TECHNICAL UNIVERSITY OF CLUJ-NAPOCA ACTA TECHNICA NAPOCENSIS
International Conference on Engineering Graphics and Design 12-13 June 2009
GEARED TRANSMISSIONS DESIGN Relly Victoria PETRESCU, Florian Ion PETRESCU Abstract: In this paper one presents shortly an original method to obtain the efficiency of the geared
transmissions in function of the cover grade of gearing. With the presented relations one can make the dynamic synthesis of geared transmissions, having in view increasing the efficiency of gearing mechanism in works. Key words: Efficiency, Force, Geared Transmission, Cover Grade, Dynamic, Tangential Velocity, Motor Velocity, Momentary Efficiency, Mechanical Efficiency, Wheel, Tooth, Four Pair in Contact.
1. INTRODUCTION In this paper one makes a brief presentation of an original method to obtain the efficiency of the geared transmissions in function of the cover grade. With the presented relations one can make the dynamic synthesis of the geared transmissions having in view increasing the efficiency of gearing mechanisms in work. 2. DETERMINING THE GEARING EFFICIENCY IN FUNCTION OF THE COVER GRADE One calculates the efficiency of a geared transmission, having in view the deed that at one moment there are more couples of teeth in contact, and not just one. The start model has got four pairs of teeth in contact (4 couple) concomitantly. The first couple of teeth in contact has the contact point i, defined by the ray ri1, and the pressure angle αi1; the forces which action in this point are: the motor force Fmi, perpendicular in i on the position vector ri1 and the transmitted force from the wheel 1 at the wheel 2 by the point i, Fτi, parallel with the gearing line and with the sense from the wheel 1 to the wheel 2, the transmitted force being practically the projection of the motor force at the gearing line; the defined velocities are similar with the
forces (having in view the original kinematics, or the precisely kinematics adopted); the same parameters will be defined for the next three points of contact, j, k, l (see the picture number one). For starting one writes the relations between the velocities (see the system 1):
vτi = vmi ⋅ cos α i = ri ⋅ ω1 ⋅ cos α i = rb1 ⋅ ω1 vτj = vmj ⋅ cos α j = rj ⋅ ω1 ⋅ cos α j = rb1 ⋅ ω1 vτk = vmk ⋅ cos α k = rk ⋅ ω1 ⋅ cosα k = rb1 ⋅ ω1
(1)
vτl = vml ⋅ cos α l = rl ⋅ ω1 ⋅ cos α l = rb1 ⋅ ω1 From the relations (1), one obtains the equality of the tangential velocities (2), and one explicit the motor velocities (see the relations 3): vτi = vτj = vτk = vτl = rb1 ⋅ ω1
v mi = v mk
rb1 ⋅ ω1 r ⋅ω ; v mj = b1 1 ; cos α i cos α j
r ⋅ω r ⋅ω = b1 1 ; v ml = b1 1 cos α k cos α l
(2)
(3)
The transmitted forces concomitantly in the four points must be the same (see the relation 4):
Fτi = Fτj = Fτk = Fτl = Fτ
(4)
K 1i = rb1 ⋅ tgα i ; K 1 j = rb1 ⋅ tgα j ;
The motor forces are (5):
K 1 k = rb1 ⋅ tgα k ; K 1l = rb1 ⋅ tgα l
Fτ Fτ Fmi = ; Fmj = ; cos α i cos α j
K 1 j − K 1i = rb1 ⋅ (tgα j − tgα i );
(5)
Fτ Fτ ; Fml = = cos α k cos α l
Fmk
The relations (7) and (8) are auxiliary relations:
K 1 j − K 1i = rb1 ⋅ tgα j = tgα i +
2 ⋅π ⇒ z1
2 ⋅π z1
K 1 k − K 1i = rb1 ⋅ (tgα k − tgα i ); K 1 k − K 1i = rb1 ⋅ 2 ⋅
O2 Fτl, vτl rb2 Fml, vml
Fτi, vτi
K2 l k
Fmi, vmi
j
rl1 rk1
tgα k = tgα i + 2 ⋅
ri1αj αi
rj1
K1
K 1l − K 1i = rb1 ⋅ 3 ⋅ tgα l = tgα i + 3 ⋅
rb1 O1
tgα j = tgα i ± Fig. 1 Four pairs of teeth in contact concomitantly.
The momentary efficiency can be written in the form (6):
ηi =
Pu P = τ = Pc Pm
Fτi ⋅ vτi + Fτj ⋅ vτj + Fτk ⋅ vτk + Fτl ⋅ vτl Fmi ⋅ v mi + Fmj ⋅ v mj + Fmk ⋅ v mk + Fml ⋅ v ml
=
4 ⋅ Fτ ⋅ rb1 ⋅ ω1 Fτ ⋅ rb1 ⋅ ω1 Fτ ⋅ rb1 ⋅ ω1 Fτ ⋅ rb1 ⋅ ω1 Fτ ⋅ rb1 ⋅ ω1 + + + cos 2 α i cos 2 α j cos 2 α k cos 2 α l =
=
4 1 1 1 1 + + + 2 2 2 cos α i cos α j cos α k cos 2 α l
=
4 4 + tg α i + tg α j + tg 2α k + tg 2α l 2
2
(6)
2 ⋅π z1
K 1l − K 1i = rb1 ⋅ (tgα l − tgα i );
i
A
2 ⋅π ⇒ z1
2 ⋅π ⇒ z1
2 ⋅π z1
2 ⋅π ; tgα k = z1
2 ⋅π 2 ⋅π = tgα i ± 2 ⋅ ; tgα l = tgα i ± 3 ⋅ z1 z1
(7)
(8)
One keeps the relations (8), with the sign plus (+) for the gearing where the drive wheel 1, has external teeth (at the external or internal gearing), and with the sign (-) for the gearing where the drive wheel 1, has internal teeth (the drive wheel is a ring, only at the internal gearing). The relation of the momentary efficiency (6), use the auxiliary relations (8) and takes the form (9): It start in the expression (9) with the relation (6) where one has four pairs in contact concomitantly, but then one generalizes the expression exchanging the 4 figure (four pairs) with E couples (exchanging figure 4 with the E variable, which represents the whole of cover grade +1), and after while one restricts the
sums expressions, one exchanges the variable E with the cover grade ε12, as well.
ηi =
4 4 + tg α i + tg α j + tg α k + tg α l 2
2
2
2
with relation (12) for the internal gearing as well.
ηm =
=
4 4 1 1 1 1 = = = = = = A1 A2 A3 A4 A5 A6 2π 2 ) + A1 = 4 + tg 2α i + (tgα i ± z1 + (tgα i ± 2 ⋅
B1 = 1 + tg 2α 0 + ⋅ (2 ⋅ ε 12 − 1) ±
2π 2 2π ) + (tgα i ± 3 ⋅ ) 2 z1 z1
+
2π + 3 ) ± 2 ⋅ tgα i ⋅ ⋅ (0 + 1 + 2 + 3) z1 2
A3 = 1 + tg 2α i + ± 2 ⋅ tgα i ⋅
4π ⋅ ∑ (i − 1) 2 ± E ⋅ z12 i =1
E 2π ⋅ ∑ (i − 1) E ⋅ z1 i =1
A4 = 1 + tg 2α 1 + ⋅
−
E
A5 = 1 + tg 2α 1 + ±
2π 2 ⋅ ( E − 1) ⋅ (2 E − 1) ± 3 ⋅ z12
2 ⋅ π ⋅ cos α 0
+
−
(11)
( z1 + z 2 ) ⋅ sin α 0 2 ⋅ π ⋅ cos α 0 a .i . 12
−
=
z e2 ⋅ sin 2 α 0 + 4 ⋅ z e + 4 2 ⋅ π ⋅ cos α 0
z i2 ⋅ sin 2 α 0 − 4 ⋅ z i + 4 2 ⋅ π ⋅ cos α 0
−
−
(12)
( z e − z i ) ⋅ sin α 0 2 ⋅ π ⋅ cos α 0
The calculations results one has been centralized in the table 1.
2π ⋅ tgα 1 ⋅ ( E − 1) z1
A6 = 1 + tg 2α 1 +
2 ⋅ π ⋅ cos α 0
−
E ⋅ ( E − 1) ⋅ (2 ⋅ E − 1) 4π ⋅ tgα 1 E ⋅ ( E − 1) ± ⋅ 6 2 E ⋅ z1
(10)
2π ⋅ tgα 0 ⋅ (ε 12 − 1) z1
z 22 ⋅ sin 2 α 0 + 4 ⋅ z 2 + 4
ε
4π 2 ⋅ E ⋅ z12
2π 2 ⋅ (ε 12 − 1) ⋅ 3 ⋅ z12
z12 ⋅ sin 2 α 0 + 4 ⋅ z1 + 4
ε 12a.e. =
4π 2 A2 = 4 + 4 ⋅ tg 2α i + 2 ⋅ (0 2 + 12 + 2 2 + z1
2
1 B1
2π 2 ⋅ (ε 12 − 1) ⋅ 3 ⋅ z12
3. CONCLUSION
2π ⋅ tgα 1 ⋅ (2 ⋅ ε 12 − 1) ± ⋅ (ε 12 − 1) z1 (9) The mechanical efficiency is more interesting than the momentary efficiency, and will be calculated approximately, exchanging the pressure angle α1, with the normal pressure angle α0, in the relation (9) which takes the form (10); where ε12 represents the cover grade of the gearing, and it will be calculated with the expression (11) for the external gearing, and
The best efficiency can be obtained with the internal gearing when the drive wheel 1 is the ring; the minimum efficiency will be obtained when the drive wheel 1 of the internal gearing has external teeth. At the external gearing, the better efficiency can be obtained when the greater wheel is the drive wheel; when one decreases the normal angle α0, the cover grade and the efficiency increase. The efficiency increase too when the number of teeth of the drive wheel, 1, increase as well (when one increases z1), see the table number 1.
4. REFERENCES
[1] Petrescu, R.V., Petrescu, F.I., Popescu, N., Determining Gear Efficiency, In Gear Solutions magazine, March 2007, USA, pp. 19-28. [2] Petrescu, F.I., Grecu, B., Comănescu, A., Petrescu, R.V., Sinteza dinamică la angrenajele cu roţi dinţate cu axe paralele. In al treilea Seminar Naţional de Mecanisme Craiova 2008, SNM’08, Proceedings, ISBN 978-973-746-910-6, 582 pag, Ed. SITECH, Craiova, p. 319-324, 2008.
[3] Petrescu, F.I., Grecu, B., Comănescu, A., Petrescu, R.V., Geared Transmissions Dynamic Synthesis, In NEW TRENDS IN MECHANISMS, Ed. Academica – Greifswald, 2008, ISBN 978-3-9402-37-101. [4] Petrescu, F.I., Grecu, B., Comănescu, A., Petrescu, R.V., Determining the Efficiency of Geared Transmissions, Proceedings of International Conference on 33Rd Automotive Engineering “Off-Road Vehicles, ORV 2008”, Military Technical Academy, Bucharest, Romania, 2008, ISBN 978-973-640-149-7. Table 1
Determining the efficiency of the geared transmissions z1
α0
z2
ε12ae
η12ae
η21ae
ε12ai
η12ai
η21ai
[grad]
42
20
126
1.799463
0.844769
0.871197
1.920902
0.83866
0.89538
46
19
138
1.875212
0.856744
0.882524
2.004616
0.850911
0.905915
52
18
156
1.964633
0.869323
0.893686
2.099177
0.864094
0.915635
58
17
174
2.062449
0.880938
0.904223
2.205675
0.87608
0.925023
65
16
195
2.173287
0.892173
0.914292
2.326635
0.887672
0.933877
74
15
222
2.301654
0.903305
0.923963
2.465121
0.899241
0.942103
85
14
255
2.449656
0.914052
0.933147
2.624685
0.910419
0.949774
98
13
294
2.620345
0.924221
0.941781
2.810001
0.920962
0.956932
115
12
345
2.822316
0.934095
0.949936
3.027795
0.931246
0.963487
137
11
411
3.062854
0.943503
0.957556
3.286764
0.941054
0.969476
165
10
495
3.351511
0.952271
0.964586
3.599006
0.950177
0.974938
204
9
510
3.687451
0.960703
0.970155
4.020382
0.958606
0.980652
257
8
514
4.097102
0.968417
0.975013
4.577099
0.966232
0.98587
336
7
672
4.666475
0.975348
0.980651
5.214391
0.973645
0.989299
457
6
914
5.427328
0.981528
0.98562
6.067023
0.980251
0.992231
657
5
1314
6.495028
0.986917
0.989899
7.264006
0.986011
0.994672
DESIGNUL TRANSMISIILOR (ANGRENAJELOR) CU ROŢI DINŢATE Rezumat: Lucrarea prezintă pe scurt o metodă originală pentru obţinerea eficienţei (randamentului) angrenajelor dinţate în funcţie de gradul de acoperire (al angrenajului respectiv). Cu relaţiile prezentate se poate face sinteza dinamică a unui angrenaj cu roţi dinţate, având în vedere creşterea eficienţei (randamentului) angrenajului în funcţionare. Petrescu Relly Victoria, PhD. Eng., Lecturer at Polytechnic University of Bucharest, GDGI Department (Department of Descriptive Geometry and Engineering Graphics), petrescurelly@yahoo.com, 0214029136; Petrescu Florian Ion, PhD. Eng. Assistant Professor at Polytechnic University of Bucharest, TMR Department (Theory of Mechanisms and Robots Department), petrescuflorian@yahoo.com, 0214029632.