A conceptual design application based on a generalized algorithm part II. Solving structures selection and evaluation.

Neagoe, Mircea ; Diaconescu, Dorin ; Jaliu, Codruta 等

1. INTRODUCTION

The qualitative schemes of the six variants, generated in the first part of the paper (Neagoe et al., 2008), are illustrated (Fig. 1) in a simplified way. Among these schemes of variants, the solving structure will be selected by the kinematical configuration (i.e. synthesis of the teeth numbers and establishment of the efficiency and the torque amplification ratio). Obviously, the solving structures of the motor-reducer function will be nominated by the variants that block the transmission when the motor is disconnected and achieve: [absolute value of i] = 100 and [eta] [greater than or equal to] 0.4; one of them will be selected by evaluation as product concept (Pahl & Beitz,1995; Ulrich & Epinger, 1995).

2. SOLVING STRUCTURES ESTABLISHMENT

In order to establish the solving structures (from the illustrated variants, Fig. 1), 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 torque 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. 1,a and b (consisting of an involute internal gear pair 1-2 and of a synchronic coupling 2-3), the condition of obtaining the transmission ratio can be written as follows (Diaconescu & Duditza, 1994,a and b):

i = [i.sup.3.sub.H,1] = [[omega].sub.H,3]/[[omega].sub.1,3] = 1/(1 - [i.sub.0]) = + 100; (1)

[i.sub.0] = [i.sup.H.sub.1,3] = [[omega].sub.1,H]/[[omega].sub.3,H] = [i.sup.H.sub.1,2] x [i.sup.H.sub.2,3] = (+[z.sub.2]/[z.sub.1]) x (+1) = +[z.sub.2]/[z.sub.1] (2)

where [i.sub.0] is the kinematical internal ratio of the planetary gears from Fig. 1,a,b (i.e. the ratio of the gearbox with fixed axes derived from planetary gear--set by motion inversion).

On the limit (for the internal involute gear pairs), admitting that [z.sub.1] = [z.sub.2] + 4, the following values are obtained from relations (1) and (2): [z.sub.1] = 400, [z.sub.2] = 396 and [i.sub.0] = +0.99.

Considering that the planetary gears from Fig. 1,a and b have the interior efficiency [[eta].sub.0] = [[eta].sup.H.sub.1,3] = [[eta].sup.H.sub.1,2] x [[eta].sup.H.sub.2,3] = 0.99, the following efficiencies are obtained (Diaconescu & Duditza, 1994,a and b):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because [[eta].sub.inv] = 0, the planetary reducers from Fig. 1,a and b don't need brake when the motor is disconnected.

As results, the variants SR1 and SR2 (Fig.1,a and b) have the following properties: a) the input angular speed is reduced 100 times: [[omega].sub.1] = + [[omega].sub.H]/100; b) the input torque is amplified 50.25 times: [T.sub.1] = -i x [eta] x [T.sub.H] = -50.25 x [T.sub.1]; c) if the motor doesn't work, transmission is blocked without brake; d) the axial overall size is reduced and the radial overall size is relatively big, e) the manufacturing technology is relatively simple, but needs high accuracy. This means that each of the variants SR1 and SR2 is a solving structure of the motor-reducer function.

[FIGURE 1 OMITTED]

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

i = [i.sup.3.sub.H,1] = [[omega].sub.H,3]/[[omega].sub.1,3] = 1/(1 - [i.sub.0]) = -100; (3)

[i.sub.0] = [i.sup.H.sub.1,3] = [[omega].sub.1,H]/[[omega].sub.3,H] = [i.sup.H.sub.1,2] x [i.sup.H.sub.2,3] = (+1) x (+[z.sub.2]/[z.sub.1]) = +[z.sub.2]/[z.sub.1]. (4)

On the limit (for the internal cycloid gear pairs with rolls), considering that [z.sub.3] = [z.sub.2] + 1, the following values 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. 1,c and d has the interior efficiency [[eta].sub.0] = [[eta].sup.H.sub.1,3] = [[eta].sup.H.sub.1,2] x [[eta].sup.H.sub.2,3] = 0.995-0.998 = 0.993, the following efficiencies are obtained:

[eta] = [[eta].sup.3.sub.H,1] = 0.584; [[eta].sub.inv] = [[eta].sup.3.sub.1,H] = 0.293.

Because [[eta].sub.inv] > 0, the planetary reducers from Fig. 1,c and d need brake when the motor is disconnected. As results, the variants SR1 and SR2 are solving structure too.

The case of the SR5 and SR6 variants. The planetary chain reducers from Fig. 1,e and f (consisting of a synchronic 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, except for the internal efficiency [[eta].sub.0] which becomes [[eta].sub.0] = 0.988. Therefore, the chain planetary reducers have the following efficiencies' values :

[eta] = [[eta].sup.3.sub.H,1] = 0.449, [[eta].sub.inv] = [[eta].sup.3.sub.1,H] = -0.212.

Because [[eta].sub.inv] < 0, the planetary reducers from Fig. 1,e and f don't need brakes when the motor is disconnected.

As results, the variants SR5 and SR6 have the following properties:

a) the input angular speed is reduced 100 times: [[omega].sub.1] = -[[omega].sub.H]/100; b) The input torque is amplified 44.9 times: [T.sub.1] = -i x [eta] x [T.sub.H] = +44.9 x [T.sub.1]; c) if the motor doesn't run, the transmission can be blocked without brake; d) the complexity is relatively reduced e) the axial overall size is relative reduced and the radial overall size is relatively big in the case of scheme e, f) the manufacturing technology is relatively simple.

This means that both variants SR5 and SR6 are solving structures of the motor-reducer function.

3. CONCLUSIONS

The principle solution or the motor-reducer concept must be identified among the previous 6 solving structures.

Therefore, the generated structures are further ordered through a technical and economical evaluation (fine evaluation, Fig. 3).

The main features of the solving structures that were previously considered are systematized in Fig. 2; because the structure SR5 has the biggest radial overall size and the minimum efficiency, this variant was eliminated.

The remained solving structures are ordered in Fig. 3, considering that the 4 criteria are following different weights: A [approximately equal to] 4B [approximately equal to] 6C [approximately equal to] 8D (fine evaluation).

With the marks from Fig. 3, it results that the principle solution is designated by the structure SR6 (Fig. 1,f).

This principle solution contains three feasible modules (a motor without brake, a chain reducer and a Schimdt semi-coupling with rolls) and represents the input entity in the embodiment design phase.

4. REFERENCES

Diaconescu, D.V.& Duditza, Fl. (1994,a). Wirkungsgradberechnung von zwanglaufigen Planetengetrieben. Teil I: Entwiklung einer neuen Methode. Antriebstechnik 33 (1994) 10, S. 70-74

Diaconescu, D.V. & Duditza, Fl. (1994,b). Wirkungsgradberechnung von zwanglaufigen Planetengetrieben. Teil II: Weitere Beispielrechnungen und Vorteile. Antriebstechnik 33 (1994) 11, S. 61-63

Neagoe, M. et al. (2008). A Conceptual Design Application Based on a Generalized Algorithm. Part I. Generation of the Solving Structural Variants, The 19th International DAAAM SYMPOSIUM "Intelligent Manufacturing & Automation: Focus on Next Generation of Intelligent Systems and Solutions", 22-25th October 2008 (accepted paper)

Pahl, G. & Beitz, W. (1995). Engineering Design, Springer, ISBN 3540504427, London

Ulrich, K. & Epinger, S. (1995). Product Design and Development, McGraw-Hill Inc. ISBN 0-07-113742-4, New York

Fig. 2. The technical characteristics of the solving structures.

Solving structure SR1 SR2

Fig.1 a b

TECHNICAL CHARACTERISTICS

1. The numbers of the gears' [z.sub.1] = 400 [z.sub.1] = 400
teeth [z.sub.2] = 396 [z.sub.2] = 396

2. The reducing ratio for the 100 100
input speed

3. The efficiency for the [eta] H1 = 0.5025 [eta] H1 = 0.5025
direct actuation [eta]

4. The efficiency for the [eta]1H=0 [eta]1H = 0
inverse actuation
[[eta].sub.inv]

5. The amplification ratio for 50.25 50.25
the input torque

Solving structure SR3 SR4

Fig.1 c d

TECHNICAL CHARACTERISTICS

1. The numbers of the gears' [z.sub.2] = 100 [z.sub.2] = 100
teeth [z.sub.3] = 101 [z.sub.3] = 101

2. The reducing ratio for the -100 -100
input speed

3. The efficiency for the [eta]H1 = 0.584 [eta]H1 = 0.584
direct actuation [eta]

4. The efficiency for the [eta]1H = 0.293 [eta]1H = 0.293
inverse actuation
[[eta].sub.inv]

5. The amplification ratio for 58.4 58.4
the input torque

Solving structure SR5 SR6

Fig.1 e f

TECHNICAL CHARACTERISTICS

1. The numbers of the gears' [z.sub.2] = 100 [z.sub.2] = 100
teeth [z.sub.3] = 101 [z.sub.3] = 101

2. The reducing ratio for the -100 -100
input speed

3. The efficiency for the [eta]H1 = 0.449 [eta]H1 = 0.449
direct actuation [eta]

4. The efficiency for the [eta]H = -0.212 [eta]1H = -0.212
inverse actuation
[[eta].sub.inv]

5. The amplification ratio for 44.9 44.9
the input torque

Fig. 3 Concept selection (SR6) by fine evaluation of the solving
structures SR1, SR2, SR3, SR4 and SR6.

 SR1

Criterion [w.sub.k] [N.sub.k] [w.sub.k] x
 [N.sub.k]

A 0.649 8 5.192
B 0.162 8 1.296
C 0.108 7 0.756
D 0.081 8 0.648
Place: Sum: 4 7.892

 SR2 SR3

Criterion [N.sub.k] [w.sub.k] x [N.sub.k] [w.sub.k] x
 [N.sub.k] [N.sub.k]

A 8 5.192 8 5.192
B 8 1.296 9 1.458
C 7 0.756 8 0.864
D 7 0.567 8 0.648
Place: 5 7.811 2 8.162

 SR4 SR6

Criterion [N.sub.k] [w.sub.k] x [N.sub.k] [w.sub.k] x
 [N.sub.k] [N.sub.k]

A 8 5.192 10 6.490
B 9 1.458 7 1.134
C 8 0.864 9 0.972
D 7 0.567 8 0.648
Place: 3 8.081 1 9.244