AN OTTO ENGINE DYNAMIC MODEL
Florian Ion Tiberiu Petrescu
Bucharest Polytechnic University, Romania
E-mail: petrescuflorian@yahoo.com
Relly Victoria Virgil Petrescu
Bucharest Polytechnic University, Romania
E-mail: petrescuvictoria@yahoo.com
Submission: 07/10/2015
Accept: 23/10/2015
ABSTRACT
Otto engine
dynamics are similar in almost all common internal combustion engines. We can
speak so about dynamics of engines: Lenoir, Otto, and Diesel. The dynamic
presented model is simple and original. The first thing necessary in the
calculation of Otto engine dynamics, is to determine the inertial mass reduced
at the piston. One uses then the Lagrange equation. Kinetic
energy conservation shows angular speed variation (from the shaft) with
inertial masses. One uses and elastic
constant of the crank shaft, k. Calculations should be made for an
engine with a single cylinder. Finally it makes a dynamic analysis of the
mechanism with discussion and conclusions. The ratio between the crank length r
and the length of the connecting-rod l is noted with landa. When landa increases the mechanism dynamics is deteriorating.
For a proper operation is necessary the reduction of the ratio landa,
especially if we want to increase the engine speed. We can reduce the
acceleration values by reducing the dimensions r and l.
Keywords: Otto engine, Dynamics, Lagrange
equation, Dynamic model, Shaft elastic constant.
1. INTRODUCTION
The dynamic study of mechanisms Otto
engine type is
most important to predict how that will
work in real engines.
Some
Otto engine dynamic models were presented by: (AMORESANO, 2013; DAWSON, 2005; DE FALCO, 2013A-B; GUZZELLA,
2004; HEYWOOD, 1988; PETRESCU, 2005, 2009, 2012a-b, 2014a-c, 2015; RAMOS, 1989).
Almost a quarter of the planet's population works directly or indirectly for the
construction of machines. Most
specialists are involved in the development and production of road vehicles.
If Otto engine production would
stop right now, they will still working
until at least about 40-50 years to complete replacement of the existing fleet
today.
Old gasoline engines carry us every day for nearly 150 years. “Old
Otto engine” (and his brother, Diesel) is today: younger, more
robust, more dynamic, more powerful,
more economical, more independent, more reliable, quieter, cleaner, more compact, more sophisticated, more stylish,
more secure, and more especially necessary and wanted. At the global
level we can manage to remove annually about 60,000 cars. But annually appear
other million cars (see the table 1).
Table 1. World cars produced
|
year |
cars produced |
|
2011 |
59,929,016 |
|
2010 |
58,264,852 |
|
2009 |
47,772,598 |
|
2008 |
52,726,117 |
|
2007 |
53,201,346 |
|
2006 |
49,918,578 |
|
2005 |
46,862,978 |
|
2004 |
44,554,268 |
|
2003 |
41,968,666 |
|
2002 |
41,358,394 |
|
2001 |
39,825,888 |
|
2000 |
41,215,653 |
|
1999 |
39,759,847 |
In full energy crisis since 1970 until today, production
and sale of
cars equipped with
internal combustion heat
engines
has skyrocketed, from some millions yearly to over sixty
millions yearly now, and the world fleet started from tens of millions reached today the
billion. As long as we produce electricity and heat by burning fossil fuels is
pointless to try to replace all thermal engines with electric motors, as loss
of energy and pollution will be even larger. However, it is well to
continuously improve the thermal engines, to reduce thus fuel consumption.
Planet supports now about one billion motor vehicles in circulation.
Otto and diesel engines
are today the best solution for the transport of our day-to-day work, together
and with electric motors.
Even in these conditions
internal combustion engines will be maintained in land vehicles (at least), for
power, reliability and especially their dynamics.
2. DETERMINING THE FIRST EQUATIONS
The first thing necessary
in the calculation of Otto engine dynamics, is to determine the inertial mass
reduced at the piston (1).
(1)
Then it derives the reduced mass to
the crank position angle (2).
(2)
Lagrange equation is written in the
form (3).
(3)
Were used for piston the next
kinematics parameters (4).
(4)
3. DYNAMIC EQUATIONS
The dynamic equation of motion of
the piston, obtained by integrating the Lagrange equation (3), takes the form
5.
(5)
Dynamic reduced velocity (6) and
dynamic reduced acceleration (7) are obtained by derivation:
(6)
(7)
Angular velocity is obtained through kinetic energy conservation (8-12).
(8)
(9)
(10)
(11)
(12)
Dynamic
velocity (13) and kinematics velocity (14) are written:
(13)
(14)
Dynamic
acceleration (15) and kinematics acceleration (16) are written:
(15)
(16)
4. NOTATIONS AND FIGURES
In the Figure 1 it presents the
crank shaft.
The relation (17) determines the
elastic constant of the crank shaft, k.
(17)
For the masses one uses the
notations (18); see the Figure 2.
the ratio between lengths of crank and rod;
the mass of the piston, with piston bolt and segments;
the mass of the rod;
Figure
1: Crank Shaft
(18)
The
parameters c1-c4 take the forms (19):
(19)
The moment of inertia can be determined with
the relation (20).
(20)
The
crank length, r, and the length of the connecting-rod, l, can be seen in the kinematics
schema of an Otto mechanism (Figure 2).
Figure
2: Otto mechanism kinematics schema
5. DYNAMIC ANALYSIS OF THE MECHANISM, DISCUSSION AND
CONCLUSION
When increases the
mechanism dynamics is deteriorating.
r=0.25
[m] l=0.3 [m]
For n=8000 [r/m] the mechanism is
working normally (see the accelerations diagram from the picture 3):
Figure
3: Dynamic and kinematics accelerations; n=8000 [r/m];
r=0.25 [m] l=0.3 [m]
At n=9000 [r/m] the mechanism work
abnormally (see the accelerations diagram from the picture 4):
Figure
4: Dynamic and kinematics accelerations; n=9000 [r/m]; r=0.25[m];l=0.3[m]
For a proper operation is necessary
reduction of the ratio, especially if we want to increase the engine speed (see the
next diagrams).
Figure
5: Dynamic and kinematics accelerations; n=12000 [r/m];
r=0.25[m];l=0.6[m]
Figure
6: Dynamic and kinematics accelerations; n=14000 [r/m];
r=0.25[m];l=0.9[m]
We can reduce the acceleration
values by reducing r and l.
Figure
7: Dynamic and kinematics accelerations; n=15000 [r/m];
r=0.05[m];l=0.15[m]
Figure
8: Dynamic and kinematics accelerations; n=50000 [r/m];
r=0.003[m];l=0.009[m]
One can reduce the acceleration
values especially if we want to increase the engine speed by reducing r and l
(the lengths of crank and rod).
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