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
.
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