Determining the Efficiency of Geared Transmissions ’’FLORIAN ION PETRESCU, BARBU GRECU, ADRIANA COMĂNESCU, *RELLY VICTORIA PETRESCU TMR (Theory of Mechanisms and Robots) and *GDGI Departments Polytechnic University of Bucharest Splaiul Independenţei 313, Sector 6, zip 060042, Bucharest ROMANIA petrescuflorian@yahoo.com, bgrecu2000@yahoo.com, adrianacomanescu@yahoo.com, petrescuvictoria@yahoo.com ’’http://iondynamics.blogspot.com
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: Geared Transmission, Cover Grade, Dynamic, Efficiency, 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.
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
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
From the relations (1), one obtains the equality of the tangential velocities (2), and one explicit the motor velocities (see the relations 3): (2) 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
(3)
The transmitted forces concomitantly in the four points must be the same (see the relation 4): (4) Fτi = Fτj = Fτk = Fτl = Fτ The motor forces are (5):
Fmi = Fmk =
K 1i = rb1 ⋅ tgα i ; K 1 j = rb1 ⋅ tgα j ;
Fτ Fτ ; Fmj = ; cos α i cos α j
(5)
Fτ Fτ ; Fml = cos α k cos α l
K 1 k = rb1 ⋅ tgα k ; K 1l = rb1 ⋅ tgα l K 1 j − K 1i = rb1 ⋅ (tgα j − tgα i ); 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 ); O2 Fτl, vτl
K 1 k − K 1i = rb1 ⋅ 2 ⋅
rb2 Fml, vml
Fτi, vτi
K2 l k
Fmi, vmi
j
rl1 rk1
i
A
ri1αj αi
rj1
tgα k = tgα i + 2 ⋅
K 1l − K 1i = rb1 ⋅ (tgα l − tgα i );
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):
P P ηi = u = 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 ⋅π z1
K 1l − K 1i = rb1 ⋅ 3 ⋅
K1
2
(6) The relations (7) and (8) are auxiliary relations:
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): One 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
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 ⋅
1 B1
B1 = 1 + tg 2α 0 + ⋅ (2 ⋅ ε 12 − 1) ±
2π 2 2π ) + (tgα i ± 3 ⋅ ) 2 z1 z1
ε
4π 2 A2 = 4 + 4 ⋅ tg α i + 2 ⋅ (0 2 + 12 + 2 2 + z1
a .e . 12
=
2
+ 3 2 ) ± 2 ⋅ tgα i ⋅
2π ⋅ (0 + 1 + 2 + 3) z1
A3 = 1 + tg 2α i +
4π ⋅ (i − 1) 2 ± 2 ∑ E ⋅ z1 i =1 2
−
E 2π ± 2 ⋅ tgα i ⋅ ⋅ ∑ (i − 1) E ⋅ z1 i =1
A4 = 1 + tg 2α 1 + ⋅
ε
4π 2 ⋅ E ⋅ z12
E ⋅ ( E − 1) ⋅ (2 ⋅ E − 1) 4π ⋅ tgα 1 E ⋅ ( E − 1) ± ⋅ 6 2 E ⋅ z1
A5 = 1 + tg 2α 1 + ±
2π 2 ⋅ ( E − 1) ⋅ (2 E − 1) ± 3 ⋅ z12
⋅ (2 ⋅ ε 12 − 1) ±
2 ⋅ π ⋅ cos α 0
+
−
(11)
( z1 + z 2 ) ⋅ sin α 0 2 ⋅ π ⋅ cos α 0 =
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 +
−
z12 ⋅ sin 2 α 0 + 4 ⋅ z1 + 4
2 ⋅ π ⋅ cos α 0
a .i . 12
−
(10)
2π ⋅ tgα 0 ⋅ (ε 12 − 1) z1
z 22 ⋅ sin 2 α 0 + 4 ⋅ z 2 + 4
+
E
2π 2 ⋅ (ε 12 − 1) ⋅ 3 ⋅ z12
3 Conclusion
2π ⋅ (ε 12 − 1) ⋅ 3 ⋅ z12 2
2π ⋅ tgα 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 with relation (12) for the internal gearing as well.
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. References: [1] Petrescu, R.V., Petrescu, F.I., Popescu, N., Determining Gear Efficiency, Gear Solutions magazine, March 2007, USA, pp. 19-28.
The table 1. Determining the efficiency of the geared transmissions z1
α0 [grad]
42
20
46
19
52
18
58
17
65
16
74
15
85
14
98
13
115
12
137
11
165
10
204
9
257
8
336 457 657
z2
ε12ae
η12ae
η21ae
ε12ai
η12ai
η21ai
126 1.799463 0.844769 0.871197 1.920902 0.83866 0.89538 138 1.875212 0.856744 0.882524 2.004616 0.850911 0.905915 156 1.964633 0.869323 0.893686 2.099177 0.864094 0.915635 174 2.062449 0.880938 0.904223 2.205675 0.87608 0.925023 195 2.173287 0.892173 0.914292 2.326635 0.887672 0.933877 222 2.301654 0.903305 0.923963 2.465121 0.899241 0.942103 255 2.449656 0.914052 0.933147 2.624685 0.910419 0.949774 294 2.620345 0.924221 0.941781 2.810001 0.920962 0.956932 345 2.822316 0.934095 0.949936 3.027795 0.931246 0.963487 411 3.062854 0.943503 0.957556 3.286764 0.941054 0.969476 495 3.351511 0.952271 0.964586 3.599006 0.950177 0.974938 510 3.687451 0.960703 0.970155 4.020382 0.958606 0.980652
514 4.097102 0.968417 0.975013 4.577099 0.966232 0.98587 7 672 4.666475 0.975348 0.980651 5.214391 0.973645 0.989299 6 914 5.427328 0.981528 0.98562 6.067023 0.980251 0.992231 5 1314 6.495028 0.986917 0.989899 7.264006 0.986011 0.994672