Florian Ion Tiberiu Petrescu
IFToMM, Romania
E-mail: fitpetrescu@gmail.com
Relly Victoria Virgil Petrescu
IFToMM, Romania
E-mail: rvvpetrescu@gmail.com
Submission: 11/12/2018
Revision: 11/28/2018
Accept: 12/6/2018
ABSTRACT
The paper presents a dynamic model that works with variable
internal damping, applicable directly to rigid memory mechanisms. If the
problem of elasticity is generally solved, the problem of system damping is not
clear and well-established. It is usually considered a constant "c"
value for the internal damping of the system and sometimes the same value c and
for the damping of the elastic spring supporting the valve. However, the
approximation is much forced, as the elastic spring damping is variable, and
for the conventional cylindrical spring with constant elasticity parameter (k)
with linear displacement with force, the damping is small and can be considered
zero. It should be specified that damping does not necessarily mean stopping
(or opposition) movement, but damping means energy consumption to brake the
motion (rubber elastic elements have considerable damping, as are hydraulic
dampers). Metal helical springs generally have a low (negligible) damping. The
braking effect of these springs increases with the elastic constant (the
k-stiffness of the spring) and the force of the spring (P0 or F0) of the spring
(in other words with the arc static arrow, x0=P0/k). Energy is constantly
changing but does not dissipate (for this reason, the yield of these springs is
generally higher). The paper presents a dynamic model with a degree of freedom,
considering internal damping of the system (c), damping for which it is
considered a special function.
More precisely, the cushioning coefficient of
the system (c) is defined as a variable parameter depending on the reduced mass
of the mechanism (m* or J reduced) and the time, ie, c depends on the
derivative of m reduced in time. The equation of the differential movement of
the mechanism is written as the movement of the valve as a dynamic response.
Keywords: Distribution
mechanism; Rigid memory mechanisms; Variable internal damping; Dynamic model;
Angular speed variation; Dynamic coefficient
1. INTRODUCTION
The
development and diversification of road vehicles and general vehicles,
especially of cars, together with thermal engines, especially internal
combustion engines (being more compact, robust, more independent, more
reliable, stronger, more dynamic etc.)., has also forced the development of
devices, mechanisms, and component assemblies at an alert pace. The most
studied are power and transmission trains.
The
four-stroke internal combustion engine (four-stroke, Otto or Diesel) comprises
in most cases (with the exception of rotary motors) and one or more camshafts,
valves, valves, and so on.
The
classical distribution mechanisms are robust, reliable, dynamic, fast-response,
and although they functioned with very low mechanical efficiency, taking much
of the engine power and effectively causing additional pollution and increased
fuel consumption, they could not be abandoned until the present. Another
problem was the low speed from which these mechanisms begin to produce
vibrations and very high noises.
Regarding
the situation realistically, the mechanisms of cam casting and sticking are
those that could have produced more industrial, economic, social revolutions in
the development of mankind. They have contributed substantially to the
development of internal combustion engines and their spreading to the detriment
of external combustion (Steam or Stirling) combustion engines.
The
problem of very low yields, high emissions and very high power and fuel
consumption has been greatly improved and regulated over the past 20-30 years
by developing and introducing modern distribution mechanisms that, besides
higher yields immediately deliver a high fuel economy also performs optimal
noise-free, vibration-free, no-smoky operation, as the maximum possible engine
speed has increased from 6000 to 30000 [rpm].
The
paper tries to provide additional support to the development of distribution
mechanisms so that their performance and the engines they will be able to
further enhance.
Particular
performance is the further increase in the mechanical efficiency of
distribution systems, up to unprecedented quotas so far, which will bring a
major fuel economy.
The
current oil and energy reserves of mankind are limited. Until the
implementation of new energy sources (to take real control over fossil fuels),
a real alternative source of energy and fuel is even "the reduction in
fuel consumption of a motor vehicle", whether we burn oil, gas and
petroleum derivatives, whether we will implement biofuels first, and later
hydrogen (extracted from water).
The
drop in fuel consumption for a given vehicle type over a hundred kilometers
traveled has been consistently since 1980 and has continued to continue in the
future.
Even
if hybrids and electric motor cars are to be multiplied, let us not forget that
they have to be charged with electricity, which is generally obtained by
burning fossil fuels, especially oil and gas, in a current planetary proportion
of about 60%. Can burn oil in large heat plants to warm up, have domestic hot
water and electricity to consume, and some of that energy is extra and we add
it to electric cars (electric vehicles), but the global energy problem is not
resolved, the crisis even deepens. This was the case when was electrified the
railroad for trains, when it were generalized trams, trolleybuses and subways,
consuming more electric power produced mainly from oil; oil consumption has
grown a lot, its price has had a huge leap, and now one looks at how the
reserves disappear quickly.
Generally,
generalizing electric cars (though it is not really ready for this), one will
give a new blow to oil and gas reserves.
Fortunately,
biofuels, biomass and nuclear power have developed very much lately (currently
based on the nuclear fission reaction). These together with the hydroelectric
power plants have managed to produce about 40% of the total energy consumed
globally. Only about 2-3% of global energy resources are produced by various
other alternative methods (despite the efforts made so far).
This
should not disarm us, and abandon the implementation of solar, wind, etc.
However,
as a first necessity to further reduce the share of global energy from oil and
gas, the first vigorous measures that will need to be pursued will be to
increase biomass and biofuels production along with the widening of the number
of nuclear power plants (despite some undesirable events, which only show that
nuclear fission power plants must be built with a high degree of safety, and in
no way eliminated from now on, and they are still the one that has been so far
"a bad evil ").
Alternative
sources will take them on an unprecedented scale, but it expects the energy
they provide to be more consistent in global percentages so that can rely on
them in a real way (otherwise, one risks that all these alternative energies
remain a sort of "fairy tale").
Hydrogen
fuel energy "when it starts when it stops" so there is no real time
now to save energy through them, so they can no longer be priority, but the
trucks and buses could even be implemented now that the storage problems have
been partially solved. The bigger problem with hydrogen is no longer the safe
storage, but the high amount of energy needed to extract it, and especially for
its bottling.
The
huge amount of electricity consumed for bottling hydrogen will have to be
obtained entirely through alternative energy sources, otherwise hydrogen
programs will not be profitable for humanity at least for the time being. The
authors thinking the immediate use of hydrogen extracted from the water with
alternative energies would be more appropriate for seagoing vessels.
Maybe
just to say that due to his energy crisis (and not just energy, from 1970 until
today), the production of cars has increased at an alert pace (but naturally)
instead of falling, and they have and were marketed and used. The world's
energy crisis (in the 1970s) began to rise from around 200 million vehicles
worldwide, to about 350 million in 1980 (when the world's energy and global
fuel crisis was declared), about 500 million vehicles worldwide, and in 1997
the number of world-registered vehicles exceeded 600 million.
In
2010, more than 800 million vehicles circulate across the planet (ANTONESCU,
2000; ANTONESCU; PETRESCU, 1985; ANTONESCU; PETRESCU, 1989; ANTONESCU et al.,
1985a; ANTONESCU et al.,1985b; ANTONESCU et al., 1986; ANTONESCU et al., 1987; ANTONESCU
et al., 1988; ANTONESCU et al., 1994; ANTONESCU et al., 1997; ANTONESCU et al.,
2000a; ANTONESCU et al.2000b ANTONESCU et al., 2001; AVERSA et al., 2017a;
AVERSA et al., 2017b; AVERSA et al., 2017c; AVERSA et al., 2017d; AVERSA et
al., 2017e; MIRSAYAR et al., 2017; PETRESCU et al., 2017a; PETRESCU et al.,
2017b; PETRESCU et al., 2017c; PETRESCU et al., 2017d; PETRESCU et al., 2017e;
PETRESCU et al., 2017f; PETRESCU et al., 2017g; PETRESCU et al., 2017h;
PETRESCU et al., 2017i, PETRESCU et al.,2015; PETRESCU; PETRESCU, 2016;
PETRESCU; PETRESCU, 2014; PETRESCU; PETRESCU, 2013a; PETRESCU; PETRESCU, 2013b;
PETRESCU; PETRESCU, 2013c; PETRESCU; PETRESCU, 2013d; PETRESCU; PETRESCU, 2011;
PETRESCU; PETRESCU, 2005a; PETRESCU; PETRESCU, 2005b; PETRESCU, 2015a; PETRESCU,
2015b, PETRESCU, 2012a; PETRESCU, 2012b; HAIN, 1971; GIORDANA et al., 1979; ANGELES;
LOPEZ-CAJUN, 1988; TARAZA et al., 2001; WIEDERRICH; ROTH, 1974; FAWCETT; FAWCETT,
1974; JONES; REEVE, 1974; TESAR; MATTHEW, 1974; SAVA, 1970; KOSTER, 1974).
2. THE STATE OF THE ART
The
Peugeot Citroën Group in 2006 built a 4-valve hybrid engine with 4 cylinders
the first cam opens the normal valve and the second with the phase shift.
Almost all current models have stabilized at four valves per cylinder to
achieve a variable distribution. In 1971, K. Hain proposes a method of
optimizing the cam mechanism to obtain an optimal (maximum) transmission angle
and a minimum acceleration at the output. In 1979, F. Giordano investigates the
influence of measurement errors in the kinematic analysis of the camel.
In
1985, P. Antonescu presented an analytical method for the synthesis of the cam
mechanism and the flat barbed wire, and the rocker mechanism. In 1988, J.
Angeles and C. Lopez-Cajun presented the optimal synthesis of the cam mechanism
and oscillating plate stick.
In
2001 Dinu Taraza analyzes the influence of the cam profile, the variation of
the angular speed of the distribution shaft and the power, load, consumption
and emission parameters of the internal combustion engine. In 2005, Petrescu
and Petrescu, present a method of synthesis of the rotating camshaft profile
with rotary or rotatable tappet, flat or roller, in order to obtain high yields
at the exit.
In
the paper (WIEDERRICH;
ROTH, 1974), there is presented a basic, single-degree,
dual-spring model with double internal damping for simulating the motion of the
cam and punch mechanism. In the paper (FAWCETT; FAWCETT, 1974) is presented
the basic dynamic model of a cam mechanism, stick and valve, with two degrees
of freedom, without internal damping.
A
dynamic model with both damping in the system, external (valve spring) and
internal one is the one presented in the paper (JONES; REEVE, 1974). A dynamic
model with a degree of freedom, generalized, is presented in the paper (TESAR; MATTHEW, 1974), in which
there is also presented a two-degree model with double damping.
In
the paper (SAVA, 1970) is
proposed a dynamic model with 4 degrees of freedom, obtained as follows: the
model has two moving masses these by vertical vibration each impose a degree of
freedom one mass is thought to vibrate and transverse, generating yet another
degree of freedom and the last degree of freedom is generated by the torsion of
the camshaft.
Also
in the paper (SAVA, 1970) is
presented a simplified dynamic model, amortized. In (SAVA, 1970) there is also showed a dynamic
model, which takes into account the torsional vibrations of the camshaft. In
the paper (KOSTER, 1974) a
four-degree dynamic model with a single oscillating motion mass is presented,
representing one of four degrees of freedom.
The
other three freedoms result from a torsional deformation of the camshaft, a
vertical bending (z), camshaft and a bending strain of the same shaft, horizontally
(y), all three deformations, in a plane perpendicular to the axis of rotation.
The sum of the momentary efficiency and the momentary losing coefficient is 1.
The work is especially interesting in how it manages to transform the four
degrees of freedom into one, ultimately using a single equation of motion along
the main axis.
The
dynamic model presented can be used wholly or only partially, so that on
another classical or new dynamic model, the idea of using deformations on
different axes with their cumulative effect on a single axis is inserted. In
works (ANTONESCU et al., 1987; PETRESCU; PETRESCU, 2005a) there is presented a
dynamic model with a degree of freedom, considering the internal damping of the
system (c), the damping for which is considered a special function. More
precisely, the damping coefficient of the system (c) is defined as a variable
parameter depending on the reduced mass of the mechanism (m* or Jreduced)
and time, ie, c, depends on the time derivative of mreduced. The
equation of differential movement of the mechanism is written as the movement
of the valve as a dynamic response.
3. DYNAMIC MODELS WITH VARIABLE INTERNAL DAMPING
Starting from the kinematic scheme of the classical
distribution mechanism (see Figure 1), the dynamic, mono-dynamic (single
degree), translatable, variable damping model (see Figure 2) is constructed,
the motion equation of which is:
(1)
Equation (1) is nothing else than the equation of Newton,
in which the sum of forces on an element in a certain direction (x) is equal to
zero.
The notations in formula (1) are as follows:
M- mass of the reduced
valve mechanism;
K- reduced elastic
constants of the kinematic chain (rigidity of the kinematic chain);
k- elastic spring valve
constant;
c - the damping
coefficient of the entire kinematic chain (internal damping of the system);
F, Ft - the elastic spring force of the
valve spring;
x - actual valve
displacement;
(the cam profile)
reduced to the axis of the valve.
Figure
1: The kinematic scheme of the classic distribution mechanism
Figure
2: Mono - dynamic model, with internal
depreciation of the variable system
The
Newton equation (1) is ordered as follows:
(2)
At
the same time the differential equation of the mechanism is also written as
Lagrange, (3), (Lagrange equation):
(3)
Equation (3), which is nothing other
than the Lagrange differential equation, allows for the low strength of the
valve (4) to be obtained by the polynomial coefficients with those of the
Newtonian polynomial (2), the reduced drive force at the valve (5), as well as
the expression of c, ie the expression of the internal damping coefficient, of
the system (6).
(4)
(5)
(6)
Thus a new formula (6) is obtained,
in which the internal damping coefficient (of a dynamic system) is equal to
half the derivative with the time of the reduced mass of the dynamic system.
The Newton motion equation (1, or
2), by replacing it with c takes the form (7):
(7)
In the case of the classical
distribution mechanism (in Figure 1), the reduced mass, M, is calculated by the
formula (8):
(8)
formula
in which or used the following notations:
m2
= stick weight;
m3
= the mass of the pushing rod;
m5
= mass of the valve;
J1
= moment of mechanical inertia of the cam;
J4
= moment of mechanical inertia of the culbutor;
= velocity of stroke imposed by cam law;
= valve speed.
If i = i25, the
valve-to-valve ratio (made by the crank lever), the theoretical velocity of the
valve (imposed by the motion law given by the cam profile) is calculated by the
formula (9):
(9)
where:
(10)
is
the ratio of the crank arms.
The following relationships are
written (11-16):
(11) (12)
(13)
(14)
(15)
(16)
where
y 'is the reduced velocity imposed by the camshaft (by the law of camshaft
movement), reduced to the valve axis.
With the previous relationships
(10), (13), (14), (16), the relationship (8) becomes (17-19):
(17)
or:
(18)
or:
(19)
We make the derivative dM/d and
result the following relationships:
(20)
(21)
(22)
Write the relationship (6) as:
(23)
which
with (22) becomes:
(24)
or
(25)
Where
was noted:
(26)
4. DETERMINATION OF MOTION EQUATIONS
With
relations (19), (12), (25) and (11), equation (2) is written first in the form
(27), which develops in forms (28), (29) and (30):
(27)
(28)
meaning:
(29)
And final form:
(30)
which can also be written in
another form:
(31)
Equation (31) can be approximated to
form (32) if we consider the theoretical input velocity y imposed by the
camshaft profile (reduced to the valve axis) approximately equal to the
velocity of the valve, x.
(32)
If the laws of entry with s, s' (low
speed), s' '(low acceleration), equation (32) takes the form (33) and the more
complete equation (31) takes the complex form (34):
(33)
(34)
5. DYNAMIC MODEL WITH FOUR DEGREES OF FREEDOM WITH
INTERNAL SYSTEM DAMPING - VARIABLE -
In
the paper (ANTONESCU et al., 1985 a) there is presented a dynamic damping model
variable as in the previous paragraph, but with four degrees of mobility.
The
hypothesis of the existence of four masses in translational motion is made at
the same time (see Figure 3).
Figure
3a shows the kinematic diagram of the classic distribution mechanism, and in
FIG. 3b is shown the corresponding dynamic pattern, with four moving masses,
thus with four degrees of freedom.
The
way in which the four dynamic masses and the corresponding elastic constants,
as well as the corresponding damping, are deduced will be presented in the
following paragraph.
The
dynamic model with four degrees of freedom (Figure 3) is considered, where the
four reduced masses of the driven element (valve) are calculated with the
formulas (35).
The
mass m1* is calculated as the mass m1 (mass of the
camshaft) that reduces to the valve axis, that is, this mass m1,
multiplies by the theoretical input speed, square, and is
divided by the square of the valve speed, the ratio
between the cam entry speed and valve velocity, and rises to
square, and this square ratio multiplies by the mass m1.
Figure
3: Dynamic model with four degrees of freedom with internal system damping -
variable -
As
the input speed must also be
reduced to the axis of the valve, instead of it write down the reduced input
velocity to the valve axis, multiplied by
the coulter transmission ratio, i, that is, we have the relation and the square
velocity, will be replaced
with, and will be
written down i2 multiplied to the mass m1 with m1'.
For mass m2*, consider the weight of the tappet, m2, plus
one third of the weight of the pushing rod, m3, and the
corresponding speed is practically the dynamic velocity of the
tappet reduced to the axis of the valve.
The
mass m3* corresponds to the pusher rod and consists of two remaining
thirds of the pushing rod weight, m3, plus half of the mass of the
stem, m4; velocity is the actual
average speed with which the pushing rod moves on the vertical axis reduced to
the valve axis, or the speed of the stopper at the point C reduced to the valve
axis.
The
mass m4* is obtained from all the summaries on the side of the
valve, ie half the mass of the valve, plus the mass m5 (which in
turn represents the sum of the valve mass and the mass of the valve pan) plus a
third of the mass of the valve spring. The speed of the valve (obviously at its
axis) was marked with.
(35)
where i = O4C / O4D (see Figure
3) represents the transmission ratio of the culbutor; m1, m2,
m3, m4, m5, m6 are in order: the
mass of the cam, the stick, the pusher rod, the stem, the valve (with the
roller) and the valve spring respectively. The following equivalent elastic
constants (see Figure 3) are reduced to the valve (36):
(36)
where k1, k2,
k3, k4, k6 are the stiffnesses (elastic constants) of the
corresponding elements. The elastic valve constant is not in question. It is
noted that F0 is the external force, known as the spring force of
the valve spring, and Fe is the balancing force at the valve,
basically the driving force. The influence of moments of mechanical inertia
(mass), weight forces and friction forces will be neglected. Following the dynamic
equilibrium for each reduced mass in part are written four equations of the
form (37-40):
(37)
(38)
(39)
(40)
The
linear displacements y1 , y2 ,
y3 , y4 =x
correspond to the reduced masses m1*, m2*,
m3*, m4*.
Assuming
that the movement y1 is known from the motion law y1
= y1 () imposed on the camshaft at the cam design, the
displacements y2, y3, x and the balance force Fe, ie the motor
force Fm, remain unknown.
In
this case it is observed that equations (38), (39) and (40) form a system of
three equations with three unknowns y2, y3, x. After calculating
the three displacements from (37), the equilibration force Fe is
obtained.
Basically,
the system is not linear because, in addition to the unknowns given by the
three displacements, we have as extra unknown the speeds and accelerations
derived from unknown movements, ie practically unknown will be ten and only
four of the system's equations.
(41)
For
the actual solution of the equation system (37) - (40), the damping
coefficients c1, c2, c3, c4 of formula (41), already known from the system with
a degree of freedom and the mass system (35), as follows (42-45):
(42)
(43)
(44) (45)
which can also be written in the
form (46-49):
(46)
(47)
(48)
(49)
Using
Relationships (46-49) and System (35), Relationships (50-53) can be obtained
immediately:
(50)
(51) (52)
(53)
Taking
into account relations (50-53), equations (37-40) are rewritten as follows
(54-57):
(54)
(55)
(56)
(57)
With
the system of equations (54-57), the dynamic model shown in Figure 3 is solved,
given that the system is nonlinear and besides the four main unknowns, y2,
y3, x, Fe, six
more unknown occur, but dependent on each other and also depend on linear
displacements, y2, y3, and x respectively.
The
system is greatly simplified if we consider the three speeds approximately
equal to each other and equal to the known entry speed; In this case, the
equation system (54-57) is considerably simplified, taking the form (58-61):
(58)
(59)
(60)
(61)
6. RESULTS; SOLVING THE DIFFERENTIAL EQUATION
In the paper was presented a dynamic model with a degree
of mobility, internal damping of the variable system, which finally leads to
the equation (54), which can be writhed in the form (62) and the simplified
equation (53), arranged now in form (63).
(62)
(63)
Differential equation (63), ie the simplified form (in
which the reduced input velocity imposed by the cam profile y' is equal to the
low dynamic velocity, x', both reduced to the valve axis) is used.
6.1.
Solving
the differential equation, through a particular solution
Equation
(63) is written as (64):
(64)
One
divides equation (64) with mS and amplify the straight term with coswt, thus obtaining the form (65):
(65)
The
following notations (66-67) are used:
(66)
(67)
Equation
(65) is written in simplified form (68):
(68)
The
particular solution of equation (68) is of the form (69):
(69)
Derivatives
1 and 2 of solution (69) are denoted by (70-71):
(70)
(71)
By
replacing values (69) and (71) in equation (68), form (72) is obtained:
(72)
The
characteristic equation is written as (73):
(73)
It is
explicit a in the form (74):
(74)
Now
write the solution X, under the forms (75), (76):
(75)
(76)
For a
more exact solution, we approximate directly in equation (74), X’’
cu y’’ cu s’’, ie , and one
arrives at the linear equation (77):
(77)
6.2.
Solving
the differential equation, through a complete private solution
Equation
(64) can be written as (78), taking into account coefficients D and D ':
(78)
One
divides equation (78) with mS.w2.D and obtain the form (79):
(79)
The
right term is amplified with (cosj+sinj) and equation (79) is written as (80):
(80)
We
note the corresponding coefficients (81-83):
(81)
(82)
(83)
Equation
(80) can now be written as (84):
(84)
The
complete particular solution of equation (84) is of the form (85), and its
derivatives according to the angle j, the derivatives
I and II, take the forms (86), respectively (87):
(85)
(86)
(87)
Introducing
solutions (85-87) in (84) one obtains equation (88):
(88)
We
identify the coefficients in the cosine and those in the sin and one obtains a
linear system of two equations with two unknown, A and B respectively:
(89)
For
the operative solving of the system (89) the first equation increases with a
and the second with (b-1), after which B is collected and then determined by A,
multiplying the first equation with (b-1) and the second one with -a, after
which it collects and obtains the system (90):
(90)
The
solution can now be written as (91), where the coefficients a, b, c are known
(81-83):
(91)
6.3.
Solving
the differential equation, with the help of Taylor series developments
Write
the relation (92), which expresses the connection between the dynamic
displacement of the valve, x, and that imposed by the cam profile, s:
(92)
The
function s(j+Dj)
was developed in a Taylor series
and retains the first 8 terms of development; now find the relationship (93):
(93)
The
relationship (93) is also written in the form (94):
(94)
By
derivation it obtains x' (relation 95):
(95)
Deriving
the second time and get x'', (relation 96):
(96)
The
differential equation used is (62), ie the complete equation, which we write in
the form (97), also taking into account the transmission function, D.
(97)
7. DISCUSSION
Dynamic
analysis for sinus law, using the relationship (97), based on Taylor series and
dynamic-A1 model, with variable internal damping, without considering the mass
m1 of the cam.
Using the relation (97) obtained
from the differential equation (62) based on the dynamic damping model of the
variable system, without considering the mass m1 of the cam, but using Taylor
series calculations with the retention of 8 consecutive terms, dynamic (A1).
For this dynamic model (A1) there is
a single dynamic diagram (Figure 4).
Figure
4: Dynamic analysis using the dynamic A1 model
The
SINus law is used, the engine speed, n = 5500 [rpm], equal ascension and
descent angles, ju=jc=750, radius of the base circle, r0
= 14 [mm]. For the maximum stroke of the tappet, hT, equal to that
of the valve, hS (i = 1), the value of h = 5 [mm] was taken. A
spring elastic constant is adopted, k = 60 [N / mm], for a valve spring
compression of x0 = 30 [mm].
Mechanical
yield is low (generally in rotary cam and punch mechanisms, mechanical
efficiency has low values, and in Module C-classical distribution mechanism
these values are even slightly lower), h=6.9%.
The
theoretical model presented and used has the advantages of simulating even the
fine vibrations of the mechanism.
8. CONCLUSIONS
The
development and diversification of road vehicles and vehicles, especially of
cars, together with thermal engines, especially internal combustion engines
(being more compact, robust, more independent, more reliable, stronger, more
dynamic etc.)., has also forced the development of devices, mechanisms, and
component assemblies at an alert pace. The most studied are power and
transmission trains.
The
four-stroke internal combustion engine (four-stroke, Otto or Diesel) comprises
in most cases (with the exception of rotary motors) and one or more camshafts,
valves, valves, and so on.
The
classical distribution mechanisms are robust, reliable, dynamic, fast-response,
and although they functioned with very low mechanical efficiency, taking much
of the engine power and effectively causing additional pollution and increased
fuel consumption, they could not be abandoned until the present. Another
problem was the low speed from which these mechanisms begin to produce
vibrations and very high noises.
Regarding
the situation realistically, the mechanisms of cam casting and sticking are
those that could have produced more industrial, economic, social revolutions in
the development of mankind. They have contributed substantially to the
development of internal combustion engines and their spreading to the detriment
of external combustion (Steam or Stirling) combustion engines.
The
problem of very low yields, high emissions and very high power and fuel
consumption has been greatly improved and regulated over the past 20-30 years
by developing and introducing modern distribution mechanisms that, besides
higher yields immediately deliver a high fuel economy) also performs optimal
noise-free, vibration-free, no-smoky operation, as the maximum possible engine
speed has increased from 6000 to 30000 [rpm].
The
paper tries to provide additional support to the development of distribution
mechanisms so that their performance and the engines they will be able to
further enhance.
Particular
performance is the further increase in the mechanical efficiency of
distribution systems, up to unprecedented quotas so far, which will bring a
major fuel economy.
The
paper presents a dynamic model that works with variable internal damping, applicable
directly to rigid memory mechanisms. If the problem of elasticity is generally
solved, the problem of system damping is not clear and well-established. It is
usually considered a constant "c" value for the internal damping of
the system and sometimes the same value c and for the damping of the elastic
spring supporting the valve.
However,
the approximation is much forced, as the elastic spring damping is variable,
and for the conventional cylindrical spring with constant elasticity parameter
(k) with linear displacement with force, the damping is small and can be
considered zero. It should be specified that damping does not necessarily mean
stopping (or opposition) movement, but damping means energy consumption to
brake the motion (rubber elastic elements have considerable damping, as are
hydraulic dampers).
Metal
helical springs generally have a low (negligible) damping. The braking effect
of these springs increases with the elastic constant (the k-stiffness of the
spring) and the force of the spring (P0 or F0) of the
spring (in other words with the arc static arrow, x0=P0/k).
Energy is constantly changing but does not dissipate (for this reason, the
yield of these springs is generally higher).
The
paper presents a dynamic model with a degree of freedom, considering internal
damping of the system (c), damping for which it is considered a special
function. More precisely, the cushioning coefficient of the system (c) is
defined as a variable parameter depending on the reduced mass of the mechanism
(m* or J reduced) and the time, ie, c depends on the derivative of m reduced in
time.
The
equation of the differential movement of the mechanism is written as the
movement of the valve as a dynamic response. Dynamic analysis for sinus law,
using the relationship (97), based on Taylor series and dynamic-A1 model, with
variable internal damping, without considering the mass m1 of the cam.
Using
the relation (97) obtained from the differential equation (62) based on the
dynamic damping model of the variable system, without considering the mass m1
of the cam, but using Taylor series calculations with the retention of 8
consecutive terms, dynamic (A1). For this dynamic model (A1) there is a single
dynamic diagram (Figure 4).
The
SINus law is used, the engine speed, n = 5500 [rpm], equal ascension and
descent angles, ju=jc=750, radius of the base circle, r0
= 14 [mm]. For the maximum stroke of the tappet, hT, equal to that
of the valve, hS (i = 1), the value of h = 5 [mm] was taken. A
spring elastic constant is adopted, k = 60 [N / mm], for a valve spring
compression of x0 = 30 [mm].
Mechanical
yield is low (generally in rotary cam and punch mechanisms, mechanical
efficiency has low values, and in Module C-classical distribution mechanism
these values are even slightly lower), h=6.9%.
The
original theoretical model presented and used has the advantages of simulating
even the fine vibrations of the mechanism.
9. ACKNOWLEDGEMENTS
This text was acknowledged and appreciated by Dr. Veturia
CHIROIU Honorific member of Technical Sciences Academy of Romania (ASTR) PhD
supervisor in Mechanical Engineering.
10. FUNDING INFORMATION
Research contract: Contract number 27.7.7/1987,
beneficiary Central Institute of Machine Construction from Romania (and
Romanian National Center for Science and Technology). All these matters are
copyrighted. Copyrights: 394-qodGnhhtej 396-qkzAdFoDBc 951-cnBGhgsHGr
1375-tnzjHFAqGF.
11. 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=smax |
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 |
x0 |
is the valve spring preload
(pretension) |
mc |
is the mass of the cam |
mT |
is the mass of the tappet |
ωm |
the nominal angular rotation
speed of the cam (camshaft) |
nc |
is the camshaft speed |
n=nm |
is the motor shaft speed nm=2nc |
w |
is the dynamic angular rotation
speed of the cam |
e |
is the dynamic angular rotation
acceleration of the cam |
r0 |
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=xc and y=yc |
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. |
12. AUTHORS’ CONTRIBUTION
All the
authors have contributed equally to carry out this work.
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