## Abstract

In the present study, the dynamic behavior of the last stage low-pressure steam turbine blade with fir-tree root at different conditions of blade root flank faces and their interfaces with rotor groove have been analyzed. Modal analysis has been done using a finite element approach to evaluate natural frequencies and evaluation of Campbell diagram generated under these conditions. For this, both healthy and defective blade have been taken. Since the variable crack size of fir-tree root flank has been taken, the excitation pattern has been evaluated due to stiffness variation of the cracked blade. This analysis provides the basis of excitation pattern of cracked blades due to inherent character and critical stressed zone. The outcome of this study forms the guidelines and checks during the fitting of blades in rotor assembly and its checks during health audit, overhaul, overspeed balancing test, and frequency turning.

## 1 Introduction

A steam turbine is a primary mover in a coal-based thermal power station to produce electric power by creating abundant torque. Adiabatically, expansion of steam flow in blade generates power in steam turbines. They weakened the strength of interior portions (blades, rotor discs, etc.) of the turbine at the flow path of the steam.

Blades are the most critical module and heart of the turbine, which undergo reversed loading and severe dynamic conditions during their lifecycle span; hence mostly failed in the fatigue model [1]. For the successive sections, blades are subjected to be steam twisting and bending forces. It is a mutual repetition to base the design on centrifugal forces alone as the centrifugal forces are so high in comparison with other steam twisting and bending forces. The blade fastening resists all these forces. Fir-tree type, T-type, straddle type, etc. are various fastening between in use. For securing a large blade, most satisfactory fastening is a fir tree as it offers an acceptable region of interaction to withstand tremendous forces [2].

This study describes the vibration behavior of low-pressure last stage steam turbine blade with a fir-tree root by using FE-based modal analysis technique. In this, different models of the blade constructed, each varied in crack size in root and fixing condition at the rotor groove. The natural frequency and mode shape of the different crack sizes in root models and variable fixing condition models were estimated and found that there are a significant variation in fundamental frequencies than the blade in healthy condition. Hence, the Campbell diagram analysis carried out to find the critical speed for the blade in that condition.

## 2 Modal Analysis of Low Pressure Steam Turbine Blade

Calculation of frequencies are complicated, in the instance, for the last stage low-pressure steam turbine blade due to its twisted and tapered construction. In this case, a 250 MW low-pressure steam turbine final phase blade of 6.3 m2 geometry has been modeled in solidworks and additionally carried out to FE based ansys solver for analysis, as shown in Fig. 1.

Blade's vibration equation of motion investigated through the support of finite element method (FEM), the numerical model's equation is as follows [10]:
$[K]{u}+[C]{u˙}+[M]{u¨}=F(t)$
(1)
where
• [K]—stiffness matrix

• [M]—mass matrix

• [C]—damping matrix, and

• {u}—node displacement vector

In case of free vibration [F(t) = 0] and for the un-damped linear elastic structure of blade, Eq. (1) can be written as
$Ku+Mu¨=0$
(2)
The structure imagined vibrating in a free harmonic form with the absence of externally applied force
$u(t)=φsin(ωt+θ)$
(3)
which leads to the Eigenvalue problem:
$[[K]−ω2[M]]{φ}={0}$
where ω is the natural frequency and φ is the structure's mode shape correspondingly. The mode shapes and natural frequencies will be attained by solving the eigenvalue problem.

A 250 MW power plant's low-pressure steam turbine last stage blade has modeled in solidworks. For modal analysis, a free-standing solitary blade with fir-tree root carried to ansys workbench, which is an FE based solver.

Blade material is X10CrNiMoV1222 and assumed as an isotropic elastic. Other properties of the material are given in Table 1.

From the shape of the object, the blade is triangular taper and highly twisted, and having small fillets and tiny edges, a continuous three-dimensional ten nodded tetrahedral elements SOLID 187 meshing method has used. By using SOLID187, all sharp corners of blade root, fillet, and sharp edges may get matched geometry without any distortion.

Boundary conditions are assigned to a pre-twisted free-standing blade's meshed model as its fixing in the rotor's open root groove. Due to the presence of centrifugal load in a steam turbine, fir-tree root blades’ six faces with rotor grooves designated as a fixed support boundary condition and two surfaces of the blade, which form collar for blade profile, assigned as frictionless sliding support during the operating state.

The formulated FE model has analyzed with the aid of ansys solver and natural frequencies and modes shapes were obtained for the first six fundamental modes. During the modal analysis of the normal blade, the effect of rotation and Coriolis component has been taken into account, and the Campbell diagram obtained.

Due to the presence of eccentric and various dynamic loading, low-pressure steam turbine last stage blade stressed at the upper flank area of fir-tree root groove; hereafter, extra prone to become the expansion of crack.

Based on the fraction of the total area of upper fir-tree root's flank area, cracks are produced in the modeled blade, with different sizes of 10%, 25%, 50%, 75%, and 90%. For modal analysis, all the geometrical parameters of blades are the same as a healthy blade except the modeled crack feature. To get a smooth model with an extreme consequence to all the sharp corners and profile radius, a continuous three-dimensional tetrahedral element with ten nodded used. The modeled cracked blade was finely meshed to minimize the inaccuracies neighbouing the cracked zone in the modeling of the cracked region of the blade. Stress stiffening effect, spin softening, and Coriolis effect were also considered for modal analysis.

For a modeled blade with a crack size of 10%, 25% 50%, 75%, and 90% of the fir-tree root upper flank area, Campbell analysis has been done and studied the critical speed of the rotor system for a blade.

### 2.3 For Variation in Fixity Condition of Fir-Tree Root Flank of Blade.

An object of a modeled healthy blade with fir-tree root has three different faces, namely, top face, middle face, and a lower face of different contact areas at both sides of blade, i.e., suction side and pressure side. Between blade fir-tree root face and rotor groove flank face, a minimal available clearance of 0.3 mm bridged during running condition due to centrifugal force. It was assumed that all the faces would remain parallel to the corresponding faces during the running state. Because of that, five different contact configurations are considered for blade root fixity and boundary condition applied on blade root as per the different root contact configuration. After the convergence of the solution for the first six modes, free vibration modes were obtained.

Different cases of variable fixing conditions on blade root interfaces are only one face (top, middle, and lower) in contact in both sides, and lower faces of suction side and pressure side individually were considered for Campbell diagram analysis to find the critical speed of the component as described in Table 2.

## 3 Result and Discussion

After carrying out the modal analysis of the last stage steam turbine blade through FE based ansys workbench, natural frequencies and six mode shape have obtained as shown in Fig. 2 out of which primary three modes are flap wise bending, chord wise bending, and torsion mode, respectively, and the remaining are higher modes known as overlapping modes.

A diagram plotted between rotational speed and frequency of the system known as “Campbell Diagram.” Campbell diagram analysis has been done to get the critical speed of the rotor blade mechanism, which can be further used to control the resonance.

To cover the entire range of events for which blade has been designed, 25% overspeed has also considered during the Campbell analysis. From the Campbell diagram as shown in Fig. 3, it is evident that the critical speed of turbine rotor for the subject low pressure (LP) blade is 3351.4 rpm, which is beyond the normal operating speed of rotor (i.e., 3000 rpm). So the blade is safe at the normal operating frequency and its designed fluctuation limit of ±5%.

The crack size of the blade increased as the natural frequency of blades decreased. Due to centrifugal loads and various dynamic loading, the free-standing blade's upper flank of fir-tree root groove is highly stressed and due to this more prone to get the development of the crack. Table 3 shows the deviation of the natural frequencies of blades concerning the magnitude of the crack.

Variation in frequency from the normal blade with the different crack size of the area of the upper flank of the fir-tree root blade as shown in Fig. 4. The natural frequencies of corresponding modes are continuously decreasing. It is evident that the least variation for different crack sizes in natural frequencies was observed in overlapping mode at mode no. 4. Campbell analysis has been done for all such cases of the cracked root at the blade from 10% crack size to 90% crack size.

Figures 57 show the Campbell diagrams for the crack size of 10%, 25%, and 50% of the area of fir-tree root upper flank areas and studied the critical speed of the rotor system for the blade as 3348.1 rpm, 3345.7 rpm, and 3291.1 rpm, respectively, which are away from the normal operating range. Similarly, Fig. 8 shows the Campbell diagram for the crack size of 75% of the area of fir-tree root upper flank areas and studied the critical speed of the rotor system for the blade is 2794.80 rpm which is below from the normal operating range.

From Fig. 9, it was seen that the critical speed had decreased as crack volume increased by comparing the critical speed of the blade and observed that up to 50% of the crack size, the critical blade speed is on the safer side, whereas in the case of 75% and 90% crack size, the critical speed is much lower than the running speed. It was evident that there is no substantial shift in the critical speed of the rotor for small and medium-size cracks. It appears in the case of large size crack at which the critical speed dropped significantly and runs below the operating range.

### 3.3 For Variation in Fixity Condition of Fir-Tree Root Flank of Blade.

Modal analysis for fir-tree root blade of last stage low-pressure steam turbine has been done for various fixing conditions, and natural frequencies and mode shapes are obtained. Table 4 shows the variation of frequencies of blades due to the various fixity conditions of fir-tree root flank of the blade with a rotor of turbine and indicated that in all the cases of fixing of the blade, there are a continuous drop in frequencies from the normal blade. The cause of this reduction in natural frequencies are the reduction of stiffness because of non-guarded surfaces on the root faces.

Variation in natural frequencies from the normal blade concerning the different fixity conditions are shown in Fig. 10 and concluded that there is a continuous drop in frequencies from the normal blade in all the cases of fixing of the blade; it is attributed to the effect of reduction in stiffness (imparted slackness) due to the non-constrained surfaces on the root faces. It is observed that the top face contributes maximum in the root stiffness and the maximum difference in frequencies takes place in the suction side and pressure side lower face in contact almost equally though, the variation in frequency from suction side contact to pressure side contact surfaces is depending on the directional flexibility imparted against the corresponding mode shape. However, it is also evident that due to the concave shape of the pressure side root face, it reveals more stiffness than the suction side convex faces.

During the study of the modal analysis, full impact deviation in dynamic response from a normal blade has been observed. As described in the previous case, a Campbell diagram analysis has been done to find the critical speed of the component. These cases are only one face (top, bottom, and lower) in contact on both sides and lower faces of suction and pressure side individually.

Figures 1113 show the Campbell diagrams for lower flank, middle flank, and top flank of fir-tree root in contact and found the critical speeds of the rotor system for the blade are 3224.9 rpm, 3294.1 rpm, and 3337.7 rpm, respectively, which are away from the normal operating range. Similarly, Figs. 14 and 15 show the Campbell diagrams when only pressure side and suction side of the lower flank of fir-tree root are in contact and found the critical speeds of the rotor system for the blade are 2977.1 rpm and 2967.1 rpm, respectively, which are nearest to the normal blade running condition, and found catastrophic in nature.

Figure 16 shows the variation in critical speed (rpm) concerning the different fixity conditions. It can be further seen when only a single side lower face in contact comes in the zone of an operating frequency of the machine, which is prone near the resonance condition.

## 4 Conclusion

A study of normal and cracked blades with different types of boundary conditions using FEM has been done and observed that cracks in blades lead to decrease in the natural frequencies, which further decreases as crack size increases. With the reduction in the number of contact faces in the fir-tree root, the natural frequency dropped and attributed to the effect of reduction in stiffness due to non-constrained surfaces on the root faces. If the blade stays only in contact with the lower face, then the blade faces the maximum drop in frequency, which may lead to catastrophic in failure.

From the Campbell diagram analysis, it can be concluded that the critical speed of the normal blade lies beyond the operating zone of the turbine, hence, a proper monitoring and maintenance to be done during the overspeed test at balancing. In case any gross variation in the lab testing condition or design value arises, blue contact matching was to be checked and if correction required, then it must be ensured that maximum contact in the upper flank of fir-tree roots are achieved.

A study of normal and cracked blade with different types of fixity condition through FEM has been performed and it has been observed that cracks in blades lead to decrease in natural frequency which further decrease with the increase of crack sizes. This study shows that the crack size above 50% of the flank area is catastrophic and liable for immediate failure. However, the impact of flow surges may cause the failure at higher rate and even for very tiny cracks which solely depend on various operating conditions. Fir-tree root flank face and rotor groove interference/contact conditions also impact the dynamic behavior of blades majorly. The study described in the paper shows that the vulnerability increases with the reduction of the contact face area and it worsens if the contact face is only at the suction side faces of roots which are due to the geometry of aerofoil shapes of blades.

Further, the Campbell diagram analysis reveals that the change in stiffness of blade root due to any of the above two reasons changes the critical speed of rotors for this blade, which may come within the operating range of blade and becomes the reason for catastrophic failure during operation. So, the following cautions/checks should be taken care of:

1. The critical speed of the normal blade lies beyond the operating zone of the turbine, so special care has to be taken during the overspeed test at balancing.

2. A proper natural frequency check of all the blades during the in situ condition has to be done during assembly. In case any gross variation in the lab testing condition or design value arises, blue contact matching has to be also checked with rotor groove and fir-tree root flank along with the magnetic particle inspection (MPI) of roots (which is a common practice for the identification of cracks) and further if correction required, then it must be ensured that maximum contact in the upper flank of fir-tree roots has to be achieved.

## 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.

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