Blade Group Fatigue Life Calculation under Resonant Stresses

TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Blade Group Fatigue Life Calculation under Resonant Stresses Zlatko Petreski, Goce Tasevski Ss. Cyril and Methodius University in Skopje, Faculty of Mechanical engineering, Karpos II bb, 000 Skopje, Republic of Macedonia Abstract The results of the simulations of the blade group resonant stresses in a FE environment and fatigue life calculation are presented in this paper. Numerical calculation for determination of natural frequencies, mode shapes and dynamic stresses, based on FEM and NISA package is used. Analyses are made on the blade group with three blades with rectangular cross section and typical turbine blades with taper, pretwist and asymmetric airfoil as well. The influence of the position of the lacing wire on the resonant stresses is analyzed. Three-dimensional finite element models of the blade group are made by using twenty node isoparametric solid elements. The number of degrees of freedom is different for each model (more than 30000 DOF). The fatigue life and consequent life prediction according the stress load history of the blades is made. The results of the investigation are given in tables and graphics. Keywords FEA, Blade group, Resonant stress, Fatigue life.. Introduction Operating practice of steam turbine shows that damages to blades constitute a considerable proportion of all the damages to steam turbine, therefore dynamic analysis of blades or blade groups is of great significance. The main reason for blade failures due to a fatigue is vibration. The dynamic loads on the blades arise from many sources. The main source is the change of forces due to a blade passing across the nozzles of the stator with a DOI: 0.842/TEM6-25 https://dx.doi.org/0.842/tem6-25 Corresponding author: Zlatko Petreski, Ss. Cyril and Methodius University in Skopje, Faculty of Mechanical engineering, Skopje, Republic of Macedonia Email: zlatko.petreski@mf.edu.mk 207 Zlatko Petreski, Goce Tasevski; published by UIKTEN. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. The article is published with Open Access at www.temjournal.com frequency given by the number of nozzles multiplied by the speed of the machine. For LP blades, whose natural frequencies are low, the engine order excitation due to a non-uniform spacing of the diaphragms is important [], [2] and [3]. The work presented in this paper is a part of a project that should contain investigations connected with turbine blade fatigue behavior under static and dynamic stress fields especially in the zone of high stress, during the critical condition of blade operating, where fatigue crack usually initiates. This paper is showing results of the investigation of the dynamic behavior, resonant stresses calculation during the startup of the turbine, for a blade group with the lacing wire. Generally, the influence of the position of the lacing wire on the blade on the resonant stress is simulated. Analyses are made on the blade group with three blades with rectangular cross section and typical turbine blades with taper, pretwist and asymmetric airfoil as well. The first three tangential bending natural frequencies and mode shapes and influence of the lacing wire on them are discussed in this paper. Fatigue life estimation of the blade group with lacing wire is a multidisciplinary problem. It is influenced by the static and dynamic stress fields onthe blade, fatigue properties of the blade material, loading history, and the environment of operation. Fatigue crack usually initiates in a region of high stress under critical conditions of operation [3], [4], and [5]. In this paper, the calculated static and resonant dynamic stresses are used for fatigue life estimation [6]. For this purpose the endurance limit and endurance limit modification factor of the material and operation condition are estimated []. 2. Critical speeds and resonant stresses 2.. Blade group natural frequencies A blade group exhibits more complex dynamic behavior than a free-standing blade [7], [8], [9]. The cantilever mode frequencies are influenced by the nearest blades, the shroud and lacing wires mass and elasticity. In addition to the cantilever modes, fixpinned modes appear in groups between the TEM Journal Volume 6 / Number / 207. 73

Frequency [Hz] nodal line max. amplitude line nodal line nodal Frequency [Hz] max. amplitude Frequency [Hz] TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. cantilever modes. For the purpose of resonant stress calculation, the influences of the first three bending modes in tangential direction are included [7], [8], [9] and [0]. The numerical calculation of the natural frequencies is done by using a complex program package NISA which is based on Finite Element Analysis method. We choose this software because it contains great number of modulus that are helping during generating the mesh of the system s model and it is giving opportunities of detail structure analysis. These programs during the dynamic system analysis are using standard FEA model: [ ][ ] [ ][ ][ ] () where [K] is stiffness matrix, [M] is mass matrix, [D] refers to displacement matrix (mode shapes) and [W] 2 is diagonal matrix containing eigenvalues (natural frequencies). The results of the calculations of the natural frequencies for the first three bending modes for different models of blade groups: blade group with one lacing wire located on various places on the blade (xh 0,5H), blade group with two lacing wires with different position of the second wire (xh 0,5H), are used in forming of the Campbell diagrams for each different model. In figure. the first bending modes for different models of blade groups are given. The variation of the first tangential bending mode and natural frequency for the group of three blades connected with one lacing wire with d=5 mm on different location are given in Table. and in Fig. 2 to Fig. 4. 50 45 40 35 30 bending mode F 0.92 0.87 0.8 0.78 0.75 0.69 0.62 0.56 0.5 0.44 0.37 0.3 Height Figure 2. Variation of the first bending mode with the height of the lacing wire 250 230 20 90 70 50 0.92 0.87 0.8 0.78 0.75 0.69 0.62 0.56 0.5 0.44 0.37 0.3 Height H Figure 3. Variation of the second bending mode with the height of the lacing wire H bending mode 2F 600 580 560 540 520 bending mode 3F F F 2F 2F 500 0.92 0.87 0.8 0.78 0.75 0.69 0.62 0.56 0.5 0.44 0.37 0.3 Height H Figure 4. Variation of the third bending mode with the height of the lacing wire Figure. Bending mode shapes of blade group with one lacing wire 74 TEM Journal Volume 6 / Number / 207.

TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Table. Variation of the first three tangential bending modes with the location of the lacing wire [Hz] 2F First three tangential bending modes mode F mode 2F mode 3F 200 50 E F2 6xn 5xn 4xn 3xn F 00 2xn.0xH 39.78 29.03 576.57 0.92xH 4.32 224.85 583.54 0.87xH 4.92 226.44 582.3 0.8xH 42.64 226.8 569.96 0.78xH 43.28 226.58 562.38 0.75xH 43.24 222.66 556.87 0.69xH 43.64 23.98 553.28 0.62xH 43.73 206.08 558.9 0.56xH 43.42 200.3 568.3 0.5xH 42.66 97.5 574.23 0.44xH 4.46 97.86 567.9 0.37xH 39.9 200.62 558.75 0.3xH 38.5 204.27 553.3 50 0 F 500 000 500 2000 2500 3000 [rpm] Figure 5. Campbell diagram The results of resonant stress calculations during turbine startup for interaction of the I, II and III harmonics with the bending modes of vibration are shown in Fig. 6 to 8. xn 2.2. Critical speeds and resonant forces The process of resonant stress calculations during turbine startup is connected with resonant frequencies determination according to the Campbell diagrams, e.g. interaction of the first three bending modes of vibration with the first six harmonics of the rotational speed [0]. An example of the Campbell diagram for one of the investigated models is shown in figure 5. The force that arises when a rotor blade passes across the nozzles of a stator, which experiences fluctuating lift and moment forces repeatedly at a nozzle passing frequency is given in equation (2), and engine order excitation due to a non-uniform spacing of the diaphragms is given in equation (3). Figure 6. Resonant stresses due to a first harmonic where: FK K F F0 cosk 2nzt K FK K F F0 cosk 2nt K z number of diaphragms, - phase, F 0 static force of the pressure, F K cosk(2nzt+ K ) dynamic force-k-th harmonic. (2) (3) Because of the complex and nonlinear damping mechanisms (friction from slipping surfaces, material damping and aerodynamic damping), the equivalent viscous damping effect is used to estimate the resonant stresses with damping ratio of 0,02 [], [2], []. Figure 7. Resonant stresses due to a second harmonic TEM Journal Volume 6 / Number / 207. 75

TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Figure 8. Resonant stresses due to a third harmonic Figure. Resonant stresses due to a first harmoniclacing wire on 0,8xH The results of resonant stress calculations due to I harmonic for the model of blade group with three blades, and a change of position of the lacing wire (xh 0,56H) are shown in Fig.9 to 4. Resonant stress calculations due to I harmonic for the model with two lacing wires, with first lacing wire fixed on position xh and a change of position of the second lacing wire (xh 0,56H) are shown in Fig.5 to 9. Figure 2. Resonant stresses due to a first harmoniclacing wire on 0,78xH Figure 9. Resonant stresses due to a first harmoniclacing wire on xh Figure 3. Resonant stresses due to a first harmoniclacing wire on 0,69xH Figure 0. Resonant stresses due to a first harmoniclacing wire on 0,92xH Figure 4. Resonant stresses due to a first harmoniclacing wire on 0,56xH 76 TEM Journal Volume 6 / Number / 207.

TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Figure 5. Resonant stresses due to a first harmonicsecond wire on 0,92xH Figure 9. Resonant stresses due to a first harmonicsecond wire on 0,56xH 3. Resonant stress analysis During the simulations of the resonant stresses, a unit resonant force excitation is submitted [0]. The results of the simulations of the resonant stresses change for the models of blade group with one lacing wire mounted on different height of the blade is given in Table 2. and Fig.20. Figure 6. Resonant stresses due to a first harmonicsecond wire on 0,8xH Table 2. Influence of the position of the lacing wire on resonant stresses due to a I harmonic of excitation a [N/mm 2 ] F F 2F 2F.0xH 5.34 4.45.96 0.55 0.92xH 5.34 3.9 2.06 0.22 0.8xH 5.26 2.33 2.75 0.25 0.78xH 5.24.8 2.73 0.6 0.69xH 5 0.83.6.6 0.56xH 4.33 0.22.55 0.63 0.5xH 3.83 0.2.69 0. Figure 7. Resonant stresses due to a first harmonicsecond wire on 0,78xH Resonant stress [N/mm 2 ] 6 5 4 3 2 F F 2F 2F 0 0,9 0,8 0,6 0,5 Height H Figure 20. Resonant stresses due to change of position of the lacing wire 0,7 Figure 8. Resonant stresses due to a first harmonicsecond wire on 0,68xH It can be seen that when the lacing wire is mounted near node of the mode the resonant stresses are increased, and in the situation when the lacing wire is TEM Journal Volume 6 / Number / 207. 77

Resonant stress [N/mm 2 ] Resonant stress [N/mm 2 ] TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. mounted near antinodes of the mode the resonant stresses are decreased. The results of the simulations of the resonant stress changes for the models of blade group with two lacing wires with second wire mounted on different height of the blade is given in Table 3. and Fig.2. Table 3. Influence of the position of the second lacing wire on resonant stresses due to a I harmonic of excitation a [N/mm 2 ] F F 2F 0 9 8 7 6 5 4 3 2 0 0,02 F F 2F 2F 0,08 0,06 0,04 0,02 0,0 Damping Figure 22. Resonant stresses due to change of damping ratio 4. Static stress analysis 6 5 4 3 2 0.0xH 5.34 4.45.96 0.92xH 4.4 3.57 2. 0.8xH 4.4 3.44.28 0.78xH 4.39 3.6.5 0.69xH 4.28 2.8.7 0.56xH 3.87 0.5.39 0.5xH 3.52 0.07.52 0,9 0,8 Figure 2. Resonant stresses due to change of position of the second lacing wire The influence of the damping ratio changes on the resonant stresses is given in Table 4. and Fig. 22. It can be seen that the decrease of the damping ratio increase the resonant stresses on the blade group. 0,7 F F 2F 0,6 0,5 Height H [%] For the calculation of the static stresses on the blade it is necessary to define loads under steady conditions such as centrifugal forces where F F F in, o in, l in, b c (4) Ao Ao F in, o Fin, l Fin, b (5) is centrifugal force of the blades and lacing wire and A o -area of the blade cross section, and static components of the excitation pressure forces in tangential and axial directions (F u and F a respectively): mo F u c u c2u z (6) mo Fa c a c2a p p2 lt z (7) where: c u, c 2u and c a, c 2a are projections of the input and output speeds of the steam in tangential and axial directions calculated from the steam speed diagram shown in Fig. 23. Table 4. Influence of the damping ratio changes on resonant stresses cu c2u a [N/mm 2 ] F F 2F 2F ca c w2 c2a u w c2 0.02 5.34 4.45.96 0.55 0.08 5.93 4.92 2.4 0.62 0.06 6.67 5.6 2.4 0.7 0.04 7.62 6.35 2.75 0.84 0.02 8.88 7.42 3.2 0.98 0.0 0.6 8.94 3.84.2 Figure 23. Steam speed diagram For the purpose of fatigue life prediction, stress calculations under steady conditions are made with unit forces F u and F a, and with loads calculated from equations (6) and (7) in the operating conditions of the turbine as well. u 78 TEM Journal Volume 6 / Number / 207.

TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. mo Fu z mo F c z c c 34,24 N u 2u c p p lt 34, N a a 2a 2 2 The results of the steady stress calculations are shown in Fig. 24 to Fig. 25. factors such as surface, k a, size, k b, reliability, k c, temperature, k d, stress concentration, k e, and miscellaneous effects, k f. The endurance limit modification factor R f = k a k b k c k d k e k f can be calculated. The characteristics of the material (40Ni3) used in the calculations are u =900N/mm 2, y =650N/mm 2, D =0,5 u =450N/mm 2 and the fatigue life characteristics are f = u +345=245 N/mm 2, b - 0,085, f =0,73, c= -0,62 []. The calculated value for the endurance limit modification factor is R f =0,29 []. The contour view fatigue life for the blade group with three blades with one lacing wire positioned on xh from the fatigue life calculations using program package ENDURE and DISPLAY III modulus is shown in Fig. 26. Figure 24. Von-Misses static stresses due to a unit force F u = Figure 26. Blade group fatigue life contour view Figure 25. Von-Misses static stresses due to a unit force F a = The family of -N curves calculated from the influence of the first and second harmonic excitation on the package of three blades with lacing wire located on various places on the blade (xh 0,5H) is shown in Fig. 27 and Fig. 28 respectively. 5. Fatigue life estimation For the fatigue life prediction, the conventional stress based approach involving von Misses theory and S-N-Mean stress diagram and ENDURE software is used. For this purpose, the mean stress from the centrifugal force and the static components of the excitation pressure forces in tangential and axial directions using program package NISA are calculated. The alternating stress from resonance during transient start-up period and stress that arises when a rotor blade passes across the nozzles of a stator is calculated as well. The fatigue strength of a material is defined in terms of endurance limit of a standard rotating beam specimen of the material. The strength of the blade can be considerably smaller due to a variety of Figure 27. S-N curves for different lacing wire positionfirst harmonic excitation TEM Journal Volume 6 / Number / 207. 79

Fatigue life e+7 [s] Fatigue life e+7 [s] TEM Journal. Volume 6, Issue, Pages 73-80, ISSN 227-8309, DOI: 0.842/TEM6-25, February 207. Calculations show that the intensity of the resonant stresses has a great influence on fatigue life of the blade group. The position (height) of the lacing wire in the blade group has influence on natural frequency changes, but also in the resonant stresses changes. The calculations show that the influence of the lacing wire position on resonant stresses, and fatigue life with the mode (F) is more significant than with the mode (F). References Figure 28. S-N curves for different lacing wire positionsecond harmonic excitation.5 9.5 7.5 5.5 3.5.5-0.5 Figure 29. Fatigue life changes for different lacing wire position-f mode E+ E+0 E+09 00000000 0000000 000000 00000 0000 000 00 0 0.9 0.9 Figure 30. Fatigue life changes for different lacing wire position-f mode The influence of the resonant stresses with the mode (F) and mode (F) on the fatigue life changes for the blade group with three blades and one lacing wire located on various places on the blade (xh 0,5H) is shown in Fig. 29 and Fig. 30 respectively. 6. Conclusion 0.8 Fatigue life 0.8 Fatigue life In this paper the methodology for fatigue life determination of blade group under resonant stresses is established. This methodology defines steps that consider calculation of natural frequencies, critical speeds, resonant stresses, stresses under static loads and fatigue life estimation F F 0.7 0.7 0.6 0.6 Heigh H [%] 0.5 Heigh H [%] 0.5 []. Rao, J. S., & Vyas, N. S. (996). Determination of the Blade Stresses under Constant Speed and Transient Conditions with Nonlinear Damping. Journal of Engineering for Gas Turbines and Power, 8(2), 424-433. [2]. Vyas, N. S., Sidharth, & Rao, J. S. (997). Dynamic Stress Analysis and A Fracture Mechanics approach to Life Prediction of Turbine Blades. Mechanism and Machine Theory, 32, 5 527. [3]. Hou, J., Wicks, B. J., & Antoniou, R. A. (2002). An Investigation of Fatigue Failures of Turbine Blades in a Gas Turbine Engine by Mechanical Analysis. Engineering Failure Analysis, 9(2), 20-2. [4]. Mahri, Z. L., & Rouabah, M. S. (2008). Calculation of Dynamic Stress using Finite Element Method and Prediction of Failure for Wind Turbine Rotor. WSEAS Transactions on Applied and Theoretical Mechanics, 3(), 28-4. [5]. Wang, W. Z., Xuan, F. Z., Zhu, K. L., & Tu, S. T. (2007). Failure Analysis of the Final Stage Blade in Steam Turbine. Engineering Failure Analysis, 4(4), 632-64. [6]. Halfpenny, A. (999). A Frequency Domain Approach for Fatigue Life Estimation from Finite Element Analysis. International Conference on Damage Assessment of Structures (DAMAS 99). [7]. Pavulori, S., & Magdum, A. (205). Design and Vibratory Analysis of High Pressure Steam Turbine Moving Blade, International Journal of Innovative Research in Science, Engineering and Technology, 4(8), 795-7958. [8]. Petreski, Z. (2009). Natural frequencies of a blade group with a lacing wire. Mechanical Engineering Scientific Journal 28(), -5. [9]. Petreski, Z., Tasevski, G., & Jovanova, J. (2009). Possible ways for correction the dynamic parameters of the blade packages at the turbomachines. Proc. AMO 9 th International Conference, Kranevo, Bulgaria, 2, 39-322. [0]. Petreski, Z., & Tasevski, G. (202). Blade group resonant stress FEM simulation. MTM Machines Technologies Materials, 8, 33-36. []. Vyas, N. S., & Rao, J. S. (994). Fatigue Life Estimation Procedure for a Turbine Blade Under Transient Loads. Journal of Engineering for Gas Turbines and Power, 6(), 98-206. 80 TEM Journal Volume 6 / Number / 207.