. Minimization of the axial size.

Afterwards, these (secondary) objectives are used as technical and economical evaluation criteria for the identification of the principle solution (among the solving structures that were generated). It is considered that the weight of the criteria is in the following correlations: A [approximately equal to] 4B [approximately equal to] 6C [approximately equal to] 8D.

3. Identification of the motor-reducer function

The motor-reducer global function is a sub-function of the rotational platform. In order to identify it, in Fig. 1 the following elements were successively represented:

* The platform global function (Fig. 1,a),

* Its structure of 1M+1E+1I order (Fig. 1,b) and

* The structure of 3M+4E+4I order (Fig. 1,c and d), derived from the previous structure.

The last structure, illustrated both in a descriptive variant (Fig. 1,c) and in a VDI symbolic variant (Fig. 1,d), has the following component sub-functions:

* FM1: connecting the material (platform) to the mechanical energy;

* FM2: the material (platform) angular displacing (rotating);

* FM3: registration of the material (platform) angular position;

* FE1: Connecting/disconnecting the voltage;

* FE2: Transformation of the electrical energy into mechanical energy and its irreversible transmission, together with the speed primary reducing;

* FE3: Transmission of mechanical energy with the speed secondary reducing (using the crown gear that is fixed to the platform);

* FE4: Ramification of the mechanical energy into two branches: useful energy and lost energy (through friction, mainly);

* FI1: Ramification of the input information into two branches: emission of the starting signal (after fulfilling the exploitation and safety instructions) and transmission of the information regarding the wanted angular position to the FI3 (sub)function;

* FI2: Commanding the execution of the starting and stopping signals; unlike the logical function FE1 (of AND type), the function FI2 is a logical function of INHIBITION type.

* FI3: Reception of the information regarding the angular positions: wanted and current and their comparison;

* FI4: Emission of the stopping signal when the two positions become equal, and registration of the final angular position.

Thus, the motor-reducer's global function is represented by the sub-function FE2, from the structure of 3M+ 4E+ 4I order (Fig. 1,c and d).

4. Detailing of the motor-reducer function

First, the motor-reducer function (see Fig. 2,a) from Fig. 1,d is extracted. The structure of functions in a symbolic variant from Fig. 2,b results by detailing this global function.

The three sub-functions of this structure have the following meanings:

* FE21: Transformation of the electrical energy into mechanical energy (a main sub-function);

* FE22: Irreversible transmission of the mechanical energy (a main sub-function as well);

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

* FE23: Modification of the mechanical energy parameters: speed reducing and, implicitly, the twisting moment amplification (a main sub-function);

* FE24: Transmission of the mechanical energy (unchanged) between the satellite-gear and central shaft (a main sub-function).

Further, based on the structure of functions from Fig. 2,b, the motor-reducer's solving structures are generated.

5. The generation of the solving structures

This stage contains two parts that are relatively distinct:

* Generation (synthesis) of the solving structural variants, and

* Establishment of the solving structures, by the kinematical configuration (synthesis) of the obtained variants and by the elimination of the variants whose technical characteristics don't fulfill quantitatively the requirements from the list.

5.1 Generation of the solving structural variants

From the research on the sources of existent solutions (catalogues of physical effects, catalogues of solutions for the functions of large use, prospects, patents, offers etc.), the following conclusions regarding the sub-functions FE21, ..., FE24 (from Fig. 2,b) solving result:

* All these sub-functions are of SPK type;

* The potential principle solutions of these sub-functions (the principles and the solving structures) are entirely known in the effects plan and are partially known in the plan of the effects carriers' configuration (a part of the found configurations can be reconfigured).

For exemplification, the results that were found are systematized in the simplified morphological matrix from Fig. 2,c. By means of this matrix, more solving structural variants can be generated by combining and by a compatible composition of the potential solutions from Fig. 2,c; obviously, from the obtained variants, there will be considered as solving structures (for the motor-reducer's FE2 function) only those whose technical characteristics fulfill the requirements from the list quantitatively.

For example, six distinct structural variants (from the obtained combinations) were illustrated in Fig. 2,d: SR1, ..., SR6; example: the variant SR1 (from Fig. 2,d) represents the resultant of the potential solutions: 1.2, 2.1, 3.1 and 4.1 (Fig. 2,c):

SR1 = 1.2 + 2.1 + 3.1 + 4.1

The qualitative schemes of the six variants are represented in Fig. 3,a, .., f in a simplified way (without representing the motor). For these schemes there are further established the numbers of teeth (from the condition [absolute value of i] = 100 [+ or -] 1.5 %) and, then, the efficiency and the moment amplification ratio is calculated for each of the variants. Obviously, the solving structures of the function FE2 will be nominated by the variants that obtain: [absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of the mechanical energy.

[FIGURE 3 OMITTED]

5.2. Establishment of the solving structures

In order to establish the solving structures (from the illustrated variants), first the synthesis of the number of teeth is made from the condition: [absolute value of i] = 100[+ or -]1.5%. Then, on the basis of the known efficiencies of the gear pairs with fixed axes, the efficiencies of the proposed reducers are calculated in the two possible actuation cases (direct and inverse) and the amplification ratio of the input moment for the direct actuation is established.

If the efficiency for the inverse actuation is null or negative, then the analyzed reducer transmits the power irreversibly and, therefore, the motor's brake becomes superfluous.

The case of the SR1 and SR2 variants. For each of the planetary reducers from Fig. 3,a and b (consisting of an involute internal gear pair 1-2 and of a homokinematical coupling 2-3), the condition of obtaining the transmission ratio can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [i.suub.0] denotes the kinematical interior ratio of the planetary gear with a sun gear from Fig. 3,a and b:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

On the limit (for the internal involute gear pairs), admitting that [z.sub.1] = [z.sub.1] + 4, the following values (see also Fig. 4) are obtained from relations (1) and (2): [z.sub.1] = 400, [z.sub.2] = 396 and [i.sub.0] = + 0.99. Considering that each of the planetary gears from Fig. 3,a and b has the interior efficiency:

[[eta].sub.0] = [[eta].sup.H.sub.13] = [[eta].sup.H.sub.12] * [[eta].sup.H.sub.23] = [0.995.sup.2] = 0.99

the following values are obtained for the efficiencies of the planetary reducers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] = 0, the planetary reducers from Fig. 3,a and b don't need motor brakes for the irreversible transmission of the power.

Thus, each of the variants SR1 and SR2 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = + [[OMEGA].sub.h]/100,

* it amplifies the input moment 50.25 times: [T.sub.1] = -i * [eta] * [T.sub.H] = - 50.25 * [T.sub.H],

* it transmits irreversibly the power without brake,

* it has a reduced axial overall size,

* the radial overall size is relatively big (being difficult to bank the satellites together),

* the manufacturing technology is relatively simple, but needs high accuracies. The first three properties show that the requirements from the list [absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of power) are fulfilled without motor's brake; therefore, each of the variants SR1 and SR2 is a solving structure of the function FE2.

The case of SR3 and SR4 variants. For each of the planetary reducers from Fig. 3,c and d (consisting of the homo-kinematical coupling 1-2 and the cycloidal gear pair with rollers 2-3), the condition of obtaining the transmission ratio can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [i.sub.0] denotes the interior kinematical ratio of the planetary gear with a sun gear (Fig. 3,c and d):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Considering that [z.sub.3] = [z.sub.2] + 1 (a property specific to the cycloidal gear pairs with rollers), the following values (see also Fig. 4) are obtained from relations (3) and (4): [z.sub.2] = 100, [z.sub.3] = 101 and [i.sub.0] = +1.01.

Starting from the premise that each of the planetary gears from Fig. 3,c and d has the interior efficiency :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the following values are obtained for the planetary reducers' efficiencies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] > 0, each of the planetary reducers from Fig. 3,c and d needs a motor brake for the irreversible transmission of power.

Thus, each of the variants SR3 and SR4 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = [[omega].sub.H]/100,

* it amplifies the input moment 58.4 times: [T.sub.1] = - i * [eta] * [T.sub.H] = +58.4 * [T.sub.H],

* it transmits the power irreversibly, by means of a motor brake,

* the radial overall size is relatively reduced (by banking the satellites together),

* it has a reduced axial overall size,

* the manufacturing technology uses special machine tools with high accuracies.

The first three properties show that the requirements from the list ([absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of power) are fulfilled; therefore, each of the variants SR3 and SR4 is also a solving structure of the function FE2.

The case of the SR5 and SR6 variants. The planetary chain reducers from Fig. 3,e and f (consisting of a homo-kinematical coupling 1-2 and of a chain transmission 2-3) have the same kinematics as the previous reducers; the efficiency relations and numerical values remain also unchanged, excepting the numerical value of the size n0 which becomes [[eta].sub.0] = 0.988. Therefore, the chain planetary reducers have the following efficiencies' values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] < 0, the planetary reducers from Fig. 3,e and f don't need motors' brakes for the irreversible transmission of power.

In conclusion, each of the variants SR5 and SR6 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = - [[omega].sub.H]/100,

* it amplifies the input moment 44.9 times: [T.sub.1] = - i * [eta] * [T.sub.H] = +44.9 * [T.sub.H],

* it transmits the power irreversibly without motor brake,

* the degree of complexity is relatively reduced,

* the radial overall size is reduced in the case of the scheme f and big in the case of the scheme e,

* it has a relative reduced axial overall size,

* the technology is relatively simple.

According to the first three properties, the requirements from the list ([absolute value of i] = 100 [+ or -] 1.5 %, [eta] [greater than or equal to] 0.4 and irreversible transmission of power) are fulfilled; therefore, each of the variants SR5 and SR6 represents a solving structure of the function FE2, too.

The principle solution or the motor-reducer concept will be identified between the previous 6 solving structures (see Fig. 4). Therefore, the generated structures are further ordered through a technical and economical evaluation.

6. Evaluation of the solving structures and principle solution establishment

The main features of the solving structures that were previously considered are systematized in Fig. 4. Further, the remained solving structures are ordered in Fig. 5, considering that the 4 criteria are of different weights (fine evaluation). With this aim, first there were established, in the Fig. 5,a the coefficients of absolute weight ([W.sub.k] = sum of the criterion's scores, see Fig. 5,a) and then the coefficients of relative weight ([W.sub.k] = [W.sub.k]/[SIGMA][W.sub.k], see Fig 5,a), knowing that: A [approximately equal to] 4B [approximately equal to] 6C [approximately equal to] 8D. With the marks from Fig. 5,b, it obviously results that the principle solution of the motor-reducer is designated by the structure SR6 (whose scheme is illustrated in Fig. 3,f).

7. Conclusions

From the exemplification of the generalized modeling algorithm proposed by the authors (Diaconescu et al., 2008) in the case of the conceptual design of a motor - reducer type product (which must drive a rotational platform), the following conclusions can be drawn:

* The proposed algorithm starts from the requirement list, containing the product requirements (main objectives) and technical ad economical criteria (optimization objectives), considered as input data in the conceptual design process.

* In the first step, the motor-reducer global function is identified and its subfunctions structure further developed.

* The next step deals with the generation of the solving variants through the morphological matrix tool; in this example, 6 solving structural variants are generated.

* The solving structures are next established based on their geometrical and kinematical parameters synthesis and eliminating the solving variants which don't fulfill the product requirements quantitatively.

* In the last step, the remained solving structures are evaluated using fine evaluation and hence the principle solution (SR6 structure) of the motor-reducer is designated. This principle solution contains three feasible modules (a motor without brake, a chain reducer and a Schimdt pin semi-coupling) and represents the input entity in the embodiment design phase.

DOI: 10.2507/daaam.scibook.2009.63

8. Acknowledgement

The authors will accomplish the design, manufacturing and testing of the speed increaser for stand-alone hydropower stations in the framework of the research project "Innovative mechatronic systems for micro hydrostations, meant to the efficient exploitation of hydrological potential from off-grid sites", ID_140. The preparation and publishing of this paper would not have been possible without the financial support of this research project.

9. References

Cross, N. (1994). Engineering Design Methods, J. Wiley & Sons, ISBN 0-4719-4228-6, New York

Diaconescu, D.; Neagoe, M.; Jaliu, C. & Saulescu, R. (2008). Products' Conceptual Design, Transilvania University Publishing House, ISBN 978-973-598-230-0, Brasov

Dieter, G. & Schmidt, L. (2009). Engineering Design, Fourth edition, McGraw Hill, ISBN 978-007-126341-2, Boston

Pahl, G.; Beitz, W.; Feldhusen, J. & Grote, K.H. (2007). Engineering Design: A systematic approach, Third edition, Springer-Verlag, ISBN 978-1-84628-318-5

Ulrich, K. & Epinger, S. (2008). Product Design and Development, Fourth edition, McGraw-Hill Inc., ISBN 978-007-125947-7, Boston

*** VDI--Verein Deutscher Ingenieure (1997). Richtlinien 2221 and 2222

This Publication has to be referred as: Neagoe, M[ircea]; Diaconescu, D[orin]; Jaliu, C[odruta] & Saulescu, R[adu] (2009). A Conceptual Design Application Based on a new Generalized Algorithm, Chapter 63 in DAAAM International Scientific Book 2009, pp. 653-664, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

NEAGOE, M.; DIACONESCU, D.; JALIU, C. & SAULESCU, R.

Authors' data: Univ.Prof. Dipl.-Eng. Dr. Neagoe, M[ircea]; Univ.Prof. Dipl.-Eng. Dr. Diaconescu, D[orin]; Univ.Prof. Dipl.-Eng. Dr. Jaliu, C[odruta]; Univ.Lecturer Dipl.-Eng. Dr. Saulescu, R[adu], Transilvania University of Brasov, Eroilor 29, 500036, Brasov, Romania, mneagoe@unitbv.ro, dvdiaconescu@unitbv.ro, cjaliu@unitbv.ro, rsaulescu@unitbv.ro.

Based on the analysis of the literature (Cross, 1994; Ulrich & Eppinger, 2008; Dieter & Schmidt, 2009; Pahl et al., 2007; VDI, 1997), regarding product conceptual design, a generalized modeling algorithm is proposed by authors in a recent published issue (Diaconescu et al., 2008).

This paper exemplifies an application of this algorithm on a product of motor-reducer type, which consists of a planetary speed reducer and an electric motor. This motor-reducer is used in a rotational platform on which a parabolic antenna is installed; the motor-reducer must drive the platform in different angular positions, under imposed conditions of accuracy and stability. The first steps of the algorithm up to the generation of solving structural variants inclusive are approached in this paper.

2. On the requirements list

The following requirements (main objectives) regarding the rotational platform and the motor-reducer come out from the requirements list of the global product formed by the parabolic antenna assembly:

1) The maximum diameter of the crown gear that is fixed to the platform: [approximately equal to] 5 m.

2) The maximum diameter of the pinion driving the crown gear: [approximately equal to] 100 mm.

3) The platform maximum total weight: [approximately equal to] 31.2 tones.

4) The platform maximum inertia moment vs. the fixed revolute axis: [approximately equal to] 9.75 * [10.sup.4] [kgm.sup.2].

5) The platform axial and radial support: through some peripheral rollers with fixed axes.

6) The platform maximum speed during the angular displacement: 0.6 rot/min.

7) The positioning accuracy: [+ or -] 5[degrees].

8) The stable maintenance of the platform angular position: the motor stopping must simultaneously induce the platform blocking.

9) The energy source: electrical.

10) The reducer must be a simple planetary gear with a kinematical transmission ratio: [absolute value of i] = 100 [+ or -] 1.5% (therefore, the reducer must reduce the motor speed 100 times).

11) The reducer admissible minimum efficiency: [[eta].sub.min] = 0.4 (thus, the reducer must amplify the motor moment at least [[eta].sub.min] * [absolute value of i] = 40 times).

The following four optimization objectives that are enumerated in their relative importance order are associated to these requirements:

A. Minimization of the production costs,

B. Reducing the friction losses,

C. Minimization of the radial size and

D. Minimization of the axial size.

Afterwards, these (secondary) objectives are used as technical and economical evaluation criteria for the identification of the principle solution (among the solving structures that were generated). It is considered that the weight of the criteria is in the following correlations: A [approximately equal to] 4B [approximately equal to] 6C [approximately equal to] 8D.

3. Identification of the motor-reducer function

The motor-reducer global function is a sub-function of the rotational platform. In order to identify it, in Fig. 1 the following elements were successively represented:

* The platform global function (Fig. 1,a),

* Its structure of 1M+1E+1I order (Fig. 1,b) and

* The structure of 3M+4E+4I order (Fig. 1,c and d), derived from the previous structure.

The last structure, illustrated both in a descriptive variant (Fig. 1,c) and in a VDI symbolic variant (Fig. 1,d), has the following component sub-functions:

* FM1: connecting the material (platform) to the mechanical energy;

* FM2: the material (platform) angular displacing (rotating);

* FM3: registration of the material (platform) angular position;

* FE1: Connecting/disconnecting the voltage;

* FE2: Transformation of the electrical energy into mechanical energy and its irreversible transmission, together with the speed primary reducing;

* FE3: Transmission of mechanical energy with the speed secondary reducing (using the crown gear that is fixed to the platform);

* FE4: Ramification of the mechanical energy into two branches: useful energy and lost energy (through friction, mainly);

* FI1: Ramification of the input information into two branches: emission of the starting signal (after fulfilling the exploitation and safety instructions) and transmission of the information regarding the wanted angular position to the FI3 (sub)function;

* FI2: Commanding the execution of the starting and stopping signals; unlike the logical function FE1 (of AND type), the function FI2 is a logical function of INHIBITION type.

* FI3: Reception of the information regarding the angular positions: wanted and current and their comparison;

* FI4: Emission of the stopping signal when the two positions become equal, and registration of the final angular position.

Thus, the motor-reducer's global function is represented by the sub-function FE2, from the structure of 3M+ 4E+ 4I order (Fig. 1,c and d).

4. Detailing of the motor-reducer function

First, the motor-reducer function (see Fig. 2,a) from Fig. 1,d is extracted. The structure of functions in a symbolic variant from Fig. 2,b results by detailing this global function.

The three sub-functions of this structure have the following meanings:

* FE21: Transformation of the electrical energy into mechanical energy (a main sub-function);

* FE22: Irreversible transmission of the mechanical energy (a main sub-function as well);

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

* FE23: Modification of the mechanical energy parameters: speed reducing and, implicitly, the twisting moment amplification (a main sub-function);

* FE24: Transmission of the mechanical energy (unchanged) between the satellite-gear and central shaft (a main sub-function).

Further, based on the structure of functions from Fig. 2,b, the motor-reducer's solving structures are generated.

5. The generation of the solving structures

This stage contains two parts that are relatively distinct:

* Generation (synthesis) of the solving structural variants, and

* Establishment of the solving structures, by the kinematical configuration (synthesis) of the obtained variants and by the elimination of the variants whose technical characteristics don't fulfill quantitatively the requirements from the list.

5.1 Generation of the solving structural variants

From the research on the sources of existent solutions (catalogues of physical effects, catalogues of solutions for the functions of large use, prospects, patents, offers etc.), the following conclusions regarding the sub-functions FE21, ..., FE24 (from Fig. 2,b) solving result:

* All these sub-functions are of SPK type;

* The potential principle solutions of these sub-functions (the principles and the solving structures) are entirely known in the effects plan and are partially known in the plan of the effects carriers' configuration (a part of the found configurations can be reconfigured).

For exemplification, the results that were found are systematized in the simplified morphological matrix from Fig. 2,c. By means of this matrix, more solving structural variants can be generated by combining and by a compatible composition of the potential solutions from Fig. 2,c; obviously, from the obtained variants, there will be considered as solving structures (for the motor-reducer's FE2 function) only those whose technical characteristics fulfill the requirements from the list quantitatively.

For example, six distinct structural variants (from the obtained combinations) were illustrated in Fig. 2,d: SR1, ..., SR6; example: the variant SR1 (from Fig. 2,d) represents the resultant of the potential solutions: 1.2, 2.1, 3.1 and 4.1 (Fig. 2,c):

SR1 = 1.2 + 2.1 + 3.1 + 4.1

The qualitative schemes of the six variants are represented in Fig. 3,a, .., f in a simplified way (without representing the motor). For these schemes there are further established the numbers of teeth (from the condition [absolute value of i] = 100 [+ or -] 1.5 %) and, then, the efficiency and the moment amplification ratio is calculated for each of the variants. Obviously, the solving structures of the function FE2 will be nominated by the variants that obtain: [absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of the mechanical energy.

[FIGURE 3 OMITTED]

5.2. Establishment of the solving structures

In order to establish the solving structures (from the illustrated variants), first the synthesis of the number of teeth is made from the condition: [absolute value of i] = 100[+ or -]1.5%. Then, on the basis of the known efficiencies of the gear pairs with fixed axes, the efficiencies of the proposed reducers are calculated in the two possible actuation cases (direct and inverse) and the amplification ratio of the input moment for the direct actuation is established.

If the efficiency for the inverse actuation is null or negative, then the analyzed reducer transmits the power irreversibly and, therefore, the motor's brake becomes superfluous.

The case of the SR1 and SR2 variants. For each of the planetary reducers from Fig. 3,a and b (consisting of an involute internal gear pair 1-2 and of a homokinematical coupling 2-3), the condition of obtaining the transmission ratio can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where [i.suub.0] denotes the kinematical interior ratio of the planetary gear with a sun gear from Fig. 3,a and b:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

On the limit (for the internal involute gear pairs), admitting that [z.sub.1] = [z.sub.1] + 4, the following values (see also Fig. 4) are obtained from relations (1) and (2): [z.sub.1] = 400, [z.sub.2] = 396 and [i.sub.0] = + 0.99. Considering that each of the planetary gears from Fig. 3,a and b has the interior efficiency:

[[eta].sub.0] = [[eta].sup.H.sub.13] = [[eta].sup.H.sub.12] * [[eta].sup.H.sub.23] = [0.995.sup.2] = 0.99

the following values are obtained for the efficiencies of the planetary reducers:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] = 0, the planetary reducers from Fig. 3,a and b don't need motor brakes for the irreversible transmission of the power.

Thus, each of the variants SR1 and SR2 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = + [[OMEGA].sub.h]/100,

* it amplifies the input moment 50.25 times: [T.sub.1] = -i * [eta] * [T.sub.H] = - 50.25 * [T.sub.H],

* it transmits irreversibly the power without brake,

* it has a reduced axial overall size,

* the radial overall size is relatively big (being difficult to bank the satellites together),

* the manufacturing technology is relatively simple, but needs high accuracies. The first three properties show that the requirements from the list [absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of power) are fulfilled without motor's brake; therefore, each of the variants SR1 and SR2 is a solving structure of the function FE2.

The case of SR3 and SR4 variants. For each of the planetary reducers from Fig. 3,c and d (consisting of the homo-kinematical coupling 1-2 and the cycloidal gear pair with rollers 2-3), the condition of obtaining the transmission ratio can be written:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where [i.sub.0] denotes the interior kinematical ratio of the planetary gear with a sun gear (Fig. 3,c and d):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

Considering that [z.sub.3] = [z.sub.2] + 1 (a property specific to the cycloidal gear pairs with rollers), the following values (see also Fig. 4) are obtained from relations (3) and (4): [z.sub.2] = 100, [z.sub.3] = 101 and [i.sub.0] = +1.01.

Starting from the premise that each of the planetary gears from Fig. 3,c and d has the interior efficiency :

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the following values are obtained for the planetary reducers' efficiencies:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] > 0, each of the planetary reducers from Fig. 3,c and d needs a motor brake for the irreversible transmission of power.

Thus, each of the variants SR3 and SR4 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = [[omega].sub.H]/100,

* it amplifies the input moment 58.4 times: [T.sub.1] = - i * [eta] * [T.sub.H] = +58.4 * [T.sub.H],

* it transmits the power irreversibly, by means of a motor brake,

* the radial overall size is relatively reduced (by banking the satellites together),

* it has a reduced axial overall size,

* the manufacturing technology uses special machine tools with high accuracies.

The first three properties show that the requirements from the list ([absolute value of i] = 100 [+ or -] 1.5%, [eta] [greater than or equal to] 0.4 and the irreversible transmission of power) are fulfilled; therefore, each of the variants SR3 and SR4 is also a solving structure of the function FE2.

The case of the SR5 and SR6 variants. The planetary chain reducers from Fig. 3,e and f (consisting of a homo-kinematical coupling 1-2 and of a chain transmission 2-3) have the same kinematics as the previous reducers; the efficiency relations and numerical values remain also unchanged, excepting the numerical value of the size n0 which becomes [[eta].sub.0] = 0.988. Therefore, the chain planetary reducers have the following efficiencies' values:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Because [[eta].sub.inv] < 0, the planetary reducers from Fig. 3,e and f don't need motors' brakes for the irreversible transmission of power.

In conclusion, each of the variants SR5 and SR6 has the following properties:

* it reduces the input angular speed 100 times: [[omega].sub.1] = - [[omega].sub.H]/100,

* it amplifies the input moment 44.9 times: [T.sub.1] = - i * [eta] * [T.sub.H] = +44.9 * [T.sub.H],

* it transmits the power irreversibly without motor brake,

* the degree of complexity is relatively reduced,

* the radial overall size is reduced in the case of the scheme f and big in the case of the scheme e,

* it has a relative reduced axial overall size,

* the technology is relatively simple.

According to the first three properties, the requirements from the list ([absolute value of i] = 100 [+ or -] 1.5 %, [eta] [greater than or equal to] 0.4 and irreversible transmission of power) are fulfilled; therefore, each of the variants SR5 and SR6 represents a solving structure of the function FE2, too.

The principle solution or the motor-reducer concept will be identified between the previous 6 solving structures (see Fig. 4). Therefore, the generated structures are further ordered through a technical and economical evaluation.

6. Evaluation of the solving structures and principle solution establishment

The main features of the solving structures that were previously considered are systematized in Fig. 4. Further, the remained solving structures are ordered in Fig. 5, considering that the 4 criteria are of different weights (fine evaluation). With this aim, first there were established, in the Fig. 5,a the coefficients of absolute weight ([W.sub.k] = sum of the criterion's scores, see Fig. 5,a) and then the coefficients of relative weight ([W.sub.k] = [W.sub.k]/[SIGMA][W.sub.k], see Fig 5,a), knowing that: A [approximately equal to] 4B [approximately equal to] 6C [approximately equal to] 8D. With the marks from Fig. 5,b, it obviously results that the principle solution of the motor-reducer is designated by the structure SR6 (whose scheme is illustrated in Fig. 3,f).

7. Conclusions

From the exemplification of the generalized modeling algorithm proposed by the authors (Diaconescu et al., 2008) in the case of the conceptual design of a motor - reducer type product (which must drive a rotational platform), the following conclusions can be drawn:

* The proposed algorithm starts from the requirement list, containing the product requirements (main objectives) and technical ad economical criteria (optimization objectives), considered as input data in the conceptual design process.

* In the first step, the motor-reducer global function is identified and its subfunctions structure further developed.

* The next step deals with the generation of the solving variants through the morphological matrix tool; in this example, 6 solving structural variants are generated.

* The solving structures are next established based on their geometrical and kinematical parameters synthesis and eliminating the solving variants which don't fulfill the product requirements quantitatively.

* In the last step, the remained solving structures are evaluated using fine evaluation and hence the principle solution (SR6 structure) of the motor-reducer is designated. This principle solution contains three feasible modules (a motor without brake, a chain reducer and a Schimdt pin semi-coupling) and represents the input entity in the embodiment design phase.

DOI: 10.2507/daaam.scibook.2009.63

8. Acknowledgement

The authors will accomplish the design, manufacturing and testing of the speed increaser for stand-alone hydropower stations in the framework of the research project "Innovative mechatronic systems for micro hydrostations, meant to the efficient exploitation of hydrological potential from off-grid sites", ID_140. The preparation and publishing of this paper would not have been possible without the financial support of this research project.

9. References

Cross, N. (1994). Engineering Design Methods, J. Wiley & Sons, ISBN 0-4719-4228-6, New York

Diaconescu, D.; Neagoe, M.; Jaliu, C. & Saulescu, R. (2008). Products' Conceptual Design, Transilvania University Publishing House, ISBN 978-973-598-230-0, Brasov

Dieter, G. & Schmidt, L. (2009). Engineering Design, Fourth edition, McGraw Hill, ISBN 978-007-126341-2, Boston

Pahl, G.; Beitz, W.; Feldhusen, J. & Grote, K.H. (2007). Engineering Design: A systematic approach, Third edition, Springer-Verlag, ISBN 978-1-84628-318-5

Ulrich, K. & Epinger, S. (2008). Product Design and Development, Fourth edition, McGraw-Hill Inc., ISBN 978-007-125947-7, Boston

*** VDI--Verein Deutscher Ingenieure (1997). Richtlinien 2221 and 2222

This Publication has to be referred as: Neagoe, M[ircea]; Diaconescu, D[orin]; Jaliu, C[odruta] & Saulescu, R[adu] (2009). A Conceptual Design Application Based on a new Generalized Algorithm, Chapter 63 in DAAAM International Scientific Book 2009, pp. 653-664, B. Katalinic (Ed.), Published by DAAAM International, ISBN 978-3-901509-69-8, ISSN 1726-9687, Vienna, Austria

NEAGOE, M.; DIACONESCU, D.; JALIU, C. & SAULESCU, R.

Authors' data: Univ.Prof. Dipl.-Eng. Dr. Neagoe, M[ircea]; Univ.Prof. Dipl.-Eng. Dr. Diaconescu, D[orin]; Univ.Prof. Dipl.-Eng. Dr. Jaliu, C[odruta]; Univ.Lecturer Dipl.-Eng. Dr. Saulescu, R[adu], Transilvania University of Brasov, Eroilor 29, 500036, Brasov, Romania, mneagoe@unitbv.ro, dvdiaconescu@unitbv.ro, cjaliu@unitbv.ro, rsaulescu@unitbv.ro.

Fig. 4. The technical characteristics of the solving structures Solving structure SR1 SR2 Figure 3 a b TECHNICAL CHARACTERISTICS 1. The numbers of gears' [z.sub.1] = 400 [z.sub.1] = 400 teeth [z.sub.2] = 396 [z.sub.2] = 396 2. The reducing ratio for in 100 100 3. The efficiency for [[eta].sub.1H] = [[eta].sub.1H] direct actuation [eta] 0.5025 0.5025 4. The efficiency for [[eta].sub.1H] = 0 [[eta].sub.1H] = 0 inverse actuation [[eta].sub.inv] 5. The amplification ratio 50.25 50.25 for input moment Solving structure SR3 SR4 Figure 3 c d TECHNICAL CHARACTERISTICS 1. The numbers of gears' [z.sub.2] = 100 [z.sub.2] = 100 teeth [z.sub.3] = 101 [z.sub.3] = 101 2. The reducing ratio for in -100 -100 3. The efficiency for [[eta].sub.H1] [[eta].sub.H1] direct actuation [eta] 0.584 0.584 4. The efficiency for [[eta].sub.H1] = [[eta].sub.H1] = inverse actuation 0.293 0.293 [[eta].sub.inv] 5. The amplification ratio 58.4 58.4 for input moment Solving structure SR5 SR6 Figure 3 e f TECHNICAL CHARACTERISTICS 1. The numbers of gears' [z.sub.2] = 100 [z.sub.2] = 100 teeth [z.sub.3] = 101 [z.sub.3] = 101 2. The reducing ratio for in -100 3. The efficiency for [[eta].sub.H1] = [[eta].sub.H1] = direct actuation [eta] 0.449 0.449 4. The efficiency for [[eta].sub.H1] = [[eta].sub.H1] = inverse actuation -0.212 -0.212 [[eta].sub.inv] 5. The amplification ratio 44.9 44.9 for input moment k Criterion/ A B C D [W.sub.k] [w. Criterion sub.k] 1 A 1 4 6 8 19 0.649 2 B 1/4 1 6/4 8/4 4.75 0.162 3 C 1/6 4/6 1 8/6 3.166 0.108 4 D 1/8 4/8 6/8 1 2.375 0.081 A [approximately [SIGMA] 29.29 1 equal to]4B [approximately equal to]6C [approximately equal to]8D Fig. 5. Relative weight coefficients [w.sub.k] establishment and solving structures ordering. SR1 SR2 Criterion [w.sub.k] [N.sub.k] [w.sub.k] [N.sub.k] [w.sub.k] * [N.sub.k] * [N.sub.k] A 0.649 8 5.192 8 5.192 B 0.162 6 0.972 6 0.972 C 0.108 5 0.540 5 0.540 D 0.081 8 0.648 7 0.567 Sum: 27 7.352 26 7.271 Place: 5 6 SR3 SR4 Criterion [w.sub.k] [N.sub.k] [w.sub.k] [N.sub.k] [w.sub.k] * [N.sub.k] * [N.sub.k] A 0.649 8 5.192 8 5.192 B 0.162 7 1.134 7 1.134 C 0.108 9 0.972 9 0.972 D 0.081 8 0.648 7 0.567 Sum: 32 7.946 31 7.865 Place: 2 3 SR5 SR6 Criterion [w.sub.k] [N.sub.k] [w.sub.k] [N.sub.k] [w.sub.k] * [N.sub.k] * [N.sub.k] A 0.649 9 5.841 10 6.490 B 0.162 5 0.810 5 0.810 C 0.108 6 0.648 8 0.864 D 0.081 6 0.486 8 0.648 Sum: 26 7.785 31 8.812 Place: 4 1

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