EFFECT OF BODY TILT ANGLE TO THE SHAKING FORCES ON THE DIESEL ENGINE

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31 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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1



Abstract

In this paper, a programming of dynamic
system calculation is de
veloped to determine the body
tilt angle in relation to minimize the shaking forces on
the diesel engine’s crankshaft. In this study, 1
-
cylinder
diesel engine is taken as an example. Position,
velocity, and acceleration of pins of the engine
mechanism dete
rmined by using vector analysis.
Masses and mass moments of inertia of the linkage
are used to generate the forces and moments.
Cartesian coordinate principle is used to form linear
equations. These equations are solved by using gauss
elimination method t
o obtain the shaking forces on the
crankshaft. Calculation result is validated by
comparing it to the polygon method and Newton
principles. B
ased on the graphs, the optimum
tilt angle
of the engine
’s
body
had

be
en

obtained

at


for mi
nimum horizontal shaking force
Newton
.

Keyword:


dynamic system
, vector analysis,
gauss elimination
,

shaking forces, tilt angle
.

I.

INTRODUCTION

Recently,
researches of diesel engines
become

more
attractive because of its
fuel
has
similar chara
cteristic with
the environment
-
friendly fuel resources, such as biodiesel. In

the
fact

that
, biodiesel can be used

to replace the
co
nventional diesel fuel and
it
is made from
the
renewable
resources
.


Biodiesel is a kind of environment
-
friendl
y resources of
fuel, clean, grown locally.
Palm Oil Methyl Esters

(POME) is

This work was supported in part by ANPCYT.

*
F.N. Balia, PhD student at Mechan
ical and Manufacturing
Engineering Faculty, UTHM. (corresponding author: e
-
mail:
fnbalia@yahoo.co.id).

**
S. Mahzan, Lecturer at Mech and Manufacturing Eng Faculty,
UTHM.

***
M.I. Ghazali, Professor at Mech and Manufacturing Eng Faculty,
UTHM.

***
Abas A
B Wahab, Professor at Mech and Manufacturing Eng
Faculty, UTHM.

one of those. Researchs on this area had been carried out by

researchers such as

Agarwal

[1], Ramadhas [2], and
Murugesan

[3].

Researches on the vibration analysis on the diesel

engines
had been carried out
by researchers
as follows: Geng, et al
[4], carried out their research on the piston
-
slap
-
induced
vibration of 6
-
cylinder diesel engine. Garlucci, at al [5],
carried out the research on the relation between injection
paramet
er variation and block vibration of the diesel engine
(FIAT, 2000 cc).
Brusa, et al [6], investigated concerned
with the effect of non
-
constant moment of inertia of torsional
vibration on the crankshaft of 4
-
cylinder Lycoming O
-
360
-
A3A propeller engine.
Guzzomi, et al [7], conducted the
study concerned with the effect of the piston friction on the
torsional natural frequency of crankshaft of a single cylinder
reciprocating engine.

This
research
is a preliminary
work
on the diesel engine
s

area

and its dev
elopment to the biodiesel engine purpose.
This
paper
emphasize

on the
programming of

a dynamic
system

on the
d
iesel engines.
Determining the body tilt angle

in relation to minimize the shaking forces on the diesel

engine’s crankshaft
is very important to
reduce the
shaking

forces
on the body.

For

illustration, 1
-
cylinder diesel engine
taken as an example.



II.

P
ROBLEM
F
ORMULATION

In analyzing the

shaking

forces due to the combustion
process in the chamber on the engine’s crankshaft can be
describe as follows
:


2.1


Kinematic Formulation

The
calculation step
s

of an engine has to be started at the
kinematic formulation
, to

calculat
e

the position, velocity, and
acceleration of
pins

and center of mass.

Vector analysis
principles are used to calculate those paramet
ers.
The
mechanism of
engine
shown below.




EFFECT OF BODY TILT ANGLE
TO THE SHAKING FORCE
S

ON THE DIESEL ENGINE
’S CRANKSHAFT


*

Fuadi Noor Balia,
**
Shahruddin bin Mahzan,
***
Mohd
Imran bin Ghazali, and
***
A
bas AB Wahab


Proceedings of
MUCEET
200
9


Malaysian T
echnical Universities Conference on Engineering and Technology



June 20
-
22, 2009, MS Garden,Kuantan, Pahang, Malaysia




































MUCEET
200
9






2



Fig
ure

1,

Mechanism of E
ngine

Fig
ure

1, shows a system of
the engine
mechanism
,

in
which

the body tilt angle is included as a parameter that can
effect to minimize the shak
ing forces on the engine’s
crankshaft.

Link 1 is cylinder block and journal bearing, l
ink 2
is crankshaft, link 3 is connecting rod, and link 4 is piston
that can move freely on the
cylinder (
body
)

axis direction.

Figure 2, shows the vector
model

of engine

mechanism
for calculating the posit
ion, velocity, and acceleration of pins

and center of mass of the linkages
.

In this modeling,

vector
r2

represent the crank
shaft
,
vector
r
3

represent the
connecting rod,
and
vector
r
4

represent
the motion line of
piston.


is
angle
of
r2

to
x
,

is angle of
r2

to
r4
,

is angle of
r3

to
x
,

is angle of
r3

to
r4
,

is angle

of
r4

to
x
.



Fig
ure

2,
Kinematic Modeling

of
E
ngine

Mechanism



From

figure 2, mathematical model can be

govern
ed as a

vector

equation

below,








……
..
(1)


This equation

can
be
derived to obtain the

velocity
of
p
oints along the line vector,

such below








……
..
(2)


F
rom this equation can be
calculat
ed

the
connecting rod
angular velocity such below,






(3)


E
quation (
2)
,
can be derived

to give t
he equation of
acceleration of points
motion along the cylinder axis
can be
written
as below,







(4)



Calculation of angular acceleration of linkage 3 can be
derived

and the result as below,



……………

(5)


where :





Through equation (1) to equation (5), the position.
velocity (linear and angular) and acceler
ation (linear and
angular) of pins and center of mass can be obtained
.


2.2


Dynamic Formulation

Calculation of s
haking forces in any of pins and
existing
forces
at center of mass of linkage, can be modeled as figure
3 below

[8]
.



Figure 3,
Dynamic
Modeling of Engine M
echanism


This model can be solved by using Cartesian coordinate
method (vector analysis for dynamic systems)

[9]
. T
his

engine m
echanism can be modeled separately as follows.


2.2.1

Crankshaft Modeling






3



Figure 4, Modeling of Crankshaft



Figure 4, shows a
physical
modeling of
C
rankshaft.
In
this modeling,
the crank
shaft

is separated into two part
s, that

are
crank and balancer. C
enter of rotation assumed

located
at

point A
, t
herefore all
of
the moments refer to that point.



The
inertia
torque

that is generated by rotation of
the crank and
located at

point
A
.

The i
nertia torque


that is generated

by
rotation of balancer

and loca
ted at
point
B. Reaction torque


is input
moment
to the shaft due to
the
reaction
of combustion and inertia
load
s.

The
vector


is the
reaction of
the crankshaft

to the
crank, while
vector

is the
reaction
force
of

crank
pin
to the
crank
.
The mass


is of crank

mass and generate the
inertia force
of

and
center
ed at point A.

The

is the mass of balancer and generate the ine
rtia force of

and
center
ed at point B. The mass


is
a

half
mass of crank
pin

to the crank
, this mass generate a half of
inertia force
and
center
ed at point C.
In this
modeling, the
dist
ributed
weig
ht
of
linkage
part
are included.

W
2

is

a

half
of
weight
of
crank
shaf
t
.

W
2a

is the weight of
crank,
W
2b
is the weight balancer.

W
2c


is
a

half
of the
weight
of crankpin



2.2.2

Connecting rod Modeling







Figure 5,

Modeling of Connecting rod



Figure
5 shows a physical modeling of C
onnecting rod.
Center of rotation assumed
to be
located at point

D
.


The inertia torque

is

generated
due to
the
rotation
of connecting rod.
The mass

is the connecting rod
mass
and generate the inertia force

and
centere
d at point
D. T
he
vector

F
3


is
the reaction
force
of crankpin to the
connecting rod, while the force of
F
4

is
reaction of piston
pin to the connecti
ng rod.


The weight

W
3

is connecting rod
weight and centered at point D.







2.2.3 Piston Modeling






Figure 6, Modeling of Piston


The vector

is the
reaction
force
of connecting rod to
th
e pin of piston
.
The mass

is the sum of piston’s pin
and piston mass itself

and generated the inertia force

and located at point E
.

The weight
W
4


is the sum of pin and
piston weight. The vector

is reaction force of cylinder to




4

the piston and located at the length of vector


z

from the
center of mass, and the vector

is a force as a result of
the com
bustion process in the cylinder to the piston.


2.3

Mathematical

Equations



Mathematical modeling can be developed by using
vector analysi
s (Cartesian coordinate) method for engine
mechanism.



2.3.1

Equation for
Crank
shaft


Crank
shaft
mechanism is modeled by assumed that the
center of rotation
loca
ted
on
point A

and mas
s of each of
crank part located at the center of each part.


Equation of the forces equilibrium
vector
of

the

Crank
shaft
,




….. (6)


Equation of moment equilibrium
vector
of
the
Crank
shaft
,






……. (7)

where:



















Equation (6) can be developed to give the force equations
in x and

y direction, it mean give two rows of equation.

Equation (7) give the moment equations to the center of
rotation, after developing it using vector analysis, this
equation give a row of moment equation.

From equation (6) and (7) can result three rows of
l
inear
equation to form matrix.














2.3.2

Equation for
Connecting rod


Equation of the forces equilibrium
vector
of Connecting rod,





…………….. (8)


Equation of the moment equilibrium

vector
of Connectin
g
rod,






…………….. (9)


Equation (8) is developed to give

two rows of

the force
equations in x and y direction, and equation (9) give a row of
moment equation.

From equation (8) and (9) can result three rows of li
near
equation to form matrix.



2.3.
3 Equation for
Piston


Equation of forces equilibrium of piston,






…………… (10)


Equation of moments equilibrium,








……….
..
….. (11
)


Equation (10) can be developed into two rows of the
force equation, in x and y direction, while equation (11) can
be developed to be a row of moment equation in x and y.

These equations give three rows of linear equation to
form matri
x.


2.
3.
4

Matrix Formation



The equation (6) to equation (1
1
)
that have nine of
linear equations and
can be solved
simultaneously
by matrix
formation,









……………. (12
)


where,

is a
coefficient
of
symmetric

matrix

(9x9
matrix element
)
,

are

the forces and torque
parameter to
be solved
(9x1 matrix

element
),

while


are

the
effective
inertia forces and
moment

(9x1 matrix

element
) due

to the
moment
and forces
of inertia.


The forces and torque in matrix

can be solved by
using Gauss
-
Jordan
Elimination
principle
for solving
Simultaneous
ly

Linear Equation

[10]
.



III.

M
ETHOD OF
S
OLUTION

This programming is
divide
d
in two category
calculation. Firstly, kinematic step and secondly, dynamic
system formulation.

Compiler
Visual C++ is used for

programming language [11
].

Briefly, f
igure 7, shows a flow chart of
programming
.






5






Figur
e 7, Flow Chart of Programming


On the kinematic step, calculation of position, velocity,
and acceleration of pins and center of mass of each link were
carried out. Vector of position was written as equation (1). In

this development
vector

and
considered as the length
of crank and connecting rod, the values are constant.

Vector
considered as the length of position between
pi
ston and
the main bearing, this

is
a
n unconstant

variable.


Eq
uation (2)

gives the linear velocity of piston

motion along the cylinder
axis
.
Rotation of the crank gives the angular velocity and
acceleration of the connecting rod,

and
, it can
be seen
in equation (3) and (
5).
For this purpose, the main bearing is
fixed,
and body tilt angle is included as a parameter to adjust
the
positio
n
of
axis of piston
to x
(same as engine’s body tilt
angle axis).

Through t
his development,
the
effect of body tilt

angle

can be seen from

the equation (3) to equation (5).


Equation (4) shows
the
linear acceleration of piston motion
along the cylinder axis.


Validation
of kinematic calculations
had been carried out
by comparing it to the hand calculation of polygon

method
(see attachment
-
1 and attachment
-
2).

On the dynamics step, calculation of forces and moments
started of physical modeling of
the
crank and balancer,
connecting rod, and piston.

Figure 3 shows the
physical
dynamics
modeling of engine
mechanism
.

Con
cept of
Cartesian coordinate is used in accordance with the vector
analysis of forces and moments.

Crankshaft model
,

as showed at figure 4, can be derived
mathematically in accordance with d’Alembert principle of
equilibrium. Equation (6) shows the vector
for force
equilibrium, while equation (7) for moment equilibrium.

Connecting rod, as showed at figure 5, can be derived into
the forces and moments (equation (8) and (9)). Piston is
considered as the body motion along the cylinder axis.
Friction is neglect
ed in this situation, because of the
complexity

of calculation.

Matrix formation is used to
collect

the ninth of equation
s

had been developed from equation (6) to equation

(11). For
detail explanation, see section 2.3 (Mathematical Equation
s
)
for every st
ep of modeling.


The data that are used for this solution is taken from

the
size of diesel engine mechanism. Piston bore size d = 82.5
mm, stroke L = 92.5 mm, connecting rod length = 15 mm.
Length of crank r
a

= 47 mm, balancer length r
b

= 78 mm.

The
inter
nal pressure force


data delivered to the equation (10)
depend on the crank angle of combustion

[12
]
.


Simultaneously l
inear equation
s

in
a
matrix form can be
solved by using Gauss Elimination procedure, by inserting
pivot point algorithm. This solution

gives the result of
parameter
s

that
are

considered as the reaction forces o
f

the
pins.


Validation
of dynamics system
had been carried out by
comparing it to
the
static equilibrium,
by

eliminating the
dynamic parameters, such the
effective
forces and mome
nts
of inertia

(see attachment
-
3 and attachment
-
4).

Validation for mass of piston carried out by getting its
weight. For connecting rod, the mass, centroid and mass
moment of inertia by
calculation and
getting its weight

and
compare it
, t
he part
s are

mod
eled as a volume combination of
the separately block
s
.

For crankshaft, calculation of mass,
centroid and mass moment of inertia carried

out by modeling
its volume as the

combination of many
block
s

(same
procedure with connecting rod)
.

IV.

R
ESULT AND
D
ISCUSSIO
N







Figure 8, Graphic of Tilt Angle to Shaking Force

Programming of this engine mechanism give
s

a

comparatively result, even for
accuracy of calculation and in
accordance with the engine’s body tilt angle
, see figure 8
.

This angle result gives the
minimum shaking force in
horizontal direction,
Rx

= 2.07 Newton
, while f
or vertical
direction,
Ry

=
-
11957.05 Newton
. For
diesel engine has
pressure 3
,
5
MPa
, so
internal pressure force
F
c

= 18709.70




6

Newton, at 10
0

of crank angle.
Rotation of engine is con
stant,

n
= 4500 rpm.
This
condition is
very important consideration
for the human body comfort characteristic.

The human body
characteristic can stand for up and
down shaking motion,
but

for horizontal shak
e must be avoided for a long period
.


The optimum

tilt angle of engine’s body obtained
.

Validation
of kinematic calculation
,

a good result was
reached. Calculation for
n

= 2000 rpm (

rad/sec),

,

gives
angular velocity for C
onnecting r
od

rad/sec for calculation and

rad/sec for

polygon method. Angular acceleration

rad/sec2 for calculation and

rad/sec2 for polygon method.

Piston acceleration

of
polygon method

ft/sec2 , for calculation

in/sec2 =
9286.3

ft/sec2 (see attachment
-
1 and

attachment
-
2).

For dynamics calculation, validation had been carried out
with a good result. The effective force a
nd moment must
be
eliminated (set to zero) to
get
static balance.

For
r
2

= 10 cm,
r
3

= 20 cm,

gives the result for manual
calculation, for
F
c

= 2500 N, obtained
F
14y

= 645.5 N,
F
2x

=
2500 N,
F
2y

=
-
645.5 N,
T
i

=
-
18090.19 N.
cm.
Th
e result of

computer calculation, for
F
c

= 2500 N, obtained
F
14y

= 645.5
N,
F
2x

= 2500 N,
F
2y

=
-
645.5 N,
T
i

=
-
18090.17 N
.
cm

(
see
attachment
-
3 and attachment
-
4
)
.

Validation for calculating the connecting rod mass and
centroid can be explained, total mass

m

= 0.79 kg,

m,

kg
-
m
2
. By getting weight
procedure, it is obtained that mass
m

= 0.75 kg,
m.

By this validation procedure, it can be
assumed that the calculation value of cranksh
aft can be
accepted. For crank
: mass

m
a

=
0.5867 kg, centroid

m, mass moment of inertia
I
zz

= 0.000494
kg
-
m
2

(one crank). For balancer: mass
m
b

= 0.8829 kg,
centroid

m, mass moment of inertia
I
zz

= 0.001028 kg
-
m
2
.


V.

C
ONCLUSION

For revolution n = 4500 rpm, and maximum internal
pressure force
F
c

= 18709.70 N, at crank angle of 10
0
,
it is
obtained
the
minimum
horizontal shaking
force
of Rx = 2.07
N, and the vertical force Ry =
-
11957.05 N.

By
this
calculation, the
optimum
body
tilt angle
can be adjusted to
be
.

This programming has reached a good accuracy for
calculating dynamic system of engine mechanism.


VI.

R
EFERENCES



[1]
A.K. Agarwal
, Biofuel (alcohols and biodiesel)
applications as fuel for internal combustion engines,
Progress in Energy and Combustion Science 33 (2007)
233
-
271.

[2]

A.S. Ramadhas
,
S. Jayaraj
,
and C. Muraleedharan
, Use
of vegetable oil as IC engine fuel: A review,
Renewable Energy 29 (2004) 727
-
742.

[
3]

A. Murugesan.
,
A. Umarani, R. Subramanian.
, and
N.
Nedunchezhian
, Bio
-
diesel as an alternative fuel for
diesel engines, Renewable and Sustainable Energy
Reviews (2007).

[4]
Z. Geng,
and
J. Chen
, Investigation into piston
-
slap
-
induced vibration

for engine condition simulation and
monitoring, Journal of Sound and Vibration 282(2005)
735
-
751.

[5]
A.P. Garlucci
,
F.F.
Chiara,
and D.
Laforgia, Analysis of
the relation between injection parameter variation and
block vibration of an internal combus
tion diesel
engine, Journal of Sound and Vibration, 295 (2006)
141
-
164.

[6]

E.
Brusa, Torsional Vibration of Crankshaft: Effects of
Non
-
Constant Moments of Inertia, Journal of Sound
and Vibration, 205(1997) 135
-
150

[7]
A.L. Guzzomi
, The effect of pisto
n friction on the
torsional natural frequency of a reciprocating engine,
Journal of Mechanical Systems and Signal Processing
(2007) 2833
-
2837.

[8]
A.R.
Holowenko, Dynamics of Machinery, John Wiley
& Sons, 1955
, pp. 184
-
237
.

[9]
F.P. Beer
, and
E.R. Jo
hnston. Jr
, Vector Mechanics for
Engineers, Dynamics,

Sixth Edition, McGraw
-
Hill,
1997
, pp. 885
-
895.

[10]
W
.H. Press,
Numerical Recipes in C++, The Art of
Scientific Computing, Second Edition, Cambridge
University Press, 2002, pp. 39
-
51.

[11
]

I. Horton
, Beginning Visual C++ 6, Wrox Press, 1998

[12
]

J.B. Heywood, Internal Combustion Engine
Fundamentals, Mc Graw Hill, 1988, pp. 491
-
561.


A
CKNOWLEDGMENT

Thanks to ANPCYT for supporting this event. Special
thanks to UTHM that has collaborated with UMP, UTeM,

and UniMAP
.


Unforgettable
special thanks to my

c
olleagues
, Muhaimin
Ismoen and
Dr. Waluyo A
.
S,
who have given much support
and
the
effort
s

for
this
research

development
.