Proceedings of MUCEET2009
Malaysian Technical Universities Conference on Engineering and Technology
June 20-22, 2009, MS Garden,Kuantan, Pahang, Malaysia
MUCEET2009
EFFECT OF BODY TILT ANGLE TO THE SHAKING FORCES
ON THE DIESEL ENGINE’S CRANKSHAFT
*
Fuadi Noor Balia, **Shahruddin bin Mahzan, ***Mohd Imran bin Ghazali, and *** Abas AB Wahab

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
parameter 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-360A3A 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 engines
area and its development to the biodiesel engine purpose.
This paper emphasize on the programming of a dynamic
system on the diesel 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.
Abstract—In this paper, a programming of
dynamic system calculation is developed 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
determined 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 to obtain the shaking
forces on the crankshaft. Calculation result is
validated by comparing it to the polygon method
and Newton principles. Based on the graphs, the
optimum tilt angle of the engine’s body had been
obtained at   87.72 0 for minimum horizontal
shaking force Rx  2.07 Newton.
II. PROBLEM FORMULATION
Keyword: dynamic system, vector analysis, gauss elimination,
shaking forces, tilt angle.
In analyzing the shaking forces due to the combustion
process in the chamber on the engine’s crankshaft can be
describe as follows:
I. INTRODUCTION
Recently, researches of diesel engines become more
attractive because of its fuel has similar characteristic with
the environment-friendly fuel resources, such as biodiesel. In
the fact that, biodiesel can be used to replace the
conventional diesel fuel and it is made from the renewable
resources.
Biodiesel is a kind of environment-friendly resources of
fuel, clean, grown locally. Palm Oil Methyl Esters (POME)
is one of those. Researchs on this area had been carried out
by researchers such as Agarwal [1], Ramadhas [2], and
2.1 Kinematic Formulation
The calculation steps of an engine has to be started at the
kinematic formulation, to calculate the position, velocity,
and acceleration of pins and center of mass. Vector analysis
principles are used to calculate those parameters. The
mechanism of engine shown below.
4
Y
3
1
This work was supported in part by ANPCYT.
* F.N. Balia, PhD student at Mechanical 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 AB Wahab, Professor at Mech and Manufacturing Eng Faculty,
UTHM.

X
2
Figure 1, Mechanism of Engine
1
Figure 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 shaking forces on the engine’s
crankshaft. Link 1 is cylinder block and journal bearing, link
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 position, velocity, and acceleration of
pins and center of mass of the linkages. In this modeling,
vector r2 represent the crankshaft, vector r3 represent the
connecting rod, and vector r4 represent the motion line of
piston.  2 is angle of r2 to x,  2 is angle of r2 to r4,  3
is angle of r3 to x,
to x.
where :
A  r2 x cos   r2 y sin  .2
2
B  (r sin   r cos  ).
2x
2y
2
2
C  (r3 x sin   r3 y cos  ).3
D  (r3 y sin   r3 x cos  )
Through equation (1) to equation (5), the position.
velocity (linear and angular) and acceleration (linear and
angular) of pins and center of mass can be obtained.
2.2
Dynamic Formulation
Calculation of shaking forces in any of pins and existing
forces at center of mass of linkage, can be modeled as figure
3 below [8].
 is angle of r3 to r4,  is angle of r4
Y
r3
4
r4
r2
Y
3
3
1

2

X
2
X
2
Figure 2, Kinematic Modeling of Engine Mechanism
Figure 3, Dynamic Modeling of Engine Mechanism
From figure 2, mathematical model can be governed as a
vector equation below,
r4  r2  r3
……..
This model can be solved by using Cartesian coordinate
method (vector analysis for dynamic systems) [9]. This
engine mechanism can be modeled separately as follows.
(1)
This equation can be derived to obtain the velocity of
points along the line vector, such below
r4  2 xr2  3 xr3
……..
2.2.1 Crankshaft Modeling
(2)
C
-F3
From this equation can be calculated the connecting rod
angular velocity such below,
 r2 x cos   r2 y sin 
3  
 r3 y sin   r3 x cos 

2


A
2
2
 r2 y .2  r2 x .2  r3 y .3  r3 x .3
cos 
(3)
O2
2
B
A BC
D
p2a
-W2a
2, 2, 2
m2b.a2b
q2a
I2b 2
-W2b
p2b
… (4)
Calculation of angular acceleration of linkage 3 can be
derived and the result as below,
3  
F2
Ti
Equation (2), can be derived to give the equation of
acceleration of points motion along the cylinder axis can be
written as below,
r4 
-W2c
m2a.a2a
I2a 2
……
m2c.a2c
……………
Figure 4, Modeling of Crankshaft
Figure 4, shows a physical modeling of Crankshaft. In
this modeling, the crankshaft is separated into two parts, that
are crank and balancer. Center of rotation assumed located
at point A, therefore all of the moments refer to that point.
(5)
2
2.2.3
Piston Modeling
The inertia torque I 2 a .2 that is generated by rotation of
F4
the crank and located at point A. The inertia torque I 2 b .2
that is generated by rotation of balancer and located at point
B. Reaction torque Ti is input moment to the shaft due to
the reaction of combustion and inertia loads.
The vector F2 is the reaction of the crankshaft to the
Fc
4
m4.a4
E
crank, while vector F3 is the reaction force of crankpin to
-W4
F14
1
m2 a is of crank mass and generate the
inertia force of m2 a .a 2 a and centered at point A. The m 2 b
the crank. The mass

is the mass of balancer and generate the inertia force of
m2b .a 2b and centered at point B. The mass m 2 c is a half
Figure 6, Modeling of Piston
mass of crankpin to the crank, this mass generate a half of
inertia force m2 c .a 2 c and centered at point C. In this
F4 is the reaction force of connecting rod to
the pin of piston. The mass m4 is the sum of piston’s pin
and piston mass itself and generated the inertia force m4 .a4
The vector
modeling, the distributed weight of linkage part are
included. W2 is a half of weight of crankshaft. W2a is the
weight of crank, W2b is the weight balancer. W2c is a half of
the weight of crankpin
2.2.2
z
and located at point E. The weight W4 is the sum of pin and
F14 is reaction force of cylinder
piston weight. The vector
Connecting rod Modeling
to the piston and located at the length of vector z from the
center of mass, and the vector
q3
Fc is a force as a result of
the combustion process in the cylinder to the piston.
p3
2.3
Mathematical Equations
F3
3 , 3, 3
Mathematical modeling can be developed by using
vector analysis (Cartesian coordinate) method for engine
mechanism.
m3.a3
D
3
I3 3
2.3.1 Equation for Crankshaft
Crankshaft mechanism is modeled by assumed that the
center of rotation located on point A and mass of each of
crank part located at the center of each part.
-W3
-F4
Equation of the forces equilibrium vector of the Crankshaft,
F2  F3  m2a .a2a  m2b .a2b  m2c .a2c ….. (6)
Figure 5, Modeling of Connecting rod
Figure 5 shows a physical modeling of Connecting rod.
Center of rotation assumed to be located at point D.
Equation of moment equilibrium vector of the Crankshaft,
q2 a xF2  p2 a xF3  Ti 
The inertia torque I 33 is generated due to the rotation
of connecting rod. The mass
E  F G H  I  J
m3 is the connecting rod mass
and generate the inertia force
m3 .a3 and centered at point
where:
D. The vector F3 is the reaction force of crankpin to the
connecting rod, while the force of F4 is reaction of piston
pin to the connecting rod. The weight W3 is connecting rod
weight and centered at point D.
E  ( p2b  q2 a ) xm2b .a2b
F  ( p2b  q2 a ) xW2b
G  q 2 a xW2
H  p2 a xW2c
I  p2a xm2c .a2c
J  ( I  I )
2a
3
2b
2
……. (7)
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 linear
equation to form matrix.
III. METHOD OF SOLUTION
This programming is divided in two category
calculation. Firstly, kinematic step and secondly, dynamic
system formulation. Compiler Visual C++ is used for
programming language [11].
Briefly, figure 7, shows a flow chart of programming.
2.3.2 Equation for Connecting rod
START
Equation of the forces equilibrium vector of Connecting rod,
F3  F4  m3 .a3  W3
……………..
Kinematic steps:
Vector model of
position, velocity, and
acceleration of pins and center of
mass of linkages
(8)
Equation of the moment equilibrium vector of Connecting
rod,
q3 xF3  p3 xF4  I 3 .3
……………..
Validation (compare
to polygon method)
No
(9)
OK
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 linear
equation to form matrix.
Dynamic steps:
Vector model of
Forces and Moments on the pins
and linkages
Validation (compare
to static balance)
2.3.3 Equation for Piston
Equation of forces equilibrium of piston,
F4  F14  m4 .a4  Fc
……………
OK
(10)
FINISH
Equation of moments equilibrium,
zxF14  0
Figure 7, Flow Chart of Programming
………... …..
(11)
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 r2 and r3 considered as the
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 matrix.
2.3.4
length of crank and connecting rod, the values are constant.
Vector r4 considered as the length of position between
piston and the main bearing, this is an unconstant variable.
Equation (2) gives the linear velocity of piston motion along
the cylinder axis. Rotation of the crank gives the angular
Matrix Formation
velocity and acceleration of the connecting rod, 3 and 3 ,
The equation (6) to equation (11) that have nine of
linear equations and can be solved simultaneously by matrix
formation,
Ax  b
No
…………….
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 position of axis of piston to x (same
as engine’s body tilt angle axis). Through this 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. Concept of
(12)
A is a coefficient of symmetric matrix (9x9
matrix element), x are the forces and torque parameter to
be solved (9x1 matrix element), while b are the effective
where,
inertia forces and moment (9x1 matrix element) due to the
moment and forces of inertia.
The forces and torque in matrix x can be solved by
using Gauss-Jordan Elimination principle for solving
Simultaneously Linear Equation [10].

4
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 neglected in this situation, because of the
complexity of calculation.
Matrix formation is used to collect the ninth of equations
had been developed from equation (6) to equation (11). For
detail explanation, see section 2.3 (Mathematical Equations)
for every step 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 ra = 47 mm, balancer length rb = 78 mm.
The internal pressure force data delivered to the equation
(10) depend on the crank angle of combustion [12].
Simultaneously linear equations in a matrix form can be
solved by using Gauss Elimination procedure, by inserting
pivot point algorithm. This solution gives the result of
parameters that are considered as the reaction forces of 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 moments
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, the parts are modeled as a volume combination
of the separately blocks. For crankshaft, calculation of mass,
centroid and mass moment of inertia carried out by modeling
its volume as the combination of many blocks (same
procedure with connecting rod).
Programming of this engine mechanism gives 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 for vertical
direction, Ry = -11957.05 Newton. For diesel engine has
pressure 3,5 MPa, so internal pressure force Fc = 18709.70
Newton, at 100 of crank angle. Rotation of engine is
constant, 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 shake must be avoided for
a long period. The optimum tilt angle of engine’s body
obtained   87.72 .
Validation of kinematic calculation, a good result was
reached. Calculation for n = 2000 rpm (   209.4
0
rad/sec),
rod
  00 ,
3  61.98
rad/sec for
 3  9936.41
gives angular velocity for Connecting
rad/sec for calculation and
3  62.0
polygon method. Angular acceleration
rad/sec2 for calculation and  3  9940
rad/sec2 for polygon method. Piston acceleration of
polygon method
a  9290 ft/sec2 , for calculation
a  111435.54 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 and moment must be
eliminated (set to zero) to get static balance. For r2 = 10 cm,
r3 = 20 cm,  2  30 gives the result for manual
calculation, for Fc = 2500 N, obtained F14y = 645.5 N, F2x =
2500 N, F2y = -645.5 N, Ti = -18090.19 N.cm. The result of
computer calculation, for Fc = 2500 N, obtained F14y = 645.5
N, F2x = 2500 N, F2y = -645.5 N, Ti = -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,
y  0.045 m, I zz  0.00314 kg-m2. By getting weight
procedure, it is obtained that mass m = 0.75 kg,
y  0.049 m. By this validation procedure, it can be
assumed that the calculation value of crankshaft can be
accepted. For crank: mass ma = 0.5867 kg, centroid
y  0.0272445 m, mass moment of inertia Izz = 0.000494
kg-m2 (one crank). For balancer: mass mb = 0.8829 kg,
y  0.03988 m, mass moment of inertia
centroid
Izz = 0.001028 kg-m2.
0
IV. RESULT AND DISCUSSION
Body Tilt Angle
4000
2000
0
-2000
Force
-4000
V. CONCLUSION
-6000
100
-8000
For revolution n = 4500 rpm, and maximum internal
pressure force Fc = 18709.70 N, at crank angle of 100, 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
90
-10000
Tilt Angle
85
-12000
Tilt
75
Rx
Ry
be
75
80
85
87.72
90
95
100
  87.72 0 .
This programming has reached a good accuracy for
calculating dynamic system of engine mechanism.
Figure 8, Graphic of Tilt Angle to Shaking Force
5
VI. REFERENCES
[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-slapinduced 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 combustion 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 piston 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. Johnston. 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.
ACKNOWLEDGMENT
Thanks to ANPCYT for supporting this event. Special
thanks to UTHM that has collaborated with UMP, UTeM,
and UniMAP.
Unforgettable special thanks to my colleagues, Muhaimin
Ismoen and Dr. Waluyo A.S, who have given much support
and the efforts for this research development.
6