“VIBRATION ANALYSIS OF A BEAM USING NEURAL NETWORK TECHNIQUE” A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF TECHNOLOGY IN MECHANICAL ENGINEERING By PARTHASARATHI BEHERA Under the Guidance of Dr. R.K. BEHERA Department of Mechanical Engineering National Institute of Technology Rourkela Rourkela – 769 008
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“VIBRATION ANALYSIS OF A BEAM USING NEURAL NETWORK TECHNIQUE”
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
BACHELOR OF TECHNOLOGY
IN
MECHANICAL ENGINEERING
By
PARTHASARATHI BEHERA
Under the Guidance of
Dr. R.K. BEHERA
Department of Mechanical Engineering National Institute of Technology Rourkela
Rourkela – 769 008
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the project entitled,” VIBRATION ANALYSIS OF A BEAM USING
NEURAL NETWORK” submitted by ‘Mr. Parthasarathi Behera’ in partial fulfillments for
the award of Bachelor of Technology Degree in Mechanical Engineering at National Institute of
Technology, Rourkela(Deemed University) is an authentic work carried out by him under my
supervision and guidance.
To the best of my knowledge, the matter embodied in the report has not been submitted to any
other University/Institute for the award of any Degree or diploma.
Date:
(Prof. R.K. Behera )
Dept. of Mechanical Engineering,
National Institute of Technology
Rourkela 769008, Orissa
ACKNOWLEDGEMENT
I wish to express my deep sense of gratitude and indebtedness to Prof. R.K. BEHERA,
Department of Mechanical Engineering, N.I.T Rourkela for introducing the present topic and for
there inspiring guidance, constructive criticism and valuable suggestion throughout this project
work.
I would like to express my gratitude to Prof. R.K. Sahoo(Head of the Department) and Prof. K.P
Maity for there valuable suggestions and encouragements at various stages of the work. I am also
thankful to all staff members of Department of Ceramic Engineering NIT Rourkela.
I am also thankful to postgraduate student of Mechanical Engineering Department working under
Prof. R.K. Behera for helping me throughout my project.
I feel a deep sense of gratitude for my father Mr. R.K. Behera and mother Mrs. R.R. Behera who
formed a part of my vision and taught me the good things that really matter in life.
Last but not least, my sincere thanks to all my friends who have patiently extended all sorts of
help for accomplishing this undertaking.
( PARTHASARATHI BEHERA)
ABSTRACT
Using changes in global dynamic characteristics for detection of cracks has been a hot research
topic now a days and is a source of attraction for civil, aerospace, and mechanical engineering
communities in recent years. Crack in vibrating components causes a change in physical
properties of a structure which in turn affects dynamic response characteristics. Therefore we
have to study the dynamic response characteristics in order to avoid any catastrophic failures and
to follow structural integrity and performance for which the parameters considered are crack
depth and its location.
In the present study, vibration analysis is carried out on a cantilever beam with two open
transverse cracks, to study the response characteristics. Its natural frequency and mode shapes
are determined by applying suitable boundary conditions. The results obtained numerically are
compared with the results obtained from the simulation. The simulations have done with the help
of ALGOR software.
Then by using Feed-forward, back propagation neural network the relationship between the
location and the depth of the crack as input and the structural eigenfrequencies as output are
studied.
At the end by performing both the simulation and computational analysis it is proved that the
presence of cracks affects the natural frequency and the mode shapes of the structure. The results
indicate that the current approach can identify both the location and the extent of damages in the
beam.
CHAPTER 1
INTRODUCTION
1. INTRODUCTION
Damage detection and location, and condition assessment of structures have always been
important subjects. Damage in a structure generally causes a local increase in flexibility, which
depends on the extent of the damage. This reduces the natural frequencies of vibration and
affects the natural mode shapes -effects which have been used, with somewhat mixed success, to
evaluate the deterioration [1].
Cracks present a serious threat to the performance of structures since most of the structural
failures are due to material fatigue. For this reason, methods allowing early detection and
localization of cracks have been the subject of intensive investigation the last two decades. As a
result, a variety of analytical, numerical and experimental investigations now exist. A review of
the state of the art of vibration based methods for testing cracked structures has been published
by Dimarogonas (1996).
The most important aspects of structural health monitoring is that the technique provides
information on the life expectancy of structures, simultaneously detects and locates structural
damage. This needs idea of the model of structures in great detail, which is always not possible.
In addition to it, dynamic systems usually posses non-linear characteristics, which causes
practical difficulties on the model-based damage detection techniques.
In the present survey a number of papers published so far have been surveyed, reviewed and
analyzed. Most of researchers studied the effect of single crack on the dynamics of structures.
However in actual practice structural members are highly susceptible to transverse cross-
sectional cracks due to fatigue. Therefore attempt has been made to monitor the dynamic
behavior of basic structures with crack systematically. Here vibration analysis on a cantilever
beam with and without crack is carried out. First the results are obtained analytically and then
they are compared with simulation results. In first phase two transverse surface cracks are
considered in developing the analytical expressions in dynamic characteristics of structures.
These cracks introduce new boundary conditions for the structures at the location of the cracks.
These boundary conditions are derived from strain energy equation using castiligiano’s theorem.
Presence of crack also causes reduction of stiffness of the structures which has been derived
from stiffness matrix. The detailed analyses of crack modeling and stiffness matrices are
presented in subsequent sections. Euler-Bernoulli beam theory is used for dynamic
characteristics of beams with transverse cracks. Modified boundary conditions due to presence of
crack have been used to find out the theoretical expressions for natural frequencies and mode
shape for the beams.
Artificial Neural Networks (ANN) has emerged as a promising tool for monitoring and
classification of fault in machine and equipment. This technique is well prepared for solving
inverse variational problems in the context of monitoring and fault detection because of their
pattern recognition and interpolation capabilities (Lopes, Jr. et al., 1997). ANN also successfully
approach and classify the problems associated with non-linearities, provided they are well
represented by input patterns, and also can avoid the complexity introduced by conventional
computational methods. Furthermore, the learning capabilities of neural networks are well suited
to process a large number of distributed sensors, which is ideal for smart structures.
In this study a feed-forward back-propagation neural network is used to learn the input (the
location and depth of a crack)-output (the structural eigen frequencies) relation of the structural
system. A neural network for the cracked structure is trained to approximate the response of the
structure by the data set prepared for various crack sizes and locations.
CHAPTER 2
LITERATURE REVIEW
2. LITERATURE REVIEW
Local flexibility are induced due to the presence of cracks in the structure which affects the
dynamic behavior of the whole structure to a considerable degree. It causes reduction in natural
frequencies and changes in mode shapes of vibrations. Any analysis of these changes makes it
possible to identify cracks.
The effect of cracks upon the dynamic behaviour of cracked beams has been studied by
many authors. Dimarogonas [ 11, Chondros [2] and Chondros and Dimarogonas [3,4]
modeled the crack as a local flexibility computed with fracture mechanics methods and
measured experimentally, and they developed a spectral method to identify cracks in
various structures relating the crack depth to the change in natural frequencies of the first
three harmonics of the structure for known crack position.
Cawley and Adams [5] have developed a technique based on experiment to estimate the
location and depth of the crack from changes in the natural frequencies. Anifantis et al. [6]
developed the spectral method for identification of earthquake-induced defects in
reinforced concrete frames by analyzing the changes in the vibration frequency spectrum.
They also showed that any localized damage, such as a crack, would affect each
vibration mode differently, for different structures, depending on the particular location,
orientation and magnitude of the crack.
Kirshmer [7], Thomson [8] and Petroski [9, lo] explained the effects of cracks on
structural response through simple reduced section models of cracked beams using energy
methods, and elaborated the effect of the size and location of the crack to the natural
frequency and vibration mode of the damaged beam.
Inagaki ef al. [ 11], in the case of transverse vibrations of cracked rotors, estimated the crack
size and position by natural vibration analysis and by static deflection analysis. Grabowski [12]
came to the conclusion that there is a strong dependence of vibrational behavior of cracked
rotors on the crack position and magnitude using modal analysis. Mayes and Davies [13]
proposed a method for the prediction of the magnitude of a rotating cracked and rotor crack
location, from analytically obtained mode shapes and frequency measurements.
Christides and Barr [ 141 assumed an exponential stress distribution in the vicinity of the crack
and applied a variational principle to study the dynamic behavior of the system. If the stress
distribution be known, it would have made this method very rational. The exponential
approximation is valid only for notches and the exponent is estimated experimentally. In fact, it
was pointed out by Warburton [ 151that, for example, for torsional vibration of rods, the local
flexibility approach could be used for the estimation of the Christides and Barr exponent.
Yuen [16] presented a systematic study of the relationship between size and damage location
and the changes in the eigenvectors and eigenvalues of a cantilever beam.
Stubbs and Kim,1996[17] proposed that to detect damage using modal based methods, the
vibration response of a structure before and after damage occurs is usually desired although a
baseline is not always required. If damage location is known in advance, such as at critical bolt
joints, an electro-mechanical impedance method advanced by Rogers et al. (e.g. Liang, Sun and
Rogers, 1996; Rogers and Giurgiatiu , 1997) has been shown to be very effective.
Wu, Ghaboussi and Garrett(1992)[18] adopted an NN model to portray the structural behavior
before and after damage in terms of the frequency response function , and then used this trained
model to detect location and extent of damages by feeding in measured dynamic response.
Masri , Ghassiakos and Caughey (1996)[19] used a multilayer perceptron NN model to monitor
the change in the dynamic characteristics of a structure - unknown system. Zhao , Ivan and
DeWolf (1998)[19] used a counter-propagation NN model to identify the damages in beams and
frames.
Klenke and Paez (1994)[20] used two probabilistic techniques , one of which involved a
probabilistic neural network model ,to detect the damages in the aerospace housing components.
The application of neural networks in the area of damage detection has also been studied by