MACHINE MOTION EQUATIONS
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: 27/03/2015
Accept: 09/04/2015
ABSTRACT
This paper presents
the dynamic, original, machine motion equations. The equation of motion of the
machine that generates angular speed of the shaft (which varies with position
and rotation speed) is deduced by conservation kinetic energy of the machine.
An additional variation of angular speed is added by multiplying by the
coefficient dynamic D (generated by the forces out of mechanism and or by the
forces generated by the elasticity of the system). Kinetic energy conservation
shows angular speed variation (from the shaft) with inertial masses, while the
dynamic coefficient introduces the variation of w with forces acting in the
mechanism. Deriving the first equation of motion of the machine one can obtain
the second equation of motion dynamic. From the second equation of motion of
the machine it determines the angular acceleration of the shaft. It shows the
distribution of the forces on the mechanism to the internal combustion heat
engines. Dynamic, the velocities can be distributed in the same way as forces.
Practically, in the dynamic regimes, the velocities have the same timing as the
forces. Calculations should be made for an engine with a single cylinder.
Originally exemplification
is done for a classic
distribution mechanism, and then even
the module B distribution mechanism of an
Otto engine type.
Keywords: Machine
motion equations, Dynamic coefficient, Classical distribution, Cam dynamic
synthesis, Otto engine.
1. INTRODUCTION
In conditions which started to magnetic motors, oil
fuel is decreasing, energy which was obtained by burning oil is replaced with
nuclear energy, hydropower, solar energy, wind, and other types of
unconventional energy, in the conditions in which electric motors have been
instead of internal combustion in public transport, but more recently they have
entered in the cars world (Honda has produced a vehicle that uses a compact
electric motor and electricity consumed by the battery is restored by a system
that uses an electric generator with hydrogen combustion in cells, so we have a
car that burns hydrogen, but has an electric motor), which is the role and
prospects which have internal combustion engines type Otto or Diesel?
Internal combustion engines in four-stroke (Otto,
Diesel) are robust, dynamic, compact, powerful, reliable, economic, autonomous,
independent and will be increasingly clean (AMORESANO; AVAGLIANO; NIOLA;
QUAREMBA,2013; ANDERSON, 1984; ANGELAS; LOPEZ-CAJUN, 1988; ANTONESCU; PETRESCU;
ANTONESCU, 2000; ANTONESCU; OPREAN; PETRESCU, 1987; BARZEGARI; ANTONESCU, 2011;
BISHOP, 1950-51; CHOI; KIM, 1994; DE FALCO; DI MASSA; PAGANO; STRANO, 2013; DE
FALCO; DI MASSA; PAGANO, 2013; GANAPATHI; ROBINSON, 2013; GIORDANA, 1979; HAIN,
1971; HEYWOOD, 1988; HRONES, 1948; KARIKALAN; CHANDRASEKARAN; SUDHAGAR, 2013;
LEIDEL, 1997; MAHALINGAM; RAMESH BAPU, 2013; NAIMA; LIAZID, 2013; NARASIMAN;
JEYAKUMAR; MANI, 2013; PETRESCU, 2012a; PETRESCU, 2012b; PETRESCU; PETRESCU
1995; PETRESCU; PETRESCU 2005a; PETRESCU; PETRESCU, 2005b; PETRESCU; PETRESCU,
2005c; PETRESCU; ANTONESCU, 2008; PETRESCU; PETRESCU, 2011; PETRESCU; PETRESCU,
2013a; PETRESCU; PETRESCU, 2013b; PETRESCU; PETRESCU, 2013c; PETRESCU;
PETRESCU, 2013d; PETRESCU; PETRESCU, 2014; RAHMANI; DRAOUI; BOUANINI;
BENACHOUR, 2013; RAVI, 2013; RONNEY; SHODA; WAIDA; DURBIN, 1994; SAMIM;
ANTONESCU, 2008; SAPATE; TIKEKAR, 2013; SETHUSUNDARAM; ARULSHRI; MYLSAMY, 2013;
SHRIRAM; ANTONESCU, 2012; TARAZA, 2002; WANG, 2011; XIANYING; ANTONESCU, 2012;
ZAHARI; ABRAS; MAT ARISHAD; ZAINAL; MUHAMAD, 2013; ZHAO; ANTONESCU, 2012).
Magnetic motors (combined with the electromagnetic)
are just in the beginning, but they offer us a good perspective, especially in
the aeronautics industry.
Probably at the beginning they will not be used to act
as a direct transmission, but will generate electricity that will fill the
battery that will actually feed the engine (probably an electric motor).
The Otto engines or those with internal combustion in
general, will have to adapt to hydrogen fuel.
Its compound (the basic, hydrogen) can be extracted
industrially, practically from any item (or combination) through nuclear,
chemical, photonic by radiation, by burning, etc... (Most easily hydrogen can
be extracted from water by breaking up into constituent elements, hydrogen and
oxygen; by burning hydrogen one obtains water again that restores a circuit in
nature, with no losses and no pollution); in this mode one can have a water car
(which works not with the water, but with the hydrogen extracted by water).
If one uses stored hydrogen, then, hydrogen must be
stored in reservoirs cell (a honeycomb) for there is no danger of explosion;
the best would be if we could breaking up water directly on the vehicle, in
which case the reservoir would feed water (and there were announced some
successful).
As a backup, there are trees that can donate a fuel
oil, which could be planted on the extended zone, or directly in the consumer
court. With many years ago, Professor Melvin Calvin, (Berkeley University),
discovered that “Euphora” tree, a rare species, contained in its trunk a liquid
that has the same characteristics as raw oil. The same professor discovered on
the territory of Brazil, a tree which contains in its trunk a fuel with
properties similar to diesel.
During a journey in Brazil, the natives driven him
(Professor Calvin) to a tree called by them "Copa-Iba". From the tree
trunk begin flow a gold liquid, which was used as indigenous raw material base
for the preparation of perfumes or, in concentrated form, as a balm. Nobody see
that it is a pure fuel that can be used directly by diesel engines.
Calvin said that after he poured the liquid extracted
from the tree trunk directly into the tank of his car (equipped with a diesel),
engine functioned irreproachable.
In Brazil the tree is fairly widespread. It could be
adapted in other areas of the world, planted in the forests, and the courts of
people.
From a jagged tree is filled about half of the tank;
one covers the slash and it is not open until after six months; it means that
having 12 trees in a courtyard, a man can fill monthly a tank with the new
natural diesel fuel.
In some countries (USA, Brazil, Germany) producing
alcohol or vegetable oils, for their use as fuel.
In the future, aircraft will use ion engines,
magnetic, laser or various micro particles accelerated. Now, and the life of
the jet engine begin to end.
Recently it was announced that occurred in Germany
car that runs on
salt water. This means that we will not
put in tank oil
or water but salt
water.
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 in the world |
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. Even if we stop totally production of heat engines, would still
need minimum 50 years to eliminate total the existing car park in the current
rate.
Electric
current is still produced in majority by combustion of hydrocarbons, making the
hydrocarbon losses to be higher when we use electric motors. When we will have
electric current obtained only from green energy or nuclear, sustainable and
renewable energy sources, it is only then that we'll be able to enter gradually
and electric motors (PETRESCU 2012; PETRESCU; PETRESCU, 2013C; PETRESCU;
PETRESCU, 2013D; RAHMANI; DRAOUI; BOUANINI; BENACHOUR, 2013; RAVI, SUBRAMANIAN,
2013; RONNEY; SHODA; WAIDA; DURBIN, 1994; SAMIM; ANTONESCU, 2008; SAPATE;
TIKEKAR, 2013).
Otto and diesel engines are today the best solution for the transport of
our day-to-day work, together and with electric motors.
For these reasons it is imperative as we can calculate exactly the
engine efficiency, in order to can increase it permanently.
Even in these conditions internal combustion engines
will be maintained in land vehicles (at least), for power, reliability and
especially their dynamics. Thermal engine efficiency is still low and, about
one third of the engine power is lost just by the distribution mechanism (AMORESANO;
AVAGLIANO; NIOLA; QUAREMBA,2013; ANDERSON, 1984; ANGELAS; LOPEZ-CAJUN, 1988;
ANTONESCU; PETRESCU; ANTONESCU, 2000; ANTONESCU; OPREAN; PETRESCU, 1987;
BARZEGARI; ANTONESCU, 2011; BISHOP, 1950-51; CHOI; KIM, 1994; DE FALCO; DI
MASSA; PAGANO; STRANO, 2013; DE FALCO; DI MASSA; PAGANO, 2013; GANAPATHI;
ROBINSON, 2013; GIORDANA, 1979; HAIN, 1971; HEYWOOD, 1988; HRONES, 1948;
KARIKALAN; CHANDRASEKARAN; SUDHAGAR, 2013; LEIDEL, 1997; MAHALINGAM; RAMESH
BAPU, 2013; NAIMA; LIAZID, 2013; NARASIMAN; JEYAKUMAR; MANI, 2013; PETRESCU,
2012a; PETRESCU, 2012b; PETRESCU; PETRESCU 1995; PETRESCU; PETRESCU 2005a;
PETRESCU; PETRESCU, 2005b; PETRESCU; PETRESCU, 2005c; PETRESCU; ANTONESCU,
2008; PETRESCU; PETRESCU, 2011; PETRESCU; PETRESCU, 2013a; PETRESCU; PETRESCU,
2013b; PETRESCU; PETRESCU, 2013c; PETRESCU; PETRESCU, 2013d; PETRESCU;
PETRESCU, 2014; RAHMANI; DRAOUI; BOUANINI; BENACHOUR, 2013; RAVI, 2013; RONNEY;
SHODA; WAIDA; DURBIN, 1994; SAMIM; ANTONESCU, 2008; SAPATE; TIKEKAR, 2013;
SETHUSUNDARAM; ARULSHRI; MYLSAMY, 2013; SHRIRAM; ANTONESCU, 2012; TARAZA, 2002;
WANG, 2011; XIANYING; ANTONESCU, 2012; ZAHARI; ABRAS; MAT ARISHAD; ZAINAL;
MUHAMAD, 2013; ZHAO; ANTONESCU, 2012). Mechanical efficiency of cam mechanisms
was about 4-8%. In the past 20 years, managed to increase to about 14-18%, and
now is the time to pick it up again at up to 60%. This is the main objective of
this paper.
2. NOMENCLATURE
_{}: is the moment of inertia (mass or mechanical) reduced to the camshaft;
_{}: is the maximum
moment of inertia (mass or mechanical) reduced to the camshaft;
_{}: is the minimum
moment of inertia (mass or mechanical) reduced to the camshaft;
_{}: is the average moment of inertia (mass or
mechanical,
reduced to the camshaft) ;
_{}: is the first
derivative of the moment of inertia (mass or
mechanical, reduced to the camshaft) in relation with the j angle;
_{}: is the momentary efficiency of the cam-pusher
mechanism;
_{}: is the mechanical yield of the cam-follower
mechanism;
t: is the transmission angle;
_{}: is the pressure angle;
s: is the movement of the pusher;
h: is the follower
stroke; h=s_{max};
s’: is the first derivative in
function of j of the tappet
movement, s;
s’’: is the second derivative in
raport of j angle of the
tappet movement, s;
s’’’: is the third derivative of the
tappet movement s, in raport of the j angle;
x:
is the real, dynamic, movement of the pusher;
x’: is the real, dynamic, reduced
tappet speed;
x’’: is the real, dynamic, reduced
tappet acceleration;
_{}: is the real, dynamic, acceleration of the tappet
(valve).
_{}: is the normal (cinematic) velocity of the tappet;
_{}: is the normal (cinematic) acceleration of the
tappet;
j: is the
rotation angle of the cam (the position angle);
K: is the elastic
constant of the system;
k: is the elastic
constant of the valve spring;
x_{0}: is the valve spring
preload (pretension);
m_{c}: is the mass of the cam;
m_{T}: is the mass of the
tappet;
ω_{m}: the nominal angular
rotation speed of the cam (camshaft);
n_{c}: is the camshaft speed;
n=n_{m}: is the motor shaft
speed; n_{m}=2n_{c};
w: is the dynamic angular rotation
speed of the cam;
e: is the dynamic angular rotation
acceleration of the cam;
r_{0}: is
the radius of the base circle;
r=r: is the radius of the cam (the
position vector radius);
q: is the position vector angle;
x=x_{c} and
y=y_{c}: are the cartesian coordinates of the cam;
_{}: is the dynamic coefficient;
_{}: is the derivative of _{} in function of
the time;
_{}: is the derivative of _{} in function of
the position angle of the camshaft, j;
Fm: is the motor force;
Fr: is the resistant force.
3. Determining the first
machine equation
One presents the dynamic, original, machine motion equations. The
equation of motion of the machine that generates angular speed of the shaft
(which varies with position and rotation speed) is deduced by conservation
kinetic energy of the machine. An additional variation of angular speed is
added by multiplying by the coefficient dynamic (generated by the forces out of
mechanism) (ANTONESCU; OPREAN; PETRESCU, 1987; PETRESCU, 2012a; PETRESCU,
2012b; PETRESCU; PETRESCU, 1995; PETRESCU; PETRESCU, 2005a; PETRESCU; PETRESCU,
2005b; PETRESCU; PETRESCU, 2005c; PETRESCU; ANTONESCU, 2008; PETRESCU;
PETRESCU, 2011; PETRESCU; PETRESCU, 2013a; PETRESCU; PETRESCU, 2013b; PETRESCU;
PETRESCU, 2013c; PETRESCU; PETRESCU, 2013d; PETRESCU; PETRESCU, 2014).
Kinetic energy conservation shows angular speed variation (from the main
shaft) with inertial masses, while the dynamic coefficient introduces the
variation of w with forces acting in the mechanism (ANTONESCU; OPREAN; PETRESCU,
1987; PETRESCU; PETRESCU, 2005; PETRESCU; PETRESCU, 2014).
In
system (1) one determines the variable rotation velocity of the (cam) shaft, in
function of the position j of
the shaft, and of rotation nominal speed w_{m}. One
starts from the equation of kinetics energy (that is conserved).
_{} (1)
The first
movement equation of the machine takes the initial forms (2):
_{} (2)
Since J* is a function
of the angle j and w_{n} is a function of n,
it follows that w is
a function of angle j and angular rotation speed n=>_{}.
An additional variation of angular speed is added by multiplying by the
coefficient dynamic (generated by the forces from mechanism and out of
mechanism). The final forms of the first movement equation of the machine can
be seen in the system (3).
_{} (3)
Generally the coefficient D has two components, the first one (Dc)
generated by the forces from couples, and the second one generated by the elasticity forces of the system (De).
Dc was already presented in more published articles, but
its influence on dynamics of classical distribution is insignificant. For this
reason in this paper we’ll present only the new coefficient, De, which
introducing the elasticity effect of the system. In
all older articles this important effect was treated separately by integrating the equations of motion Lagrange
or Newton.
Dynamic equation obtained for the system elasticity effect was
complicated. But the
main reason for introducing this coefficient
is that the authors want all the dynamic influences
of system can be
introduced as a dynamic
factor. Even the
effect of inertial forces (the masses system)
appears as a dynamic
factor (see the system 4). In this way all the
dynamic influences appear as a change produced to the input angular
velocity.
_{} (4)
4. Determining the second
machine equation
The second machine equation is determined by the derivation of the first
machine equation in function of the time (see the system 5).
_{} (5)
5. Application to the Otto
Engine CLASSICAL DISTRIBUTION
One determines now the dynamics of the classical distribution at an Otto
engine (Figure 1) (ANTONESCU; OPREAN; PETRESCU, 1987; PETRESCU; PETRESCU,
2005; PETRESCU; PETRESCU, 2014) by an original method.
Figure 1: Classical distribution
First of all one determines the motor and resistant forces (system 6)
reduced at the axis of the follower.
_{} (6)
Now identify the
reduced moment of systems 5 and 6 and we obtain the relations of the system 7 (with D_{c}=1 and D_{c}’=0).
_{} (7)
For
the classical distribution we still use simplified dynamic relations (system
8).
_{} (8)
It also uses and relationships already known (system 9).
_{} (9)
For dynamic calculations must be determined:_{},_{}, _{}, _{}, _{} (relations
system 10).
_{} (10)
The normal angular velocity of the cam (camshaft) _{} may be
determined with the relation (11).
_{} (11)
The dynamic acceleration of the follower can be seen in the figure 2
(h=0.006 [m]; r_{0}=0.013 [m]; j_{0}=p/2
[rad]; m_{c}=0.2 [kg]; m_{T}=0.1 [kg]; n=n_{m}=5000
[rpm]; h=0.305; k=150000
[N/m]; x_{0}=0.03 [m]; a=4s). It utilizes cosine law of motion.
The cam profile may be seen in the Figure 3.
Figure 2: Classical distribution tappet dynamic
acceleration; a=4s.
Figure 3: Classical distribution cam profile for a=4s;
h=30%.
A more accurate determination of the dynamic acceleration
of plunger involves using an old dynamic model that takes into account the elasticity system
(relation 12-15) (ANTONESCU; OPREAN;
PETRESCU, 1987; PETRESCU;
PETRESCU, 2005; PETRESCU; PETRESCU, 2014).
_{} (12)
Where x is the dynamic movement of the pusher, while s is its normal,
kinematics movement. K is the spring constant of the
system, and k is the spring constant of the tappet spring. It note,
with x_{0} the tappet spring preload, with m_{T} the mass of
the tappet, with ω the angular rotation speed of the cam (or camshaft), where
s’ is the first derivative in function of j of the tappet movement, s (ANTONESCU; OPREAN; PETRESCU, 1987; PETRESCU; PETRESCU, 2005; PETRESCU; PETRESCU,
2014).
_{} (13)
_{} (14)
Further the acceleration of the tappet can be determined directly real
(dynamic) using the relation (15) (ANTONESCU; OPREAN; PETRESCU,
1987; PETRESCU; PETRESCU, 2005; PETRESCU; PETRESCU, 2014) (Figure
4).
_{} (15)
Figure
4: Classical distribution tappet real dynamic acceleration obtained with the
old dynamic systems: h=0.006 [m]; r_{0}=0.013 [m]; j_{0}=p/2
[rad]; m_{c}=0.2 [kg]; m_{T}=0.1[kg]; n=n_{m}=5000
[rpm]; h=0.305; k=150000
[N/m]; x_{0}=0.03[m].
In this
moment, we must leave the classical module C (Figure 1), and take the module B
(Figure 5) which may increases further the yield of the distribution mechanism.
6. Determining of
Momentary Dynamic Efficiency of the Rotary Cam and Translated Follower with
Roll
The pressure angle d (Figure
5), is determined by relations (6.5-6.6). One can write the next forces, speeds
and powers (6.13-6.18). F_{m}, v_{m}, are perpendicular to the
vector r_{A} at A. F_{m} is divided into F_{a} (the
sliding force) and F_{n} (the normal force). F_{n} is divided
also, into F_{i} (the bending force) and F_{u} (the useful
force). The
momentary dynamic efficiency can be obtained from relation (6.18).
Figure
5: Forces and speeds to the cam with translated follower with roll
The written relations are the following.
_{} (6.1)
_{} (6.2)
_{} (6.3)
_{} (6.4)
_{} (6.5)
_{} (6.6)
_{} (6.7)
_{} (6.8)
_{} (6.9)
_{} (6.10)
_{} (6.11)
_{} (6.12)
_{} (6.13)
_{} (6.14)
_{} (6.15)
_{} (6.16)
_{} (6.17)
_{} (6.18)
The used law is the classical law, cosine law.
The synthesis of the cam profile can be made with the relationships
(6.19) when the cam rotates clockwise and with the expressions from the
system (6.20) when the cam rotates counterclockwise (trigonometric).
_{} (6.19)
_{} (6.20)
The r_{0} (the radius of the base circle of the cam) is 0.013
[m]. The h (the maximum displacement of the tappet) is 0.020 [m]. The angle of
lift, j_{u} is p/3
[rad]. The radius of the tappet roll is r_{b}=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can
be seen in the Figure 6.
Figure 6:
The cosine profile at the cam with translated follower with roll; r_{0}=13[mm],
h=20[mm], j_{u}=p/3[rad], r_{b}=2[mm],
e=0[mm].
The obtained mechanical yield (obtained by integrating the instantaneous efficiency
throughout the climb
and descent) is
0.39 or h=39%. The
dynamic diagram can be seen in the Figure 7 (the dynamic setting are partial
normal). Valve spring preload 9 cm no longer poses today. Instead, achieve a long arc very hard (k=500000[N/m]), require special
technological knowledge.
Figure 7:
The dynamic diagram at the cosine profile at the cam with translated follower with roll; r_{0}=13[mm];
h=20[mm]; j_{u}=p/3[rad]; r_{b}=2[mm];
e=0[mm]; n=5500[rpm]; x_{0}=9[cm]; k=500[kN/m]
It tries increase the yield; angle of climb is halved j_{u}=p/6[rad] (see the profile in the Figure 8).
The r_{0} (the radius of the base circle of the cam) is 0.015
[m]. The h (the maximum displacement of the tappet) is 0.010 [m]. The angle of
lift, j_{u} is p/6
[rad]. The radius of the tappet roll is r_{b}=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can
be seen in the Figure 8.
Figure 8:
The cosine profile at the cam with translated follower with roll; r_{0}=15[mm],
h=10[mm], j_{u}=p/6[rad], r_{b}=2[mm],
e=0[mm].
The obtained mechanical yield (obtained by integrating the instantaneous efficiency
throughout the climb
and descent) is
0.428 or h=43%. The
dynamic diagram can be seen in the Figure 9 (the dynamic setting are not
normal). Valve spring preload 20 cm no longer poses today. Instead, achieve a long arc very-very hard (k=1500000[N/m]), require
special technological knowledge.
Figure 9:
The dynamic diagram at the cosine profile at the cam with translated follower with roll; r_{0}=15[mm];
h=10[mm]; j_{u}=p/6[rad]; r_{b}=2[mm];
e=0[mm]; n=5500[rpm]; x_{0}=20[cm]; k=1500[kN/m]
Camshaft runs at a shaft speed halved (n_{c}=n/2). If we more reduce camshaft speed by three times (n_{c}=n/6), we
can reduce and the preload of the valve spring (x_{0}=5[cm]); see the dynamic diagram in the Figure 10. However, in this case, the cam
profile should be tripled (see the Figure 11).
Figure 10: The dynamic
diagram at the cosine tripled profile at the cam with translated follower with roll; r_{0}=15[mm];
h=10[mm]; j_{u}=p/6[rad]; r_{b}=2[mm];
e=0[mm]; n=5500[rpm]; x_{0}=5[cm]; k=1500[kN/m]
Figure 11:
The cosine tripled profile at the cam with translated follower with roll; r_{0}=15[mm],
h=10[mm], j_{u}=p/6[rad], r_{b}=2[mm],
e=0[mm].
It tries increase the yield again; angle of climb is reduced to the
value j_{u}=p/8[rad]. The r_{0} (the radius of the base circle of
the cam) is 0.013 [m]. The h (the maximum displacement of the tappet) is 0.009
[m]. The angle of lift, j_{u} is p/8 [rad]. The radius of the tappet roll is r_{b}=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can
be seen in the Figure 12.
Figure 12:
The cosine profile at the cam with translated follower with roll; r0=13[mm],
h=9[mm], ju=p/8[rad], rb=2[mm],
e=0[mm].
The obtained mechanical yield (obtained by integrating the instantaneous efficiency
throughout the climb
and descent) is
0.538 or h=54%. The
dynamic diagram can be seen in the Figure 13 (the dynamic setting are not
normal). Valve spring preload 30 cm no longer poses today. Instead, achieve a long arc very-very hard (k=1600000[N/m]), require
special technological knowledge.
Figure 13:
The dynamic diagram at the cosine profile at the cam with translated follower with roll; r0=13[mm];
h=9[mm]; ju=p/8[rad]; rb=2[mm];
e=0[mm]; n=5000[rpm]; x0=30[cm]; k=1600[kN/m]
Camshaft runs at a shaft speed halved (n_{c}=n/2). If we more reduce camshaft speed by four times (n_{c}=n/8), we
can reduce and the preload of the valve spring, x_{0}=9[cm] and the elastic constant of the valve spring, k=15000[N/m]; see the
dynamic diagram in the Figure 14. However, in this case, the cam profile should
be fourfold (see the Figure 15).
Figure 14:
The dynamic diagram at the cosine fourfold profile at the cam with translated
follower with roll; r0=13[mm]; h=9[mm]; ju=p/8[rad];
rb=2[mm]; e=0[mm]; n=5000[rpm]; x0=9[cm]; k=15[kN/m]
Figure 15:
The cosine fourfold profile at the cam with translated follower with roll; r0=13[mm],
h=9[mm], ju=p/8[rad], rb=2[mm],
e=0[mm].
With the same angle of climb j_{u}=p/8[rad], can increase performance even further, if the size
tappet race take a greater value (h=12[mm]). The r_{0} (the radius of
the base circle of the cam) is 0.013 [m].
The h (the maximum displacement of the tappet) is 0.012 [m]. The angle
of lift, j_{u} is p/8
[rad]. The radius of the tappet roll is r_{b}=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can be seen in the Figure 16.
Figure 16:
The cosine profile at the cam with translated follower with roll; r0=13[mm],
h=12[mm], ju=p/8[rad], rb=2[mm],
e=0[mm].
For correct operation it is necessary to decrease the speed of the
camshaft four times, and all four times multiplication of the cam profile. Camshaft runs at a shaft speed halved (n_{c}=n/2).
If we more reduce camshaft speed by four times (n_{c}=n/8),
we can reduce and the preload of the valve spring, x_{0}=9[cm]. The elastic constant
of the valve spring is k=1500000[N/m]. See the dynamic diagram in the Figure 17.
However, in this case, the cam profile should be fourfold. The obtained mechanical yield is
0.60 or h=60%.
Figure 17.
The dynamic diagram at the cosine fourfold profile at the cam with translated follower with roll; r0=13[mm];
h=12[mm]; ju=p/8[rad]; rb=2[mm];
e=0[mm]; n=5000[rpm]; x0=9[cm]; k=1500[kN/m]
For now is necessary to stop here.
If we increase
h, or decrease the angle j_{u}, then is tapering
cam profile very much. We must stop now
at a yield value, h=60%.
The distribution mechanisms work with small efficiency
for
about 150 years;
this fact affects the total yield of the internal heat engines. Much of the mechanical energy of an
engine
is lost through the
mechanism of distribution. Multi-years the yield of the distribution mechanisms was only 4-8%. In the past 20 years it
has managed a lift
up to the value of 14-18%;
car pollution has decreased and
people have better breathing again. Meanwhile the number of vehicles has tripled and the pollution increased
again.
Now, it’s the time when we must try again to grow the
yield of the distribution mechanisms.
The paper presents an original method to increase the
efficiency of a mechanism with cam and follower, used at the distribution
mechanisms.
This paper treats only two modules: the mechanism with
rotary cam and plate translated follower (the classic module C) and the
mechanism with rotary cam and translated follower with roll (the modern module
B).
At the classical module C we can increase again the yield
to
about 30%. The
growth
is difficult. Dimensional parameters of the cam must be
optimized; optimization and synthesis of the cam profile are made dynamic, and
it must set the elastic (dynamic) parameters of the valve (tappet) spring: k
and x0.
The law used is not as important as the module used,
sizes and settings used. We take the classical law cosine; dimensioning the
radius cam, lift height, and angle of lift.
To grow the cam yield again we must leave the classic
module C and take the modern module B. In this way the efficiency can be as high as 60%.
Yields went increased from 4% to 60%, and we can consider
for the moment that we have gain importance, since we work with the cam and
tappet mechanisms.
If we increase h, or
decrease the angle ju, then is tapering cam profile very much. We must stop now at a yield value, h=60%.
In this work one
presents the dynamic, original, machine motion equations. The equation of
motion of the machine that generates angular speed of the shaft (which varies
with position and rotation speed) is deduced by conservation kinetic energy of
the machine. An additional variation of angular speed is added by multiplying
by the coefficient dynamic (generated by the forces out of mechanism). Kinetic
energy conservation shows angular speed variation (from the main shaft) with
inertial masses, while the dynamic coefficient introduces the variation of w
with forces acting in the mechanism. The first movement equation of the machine
takes the initial forms (system 2) and the final forms (systems 3-4).
The second machine equation is determined by the
derivation of the first machine equation in function of the time (system 5).
One determines then the dynamics of the classical
distribution at an Otto engine.
An important
way
to reduce losses of heat
engines
is how to achieve a good camshaft (a good distribution) mechanism. The profile
synthesis must be made dynamic. The presented relationships permit this.
AMORESANO, A.;
AVAGLIANO, V.; NIOLA, V.; QUAREMBA, G. (2013)
The Assessment of the
in-Cylinder Pressure by Means of the Morpho-Dynamical Vibration Analysis –
Methodology and Application, in IREME Journal, v. 7, n. 6,
September, p. 999-1006.
ANDERSON, R. B. (1984) The Fischer-Tropsch Synthesis, Academic
Press.
ANGELAS, J.; LOPEZ-CAJUN, C. (1988) Optimal
synthesis of cam mechanisms with oscillating flat-face followers. Mechanism
and Machine Theory 23, n. 1, p. 1-6.
ANTONESCU, P.; PETRESCU, F. I.; ANTONESCU, O.
(2000) Contributions to the Synthesis of The Rotary Disc-Cam Profile, In
VIII-th International Conference on the Theory of Machines and Mechanisms, Liberec, Czech
Republic, p. 51-56.
ANTONESCU, P.; OPREAN, M.;
PETRESCU, F. I. (1987) Analiza dinamică a mecanismelor de
distribuţie cu came. În al VII-lea
Simpozion Naţional de Roboţi Industriali, MERO'87, Bucureşti, v. III, p.
126-133.
BARZEGARI, A.; ANTONESCU,
O. (2011), A New Algorithm Based on Particle Swarm
Optimization for Solving Power Economic Dispatch Considering Valve-Point
Effects and Emission Constraints, I.RE.MO.S. Journal, June (Part B), v.
4, n. 3, p. 1303-1311.
BISHOP, J. L. H. (1950-51), An
analytical approach to automobile valve gear design. Inst. of Mech. Engrs.
Auto-Division Proc. 4, p. 150-160.
CHOI, J. K.; KIM, S. C. (1994) HYUNDAI
MOTOR CO. KOREA, An Experimental Study on the Frictional Characteristics in the
Valve Train System. (945046), In FISITA CONGRESS, 17-21 October, Beijing,
p. 374-380.
DE FALCO, D.; DI MASSA, G.; PAGANO, S.;
STRANO, S. (2013), Motorcycle Handlebar Dynamic Response: Theoretical and
Experimental Investigation, in IREME Journal, v. 7, n. 5, July, p.
795-801.
DE FALCO, D.; DI MASSA, G.; PAGANO, S.
(2013) A Full Scale Motorcycle Dynamic Rig, in IREME Journal, v. 7, n.
3, March, p. 519-526.
GANAPATHI, P.; ROBINSON, Y. (2013) Experimental
Investigation on the Performance, Emission and Combustion Characteristics of a
Diesel Engine Fuelled with Polymer Oil – Ethanol Blends, in IREME Journal,
v. 7, n. 5, July, p. 919-924.
GIORDANA, F. (1979), On the influence of
measurement errors in the Kinematic analysis of cam. Mechanism and Machine
Theory 14, n. 5, p. 327-340.
HAIN, K. (1971) Optimization of a cam mechanism
to give goode transmissibility maximal output angle of swing and minimal
acceleration. Journal of Mechanisms, v. 6, n. 4, p.419-434.
HEYWOOD, J. B. (1988) Internal
Combustion Engine Fundamentals, McGraw-Hill.
HRONES, J. A. (1948) An analysis of
Dynamic Forces in a Cam-Driver System, Trans. ASME, n. 70, p. 473-482.
KARIKALAN, L.; CHANDRASEKARAN, M.;
SUDHAGAR, K. (2013) Comparative Studies on Vegetable Oil Usage in CI Engines as
an Alternative to Diesel Fuel, in IREME Journal, v. 7, n. 4, May, p.
705-715.
LEIDEL, J. A. (1997) An Optimized
Low Heat Rejection Engine for Automotive Use, SAE paper No. 970068.
MAHALINGAM, S.; RAMESH BAPU, B.R.
(2013) Experimental and Emission Analysis of Rubber Seed Oil and Jatropha Oil
Blends with Diesel in Compression Ignition Engine, in IREME Journal, v.
7, n. 5, July, p. 955-959.
NAIMA, K.; LIAZID, A. (2013) Numerical
Investigation on Combustion Behaviors of Direct-Injection Spark Ignition Engine
Fueled with CNG-Hydrogen Blends, in IREME Journal, v. 7, n. 4, May, p.
652-663.
NARASIMAN, V.; JEYAKUMAR, S.; MANI, M.
(2013) Optimizing the Compression Ratio of C.I. Engine Fuelled in Sardine Oil
Ethyl Ester, in IREME Journal, v. 7, n. 3, p. 463-467.
PETRESCU, F. I. (2012a) Bazele
analizei și optimizării sistemelor cu memorie rigidă – curs și aplicații,
Create Space publisher, USA, ISBN 978-1-4700-2436-9, 164 pages, Romanian
edition.
PETRESCU, F. I. (2012b) Teoria
mecanismelor – Curs si aplicatii (editia a doua), Create Space publisher,
USA, September, ISBN 978-1-4792-9362-9, 284 pages, Romanian version.
PETRESCU, F. I. ; PETRESCU R. V.
(1995) Contributii la sinteza mecanismelor de distributie ale motoarelor cu
ardere internã. In E.S.F.A.’95, v. 1, p.
257-264, Bucuresti, May.
PETRESCU, F. I.; PETRESCU R. V. (2005a) Contributions
at the dynamics of cams. In the Ninth
IFToMM International Symposium on Theory of Machines and Mechanisms, SYROM
2005, Bucharest, Romania, v. I, p. 123-128.
PETRESCU, F. I.; PETRESCU, R. V. (2005b)
Determining the dynamic efficiency of
cams. In the Ninth IFToMM
International Symposium on Theory of Machines and Mechanisms, SYROM 2005,
Bucharest, Romania, v. I, p. 129-134.
PETRESCU, F. I.; PETRESCU, R. V. (2005c)
Popescu N., The efficiency of cams.
In the Second International Conference Mechanics and Machine Elements,
Technical University of Sofia, November 4-6, Sofia, Bulgaria, v. II, p.
237-243.
PETRESCU, F. I.; ANTONESCU, O. (2008) Cams Dynamic Efficiency
Determination. In New Trends in Mechanisms, Ed. Academica
- Greifswald, I.S.B.N. 978-3-940237-10-1, p. 49-56.
PETRESCU, F. I.; PETRESCU, R. V.
(2011) Dinamica mecanismelor de distributie, Create Space publisher,
USA, December, ISBN 978-1-4680-5265-7, 188 pages, Romanian version.
PETRESCU, F. I.; PETRESCU, R. V. (2013a) An
Algorithm for Setting the Dynamic Parameters of the Classic Distribution
Mechanism, In IREMOS Journal, ISSN: 1974-9821, v. 6, n. 5,
Part B, October, p. 1637-1641.
PETRESCU, F. I.; PETRESCU, R. V. (2013b) Dynamic
Synthesis of the Rotary Cam and Translated Tappet with Roll, In IREMOS
Journal, ISSN: 1974-9821, v. 6, n. 2, Part B, April, p. 600-607.
PETRESCU, F. I.; PETRESCU, R. V.
(2013c) Forces and Efficiency of Cams, in IREME Journal, v. 7, n. 3,
March, p. 507-511.
PETRESCU, F. I.; PETRESCU, R. V.
(2013d) Cams with High Efficiency, in IREME Journal, v. 7, n. 4, May, p.
599-606.
PETRESCU, F. I.; PETRESCU, R. V. (2014) Cam
Gears Dynamics in the Classic Distribution. In Independent Journal of
Management & Production, ISSN: 2236-269X, v. 5, n. 1, January, p.
166-185.
RAHMANI, L.; DRAOUI, B.; BOUANINI, M.;
BENACHOUR, E. (2013) CFD Study on Heat Transfer to Bingham Fluid During with
Gate Impeller, in IREME Journal, v. 7, n. 6, September, p. 1074-1079.
RAVI, S. (2013) Subramanian, R.,
Diesel Fuel Additives: an Overview, in IREME Journal, v. 7, n. 4, May,
p. 698-704.
RONNEY, P. D.; SHODA, M.; WAIDA, S. T.;
DURBIN, E. J. (1994) Throttleless Premixed-Charge Engines: Concept and
Experiment, in Journal of Automobile Engineering, v. 208, p. 13-24.
SAMIM, Y.; ANTONESCU, O. (2008) Analytical
Dynamic Response of Elastic Cam-Follower Systems with Distributed Parameter
Return Spring, Journal of Mechanical Design (ASME), v. 115, n. 3,
(online June), p. 612-620.
SAPATE, K. D.; TIKEKAR, A. N. (2013) Engine
Mapping for Improvement in Fuel Efficiency of Two Stroke SI Engine, in IREME
Journal, v. 7, n. 3, March, p. 392-394.
SETHUSUNDARAM, P. P.; ARULSHRI, K. P.;
MYLSAMY, K. (2013) Biodiesel Blend, Fuel Properties and its Emission
Characteristics Sterculia Oil in Diesel Engine, in IREME Journal, v. 7, n.
5, July, p. 925-929.
SHRIRAM, R.; ANTONESCU, O. (2012) Design and
Development of Camless Valve Train for I.C. Engines, IREME
Journal,
July, v. 6, n. 5, p. 1044-1049.
TARAZA D. (2002) Accuracy Limits of IMEP
Determination from Crankshaft Speed Measurements, SAE Transactions, Journal of Engines, n. 111, p. 689-697.
WANG, W. (2011) Creation Design of Cam
Mechanism Based on Reverse Engineering, Advanced Materials Research Journal,
v. 230-232, p. 453-456.
XIANYING, F.; ANTONESCU, O. (2012) Meshing
Efficiency of Globoidal Indexing Cam Mechanism with Steel Ball, Advanced
Materials Research Journal, v. 413, p. 414-419.
ZAHARI, I.; ABRAS, M. A.; MAT ARISHAD,
N. I.; ZAINAL, S. F.; MUHAMAD, M. F. (2013) Experimental Study to Identify
Common Engine Part Load Conditions between Malaysian City Driving and NEDC
Test, in IREME Journal, v. 7, n. 6, September, p. 1152-1158.
ZHAO H. D.; ANTONESCU, O. (2012), Research on
Dynamic Behavior of Disc Indexing Cam Mechanism Based on Virtual Prototype
Technology, Key Engineering Materials Journal, v. 499, p. 277-282.