Florian Ion Tiberiu Petrescu
Bucharest Polytechnic University, Romania
Email: petrescuflorian@yahoo.com
Relly Victoria Virgil Petrescu
Bucharest Polytechnic University, Romania
Email: petrescuvictoria@yahoo.com
Submission: 10/20/2015
Revision: 10/30/2015
Accept: 11/28/2018
ABSTRACT
The paper
presents an original algorithm composed by authors in order to determine
through exact calculations the efficiency value of a simple planetary
mechanism, increasingly used in aerospace, robotics, mechatronics, machine
building, and various automation. The calculation program is written in excel
and for its simple automation four signtype switches, are used plus or minus
1, and a logic function for checking the status and choosing the corresponding
value. In this way, the program is generalized to be used for any type of
simple planetary mechanism for the purpose of accurately determining its yield.
Keywords: Computer science; Calculation program;
Automation; Logic function
1. INTRODUCTION
A mechanical transmission is a
mechanical device used for the purpose of transmitting movement from one side
to the other usually by means of a mechanism having elements and kinematic
couplings. The mechanical transmission must adjust the torque and speed between
the drive (input) and traction element (s) (CAO et al., 2013).
Motion transmission is one of the
most common functions of general mechanical components, mechanical devices
designed to replace the human hand (GARCIA et al., 2007).
According to the mechanisms, the
transmission is dimensioned according to considerations related to (LEE, 2013):
•
Position of a part of the mechanism
• The
desired movement
•
Force or torque
• The
power
Epicyclic
transmission is a mechanical transmission device. It has the particularity of
having two degrees of mobility, such as a differential, that is, the
association of three trees at different rotational speeds with a mathematical
relationship: The speeds of two trees are thus correlated to the third.
These trains are often used to
reduce speed due to the high reduction ratios that this configuration allows to
be uniformly compact with a single wheel. These mechanisms are especially found
in automatic gearboxes, the Toyota Hybrid Synergy Drive, the integrated bicycle
knobs, the electric motors on wheels, the winch and the doublerobotic clutch.
The term epicycloid comes from the
trajectory that follows an epicicloid from a point of
the satellites observed on the inner planetary. However, a hypocicloid
is observed if the motion reference is external to the planetary, often fixed
in the reducers. This corresponds exactly to what the observer observes when
viewing a satellite.
Configuration adopted in the machine
differential. The axis of rotation of the satellites (often in pairs) is
perpendicular to the planet. As a result, the gears are conical.
If the satellite stays fixed on the
planetary gears, the two planets have the same rotation speed. When planets
move at different speeds, the satellite rotates while transmitting power:
•
Applications
•
Blaise Pascal computing machine: Pascaline
•
Integrated blade for bicycle tools
•
Differences in the car
• Most
automatic automatic gearboxes and more mechanical
gearboxes (for example, the Ford T model)
• The
electromechanical transmission of the Toyota Prius hybrid and, in general,
Toyota and Lexus (the HSD system)
•
Frequency multipliers in wind turbines
•
Garden tools (e.g. plant crushers)
•
Helicopters
It is mainly used for epicyclic trains. They are present in automatic gearboxes
and in many gearboxes coupled to electric motors. They appear in the same
catalogs as the latter. Their geometry provides a coaxial output shaft with the
input shaft, which facilitates its implementation. Finally, they have a great
ability to reduce speed. In general, three satellites are placed on the
satellite carrier. Thus, the forces in the gear are not taken over by the
bearings. As a result, these gearboxes are very suitable for high torque
transmission.
These devices are sometimes used as
multipliers, as with wind turbines. Again, here is their compactness and the
absence of the radial force induced in the camps of incoming and outgoing trees
that justify their use.
From the simplest train (Type I), mobility
is eliminated by fastening the outer gear, also called the crown.
The input tree, while rotating,
forces the satellite to roll inside the crown. In her movement, she drives the
satellite door as if it were a maneuver. The planetary support is the output
shaft of the device. In this configuration, the output rotates in the same
direction and slower than the input.
Planetary transmissions have a
number of advantages compared to fixed axle transmission. Under similar
operating conditions, planetary transmissions operate longer and produce less
noise compared to a fixed shaft transmission [16]. The gearboxes are generally
used to reduce the rotational speed of the lucrative machine, such as a
turbine, relative to the engine speed used. Easy to achieve performance up to
110 MW or turbine rotation speeds of approximately 60,000 rpm. Parallel shaft
transmission mechanisms are typically used in these applications. However, some
manufacturers of packaging and generator manufacturers use planetary gears for gas
and steam turbine systems. Most planetary gear manufacturers are located in
Europe and the United States. Generally, the synthesis of classical planetary
mechanisms is based on the kinematic relations given by formula 1 (ANDERSON; LOEWENTHAL,
1986; PETRESCU, 2019), considering mainly the transmission ratio achieved. The
most commonly used model is the differential planetary mechanism shown in
Figure 1 with two degrees of freedom. Formula 1 is obtained through the
wellknown Willis relationship.
(1)
where,
is the ratio of transmission input output,
corresponding to the mechanism with fixed axis (when the planetary carrier H is
fixed) and is determined in function of the cinematic schematic of planetary
gear used.
Figure
1: Kinematic schematic of a differential planetary mechanism (M = 2)
In general, planetary mechanisms are
less synthesized based on their mechanical efficiency developed during
operation, although this is basically the main engineering criterion of
performance.
Even when considering the efficiency
criterion, the determination of the planetary yield is made only with
approximate relationships (MARTIN, 1981; ANTONESCU, 1979; PELECUDI et al.,
1985; PENNESTRI; FREUDENSTEIN, 1993) or with a particular character (not
generalized) del (CASTILLO, 2002; CHO et al., 2006). This paper aims to
determine the real efficiency of planetary trains by means of an exact,
generalized calculation algorithm that takes into account the efficiency of the
mechanism determined by an exact original method, the algorithm being also an
original one (PETRESCU; PETRESCU, 2011; PETRESCU, 2012; PETRESCU et al., 2016).
2. MATERIALS AND METHODS
Classically, the synthesis of
planetary mechanisms is generally achieved through kinematic relations,
especially taking into account the ratio of transmission made from input to
output. The most commonly used model is the differential planetary mechanism as
shown in Figure 1.
Generally, in practice, simple
planetary mechanisms with only one degree of mobility are used, except for the
differential mechanism. For the differential mechanism to achieve a single
degree of mobility, remaining in use with a single input and a unique output,
it is necessary to reduce the mobility of the two to one mechanism, a reduction
that can be achieved by connecting in series or parallel two or several
planetary mechanisms by linking to fixed axle mechanisms or by immobilizing one
of its movable elements; element 1 in the case shown in Figure 2 (in which case
the wheel 1 is identified with the fixed element 0).
The entrance to the simple planetary
mechanism shown in Figure 2 is made by the planetary support (H), and the
output is made by means of the movable kinematic element (3), the wheel (3).
The kinematic ratio between inputoutput (H3) can be written as shown in
relation 1; is
the ratio of transmission input output of the model in Figure 2 and is
determined by relation 2, depending on the number of teeth of wheels 1, 2, 2'
and 3:
(2)
Figure 2: Kinematic schema of a simple planetary
mechanism (M=1)
For the stiffening of the various
kinematic planetary systems represented in figure 3, where the input is made by
the planetary support (H) and the output is obtained by the final element (f),
the initial element that is usually immobilized will be used for kinematic
calculations in the generalized relations 1 and 2; thus the relation 1 takes
the general form 3 and the relation 2 is written in one of the custom 4 forms
for each separately presented scheme used; where i it
becomes 1 and f takes the value 3 or 4 as the case (PETRESCU AND PETRESCU,
2011; PETRESCU, 2012; PETRESCU et al., 2016).
(3)
(4)






Figure 3: Planetary systems
(5)
For the ratio between the input and
output of the transmission of the models in Figure 3, one of the system 5
relations is chosen, according to the situation used.
For a normally planetary system (Figure
2) the mechanical efficiency can be determined based on system 5, which shows
that the power between inputoutput is conserve. With Willis one determines ω_{3}.
It uses then the conservation of power between inputoutput for a simple
planetary gear apparently fixed axis (1input, 3output). For calculating the
efficiency of the simple planetary mechanism, the original relationships given
by system 6 are used.
(6)
It also uses the auxiliary original
calculation relationships 79 (PETRESCU; PETRESCU, 2014 ab).
(7)
(8)
(9)
3. RESULTS AND DISCUSSION
To calculate the yield of a simple
planetary mechanism using original relationships 69, one writes an original
calculation algorithm using the familiar excel program shown below in the table
1.
Table 1: An
original calculation algorithm written in excel

A 
B 
1 
_{H3} 
=B36 
2 
i_{13H} 
=B33 
3 
x 
=IF(B33<=1,1,1) 
4 
j[deg] 
400 
5 
z_{1} 
20 
6 
z_{2} 
50 
7 
z_{2'} 
30 
8 
z_{3} 
40 
9 
a_{012}
[deg] 
=200.01*B4 
10 
a_{023}
[deg] 
=20+0.01*B4 
11 
b_{12}
[deg] 
=10+0.05*B4 
12 
b_{23}
[deg] 
=10+0.06*B4 
13 
sign z_{1} 
1 
14 
sign z_{2'} 
1 
15 
Sign 12 
1 
16 
Sign 23 
1 
17 


18 
a_{012}
[rad] 
=B9*PI()/180 
19 
a_{023}
[rad] 
=B10*PI()/180 
20 
b_{12}
[rad] 
=B11*PI()/180 
21 
b_{23}
[rad] 
=B12*PI()/180 
22 
cos(b_{12}) 
=COS(B20) 
23 
tan(a_{012}) 
=TAN(B18) 
24 
tan(b_{12}) 
=TAN(B20) 
25 
_{12} 
=(1+B24^2)/2/PI()*(SQRT(((B5+2*B22)*B23)^2+4*B22^3*(B5+B22))+B15*SQRT(((B6+B15*2*B22)*B23)^2+B15*4*B22^3*(B6+B15*B22))(B5+B15*B6)*B23) 
26 
_{12} 
=B5^2*B22^2/(B5^2*(B23^2+B22^2)+2/3*PI()^2*B22^4*(B251)*(2*B251)+B13*2*PI()*B23*B5*B22^2*(B251)) 
27 
cos(b_{23}) 
=COS(B21) 
28 
tan(a_{023}) 
=TAN(B19) 
29 
tan(b_{23}) 
=TAN(B21) 
30 
_{23} 
=(1+B29^2)/2/PI()*(SQRT(((B7+2*B27)*B28)^2+4*B27^3*(B7+B27))+B16*SQRT(((B8+B16*2*B27)*B28)^2+B16*4*B27^3*(B8+B16*B27))(B7+B16*B8)*B28) 
31 
_{23} 
=B7^2*B27^2/(B7^2*(B28^2+B27^2)+2/3*PI()^2*B27^4*(B301)*(2*B301)+B14*2*PI()*B28*B7*B27^2*(B301)) 
32 


33 
i_{13H} 
=B6*B8/B5/B7 
34 
_{3H} 
=B26*B31 
35 
x 
=IF(B33<=1,1,1) 
36 
_{H3} 
=B34^B35*(1B33)/(1B34^B35*B33) 
The tooth mark in lines 1314, take
1 for outer teeth, and 1 for inner teeth.
Gear markings in lines 1516, are 1
for external gearing and 1 for internal gearing.
In system 6 there is an x exponent
that can take the value +1 or 1. It is identified in the computing program
through a logical function (see the line 35).
Next,
some examples of calculation are presented in the four Figure 47. It analyzed
the mechanism model shown in Figure 2. Inputoutput gear ratio achieved, iH3,
is: 11 (Figure 4), 10 (Figure 5), 5 (Figure 6), 0.2 (Figure 7). On the left
side of a Fig. using standard pressure angle of 20 degrees [deg]
and the right of every Fig. uses a pressure angle decreased to 10 degrees [deg].


Figure
4: i_{13}^{H}=1.1 
Figure
5: i_{13}^{H}=0.9(09) 


Figure
6: i_{13}^{H}=1.25 
Figure
7: i_{13}^{H}=1.(6) 
It was considered for all cases a
tilt teeth angle of 15 degrees [deg].
4. CONCLUSIONS
The
planetary system efficiency given by the original exact presented formula has
the great advantage to be determined easy and for any type of simple planetary
gear.
By this we have now high precision
in any asked case.
Planetary mechanisms have a
significant number of advantages over fixed axis transmission. Under similar
operating conditions, the planetary transmissions have a longer life and
produce less noise compared to a fixed shaft transmission.
The efficiency of the planetary
system obtained through the original formula presented has the great advantage
of being easily determined and can at the same time be used for any type of
simple planetary mechanisms. In other words, the great advantage of this
formula is its generality.
A second great advantage of this formula
is its precision.
Analyzing the previous calculation
examples, it can be noticed that the yield (of a simple planetary gear)
increases when the sun gear transmission ratio inputoutput decreases and also
when alfa0 angle is decreased.
References
Anderson, N. E.; S.H. Loewenthal, S. H.
(1986) Efficiency of nonstandard and high contact ratio involute spur gears. J.
Mech. Trans. Automat. Des., n. 108, p. 424432. DOI: 10.1115/1.3260774.
Antonescu P. (1979) Book: Mecanisme, calculul structural şi cinematic,
Editura IPB, Bucureşti.
Cao, W.; Ding, H.; Zi, B.; Chen, Z, (2013) New
structural representation and digitalanalysis platform for symmetrical
parallel mechanisms. Int. J. Adv. Robotic Syst., n. 10, p. 243255.
http://cdn.intechopen.com/pdfs/44679.pdf
Cho, S.; Ahn, K.; Lee,
J. M. (2006) Efficiency of the planetary gear
hybrid powertrain. SAGE Jo. Automobile Eng., n. 220, p. 14451454. DOI:
10.1243/09544070JAUTO176.
del Castillo, J. M. (2002) The analytical expression of the efficiency of planetary gear trains. Mech.
Machine Theory, n. 37, p. 197214. DOI: 10.1016/S0094114X(01)000775.
GARCIA, E.; JIMENEZ, M. A.; DE SANTOS, P. G.; ARMADA, M.
(2007) The evolution of robotics research. IEEE Robot. Automat. Magazine,
n. 14, p. 90103. DOI: 10.1109/MRA.2007.339608.
Lee, B. J. (2013) Geometrical derivation of differential kinematics to calibrate
model parameters of flexible manipulator. Int. J. Adv. Robot. Syst., n. 10,
p. 106115. DOI: 10.5772/55592.
Martin, K. F. (1981) The efficiency of involute spur gears. ASME J. Mech. Des.,
n. 103, p. 160169. DOI: 10.1115/1.3254855.
Pelecudi, C. H. R.; Maroş,
D.; Merticaru, v.; Pandrea, n.; Simionescu, I. (1985) Mecanisme, Editura Didactică şi Pedagogică,
Bucureşti.
PENNESTRI, E. E.; FREUDENSTEIN, F. F. (1993) The
mechanical efficiency of epicyclic gear trains. ASME.
J. Mech. Des., n. 115, p. 645651. DOI: 10.1115/1.2919239.
Petrescu, F. I. T.; Petrescu, R. V. (2011) Trenuri
Planetare. 1 Edn., CreateSpace Independent Publishing Platform,
ISBN10: 1468030418, p. 204.
Petrescu, F. I. T. (2012) Teoria Mecanismelor: Curs si Aplicatii. 1st Edn., CreateSpace Independent Publishing Platform,
ISBN10: 1479293628, p. 304.
PETRESCU, FIT., 2019. About the nuclear particles’
structure and dimensions. Comp. Part. Mech. Volume 6, Issue 2, Pages
191194. https://doi.org/10.1007/s4057101802067
Petrescu, R. V.; Aversa, R.; Apicella, A.;
Mirsayar, M. M.; Petrescu, F. I. T.
(2016) About the Gear Efficiency to a Simple Planetary Train. American
Journal of Applied Sciences, v. 13, n. 12, p. 14281436. DOI:
10.3844/ajassp.2016.1428.1436. (2016)
Petrescu, F. I. T.; Petrescu, R. V. (2014a) High efficiency gear. Facta Universitatis, Series:
Mech. Eng., n. 12, p. 5160.
Petrescu, F. I. T.; Petrescu, R. V. (2014b) High efficiency gears
synthesis by avoid the interferences. Independent J. Manage. Product., n.
5, p. 275298.