Pitting Corrosion Investigation of Cantilever Beams Using F. E. Method
DAMAGE DETECTION IN CANTILEVER BEAMS …...DAMAGE DETECTION IN CANTILEVER BEAMS USING ARTIFICIAL...
Transcript of DAMAGE DETECTION IN CANTILEVER BEAMS …...DAMAGE DETECTION IN CANTILEVER BEAMS USING ARTIFICIAL...
DAMAGE DETECTION IN CANTILEVER BEAMS USING ARTIFICIAL NEURAL
NETWORKS
LOKESH KONDRU1 & M R NARASINGA RAO
2
1Department of Mechanical Engineering, KL University, Guntur, Andhra Pradesh, India
2Department of Computer Science and Engineering, KL University, Guntur, Andhra Pradesh, India
ABSTRACT
In Cantilever Beams when dynamic loading is applied that is varying the load with respect to time, which may
result in cracks. It is a challenge to know presence of cracks.
Hence the varying frequencies are obtained from ANSYS. We use these results to train a Neural Network and find
the presence of cracks. The presences of damages change the physical characteristics of a structure which in turn alter its
dynamic response characteristics. Therefore there is need to understand dynamics of damaged structures. Damage depth
and location are the main parameters for the vibration analysis.
The application of these beams is in gas turbine blades but it is a tough task to create blade and then to do analysis
so we will consider a cantilever beam which resembles the turbine blade and do analysis by considering some notches and
then find the varying natural frequencies.
KEYWORDS: Cantilever Beam, Dynamic Response, ANSYS, Neural Networks
INTRODUCTION
Propagating fatigue cracks can have detrimental effects on the reliability of rotating machinery [1]. Mechanical
structures in service life are subjected to combined or separate effects of the dynamic load, temperature, corrosive medium
and other type of damages. The importance of an early detection of cracks appears to be crucial for both safety and
economic reasons because fatigue cracks are potential source of catastrophic structural failure. So, an early crack warning
can considerably extend the durability of these expensive machines, increasing their reliability at the same time. Although
researches are going on to develop various online and offline techniques for crack detection, vibration characteristics of
cracked structures can be use ful for an on-line detection of cracks (non-destructive testing) without actually dismantling
the structure or without reaching at the location of the crack (effective when the location of the crack is not accessible) [2].
A crack in a structure changes its dynamic characteristics such as natural frequencies. By observing the changes in
the dynamic characteristics the crack can be detected successfully. As machine components such as turbine blades (where
there is a possibility of fatigue crack) can be treated as a cantilever beam, once we can study the effect of crack in the
dynamic response of cantilever beam it can be extended to develop online crack detection of such machine components.
Hence the present work focuses on crack detection in a cantilever beam using Neural Networks. This is
performed in two steps. First the finite element model of the cracked cantilever beam is established with the help of
ANSYS Software. Finite Element Analysis is the most powerful tool which gives the results for complicated on line
working assemblies for dynamic analysis [3, 4]. The beam is discretised into number of elements. Crack is assumed to be
in different locations. Next, for each location crack depth is varied. Natural frequencies been evaluated for different crack
International Journal of Mechanical and Production
Engineering Research and Development (IJMPERD)
ISSN 2249-6890
Vol. 3, Issue 1, Mar 2013, 259-268
©TJPRC Pvt. Ltd.
260 Lokesh Kondru & M R Narasinga Rao
depth and its location. For validating the analytical results these results are given to Neural Network and make the network
perfectly trained so that when we give frequencies as input values we get the place and depth of crack in beams.
Geometry Considered
Figure 1: Cantilever Beam with Crack
The objective of the present work is to evaluate the natural frequencies of a cantilever beam to study the effect of
crack on the dynamic behavior of the beam. Putting appropriate boundary condition for cantilever beam the general Eigen
value problem will be specified for the proposed case. Fig. 1 shows a cantilever beam with a crack. To find out Eigen
value and Eigen vectors finite element method (FEM) has been used [5]. The analysis has been carried out by ANSYS
Software. This code is a general purpose Software on finite element analysis. It contains a library of different types of
elements and different types of analysis. To solve the present problem, 4-node shell element (shell 63) has been used.
Thin beam and thick beam with and without crack have been modeled. For thin beam, a U shaped slot has been made by
saw cut. For thick beam at first a V-notch is made and then by reversed loading fatigue crack has been generated. Keeping
this in view for thin beam crack is modeled as U-notch and for thick beam as V-notch with a narrow opening.
METHODOLOGY
Modal Analysis by ANSYS
ANSYS is a powerful multipurpose code for finite element analysis and design. It can be used in a wide variety
of analyses like structural analysis (includes modal analysis), thermal analysis, fluid analysis, coupled field analysis etc.
Modal analysis of ANSYS is used to determine the natural frequencies and mode shapes, which are important
parameters in the design of a structure for dynamic loading conditions. They are also required for spectrum analysis or for
a mode superposition harmonic transient analysis.
Modal analysis in ANSYS program is linear analysis. Any nonlinearities such as plasticity and contact element
will be ignored even if they are defined there are four mode extraction methods Block Lanczos (default), subspace, Power
Dynamics, reduced, unsymmetric, damped, and QR damped. The damped and QR damped methods allow to include
damping in the structure.
MODAL ANALYSIS IS DONE IN FOUR STEPS
Building the Model
In this step job name and analysis title is specified. Then, PREP7 preprocessor is used to define the element
types, element real constants, material properties, and the model geometry. Material properties can be linear, isotropic or
orthotropic, and constant or temperature-dependent. Young's modulus (EX) (or stiffness in some form) and density
Damage Detection In Cantilever Beams Using Artificial Neural Networks 261
(DENS) (or mass in some form) for a modal analysis should be defined. Nonlinear properties are ignored. For the specific
element type required real constants should be applied [6, 7].
Applying Loads & Obtaining Solution
In this step, analysis type & option is defined, loads are applied, load step option is specified and finite element
solutions for the natural frequencies are initiated.
Expanding Modes
In its strictest sense the term expansion means expanding the reduced solution to the full DOF set. In modal
analysis however the term expansion simply means writing mode shape to the result file. That is, expanding modes applies
not just to reduce mode shape from the reduced mode extraction method but to full mode shapes from the other mode
extraction method as well.
Reviewing the Results
Results from a modal analysis are written to the structural results file, Jobname .RST. They consist of natural
frequencies, expanded mode shapes and relative stress & force distribution.
The element type used here is a 4-node shell element (Shell63). Shell 63 has both bending and membrane
capabilities. Both in plane and normal loads are permitted. The element has six degrees of freedom at each node,
transmissions in the nodal X, Y & Z directions and rotations about the three axes. The geometry, node locations and the
coordinate system for this element are shown in the Fig. 2. The element is defined by four nodes, four thickness an elastic
foundation stiffness & the orthotropic material properties. The element X-axis may be rotated by an angle Theta (in
degrees). The thickness is assumed to vary smoothly over the area of the element with the thickness input at the four
nodes. If the element has a constant thickness only one thickness (TK (I)) need be input. If the thickness is not constant all
four must be input.
The elastic foundation thickness (EFS) is defined as a pressure required to produce a unit normal deflection of the
foundation. The elastic foundation thickness is by passed if EFS is less than or equal to zero.
Figure 2: A Four Node Shell Element (Shell 63)
Material Properties
Material is assumed to be elastic and isotropic.
Both the thin beam and thick beams are made of mild steel for which required properties are given in the table
below:
262 Lokesh Kondru & M R Narasinga Rao
Table 1: Properties of Material
Modulus of Elasticity EXX (N/m2) 2.1e11
Poisson’s Ratio NUXY 0.3
Density DENS (Kg/m3) 7.85e3
Boundary Conditions
The boundary condition for both the beam is shown in Fig. 3. For shell 63 (4 node shell element) every node is
having 6 degrees of freedom, 3 translations Ux, Uy, Uz & 3 rotations Rotx, Roty, Rotz. Now, all the degrees of freedom of
the nodes, which are at fixed end, are restrained to simulate the condition of cantilever beam.
Figure 3: ANSYS Model with Boundary Condition
Back-Propagation Algorithm using Neural Networks
The application of the back-propagation algorithm involves two phases: [8]
During the first phase the input x is presented and propagated forward through the network to compute the output
for each input unit. This output is compared with its desired value resulting in an error signal δp for each output unit.
The second phase involves a backward pass through the network during which the error signal is passed to each
unit in the network and appropriate weight changes are calculated. The weight of a connection is adjusted by an amount
proportional to the product of an error signal δ, on the unit k receiving the input and the output of the unit j sending this
signal along the connection.
If the unit is an output unit, the error signal is given by
Take as the activation function F the 'sigmoid' function as defined,
In this case the derivative is equal to
Damage Detection In Cantilever Beams Using Artificial Neural Networks 263
such that the error signal for an output unit can be written as:
The error signal for a hidden unit is determined recursively in terms of error signals of the units to which it
directly connects and the weights of those connections. For the sigmoid
activation function is given by,
Methodology in Neural Networks
We have considered four inputs, and three output samples were considered in this research. There will be
interactive session between the user and the neural network model, in which the user has to enter various values for neural
network parameters.
These parameters include, Momentum Rate (MR), Learning Rate (LR), Maximum Total Error (MTE),
Maximum Individual Error (MIE), Maximum Number of Iterations (MIT), Number of hidden layers (NHL), Number of
units in the hidden layer (NUHL) and finally the number of samples (NS), After entering all these values, the network
generates an error file which are called Normalized System Errors (NSE)
Figure 4: The Discretised Model of Thin Cantilever Beam with U-Notch
RESULTS
Identifying Inputs from ANSYS
Using ANSYS Software, the natural frequencies of the cracked cantilever beams are obtained for cracks located at
normalized distance (c/l) from the fixed end with a normalized depth (a/h), for both thin beam and thick beam [9]. The
normalized natural frequencies Fr1, Fr2, Fr3 are defined as the ratio of frequency for cracked and un-cracked beam in 1st,
2nd & 3rd mode respectively.
Figure 4 shows the discretised model (zoomed near the position of crack) of thin beam with U-notch.
264 Lokesh Kondru & M R Narasinga Rao
Parametric studies have been carried out for thin beam having length (l) = 260 mm, width (w) = 25 mm and
thickness (h) = 4.4 mm. The breadth (b) of U notch has been kept as 0.32 mm. Crack depth (a) has been varied from 0.6
mm to 3 mm in steps of 0.6 mm. The crack location (c) has been varied from 30 mm to 230 mm in steps of 20 mm.
Results of Modal Analysis
The following are different tables depicting the frequencies obtained by considering crack depth, and their lengths
as inputs.
Table 2: Frequencies Obtained at Crack Depth of 0.0003
Table 3: Frequencies Obtained at Crack Depth of 0.0006
Table 4: Frequencies Obtained at Crack Depth of 0.0009
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Table 5: Frequencies Obtained at Crack Depth of 0.0012
Table 6: Frequencies Obtained at Crack Depth of 0.0015
Table 7: Frequencies Obtained at Crack Depth of 0.0018
Table 8: Frequencies Obtained at Crack Depth of 0.0021
266 Lokesh Kondru & M R Narasinga Rao
Table 9: Frequencies Obtained at Crack Depth of 0.002
Table 10: Frequencies Obtained at Crack Depth of 0.003
Results for Damage Detection Process Using Back Propagation Technique in Neural Networks
Serial
No MR LR MTE MIE MIT NHL NUHL NS NSE
1 0.9 0.5 0.01 0.001 10000 1 1 17 0.1008
2 0.9 0.5 0.01 0.001 10000 1 1 34 0.1006
3 0.9 0.5 0.01 0.001 10000 1 1 51 0.0905
4 0.9 0.5 0.01 0.001 10000 1 1 68 0.0905
5 0.9 0.5 0.01 0.001 10000 1 1 85 0.0905
6 0.9 0.5 0.01 0.001 10000 1 1 102 0.0905
7 0.9 0.5 0.01 0.001 10000 1 1 119 0.1007
8 0.9 0.5 0.01 0.001 10000 1 1 136 0.0906
9 0.9 0.5 0.01 0.001 10000 1 1 153 0.0932
10 0.9 0.5 0.01 0.001 10000 1 1 163 0.0923
Comparison of the Results Generated with ANSYS and Neural Networks
Inputs
(Frequencies)
Expected Outputs Obtained Outputs
Damage
Depth
Length at which
Damage Occurred
Damage
Depth
Length at which Damage
Occurred
1,7.861,48.46,134.56 0.003 0.52,0.56 0.0016 0.32,0.36
1,7.673,48.65,137.12 0.0024 0.04,0.08 0.00175 0.029,0.058
1,7.8725,49.284,137.78 0.0018 0.68,0.72 0.0013 0.49,0.52
1,7.8431,49.33,137.84 0.0012 0.16,0.2 0.0008 0.117,0.146
Damage Detection In Cantilever Beams Using Artificial Neural Networks 267
CONCLUSION OF RESULTS
Results obtained from Neural Networks are compared with the ANSYS results and the accuracy obtained for
Damage depth is 73.33% and for Length at which damage occurred is 73.2%.
CONCLUSIONS
A method for identifying crack parameters (crack depth and its location) in a cantilever beam using Neural
Networks has been attempted in the present work. Parametric studies have been carried out using ANSYS Software to
evaluate modal parameters (natural frequencies) for different crack parameters. The identification procedure presented in
this study is believed to provide a useful tool for detection of crack in a beam. If the network is further trained perfectly we
can achieve more accurate results and this procedure helps in detecting cracks efficiently. This developed model can also
be used for different varied applications of engineering, given the predefined values as input to the system.
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