research article comparison of cfbp, ffbp, and rbf

Research ArticleComparison of CFBP FFBP and RBF Networks inthe Field of Crack Detection

Dhirendranath Thatoi1 Punyaslok Guru1 Prabir Kumar Jena2

Sasanka Choudhury1 and Harish Chandra Das1

1 Department of Mechanical Engineering Institute of Technical Education and Research SOA University BhubaneswarOdisha 751030 India

2 VSSUT Burla Odisha 768018 India

Correspondence should be addressed to DhirendranathThatoi dnthatoigmailcom

Received 31 July 2013 Revised 26 October 2013 Accepted 22 November 2013 Published 4 February 2014

Academic Editor Chia-Feng Juang

Copyright copy 2014 DhirendranathThatoi et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The issue of crack detection and its diagnosis has gained a wide spread of industrial interestThe crackdamage affects the industrialeconomic growth So early crack detection is an important aspect in the point of view of any industrial growth In this paper adesign tool ANSYS is used to monitor various changes in vibrational characteristics of thin transverse cracks on a cantilever beamfor detecting the crack position and depth andwas compared using artificial intelligence techniquesThe usage of neural networks isthe key point of development in this paperThe three neural networks used are cascade forward back propagation (CFBP) networkfeed forward back propagation (FFBP) network and radial basis function (RBF) network In the first phase of this paper theoreticalanalysis has been made and then the finite element analysis has been carried out using commercial software ANSYS In the secondphase of this paper the neural networks are trained using the values obtained from a simulated model of the actual cantilever beamusing ANSYS At the last phase a comparative study has been made between the data obtained from neural network technique andfinite element analysis

1 Introduction

The presence of cracks in a mechanical member has catas-trophic effects on its functionality and the failure in detectingthe said cracks may result in failure of the mechanical mem-berHence studies related to crack detection has been amajorfield of interest among researchers and scholars Crack detec-tion using vibrational analysis is considered the most prolificand efficient technique in crack detection but the integrationof vibrational analysis with artificial intelligence and softcomputing techniques has provided new dimensions to crackdetection and analysis

There have been many papers suggesting different meth-ods for detecting cracks in a simple cantilever beam byusing vibrational analysis The inverse measurement of crackparameters from vibrational analysis is the basis on which thecrack analysis is done in this paper and the relevant literaturein this regard is described as follows Lifshitz and Rotem[1] pioneered the proposed damage detection via vibration

measurements-inverse measurement of crack parametersfrom vibrational parameters They look at the change inthe dynamic moduli which can be related to the frequencyshift as indicating damage in particle-filled elastomers Thedynamic moduli which are the slopes of the extensional androtational stress-strain curves under dynamic loading arecomputed for the test articles from a curve-fit of themeasuredstress-strain relationships at various levels of filling Byconsidering the change ofmodal frequencies as the key factorof damage various researchers have used the inverse methodfor examining the changes of modal frequencies [2ndash13]

Many times it is quite difficult to determine the crackparameters by presenting a physical model of the can-tilever beam along with the crack In order to examineand analyze the various parameters and simulate a simplecrack on a cantilever beam efforts have been taken withthe help of designing software Apart from using a simulationsoftware researchers have used different means of analyzingthe entire system Sahoo et al [14] have used ultrasonic

Hindawi Publishing CorporationModelling and Simulation in EngineeringVolume 2014 Article ID 292175 13 pageshttpdxdoiorg1011552014292175

2 Modelling and Simulation in Engineering

signals to monitor the various vibrational changes andanomalies

After the successful monitoring of these changes andanomalies a conclusion was made about the various crackparameters In general if a person tries to reach such a con-clusion by following a theoretical process it would take hima longer period of time as well as decrease the effectivenessand efficiency of the obtained result Hence various methodsapart from the conventional technique are utilized to obtaina good and approximately exact result Suresh et al [15]presented a method by considering the flexural vibration ina cantilever beam with a transverse crack They computedmodal frequency parameters analytically for various cracklocations and depths After obtaining these values the param-eters so obtained were used to train the neural network toidentify the damage location and size Saridakis et al [16]proposed fuzzy logic genetic algorithm (GA) and neuralnetwork for considering the dynamic behavior of a shaft withtwo transverse cracks characterized by position depth andrelative angle Little and Shaw [17] solved exactly a linearisedversion of the model They explicitly show that the capacityof the memory is related to the number of synapses and notthe number of neurons Amethod was proposed by Loutridiset al [18] ldquoforced vibration behavior and crack detectionof cracked beams using instantaneous frequencyrdquo to detectthe crack in beam depending on instantaneous frequency(IF) The study monitored the dynamic behavior for thebeam with breathing crack They carried out the researchunder harmonic excitation with experimental and theoreticalresultsThe observed data of the simulation and experimentaltest were analyzed using MATLAB A relation between thedepth of crack and the main difference of instantaneousfrequency was established The instantaneous frequency wasfound to be a good indicator for the size of crack

Thatoi et al [19] have reviewed the variousmethodologiesused for crack detection and fault diagnosis Doebling et al[20] have reviewed different techniques of detection locationand characterization of structural damage and faults Theanalysis included changes in modal frequency changes inmode shapes and changes in flexibility coefficients

Qian et al [21] used a finite element model to predict thebehavior of a beam with an edge crack by taking an elementstiffnessmatrix which is derived from an integration of stressintensity factor Kim and Stubbs [22] located the position andestimated the size of a crack in beam type structures by takingthe changes in natural frequencies which is formulated byrelating fractional changes in modal energy due to the pres-ence of crack Ihn and Chang [23] have proposed a methodfor detecting and monitoring hidden fatigue crack growthThey have used diagnostic signals generated from nearbypiezoelectric actuators built into the structure to detect thecrack growth A complete built-in diagnostic system for thetests was developed including a sensor network hardwareand the diagnostic software Lele and Maiti [24] have con-sidered the effect of shear deformation and rotational inertiathrough the Timoshenko beam theory for detection andmeasurement of crack extension of short beams Quek et al[25] have inspected the sensitivity of wavelet technique in thedetection of cracks in beam-like structure In this aspect they

X

L

O

Y

Z

T

W

b1P2

P1

Lc

Figure 1 Beam model

have compared the results of two wavelets such as Haar andGabor wavelets Zagrai and Giurgiutiu [26] have describedthe utilization of electromechanical impedance method forcrack detection in thin plates This method allows the directidentification of structural dynamics by obtaining its electro-mechanical impedance

Crack detection in the beam-like structures having singleas well as multicracks was modeled using artificial neuralnetwork (ANN) and the various characteristics were studied[27ndash29]

In the present study a methodical approach to analyzeand detect the crack parameters that is the crack locationand relative crack depth from a set of frequency valuesobtained from a simulated model of the cantilever beamcontaining a thin transverse crack using ANSYS Using thedata cascade forward back propagation (CFBP) feed forwardback propagation (FFBP) and radial basis function (RBF)neural network models are developed and finally result thusobtained are compared

2 Mathematical Formulation

21 Computation of Flexibility Matrix of a Damaged BeamSubjected to Complex Loading A beam with cracks hassmaller stiffness than a normal beam This decreased localstiffness can be formulated as a matrix The dimension ofthe matrix would depend on the degrees of freedom inthe problem Figure 1 shows a cantilever beam of width 119882and height 119879 having a transverse surface crack of depth 119887

1

The beam experiences combined longitudinal and transversemotion due to the axial force 119875

1and bending moment 119875

2

Here two degrees of freedom are considered leading to a 2times2local stiffness matrix

The relationship between strain energy release rate 119869(119887)and stress intensity factors (119866

119868119894) at the crack section is given

by Tada et al [30] as

119869 (119887) =1

1198641015840(1198661198681+ 1198661198682)2

(1)

where 1198641015840 = 119864(1minus]2) for plane strain condition and 1198641015840 = 119864for plane stress119866

1198681= stress intensity factor for openingmode

119868 due to load 1198751 and 119866

1198682= stress intensity factor for opening

mode 119868 due to load 1198752

Modelling and Simulation in Engineering 3

From earlier studies (Tada et al [30]) the values of stressintensity factors are

1198661198681=

1198751

119882119879

radic120587119887(1198651(119887

119879))

1198661198682=61198752

1198821198792radic120587119887(119865

2(119887

119879))

(2)

where 0 le 119887 le 1198871and the experimentally determined

functions 1198651and 119865

2are expressed as follows

1198651(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times (0752 + 202 (119887

119879)

+ 037(1 minus sin(1205871198872119879))

3

)

times (cos(1205871198872119879))

minus1

1198652(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times 0923 + 0199(1 minus sin (1205871198872119879))4

cos (1205871198872119879)

(3)

The strain energy release rate (also called strain energydensity function) at the crack location is defined as

119869 (119887) =120597119880119905

120597 (119887 times119882) (4)

where (119887 times 119882) is the newly created surface area of the crack

119889119880119905= 119869 (119887) 119889 (119887 times119882) = 119882119869 (119887) 119889119887 (5)

since the width of the cross section of the beam is constant

119880119905= 119882int

1198871

0

119869 (119887) 119889119887 (6)

So the strain energy release (119880119905) due to the crack of depth 119887

1

is calculatedThen fromCastiglianorsquos theorem the additional displace-

ment along the force 119875119894is given as

119878119894=120597119880119905

120597119875119894

(7)

Using (1) and (6)

119878119894=

120597

120597119875119894

[119882int

1198871

0

119869 (119887) 119889119887] = 119882120597

120597119875119894

[int

1198871

0

119869 (119887) 119889119887] (8)

119862119894119895=120597119878119894

120597119875119895

=120597

120597119875119895

[119882120597

120597119875119894

int

1198871

0

119869 (119887) 119889119887]

= 1198821205972

120597119875119894120597119875119895

int

1198871

0

119869 (119887) 119889119887

(9)

The flexibility influence coefficient 119862119894119895as per definition will

be as followsBy substituting (1) in (9)

119862119894119895=119882

1198641015840

1205972

120597119875119894120597119875119895

int

1198871

0

(1198661198971+ 1198661198972)2

119889119887 (10)

Putting 120575 = (119887119879) 119889120575 = 119889119887119879 we get 119889119887 = 119879119889120575 and when119887 = 0 120575 = 0 119887 = 119887

1 120575 = 119887

1119879 = 120575

1

From the above condition (10) converts to

119862119894119895=119882119879

1198641015840

1205972

120597119875119894120597119875119895

int

1205751

0

(1198661198971+ 1198661198972)2

119889120575 (11)

Equation (11) will give different expressions of flexibilityinfluence coefficient 119862

119894119895

119862119894119895= flexibility influence coefficient in 119894 direction (119883-

direction or 119884-direction) due to the load in 119895 direction (1198751

or 1198752)The expressions for119862

11 11986212(=11986221) and119862

22are obtained

as follows

11986211=119882119879

1198641015840int

1205751

0

120587119887

1198822119879221198651

2(120575) 119889120575

=2120587

1198821198641015840int

1205751

0

1205751198651

2(120575) 119889120575

11986212= 11986221=

12120587

1198641015840119879119882int

1205751

0

1205751198651(120575) 1198652(120575) 119889120575

11986222=

72120587

11986410158401198821198792int

1205751

0

1205751198652

2(120575) 119889120575

(12)

The local stiffness matrix can be obtained by taking theinversion of compliance matrix That is

119870 = [11987011

11987012

11987021

11987022

] = [11986211

11986212

11986221

11986222

]

minus1

(13)

Converting the influence coefficient into dimensionless form

11986211= 11986211

1198821198641015840

2120587 119862

12= 11986212

1198641015840119879119882

12120587= 11986221

11986222= 11986222

11986410158401198821198792

72120587

(14)

22 Governing Equations for Vibration Mode of the CrackedBeam The cantilever beam as mentioned in Section 21 isconsidered here for free vibration analysis A cantilever beamof length ldquo119871rdquo width ldquo119882rdquo and depth ldquo119879rdquo with a crack of depthldquo1198871rdquo at a distance ldquo119871

119888rdquo from the fixed end is considered

as shown in Figure 1 Taking 1198781(119909 119905) and 119878

2(119909 119905) as the

amplitudes of longitudinal vibration for the sections beforeand after the crack position and 119881

1(119909 119905) 119881

2(119909 119905) are the

amplitudes of bending vibration for the same sections asshown in Figure 2


L

S1 S2

V1 V2

Lc

Figure 2 Beam model with deflection

The free vibration of an Euler-Bernoulli beam of a con-stant rectangular cross section is given by the following dif-ferential equations as

12059721198781205971199052= (119864120588)120597

21198781205971199092 for longitudinal vibration

and119864119868(12059741198811205971199094) minus 120588120596

2119881 = 0 for lateral vibration of

beam

Thenormal functions for the cracked beam in nondimen-sional form for both the longitudinal and bending vibrationsin steady state can be defined as

1198781(119909) = 119861

1cos (119867

119904119909) + 119861

2sin (119867

119904119909) (15a)

1198782(119909) = 119861

3cos (119867

119904119909) + 119861

4sin (119867

119904119909) (15b)

1198811(119909) = 119861

5cosh (119867V119909) + 1198616 sinh (119867V119909)

+ 1198617cos (119867V119909) + 1198618 sin (119867V119909)

(15c)

1198812(119909) = 119861

9cosh (119867V119909) + 11986110 sinh (119867V119909)

+ 11986111cos (119867V119909) + 11986112 sin (119867V119909)

(15d)

where 119909 = 119909119871 119878 = 119878119871 119881 = 119881119871 120572 = 119871119888119871

119867119904=120596119871

119863119904

119863119904= (

119864

120588)

12

119867119881= (

1205961198712

119863V)

12

119863V = (119864119868

120583)

12

120583 = 119860120588

(16)

119861119894 (119894 = 1 12) constants are to be determined from boundary

conditions The boundary conditions of the cantilever beamin consideration are

1198781(0) = 0 119881

1(0) = 0 119881

1015840

1(0) = 0

1198781015840

2(1) = 0 119881

10158401015840

2(1) = 0 119881

101584010158401015840

2(1) = 0

(17)

At the cracked section

1198781(120572) = 119878

2(120572) 119881

1(120572) = 119881

2(120572)

11988110158401015840

1(120572) = 119881

10158401015840

2(120572) 119881

101584010158401015840

1(120572) = 119881

101584010158401015840

2(120572)

(18)

Also at the cracked section

1198601198641198891198781(119871119888)

119889119909= 11987011(1198782(119871119888) minus 1198781(119871119888))

+ 11987012(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(19)

Multiplying both sides of the above equation by1198601198641198711198701111987012

119873111987321198781015840

1(120572) = 119873

2(1198782(120572) minus 119878

1(120572)) + 119873

1(1198811015840

2(120572) minus 119881

1015840

1(120572))

(20)

Similarly

11986411986811988921198811(119871119888)

1198891199092= 11987021(1198782(119871119888) minus 1198781(119871119888))

+ 11987022(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(21)


1198733119873411988110158401015840

1(120572) = 119873

3(1198782(120572) minus 119878

1(120572))

+ 1198734(1198811015840

2(120572) minus 119881

1015840

1(120572))

(22)

where 1198731= 119860119864119871119870

11 1198732= 119860119864119870

12 1198733= 119864119868119871119870

22 1198734=

119864119868119871211987021

The normal functions (15a) (15b) (15c) and (15d) alongwith the boundary conditions as mentioned above yield thecharacteristic equation of the system as

|119876| = 0 (23)

where119876 is a 12times12matrix as given belowwhose determinantis a function of natural circular frequency (120596) the relativelocation of the crack (120572) and the local stiffness matrix (119870)which in turn is a function of the relative crack depth 120575

1=

(1198871119879)


Figure 3 Finite element mesh and mesh zoom of the crack

119876 =

[[[[[[[[[[[[[[[[[[

[

1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1198663

1198664minus1198667minus1198668

0 0 0 0

0 0 0 0 1198664

1198663

1198668minus1198667

0 0 0 0

11986611198662minus1198665minus1198666minus1198661minus11986621198665

1198666

0 0 0 0

11986621198661


0 0 0 0

11986611198662

1198665


0 0 0 0

11987811198782

1198783

1198784

minus1198662minus11986611198666minus1198665

1198785

1198786

1198787

1198788

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 minus11987981198797

0 0 0 0 0 0 0 0 minus11987961198795

1198796minus1198795

119878911987810

11987811

11987812

11987813

11987814

11987815

11987816

11987817

11987818

minus1198795minus1198796

]]]]]]]]]]]]]]]]]]

]

(24)

where

1198781= 1198662+ 1198733119867V1198661 119878

2= 1198661+ 1198733119867V1198662

1198783= minus1198666minus 1198733119867V1198665 119878

4= 1198665minus 1198733119867V1198666

1198785=11987334

119867V1198795 119878

6=11987334

119867V1198796 119878

7=minus11987334

119867V1198795

1198788=minus11987334

119867V1198796 119878

9= 11987312119867V1198662

11987810= 11987312119867V1198661 119878

11= minus11987312119867V1198666

11987812= 11987312119867V1198665 119878

13= minus11987312119867V1198662

11987814= minus11987312119867V1198661 119878

15= 11987312119867V1198666

11987816= minus11987312119867V1198665 119878

17= 1198795minus 11987311198671199041198796

11987818= 1198796+ 11987311198671199041198795

1198661= Cosh (119867V120572) 119866

2= Sinh (119867V120572)

1198663= Cosh (119867V) 119866

4= Sinh (119867V)

1198665= Cos (119867V120572) 119866

6= Sin (119867V120572)

1198667= Cosh (119867V) 119866

8= Sin (119867V)

1198795= Cos (119867

119904120572)

1198796= Sin (119867

119904120572) 119879

7= Cos (119867

119904) 119879

8= Sin (119867

119904)

11987312=1198731

1198732

11987334=1198733

1198734

(25)

3 Finite Element Model Using ANSYS

The vibrational analysis of a continuous beam by analyticalprocedures is quite appropriate and less complicated How-ever with the introduction of crack in a beam the analysisof the beam for its vibrational characteristics becomes morecomplicated Since the equation of motion of the continuousbeam is a partial differential equation and we have with usvarious initial and boundary conditions we use the finiteelement method (FEM) which translates the complex partialdifferential equations into linear algebraic equations andhence the mode of solution becomes simpler

In the present research the ANSYS is used as a tool tomodel and simulate a beam with a crack to monitor the vari-ation in its vibrational characteristics The beam is modeledusing design software such as solid work and it is imported toANSYS for the analysis

Now after importing the model file its geometry wasmodified and divided the entire structure into meshes(Figure 3) by using FEM and has been solved for the modesof frequencies The meshing size should be increased so thatit uniformly covers the entire structure After the modelis properly meshed and solved by using FEM the variousfrequency values were obtained for a particular combinationof crack location and depth

The above procedure is elaborated as follows

(1) Double click on workbench Import geometry fromsolid works file saved in solid works as igs file

(2) Modify geometry click on mesh and increase themeshing size

(3) Provide support and make one end fixed(4) Click on solve(5) Click on displacement


4 The Inverse Tool Artificial Neural Network

In order to determine the crack parameters from the fre-quency data we take the help of artificial intelligence in theform of neural network The structure of a neural net is verysimilar to the exact biological structure of a human braincell In order to be precise neural network can be statedas a network model whose functionality is similar to thatof the brain In other words a neural network is at firsttrained to recognize a predefined pattern or an already knownrelationship from certain prefound values It works by takingcertain number of inputs and computing the output aftercarefully adjusting the weights which are attached with theinput values to differentiate these input values on the basis ofimportance and priority in processing These weight valuesare utilized to obtain the final output For example if we havetwo inputs 119883

1and 119883

2 then a simple neural network can be

designed and the net input can be found out as

119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909

2are the activations of the input neurons 119883

1

and1198832 that is the output of the input signalsThe output119910 of

the output neuron 119884 can be obtained by applying activationsover the net input that is the function of the net input

119910 = 119891 (119910in) (27)

Output = Function (net input calculated)The function to be applied over the net input is called an

activation functionA neural network is classified on the basis of the modelrsquos

synaptic interconnections the learning rule adapted and theactivation functions used in the neural net Based on thesynaptic interconnections we choose a multilayer perceptronmodel for our research purpose Now depending on theprocess of learning a neural network it is classified assupervised learning network unsupervised learning net-work and reinforced learning network Supervised learningprocess requires a set of already known values to trainthe network and hence find out the output From the setof values obtained after monitoring the vibrational char-acteristics of the cracked beam and subjecting it to finiteelement modeling the corresponding values are trained tothe network The tansigmoid hyperbolic function is chosenas the activation function Finally the cascade forward backpropagation (CFBP) network model the feed forward backpropagation (FFBP) network model and the radial basisfunction (RBF) network model are used and the results areanalysed

41 The CFBP Network As stated earlier in the present studya CFBP network (Figure 4) is used This network is verysimilar to the feed forward back propagation networks withthe difference being that the input values calculated afterevery hidden layer are back-propagated to the input layerand the weights adjusted subsequently The input values aredirectly connected to the final output and a comparisonoccurs between the values obtained from the hidden layers

Hidden layer

Output layerInput layer

Out

put

Inpu

t

Figure 4 Structure of CFBP network

and the values obtained from the input layers and weights areadjusted accordingly

Sahoo et al [14] and Gopikrishnan et al [31] observedthat the results obtained fromCFBP networks aremuchmoreefficient than the FFBP networks Badde et al [32] suggestedthat CFBP networks show better and efficient results in mostcases

The algorithm followed in the present paper is given asfollows

(1) Initialize the predefined input matrix(2) Initialize the desired output or target matrix(3) Initialize the network by using the net = newcf

(Input Output Hidden layers Transfer FunctionTraining algorithm Learning Function PerformanceFunction)

(4) Define the various training parameters such as num-ber of epochs number of validation checks andmaximum and minimum gradient

(5) Test the new found weights and biases for accuracy(6) Using the weights and biases determine the unknown

results

The initial weight and bias values are taken as 0 (zero)In Figure 4 the inputs are connected to the hidden layer

as well as the output layer

42 The FFBP Network Another network that we are usingfor our comparative study in the detection of cracks in acantilever beam is the feed forward back propagation (FFBP)network (Figure 5) This network differs from the CFBPnetwork on the basis that each subsequent layer has a weightcoming from the previous layer and no connection is madebetween the layers and the first layer All layers have biasesThe last layer is the network output

In this study relevant information comparing the resultsof both networks as well as the result from a third network ispresentedThe algorithm used for CFBP network is also usedin case of the FFBP network except for the network creationmode which uses the keyword newff


Hidden layerInput layer Output layer

Out

put

Inpu

t

Figure 5 Structure of a FFBP network

43 The RBF Network The radial basis function (RBF) net-work (Figure 6) is basically used to find the least numberof hidden layers or neurons in a single hidden layer untila minimum error value is reached The RBF networks canbe used to approximate functions For network creation thekeyword newrb adds neurons to the hidden layer of a radialbasis network until it meets the specified mean squared errorgoal

5 Results and Discussions

As discussed earlier the motive of this paper is to detect thelocation of a single crack formed on a cantilever beam Amethodology involving ANN and ANSYS was followed toget the results and was discussed To begin with a transversethin crack is formed on a cantilever beam Finite elementmodeling is applied in the cracked beam using ANSYS tomonitor the various frequencies in different modes of crackformation Based on this sample data a neural network hasbeen trained to find out the crack depth and location of anyarbitrary transverse crack on a cantilever beam

Hence there are two steps in present mode of solution

(1) monitoring the vibrational changes using ANSYS(2) creating an ANN and training it to provide us with an

approximate result

A thin transverse crack is present on a cantilever beam oflength 26 cm and with this crack a beam model is designedusing ANSYS to obtain a set of natural frequencies for thefirst second and thirdmodes at different crack locationswithvarying crack depths

Cascade forward back propagation network feed forwardback propagation network and radial basis function networkwere used to estimate the crack location and relative crackdepth A single hidden layer was used with 13 neurons TheTANSIG hyperbolic tangent sigmoid transfer function oractivation function was used to calculate the final output ofthe neural net

The TANSIG transfer function is given as

tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn

Figure 6 Structure of RBF neural network

0 200 400 600 800 1000

Best validation performance is 0013719 at epoch 819

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs

Figure 7 Performance graph of CFBP network

The Levenberg-Marquardt (trainlm) training process wasfollowed to train the neural network Training stopswhen anyof these conditions occurs

(1) The maximum number of epochs (repetitions) isreached

(2) The maximum amount of time has been exceeded(3) Performance has been minimized to the goal

The division of training data was done using the random(Dividerand) method

The number of iterations provided was 200 The gradientwas set at amaximum value of 1 andminimum value of 0Theperformance or goal or maximum number of error checkswas set to be 150 There was no constraint on the amount oftime for which the training program ran Since the number ofvalues employed for testing was large in number hence fewvalues were taken to depict the efficiency of the ANNmodelThe tested data for a relative crack depth of 01 and varyinglocationswas tabulated and a comparisonwith the actual datais shown in Table 1

The network formed generating a performance plotwhich is shown in Figure 7 by mentioning the relationship


5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target

Validation R = 099991

FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021

(d)Figure 8 Regression plot of CFBP network

010

2030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Location obtained from ANSYSLocation obtained from ANN

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 9 Comparison of result of location between ANSYS andCFBP network

000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Relative crack depth from ANSYSRelative crack depth from ANN

1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 10 Comparison of result of relative crack depth betweenANSYS and CFBP network

0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900

Figure 11 Performance graph of FFBP network

between the trained tested and validated points against athreshold value

A regression plot is also generated which is shown inFigure 8 for the individual results obtained


5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)

Figure 12 Regression plot of FFBP network

On plotting the values obtained from ANSYS and ANNand comparing both it was observed that minimum differ-ence was obtained between both of the values thus validatingour training processThe plots of crack location for the valuesrepresented in Table 1 are shown in Figure 9

The plot of Figure 10 shows the comparison of results forvarius crack depths obtained fromANSYS andCFBPnetworkby fixing the crack location

It was observed that after running the CFBP network fora particular number of iterations a certain value of errorbetween the ANSYS generated values and the network gen-erated values was obtained It is essential to mention that theerror obtained here is of the order of 10minus2 and in general forany number of iterations run over any period of time andkeeping the abovementioned parameters (except the numberof epochs) constant the error obtained is of the same orderand varying upto 10minus4 under a uniform testing training andvalidation situation

After observing the results of CFBP network the FFBPnetwork is introduced for the comparative study In thisnetwork 13 hidden neurons and a single hidden layer havebeen taken The activation function training function thedivision of data function number of iterations gradient valueand all the parameters are similar to those used for the CFBP

network After training the data to FFBP network with thesimilar process (by keeping relative crack depth constantand varying the location) the results are obtained and thecomparison with the results of actual data is presented inTable 2

Similar to CFBP network the performance plot and theregression plot of FFBP network obtained from the traineddata are shown in Figures 11 and 12 respectively

On plotting the values obtained from ANSYS and ANNand comparing both it has been observed that minimumdifference was obtained between both of the values thusvalidating our training processThe plots of crack location forthe values represented in Table 2 are shown in Figure 13

Figure 14 shows the comparison of results for varius crackdepths obtained fromANSYS and FFBPnetwork by fixing thecrack location

In comparison with CFBP network it is observed that theFFBPnetwork is showing the better results For the above casethe CFBP network gives the best validation performance interms of root mean square error (RMSE) value of 0018464whereas the FFBP network gives 0041058 which shows thatthe mean squared error calculated in case of the CFBPnetwork is lower than that of the FFBP network


000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Relative crack depth obtained from ANSYSRelative crack depth obtained from FFBP net

1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 13 Comparison of result of relative crack depth betweenANSYS and FFBP network

0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Location obtained from ANSYSLocation obtained from FFBP net

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Figure 14 Comparison of result of crack location between ANSYSand FFBP network

0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce

Performance is 000506572 goal is 0

Figure 15 Performance graph of RBF network

The RBF network is used then to complete the compara-tive study The RBF network takes the input and output dataused for training which we get from ANSYS The defaultvalue of the mean squared error goal set in the RBF networkis 00 and the spread value is also taken as the default valuewhich is 10 The more the spread the smoother the functionapproximation Basically the newrb keyword creates a two-layered network The results obtained and the comparisonswith the results of actual data are shown in Table 3

The RBF network generated a performance plot which isdifferent from the CFBP and FFBP networks on the groundsthat in an RBF network a particular goal is set to be reachedbut the number of iterations is not fixed For this particularcase it was observed that the errors generated in all cases are ofthe order of 10minus3and the RBF network certainly yields a better

000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Relative crack depth obtained from ANSYSRelative crack depth obtained from RBF net

1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 16 Comparison of result of relative crack depth betweenANSYS and RBF network

0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Location obtained from ANSYSLocation obtained from RBF net

1 2 3 4 5 6 7 8 9 10 11 12 13

Figure 17 Comparison of result of crack location between ANSYSand RBF network

result at more frequent intervals than the CFBP and the FFBPnetworks The performance plot is shown in Figure 15

In similarity to CFBP and FFBP network a comparisonplot has been obtained from the various results of actual dataand RBF network for relative crack depth and crack locationand is shown in Figures 16 and 17 respectively

After observing the results of various networks used(CFBP FFBP and RBF) a comparison of results has beenmade and is shown in Table 4 In the following table acomparative study has been made between the three neuralnetwork techniques in comparison with ANSYS and also thepercentage of error has been presented It was observed thatfor some cases RBF network result outperforms the resultsof the other two networks But in general CFBP was foundto be more efficient in terms of error and computationalcomplexity

Ultimately after observing the training process of all threenetworks and individually comparing the results of each ofthe networks the three networks combined together withrespect to the relative crack depth and crack location Wefind that the result obtained from all three networks is verysimilar with marginal differences at certain particular pointsBy comparing the results of relative crack depth and cracklocation for the three networks a comparison plot amongthree networks have been drawn and these are shown inFigure 18 and Figure 19 The point to note is that overalldeductions suggest that RBF network gives the optimumresult The CFBP net is much more flexible on the groundsthat we can alter its efficiency by varying the number and sizeof the hidden layers

Table 5 shows the various properties of the material usedfor the above study


Table 1 Input data and results for CFBP network

Test points First three natural frequencies (Hz) CFBPrcd Location 1st mode 2nd mode 3rd mode rcd Location0100 2000 48209 302187 846297 0103 23290100 4000 48214 302270 846630 0125 38360100 5000 48216 302305 846758 0129 48090100 6000 48219 302340 846845 0126 58090100 8000 48221 302400 847003 0127 8115

Table 2 Input data and results for FFBP network

Test points First three natural frequencies (Hz) FFBPrcd Location 1st mode 2nd mode 3rd mode rcd Location0100 2000 48209 302187 846297 0106 20300100 4000 48176 302010 845879 0096 37650100 5000 48134 301797 845399 0098 49520100 6000 48086 301549 844821 0096 61480100 8000 48031 301261 844155 0095 8093

Table 3 Input data and results for RBF network

Test points First three natural frequencies (Hz) RBFrcd Location 1st mode 2nd mode 3rd mode rcd Location0100 2000 48209 302187 846297 0099 19640100 4000 48176 302010 845879 0102 46780100 5000 48134 301797 845399 0150 50120100 6000 48086 301549 844821 0098 61760100 8000 48031 301261 844155 0101 8131

Table 4 Comparison of results among CFBP RBF and FFBP network

Test points First three natural frequencies (Hz) CFBP FFBP RBFrcd Location 1st mode 2nd mode 3rd mode rcd Location rcd Location rcd Location0100 2000 48209 302187 846297 0103 2329 0106 2030 0099 19640100 4000 48176 302010 845879 0125 3836 0096 3765 0102 46780100 5000 48134 301797 845399 0129 4809 0098 4952 0150 50120100 6000 48086 301549 844821 0126 5809 0096 6148 0098 61760100 8000 48031 301261 844155 0127 8115 0095 8093 0101 81310100 10000 48226 302448 847068 0150 10095 0089 9802 0099 99820100 12000 48229 302481 847051 0141 11925 0088 11725 0101 120980100 14000 48234 302503 846976 0094 13908 0082 13918 0100 139780100 15000 48236 302511 846928 0135 15248 0079 14767 0099 14908

Table 5 Material properties of aluminum-alloy 2014-1198794

Youngrsquosmodulus

119864

Density120588

Poissonrsquosratio120583

Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion

The effects of transverse cracks on the vibrating uniformcracked cantilever beam have been presented in this paperThe main purpose of this research work has been to developa proficient technique for diagnosis of crack in a vibratingstructure in short span of time The vibration analysis has

been done using theoretical and also it has been carried outthrough using finite element method as per ANSYS In thisanalysis natural frequency plays an important role in theidentification of crack Crack has been identified in terms ofcrack depth and crack location The results obtained fromANSYS are used to develop artificial intelligence techniquesusing three neural networks (FFBP RBF and CFBP) TheCFBP network shows a better result than the FFBP networkthe CFBP network gives the best validation performanceof 0018464 whereas the FFBP network gives 0041058It was observed that for some cases RBF network resultoutperforms the results of the other two networks But ingeneral CFBPwas found to bemore efficient in terms of errorand computational complexity As it was observed that the


008

009

010

011

012

013

014

015

016

1 2 3 4 5

Relative crack depth obtained from ANSYSRelative crack depth obtained from CFBP networkRelative crack depth obtained from FFBP networkRelative crack depth obtained from RBF network

Figure 18 Overall comparison of result of relative crack depthamong ANSYS CFBP FFBP and RBF networks

0123456789

1 2 3 4 5

Crack location obtained from ANSYSCrack location obtained from CFBP networkCrack location obtained from FFBP network

Figure 19 Overall comparison of result of crack location amongANSYS CFBP FFBP and RBF networks

predicted results of neural network technique are reasonablyadequate and in agreement with the theoretical result thedeveloped models can be efficiently used for crack detectionproblems

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

References

[1] J M Lifshitz and A Rotem ldquoDetermination of reinforcementunbonding of composites by a vibration techniquerdquo Journal ofComposite Materials vol 3 no 3 pp 412ndash423 1969

[2] R D Adams P Cawley C J Pye and B J Stone ldquoA vibrationtechnique for non-destructively assessing the integrity of struc-turesrdquo Journal of Mechanical Engineering Science vol 20 no 2pp 93ndash100 1978

[3] W Wang and A Zhang ldquoSensitivity analysis in fault vibrationdiagnosis of structuresrdquo in Proceedings of the 5th InternationalModal Analysis Conference pp 496ndash501 1987

[4] N Stubbs T H Broome and R Osegueda ldquoNondestructiveconstruction error detection in large space structuresrdquo AIAAjournal vol 28 no 1 pp 146ndash152 1990

[5] G Hearn and R B Testa ldquoModal analysis for damage detectionin structuresrdquo Journal of Structural Engineering vol 117 no 10pp 3042ndash3063 1991

[6] M H Richardson and M A Mannan ldquoRemote detection andlocation of structural faults using modal parametersrdquo in Pro-ceedings of the 10th International Modal Analysis Conference pp502ndash507 1992

[7] D R Sanders Y I Kim and N Stubbs ldquoNondestructive eval-uation of damage in composite structures using modal para-metersrdquo Experimental Mechanics vol 32 no 3 pp 240ndash2511992

[8] Y Narkis ldquoIdentification of crack location in vibrating simplysupported beamsrdquo Journal of Sound and Vibration vol 172 no4 pp 549ndash558 1994

[9] R Brincker P H Kirkegaard P Anderson andM E MartinezldquoDamage detection in an offshore structurerdquo in Proceedings ofthe 13th International Modal Analysis Conference pp 661ndash6671995

[10] L BalisCremaA Castellani andGCoppotelli ldquoGeneralizationof non destructive damage evaluation usingmodal parametersrdquoin Proceedings of the 13th International Modal Analysis Confer-ence pp 428ndash431 1995

[11] P S Skjaerbaek S R KNielsen andA S Cakmak ldquoAssessmentof damage in seismically excited RC-structures from a singlemeasured responserdquo in Proceedings of the 14th InternationalModal Analysis Conference pp 133ndash139 1996

[12] A AL-Qaisia and U Meneghetti ldquoCrack detection in platesby sensitivity analysisrdquo in Proceedings of the 15th InternationalModal Analysis Conference (IMAC rsquo97) pp 1831ndash1837 OrlandoFla USA February 1997

[13] I Villemure C E Ventura and R G Sexsmith ldquoImpact andambient vibration testing to assess structural damage in rein-forced concrete beamsrdquo in Proceedings of the 14th InternationalModal Analysis Conference Dearborn Michigan 1996

[14] A K Sahoo Y Zhang andM J Zuo ldquoEstimating crack size andlocation in a steel plate using ultrasonic signals andCFBPneuralnetworksrdquo in Proceedings of the IEEE Canadian Conference onElectrical and Computer Engineering (CCECE rsquo08) pp 1751ndash1754 May 2008

[15] S Suresh S N Omkar R Ganguli and V Mani ldquoIdentificationof crack location and depth in a cantilever beam using a mod-ular neural network approachrdquo Smart Materials and Structuresvol 13 no 4 pp 907ndash915 2004

[16] K M Saridakis A C Chasalevris C A Papadopoulos and AJ Dentsoras ldquoApplying neural networks genetic algorithms andfuzzy logic for the identification of cracks in shafts by usingcoupled response measurementsrdquo Computers and Structuresvol 86 no 11-12 pp 1318ndash1338 2008

[17] W A Little and G L Shaw ldquoAnalytic study of the memorystorage capacity of a neural networkrdquoMathematical Biosciencesvol 39 no 3-4 pp 281ndash290 1978


[18] S Loutridis E Douka and L J Hadjileontiadis ldquoForced vibra-tion behaviour and crack detection of cracked beams usinginstantaneous frequencyrdquo NDT and E International vol 38 no5 pp 411ndash419 2005

[19] D NThatoi H C Das and D R Parhi ldquoReview of techniquesfor fault diagnosis in damaged structure and engineeringsystemrdquo Advances in Mechanical Engineering vol 2012 ArticleID 327569 11 pages 2012

[20] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and healthmonitoring of structural andmechanical systems from changes in their vibration character-istics a literature reviewrdquo Tech Rep LA-13070-MS Los AlamosNational Laboratory Los Alamos NM USA 1996

[21] G-L Qian S-N Gu and J-S Jiang ldquoThe dynamic behaviourand crack detection of a beam with a crackrdquo Journal of Soundand Vibration vol 138 no 2 pp 233ndash243 1990

[22] J-T Kim and N Stubbs ldquoCrack detection in beam-type struc-tures using frequency datardquo Journal of Sound and Vibration vol259 no 1 pp 145ndash160 2003

[23] J-B Ihn and F-K Chang ldquoDetection and monitoring of hid-den fatigue crack growth using a built-in piezoelectric sen-soractuator network II Validation using riveted joints andrepair patchesrdquo Smart Materials and Structures vol 13 no 3pp 621ndash630 2004

[24] S P Lele and S K Maiti ldquoModelling of transverse vibrationof short beams for crack detection and measurement of crackextensionrdquo Journal of Sound and Vibration vol 257 no 3 pp559ndash583 2002

[25] S-TQuekQWang L Zhang andK-KAng ldquoSensitivity anal-ysis of crack detection in beams by wavelet techniquerdquo Interna-tional Journal of Mechanical Sciences vol 43 no 12 pp 2899ndash2910 2001

[26] A N Zagrai and V Giurgiutiu ldquoElectro-mechanical impedancemethod for crack detection in thin platesrdquo Journal of IntelligentMaterial Systems and Structures vol 12 no 10 pp 709ndash7182001

[27] P R Baviskar and V B Tungikar ldquoMultiple cracks assessmentusing natural frequency measurement and prediction of crackproperties by artificial neural networkrdquo International Journal ofAdvanced Science and Technology vol 54 pp 23ndash38 2013

[28] A Das S Sivaprasad M Tarafder S K Das and S TarafderldquoEstimation of damage in high strength steelsrdquo Applied SoftComputing vol 13 no 2 pp 1033ndash1041 2013

[29] D N Thatoi and P K Jena ldquoInverse analysis of crack in fixed-fixed structure by neural network with the aid of modal analy-sisrdquo Advances in Artificial Neural Systems vol 2013 Article ID150209 p 8 2013

[30] H Tada P C Paris andG R IrwinTheStress Analysis of CracksHand Book Del Research Corporation Hellertown Pennsylva-nian 1973

[31] M Gopikrishnan and T Santhanam ldquoEffect of different neuralnetworks on the accuracy in iris patterns recognitionrdquo Interna-tional Journal of Reviews in Computing vol 7 pp 22ndash28 2011

[32] D S Badde A K Gupta and V K Patki ldquoCascade and feedforward back propagation artificial neural network models forprediction of compressive strength of readymix concreterdquo IOSRJournal ofMechanical and Civil Engineering vol 3 pp 1ndash6 2013

International Journal of

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RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



signals to monitor the various vibrational changes andanomalies

After the successful monitoring of these changes andanomalies a conclusion was made about the various crackparameters In general if a person tries to reach such a con-clusion by following a theoretical process it would take hima longer period of time as well as decrease the effectivenessand efficiency of the obtained result Hence various methodsapart from the conventional technique are utilized to obtaina good and approximately exact result Suresh et al [15]presented a method by considering the flexural vibration ina cantilever beam with a transverse crack They computedmodal frequency parameters analytically for various cracklocations and depths After obtaining these values the param-eters so obtained were used to train the neural network toidentify the damage location and size Saridakis et al [16]proposed fuzzy logic genetic algorithm (GA) and neuralnetwork for considering the dynamic behavior of a shaft withtwo transverse cracks characterized by position depth andrelative angle Little and Shaw [17] solved exactly a linearisedversion of the model They explicitly show that the capacityof the memory is related to the number of synapses and notthe number of neurons Amethod was proposed by Loutridiset al [18] ldquoforced vibration behavior and crack detectionof cracked beams using instantaneous frequencyrdquo to detectthe crack in beam depending on instantaneous frequency(IF) The study monitored the dynamic behavior for thebeam with breathing crack They carried out the researchunder harmonic excitation with experimental and theoreticalresultsThe observed data of the simulation and experimentaltest were analyzed using MATLAB A relation between thedepth of crack and the main difference of instantaneousfrequency was established The instantaneous frequency wasfound to be a good indicator for the size of crack

Thatoi et al [19] have reviewed the variousmethodologiesused for crack detection and fault diagnosis Doebling et al[20] have reviewed different techniques of detection locationand characterization of structural damage and faults Theanalysis included changes in modal frequency changes inmode shapes and changes in flexibility coefficients

Qian et al [21] used a finite element model to predict thebehavior of a beam with an edge crack by taking an elementstiffnessmatrix which is derived from an integration of stressintensity factor Kim and Stubbs [22] located the position andestimated the size of a crack in beam type structures by takingthe changes in natural frequencies which is formulated byrelating fractional changes in modal energy due to the pres-ence of crack Ihn and Chang [23] have proposed a methodfor detecting and monitoring hidden fatigue crack growthThey have used diagnostic signals generated from nearbypiezoelectric actuators built into the structure to detect thecrack growth A complete built-in diagnostic system for thetests was developed including a sensor network hardwareand the diagnostic software Lele and Maiti [24] have con-sidered the effect of shear deformation and rotational inertiathrough the Timoshenko beam theory for detection andmeasurement of crack extension of short beams Quek et al[25] have inspected the sensitivity of wavelet technique in thedetection of cracks in beam-like structure In this aspect they

X

L

O

Y

Z

T

W

b1P2

P1

Lc

Figure 1 Beam model

have compared the results of two wavelets such as Haar andGabor wavelets Zagrai and Giurgiutiu [26] have describedthe utilization of electromechanical impedance method forcrack detection in thin plates This method allows the directidentification of structural dynamics by obtaining its electro-mechanical impedance

Crack detection in the beam-like structures having singleas well as multicracks was modeled using artificial neuralnetwork (ANN) and the various characteristics were studied[27ndash29]

In the present study a methodical approach to analyzeand detect the crack parameters that is the crack locationand relative crack depth from a set of frequency valuesobtained from a simulated model of the cantilever beamcontaining a thin transverse crack using ANSYS Using thedata cascade forward back propagation (CFBP) feed forwardback propagation (FFBP) and radial basis function (RBF)neural network models are developed and finally result thusobtained are compared

2 Mathematical Formulation

21 Computation of Flexibility Matrix of a Damaged BeamSubjected to Complex Loading A beam with cracks hassmaller stiffness than a normal beam This decreased localstiffness can be formulated as a matrix The dimension ofthe matrix would depend on the degrees of freedom inthe problem Figure 1 shows a cantilever beam of width 119882and height 119879 having a transverse surface crack of depth 119887

1

The beam experiences combined longitudinal and transversemotion due to the axial force 119875

1and bending moment 119875

2

Here two degrees of freedom are considered leading to a 2times2local stiffness matrix

The relationship between strain energy release rate 119869(119887)and stress intensity factors (119866

119868119894) at the crack section is given

by Tada et al [30] as

119869 (119887) =1

1198641015840(1198661198681+ 1198661198682)2

(1)

where 1198641015840 = 119864(1minus]2) for plane strain condition and 1198641015840 = 119864for plane stress119866

1198681= stress intensity factor for openingmode

119868 due to load 1198751 and 119866

1198682= stress intensity factor for opening

mode 119868 due to load 1198752



1198661198681=

1198751

119882119879

radic120587119887(1198651(119887

119879))

1198661198682=61198752

1198821198792radic120587119887(119865

2(119887

119879))

(2)




1198651(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times (0752 + 202 (119887

119879)

+ 037(1 minus sin(1205871198872119879))

3

)

times (cos(1205871198872119879))

minus1

1198652(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times 0923 + 0199(1 minus sin (1205871198872119879))4

cos (1205871198872119879)

(3)


119869 (119887) =120597119880119905

120597 (119887 times119882) (4)


119889119880119905= 119869 (119887) 119889 (119887 times119882) = 119882119869 (119887) 119889119887 (5)


119880119905= 119882int

1198871

0

119869 (119887) 119889119887 (6)


1



119878119894=120597119880119905

120597119875119894

(7)

Using (1) and (6)

119878119894=

120597

120597119875119894

[119882int

1198871

0

119869 (119887) 119889119887] = 119882120597

120597119875119894

[int

1198871

0

119869 (119887) 119889119887] (8)

119862119894119895=120597119878119894

120597119875119895

=120597

120597119875119895

[119882120597

120597119875119894

int

1198871

0

119869 (119887) 119889119887]

= 1198821205972

120597119875119894120597119875119895

int

1198871

0

119869 (119887) 119889119887

(9)



119862119894119895=119882

1198641015840

1205972

120597119875119894120597119875119895

int

1198871

0

(1198661198971+ 1198661198972)2

119889119887 (10)


1 120575 = 119887

1119879 = 120575

1


119862119894119895=119882119879

1198641015840

1205972

120597119875119894120597119875119895

int

1205751

0

(1198661198971+ 1198661198972)2

119889120575 (11)


119894119895




11 11986212(=11986221) and119862

22are obtained

as follows

11986211=119882119879

1198641015840int

1205751

0

120587119887

1198822119879221198651

2(120575) 119889120575

=2120587

1198821198641015840int

1205751

0

1205751198651

2(120575) 119889120575

11986212= 11986221=

12120587

1198641015840119879119882int

1205751

0

1205751198651(120575) 1198652(120575) 119889120575

11986222=

72120587

11986410158401198821198792int

1205751

0

1205751198652

2(120575) 119889120575

(12)


119870 = [11987011

11987012

11987021

11987022

] = [11986211

11986212

11986221

11986222

]

minus1

(13)


11986211= 11986211

1198821198641015840

2120587 119862

12= 11986212

1198641015840119879119882

12120587= 11986221

11986222= 11986222

11986410158401198821198792

72120587

(14)




2(119909 119905) as the


1(119909 119905) 119881

2(119909 119905) are the



L

S1 S2

V1 V2

Lc



12059721198781205971199052= (119864120588)120597


and119864119868(12059741198811205971199094) minus 120588120596


beam


1198781(119909) = 119861

1cos (119867

119904119909) + 119861

2sin (119867

119904119909) (15a)

1198782(119909) = 119861

3cos (119867

119904119909) + 119861

4sin (119867

119904119909) (15b)

1198811(119909) = 119861

5cosh (119867V119909) + 1198616 sinh (119867V119909)

+ 1198617cos (119867V119909) + 1198618 sin (119867V119909)

(15c)

1198812(119909) = 119861

9cosh (119867V119909) + 11986110 sinh (119867V119909)

+ 11986111cos (119867V119909) + 11986112 sin (119867V119909)

(15d)

where 119909 = 119909119871 119878 = 119878119871 119881 = 119881119871 120572 = 119871119888119871

119867119904=120596119871

119863119904

119863119904= (

119864

120588)

12

119867119881= (

1205961198712

119863V)

12

119863V = (119864119868

120583)

12

120583 = 119860120588

(16)



1198781(0) = 0 119881

1(0) = 0 119881

1015840

1(0) = 0

1198781015840

2(1) = 0 119881

10158401015840

2(1) = 0 119881

101584010158401015840

2(1) = 0

(17)


1198781(120572) = 119878

2(120572) 119881

1(120572) = 119881

2(120572)

11988110158401015840

1(120572) = 119881

10158401015840

2(120572) 119881

101584010158401015840

1(120572) = 119881

101584010158401015840

2(120572)

(18)


1198601198641198891198781(119871119888)

119889119909= 11987011(1198782(119871119888) minus 1198781(119871119888))

+ 11987012(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(19)


119873111987321198781015840

1(120572) = 119873

2(1198782(120572) minus 119878

1(120572)) + 119873

1(1198811015840

2(120572) minus 119881

1015840

1(120572))

(20)

Similarly

11986411986811988921198811(119871119888)

1198891199092= 11987021(1198782(119871119888) minus 1198781(119871119888))

+ 11987022(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(21)


1198733119873411988110158401015840

1(120572) = 119873

3(1198782(120572) minus 119878

1(120572))

+ 1198734(1198811015840

2(120572) minus 119881

1015840

1(120572))

(22)

where 1198731= 119860119864119871119870

11 1198732= 119860119864119870

12 1198733= 119864119868119871119870

22 1198734=

119864119868119871211987021


|119876| = 0 (23)


1=

(1198871119879)



119876 =

[[[[[[[[[[[[[[[[[[

[

1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1198663

1198664minus1198667minus1198668

0 0 0 0

0 0 0 0 1198664

1198663

1198668minus1198667

0 0 0 0


1198666

0 0 0 0

11986621198661


0 0 0 0

11986611198662

1198665


0 0 0 0

11987811198782

1198783

1198784


1198785

1198786

1198787

1198788

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 minus11987981198797

0 0 0 0 0 0 0 0 minus11987961198795

1198796minus1198795

119878911987810

11987811

11987812

11987813

11987814

11987815

11987816

11987817

11987818


]]]]]]]]]]]]]]]]]]

]

(24)

where

1198781= 1198662+ 1198733119867V1198661 119878

2= 1198661+ 1198733119867V1198662

1198783= minus1198666minus 1198733119867V1198665 119878

4= 1198665minus 1198733119867V1198666

1198785=11987334

119867V1198795 119878

6=11987334

119867V1198796 119878

7=minus11987334

119867V1198795

1198788=minus11987334

119867V1198796 119878

9= 11987312119867V1198662

11987810= 11987312119867V1198661 119878

11= minus11987312119867V1198666

11987812= 11987312119867V1198665 119878

13= minus11987312119867V1198662

11987814= minus11987312119867V1198661 119878

15= 11987312119867V1198666

11987816= minus11987312119867V1198665 119878

17= 1198795minus 11987311198671199041198796

11987818= 1198796+ 11987311198671199041198795

1198661= Cosh (119867V120572) 119866

2= Sinh (119867V120572)

1198663= Cosh (119867V) 119866

4= Sinh (119867V)

1198665= Cos (119867V120572) 119866

6= Sin (119867V120572)

1198667= Cosh (119867V) 119866

8= Sin (119867V)

1198795= Cos (119867

119904120572)

1198796= Sin (119867

119904120572) 119879

7= Cos (119867

119904) 119879

8= Sin (119867

119904)

11987312=1198731

1198732

11987334=1198733

1198734

(25)












1and 119883



119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909


1



119910 = 119891 (119910in) (27)





Hidden layer


Out

put

Inpu

t









results







Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











1198661198681=

1198751

119882119879

radic120587119887(1198651(119887

119879))

1198661198682=61198752

1198821198792radic120587119887(119865

2(119887

119879))

(2)




1198651(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times (0752 + 202 (119887

119879)

+ 037(1 minus sin(1205871198872119879))

3

)

times (cos(1205871198872119879))

minus1

1198652(119887

119879) = (

2119879

120587119887tan(120587119887

2119879))

05

times 0923 + 0199(1 minus sin (1205871198872119879))4

cos (1205871198872119879)

(3)


119869 (119887) =120597119880119905

120597 (119887 times119882) (4)


119889119880119905= 119869 (119887) 119889 (119887 times119882) = 119882119869 (119887) 119889119887 (5)


119880119905= 119882int

1198871

0

119869 (119887) 119889119887 (6)


1



119878119894=120597119880119905

120597119875119894

(7)

Using (1) and (6)

119878119894=

120597

120597119875119894

[119882int

1198871

0

119869 (119887) 119889119887] = 119882120597

120597119875119894

[int

1198871

0

119869 (119887) 119889119887] (8)

119862119894119895=120597119878119894

120597119875119895

=120597

120597119875119895

[119882120597

120597119875119894

int

1198871

0

119869 (119887) 119889119887]

= 1198821205972

120597119875119894120597119875119895

int

1198871

0

119869 (119887) 119889119887

(9)



119862119894119895=119882

1198641015840

1205972

120597119875119894120597119875119895

int

1198871

0

(1198661198971+ 1198661198972)2

119889119887 (10)


1 120575 = 119887

1119879 = 120575

1


119862119894119895=119882119879

1198641015840

1205972

120597119875119894120597119875119895

int

1205751

0

(1198661198971+ 1198661198972)2

119889120575 (11)


119894119895




11 11986212(=11986221) and119862

22are obtained

as follows

11986211=119882119879

1198641015840int

1205751

0

120587119887

1198822119879221198651

2(120575) 119889120575

=2120587

1198821198641015840int

1205751

0

1205751198651

2(120575) 119889120575

11986212= 11986221=

12120587

1198641015840119879119882int

1205751

0

1205751198651(120575) 1198652(120575) 119889120575

11986222=

72120587

11986410158401198821198792int

1205751

0

1205751198652

2(120575) 119889120575

(12)


119870 = [11987011

11987012

11987021

11987022

] = [11986211

11986212

11986221

11986222

]

minus1

(13)


11986211= 11986211

1198821198641015840

2120587 119862

12= 11986212

1198641015840119879119882

12120587= 11986221

11986222= 11986222

11986410158401198821198792

72120587

(14)




2(119909 119905) as the


1(119909 119905) 119881

2(119909 119905) are the



L

S1 S2

V1 V2

Lc



12059721198781205971199052= (119864120588)120597


and119864119868(12059741198811205971199094) minus 120588120596


beam


1198781(119909) = 119861

1cos (119867

119904119909) + 119861

2sin (119867

119904119909) (15a)

1198782(119909) = 119861

3cos (119867

119904119909) + 119861

4sin (119867

119904119909) (15b)

1198811(119909) = 119861

5cosh (119867V119909) + 1198616 sinh (119867V119909)

+ 1198617cos (119867V119909) + 1198618 sin (119867V119909)

(15c)

1198812(119909) = 119861

9cosh (119867V119909) + 11986110 sinh (119867V119909)

+ 11986111cos (119867V119909) + 11986112 sin (119867V119909)

(15d)

where 119909 = 119909119871 119878 = 119878119871 119881 = 119881119871 120572 = 119871119888119871

119867119904=120596119871

119863119904

119863119904= (

119864

120588)

12

119867119881= (

1205961198712

119863V)

12

119863V = (119864119868

120583)

12

120583 = 119860120588

(16)



1198781(0) = 0 119881

1(0) = 0 119881

1015840

1(0) = 0

1198781015840

2(1) = 0 119881

10158401015840

2(1) = 0 119881

101584010158401015840

2(1) = 0

(17)


1198781(120572) = 119878

2(120572) 119881

1(120572) = 119881

2(120572)

11988110158401015840

1(120572) = 119881

10158401015840

2(120572) 119881

101584010158401015840

1(120572) = 119881

101584010158401015840

2(120572)

(18)


1198601198641198891198781(119871119888)

119889119909= 11987011(1198782(119871119888) minus 1198781(119871119888))

+ 11987012(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(19)


119873111987321198781015840

1(120572) = 119873

2(1198782(120572) minus 119878

1(120572)) + 119873

1(1198811015840

2(120572) minus 119881

1015840

1(120572))

(20)

Similarly

11986411986811988921198811(119871119888)

1198891199092= 11987021(1198782(119871119888) minus 1198781(119871119888))

+ 11987022(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(21)


1198733119873411988110158401015840

1(120572) = 119873

3(1198782(120572) minus 119878

1(120572))

+ 1198734(1198811015840

2(120572) minus 119881

1015840

1(120572))

(22)

where 1198731= 119860119864119871119870

11 1198732= 119860119864119870

12 1198733= 119864119868119871119870

22 1198734=

119864119868119871211987021


|119876| = 0 (23)


1=

(1198871119879)



119876 =

[[[[[[[[[[[[[[[[[[

[

1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1198663

1198664minus1198667minus1198668

0 0 0 0

0 0 0 0 1198664

1198663

1198668minus1198667

0 0 0 0


1198666

0 0 0 0

11986621198661


0 0 0 0

11986611198662

1198665


0 0 0 0

11987811198782

1198783

1198784


1198785

1198786

1198787

1198788

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 minus11987981198797

0 0 0 0 0 0 0 0 minus11987961198795

1198796minus1198795

119878911987810

11987811

11987812

11987813

11987814

11987815

11987816

11987817

11987818


]]]]]]]]]]]]]]]]]]

]

(24)

where

1198781= 1198662+ 1198733119867V1198661 119878

2= 1198661+ 1198733119867V1198662

1198783= minus1198666minus 1198733119867V1198665 119878

4= 1198665minus 1198733119867V1198666

1198785=11987334

119867V1198795 119878

6=11987334

119867V1198796 119878

7=minus11987334

119867V1198795

1198788=minus11987334

119867V1198796 119878

9= 11987312119867V1198662

11987810= 11987312119867V1198661 119878

11= minus11987312119867V1198666

11987812= 11987312119867V1198665 119878

13= minus11987312119867V1198662

11987814= minus11987312119867V1198661 119878

15= 11987312119867V1198666

11987816= minus11987312119867V1198665 119878

17= 1198795minus 11987311198671199041198796

11987818= 1198796+ 11987311198671199041198795

1198661= Cosh (119867V120572) 119866

2= Sinh (119867V120572)

1198663= Cosh (119867V) 119866

4= Sinh (119867V)

1198665= Cos (119867V120572) 119866

6= Sin (119867V120572)

1198667= Cosh (119867V) 119866

8= Sin (119867V)

1198795= Cos (119867

119904120572)

1198796= Sin (119867

119904120572) 119879

7= Cos (119867

119904) 119879

8= Sin (119867

119904)

11987312=1198731

1198732

11987334=1198733

1198734

(25)












1and 119883



119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909


1



119910 = 119891 (119910in) (27)





Hidden layer


Out

put

Inpu

t









results







Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










L

S1 S2

V1 V2

Lc



12059721198781205971199052= (119864120588)120597


and119864119868(12059741198811205971199094) minus 120588120596


beam


1198781(119909) = 119861

1cos (119867

119904119909) + 119861

2sin (119867

119904119909) (15a)

1198782(119909) = 119861

3cos (119867

119904119909) + 119861

4sin (119867

119904119909) (15b)

1198811(119909) = 119861

5cosh (119867V119909) + 1198616 sinh (119867V119909)

+ 1198617cos (119867V119909) + 1198618 sin (119867V119909)

(15c)

1198812(119909) = 119861

9cosh (119867V119909) + 11986110 sinh (119867V119909)

+ 11986111cos (119867V119909) + 11986112 sin (119867V119909)

(15d)

where 119909 = 119909119871 119878 = 119878119871 119881 = 119881119871 120572 = 119871119888119871

119867119904=120596119871

119863119904

119863119904= (

119864

120588)

12

119867119881= (

1205961198712

119863V)

12

119863V = (119864119868

120583)

12

120583 = 119860120588

(16)



1198781(0) = 0 119881

1(0) = 0 119881

1015840

1(0) = 0

1198781015840

2(1) = 0 119881

10158401015840

2(1) = 0 119881

101584010158401015840

2(1) = 0

(17)


1198781(120572) = 119878

2(120572) 119881

1(120572) = 119881

2(120572)

11988110158401015840

1(120572) = 119881

10158401015840

2(120572) 119881

101584010158401015840

1(120572) = 119881

101584010158401015840

2(120572)

(18)


1198601198641198891198781(119871119888)

119889119909= 11987011(1198782(119871119888) minus 1198781(119871119888))

+ 11987012(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(19)


119873111987321198781015840

1(120572) = 119873

2(1198782(120572) minus 119878

1(120572)) + 119873

1(1198811015840

2(120572) minus 119881

1015840

1(120572))

(20)

Similarly

11986411986811988921198811(119871119888)

1198891199092= 11987021(1198782(119871119888) minus 1198781(119871119888))

+ 11987022(1198891198812(119871119888)

119889119909minus1198891198811(119871119888)

119889119909)

(21)


1198733119873411988110158401015840

1(120572) = 119873

3(1198782(120572) minus 119878

1(120572))

+ 1198734(1198811015840

2(120572) minus 119881

1015840

1(120572))

(22)

where 1198731= 119860119864119871119870

11 1198732= 119860119864119870

12 1198733= 119864119868119871119870

22 1198734=

119864119868119871211987021


|119876| = 0 (23)


1=

(1198871119879)



119876 =

[[[[[[[[[[[[[[[[[[

[

1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1198663

1198664minus1198667minus1198668

0 0 0 0

0 0 0 0 1198664

1198663

1198668minus1198667

0 0 0 0


1198666

0 0 0 0

11986621198661


0 0 0 0

11986611198662

1198665


0 0 0 0

11987811198782

1198783

1198784


1198785

1198786

1198787

1198788

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 minus11987981198797

0 0 0 0 0 0 0 0 minus11987961198795

1198796minus1198795

119878911987810

11987811

11987812

11987813

11987814

11987815

11987816

11987817

11987818


]]]]]]]]]]]]]]]]]]

]

(24)

where

1198781= 1198662+ 1198733119867V1198661 119878

2= 1198661+ 1198733119867V1198662

1198783= minus1198666minus 1198733119867V1198665 119878

4= 1198665minus 1198733119867V1198666

1198785=11987334

119867V1198795 119878

6=11987334

119867V1198796 119878

7=minus11987334

119867V1198795

1198788=minus11987334

119867V1198796 119878

9= 11987312119867V1198662

11987810= 11987312119867V1198661 119878

11= minus11987312119867V1198666

11987812= 11987312119867V1198665 119878

13= minus11987312119867V1198662

11987814= minus11987312119867V1198661 119878

15= 11987312119867V1198666

11987816= minus11987312119867V1198665 119878

17= 1198795minus 11987311198671199041198796

11987818= 1198796+ 11987311198671199041198795

1198661= Cosh (119867V120572) 119866

2= Sinh (119867V120572)

1198663= Cosh (119867V) 119866

4= Sinh (119867V)

1198665= Cos (119867V120572) 119866

6= Sin (119867V120572)

1198667= Cosh (119867V) 119866

8= Sin (119867V)

1198795= Cos (119867

119904120572)

1198796= Sin (119867

119904120572) 119879

7= Cos (119867

119904) 119879

8= Sin (119867

119904)

11987312=1198731

1198732

11987334=1198733

1198734

(25)












1and 119883



119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909


1



119910 = 119891 (119910in) (27)





Hidden layer


Out

put

Inpu

t









results







Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119876 =

[[[[[[[[[[[[[[[[[[

[

1 0 1 0 0 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0 0

0 0 0 0 1198663

1198664minus1198667minus1198668

0 0 0 0

0 0 0 0 1198664

1198663

1198668minus1198667

0 0 0 0


1198666

0 0 0 0

11986621198661


0 0 0 0

11986611198662

1198665


0 0 0 0

11987811198782

1198783

1198784


1198785

1198786

1198787

1198788

0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 minus11987981198797

0 0 0 0 0 0 0 0 minus11987961198795

1198796minus1198795

119878911987810

11987811

11987812

11987813

11987814

11987815

11987816

11987817

11987818


]]]]]]]]]]]]]]]]]]

]

(24)

where

1198781= 1198662+ 1198733119867V1198661 119878

2= 1198661+ 1198733119867V1198662

1198783= minus1198666minus 1198733119867V1198665 119878

4= 1198665minus 1198733119867V1198666

1198785=11987334

119867V1198795 119878

6=11987334

119867V1198796 119878

7=minus11987334

119867V1198795

1198788=minus11987334

119867V1198796 119878

9= 11987312119867V1198662

11987810= 11987312119867V1198661 119878

11= minus11987312119867V1198666

11987812= 11987312119867V1198665 119878

13= minus11987312119867V1198662

11987814= minus11987312119867V1198661 119878

15= 11987312119867V1198666

11987816= minus11987312119867V1198665 119878

17= 1198795minus 11987311198671199041198796

11987818= 1198796+ 11987311198671199041198795

1198661= Cosh (119867V120572) 119866

2= Sinh (119867V120572)

1198663= Cosh (119867V) 119866

4= Sinh (119867V)

1198665= Cos (119867V120572) 119866

6= Sin (119867V120572)

1198667= Cosh (119867V) 119866

8= Sin (119867V)

1198795= Cos (119867

119904120572)

1198796= Sin (119867

119904120572) 119879

7= Cos (119867

119904) 119879

8= Sin (119867

119904)

11987312=1198731

1198732

11987334=1198733

1198734

(25)












1and 119883



119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909


1



119910 = 119891 (119910in) (27)





Hidden layer


Out

put

Inpu

t









results







Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1and 119883



119884in = 11990911199081 + 11990921199082 (26)

where 1199091and 119909


1



119910 = 119891 (119910in) (27)





Hidden layer


Out

put

Inpu

t









results







Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Out

put

Inpu

t







approximate result




tansig (119899) = 2

1 + 119890minus2119899minus 1 (28)

Input layer

Hidden layer

Output layer

Out

put

Inpu

t

H1

H2

H3

Hn


0 200 400 600 800 1000


Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

102

101

100

10minus1

10minus2

10minus3

1000 epochs









5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5

10

15

20

25

5 10 15 20 25

Target

Training R = 099999

FitData

Y = T

Out

put≃

1lowast

targ

et+000021

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0011

(b)

Test R = 099977

5

10

15

20

25

5 10 15 20 25

Target

FitData

Y = T

Out

put≃

1lowast

targ

et+minus000074

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099993

FitData

Y = T

Out

put≃

1lowast

targ

et+00021


010

2030


1 2 3 4 5 6 7 8 9 10 11 12 13 14


000020040060

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0 200 400 600 800 1000

Mea

n sq

uare

d er

ror (

mse

)

TrainValidation

TestBest

103

102

101

100

10minus1

10minus2

10minus3


1000 epochs100 300 500 700 900





5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










5

10

15

20

25

5 10 15 20 25

Target

Training R = 099998

FitData

Y = T

Out

put≃

1lowast

targ

et+000031

(a)

5

10

15

20

25

5 10 15 20 25

Target


FitData

Y = T

Out

put≃

1lowast

targ

et+0027

(b)

5

10

15

20

25

5 10 15 20 25

Target

Test R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+minus00011

(c)

5

10

15

20

25

5 10 15 20 25

Target

All R = 099992

FitData

Y = T

Out

put≃

1lowast

targ

et+00053

(d)












000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










000010020030040050

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16


1 2 3 4 5 6 7 8 9 10 11 12 13 14


0

TrainValidationTest

102

101

100

10minus1

10minus2

10minus3

300 epochs50 100 150 200 250 300

Perfo

rman

ce





000020040060


1 2 3 4 5 6 7 8 9 10 11 12 13


0102030


1 2 3 4 5 6 7 8 9 10 11 12 13

















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















Youngrsquosmodulus

119864

Density120588


Length119871

Diameter119863

724GPa 28 gmcc 033 800mm 150mm

6 Conclusion




008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










008

009

010

011

012

013

014

015

016

1 2 3 4 5



0123456789

1 2 3 4 5






References




































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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