Static fracture and modal analysis simulation of a gas turbine compressor blade and bladed disk system
- Ralston Fernandes^{1},
- Sami El-Borgi^{2}Email author,
- Khaled Ahmed^{3},
- Michael I. Friswell^{4} and
- Nidhal Jamia^{4}
https://doi.org/10.1186/s40323-016-0083-7
© The Author(s) 2016
Received: 30 June 2016
Accepted: 14 November 2016
Published: 24 November 2016
Abstract
This paper presents a methodology for conducting a 3-D static fracture analysis with applications to a gas turbine compressor blade. An open crack model is considered in the study and crack-tip driving parameters are estimated by using 3-D singular crack-tip elements in ANSYS\(\circledR \). The static fracture analysis is verified with a special purpose fracture code (FRANC3D). Once the crack front is perfectly defined and validated, a free vibration study is conducted by analyzing the natural frequencies and modeshapes for both a single blade and bladed disk system. Taking advantage of high performance computing resources, a high fidelity finite element model is considered in the parametric investigation. In the fracture simulation, the influence of the size of a single edged crack as well as the rotational velocity on fracture parameters (stress intensity factors and J-Integral) are evaluated. Results demonstrate that for the applied loading condition, a mixed mode crack propagation is expected. In the modal analysis study, increasing the depth of the crack leads to a decrease in the natural frequencies of both the single blade and bladed disk system, while increasing the rotational velocity increases the natural frequencies. The presence of a crack also leads to mode localization for all mode families, a phenomenon that cannot be captured by a single blade analysis.
Keywords
Single blade Bladed disk system Crack Singular elements Static and modal analysis Finite element analysisBackground
Gas turbine compressor blades are an integral component to a gas turbine assembly and their failures can lead to catastrophic downstream damage. Meher-Homji and Gabriles [1] identified and listed the predominant failure mechanisms for gas turbine blades such as high cycle fatigue (HCF), low cycle fatigue (LCF), thermo-mechanical fatigue (TMF), environmental attack (oxidation, sulfidation, corrosion), damage due to creep, erosion and wear, impact damage (due to foreign object damage (FOD), domestic object damage (DOD) or clash/clang of compressor blades owing to surge) or a combination of above failure mechanisms. According to this study [1], HCF is mostly caused by aerodynamic excitations (like nozzle and vane passing frequencies, strut pass frequencies) or by self-excited vibration and flutter. If a periodic force acts at the blade natural frequency, then resonance can take place. Therefore, they explain that resonant fatigue is an important failure mechanism and if damping is insufficient to adequately absorb the periodic input energy, stresses can grow until failure occurs by overstress or through propagation of a fatigue crack. They also note that although HCF stresses themselves may not be very high, the magnitude of stresses can increase quite dramatically at resonance.
Given the severity of fatigue induced cracks, several numerical and experimental studies have been conducted to study the effect of crack initiation and propagation as well as the effect of cracks on the dynamic response of the rotor and bladed disk. While significant attention has been devoted to failure analysis of cracked turbine blades in recent years, with advances in three-dimensional fracture analysis, efforts have been made to characterize the crack propagation in cracked blades. Witek [2] proposed a hybrid procedure of crack growth dynamic estimation in an aero-turbine compressor blade subjected to resonant vibrations. Poursaeidi and Bakhtiari [3] used the fracture analysis code FRANC3D [4] to simulate fatigue crack growth in a gas turbine compressor blade, while using the Paris and Forman-Newman-De Koning model to predict fatigue life. The two studies use the Raju-Newman analytical solution [5] to calculate stress intensity factors for a semi-elliptical flaw in a turbine blade assumed as a flat plate. Kirthan et al. [6] used the finite element code ANSYS® to calculate stress intensity factors for a simplified approximation of a gas turbine blade while performing a fatigue crack growth simulation using the Paris-Erdogan crack growth model. While most of the above studies focus on a quasi-static fracture analysis, in reality, cracks in rotating bodies demonstrate a breathing effect due to the opening and closing of the crack. To that effect, Liu and Jiang [7] presented a dynamic crack simulation in rotating blades using a hexahedral finite element method.
Vibration based structural health monitoring is a well-established method of crack detection in rotor systems and has therefore received considerable attention over the years. The cyclic symmetry of perfectly tuned bladed disk structures can be exploited to reduce finite element analysis computational costs by just studying one sector of a bladed disk, provided the necessary phase relations are met. In reality however, defects in blades can destroy the cyclic symmetry, leading to a phenomenon known as mistuning. Mistuning in bladed disks due to the presence of a crack or multiple cracks in blades can cause vibration localization about a few blades [8]. The dynamic response of a cracked blade has therefore received considerable interest in literature and this paragraph briefly outlines some numerical simulation studies conducted to characterize the effect of cracks on the dynamic response of the blade and the bladed disk. Shukla and Harsha [9] conducted an experimental and FEM comparative study of the modal response of a blade alone cracked and uncracked steam turbine blade and observed a frequency shift in the frequency response due to the presence of a crack. Kuang and Huang [8, 10], demonstrated the mode localization effect by studying both the free and forced response analysis of bladed disks. Each blade was modeled as an Euler–Bernoulli beam and the crack effect was treated as local disorder of the system. Fang et al. [11] expressed the local stiffness loss of a bladed disk system due to a crack based on a fracture mechanics model. By modeling the blades as Euler-Bernoulli beams, a parametric study was conducted to determine the effect of various parameter such as internal coupling factor, crack severity, engine order of excitation, and number of blades of the mode localization effect. Saito et al. [12] employed a nonlinear crack model to capture the breathing crack phenomenon and used a hybrid-interface component mode synthesis (CMS) modeling approach to reduce the number of degrees of freedom in the bladed disk model. They showed that when compared with the linear model of an open crack, the frequency shift in the forced response for a nonlinear breathing crack is smaller. Shiryayev et al. [13] compared the power spectral densities (PSDs) of simulated steady-state vibration data of a bladed disk with a surface crack on the disk for different amplitudes of excitation, measurement location and size of the crack. Using appropriate signal processing techniques, they observed harmonics in the power spectrum due to the nonlinearity caused by the crack.
From the cited literature, it can be concluded that most authors either use a simplified model of a bladed disk or have adopted special purpose model reduction techniques to capture the response of the global system in the presence of a crack. However, with tremendous advances in high performance computing capabilities, it is now possible to perform high fidelity simulations of an entire bladed disk. The main objective of this paper is therefore to develop a high fidelity model capable of studying the localized response of a cracked blade as well the global dynamic response. The localized response is characterized via a detailed 3-D static fracture analysis around the crack-tip. With the crack modeled from a fracture mechanics approach, the global dynamic response is then characterized via a free vibration study of the single cracked blade as well as a bladed disk system with a cracked blade. A comparison of the single blade modal analysis and the bladed disk modal analysis is then conducted to study the efficacy of using a cracked single blade analysis alone for purposes of structural health monitoring. The paper is organized as follows. “Finite element modeling of a single blade and a system of blades” section describes the model under study and the meshing strategy. A validation study to determine the effectiveness of the meshing strategy is conducted in “Crack modeling and meshing” section and “Static fracture analysis” section outlines the results of the fracture analysis for a single blade. “Modal analysis” section compares the modal analysis study for a cracked and uncracked single and system of blades. The results are finally summarized in “Conclusions” section.
Finite element modeling of a single blade and a system of blades
This section describes the finite element mesh used to model the compressor rotor blade and the applied boundary conditions used to constrain the model. The geometry of a first stage gas turbine compressor blade is obtained from US patent US 7,520,729 B2 [14] and is used for both the single blade analysis and the bladed disk analysis. The bladed disk consists of 20 identical equally spaced compressor blades mounted on a disk of outer diameter of 700 mm and inner diameter of 530 mm. For the single blade analysis, a fixed support is added to the faces where the dovetail fits into the bladed disk and for the bladed disk analysis. This boundary condition is idealized via a bonded contact between the faces of the dovetail and the bladed disk. Three values of the blade rotor speed are assumed: 500, 1000 and 2000 rad/s. The speed is applied at a distance of 200 mm from the root of the blade in the Y direction and about the Z axis. The material is assumed to be linear elastic and isotropic for both the blade and the disk with the following mechanical properties: Young’s Modulus \(\textit{E}=\) 206 GPa, Poisson’s ratio \(\upupsilon =\) 0.3 and density \(\uprho =\) 7850 kg/m\(^{\mathrm {3}}\).
The geometry of both the single blade and the bladed disk was created in SolidWorks and the three dimensional finite element mesh was generated using the ANSYS® finite element package. The finite element mesh for the uncracked single blade is shown in Fig. 1a and for bladed disk is shown in Fig. 1b. The single blade is divided into separate regions to apply the required mesh controls in order to refine the mesh in regions of interest (such as high stress concentration and/or around the crack front). Since high mesh refinement is introduced around the region of the crack front, the size of the finite element model increases once the crack is introduced. For the fracture analysis study of single cracked blades, 20 noded solid brick elements with quadratic interpolation were used away from the crack and 15 noded quarter-point singular wedge elements around the crack front. The ANSYS finite element model with the associated boundary conditions for the uncracked single blade was imported into FRANC3D.
Crack modeling and meshing
Once the location of the crack and the size of the crack is identified, the region around the crack needs to be meshed appropriately to accurately calculate fracture parameters such as stress intensity factors. The following sections first outline the theory behind both programs employed in this paper to carry out the fracture analysis, describe the meshing strategy employed in both programs and a validation study is conducted to determine the effectiveness of both approaches.
M-integral for computing individual stress intensity factors
Fracture meshing in ANSYS®
Similar to FRANC3D, ANSYS\(\circledR \) also employs an adaptive mesh generation technique but the software is limited to the use of semi-elliptical cracks. Since through the thickness edge cracks are used in this study, one cannot take advantage of this feature. An alternate strategy is therefore employed, where the crack has to be meshed manually by the user.
Static fracture analysis
Appendix A summarizes a mesh sensitivity study in which the mesh density around the turbine blade crack-tip for the ANSYS model is examined with five configurations; normal mesh density, which is close to that used by FRANC3D, two coarser mesh densities and two finer mesh densities. As reported in the appendix, this study indicates that mesh configuration around crack-tip has shown that the stress intensity factors predicted by ANSYS show convergence starting from the coarse mesh around the crack-tip. Therefore, in the remainder of this section, results obtained by ANSYS correspond to those obtained by the normal mesh similar to that of FRANC3D.
Effect of crack depth on fracture parameters
Figures 7 and 8 compare the Mode I and Mode II stress intensity factors for cracks with depths of 8 and 16 mm respectively from the leading edge of the blade. It is quite clear that for each mode of crack propagation as the size of the crack increases, so does the corresponding value of the stress intensity factor.
Effect of rotational velocity on fracture parameters
It is clear that from the results of the parametric study, for all cases, the dominant mode of crack propagation is Mode I, while there is an appreciable contribution from Mode II. A mixed mode of crack propagation is therefore a likely scenario based on the applied boundary conditions.
Modal analysis
Comparison of calculated natural frequencies using linear and quadratic interpolation
Natural frequency with linear elements | Natural frequency with quadratic elements | Percentage difference (%) |
---|---|---|
310.4 | 310.3 | 0.03 |
970.5 | 969.6 | 0.09 |
1077.3 | 1076.8 | 0.05 |
2084.3 | 2081.7 | 0.12 |
2246.9 | 2243.3 | 0.16 |
2722.7 | 2718.7 | 0.15 |
Single blade
In the ANSYS Workbench\(\circledR \), the model is pre-stressed by using the same kinematic boundary conditions as in the fracture simulation. In each parametric study, the first six natural frequencies of the blade are calculated. Figure 10a–c plot the first three mode shapes of vibration, namely, the first bending mode, the second bending mode and the first torsional mode.
Effect of crack depth on the free vibration response of a blade alone
First six frequencies for uncracked and cracked single blade for rotational velocity \(=\) 500 rad/s
Mode number | Uncracked | 4 mm crack depth | 8 mm crack depth | 16 mm crack depth |
---|---|---|---|---|
1 | 311.6 | 310.4 | 305.7 | 298.9 |
2 | 984.1 | 970.5 | 939.4 | 896.2 |
3 | 1077.8 | 1077.3 | 1076.3 | 1071.2 |
4 | 2085.7 | 2084.3 | 2055.2 | 1997.7 |
5 | 2250.4 | 2246.9 | 2213.3 | 2181.0 |
6 | 2727.3 | 2722.7 | 2704.3 | 2670.1 |
First six frequencies for uncracked and cracked single blade for rotational velocity \(=\) 1000 rad/s
Mode number | Uncracked | 4 mm crack depth | 8 mm crack depth | 16 mm crack depth |
---|---|---|---|---|
1 | 404.3 | 403.5 | 398.1 | 390.8 |
2 | 1061.0 | 1057.4 | 1031.9 | 992.5 |
3 | 1122.8 | 1120.6 | 1111.8 | 1103.1 |
4 | 2151.4 | 2145.7 | 2104.5 | 2054.7 |
5 | 2308.5 | 2309.8 | 2260.3 | 2219.5 |
6 | 2795.1 | 2800.9 | 2769 | 2739.7 |
First six frequencies for uncracked and cracked single blade for rotational velocity \(=\) 2000 rad/s
Mode number | Uncracked | 4 mm crack depth | 8 mm crack depth | 16 mm crack depth |
---|---|---|---|---|
1 | 637.9 | 636.5 | 630.1 | 622.0 |
2 | 1145.4 | 1143.8 | 1137.0 | 1124.4 |
3 | 1457.4 | 1432.2 | 1401.5 | 1365.8 |
4 | 2257.9 | 2251.6 | 2241.8 | 2250.2 |
5 | 2556.7 | 2439.2 | 2384.8 | 2335.3 |
6 | 3005.6 | 2971.4 | 2929.3 | 2955.9 |
Bladed disk
The underlying assumption of the uncracked bladed disk is that each blade is perfectly tuned, both numerically and physically. This is done by ensuring that each blade has the same number and type of finite elements and subject to identical boundary conditions. For the bladed disks with a single cracked blade, cracks at three different distances from the leading edge of the blade are used.
Effect of crack depth on the free vibration response of a bladed disk
Figure 12a–c show the first modes of the first three mode families of a bladed disk with a crack at a depth of 4mm from the leading edge. Figure 13a–c show the first modes of the first three mode families of a bladed disk with a crack at a depth of 8mm from the leading edge. It is apparent that the presence of a small crack can lead to mode localization of the cracked blade in the bladed disk system. From the point of view of structural health monitoring of turbines, this is significant because mode localization makes it easier for contactless methods of crack detection to detect anomalies in the vibrational response during run up or run down. However, if one examines Fig. 12c which corresponds to the first torsional mode for a crack of depth 4 mm and compares it with the first torsional mode for a crack of depth 8 mm (Fig. 13c) it is clear that the mode is not localized around just the cracked blade, rather, the energy is distributed to blades in the vicinity. This suggests that for certain modes, the size of the crack can also affect mode localization.
The combined single blade fracture and modal analysis calculations were performed on the RAAD supercomputer of Texas A&M University at Qatar and took about 4 h of computational time. The modal analysis bladed disk simulations were performed on a high performance computing workstation connected to a PowerEdge 815 with 24 cores AMD Opteron 2.4, GHz and 128 GB RAM server and took around 2 h of computational time.
Conclusions
The present study incorporates a fracture mechanics approach to characterize an open crack in a bladed disk structure and analyze the effect of a crack on both the vibrational response of a compressor single blade and bladed disk. A fracture analysis study is carried out in ANSYS\(\circledR \) by manually meshing the region around the crack front and compared with results from a special purpose fracture code, FRANC3D. A good agreement is achieved for the Mode I and Mode II stress intensity factors, while there is a significant difference for the Mode III stress intensity factor. Based on the applied boundary conditions, the crack is likely to propagate in a mixed mode of propagation. Modal analysis of the single blade revealed that the natural frequency decreases as the size of the crack increases and increases as the rotational speed increases. A similar trend is obtained within different modal families of the bladed disk, but for certain frequencies within modal families, the bladed disk mode is localized about the cracked blade. When studying the efficacy of using a single blade analysis for purposes of structural health monitoring, it is found that for certain vibration modes, there is a significant difference between the natural frequencies of the single blade and the bladed disk, suggesting that a single bladed analysis alone may not suffice.
Declarations
Author's contributions
RF performed the fracture simulations using ANSYS and FRANC3D and the modal analysis simulations suing ANSYS. SEB worked on comparing the results obtained for the fracture simulations in ANSYS and FRANC3D. KA developed the manual zoning and meshing of the cracked blades using ANSYS and performed the modal analysis simulations of the bladed disk system on a HPC work-station. MF and NJ proofread the document and added their advice on presenting the results of the modal analysis simulation. All authors read and approved the final manuscript.
Acknowledgements
The authors gratefully acknowledge the support of the Qatar National Research Fund through Grant number NPRP 7-1153-2-432. The authors also thank Texas A&M at Qatar’s Advanced Scientific Computing (TASC) for access to the RAAD Supercomputer.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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