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ENGINE DYNAMICS AND STRUCTURAL VIBRATION

Joseph T. Francis

Fairbanks Morse Engine

701 White Avenue, Beloit, WI

Joe.francis@fairbanksmorse.com

Abstract

The reciprocating internal combustion engine is inherently a vibrating machine. Two of

the major sources of excitations are the reciprocating forces of the crank mechanism and

the gas harmonic torques from combustion. For multi-cylinder engines, even though the

reciprocating forces are internally balanced for the engine as a whole, the longitudinal

pitching couples produced by these forces could rock the engine back and forth. The gas

harmonic torques predominantly excites the torsional vibration of the drive train, but

could also excite structural modes, especially if the equipment is flexibly mounted. Case

histories are presented where reciprocating and gas harmonic force excitations resulted in

structural vibration of engine support skids. The presentation will show how prudent use

of vibration troubleshooting tools including finite element modeling, experimental modal

testing, and Operating Deflected Shape (ODS) can be used for finding effective fixes.

Key Words: Internal Combustion Engine, Reciprocating Force, Pitching Couple, Yaw

Couple, Gas Harmonic Torques.

Introduction

The principal source of excitation forces in an internal combustion engine are: 1)

centrifugal forces and couples caused by the out-of-balance rotating parts. 2) Inertia

forces and couples caused by the acceleration of the reciprocating parts. 3) Torque

reaction couples caused by the harmonic components of the engine torque. The first two

are classified as mass forces and the third is grouped under gas forces.

The centrifugal forces are due to mass offset of the crank pin, crank web, and part of the

large end of the connecting rod from the center of rotation of the main bearings. Rotating

imbalance manifests as one order vibration along the line of stroke and in transverse

directions. The inertia forces are due to the oscillating parts of the engine that include the

piston, piston pin, and small end of connecting rod and introduce mainly 1 and 2 order

vibration along the line of stroke. Additionally, in multi-cylinder engines with cylinder

centers separated in space, the couple due to rotating imbalance can produce pitching and

yaw vibrations and reciprocating forces induced couple can cause pitching vibration. The

cyclic firing pressures present during the combustion cycle of an internal combustion

engine create harmonic torque excitations that can cause torsional and roll vibration of

the crankshaft and the engine frame. For rotational rigid body definitions see Figure 1.

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1. Single Cylinder Engine

To better understand the dynamics of the multi-cylinder engine, one has to first study a

single cylinder engine. In a single cylinder engine, the crank executes a pure rotary

motion while the piston slides back and forth, executing a pure reciprocating motion.

Whereas, the connecting rod is a floating body and undergoes a combination of rotary

and reciprocating motion.

One could simplify the dynamics of a single cylinder engine by assuming it to be a slider

crank mechanism as shown in Figure 2. In order to get there some assumptions need to

need to be made. First, lump an apportioned rotating mass (Mrot) at the crank pin. These

include the crankpin, web, and a portion of the connecting rod. Similarly, lump the

reciprocating mass (Mrec) at the piston including the piston and remaining portion of the

connecting rod. The unbalanced rotating mass Mrot rotating at crank radius R generates

centrifugal force (Frot) at the crank pin which is reacted by the main bearing. This force

will generate one order vibration along the line of stroke and in lateral directions. An

order is the number of cycles the event completes per revolution. Fortunately, the rotating

imbalance can easily be addressed by adding counter weights to the web. Therefore,

going forward it will be assumed that there are no unbalanced rotating forces.

What is left is the reciprocating inertia forces due to the periodic acceleration and

deceleration of the reciprocating mass that acts along the axis of the cylinder. The

reciprocating forces can be derived from the equation of motion of the piston and can be

approximated as Eq. (1).

[ ( )]

[ ( )

]

Figure 2: Slider Crank Mechanism

cos() +

cos(2)

Equation (1)

Equation (2)

Equation (3)

Figure 1: Typical Rotational

Modes of an Engine due to

Internal Excitation.

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Where R is the crank radius, L is the length of the connecting rod, n is the ratio of L/R,

and is the angular velocity of the crank in rad/s. The reciprocating force can be

decomposed into two parts: the primary force (Eq. (2)) which repeats itself once per

revolution of the crankshaft and a secondary component (Eq. (3)) that repeats twice per

revolution. There are higher orders present, but they are small except for very high

speeds. The secondary component is a function of n and for infinitely long connecting

rods it becomes zero.

The primary reciprocating force can be represented as shown in Figure 3 by lumping the

reciprocating mass at the crankpin and rotating at a velocity of . Then, the primary

reciprocating force is the component of the force acting along the axis of the cylinder.

Similarly, the secondary reciprocating force can be represented as shown in Figure 4 by

lumping a mass of at the crank pin, rotating at twice the crank speed. This kind

of representation will later facilitate studying the balance of multi-cylinder engines.

One way to balance the primary force is to attach the Mrec mass to the crank web in

addition to the Mrot, as shown in Figure 5. This is not a complete solution, since it

introduces a new undesirable force s ( ) in the direction perpendicular to the

line of action of the piston. Figure 6 is an illustration of another way to balance primary

force by using two balancer shafts that are driven off the crankshaft. Here, half the

reciprocating mass appropriately positioned on two balancer shafts rotating in the

opposite directions neutralizes the one order reciprocating force completely. The vertical

components cancel each other. Thinking along the same line, secondary forces can also

be balanced by properly adding additional sets of gears running at twice the speed and

mass positioned appropriately.

Figure 3: Primary Force Representation Figure 4: Secondary Force Representation

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Figure 5: Balancing of Primary Reciprocating Force Using Balance Mass

2. Balancing Reciprocating Forces in Multi-cylinder Engines.

Fortunately in multi-cylinder engines, by properly arranging the cylinders around the

crank circle, even though the reciprocating forces generated by the individual cylinders

are not balanced, they are collectively balanced for the engine as a whole. And drastic

measures like adding balancer shafts need not be made.

Figure 7 shows the schematic of a two-stroke, 4-cylinder engine. The four cranks throws

are arranged evenly around the crank circle with an angle between consecutive firings of

360/4=90 degrees. Also shown is the end view with the reciprocating mass placed at each

of the crankpins. This is called the crank diagram. As the crankshaft rotates clockwise,

cylinder 1 reaches top dead center (TDC) and it fires first, after 90 degree rotation

cylinder 2 reaches TDC and it fires and so forth. Thus the firing order is for this engine is

1-2-3-4. The crank diagram is useful in investigating the total balance of multi-cylinder

engines.

Figure 6: Balancing of Primary Reciprocating Force using Balancer Shafts

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Figure 7: Crank Diagram of a Four Cylinder Engine

The state of balance of multi-cylinder engines can be examined analytically by drawing

the crank diagrams and resolving the reciprocating forces from different cylinders along

the line of action. It can also be done graphically using vector diagrams. Figure 8 shows

an example of how vector diagrams can be used to check the state of balance of primary

forces for 2-, 3-, and 4-cylinder engines. The length of the vector represents the

maximum primary reciprocating force and the angle represents the phasing and is drawn

in the direction of the lumped reciprocating masses in the crank diagram. The forces form

a closed polygon meaning that the primary reciprocating forces are balanced for the

engine as a whole. When the forces are balanced, their component along the cylinder axis

is also balanced. Thus for engines with cranks evenly spaced around the crank circle, the

primary reciprocating forces are balanced for the engine as a whole.

To investigate the secondary reciprocating force balance, one could draw similar

secondary crank diagrams by doubling the crank angle, keeping in mind that the

secondary forces rotate at twice the speed of the crank. Figure 9 shows the secondary

crank diagrams and force polygons for two-stroke, 2-, 3-, and 4-cylinder engines. Here,

the 2 cylinder engine has high secondary force imbalance. In general for multi-cylinder

engines having two or more evenly arranged cranks, the primary forces are balanced and

for engines having three or more cylinders, the secondary forces are balanced too. Four-

stroke 4-cylinder engines which are very popular in automotive industry with 1-3-4-2

firing order (uneven cranks) have very high secondary unbalanced force. These engines

employ flexible mounts and balancer shafts as described in the previous section to

smoothen out the high vibratory force along the line of stroke.

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Figure 8: Primary Force Imbalance Check

Figure 9: Secondary Force Imbalance Check

It was found that by arranging the cranks evenly around the crankshaft, the total primary

and secondary forces are neutralized for most multi-cylinder engines. But this is not true

for moments created by these forces. When the line of action of individual cylinders is

separated, they produce couples that tend to pitch the engine back and forth and are not

necessarily balanced for the engine as a whole. The magnitude of the residual couple is a

function of firing order and the spacing between cylinders. When the forces are balanced,

the longitudinal couple can be calculated by taking moment about any axis normal to the

crank line. Figure 10 & Figure 11 show an example of how couple balance can be

checked for a 4-cylinder engine graphically using vector summation. The moment

diagrams are created by taking moment of individual cylinders about the reference axis

and drawing vectors taking into consideration the phasing of cylinders. It can be seen that

for this crank arrangement, both primary and secondary moments are out of balance and

would produce a net pitching moment rocking the engine back and forth. An in depth

treatment on the subject of multi-cylinder engine balance including V- engines can be

found in [6].

The choice of crank throw arrangement (firing order) has big influence on reciprocating

force and moment balance. Four-stroke engines have complete primary and secondary

balance of both forces and couples, provided they have 6 or more even number of

cylinders and are arranged evenly around the crank circle with mirror symmetry about the

mid-length of the crankshaft. Figure 12 shows a six cylinder engine crankshaft that fits

the above criteria.

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Figure 10: Primary Moment Diagram of 4-Cylinder Engine

Figure 11: Secondary Moment Diagram for 4-Cylinder Engine

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Figure 12: Typical 6 Cylinder evenly arranged crank throws, symmetric about center

The following case history talks about how the unbalanced reciprocating moments in a 9

cylinder inline engine (9 cylinder engine does not meet the above criteria) lead to engine

vibration.

3. Case History 1

The engine in question is a 9 cylinder, four-stroke, in-line engine coupled to a generator

producing 9.5 MW @ 514 RPM, with firing order 1-6-3-2-8-7-4-9-5. The engine is

mounted on vibration isolators and the generator is hard mounted. The predicted rigid

body modes during the design stage ranged from 4 to 16 Hz. After installation in the

field, the engine experienced higher than normal one order vibration (8.6 Hz) in the yaw

direction (rotation about the vertical axis). Vibration measurements were made by

coasting down the engine from 540 to 300 rpm. The Figure 13 shows the coastdown plot

of the of the one order vibration in lateral and vertical directions. Two resonances can be

observed close to the running speed.

In order to determine the mode shape of these resonances, ODS measurements were

made around the side and the base of the engine while operating the engine at resonant

speeds of 450 and 519 rpm. Figure 14 and Figure 15 show the mode shapes of these two

resonances. The first mode was the vertical bouncing mode and the second mode was the

yaw mode. With the second mode critical speed very close to the operating speed, there

was convincing evidence that the source of vibration was the one order excitation

generated within the engine. Being one order, the usual suspect was imbalance in the

flywheel of the engine, and the engine has a heavy flywheel. Balancing the flywheel in

place to minimize the yaw had only mixed results with vertical vibration going up most

of the time.

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Figure 13: Coastdown plot of one order vibration

Figure 14: ODS of the Engine at 450 RPM

Figure 15: ODS of the Engine at 519 RPM

It was decided to look closely into the engine dynamics and reciprocating force balance

of this engine. The engine has a fully counterweighted crankshaft, thus there are no

rotating imbalances present. Figure 16 shows the crank diagram, the primary

reciprocating force, and the moment balance of the engine. Being closed polygon, the

primary forces are balanced for the engine as a whole. But the primary couples are not in

balance for the engine as a whole. This couple acts in the plane of the cylinders and can

excite the pitching mode of the engine block.

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Figure 16: Primary Force & Moment Diagram of One Order

Further consultation with the engine licensor revealed that in order to reduce the impact

of the one order pitching couple, a counter couple is created by adding a weight on the

crankshaft damper and an equal weight on the flywheel at the opposite end of the

crankshaft. The weights are placed 180 degrees out-of-phase and sized to neutralize the

pitching moment. But being a rotating mass, it introduced a yaw couple in the

perpendicular plane which is otherwise not there in a normal engine. This is what was

exciting the yaw mode of the engine.

The licensor routinely added these weights for engines on soft mounts as a protection

against large pitching vibration. The predicted yaw mode frequency calculation turned

out to be incorrect due to the fact that for these long engine blocks, the yaw mode was not

fully rigid but involved some flexing of the block in the longitudinal bending direction.

Since in reality, the pitching frequency at 11.3 Hz was well separated from the running

speed and the normal pitching amplitude (non-resonant) from the primary reciprocating

force was small, there wasnt really a need to add these counter weights.

Calculations done by reconfiguring the mounts layout and changing the rubber stiffness

did not succeed due to the limited number of mount options and the proximity of other

resonant frequencies and the multitude of excitation frequencies present. It was

recommended to remove these couple cancelling weights.

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4. Gas Force Induced Vibration

Due to the cyclic firing pressure present in the

working cycle of an IC engine, it generates in

addition to the mean torque, many harmonic

components. The tangential effort curve of an

internal combustion engine repeats itself after

every complete working cycle. For a four-

stroke engine, the interval of repetition is two

revolutions of the crankshaft and for a two-

stroke engine it is one revolution. Tangential

effort is the normalized instantaneous torque

at the crank pin so that it is independent of the

cylinder dimensions and is obtained by

dividing the instantaneous torque by crank

radius and piston area.

This curve can be represented by a Fourier

series containing a constant term which

represents the mean torque and a series of sine

components. It is convenient to label these

harmonics in terms of the number of cycles it

complete per revolution, also known as

orders. Thus a four-stroke engine, has

order harmonic because it completes one cycle per two revolutions or a cycle per

revolution. It also has higher order components at 1, 1, 2, 2 etc. Figure 17 is the

tangential effort curve for a four stroke engine and the first 10 harmonic components. The

mean driving torque overcomes the external resisting torque. The harmonically varying

components can set-off torsional vibration of the crankshaft and vibration of the engine

frame and mounting.

In a multi-cylinder engine the phasing of an order between cylinders and the mode shape

of the natural frequency being excited determines the severity of that order in exciting

that mode. The firing order and # of strokes per cycle (4-stroke vs. 2-stroke) determines

the phase relationship of a harmonic order between different cylinders of an engine.

Figure 18 shows an example of how to obtain the phasing of torsional orders between

cylinders in a multi-cylinder engine. The upper section shows the firing sequence or the

crank diagram of a 6 cylinder four-stroke engine with firing order 1-5-3-6-2-4. In the case

of four stroke engines, since the working cycle occupies 720 crankshaft degrees, the

combustion pulses spaced at 720/6=120 degrees around the crank circle. In order to build

Figure 17: Tangential Effort Curve for a 4 Stoke,

Single cylinder Engine. Illustration taken from [6]

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the phase diagram, it is convenient to regard the 360 degree phase diagram of a four-

stroke engine to be equivalent to 720 degrees or two revolutions of the crankshaft. Thus

the order phase diagram for a 6 cylinder engine consists of six vectors evenly spaced at

60 degrees in the same sequence of the firing order. The phase diagram for higher orders

are obtained by multiplying all the angles of the order phase diagram by 2, 3, 4, etc. In

Figure 18 (lower section) shows phasing of orders grouped into 6 groups having the same

phasing.

For a given mode of vibration, the input energy from cylinders of multi-cylinder engine is

proportional to Tnan, where Tn is the nth

order harmonic torque and an is the normalized

Eigen vector or mode shape of the crankshaft. Tn being a constant, the vector summation

of the normalized mode shape across all cylinders taking into consideration the phasing

of the harmonic torques between cylinders can be used as a measure of the relative merits

of different firing orders. This can also be used to judge if an order will excite a particular

mode with respect to torsional vibration.

Figure 19 is an example of a four stroke, 6 cylinder engine (Firing order 1-4-2-6-3-5)

coupled to a generator through a soft coupling. The 1st two normalized mode shapes of

drivetrain are shown. The first mode is the rigid body mode of the crankshaft and the

second mode is the 1st torsional mode of crankshaft with a node in the middle. In order to

find the vector sum across the cylinders, vectors proportional to the normalized

amplitudes are drawn for each cylinder with vectors pointing in the direction of harmonic

torque components. Figure 20 shows how this can be accomplished graphically. From the

Figure 18: Crank Diagram and Phasing of Orders 1 to 10 for a six cylinder, four

stroke engine.

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vector summation for the 1st mode (Figure 20 middle), it can be seen that with the

amplitudes nearly the same for all cylinders, the major orders (3, 6, 9) can strongly excite

the 1st mode, if frequency coincidence occurs. The vector summation of other orders are

zero or relatively smaller in comparison. Whereas, for the second mode (Figure 20

bottom), with a node in the middle of the crankshaft, the major orders add to zero while

the input energy of minor orders (1, 4, 7, etc.) become predominant because of the

180 degree phasing. Once again other orders sum to zero or are very small.

Figure 20: Vector Summation of Torsional Amplitudes for 1st and 2

nd Mode.

Figure 19: 1st and 2

nd Normalized Torsional Mode Shape of a Soft Coupled Genset.

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5. Torque reaction on engine frame

The nominal torque and the harmonic torque components on the

crankshaft are reacted by the main bearings and the cylinder walls

forming an opposing couple. Thus, the engine frame experiences a

torque equal to that of the crankshaft but opposite in direction. This

couple tend to roll the engine about a longitudinal axis of the

crankshaft when mounted on flexible mounts. See Figure 21.

For four-stroke in-line engines with evenly spaced firing intervals, the

harmonic orders m, m, 1m, 2m etc. and for two-stroke engines m,

2m, 3m, 4m etc., are completely in-phase and add across the cylinders.

All the other orders add to zero when added for the engine as a whole,

considering the phasing. An in depth discussion on torsional vibration

characteristics of multi-cylinder engines can be found in [1].

Up to now this discussion dealt with in-line engines. V-engines are essentially an

extension of in-line engines two side-by-side in-line banks separated by the V-angle The

next case history is about 16 V-engine but will be treated as 8 cylinder in-line engine for

discussion purposes.

6. Case History 2

Figure 22: 3D Model of the engine and generator on soft mounted supports.

Figure 22 is the 3D model of a medium speed, 16 V, four-stroke engine running at 900

rpm driving a generator producing 4.4 MW. The engine and generator are mounted on a

common skid which in turn is supported on flexible mounts. During start-up the engine

Figure 21: Torque

Reaction on Engine

Frame.

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experienced strong vibration at 1 order in vertical and lateral directions. The engine

could not be run above 25% load due to the severity of vibration. Figure 23 shows the

vibration spectra at 25% load in all three directions measured on the skid.

One could see strong 1 order vibration (22.5 Hz) in both vertical and lateral directions.

Also seen are the responses at order multiples as expected. A coastdown run was made

operating the engine from 900 to 300 rpm taking measurements at 50 rpm intervals. With

harmonic excitations present at every order, a four-stroke engine is an ideal shaker.

The waterfall plot can be found in Figure 24. The waterfall was convincing evidence that

the 22.5 Hz was a structural resonance with the mode being excited by 1, 2, and 2

orders at their respective critical speeds. Figure 25 is a plot of the 1 order slice

confirming the strong resonance near operating speed of 916 rpm.

Figure 25: 1 Order Vibration Data Measured on the

Engine. Figure 24: Waterfall plot of Lateral Vibration

Figure 23: Vibration Spectra Measured on the Engine Skid at 25 % Load.

22.5 Hz

line

1

2 2

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An ODS vibration test was made at the resonant speed of 916 rpm to determine the

structural mode that was being excited. ODS mode shape at 23 Hz can be seen in Figure

26. A finite element model had been made during the design stage for stress analysis and

frequency prediction purposes. Figure 28, column 2 lists the two closest predicted

frequencies and Figure 27 gives the shape of the same mode from the finite element

prediction. Comparing the ODS and finite element results, it is satisfying to note how

well the predicted and test frequencies as well as mode shapes matched for such a

complex structure.

Figure 28: Frequency Comparison from FEA and Modal Test.

Mode

Number

Original FEA

Predicted

Frequencies

(Hz)

Original

Measured

Frequencies

(Hz)

FEA Modified

Model Results (1)

(Hz)

Modal Test

Result

(Hz)

Generator

Rocking

Mode

21.9 19.0 20.2 19.5

Torsion Mode

of the Engine 23.2 23.3 26.2 25.5

Note 1) The original FEA model was tweaked to match the measured frequencies before the

stiffening plates were added.

Figure 29 shows the crank sequence, phasing of the 1 order and the vector summation

of the 1st four and last four cylinders of one bank. The 1 order phasing is obtained by

multiplying the order phase angles by 3. The 1 order harmonic torque of the first four

cylinders and the last four cylinders are 180 degrees out-of-phase and imposes a twisting

couple on the engine structure at nearly the same frequency as the twisting mode of the

structure. The root cause becomes clear if one compares the phasing of the 1 order

harmonic and the mode shape of the 23 Hz structural mode of the engine skid in Figure

Figure 26: Opertion Deflected Shape of Engine Figure 27: FEA Mode Shape @ 23.1 Hz

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26 & Figure 27. Figure 30 shows the common skid (left) and the original rail cross

section (upper right) under the engine. As it can be seen, being an open section, the rail

under the engine is weak in torsion. The solution to the problem was to increase the

torsional frequency of the support skid.

Figure 29: Crank Sequence & Vector Summation of 1 order Harmonic.

Figure 30: Generator Skid Model and Original & Modified Cross Section

Original Skid Cross Section

Modified Skid Cross Section

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Cross bracing the mid-section between the engine and the generator was one suggested

solution and the other was to make the skid section underneath the engine into a box-like

structure, adding torsional rigidity. The finite element model became very handy for

studying possible alternate solutions. Trial and error on the actual equipment is very

expensive and time consuming. Two solutions were tried on the finite element model.

The first solution in addition to raising the problem frequency, also raised the next lowest

mode closer to 1 order. The second alternative proved to be the optimal solution. Plates

were added between gussets, underneath the engine to convert the engine rail into a box

section thus increasing the torsional rigidity. The modified section of the skid can be seen

in Figure 30, lower right.

A modal hammer test was conducted after implementing the modifications. The modal

frequencies from the test showed that the finite element model predictions matched well.

Figure 31 shows the vibration measurements after the modifications at 100% load. At 12

mm/s, the vibration limits now are well within the vibration limits of the engine.

7. Conclusions

The major sources of dynamic excitations present in combustion engines are

presented. Also discussed was how these forces and moments can excite the engine

frame and foundation and the method of using phasing diagrams in determining the

severity of one order over the other. Case histories were presented where engines

mounted on flexible mounts experienced vibration due to these forces and how they

were solved using finite element analysis and vibration testing techniques including

ODS and modal testing.

Figure 31: Vibration Spectra Measured on the Engine After Modification.

Axial

Lateral

Vertical

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References:

1. Wilson, W. Ker, Practical Solution of Torsional Vibration Problems, Volume 2,

Amplitude Calculations, Third Edition, Chapman & Hall LTD., 1963

2. Wilson, W. Ker, Vibration Engineering, A Practical Treatise on the balancing of

Engines, Mechanical Vibration and Vibration Isolation. , Charles Griffin &

Company Limited, 1959

3. Hartog J.P. Den, Mechanical Vibrations, 4th Edition, Dover Publications, 1956

4. Rangawala, Abdulla S., Reciprocating Machinery Dynamics, Marcel Dekker, Inc.

2001

5. Taylor Charles Fayette, The Internal Combustion Engine in Theory and Practice,

Volume 2, The M.I.T. Press, 1985

6. Heisler Heinz, Advanced Engine Technology, SAE The International Society for

Advancing Mobility Land Sea Air and Space International, 1995

7. A Handbook on Torsional Vibration, Compiled by E.J. Nestorides of the

B.I.C.E.R.A Research Laboratory, Cambridge at the University Press, 1958

8. Tienhaara, Hannu, Guideline to engine dynamics and vibration, Wartsila

Maritime News, 2-2004