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:


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


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.


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






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


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.

REFERENCES

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assess wellbeing in diabetes”, JAPI volume 57, February 2009.

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