## Abstract

Steam turbine blades are the important components in power system shaft lines subjected to severe temperatures, leading to low/high cycle fatigue failures. The transient conditions occurring during startup and shutdown events generate alternative stresses causing the fracture at the blade roots. The present work deals with the effect of localized damage on the vibration characteristics and damage identification study in the last stage low-pressure (LP) steam turbine blade. Initially, free vibration studies and transient analysis of the last row LP blade section are conducted using the finite element model. A crack near the root region is modeled by a torsional spring, whose stiffness is expressed in terms of crack depth ratio. Effects of crack depth ratio and location near the roots on the natural frequencies and transient response amplitudes are studied in detail. The relationship between the damage parameters and blade frequencies is established through the backpropagation neural network model.

## 1 Introduction

Steam turbine blades are important components of the steam turbine as they are exposed to severe temperatures and pressures that fluctuate over time and develop fluctuating stress. These forces vary during the operation based on the steam conditions and the operating speed and hence subjected to repeated loading resulting in low cycle fatigue (LCF). Mainly during startup and shutdown, steam turbine blades are subjected to LCF, causing alternating stresses and resulting in fracture. Cracks are developed in the root part of the blade due to fatigue. The crack length varies with the operating speed, and it can result in catastrophic failure if not investigated at the early stages of crack generation. During the normal operation of the steam turbine, periodic variation occurs in the steam forces at different operating speed frequencies resulting in the vibration of the rotor disk. The stator-rotor interactions are quite small. A thumb rule to identify rotor blade damage is that the impulsive load caused by a damaged rotor blade on the stator blades results in an increase in shaft speed harmonics. The most common source of excitation is due to pressure disturbances, non-uniform flow, and mistuning effects. As the amplitude of vibration increases at the resonance, often it is recommended to operate in sub-resonance regions to minimize further damage propagation.

The modeling of rotating blades has a long history. In the case of modal analysis, pioneering works of earlier researchers during 1960–1980 have given quite interesting modeling tools for large steam turbine blades [15]. Bhagi et al. [6] used finite element method (FEM) techniques to do a modal analysis followed by a dynamic stress study of the steam turbine blade. Roastard and Lester [7] used Galerkin FEM to calculate the free vibration frequencies of twisted cantilever beams. The effect of several parameters on the vibration behavior of rotating blades was investigated, including tapered angle [8,9], blade tip mass [10], and various combined effects [1116]. In the case of the kineto-static analysis two- and three-dimensional, frictionless or frictional, models of the rotating blades and their roots were developed [1721]. These works give some hints on how to build geometrical and physical models of the rotating blades properly, also in the case of the modal analysis.

Fatigue crack growth and initiation for different types of loading in steam turbine blades were studied [2935]. Due to inherent character and crucial stressed zone, the foregoing analysis offers the basis for the excitation pattern of broken blades. The findings of the studies are used to create standards and checks for blade fitting in rotor assemblies, as well as health audits, overhauls, overspeed balancing tests, and frequency turning. The use of neural networks [36] and optimization tools [37,38] are found in the literature for the prediction of fatigue life and in blade root design applications.

From the aforementioned literature, it can be deduced that a lot of work has been done concerning vibration analysis and traditional damage identification in the steam turbine blades; however, the study of crack parameters of the blade using non-destructive computation methods via artificial intelligence has not been carried out to a great extent. In this paper, FEM has been used to evaluate the vibration behavior of normal and cracked steam turbine blades. The crack is modeled as an element stiffness loss factor given by an empirical formula with crack depth and crack location as governing parameters. For different values of crack depth and location from the fixed end of the blade, the natural frequencies are recorded. Furthermore, a machine learning methodology based on a multi-layer perceptron neural network model is developed to map the relationship between inputs and outputs to predict the crack parameters. The main novelty of the present research consists of the application of the existing artificial neural network (ANN) model to determine the inverse relation between the blade frequencies and the damage parameters. The following is a breakdown structure of this paper: The technique used in this study, as well as mathematical modeling and finite element solution, are all covered in Sec. 2. The results and discussion are described in Sec. 3, and Sec. 4 gives the conclusions of the works.

### 2.1 Mathematical Model.

The airfoil blade under consideration is modeled using coupled bending-twisting beam elements. The equations of motion are obtained in terms of bending degrees-of-freedom (DOF) (ux, uy, ux′, uy′) and torsional angular deflection (Ψ). Two bending deflections come from the flapwise and chordwise directions while coupling with torsion occurs due to asymmetric airfoil blade section. The dynamic equations of motion are derived from the expressions of kinetic energy and potential energy equations for a thin pre-twisted airfoil beam [5]. As the analysis of a three-dimensional model requires enormous computational resources and consumes a large time, it is planned to employ the proposed bending-twisting coupled beam element model in the finite element analysis. The assumptions made in modeling the blade are as follows:

1. The blade’s material is assumed to be homogenous and isotropic.

2. The impact of rotary inertia and shear deformation has been neglected due to the slender shape of the blade.

Figure 1 depicts the low-pressure last stage steam turbine blade configuration.

Fig. 1
Fig. 1
Close modal
Kinetic energy
$T=12∫0lρ[A(u˙x2+u˙y2)+ITψ˙2]dz+12∫0lρ(Iyu˙x′2+Ixu˙y′2)dz+∫0lρIxyu˙x′u˙y′dz$
(1)
Potential energy
$V=12∫0lE(Ixuy″2+Iyux″2+2Ixyux″uy″)dz+12∫0l(GITψ′2−ρAΩ2uy2)dz+12∫0lμΩ2(ux′2+uy′2)dz−12∫0lρΩ2(Iy−Ix)ψ2dz$
(2)
where,
$μ=∫ile+zLρA(Rh+ile+z)dzfor(i+1)thelement$
(3)
$Ix=IYsin2(φ)+IXcos2(φ)$
(4)
$Iy=IYcos2(φ)+IXsin2(φ)$
(5)
$Ixy=12(IY−IX)sin(2φ)$
(6)
Here, IT is the polar moment of inertia, φ is the twist angle, Rh is the hub radius, A represents the blade area, L is the total length of the blade, le is the length of each blade element Ω is the speed of the blade in rad/s. Also, E, G, and ρ are elastic, shear modulus, and density respectively of blade material. IX, IY, IXY are the second moments of area about x, y axes at the root of the blade, and the product of inertia at a section with respect to the origin as the centroid of the cross section. The over dot denotes differentiation with respect to time and the tick (prime sign) denotes differentiation with respect to z, respectively. The dynamic model is established with finite element analysis.

### 2.2 Free Vibration Analysis.

By considering the bending deflections ux and uy as cubic polynomials and torsion ψ as a linear polynomial and substituting in energy expressions, the element stiffness and mass matrices are obtained [39,40]. A consistent mass approach [40] has been considered for developing the element mass matrix in the current work. As shown in Fig. 2, each beam element has ten degrees-of-freedom.

Fig. 2
Fig. 2
Close modal
The vector of element nodal degrees-of-freedom (DOFs)
$[d]eT=[d1d2d3…d10]$
(7)

The global mass and stiffness matrices are generated by assembling the element matrices and imposing the boundary constraints.

Resulting free vibration equation becomes
$([K]−ω2[M]){D}=0$
(8)
where the global stiffness matrix is given as [K] = [Ks] + [Kg] and [M] is the global mass matrix. [Ks] and [Kg] are structural stiffness and geometric stiffness matrices, {D} is the global displacement vector, and ɷ represents the natural circular frequency of the blade in rad/s. As the blade rotates, it is subjected to centrifugal force, which is a tensile force and acts along the length of the blade. When the tensile axial force is present in the member, the bending stiffness of the member increases, and this results in the variation of the stiffness matrix. The stiffness matrix that includes this kind of stress-stiffening effect is called a geometric stiffness matrix. The modal analysis of the blade can be computed by solving Eq. (8) numerically.

### 2.3 Forced Vibration Analysis.

A blade undergoes various types of excitations such as pressure fluctuations, change in velocity, and angle of steam flow, causing periodic fluctuations of force. This may be due to the change in the direction of steam flow, especially from axial to radial at the exit from moving blades. Also, cyclic effects may be present due to manufacturing constraints or structural features such as leakage through diaphragm discs and gaps in stationary blade shrouds. Such a source gives rise to excitation frequencies at the rotational frequency or lower harmonics of that frequency. Finally, aerodynamic fluctuations result in nozzle wake excitation encountered by the rotating blade as it passes each stationary blade. Such excitation is encountered by the rotating blades as excitation at the nozzle passing frequency and its multiples. The dynamic equation of motion of forced vibrations on a single blade is given as
$[M]{D¨}+[C]{D˙}+[K]{D}={F(t)}$
(9)
where [C] is the global damping matrix which can be obtained using the Caughey damping series in terms of products of the flexibility matrix [41]. The force vector {F(t)} in the present work includes mainly a combination of steady-state forces Fb and nozzle wake excitation forces Fi, which are unsteady forces, and the combined effect of other forces Fc. This force is considered as sinusoidal pulse acting across the pitch of one nozzle during the rotation of a rotor blade for a time period of t0. This pulse can be considered as a combination of dual sinewaves. This is because of two types of time responses, of which one occurs during time t < t0 and the second during time t > t0 as given in Fig. 3.
Fig. 3
Fig. 3
Close modal
The excitation force acting on the ith blade is written as
$Fi=FT+Fmsin(ωt+∅i),i=1,2,…,Nb$
(10)
Fi is considered to be consists of two components namely transient and harmonic. The transient force (FT) will occur only for some instance of time and is assumed to be a rectangular pulse in this analysis. The value of Fm was assumed to be 1000 N. The combined effect of two force components on the blade can be noticed from the time domain response of the blade.
The phase angle of the excitation force is
$∅i=2π(m)(i−1)Nb,i=1,2,…..Nb$
(11)
where m is the engine order of the excitation, Nb is the total number of blades, and ɷ is the forcing frequency. A pulsating pressure field is produced when the working fluid interacts with the rotating turbine blades during operation. When this pressure field is circumferentially expanded, a harmonic series whose coefficients are referred to as engine orders is produced. An engine order essentially refers to the number of sine waves traveling along the circumference of the rotor. The dynamic equation of motion Eq. (9) of the blade was solved numerically using Houbolt’s time integration method [42], and the time and frequency domain responses of the blade were obtained numerically.

### 2.4 Crack Analysis.

Steam turbine blades are subjected to high temperatures and pressures that fluctuate over time and produce alternative stresses. The last stage steam turbine has huge, mounted blades that are positioned around the rotor disk’s edge. Blade dynamics can be considered similar to that of the cantilever beam with a fixed end as a critical region. The cracks originate in the fir-tree root of the blade at the trailing edge side of the blade root. It was observed that the crack propagates from the corner to adjacent faces. These blades are fractured mostly due to the crack formation in the root of the blade as shown in Fig. 4 due to variable loads experienced during starting and shutdown.

Fig. 4
Fig. 4
Close modal

The crack length varies with the operating speed, and it can result in catastrophic failure if not detected at the early stages of crack generation. As shown in Fig. 5, the crack model can be represented as two unbroken beams connected by a torsional spring of stiffness K. This method of modeling the crack as a massless rotational spring is known as the rotational spring model method. This method assumes that in the case of transverse vibration of beams, there is an extra angular rotation at the crack location proportional to the bending moment at the section. The effects of discontinuities in axial displacement and transverse displacement are considered to be negligible compared with that of the discontinuity in bending slope. The method separates the beam into two segments having different deflection patterns. The infinite stiffness value of the spring represents no crack whereas zero stiffness shows the complete separation of the member. The magnitude between zero and infinite defines the presence of a crack with a certain severity, the inverse function of spring stiffness. Hence, decreasing the stiffness value indicates increasing the crack severity. It is assumed that the crack is perpendicular to the beam surface and is always open having a uniform depth in width [43].

Fig. 5
Fig. 5
Close modal
The stiffness of the torsional spring K depends on the crack depth and position of the crack from the fixed end and the beam material properties. The expression for the stiffness K can be obtained from flexibility using the equation
$K=1S$
(12)
where S is the crack flexibility which is given as follows [44]:
$(1−υ2)KI2E=Mb22dSdCd$
(13)
Where E is the modulus of elasticity, Cd is the depth of the crack, ν is the Poisson’s ratio of the material of the blade, KI is the stress intensity factor (SIF) under mode-I bending load, and Mb is the bending moment at the cracked section.
The magnitude of SIF can be obtained using the Lagrange interpolation method as
$KI=6Mbπhbσhb2f(σ),σ=Cdhb,σ≤0.7$
(14)
Where hb is the thickness, and f(σ) is a local compliance function. Substituting Eq. (14) into Eq. (13) leads to
$S=∫0σ72π(1−υ2)σf2Ehb2dσ$
(15)
Substituting Eq. (15) into Eq. (12) and on further simplification, Eq. (16) is obtained, which is given as follows [27]:
$K=Ebhb272π(L−Cl)2(1−υ2)f$
(16)
Where Cl represents the crack position with respect to the fixed end of the beam and L is the total length of the beam. This stiffness corresponding to the given crack can also be determined experimentally through the deflection method or inverse vibration analysis.
The function f is computed from the strain energy density function and is given as [28]
$f=0.629(Cdhb)2−1.047(Cdhb)3+4.602(Cdhb)4−9.975(Cdhb)5+20.295(Cdhb)6−32.993(Cdhb)7+47.041(Cdhb)8−40.693(Cdhb)9+19.6(Cdhb)10$
(17)

Further, this torsional stiffness is incorporated into the finite element model to consider the effect of the crack at a particular element. The calculated torsional stiffness K was added to the torsional degrees-of-freedom of the element in which the crack is present. In order to incorporate this model of a cantilever beam in the airfoil blade, the chord length and thickness of the blade are approximated in terms of beam width and beam height. The value of b = 0.85t and hb = 0.75c are considered to evaluate the torsional stiffness of the spring, where t is the maximum thickness, and c is the chord length of the airfoil section of the blade. The effect of crack depth and crack location from the fixed end on the blade dynamics is studied.

### 2.5 Implementation of Artificial Neural Networks.

ANNs are new technologies employed in several industrial scenarios. Thus, they provide an interactive approach to deal with complex problems. Neural networks mimic the human behavior of thinking and acting. The basic element of ANN is an artificial neuron, which comprises of several inputs and one output obtained as a function of the weighted sum of all the inputs reaching the neuron. Several such neurons arranged in a distributed manner generate power to the neural network. Of various neural network architectures, the multi-layer perceptron (MLP) model acquired wide attention due to its several benefits. ANN concept is employed in multi-input, multi-output mapping studies, and the relationship between the inputs and outputs is established in a non-parametric manner. The architecture of multi-layer ANN with error-back propagation is shown in Fig. 6 [45].

Fig. 6
Fig. 6
Close modal

Multi-layer perceptron network consists of an input layer, an output layer, and one or more intermediate or hidden layers, which are all fully connected in a feed-forward manner. Each connection is weighed with initial random numbers and adjusted according to the input-output data supplied to the network using the error-back propagation learning rule. The process of updating the weights in each learning cycle is called training. The error computed at the end of each cycle is referred to as mean square error (MSE) which is the square of the difference between the target outputs and calculated output values. The inputs to the network are the first three fundamental frequencies in Hz, and the outputs are the crack parameters (Cd) and (Cl), respectively. The set of weights is used to store the knowledge in the neural network. These connection weights are modified using the back propagation learning method and are known as training of the network. ANN was trained with the numerically obtained data to establish the relationship between the crack parameters and the fundamental frequencies of the blade.

## 3 Results and Discussion

In order to identify the correct set of frequencies for different crack parameters, several initial studies are conducted.

### 3.1 Modal Analysis.

Modal analysis and dynamic response simulations are carried out using in-house computer algorithms written in matlab. The results of the analysis are initially validated with the 3D model of the uncracked blade [9]. NACA64A410 airfoil is considered for blade tip profile. The material properties and blade dimensions are provided in Table 1.

Table 1

Density (ρ)4429 kg/m3
Poisson’s ratio (ν)0.31
Young’s modulus (E)104.8 GPa
Thermal conductivity6.7 W/mK
Tensile strength1050 MPa
The thickness of the blade section (t)36 mm
Chord length of the blade section (c)120 mm
Twist angle (φ)20 deg
Fir-tree root thickness and height50 mm
Density (ρ)4429 kg/m3
Poisson’s ratio (ν)0.31
Young’s modulus (E)104.8 GPa
Thermal conductivity6.7 W/mK
Tensile strength1050 MPa
The thickness of the blade section (t)36 mm
Chord length of the blade section (c)120 mm
Twist angle (φ)20 deg
Fir-tree root thickness and height50 mm

The first four natural frequencies obtained are depicted in Table 2. The blade configuration is analyzed simultaneously using a solid model in both Solidworks and ansys software using default tetrahedral mesh and with SOLID185 elements in ansys. The 3D model of the uncracked blade has been finely meshed into 18,221 tetrahedron elements comprising of 30,084 nodes with an element size of 9.649 mm using global mesh controls.

Table 2

First four natural frequencies of the blade

Number of elementsTotal system degrees-of-freedomNatural frequencies (Hz)
1st mode2nd mode3rd mode4th mode
21567.2149.2412.2896.6
42567.1150.7408.8901.1
63567.1151.0408.7900.4
84567.1151.1408.8900.3
105567.1151.2408.9900.2
126567.1151.2408.9900.2
ANSYS1566.8178.1663.5828.7
SolidWorks65.9174.3660.2825.4
Number of elementsTotal system degrees-of-freedomNatural frequencies (Hz)
1st mode2nd mode3rd mode4th mode
21567.2149.2412.2896.6
42567.1150.7408.8901.1
63567.1151.0408.7900.4
84567.1151.1408.8900.3
105567.1151.2408.9900.2
126567.1151.2408.9900.2
ANSYS1566.8178.1663.5828.7
SolidWorks65.9174.3660.2825.4

It is observed from the current program that as the number of elements reaches 12, frequency convergence is noticed. The validation of the developed finite element code was done by comparing the results with published literature [5]. While the fundamental frequency is matching with the 3D solution, the deviation from higher order modes is due to the ten degrees-of-freedom element configuration in the proposed analysis. The 3D finite element solvers employ only three translational degrees-of-freedom per node. The deviations in the natural frequencies between the present model and commercial software solutions also occur due to the following reasons: (i) Software 3D analysis does not account for any rotational degrees-of-freedom in element formulation, so the torsional degree concept doesn’t arise and (ii) while developing the 3D model, curvature, non-uniform twist angle, and fir-tree root effects were automatically accounted for, while they are not considered in the present model. Figure 7 shows the first three mode shapes of the blade obtained using simulation software.

Fig. 7
Fig. 7
Close modal

### 3.2 Forced Vibration Response of the Blade.

The blade is subjected to the combination of forces as given in the forced vibration Sec. 2.4. Houbolt’s time integration method was employed to numerically solve the dynamic equation of motion of the blade. The time response of the blade is given in Fig. 8, which is the plot of the time versus the resultant displacement of the blade tip. The corresponding frequency spectrum is illustrated in Fig. 9.

Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal

The two dominant peaks indicate the first two fundamental frequencies of the system.

### 3.3 Effect of Crack Parameters.

The cracked blade finite element model is further employed to predict the effects of depth of crack on the blade frequencies. The effect of root crack depth on the first three natural frequencies of the blade is depicted in Fig. 10.

Fig. 10
Fig. 10
Close modal

In order to validate the frequency variation, the results of the 3D model of the blade with different root crack depths are also presented for the fundamental mode. It is noticed that the crack depth effect from the present formulation and 3D simulation are quite similar. The percentage accuracy of predictions of the fundamental frequency is found to be 97% relative to the 3D model result. Furthermore, the effect of crack location from blade root on the first three modes is presented in Table 3 using the present finite element formulation. It is seen that as the crack location is away from the blade root natural frequencies are slowly increasing, which was proved earlier in the literature. On validation of the proposed code, it is further employed to generate the dynamic properties as a function of crack parameters.

Table 3

Variation of fundamental frequencies of the blade with different crack locations at a crack depth of 10 mm

Distance from the root Cl (mm)Natural frequencies of the blade (Hz)
Mode 1Mode 2Mode 3
064.1120.9260.7
2064.2122.2265.5
4064.4123.6270.3
6064.6124.9275.3
8064.7126.2280.5
10064.8127.6285.7
15064.2130.8299.4
20065.5133.9313.6
25065.8136.8328.2
Distance from the root Cl (mm)Natural frequencies of the blade (Hz)
Mode 1Mode 2Mode 3
064.1120.9260.7
2064.2122.2265.5
4064.4123.6270.3
6064.6124.9275.3
8064.7126.2280.5
10064.8127.6285.7
15064.2130.8299.4
20065.5133.9313.6
25065.8136.8328.2

The frequency spectra of the normal (uncracked) and the cracked blade are shown in Figs. 11 and 12, respectively.

Fig. 11
Fig. 11
Close modal
Fig. 12
Fig. 12
Close modal

It can be seen that the fast Fourier transform (FFT) of the cracked blade response has abnormal peaks as compared to that of a normal blade.

### 3.4 Identification of Crack Parameters by Artificial Neural Networks.

Figure 13 shows the variation of the first three natural frequencies of the blade as a function of the crack position (Cl) and depth (Cd). It can be seen that the parameters have a vital influence. Cracks of different depths at different positions change the frequencies considerably. The effect of crack location depends on the vibration mode.

Fig. 13
Fig. 13
Close modal

It is seen that natural frequencies vary irregularly with crack depth and location. Therefore, the artificial neural network can be effectively employed here to develop a relationship between the crack depth and location with frequencies.

In the present work, multi-layer ANN has been used to identify the crack parameters based on the frequency data. ANN was trained to establish the relationship between fundamental frequencies and crack parameters. Once the ANN is trained, then based on the variation in the frequency data, the presence of a crack can be identified. Three inputs to the input layer are the first three natural frequencies while the two outputs are the crack depth and crack distance from the root respectively. Out of the total data set, 70% of the data was used for training, and the remaining for validation and test data sets. ANN was trained rigorously with several sets of data values representing the combined effects of crack parameters on the fundamental frequencies of the blade. The part of the training data set is illustrated in Table 4.

Table 4

Part of the training data set used for training the artificial neural network

Natural frequencies of the blade (Hz)Crack depth Cd (mm)Distance from the root Cl (mm)
Mode 1Mode 2Mode 3
67.1151.2408.900
64.1120.8260.514200
65.3131.1301.1820
58.387.8177.422140
66.2140.4347.6660
58.186.9175.7180
52.869.8147.22410
66.7146.5382.1480
64.9127.6285.710100
57.383.9170.12040
Natural frequencies of the blade (Hz)Crack depth Cd (mm)Distance from the root Cl (mm)
Mode 1Mode 2Mode 3
67.1151.2408.900
64.1120.8260.514200
65.3131.1301.1820
58.387.8177.422140
66.2140.4347.6660
58.186.9175.7180
52.869.8147.22410
66.7146.5382.1480
64.9127.6285.710100
57.383.9170.12040

The sigmoidal transfer function is employed between the hidden layers and the linear transfer function is used in the output layer. The number of neurons in the hidden layer was computed based on the number of neurons in the output and input layers. Before using a neural network, the normalization process is applied to the output data. The validation data set which is separate from the training data set is used to validate the model performance during training. This validation process gives information that helps us tune the model’s hyperparameters and configurations accordingly. The hyper-parameter tuning (e.g., number of neurons) of the model was done with this validation data set. A part of the data set which is used as the validation data set is given in Table 5 respectively.

Table 5

Part of the validation data set

Natural frequencies of the blade (Hz)Crack depth Cd (mm)Distance from the root Cl (mm)
Mode 1Mode 2Mode 3
61.2110.6220.316220
67.5142.4350.2630
59.892.7190.522190
69.4150.2387.52100
65.196.8200.1120
Natural frequencies of the blade (Hz)Crack depth Cd (mm)Distance from the root Cl (mm)
Mode 1Mode 2Mode 3
61.2110.6220.316220
67.5142.4350.2630
59.892.7190.522190
69.4150.2387.52100
65.196.8200.1120
After the training process is completed, the test data are introduced into the ANN model and the accuracy of the model was then examined. The best way to validate the accuracy of an ANN is to estimate the errors between the outputs and the target data. For this, the MSE is plotted as a function of the number of epochs. This validation process gives information that helps us to tune the model’s hyperparameters and configurations accordingly. The hyper-parameter tuning (e.g., number of neurons) of the model was done with this validation data set. Further, the number of neurons in the hidden layer was varied to get the best regression value (R2). To obtain the optimal ANN architecture, the number of hidden layers is varied to obtain the highest value of the coefficient of determination R2 which is given as
$R2=1−∑i=1n(oi−yi)2∑i=1n(oi−o¯)2$
(18)
Where, $∑i=1n(oi−yi)2$ is the residual sum of squares and $∑i=1n(oi−o¯)2$ is the total sum of squares with oi and $o¯$ as the target set and their mean respectively, while yi is the predicted output values. The values of R2 and MSE for the different numbers of neurons in the hidden layer are tabulated in Table 6.
Table 6

Performance measures

Number of neurons in the first hidden layerR2MSE
40.99984.685 × 10−6
60.99994.1638 × 10−7
80.99942.675 × 10−5
100.99808.9254 × 10−5
Number of neurons in the first hidden layerR2MSE
40.99984.685 × 10−6
60.99994.1638 × 10−7
80.99942.675 × 10−5
100.99808.9254 × 10−5

The training trend and regression plot which is a plot between the target parameter and predicted outputs of the 3-6-2 ANN architecture is shown in Fig. 14.

Fig. 14
Fig. 14
Close modal

The ANN testing step is crucial for verifying the accuracy and performance. The testing process introduces three input vectors and then using the trained ANN, the outputs are estimated. After outputs are calculated, a comparison with true data is made. The comparison of predicted values of crack depth and crack location with the actual values is given in Table 7, respectively.

Table 7

Comparison of actual and predicted parameters

NumberParameterActual (FEM)Predicted (ANN)Error (%)
1Cd88.05−0.6
Cl2019.890.5
2Cd43.951.25
Cl8079.830.2
3Cd1010.12−1.2
Cl10099.870.13
4Cd1414.16−1.1
Cl200199.640.18
5Cd66.1−1.7
Cl6060.23−0.38
NumberParameterActual (FEM)Predicted (ANN)Error (%)
1Cd88.05−0.6
Cl2019.890.5
2Cd43.951.25
Cl8079.830.2
3Cd1010.12−1.2
Cl10099.870.13
4Cd1414.16−1.1
Cl200199.640.18
5Cd66.1−1.7
Cl6060.23−0.38

The predicted depth and location are very close to the test data, which signifies that the ANN model is well trained; therefore, the model can be used as an alternative approach to predict the depth and location of cracks.

## 4 Conclusions

This study provided an alternative methodology for predicting the crack parameters in steam turbine blades. The vibration behavior of the last stage low-pressure (LP) cracked and uncracked steam turbine blade was studied through the finite element model and artificial neural network. FEM was employed to develop an efficient 1D model of the blade. It was found that the results obtained from the model for lower modes were fairly in accordance with the results obtained from modeling software. Due to the additional geometric effects like curvature, fit tree root and non-uniform twist the higher mode of the 3D model have considerable deviation; however, the results are close to the earlier works in the literature.

Further, it was deduced that the effect of the crack is predominant when the crack generates at the root and the dynamic behavior of the blade changes in the presence of the crack. Natural frequencies were chosen as control parameters to localize the cracks. ANN was employed to identify crack parameters based on the natural frequency data. The obtained ANN results showed that the forecasted results were in good agreement with the actual data. Thus, this work can be used to identify the localized damage in the last stage steam turbine blade based on the natural frequencies of the blade. This will help to prevent the catastrophic failure of the blade.

## Acknowledgment

The authors would like to thank the Department of Mechanical Engineering, National Institute of Technology Rourkela, for extending the facilities for this research work.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## References

1.
Carnegie
,
W.
,
1959
, “
,”
Proc. Inst. Mech. Eng.
,
173
(
1
), pp.
343
374
.
2.
Subrahmanyam
,
K. B.
,
Kulkarni
,
S. V.
, and
Rao
,
J. S.
,
1981
, “
Coupled Bending–Bending Vibration of Pre-Twisted Cantilever Blading Allowing for Shear Deflection and Rotary Inertia by the Reissner Method
,”
Int. J. Mech. Sci.
,
23
(
9
), pp.
517
530
.
3.
Houbolt
,
J. C.
, and
Brooks
,
G. W.
,
1958
, “
Differential Equations of Motion for Combined Flapwise Bending Chordwise Bending and Torsion of Twisted Non-Uniform Rotor Blades
,”
NACA Report
, pp.
13
46
4.
Yardimoglu
,
B.
, and
Yildirim
,
T.
,
2004
, “
Finite Element Model for Vibration Analysis of Pre-Twisted Timoshenko Beam
,”
J. Sound Vib.
,
273
(
4–5
), pp.
741
754
.
5.
,
V.
,
1984
, “
Dynamic Analysis of Practical Blades With Shear Center Effect
,”
J. Sound Vib.
,
92
(
4
), pp.
471
490
.
6.
Bhagi
,
L. K.
,
Rastogi
,
V.
,
Gupta
,
P.
, and
,
S.
,
2018
, “
Dynamic Stress Analysis of L-1 Low Pressure Steam Turbine Blade: Mathematical Modelling and Finite Element Method
,”
Mater. Today: Proc.
,
5
(
14
), pp.
28117
28126
.
7.
Rosard
,
D.
, and
Lester
,
P.
,
1999
, “
Natural Frequency of Twisted Cantilever Beam
,”
ASME J. Appl. Mech.
,
20
(
2
), pp.
241
244
.
8.
Chen
,
Y.
,
Zhang
,
J.
, and
Zhang
,
H.
,
2017
, “
Free Vibration Analysis of Rotating Tapered Timoshenko Beams Via Variational Iteration Method
,”
J. Vib. Contr.
,
23
(
2
) pp.
220
234
.
9.
Shetkar
,
K. A. D. R.
, and
Srinivas
,
J.
,
2021
, “
Analytical Modeling and Vibration Analysis of the Last-Stage LP Steam Turbine Blade Made of Functionally Graded Material
,”
Arabian J. Sci. Eng.
,
46
(
8
), pp.
7363
7377
.
10.
Hoa
,
S. V.
,
1979
, “
Vibration of a Rotating Beam With Tip Mass
,”
J. Sound Vib.
,
67
(
3
), pp.
369
381
.
11.
Sarkar
,
K.
, and
Ganguli
,
R.
,
2014
, “
Modal Tailoring and Closed-Form Solutions for Rotating Non-Uniform Euler–Bernoulli Beams
,”
Int. J. Mech. Sci.
,
88
, pp.
208
220
.
12.
Sina
,
S.
, and
,
H.
,
2014
, “
Axial-Torsional Vibrations of Rotating Pretwisted Thin Walled Composites Beams
,”
Int. J. Mech. Sci.
,
80
, pp.
93
101
.
13.
Chandiramani
,
N.
,
Shete
,
C.
, and
Librescu
,
L.
,
2003
, “
Vibration of Higher-Order-Shearablepretwisted Rotating Composite Blades
,”
Int. J. Mech. Sci.
,
45
(
12
), pp.
2017
2041
.
14.
Yoo
,
H. H.
,
Kwak
,
J. Y.
, and
Chung
,
J.
,
2001
, “
Vibration Analysis of Rotating Pre-Twisted Blades With a Concentrated Mass
,”
J. Sound Vib.
,
240
(
5
), pp.
891
908
.
15.
Na
,
S.
,
Librescu
,
L.
, and
Shim
,
K.
,
2003
, “
Modeling and Bending Vibration Control of Non-Uniform Thin-Walled Rotating Beams Incorporating Adaptive Capabilities
,”
Int. J. Mech. Sci.
,
45
(
8
), pp.
1347
1367
.
16.
Chen
,
C. K.
, and
Ho
,
S. H.
,
1999
, “
Transverse Vibration of a Rotating Twisted Timoshenko Beams Under Axial Loading Using Differential Transform
,”
Int. J. Mech. Sci.
,
41
(
11
), pp.
1339
1356
.
17.
Chan
,
S. K.
, and
Tuba
,
I. S.
,
1971
, “
A Finite Element Method for Contact Problems for Solid Bodies—Part II: Applications to Turbine Blade Fastenings
,”
Int. J. Mech. Sci.
,
13
(
7
), pp.
627
639
.
18.
Zboiński
,
G.
,
1993
, “
Finite Element Computer Program for Incremental Analysis of Large Three-Dimensional Frictional Contact Problems of Linear Elasticity
,”
Comp. Struc.
,
46
(
4
), pp.
679
687
.
19.
Zboinski
,
G.
,
1995
Physical and Geometrical Non-Linearities in Contact Problems of Elastic Turbine Blade Attachments
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
,
209
(
4
), pp.
273
286
.
20.
Papanikos
,
P.
,
Meguid
,
S. A.
, and
Stjepanovic
,
Z.
,
1998
, “
Three Dimensional Nonlinear Finite Element Analysis of Dovetail Joints in Aero-Engine Discs
,”
Finite Elem. Anal. Des.
,
29
(
3/4
), pp.
173
186
.
21.
Meguid
,
S. A.
,
Kanth
,
P. S.
, and
Czekanski
,
A.
,
2000
, “
Finite Element Analysis of fir-Tree Region in Turbine Disk
,”
Finite Elem. Anal. Des.
,
35
(
4
), pp.
305
317
.
22.
Dhar
,
D.
,
Sharan
,
A. M.
, and
Rao
,
J. S.
,
2004
, “
Transient Stress Analysis and Fatigue Life Estimation of Turbine Blades
,”
ASME J. Vib. Acoust.
,
126
(
4
), pp.
485
495
.
23.
Shukla
,
A.
, and
Harsha
,
S. P.
,
2016
, “
Vibration Response Analysis of Last Stage LP Turbine Blades for Variable Size of Crack in Root
,”
Proc. Tech.
,
23
, pp.
232
239
.
24.
Wang
,
W. Z.
,
Xuan
,
F. Z.
,
Zhu
,
K. L.
, and
Tu
,
S. T.
,
2007
, “
Failure Analysis of the Final Stage Blade in Steam Turbine
,”
Eng. Fail Anal.
,
14
(
4
), pp.
632
641
.
25.
Mazur
,
Z.
,
Illescas
,
R. G.
,
Romano
,
J. A.
, and
Rodriguez
,
N. P.
,
2008
, “
,”
Eng. Fail. Anal.
,
15
(
1–2
), pp.
129
141
.
26.
Yashar
,
A.
,
Ferguson
,
N.
, and
Tehrani
,
M. G.
,
2018
, “
Simplified Modelling and Analysis of a Rotating Euler-Bernoulli Beam With a Single Cracked Edge
,”
J. Sound Vib.
,
420
, pp.
346
356
.
27.
Dimarogonas
,
A. D.
,
Paipetis
,
S. A.
, and
Chondros
,
T. G.
,
2013
,
Analytical Methods in Rotor Dynamics
,
Springer
,
Dordrecht, The Netherlands
.
28.
Liu
,
W.
, and
Barkey
,
M. E.
,
2018
, “
Effects of Breathing Behaviour on Crack Growth of a Vibrating Beam
,”
J. Shock Vib.
,
2018
, pp.
1
12.
.
29.
Cunningham
,
B. M. D.
,
Evangelou
,
A.
,
You
,
C.
,
Morris
,
A.
,
Wise
,
J.
,
Philippa
,
A. S.
,
Andrew
,
R.
, and
Hamilton
,
R.
,
2021
, “
Fatigue Crack Initiation and Growth Behavior in a Notch With Periodic Overloads in the Low-Cycle Fatigue Regime of FV566 Ex-Service Steam Turbine Blade Material
,”
Fatigue Fract. Eng. Mater. Struct.
,
45
(
2
), pp.
546
564
30.
Sing
,
S.
,
Kharub
,
M.
,
Singh
,
J.
, and
Jangid
,
V.
,
2021
, “
Brief Survey on Mechanical Failure and Preventive Mechanism of Turbine Blades
,”
Mater. Today: Proc.
,
38
, pp.
2515
1524
.
31.
Skorupa
,
M.
,
1998
, “
Load Interaction Effects During Fatigue Crack Growth Under Variable Amplitude Loading—A Literature Review. Part I: Empirical Trends
,”
Fatigue Fract. Eng. Mater. Struct.
,
21
(
8
), pp.
987
1006
.
32.
Zhou
,
X.
,
Gaenser
,
H.-P.
, and
Pippan
,
R.
,
2016
, “
The Effect of Single Overloads in Tension and Compression on the Fatigue Crack Propagation Behaviour of Short Cracks
,”
Int. J. Fatigue
,
89
, pp.
77
86
.
33.
Ward-Close
,
C. M.
,
Blom
,
A. F.
, and
Ritchie
,
R. O.
,
1989
, “
Mechanisms Associated With Transient Fatigue Crack Growth Under Variable-Amplitude Loading: An Experimental and Numerical Study
,”
Eng. Fract. Mech.
,
32
(
4
), pp.
613
638
.
34.
Bhamu
,
R. K.
,
Shukla
,
A.
,
Sharma
,
S. C.
, and
Harsha
,
S. P.
,
2021
, “
Dynamic Analysis of Low-Pressure Steam Turbine Last Stage fir-Tree Root Blade
,”
ASME J. Nondestr. Eval., Diagn. Progn. Eng. Syst.
,
4
(
2
), p.
021002
.
35.
Taheri
,
F.
,
,
D.
, and
Pegg
,
N.
,
2003
, “
Experimental and Analytical Investigation of Fatigue Characteristics of 350WT Steel Under Constant and Variable Amplitude Loadings
,”
Mar. Struct.
,
16
(
1
), pp.
69
91
.
36.
Rodriguez
,
J. A.
,
Hamzaoui
,
Y. E.
,
Hernandez
,
J. A.
,
Garcia
,
J. C.
,
Flores
,
J. E.
, and
Tejeda
,
A. L.
,
2013
, “
The Use of Artificial Neural Network (ANN) for Modeling the Useful Life of the Failure Assessment in Blades of Steam Turbines
,”
Eng. Failure Anal.
,
35
, pp.
562
575
.
37.
Quintanar-Gago
,
D. A.
,
Nelson
,
P. F.
,
Díaz-Sánchez
,
A.
, and
Boldrick
,
M. S.
,
2021
, “
Assessment of Steam Turbine Blade Failure and Damage Mechanisms Using a Bayesian Network
,”
Reliab. Eng. Syst. Safety.
,
207
, p.
107329
.
38.
Song
,
W.
,
Keane
,
A. J.
,
Rees
,
J.
,
,
A. and
Bagnall
,
S.
,
2002
, “
Turbine Blade fir-Tree Root Design Optimization Using Intelligent CAD and Finite Element Analysis
,”
Comput. Struct.
,
80
(
24
), pp.
1853
1867
.
39.
Przemieniecki
,
J. S.
,
1968
,
Theory of Matrix Structural Analysis
,
McGraw-Hill Book Company, Inc.
,
New York
.
40.
Griffiths
,
D. V.
,
1989
, “
Advantages of Consistent Over Lumped Methods for Analysis of Beams on Elastic Foundations
,”
Commun. Appl. Numer. Meth.
,
5
(
1
), pp.
53
60
.
41.
Lanzi
,
A.
, and
Luco
,
J.
,
2017
, “
Caughey Damping Series in Terms of Products of the Flexibility Matrix
,”
J. Eng. Mech.
,
143
(
9
).
42.
Chung
,
J.
, and
Hulbert
,
G. M.
,
1994
, “
A Family of Single-Step Houbolt Time Integration Algorithms for Structural Dynamics
,”
Comput. Meth. Appl. Mech. Eng.
,
118
(1–2), pp.
1
11
.
43.
Kagdi
,
F.
, and
Mungla
,
M.
,
2019
, “
Review Paper on Crack Modeling and Crack Detection Methods
,”
Int. J. Eng. Res. Tech.
,
8
(
07
).
44.
Broek
,
D.
,
1986
,
Elementary Engineering Fracture Mechanics
,
Martinus Nijhoff Publishers
,
Dordrecht, The Netherlands
.
45.
Cabán
,
P.
,
Masters
,
F. J.
, and
Phillips
,
B. M.
,
2018
, “
Predicting Roof Pressures on a Low-Rise Structure From Freestream Turbulence Using Artificial Neural Networks
,”
Front. Built Environ.
, online.