HIGH
EFFICIENCY GEARS SYNTHESIS BY AVOID THE INTERFERENCES
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:
27/09/2013
Revision:
19/10/2013
Accept:
20/10/2013
ABSTRACT
The paper presents an original
method to determine the efficiency of the gear, the forces of the gearing, the
velocities and the powers. It is analyzing the influence of a few parameters
concerning gear efficiency. These
parameters are: z_{1 }  the number of teeth for the primary wheel
of gear; z_{2 }  the number of
teeth of the secondary wheel of gear; alpha_{0}  the normal pressure
angle on the divided circle; beta  the inclination angle. With the relations
presented in this paper, it can synthesize the gear’s mechanisms. Today, the
gears are present everywhere, in the mechanical’s world (In vehicle’s
industries, in electronics and electrotechnique equipments, in energetically
industries, etc). Optimizing this mechanism (the gears mechanism), we can
improve the functionality of the transmissions with gears. At the gear
mechanisms an important problem is the interference of the teeth. To avoid the
interference between teeth, we must know the minimum number of teeth of the
driving wheel, in function of the pressure angle (normal on the pitch circle,
alpha0), in function of the tooth inclination angle (beta), and in function of
the transmission ratio (i).
Keywords: gears; gear efficiency; forces; velocities; powers; number of teeth; avoid the
interference.
1.
INTRODUCTION
Gears, broke today in all fields. They have the
advantage of working with very
high efficiency. Additionally gears can transmit large
loads. Regardless of
their size, gear must
be synthesized carefully considering the specific conditions. This
paper tries to present the main
conditions that must be met for
correct synthesis of a gear (PETRESCU, 2012; PETRESCU;
PETRESCU, 2002; PETRESCU; PETRESCU, 2003; PETRESCU; PETRESCU; POPESCU, 2007).
Top of the use of sprocket mechanisms
must be sought in ancient Egypt with at least a thousand years before Christ.
Here were used for the first time, transmissions wheeled "spurred" to
irrigate crops and worm gears to the cotton processing (LIN C., 2011; LEI
X., 2011). With 230 years BC, in the city of Alexandria in Egypt, they have been
used the wheel with more levers and gear rack. Such gears have been constructed
and used beginning from the earliest times, to the top for lifting the heavy
anchors of vessels and for claim catapults used on the battlefields. Then, they were introduced in cars with wind and
water (as a reducing or multiplying at the
pump from windmills or water), (see Figure 1).
Figure 1: Transmissions wheeled
"spurred" to irrigate crops and worm gears to the cotton processing
The
Antikythera Mechanism is the name given to an astronomical calculating device,
measuring about 32 by 16 by 10 cm, which was discovered in 1900 in a sunken
ship just off the coast of Antikythera, an island between Crete and the Greek
mainland. Several kinds of evidence point incontrovertibly to around 80
B.C. for the date of the shipwreck. The device, made of bronze gears fitted in
a wooden case, was crushed in the wreck, and parts of the faces were lost,
"the rest then being coated with a hard calcareous deposit at the same
time as the metal corroded away to a thin core coated with hard metallic salts
preserving much of the former shape of the bronze" during the almost 2000 years
it lay submerged. (See Antikythera 1, in Figure 2).
Figure
2: The Antikythera Mechanism is the name given
to an astronomical calculating device
Modern adventure began
with the gear wheel spurred of Leonardo da Vinci, in the fifteenth century. He founded the new kinematics and dynamics
stating inter alia the principle of superposition of independent movements (Figure 3).
Figure
3: The modern adventure began with the gear
wheel spurred of Leonardo da Vinci, in the fifteenth century
Benz had engine with
transmissions sprocket gearing and Gear chain (patented after 1882, Figure 4), but the first gearing patent (the
drawings of a patent first gear transmission) and of gearing wheels with chain was
made in 1870 by the British Starley
& Hillman (REY, 2013).
Figure
4: The Benz patent
After 1912, in
Cleveland (USA), begin to produce industrial specialized wheels and gears
(cylindrical, worm, conical, with straight teeth, inclined or curved; see Figure 5).
Figure
5: In Cleveland, after 1912 begin to produce industrial specialized wheels
Today, the gears are present
everywhere, in the mechanical’s world (In vehicle’s industries, in electronics
and electrotechnique equipment’s, in energetically industries, etc.; Figure
6).
Figure 6. Gearing today
2.
GEARS SYNTHESIS
The calculating relations (PETRESCU,
2012), are the next (2.12.21).
2.1.
Determining
the momentary dynamic (mechanical) efficiency, the forces of the gearing, and
the velocities
The forces, velocities, powers and
efficiency can be obtained with the relationships 2.12.6; the forces from
gearing may be seen in the Figure 7.
Figure
7: The forces and the velocities of the
gearing
_{} (2.1)
with:
_{} the motive force (the driving force);
_{} the transmitted force (the useful force);
_{} the slide force (the lost force);
_{} the velocity of element 1, or the speed of wheel 1 (the
driving wheel);
_{} the velocity of element 2, or the speed of wheel 2 (the
driven wheel);
_{} the relative speed of the wheel 1 in relation with the
wheel 2 (this is a sliding speed).
The consumed power (in this case the driving power):
_{} (2.2)
The
useful power (the transmitted power from the profile 1 to the profile 2) will
be written:
_{} (2.3)
The
lost power will be written:
_{} (2.4)
The
momentary efficiency of couple will be calculated directly with the next
relation:
_{} (2.5)
The
momentary losing coefficient (PETRESCU, 2012), will be written:
_{} (2.6)
It
can easily see that the sum of the momentary efficiency and the momentary
losing coefficient is 1.
Now,
one can determine the geometrical elements of gear. These elements will be used
in determining the couple efficiency, η.
2.2.
The geometrical elements of the
gear
We
can determine the next geometrical elements of the external gear, (for the
right teeth, b=0), (PETRESCU,
2012).
The radius of the basic circle of the
wheel 1 (of the driving wheel), (2.7).
_{} (2.7)
The
radius of the outside circle of wheel 1 (2.8):
_{} (2.8)
It determines now the maximum
pressure angle of the gear (2.9):
_{} (2.9)
And
now one determines the same parameters for the wheel 2, the radius of basic
circle (2.10) and the radius of the outside circle (2.11) for the wheel 2:
_{} (2.10)
_{} (2.11)
Now
it can determine the minimum pressure angle of the external gear (2.12, 2.13):
_{} (2.12)
_{} (2.13)
Now
we can determine, for the external gear, the minimum (2.13) and the maximum
(2.9) pressure angle for the right teeth. For the external gear with bended teeth
(b¹0) it uses the relations (2.14, 2.15 and 2.16):
_{} (2.14)
_{} (2.15)
_{} (2.16)
For
the internal gear with bended
teeth (b¹0) it uses the relations (2.14
with 2.17, 2.18A, or with 2.19, 2.20B):
A.
When
the driving wheel 1, has external teeth:
_{}
(2.17)
_{} (2.18)
B.
When the driving wheel 1, have internal teeth:
_{} (2.19)
_{} (2.20)
2.3.
Determining
the efficiency of the gear
The
efficiency of the gear will be calculated through the integration of momentary
efficiency on all sections of gearing movement, namely from the minimum
pressure angle to the maximum pressure angle, the relation (2.21), (PETRESCU,
2012).
_{} (2.21)
2.4.
External and internal gearing
Exterior gearing
are the most common, not because it would be best but because they are easier
to designed and constructed (Figure 8). Interior
gearing may be more efficient and more reliable when they are projected
properly (Figure 9).


Figure
8. An external gearing 
Figure
9. An internal
gearing 
To an external
gearing contact between profiles shall only be made to a single point, while at
the internal gearing the contact between profiles is by winding each other (see
the Figure 10 and (PETRESCU, 2012)).
Figure 10. Contact between profiles
3.
GEARS SYNTHESIS BY
AVOID THE INTERFERENCES
In order to avoid interference
phenomenon, point A must lie between C and K_{1} (the addendum circle
of the wheel 2, C_{a2} need to cut the line of action between points C
and K_{1}, and under no circumstances does not exceed the point K_{1}).
Similarly, C_{a1} addendum circle must cut the action line between
points C and K_{2}, resulting point E, which in no circumstances, does
not exceed the point K_{2}.
The conditions to avoid the
phenomenon of interference can be written with the relations (3.1).
The basic
conditions of interference, are the same (CA<K_{1}C; CE<K_{2}C),
but the originality of this new presented method consist in the mode in which
it was solved the classical relationship (see the system 3.1) (PETRESCU,
2012), (see the Figure 11).
Figure 11.
Line of action (tt’) at an external gearing
_{} (3.1)
Relationship which generates _{} always gives lower values than the relationship which generates _{} so it is sufficient
the condition (3.2) for finding the minimum number of teeth of the wheel 1,
necessary to avoid interference (PETRESCU, 2012).
_{} (3.2)
When we
have inclined teeth, one takes z_{min}àz_{min}/cosb, and a_{0}àa_{0t}, and the relationship (3.2) takes the form (3.3). The
minimum number of teeth of the driving wheel 1, is a function on some
parameters: the pressure angle (normal on the pitch circle, a_{0}),
the tooth inclination angle (b),
and the transmission ratio (i=i_{12}=z_{2}/z_{1}=z_{2}/z_{1}),
(see the relationship 3.3, and (MAROŞ, 1958); STOICA, 1977).
_{} (3.3)
The
system (3.3) is a simple, unitar and general relationship which can give the
solutions of the minimum number of teeth of the wheel 1 (the driving wheel), to
avoid the interference. In the following tables (115) is chosen an alpha0
value (35 [deg]), and successively increased beta angle values (from 0 [deg]
to 40 [deg]) and the transmission ratio i (from 1, to 80), and one gets the
minimum numbers of teeth. Then, we will decrease successively the value of the
angle alpha0 (from 35 [deg] to 5 [deg]). See the tables (115).
At the
internal gearing the condition to avoid the interference is the same like at
the external gearing (relation 3.3).
In
addition it can write and the condition of the existence of the wheel with the
internal teeth (systems 3.4 and 3.5).
_{} (3.4)
_{} (3.5)
They
were used and additional relations (3.6).
_{} (3.6)
Table 1 a0=35
[deg], b=0 [deg]
Table
2 a_{0}=35 [deg], b=10
[deg]
Table 3 a0=35 [deg], b=20 [deg]
Table 4 a0=35 [deg], b=30 [deg]
Table 5 a0=35 [deg], b=40 [deg]
Table 6 a0=20 [deg], b=0 [deg]
Table 7 a0=20 [deg], b=10 [deg]
Table 8 a0=20 [deg], b=20 [deg]
Table 9 a0=20 [deg], b=30 [deg]
Table 10 a0=20 [deg], b=40 [deg]
Table 11 a0=5 [deg], b=0 [deg]
Table 12 a0=5 [deg], b=10 [deg]
Table 13 a0=5 [deg], b=20 [deg]
Table 14 a0=5 [deg], b=30 [deg]
Table 15 a0=5 [deg], b=40 [deg]
The presented method has the great
advantage to optimize the number of teeth for a gear before to make its
synthesis. In this mode the constructor may elect the minimum number of teeth,
for an imposed transmission ratio, i.
Classical to realize an i=2, the
constructor can select between 18 or 33 teeth to the driving wheel 1, which
means a 36 or 66 teeth for the driven wheel 2. With the aid of the presented tables, he
can make a multiple selection.
The
engineer can select for the driving wheel 1 a number of teeth z_{1}=6,
with an alpha_{0}=35 [deg], and a beta=0 [deg]. He may do this not only
for a transmission ratio i=2, but and for the domain from i=1.25 to i=25 (see
the table 1).
If
he elect alpha_{0}=35 [deg] and beta=40 [deg], then he can take a
number of teeth for the driving wheel 1 of z_{1}=4, and can do this for
the entire domain from i=2 to i=80 (see the table 5). The constructor may do
this when it is necessary a minimum number of teeth, but with an efficiency of
the gear decreasing.
Contrary,
when we wish a great efficiency, one must increase the number of teeth and
decrease the angles alpha_{0} and beta. With an alpha_{0}=5 [deg] and
beta=0 (table 11), we can take the number of teeth at the driving wheel 1, from
z_{1}=189 (for i=1.25), to z_{1}=262 (for i=80).
For the known classical alpha_{0}=20
[deg] and beta=0 [deg], when the ratio i vary from 1 to 80, z_{1} takes
the values from 13 to 18 (see the table 6). With the
classical method it was taken only the minimum value 18 to the minimum number
of teeth (PETRESCU, 2012).
4.
CONCLUSIONS
The presented method manages to synthesize
(in theory) the best option parameters for any desired
gear.
Relationships shown have the
great advantage of donating optimal
solutions for any situation
you want without the need for difficult calculations, experimental building, or
specialized software. All relationships
have been calculated and checked with programs written in excel.
Comparisons made with specialized
software (Inventor) showed a precision
(matching) perfect. Workload and
procedures could be so much smaller.
The parallel drawn between the software "Autodesk
Inventor" and the presented calculation
relationships will be highlighted in the following paper (as handle a
large volume), (OLIVEIRA; LIMA, 2003).
Applied in the automotive
industry, at the transmission mechanisms, these changes
may decrease overall fuel consumption further, and pollutant emissions (NOGUEIRA; REAL, 2011).
Gears can be designed to operate
without noise (PELECUDI
et al., 1985), (PETRESCU;
PETRESCU; POPESCU, 2007; MAROŞ, 1958),
(LENI MATOS, 2011; NOGUEIRA;
REAL, 2011).
But, the applications will be spectacular
in the automatic
transmissions used in aerospace, in robotics and mechatronics.
When the number of teeth of the
wheel 1 increases, it can decrease the normal pressure angle, a_{0}. One shall see that for z_{1}=90
it can take less for the normal pressure angle (for the pressure angle of
reference), a_{0}=8^{0}.
The efficiency (of the gear)
increases when the number of teeth for the driving wheel 1, z_{1},
increases as well, and when the pressure angle, α_{0},
diminishes; z_{2} or i_{12} are not so much influence
about the efficiency value;
It can easily see that for the value
α_{0}=20^{0},
the efficiency takes roughly the value η≈0.89 for any values of the others parameters (this
justifies the choice of this value, α_{0}=20^{0}, for the standard
pressure angle of reference).
The
better efficiency may be obtained only for a α_{0}≠20^{0}.
But the pressure angle of reference,
α_{0}, can
be decreased the same time the number of
teeth for the driving wheel 1, z_{1}, increases, to increase the gears’
efficiency;
Contrary, when we desire to create a
gear with a low z_{1} (for a less gauge), it will be necessary to
increase the α_{0}
value, for maintaining a positive value for α_{m} (in
this case the gear efficiency will be diminished);
When β increases, the
efficiency, η,
increases too, but the growth is insignificant.
The module of the gear, m, has
not any influence on the gear’s efficiency value.
When
α_{0} is diminished it can take a higher normal module, for increasing
the addendum of teeth, but the increase of the module m at the same time with
the increase of the z_{1 }can lead to a greater gauge.
The gears’ efficiency, η, is really a
function of α_{0} and
z_{1}: η=f(α_{0},z_{1});
α_{m} and α_{M } are just the intermediate parameters.
For a good projection of the gear,
it’s necessary a z1 and a z2 greater than 3060; but this condition may
increase the gauge of mechanism.
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.
For
the external gearing, the best efficiency is obtained when the bigger wheel is
the drive wheel; when we decrease the normal angle a0, the
contact ratio increases and the efficiency increases as well.
The efficiency increases too, when the
number of teeth of the drive wheel 1 increases (when z1 increases).
Optimizing this mechanism (the gears
mechanism), we can improve the functionality of the transmissions with gears.
At
the gear mechanisms an important problem is the interference of the teeth. To
avoid the interference between teeth, we must know the minimum number of teeth
of the driving wheel, in function of the pressure angle (normal on the pitch
circle, alpha0), in function of the tooth inclination angle (beta), and in
function of the transmission ratio (i).
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