*Periodicity.:*

**October - January 2014***e-ISSN......:*

**2236-269X**### CAM GEARS DYNAMICS IN THE CLASSIC
DISTRIBUTION

*Florian
Ion Petrescu*

*Bucharest
Polytechnic University, Romania*

*E-mail:
petrescuflorian@yahoo.com*

*Relly
Victoria Petrescu*

*Bucharest
Polytechnic University, Romania*

*E-mail:
petrescuvictoria@yahoo.com*

*Submission:
03/09/2013*

*Revision:
17/09/2013*

*Accept:
23/09/2013*

ABSTRACT

The paper presents an original
method to determine the general dynamics of mechanisms with rotation cams and
followers, particularized to the plate translated follower. First, one presents
the dynamics kinematics. Then one solves the Lagrange equation and with an
original dynamic model with one degree of freedom, with variable internal
amortization, it makes the dynamic analysis of two models.

## Keywords: cam mechanisms; rotary cam; plate
translated follower; dynamics; classic distribution

**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.

The Otto engines or those with internal combustion in
general, will have to adapt to hydrogen fuel. It is composed of the basic
hydrogen which can be extracted industrially, practically from any item (or
combination) through nuclear, chemical, photonic by radiation, by burning
process, 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). Hydrogen must be stored in reservoirs cell (a honeycomb) to ensure
that 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".

At the time of boring the tree trunk, from it to 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), the 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.

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. In the future the jet will be accomplished with atomic particle
or nuclear.

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. We must improve the distribution mechanism.

The paper proposes an original
dynamic model illustrated for the rotating cam with plate translated follower.
One presents the dynamic kinematics (the original kinematics); the variable
velocity of the camshaft obtained by an approximate method is used with an
original dynamic system having one degree of freedom and** **a** **variable internal
amortization; it tests two movement laws, one classic and an other original.

**2.
****DYNAMIC OF THE
CLASSIC DISTRIBUTION MECHANISM**

**2.1.
****Precision Kinematics in the
Classic Distribution Mechanism**

In the fig. number one, it
presents the kinematic schema of the classic distribution mechanism, in two
consecutive positions (HAIN K., 1971; GIORDANA F., 1979); with an interrupted line is represented the particular
position when the follower is situated in the lowest possible plane, (s=0), and
the cam which has a clockwise rotation, with constant angular velocity, w, is situated in the point A^{0},
(the fillet point between the base profile and the** **rise profile), a** **particular
point that marks the beginning** **of
the rise movement of the follower (ANGELAS J. and
Lopez-Cajun C., 1988), imposed by the cam-profile (DAN K.,
2009); with a continue line is represented the higher joint in a certain
position of the rise phase (PETRESCU F. I., 1987).

Figure 1. The kinematics of the classic distribution
mechanism

The point A^{0},
which marks the initial higher pair, represents in the same time the contact
point between the cam and** **the** **follower in the first position. The
cam is rotating with the angular velocity, w (the camshaft angular
velocity), describing the angle j, which shows how the base
circle has rotated clockwise (together with the camshaft); this rotation can be
seen on the base circle between the two particular points, A^{0} and A^{0i}.

In this time the
vector r_{A}=OA (which represents the distance between the centre of
cam O, and the contact point A), has rotated anticlockwise** **with the angle t. If one measures the angle q, which positions the general
vector, r_{A}, in function of the particular vector, r_{A0},
one obtains the relation (0):

_{} (0)

where r_{A} is the module of the
vector _{},
and q_{A} represents the phase angle of
the vector _{}.

The angular velocity
of the vector _{} is _{} which is a function of the angular velocity of
the camshaft, w, and of the angle j (by the movement laws s(j), s’(j), s’’(j)).

The follower isn’t
acted directly by the angle j and the angular velocity w; it’s acted by the vector _{},
which has the module r_{A}, the position angle q_{A} and the angular velocity _{}.
From here we deduce a particular (dynamic) kinematics, the classical kinematics
being just static and approximate kinematics (ANTONESCU
P., 1985).

Kinematics we define the next velocities
(Fig. 1): _{}=the
cam’s** **velocity; which is the
velocity of the vector _{},
in the point A; now the classical
relation (1) becomes an approximate relation, and the real relation takes the
form (2)._{}

_{} (1)

_{} (2)

The
velocity _{} is separating into the velocity _{}=BC
(the follower’s velocity which acts on its axe, vertically) and _{}=AB
(the slide velocity between the two profiles, the sliding velocity between the
cam and the follower, which works along the direction of the commune tangent
line of the two profiles in the contact point).

Because
usually the cam profile is synthesis for the classical module C with the AD=s’
known, it can write the relations:

_{} (3)

_{} (4)

_{} (5)

_{} (6)

_{} (7)

Now,
the follower’s velocity isn’t _{} (_{}),
but it’s given by the relation (9).** **In
the case of the classical distribution mechanism the transmitting function D is
given by the relations (8):

_{} (8)

_{} (9)

The
determining of the sliding velocity between the profiles is made with the
relation (10):

_{} (10)

The
angles t and q_{A} will be determined, and also their
first and second derivatives._{}_{}

The
t
angle has been determined from the triangle ODA^{i} (Fig.1) with the
relations (11-13):

_{} (11)

_{} (12)

_{} (13)

One
derives (11) in function of j angle and obtains (14):

_{} (14)

The
relation (14) will be written in the form (15):

_{} (15)

From
the relation (12) one extracts the value of cost, which will be introduced in
the left term of the expression (15); then one reduces s’’.s’^{2} from
the right term of the expression (15) and obtains the relation (16):

_{} (16)

After
some simplifications the relation (17), which represents the expression of t’, is finally obtained:

_{} (17)

Now, when t’ has been explicitly**
(**deduced), the next derivatives can be determined. The expression (17) will be derived directly
and obtains for the beginning the relation (18):

_{} (18)

The
terms from the first bracket of the numerator (s’.s’’) are reduced, and then
one draws out s’ from the fourth bracket of the numerator and obtains the
expression (19):

_{} (19)

Now
it can calculate q_{A}, with its first two derivatives, _{} and _{}.
We will write *q**
*instead of *q** _{A}*, to simplify the notation. Now
one can determine (20) which is the same of (0):

_{} (20)

One derives (20) and obtains the
relation (21):

_{} (21)

One derives twice (20), (or derives 21)
and obtains (22), with (relation 24) and with w=ct.:

_{} (22)

One can write now
the transmission functions, D and D’ (for the classical module, C), in the
forms (23-24):

_{} (23)

_{} (24)

To calculate the
follower’s velocity (25) we** **need the
expression of the transmission function, D (where w is the general rotation speed
of the cam, the average value, or the input value, which is a constant, and w
is the momentary rotation speed of the cam which is a variable value).

_{} (25)

where

_{} (26)

For the classical
distribution mechanism (Module C), the variable w is the same as _{} (see the relation 25). But in the case of** **B and F modules (at the cam gears
where the follower has a roll), the transmitted function D and w take complex
forms.

Now, one can determine the acceleration
of the follower (27).

_{} (27)

Figure 2 represents** **the classical and dynamic kinematics;
the velocities (a), and the accelerations (b).

a

b

Figure 2. The classical and
dynamic kinematics; a-velocities and b-accelerations of the follower

To determine the
acceleration of the follower, are necessary to be known s’ and s’’, D and D’, t’ and t’’.

The dynamic
kinematics diagrams of v_{2} (obtained with relation 25, see Fig. 2a),
and a_{2} (obtained with relation 27, see Fig. 2b), have a more dynamic
aspect than one kinematic. One has used the movement law SIN, a rotational
speed of** **the crankshaft n=5500 rpm,
a rise angle j_{u}=75^{0}, a fall angle j_{d}=75^{0} (identically with the
ascendant angle), a ray of** **the basic
circle of the cam, r_{0}=17 mm and a maxim stroke of the follower, h_{T}=6
mm.

Anyway,
the dynamics is more complex (NORTON R. L., 1999), having in
view the masses and the inertia moments, the resistant and motor forces, the
elasticity constants and the amortization coefficient of the kinematic** **chain, the inertia forces of the
system, the angular** **velocity of the
camshaft and the variation of the camshaft’s angular** **velocity, w,
with the cam’s position, j,
and with the rotational speed of the crankshaft, n.

**2.2.
****Solving Approximately the
Lagrange Movement Equation**

In the
kinematics and the static forces study of the mechanisms one considers the
shaft’s angular velocity constant, _{}=constant, and the angular acceleration null, _{}. In reality, this angular velocity w isn’t
constant, but it is variable with the camshaft position, j.

In
mechanisms with cam and follower the camshaft’s angular velocity is variable as
well. One shall see further the Lagrange equation, written in the differentiate
mode and its general solution.

The
differentiate Lagrange equation has the form (28):

_{} (28)

Where J* is the inertia
moment (mass moment, or mechanic moment) of the mechanism, reduced at the
crank, and M* represents the difference between the motor moment reduced at the
crank and the resistant moment reduced at the crank; the angle j
represents the rotation angle of the crank (crankshaft). J*^{I}
represents the derivative of the mechanic moment in function of the rotation
angle j
of the crank (29).

_{} (29)

Using
the notation (29), the equation (28) will be written in the form (30):

_{} (30)

We
divide the terms** **by J* and (30)
takes the form (31):

_{} (31)

The term with _{} will be moved to the
right side of the equation and the form (32) will be obtained:

_{} (32)

Replacing
the left term of the expression (32) with (33) one obtains the relation (34):

_{} (33)

_{} (34)

Because,
for an angle j,
w
is different from the nominal constant value w_{n,}
one can write the relation (35), where dw represents the momentary
variation for the angle j;
the variable dw
and the constant w_{n}
lead us to the needed variable, w:

_{} (35)

In
the relation (35), w
and dw
are functions of the angle j,
and w_{n}
is a constant parameter, which can take different values in function of the
rotational speed** **of the drive-shaft,
n. At a moment, n is a constant and w_{n}
is a constant as well (because w_{n}
is a function of n). The angular velocity, w,
becomes a function of n too (see the relation 36):

_{} (36)

With
(35) in (34), one obtains the equation (37):

_{} (37)

The relation (37) takes the form (38): _{} (38)

The
equation (38) will be written in the form (39):

_{} (39)

The
relation (39) takes the form (40):

_{} (40)

The
relation (40) is an equation of the second degree in dw.
The discriminate of the equation (40) can be written in the forms (41) and
(42):

_{} (41)

_{} (42)

One
keeps for dw
just the positive solution, which can generate positives and negatives normal
values (43), and in this mode only normal values will be obtained for w;
for _{} one considers** **dw=0
(this case must be not seeing if the equation is correct).

_{} (43)

Observations: For
mechanisms with rotative cam and follower, using the new relations, with M^{*}
(the reduced moment of the mechanism) obtained by the writing of the known** **reduced resistant moment and by the** **determination of the reduced motor
moment by the integration of the resistant moment one frequently obtains some
bigger values for dw,
or zones with D
negative, with complex solutions for dw. This fact gives us the
obligation to reconsider the method to determine the reduced moment.

If we take into
consideration** **M*_{r} and M*_{m},
calculated independently (without integration), one obtains for the mechanisms
with cam and follower normal values for dw, and _{}.

In
paper (PETRESCU, F.I., Petrescu, R.V. 2005) one presents the relations to
determine** **the resistant force (44)
reduced to the** **valve, and the motor
force (45) reduced to the ax of the valve:

_{} (44)

_{} (45)

The
reduced resistant moment (46), or the reduced motor moment (47), can be
obtained** **by the resistant or motor
force multiplied by the reduced velocity, x’.

_{} (46)

_{} (47)

**2.3.
****The Dynamic Relations Used**

The dynamics
relations used (48-49), have been deduced in the paper [PETRESCU, F.I.,
Petrescu, R.V. (2005)]:

_{} (48) _{} (49)

**3.
****THE DYNAMIC ANALYSIS**

The dynamic analysis or the
classical movement law sin (BAGEPALLI B. S., 1991), can be seen in the diagram
from figure 3, and in figure** **4 one
can see the diagram of an original movement law (C4P) (module C) (SATYANARAYANA
M., 2009). It was considered a motor shaft speed of 5000 [rpm] (rot. per minute).

The angle of climb had been taken 75
[deg] (degrees sexazecimale). Constant spring valve spring was considered to be
20 [N/mm]. The basic circle radius of the cam is 14 [mm]. Valve spring
pretension is 40 [mm] (GE Z., 2011). Valve travel is 6 [mm]. Tappet race all 6
[mm]. Use has been made of a transmission ratio of rocker arm equal to the
unit.

The program of calculation used was
written in excel that is extremely affordable.

Figure 3. The dynamic analysis of the law sin, Module C, j_{u}=75^{0}, n=5000 rpm

Figure 4. The dynamic analysis of the new law, C4P, Module
C, j_{u}=45^{0}, n=10000 rpm

Using a law of movement of original
tappet can increase engine speed to superior (TARAZA
D., 2002). Increasing
the speed can be increased more than if we use another module cam follower,
namely the module B (see the Figure 5-6).

Figure 5. Law C4P1-5, Module B, j_{u}=80^{0}, n=40000 rpm

Figure 6. Law C4P3-2, Module F, j_{u}=85^{0}, n=40000 rpm

**4. ****CONCLUSIONS**

If we improve the cam
dynamics, it can synthesize high-speed cam, or high-performance camshafts (PETRESCU F. I. and Petrescu R. V., 2005, 2008).

In this paper one presents an original method to determine
the dynamic parameters at the camshaft (the distribution mechanisms). It makes
the dynamics, of the rotary cam and plate tappet with translational motion,
with a great precision.

The presented method is the most elegant and direct method
to determine the kinematics and dynamic parameters. The dynamic synthesis can
generate a cam profile which will work without vibrations. Processes robotization increasingly
determine and influence the emergence of
new industries, applications in
specific environmental conditions,
approach new types of technological operations, handling of objects in outer space, leading teleoperator in
disciplines such as medicine, robots that covers a whole
larger service benefits
our society, modern and computerized. In
this context, this paper seeks to contribute to the scientific and technical
applications in dynamic analysis and synthesis of cam mechanisms.

Using the classical movement
laws, the dynamics of the distribution cam-gears depreciate rapidly at the
increasing of the rotational speed of the shaft. To support a high rotational
speed** **it** **is necessary the synthesis of the cam-profile by new movement
laws, and for the new Modules.

A new and original movement law is
presented in the pictures number 4, 5 and 6; it allows the increase** **of the rotational speed to the values:
10000-20000 rpm**,** in the classical
module C presented (Fig. 4). With others modules (B, F) one can obtain
30000-40000 rpm (see Figs. 5, 6).** **

Dynamic
module B (CHANG Z., 2001, 2011) is similar to that of a
conventional module C presented in work. But the geometry module B changes, as
well as cinematic joined forces, dynamic cinematic (precision), in such a way
that particular dynamic module B is changed. In Figure 7 is presented the
module B. Why we must have high speeds? For
construction of an internal combustion modern engine, must be modified the main
mechanism, built an engine compact and high speeds. This goal can be achieved
with virtually a new module for distribution, such as the module B. On the
other hand all module B is the one with which it may be possible to make the
timing mechanisms in high yields in operation (PETRESCU
F.I., PETRESCU R.V., 2013a,b,c).

Figure 7. The geometry and the
forces at the module B.

High yields for the mechanisms of
distribution, as a matter of indifference laws of movement imposed tappet. It
is important module type chosen (in our case B) and adjustments valve spring.

The used law is the classical law,
cosine law. The synthesis of the cam profile can be made with the relationships
(50) when the cam rotates clockwise and with the expressions from the system (51) when the cam
rotates counterclockwise (trigonometric).

_{} (50)

_{} (51)

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 fig. 8.

Figure 8. 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 fig. 9 (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
9. 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]

The yield
distribution mechanisms can be increased even more, reaching even to rival
gears (PETRESCU F.I., PETRESCU R.V., 2013a).

5. REFERENCES

ANGELAS
J., LOPEZ-CAJUN C. (1988). ** Optimal synthesis of cam mechanisms with
oscillating flat-face followers**. Mechanism and Machine Theory
23,(1988), Nr. 1., p. 1-6..

ANTONESCU,
P., PETRESCU, F., 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, Vol. III, p. 126-133.

BAGEPALLI,
B.S., a.o. (1991). *Generalized Modeling of Dynamic Cam-follower Pairs in Mechanisms**.* Journal of Mechanical Design, June
1991, Vol. 113, Issue 2, p. 102-109.

CHANG, Z.,
a.o. (2001). ** A study on dynamics of roller gear cam system considering clearances**.
Mechanism and Machine Theory, January 2001, Vol. 36, N. 1, p. 143-152.

CHANG, Z., a.o. (2011). ** Effects of clearance on dynamics
of parallel indexing cam mechanism**. ICIRA’11 Proceedings of the 4th
international conference on Intelligent Robotics and Applications – Volume,
Part I, 2011, p. 270-280.

DAN,
K., a.o. (2009). ** Research on Dynamic Behavior Simulation Technology for Cam-Drive
Mechanism in Single-cylinder Engines**. SAE Technical Paper, paper number
2009-32-0089.

GE,
Z., a.o. (2011). *Mechanism Design amd Dynamic Analysis of Hybrid Cam-linkage Mechanical
Press**.* Key Engineering
Materials Journal, Vol. 474-476, p. 803-806.

GE,
Z., a.o. (2011). ** CAD/CAM/CAE for the Parallel Indexing Cam Mechanisms**. Applied
Mechanics and Materials Journal, Vol. 44-47, p. 475-479.

GIORDANA
F., s.a. (1979) ** On the influence of measurement errors in the Kinematic analysis of cam**.
Mechanism and Machine Theory 14, nr. 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
6, Nr. 4., p.419-434.

NORTON,
R.L., a.o. (1999). *Effect of Valve-Cam Ramps on Valve Train Dynamics**.* SAE, International Congress &
Exposition, Paper Number 1999-01-0801.

PETRESCU,
F.I., PETRESCU, R.V. (2005). *Contributions at the dynamics of cams**.* In the Ninth IFToMM International
Symposium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania,
Vol. I, p. 123-128.

PETRESCU F.I., ş.a. (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., (2013a). *Cams with High Efficiency**.* International Review of Mechanical
Engineering (I.RE.M.E. Journal), May 2013, Vol. 7, N. 4, ISSN: 1970-8734.

PETRESCU,
F.I., PETRESCU, R.V., (2013b). *Dynamic Synthesis of the Rotary Cam and
Translated Tappet with Roll**.*
International Review on Modelling and Simulations (I.RE.MO.S. Journal), April
2013, Vol. 6, N. 2, Part B, ISSN: 1974-9821, p. 600-607.

PETRESCU,
F.I., PETRESCU, R.V., (2013c). *Forces and Efficiency of Cams**.* International Review of Mechanical
Engineering (I.RE.M.E. Journal), March 2013, Vol. 7, N. 3, ISSN: 1970-8734, p.
507-511.

SATYANARAYANA, M., a.o. (2009). *Dynamic
Response of Cam-Follower Mechanism**.*
SAE Technic Paper, paper number 2009-01-1416.

TARAZA, D. (2002). ** Accuracy
Limits of IMEP Determination from Crankshaft Speed Measurements**.

*SAE Transactions, Journal of Engines*111, p. 689-697.

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