SOME
DYNAMIC MODELS OF RIGID MEMORY MECHANISMS
Florian
Ion Tiberiu Petrescu
IFToMM, Romania
E-mail: fitpetrescu@gmail.com
Relly
Victoria Virgil Petrescu
IFToMM, Romania
E-mail: rvvpetrescu@gmail.com
Submission: 12/9/2018
Revision: 2/8/2019
Accept: 9/19/2019
ABSTRACT
Rigid memory
mechanisms have played an important role in the history of mankind,
contributing greatly to the industrial, economic, social changes in society,
thus leading to a real evolution of mankind. Used in automated tissue wars, in
cars as distribution mechanisms, automated machines, mechanical transmissions,
robots and mechatronics, precision devices, and medical devices, these
mechanisms have been real support for mankind along the time. For this reason,
it considered being useful this paper, which presents some dynamic models that
played an essential role in designing rigid memory mechanisms.
Keywords: Distribution mechanism; Rigid memory
mechanisms; Variable internal damping; Dynamic model; Angular speed variation;
Dynamic coefficient.
1.
INTRODUCTION
Mechanisms with rigid memory or as
are commonly known cam and punch mechanisms have played an essential role in
technology and industry, managing at least two industrial revolutions followed
by major changes across society. They were the first mechanical transmissions
used in the wars of tissue that changed the face of the industry when they were
introduced into massive work.
All rigid memory devices have also
played an essential role in transport since the introduction of Otto's internal
combustion engine. Rigid memory mechanisms have been indispensable in clocks,
clocks, fine mechanics, small device mechanisms, and the medical industry,
which today plays an essential role in engineering medicine. Today rigid memory
mechanisms are widely used in the machine building industry, in robotics and
mechatronics, power engineering, as mechanical transmissions.
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.
In 1680, Dutch physicist Christian
Huygens designs the first internal combustion engine.
In 1807, the Swiss Francois Isaac de
Rivaz invented an internal combustion engine that
uses a liquid mixture of hydrogen and oxygen as fuel. However, Rivaz's engine for its new engine has been a major failure,
so its engine has passed to the deadline, with no immediate application.
In 1824, English engineer Samuel
Brown adapted a steam engine to make it work with gasoline.
In 1858, Belgian engineer Jean Joseph
Etienne Lenoir invented and patented two years later, the first real-life
internal combustion engine with spark-ignition, liquid gas (extracted from
coal), a two-stroke engine . In 1863, all Belgian Lenoir is adapting a
carburetor to his engine by making it work with oil (or gasoline).
In 1862, the French engineer
Alphonse Beau de Rochas first patented the
four-stroke internal combustion engine (but without building it).
It is the merit of German engineers
Eugen Langen and Nikolaus
August Otto to build (physically, practically the theoretical model of the
French Rochas), the first four-stroke internal
combustion engine in 1866, with electric ignition, charging and distribution in
a form Advanced.
Ten years later (in 1876), Nikolaus August Otto patented his engine.
In the same year (1876), Sir Dougald Clerk, arranges the two-time engine of the Belgian
Lenoir, (bringing it to the shape known today).
In 1885, Gottlieb Daimler arranges a
four-stroke internal combustion engine with a single vertical cylinder and an
improved carburetor.
A year later, his compatriot Karl
Benz brings some improvements to the four-stroke gasoline engine. Both Daimler
and Benz were working new engines for their new cars (so famous).
In 1889, Daimler improves the
four-stroke internal combustion engine, building a "two cylinder in
V", and bringing the distribution to today's classic form, "with
mushroom-shaped valves."
In 1890, Wilhelm Maybach, builds the
first four-cylinder four-stroke internal combustion.
In 1892, German engineer Rudolf
Christian Karl Diesel invented the compression-ignition engine, in short the
diesel engine.
In 2010, more than 800 million
vehicles circulated 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., 2000 a;
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. Hain
(1971) proposes a method of optimizing
the cam mechanism to obtain an optimal (maximum) transmission angle and a
minimum acceleration at the output. Giordano (1979) 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. Angeles and Lopez-Cajun (1988)
presented the optimal synthesis of the cam mechanism and oscillating plate
stick. Taraza (2001) 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.
Petrescu
and Petrescu (2005), 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.
Wiederrich
and 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 and
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, i.e, 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.
MATERIALS AND METHODS
3.1.
Dynamic model with a degree of
freedom with double internal damping
Wiederrich and Roth (1974) presented
a basic single-degree model with two springs and double internal damping to
simulate the movement of the cam and punch mechanism (see Figure 1) and the
relationships (1-2).
(1)
(2)
The motion equation of the proposed system (1) uses the notations (relations) in the system (2); w1 and w2 represents the system's own pulses and is calculated from the relationship system (2), depending on the elasticities K1 and K2 of the system in Figure 1 and the reduced mass M of the system.
Figure 1: Dynamic model with a degree
of freedom with double internal damping
3.2.
Dynamic model with two degrees of
freedom without internal damping
Fawcett
and Fawcett (1974) presented the basic dynamic model of a mechanism with
cam, barrel and valve, with two degrees of freedom, without internal damping
(see Figure 2, eq. 3-5).
(3)
(4)
(5)
Figure 2: Dynamic model with two
degrees of freedom without internal damping
3.3.
Dynamic model with a degree of
freedom with internal and external damping
A dynamic model with both system damping, external (spring valve) and internal damping has been presented in the paper (JONES; REEVE, 1974), (see Figure 3).
Figure 3: Dynamic model with a degree
of freedom with internal and external damping
3.4.
Dynamic model with a degree of
freedom, taking into account the internal damping of the valve spring
A dynamic model with a generalized degree of freedom is presented by Tesar and Matthew (1974) (see Figure 4):
Figure 4: Dynamic model with a degree
of freedom, taking into account the internal damping of the valve spring
The motion equation is written as (6):
(6)
Using the known relation (7), equation (6) takes the form (8):
(7)
(8)
where the coefficients m have the form (9):
(9)
The vertical reaction has the form (10):
(10)
3.5.
Dynamic two-degree, dual damping
model
Tesar and Matthew (1974) presented also the model with two degrees of freedom (see Fig. 5) with double damping:
Figure 5: Dynamic two-degree, dual
damping model
The calculation relationships used are (11-16):
(11)
(12)
(13)
(14)
(15)
(16)
3.6.
Dynamic model with four degrees of
freedom, with torsional vibrations
In the paper Sava (1970) a dynamic model with 4 degrees of freedom is proposed, 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 torsional torsion of the camshaft (see Figure 6).
The calculation relationships are (17-20).
The first two equations resolve normal vertical vibrations, the third equation takes into account the camshaft torsional vibration, and the last equation (independent of the others), the fourth, deals only with the transverse vibration of the system.
(17)
(18)
(19)
(20)
Figure 6: Dynamic model with four
degrees of freedom, with torsional vibrations
3.7.
Mono-dynamic damped dynamic model
Also in the paper Sava (1970), has presented a simplified dynamic model, amortized mono-mass (see figure 7).
The motion equation used has the form (21):
(21)
Which
can be written more conveniently as (22):
(22)
Where the coefficients A1, w12 and F are calculated with the expressions given in relation (23):
(23)
Figure 7: Mono-dynamic damped dynamic
model
3.8.
Dynamic damped two-mass model
In Figure 8 the bi-mass model proposed in the paper (SAVA, 1970) is presented.
The mathematical model is written (24, 25):
(24)
(25)
Equations (24-25) can be written as:
(26)
(27)
where the notations (28) were used:
the ratio of the two masses,
the self dimensional pulse of
the mass m,
(28)
Figure 8: Dynamic damped two-mass model
3.9.
A dynamic model with a single mass
with torsional vibrations
In Figure 9 one can see a dynamic mono-mass model that also takes into account the torsional vibrations of the camshaft (SAVA, 1970).
The study points out that camshaft torsional vibrations has a negligible influence and can, therefore, be excluded from dynamic calculation models.
The
same conclusion results from the work (SAVA, 1971) where the
torsion model is studied in more detail.
Figure 9: A dynamic model with a single
mass with torsional vibrations
3.10.
Influence of transverse vibrations
Tappet elasticity, variable length of the camshaft during cam operation, pressure angle variations, camshaft eccentricity, kinetic coupler friction, translation wear, technological and manufacturing errors, system gaming, and other factors are factors that favor the presence of a transverse vibration of the rod weight (SAVA, 1970).
In the case of high amplitude vibrations, the response parameters to the last element of the tracking system will be influenced. Following Figure 10, it can be seen that if the curve a is the trajectory of the tip A, the point A will periodically reach point A', in which case the actual stroke of the yr bar will change according to the law: yr = y-yv= y-u.tgv, where y is the longitudinal displacement of the tappet, u represents the transverse displacement of the mass m, of the tappet, and v is the pressure angle. The actual stroke, yr, will change after the law (29):
(29)
The motion equation (dimensional) is written (30):
(30)
where were denoted by (31) the non-dimensional constants:
(31)
Figure 10: Influence of transverse
vibrations
Also in the work (SAVA, 1970) the influence of the diameter of the rod, the lifting interval, the maximum length outside the tiller guides, the maximum lifting stroke and the various cam profiles on the A trajectory are analyzed.
Some conclusions:
It is noted that the reduction of the diameter of the rod of the tappet leads to the increase of the amplitude and the decrease of the average frequency of the transverse vibrations. Reducing the diameter of 1.35 times leads to an increase in amplitude of almost three times, and the average frequency decreases sensitively. Initial amplitudes are higher at the beginning of the interval, decreasing to the midpoint of the lifting interval, oscillation becoming insignificant, and towards the end of the rise due to the reduction of the length a by decreasing the y stroke the frequency increases and consequently the amplitude decreases from double to simple the beginning of the interval. Increasing the stick length beyond its 2.2 to 3 cm guides leads to an increase in vibration amplitude of about 25 times.
The law of motion without leaps in the input acceleration curve reduces the amplitude of the transverse vibration of the tappet. The author of the paper (SAVA, 1970) mentions that whatever the influence of the listed parameters is, for the cases considered, the amplitude values remain fairly small, and in case of reduced friction in the upper coupler, they can decrease even more. Consequently, the author of the paper (SAVA, 1970) concludes that the transverse vibrations of the follower exist and must draw the attention of the constructor only in the case of exaggerated values of the constants that characterize these vibrations. Regarding the distribution of internal combustion engines, the transverse vibration can be neglected without affecting the response parameters made at the valve.
3.11.
Dynamic model with four degrees of
freedom, with bending vibrations
In the paper Koster (1974), has presented a four-degree dynamic model with a single oscillating motion mass, 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 (see Figure 11). The sum of the momentary efficiency and the momentary losing coefficient is 1.
The work (KOSTER, 1974) is extremely interesting by the model it proposes (all types of deformations are being studied), but especially by the hypothesis it advances, namely: the cams speed is not constant but variable, the angular velocity of the cam w=f(b) being a function of the position of the cam (the cam angle of rotation b).
Figure 11: Dynamic model with four degrees of freedom, with bending vibrations
The angular velocity of the cam is a function of the position angle b (which we usually mark with j), and its variation is caused by the three deformations (torsion and two bends) of the shaft, as well as by the angular gaps existing between the source motors (drive) and camshaft.
The mathematical model taking into account the flexibility of the camshaft is the following; the rigidity of the cam between the cam and the cam is a function of the position b (cam angle of rotation), see the relationship (32):
(32)
(33)
Where 1 / Cc see (33) is
a constant rigidity given by the rigidity of the tappet (Cx) and the
cam (Cz) in the direction of the tappet.
(34)
And 1/Ctan (b) see (34) represents the tangential stiffness, Cβ
being the torsional stiffness of the cam and Cy the flexural stiffness at the y axis of
the cam, with Cb (b) given by the relation (35).
(35)
With (33) and (34) the relation (32)
is rewritten in the form (36):
(36)
Where a is the pressure angle, which is generally a
function of b, and at flat tachets
(used in distribution mechanisms), it has the constant value (zero): a = 0.
The motion equation is written as
(37):
(37)
where h(b) is the motion law imposed by the cam.
The pressure angle, a, thus influences (38):
(38)
Where R(b) is the current radius, which gives the cam
position (distance from the center of the cam to the cam contact point) and
approximates by the mean radius R1/2. The relation (38) can be put
in the form (39); Where the average radius, R1/2, is obtained with
the formula (40):
(39)
(40)
Rb is the radius of the
base circle, and hm is the maximum projected stroke of the tappet. This
produces an average radius, which is used in the calculations for
simplifications; ws = machine angle, constant, given by
machine speed. The equation (37) can now be written (41):
(41)
The solution of equation (41) is made for a=0, with the following notations:
The period of natural vibration is determined with relation (42):
(42)
The period of the natural vibration period is obtained by the formula (43):
(43)
The slope during the lifting of the cam (44) is:
(44)
The shaft stiffness factor is
obtained by the formula (45):
(45)
With dimensional parameters given by (46);
(46)
The motion equation is written in the form (47):
(47)
The nominal curve of the cam is known (48) and (49):
(48)
(49)
With (47), (48) and (49) the dynamic response is calculated by a numerical method.
The author of the paper (KOSTER, 1974) gives a numerical example for a motion law, corresponding to the cycloid cam (50):
(50)
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.
4.
RESULTS AND DISCUSSION
4.1.
A dynamic model with variable
internal damping
Starting from the kinematic scheme
of the classical distribution mechanism (see Figure 12), the dynamic,
mono-dynamic (single degree), translatable, variable damping model (see Figure
13) is constructed, the motion equation of which is (PETRESCU, 2008):
Figure 12: The
kinematic scheme of the classic model, with distribution mechanism
(51)
Equation (51) 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 (51) 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 º F0 -
the elastic spring force of the valve spring;
x
- actual valve displacement;
(the
cam profile) reduced to the axis of the valve.
Figure
13: Mono - dynamic model, with internal depreciation of the variable system
The Newton equation (51) is ordered
as follows:
(52)
At the same time the differential
equation of the mechanism is also written as Lagrange, (53), (Lagrange equation):
(53)
Equation (53), which is nothing other than the Lagrange differential equation, allows for the low strength of the valve (54) to be obtained by the polynomial coefficients with those of the Newtonian polynomial (52), the reduced drive force at the valve (55), as well as the expression of c, ie the expression of the internal damping coefficient, of the system (56).
(54)
(55)
(56)
Thus a new formula (56) 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 (51, or 52), by replacing it with c takes the form (57):
(57)
In the case of the classical distribution mechanism (in Figure 12), the reduced mass, M, is calculated by the formula (58):
(58)
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 (59):
(59)
where:
(60)
is the ratio of the crank
arms.
The following relationships are written (61-66):
(61)
(62)
(63)
(64)
(65)
(66)
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 (60), (63), (64), (66), the relationship (58) becomes (67-69):
(67)
or:
(68)
or:
(69)
We make the derivative dM/dj and result the following relationships:
(70)
(71)
(72)
Write the relationship (56) as:
(73)
which with (72) becomes:
(74)
or
(75)
Where was noted:
(76)
4.2.
Determination of motion equations
With relations (69), (62), (75) and (61), equation (52) is written first in the form (77), which develops in forms (78), (79) and (80):
(77)
(78)
meaning:
(79)
And final form:
(80)
which can also be written in another form:
(81)
Equation (81) can be approximated to
form (82) 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.
(82)
If the laws of entry with s, s' (low
speed), s' '(low acceleration), equation (82) takes the form (83) and the more
complete equation (81) takes the complex form (84):
(83)
(84)
4.3.
Solving the differential equation in
two steps
The
known differential equation, written in one of the above-mentioned forms, for
example in form (85) (the equation was obtained through Taylor series
developments), is solved twice. The first time is used for x' the value s' and
for x'' the value s''. In this way, the value x(0), i.e. the dynamic
displacement of the valve at step 0, is obtained. This displacement is derived
numerically and x'(0) and x''(0) are obtained. The values thus obtained are
introduced into the differential equation (which is used for the second
consecutive time) and we obtain x(1), i.e. the dynamic displacement of the
sought valve, x, which is considered to be the final value. If one try to
delete this process (for several steps), we will notice the lack of convergence
towards a unique solution and the amplification of values at each pass
(iteration). It is considered to solve the non-iterative equation, in two
steps, but exactly and directly, the one-step solution, the second one, the
first step being in fact a necessary mediation for the approximate
determination of x’ and x’’ values [54].
(85)
4.4.
The presentation of a differential
equation, (dynamic model), which takes into account the mass of cam
Starting
from the dynamic model presented, a new differential equation, describing the
dynamic operation of the distribution mechanism from four-stroke internal
combustion engines, will be obtained.
Practically,
the formula expresses the reduced mass of the whole kinematic chain, and then
changes the internal damping of the system, c, and automatically changes the
entire dynamic (differential) equation, which entitles us to say that we are
dealing with a new dynamic model, who also takes into consideration the chassis
table.
The
reduced mass M of the entire kinematic chain is now written in the form (86):
(86)
The
damping constant of the system now takes shape (87):
(87)
For
the classical distribution mechanism the given value of (88) is found and is
entered in the relation (87), which takes the form (89):
(88)
(89)
The
differential equation (90) is still used:
(90)
The
mass M, determined by (86) and the damping coefficient c, obtained with (89) in
equation (90), is then introduced and one obtains a new differential equation
(91), which is actually a new model dynamic base.
(91)
The
differential equation (91) is written in the form (92) after the two identical
terms that contain it on J1 are reduced:
(92)
Using
the transmission function, D and its first derivative, D', the differential
equation (92), becomes equation (93):
(93)
Equation (93) is arranged in the form (94):
(94)
Note x with s+Dx, (95):
(95)
With
(95), equation (94) takes the form (96), where Dx
represents the difference between the dynamic displacement x and the imposed s,
both reduced to the valve axis:
(96)
To
approximate the x' and x” values, one use relations (97-100) and finally
(99-100):
(97)
(98)
(99)
(100)
With
relations (99) and (100), but also with approximation, equation (96) is written in the form (101):
(101)
Equation
(101) is ordered as (102):
(102)
Calculate
Dx twice,
Dx(0) and
Dx.Dx(0)
gathered to generate x(0), which is used to determine the variable angular
velocity, w.
In
Dx(0)
equation w=wn =
constant.
In
the second equation Dx, w is determined
using the first equation; for low speed x' and low acceleration x'', one now
have two variants: either it can introduce directly all approximate values
calculated with relations (99-100), or it can use x'(0) and x''(0) by direct
(numerical) derivation of x(0), which otherwise will only be used to find
variable angular velocity, w.
With
Dx
gathered can to obtain the exact value of x, which one derive numerically and
obtains the final (exact) values for reduced speed, x' and reduced
acceleration, x''.
4.5.
Dynamic analysis for the sinus law,
using the relationship (102), for the dynamic model considering the mass m1 of
the cam
For
this dynamic model (A3) there is a single dynamic diagram (Figure 14): Using
the relation (102) obtained from the dynamic damping model of the variable
system, considering the mass m1 of the cam, results the dynamic model A3 apply
in the dynamic analysis presented in the diagram in Figure 14.
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, hT, equal to that of the valve, hS (i = 1), take the
value of h = 5 [mm]. A spring elastic constant, k = 60 [N / mm] a valve spring
compression, x0 = 30 [mm]. The mechanical yield is, h=6.9%.
The
original model presented has the great advantage of accurately capturing
vibrations within the analyzed system.
Figure 14:
Dynamic analysis using the A3 dynamic model
5.
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.
Rigid memory mechanisms have played
an important role in the history of mankind, contributing greatly to the
industrial, economic, social changes in society, thus leading to a real
evolution of mankind. Used in automated tissue wars, in cars as distribution
mechanisms, automated machines, mechanical transmissions, robots and
mechatronics, precision devices, and medical devices, these mechanisms have
been real support for mankind along the time. For this reason, it considered
useful this paper, which presents some dynamic models that played an essential
role in designing rigid memory mechanisms.
The original model presented has the
great advantage of accurately capturing vibrations within the analyzed system.
6.
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.
7.
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.
8.
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 |
|
|
is the derivative of |
|
Fm |
is the motor force |
|
Fr |
is the resistant force. |
9.
AUTHORS’ CONTRIBUTION
All the authors have
contributed equally to carry out this work.
REFERENCES
ANGELES, J.; LOPEZ-CAJUN, C. (1988) Optimal synthesis
of cam mechanisms with oscillating flat-face followers. Mechanism Mach. Theory,
v. 23, n. 1, p. 1-6. doi:10.1016/0094-114X(88)90002-X
ANTONESCU, P. (2000) Mechanisms and Handlers, Printech Publishing House. Bucharest.
ANTONESCU, P.; PETRESCU, F. I. T. (1985) Analytical
method of synthesis of cam mechanism and flat stick. Proceedings of the 4th
International Symposium on Mechanism Theory and Practice (TPM’ 85),
Bucharest.
ANTONESCU, P.; PETRESCU, F. I. T. (1989) Contributions
to cinetoelastodynamic analysis of distribution
mechanisms. SYROM’89, Bucharest.
ANTONESCU, P.; OPREAN, M.; PETRESCU, F. I. T. (1985a)
Contributions to the synthesis of oscillating cam mechanism and oscillating
flat stick. Proceedings of the 4th International Symposium on Theory and
Practice of Mechanisms, (TPM’ 85), Bucharest.
ANTONESCU, P.; OPREAN, M.; PETRESCU, F. I. T. (1985b)
At the projection of the oscillante cams, there are
mechanisms and distribution variables. Proceedings of the V-Conference for
Engines, Automobiles, Tractors and Agricultural Machines, I-Engines and
Automobiles, (AMA’ 85), Brasov.
ANTONESCU, P.; OPREAN, M.; PETRESCU, F. I. T. (1986)
Projection of the profile of the rotating camshaft acting on the oscillating plate
with disengagement. Proceedings of the 3rd National Computer Assisted
Designing Symposium in Mechanisms and Machine Bodies, (MOM’ 86), Brasov.
ANTONESCU, P.; OPREAN, M.; PETRESCU, F. I. T. (1987)
Dynamic analysis of the cam distribution mechanisms. Proceedings of the
Seventh National Symposium of Industrial Robots and Spatial Mechanisms
(IMS’ 87), Bucharest.
ANTONESCU, P.; OPREAN, M.; PETRESCU, F. I. T. (1988)
Analytical synthesis of Kurz profile, rotating flat
cam cam. Machine Build. Rev., Bucharest.
ANTONESCU, P.; PETRESCU, F. I. T.; ANTONESCU, O. (1994)
Contributions to the synthesis of the rotating cam mechanism and the tip of
the balancing tip. Brasov.
ANTONESCU, P.; PETRESCU, F. I. T.; ANTONESCU, O. (1997)
Geometrical synthesis of the rotary cam and balance tappet mechanism.
Bucharest.
ANTONESCU, P.; PETRESCU, F. I. T.; ANTONESCU, O.
(2000a) Contributions to the synthesis of the rotary disc-cam profile. Proceedings
of the 8th International Conference on Theory of Machines and Mechanisms,
(TMM’ 00), Liberec, Czech Republic, p. 51-56.
ANTONESCU, P.; PETRESCU,
F. I. T.; ANTONESCU, O. (2000b) Synthesis of the rotary cam profile with balance follower. Proceedings
of the 8th Symposium on Mechanisms and Mechanical Transmissions (MMT’ 000),
Timişoara, pp: 39-44.
ANTONESCU, P.; PETRESCU, F. I. T.; ANTONESCU, O. (2001)
Contributions to the synthesis of mechanisms with rotary disc-cam. Proceedings
of the 8th IFToMM International Symposium on Theory
of Machines and Mechanisms, (TMM’ 01), Bucharest, ROMANIA, pp: 31-36.
AVERSA, R.; PETRESCU, R. V.; APICELLA, A.; PETRESCU,
F. I. T.; CALAUTIT, J. K.; MIRSAYAR, M. M.; BUCINELL, R.; BERTO, F.; AKASH, B.
(2017a) Something about the V Engines Design. Am. J. Appl. Sci., v. 4,
n. 1, p. 34-52. doi:10.3844/ajassp.2017.34.52
AVERSA, R.; PETRESCU, R. V.; AKASH, B.; BUCINELL, R.;
CORCHADO, J.; BERTO, F.; MIRSAYAR, M.
M.; CHEN, G.; LI, S.; APICELLA, A.; PETRESCU, F. I. T. (2017b) Something about
the Balancing of Thermal Motors. Am. J. Eng. Appl. Sci., v. 10, n. 1, p.
200-217. doi:10.3844/ajeassp.2017.200.217
AVERSA, R.; PETRESCU, R. V.; APICELLA, A.; PETRESCU,
F. I. T.; 2017c. A Dynamic Model for Gears. Am. J. Eng. Appl. Sci., v. 10,
n. 2, p. 484-490. doi:10.3844/ajeassp.2017.484.490
AVERSA, R.; PETRESCU, R. V.; PETRESCU, F. I. T.;
APICELLA, A.; 2017d. Smart-Factory: Optimization and Process Control of
Composite Centrifuged Pipes. Am. J. Appl. Sci., v. 13, n. 11, p. 1330-1341.
doi:10.3844/ajassp.2016.1330.1341
AVERSA, R.;TAMBURRINO, F.; PETRESCU, R. V.; PETRESCU,
F. I. T.; ARTUR, M.; CHEN, G.; APICELA, A. (2017e) Biomechanically Inspired
Shape Memory Effect Machines Driven by Muscle like Acting NiTi
Alloys. Am. J. Appl. Sci., v. 13, n. 11, p. 1264-1271.
doi:10.3844/ajassp.2016.1264.1271
FAWCETT, G. F.; FAWCETT, J. N. (1974) Comparison of
polydyne and non polydyne
cams. In: Cams and cam mechanisms.
Ed. J. Rees Jones, MEP, London
and Birmingham, Alabama, 1974.
GIORDANA, F.; ROGNONI, V.; RUGGIERI, G. (1979) On the
influence of measurement errors in the Kinematic analysis of cam. Mechanism
Mach. Theory, v. 14, n. 5, p. 327-340. doi:10.1016/0094-114X(79)90019-3
HAIN, K. (1971) Optimization of a cam mechanism to
give good transmissibility maximal output angle of swing and minimal
acceleration. J. Mechanisms, v. 6, n. 4, p. 419-434.
doi:10.1016/0022-2569(71)90044-9
JONES, J. R.; REEVE, J. E. (1974) Dynamic response
of cam curves based on sinusoidal segments. In: Cams and cam mechanisms.
Ed. J. Rees Jones, MEP, London and Birmingham, Alabama.
KOSTER, M. P. (1974) The effects of backlash and
shaft flexibility on the dynamic behavior of a cam mechanism. In: Cams and
cam mechanisms, Ed. J. Rees Jones, MEP, London and Birmingham, Alabama, 1974.
MIRSAYAR, M. M.; JONEIDI, V. A.; PETRESCU, R. V.; PETRESCU,
F. I. T.; BERTO, F. (2017) Extended MTSN criterion for fracture analysis of
soda lime glass. Eng. Fract. Mech., v. 178, p.
50–59. doi:10.1016/j.engfracmech.2017.04.018
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.;
CORCHADO, J.; CALAUTIT, J. K.; APICELA,
A.; PETRESCU, F. I. T. (2017a) Yield at Thermal Engines Internal Combustion. Am.
J. Eng. Appl. Sci., v. 10, n. 1, p. 243-251.
doi:10.3844/ajeassp.2017.243.251
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.;
CORCHADO, J.; CALAUTIT, J. K.; APICELA,
A.; PETRESCU, F. I. T. (2017b) Forces at Internal Combustion Engines. Am. J.
Eng. Appl. Sci., v. 10, n. 2, p. 382-393. doi:10.3844/ajeassp.2017.382.393
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.;
CORCHADO, J.; APICELA, A.; PETRESCU, F.
I. T. (2017c) Gears-Part I. Am. J. Eng. Appl. Sci., v. 10, n. 2, p. 457-472.
doi:10.3844/ajeassp.2017.457.472
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.;
CORCHADO, J.; APICELLA, A.; PETRESCU, F. I. T. (2017d)
Gears-Part II. Am. J. Eng. Appl. Sci., v. 10, n. 2, p. 473-483.
doi:10.3844/ajeassp.2017.473.483
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.;
CORCHADO, J.; APICELA, A.; PETRESCU, F. I. T. (2017e)
Cam-Gears Forces, Velocities, Powers and Efficiency. Am. J. Eng. Appl. Sci.,
v. 10, n. 2, p. 491-505.doi: 10.3844/ajeassp.2017.491.505
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.; CORCHADO,
J.; KOSAITIS, S.; ABU-LEBDEH, T.;
APICELA, A.; PETRESCU, F. I. T. (2017f) Dynamics of Mechanisms with Cams
Illustrated in the Classical Distribution. Am. J. Eng. Appl. Sci., v. 10,
n. 2, 551-567. doi:10.3844/ajeassp.2017.551.567
PETRESCU, R. V.; AVERSA, R.;AKASH, B.; BUCINELL, R.; CORCHADO,
J.; KOSAITIS, S.; ABU-LEBDEH, T.; APICELA, A.;
PETRESCU, F. I. T. (2017g) Testing by Non-Destructive Control, Am. J. Eng.
Appl. Sci., v. 10, n. 2, p. 568-583. doi:10.3844/ajeassp.2017.568.583
PETRESCU, R. V.; AVERSA, R.;APICELA, A.; PETRESCU, F.
I. T. (2017h) Transportation Engineering. Am. J. Eng. Appl. Sci., v. 10,
n. 3, p. 685-702. doi:10.3844/ajeassp.2017.685.702
PETRESCU, R. V.; AVERSA, R.; KOZAITIS, S.; APICELA,
A.; PETRESCU, F. I. T. (2017i) The Quality of Transport and Environmental
Protection, Part I. Am. J. Eng. Appl. Sci., v. 10, n. 3, p. 738-755.
doi:10.3844/ajeassp.2017.738.755
PETRESCU, F. I. T.; CALAUTIT, J. K.; MARINKOVIC, D. (2015)
Structural Dynamics of the Distribution Mechanism with Rocking Tappet with
Roll. Am. J. Eng. Appl. Sci. v. 8, n. 4, p. 589-601.
doi:10.3844/ajeassp.2015.589.601
PETRESCU, F. I. T.; PETRESCU, R. V. (2016) Otto motor
dynamics. Geintec-Gestao Inovacao e Tecnologias, v. 6,
n. 3, p. 3392-3406. doi:10.7198/geintec.v6i3.373
PETRESCU, F. I. T.; PETRESCU, R. V. (2014) Cam Gears
Dynamics in the Classic Distribution. Ind. J. Manag.Prod.,
v. 5, n. 1, p. 166-185. doi:10.14807/ijmp.v5i1.133
PETRESCU, F. I. T.; PETRESCU, R. V. (2013a) An
Algorithm for Setting the Dynamic Parameters of the Classic Distribution
Mechanism. Int. Rev. Model. Simul., v. 6, n. 5, p. 1637-1641.
PETRESCU, F. I. T.; PETRESCU, R. V. (2013b) Dynamic
Synthesis of the Rotary Cam and Translated Tappet with Roll. Int. Rev.
Model. Simul., v. 6, n. 2, p. 600-607.
PETRESCU, F. I. T.; PETRESCU, R. V. (2013c) Cams with
High Efficiency. Int. Rev. Mech. Eng., v. 7, n. 4, p. 599-606.
PETRESCU, F. I. T.; PETRESCU, R. V. (2013d) Forces and
Efficiency of Cams. Int. Rev. Mech. Eng., v. 7, n. 3, p. 507-511.
PETRESCU, F. I. T.; PETRESCU, R. V. (2011) Dinamica mecanismelor
de distribute. Create Space publisher, USA, (Romanian version). ISBN 978-1-4680-5265-7
PETRESCU, F. I. T.; PETRESCU, R. V. (2005a) Contributions
at the dynamics of cams. In: Proceedings of the Ninth IFToMM
International Symposium on Theory of Machines and Mechanisms, Bucharest,
Romania, v. I, p. 123-128.
PETRESCU, F. I. T.; PETRESCU, R. V. (2005b)
Determining the dynamic efficiency of cams. In: Proceedings of the Ninth IFToMM International Symposium on Theory of Machines and
Mechanisms, Bucharest, Romania, v. I, p. 129-134.
PETRESCU, F. I. T. (2015a) Geometrical Synthesis of
the Distribution Mechanisms. Am. J. Eng. Appl. Sci., v. 8, n. 1, p. 63-81.
doi:10.3844/ajeassp.2015.63.81
PETRESCU, F. I. T. (2015b) Machine Motion Equations at
the Internal Combustion Heat Engines. Am. J. Eng. Appl. Sci., v. 8, n. 1,
p. 127-137. doi:10.3844/ajeassp.2015.127.137
PETRESCU, F. I. T. (2012a) Bazele
analizei și optimizării sistemelor cu memorie rigidă – curs și aplicații.
Create Space publisher, USA, (Romanian edition), 2012. ISBN 978-1-4700-2436-9
PETRESCU, F. I. T. (2012b) Teoria
mecanismelor – Curs si aplicatii (editia a doua). Create Space publisher, USA, (Romanian version),
2012. ISBN 978-1-4792-9362-9
PETRESCU, F. I. T. (2008) Theoretical and Applied
Contributions About the Dynamic of Planar Mechanisms with Superior Linkages,
Ph.D. thesis.
SAVA, I. (1970) Contributions to Dynamics and
Optimization of Income Mechanism Synthesis. Ph.D. Thesis, I.P.B., 1970.
SAVA, I. (1971) Regarding the dynamic functioning of
the internal combustion engine valve control valve. In C.M., n. 12.,
Bucharest.
TARAZA, D.; HENEIN, N. A.; BRYZIK, W. (2001) The
Frequency Analysis of the Crankshaft's Speed Variation: A Reliable Tool for
Diesel Engine Diagnosis. J. Eng. Gas Turbines Power, v. 123, n. 2, p. 428-432.
doi:10.1115/1.1359479
TESAR, D.; MATTHEW, G. K. (1974) The design of
modeled cam systems. In: Cams and cam mechanisms, Ed. J. Rees Jones, MEP,
London and Birmingham, Alabama, 1974.
WIEDERRICH, J. L.; ROTH, B. (1974) Design of low
vibration cam profiles. In: Cams and Cam Mechanisms. Ed. J. Rees Jones, MEP,
London and Birmingham, Alabama, 1974.