Romuald Rzdkowski, Artur Maurin
Institute of Fluid Flow Machinery, Polish Academy of ScienceMULTISTAGE COUPLING – FREE VIBRATION
OF MISTUNED BLADED DISCS
The manuscript delivered: April 2013
Abstract: Considered here was the effect of multistage coupling on the dynamics of an aircraft rotor consisting of eight mistuned bladed discs on a drum-disc shaft. Each bladed disc had a different number of rotor blades. Free vibrations were examined using finite element representations of rotating single blades, bladed discs, and the entire rotor. In this study the global rotating mode shapes of eight flexible mistuned bladed discs on shaft assemblies were calculated, taking into account rotational effects such as centrifugal stiffening. The thus obtained natural frequencies of the blade, shaft, bladed disc and entire shaft with discs were carefully examined to discover resonance conditions and coupling effects. This study found that mistuned system has far more intensive multistage coupling than tuned one.
Keywords: multistage coupling, mistuning, bladed disc
1. INTRODUCTION
Multistage coupling effects and disc flexibility on mistuned bladed disc dynamics were presented for the first time by Bladh et al. [1], who studied single-stage and two-stage rotors of a very simple stage geometry. Shahab and Thomas [2] presented disc flexibility coupling effects on the dynamic behaviour of a multi disc-shaft system. Rzadkowski et al. [3] showed that the coupling of three identical industrial bladed discs on a shaft segment changes the mode shapes of shrouded bladed discs up to the seventh node diameter. Sharma et al. [4] analyzed a turbine rotor with 16 discs, with only one of them being bladed, under earthquake-force excitation, but they did not investigate couplings between the shaft and bladed discs. Sinha [5] carried out an analysis of two mistuned bladed discs connected together with the stiffness coefficient. Here the bladed disc was replaced by a system of masses and springs. Laxalde et al. [6] used the multistage cyclic symmetry method to show the coupling of two bladed discs, each with a different number of blades, mounted on a flexible shaft. Rzadkowski and Drewczynski [7, 8] performed an analysis of eight tuned bladed discs on a shaft with an equal number of blades on each disc to show that coupling between particular bladed discs was visible up to modes with two nodal diameters. Similar analysis [9] was also performed for a rotor equipped with different
number of blades on each bladed disc. The results showed that taking into account different (real) amount of blades is the key feature in coupling analysis. In [10] Rzadkowski and Maurin researched influence of shaft flexibility on the dynamic characteristics of 4% mistuned bladed discs in comparison with 1% mistuned system. Free vibration analysis showed that it is important to include the shaft when investigating several mistuned bladed discs since this can considerably change the spectrum of frequencies and mode shapes with zero, one, two and more nodal diameters.
2. DESCRIPTION OF THE MODEL
Figure 1 presents the analyzed aircraft engine rotor model. The main dimensions are as follows: the outer diameter of largest - turbine disc is 0.512 m, the shaft length is 1.166 m. The number of blades in each disc is different in the same proportions as in real systems. Therefore there were 28 rotor blades in compressor stage I, 41 in stage II, 41 in stage III, 47 in stage IV, 57 in stage V, 47 in stage VI, 49 in stage VII and 83 in the turbine stage.
Fig. 1. FEM model of the aircraft engine rotor
Ansys Solid45 - isoparametric, extended brick elements with 8 nodes were used. The results obtained for these elements are equivalent to those of the Abaqus 20 node isoparametric brick element. The effects of meshing size were considered. Different mesh densities were applied. A meshing convergence test based on natural frequencies was conducted. For the final calculations the smallest reasonable mesh density was applied. The final FE model in its entirety had just over a 1.5 million DOFs.
Two bearing types were modeled as springs with a stiffness of kxx=kyy= kxy = kyx =1400N/Ðm for two roller bearings and kxx=kyy= kxy = kyx =1000N/Ðm for ball bearing.
In this paper the influence of mistuning effect will be investigated, based on comparison of numerical calculations and experiment conducted in Air Force Institute of Technology. Obtained results will be subsequently compared with the steam turbine analysis results [10]. This will allow to conclude on similar bladed rotor systems.
For the numerical analysis two rotor models were developed: tuned and mistuned. The mistuning was applied by modifying the Young modulus for every blade in each stage according to blade frequencies measured on real aircraft rotor. A free vibration analysis for both models was considered. All the calculations were made with the discs rotating 15000 rpm.
3. MODE SHAPE ANALYSIS
The natural frequencies of the rotating mistuned bladed discs were calculated using the ANSYS code. In case of both models (further considered as case1 for tuned and case 2 for mistuned system) natural frequencies for all mistuned bladed discs were computed. The modes of the mistuned bladed disc were classified in an approximate way to those of the tuned bladed disc. In the case of mistuned systems, diameter modes up to two nodes may be analyzed using nodal line descriptions. However, in the case of larger nodal diameter modes such descriptions are no longer possible, because of the mistuned blades distorting the nodal lines of the mode shapes. The natural frequencies of tuned bladed discs on the shaft are divided into the modes dominated by the bladed discs and the modes dominated by the shaft with the discs.
The natural frequencies of the mistuned cantilever blades, single mistuned bladed discs and the complete flexible shaft with eight mistuned discs were carefully examined to find resonance conditions and coupling effects.
Fig. 2, 5 and 7 present the natural frequencies of the tuned system (left hand-side) and mistuned system (right hand-side) corresponding respectively to zero, one and two nodal diameters. The right axis indicates the natural frequencies of the eight discs (without blades) on the shaft, the middle axis shows the natural frequencies of the mistuned bladed discs on the shaft, while the left axis presents the uncoupled natural frequencies of single cantilever mistuned bladed discs corresponding to an appropriate nodal diameter. The uncoupled modes single blades and bladed discs are calculated separately. For example: only bladed disc VIII and all of its single mistuned blades. On Fig. 2, 5 and 7 the number or letter ‘T’ placed next to the frequency denotes the rotor stage. Letter ‘b’ indicates single blade vibrations of the certain stage. The longitudinal modes are presented in black, the torsional modes in green, the bending bladed disc modes in red and the bending shaft dominated modes in blue. Purple lines indicate couplings between single blades and whole stages. The lines connecting the natural frequencies are divided into two types: solid lines indicating strong coupling and dashed lines for weaker coupling. Strong coupling occurs when the amplitudes of particular blades are very visible, whereas weak coupling occurs when the amplitude is relatively small. In the latter case we observe the shaft and disc vibrations without the blades. The frequencies and mode shapes of mistuned bladed discs on a solid shaft were analyzed, started from zero nodal diameter modes.
2.1. ZERO NODAL DIAMETER MODES
As shown in Fig. 2, the first frequency mode corresponding to the zero nodal diameter (132.997 Hz in case 1 and 132.973 Hz in case 2) were connected with a torsional rotor mode of 180.338 Hz, causing vibration of all bladed discs, similar in both cases. The next frequency mode (477.789 Hz in case 1 and 477.649 Hz in case 2), presented on Fig. 3, connects the vibrations of first stage bladed disc and turbine. Those vibrations were also visible for unbladed rotor, and thus there is line connection with the rotor frequency of 604.336 Hz. The observed mode shape (Fig.3) for the first bladed disc reveals the influence of mistuning – in case 2, the significant difference in blade displacement can be observed.
Similar mode couplings and distortions also occurred in subsequent zero nodal diameter modes, up to frequency of 1188 Hz (see Fig. 2). Due to applied mistuning in case 2 some of the mode shapes observed in case 1 (i.e. at 671.907 Hz) did not appeared.
Fig. 2. Natural frequencies of mistuned bladed discs on the shaft corresponding to zero nodal diameter modes for both cases
Generally we may conclude that the mistuning causes diversity in particular blades amplitudes. For stage I presented in Fig 3, each blade has a slightly different amplitude, but the modes is almost zero nodal diameter. However, this rule may not hold for other modes.
499.218 1 572.994 T 670.514 2 784.799 7 795.630 6 836.068 5 930.219 4 1097.753 3 1130.979 4 1213.751 3 1278.911 T 1382.530 1 1387.650 5 1597.495 6 1635.099 7 1647.776 T 1759.037 4 1802.194 3 1837.078 2 1874.157 1 1921.100 1 180.338 604.336 all 795.075 6 809.248 7 829.780 all 916.050 4 1030.789 3 1250.254 all 1537.390 all 1793.461 1,2 1893.427 1508.464 T 132.997 T 477.789 T+1 504.565 1 816.684 6 848.288 1,2,5,7 913.930 T+4 948.439 T+4,1,2+3'' 1061.183 3 1129.139 4 1188.157 3+1'+2,4,5'' 1234.445 3+1,4,5,6,7' 1377.184 5+1'+3'' 1437.569 1,5+6,7' 1550.345 t 1585.192 6,7 1623.895 1,2,6,7+3,4,5'' 1634.950 3,7+1,6'+2,4,5'' 1702.859 4,2+3,6,7'+1,5'' 1750.999 4,3+1,2,5,6,7'' 1811.049 all 1882.121 1 2014.042 1+2'+3,5,6,7'' 180.338 604.336 all 795.075 6 809.248 7 829.780 all 916.050 4 1030.789 3 1250.254 all 1537.390 all 1793.461 1,2 1893.427 132.973 T+all 477.647 T+1+2” 498.203 1 816.872 6 848.483 1,2,5,7 913.004 T+4 947.177 T+4,1,2+3'' 1061.206 3 1097 4 1188.300 3+1'+ b 4,5,T 1228.740 3+4 +b 5 T 6,7 1611.968 1,2 +b6,7 1632.605 3+1'+b4,6,7 1695.338 4+b6 497.835 1 575.125 T 618.828 2 783.564 7 808.321 6 832.784 5 930.192 4 1096.530 3 1130.979 4 1213.751 3 1278.911 T 1403.309 1 1209.716 5 1585.114 6 1604.498 7 1508.464 T 1751.895 4 1766.978 3 1829.518 2 1865.816 1 781.288 T 1421.587 T +1'',b 5 1015.379 T 671.907 2
Fig. 3. Comparison of mode shapes of tuned (left – 477.789 Hz) and mistuned (right – 477.647 Hz ) mistuned bladed discs on shaft corresponding to the zero nodal diameter
Fig. 4. Comparison of mode shapes of tuned (left – 1189 Hz) and mistuned (right - 1188 Hz ) mistuned bladed discs on shaft corresponding to the zero nodal diameter
In the 1189 Hz mode of case 1 and the 1188 Hz mode of case 2 (see Fig. 2 and Fig. 4) we may see that zero nodal diameter is strongly distorted in mistuned system. In tuned system the coupling between all compressor stages is visible, in mistuned system the slightly distorted zero nodal diameter in stage III couples with vibrations of single blades of stage IV and single blade of turbine stage.
2.2. ONE NODAL DIAMETER MODES
Figure 5 presents the natural frequencies of mistuned bladed discs on the shaft (cases 1 and 2), corresponding to a one nodal diameter. The split in the one nodal diameter double frequencies appears in the mistuned system as an effect of blade mistuning. The bending shaft dominated modes are presented in blue. The 371.489 Hz mode in case 1 and 371.414 in case 2 is dominated by the bending motion of the shaft caused by one nodal diameter mode shape of turbine disc at 455.871 Hz. Cantilevered turbine disc vibrates at 488.417 Hz
in case 1 and 533.554(603) Hz in case 2. Next, only the 1th bladed disc vibrates, with modes of 504.332 Hz in case 1 and 500.781 Hz and 500.952 in case 2.
Mode couplings and distortions can be observed in subsequent one nodal diameter modes, up to frequency of 1790 Hz (see Fig. 5). Due to applied mistuning in case 2 some of the mode shapes observed in case 1 (i.e. at 1238 Hz) did not occurred.
Fig. 5. Natural frequencies of mistuned bladed discs on the shaft corresponding to one nodal diameter modes for both cases
Figure 6 present the multistage coupling of all rotor stages at frequency of 1790.516 Hz in case 1 (see Fig. 5) whereas in case 2 at 1801.280 Hz there is only slight 1-nodal diameter mode coupling between stage III and IV along with the vibration of single blades on stages II, VI and VII. We know that mistuning distorts nodal diameters, but here we see that multistage coupling distortion is so considerable that the one nodal diameter mode shape of the turbine disc (along with the bending mode of connecting shaft) is cancelled. Similar situation occurs in subsequent modes at 1835.190 (1840.420) Hz and 1862.212 (1864.135) Hz (case 2).
We may conclude that the higher the modes, the mistuning induced greater changes. So far our analysis has concerned the 0 and 1 nodal diameter modes (Fig. 2 and Fig. 5).
499.417 1 523.866 T 670.563 2 732.379 T 1128.304 4 1210.591 3 1278.911 T 1357.775 1 1374.704 5 1378.744 6 1430.596 7 1542.489 4 1611.601 6 1654.430 7 1773.896 3 1822.970 4 1845.199 1 1890.918 3 1907.228 1 1954.483 2 2069.735 T 455.871 971.089 1265.660 1371.759 1459.801 1513.959 1673.870 1699.376 2069.727 371.489 T 504.332 1 670.090 2 833.792 T,1,2+3,4' 1000.484 1+2+all' 1122.082 4 1193.663 3 1298.077 5+1'+4,6,7'' 1344.326 T+5' 1365.098 5+1,6,7,t' 1389.4086 5+1,6'+4'' 1452.926 5+1' 1467.796 7+!1+5'' 1472.091 7 +1'' 1496.833 1+5' 1524.0127 4+6'+7'' 1621.155 6+1'+2,3,7'' 1636.963 6,7 1638.227 3+1,2,4,6,7' 1672.780 7+1,2,3,4'+t,6'' 1743.121 4+7'+6'' 1790.516 all 1849.776 1+2'+3,4'' 1866.210 1+2,3,4'+5,6,7'' 1902.9192 1+2'' 2099.322 2,6,7+1,3,4' 500.535 1 533.554 T 622.865 2 1099.838 4 1155.427 3 1027.839 T 1375.589 1 1272.316 5 1396.438 6 1428.760 7 1541.108 4 1594.403 6 1624.034 7 1710.832 3 1816.351 4 1873.566 1 1865.879 3 1907.228 1 1843.655 2 2090.586 T 455.871 971.089 1265.660 1371.759 1459.801 1513.959 1673.870 1699.376 2069.727 371.414 T 500.781 1 618.906 2 832.626 T,1,2+3,4' 999.813 1+2+all' 1101 4 1191.825 3 stt 1387.555 6 st5 1465.878 7+1' st7 1522.128 1+2,3,4,6'' 1520.552 4,6,7 1573.814 6,7 1635.813 3+1,2,4,6,7' 1665.383 7,3+2,4'+1,T'' 1735.728 4 1801.280 3,4 1835.190 1, 2+3,4'' 1862.212 3,4 1377.032 371.520 1882.347 533.603 1163.223 1718.938 1878.030 1542.868 1817.990 1397.542 1607.024 1430.140 1637.759 833.264 1001.100 1192.058 1470.007 1502.780 1523.550 4,6,7 1607.224 1636.423 1666.535 1736.216 1840.420 1864.135 500.554 627.624 1280.461 1063.899 2094.055 1101.724 1852.402 500.952 622.949 1104
Fig. 6. Comparison of mode shapes of tuned (left – 1791 Hz) and mistuned (right - 1801 Hz ) mistuned bladed discs on shaft corresponding to the one nodal diameter
2.3. TWO NODAL DIAMETER MODES
Figure 7 presents the natural frequencies of mistuned bladed discs on the shaft (cases 1 and 2), corresponding to a two nodal diameter mode shapes. The split in the two nodal diameter double frequencies appears in the mistuned system as an effect of blade mistuning.
Fig. 7. Natural frequencies of mistuned bladed discs on the shaft corresponding to two nodal diameter modes for both cases
1006.446 t 1636.462 t 499.665 1 643.029 T 670.699 2 1139.299 4 1225.620 3 1314.557 T 1401.988 1 1404.590 5 1644.695 6 1662.288 7 1873.013 4 1874.812 1 1914.288 3 1969.872 1 1975.063 2 499.489 1 640.864 T 670.191 2 1133.891 4 1212.037 3 1314.525 t 1357.283 1 1365.098 5+1,6,7'' 1392.581 5 1565679 6,7 1759.528 3,4+6,7'' 1835.931 2+1,3,4'' 1843.747 1+3,4'' 1846.858 1,2,3,4 1912.902 1 1006.446 T 1636.462 T 502.536 1 640.500 T 632.409 2 1107.841 4 1165.577 3 1067.219 T 1424.009 1 1295.239 5 1623.568 6 1645.954 7 1865.639 4 1893.733 1 1882.290 3 1869.221 2 st1 637.626 T 627.697 2 1112 4 1210.209 3+ b 5 stT 1375.553 1 1365.098 5+1,6,7'' st5 1551.101 6,7 + b 5 1755.903 3,4+ b 6,7 1842.545 3,4+ b 2,6,7 st1 1429.215 640.575 1884.534 1867.374 1637.428 637.667 1211.937 3+ b T 1377.05 1 + b 5 1558.059 6,7 +b 5,1 1756.524 1775.010 2 + b 4,6,7 1842.724 3,4+ b 1,2,6,7 502.596 632.483 1296.591 1071.242 1114.465 1170.289 1648.870 1894.530 1876.589 632.426 1117
In the 499.489 Hz mode in case 1 only the 1st bladed disc vibrates with two nodal diameters. Due to mistuning in case 2 the equivalent mode shape cannot be found. In the 640.864 Hz mode in case 1 and 637.626 (667) Hz modes in case 2 the turbine bladed disc vibrates.
Similar mode couplings can be observed in subsequent one nodal diameter modes, up to frequency of 1760 Hz (see Fig. 7). Due to applied mistuning in case 2 some of the mode shapes observed in case 1 (i.e. at 670 Hz) did not occurred.
Fig. 8. Comparison of mode shapes of tuend (left – 1760 Hz) and mistuned (right - 1756 Hz) mistuned bladed discs on shaft corresponding to the two nodal diameter
In the 1759.528 Hz mode of case 1 and the 1755.903 and 1756.524) Hz modes of case 2 (Fig. 7, Fig. 8) strong, two-nodal diameter coupling occurs between bladed disc 4th and 5th. In case 1 additional weak coupling can be observed between 6th and 7th bladed disc, whereas in case 2 only single blades vibrate. Here it is important to note that, due to mistuning (case 2), the overall strength of the multistage coupling has weakened. Similar multistage coupling also occurred at other frequencies: 1842.545 and 1842.724 Hz.
3. CONCLUSIONS
Free vibration analysis of an aircraft rotor with 8 mistuned bladed discs showed that mistuning considerably increases multistage coupling when compared with tuned rotor system.
The mistuning distorted the nodal diameters of mode shapes and caused multistage coupling between single blades.
Free vibration analysis has shown that mistuning changes the mode shapes and number of nodal diameters in particular mistuned bladed discs.
Due to mistuning the number of resonances and couplings in the system may change. Free vibration analysis, has not sufficiently explained multistage coupling in tuned bladed discs. To better understand this problem, forced vibration analysis had to be carried out.
References
1. Bladh R., Castanier M., C. Pierre C.: Effects of Multistage Coupling and Disc Flexibility on Mistuned Bladed Disk Dynamics, Journal of Engineering for Gas Turbines and Power 125, 2003, 121-130. 2. Shahab A.A.S., Thomas J.: Coupling effects of disc flexibility on the dynamics behaviour of multi
disc-shaft system, Journal of Sound and Vibration 114(3), 1987, 435-452.
3. Rzadkowski R., Kwapisz L., Sokoowski J., Karpiuk R., Ostowski P., Radulski W.: Natural Frequencies and Mode Shapes of Rotating Three Shrouded Blades Discs Placed on the Part of the Shaft, Proceedings of the Second International Symposium On Stability Control of Rotating Machinery (ISCORMA-2), Gdask, 4-8 August 2003, 381-392.
4. Sharma B.K., Devadig H.V., Singh A.K.: Modal Time History Analysis of a Steam Turbine Rotor to an Earthquake Excitation – A 3D Approach, Advances in Vibration Engineering. The Scientific Journal of the Vibration Institute of India, Vol.4, No.4, 2005, 351-359.
5. Sinha A.: Reduced-Order Model of Mistuned Multi-Stage Bladed Rotor, Proc. of ASME Turbo Expo 2007: Power for Land, Sea and Air, May 14-17, Montreal, Canada.
6. Laxalde D., Lombard J-P., Thouverez F.: Dynamics of multi-stage bladed disks systems, Proc. Of ASME Turbo Expo 2007: Power for Land, Sea and Air, May 14-1, Montreal, Canada.
7. Rzadkowski R., Drewczynski M.: Forced Vibration of Several Bladed Discs on the Shaft Proc. Of ASME Turbo Expo 2006: Power for Land, Sea and Air, May 8-11, Barcelona, Spain.
8. Rzadkowski R., Drewczynski M.: Coupling of Vibration of Several Bladed Discs on the Shaft, Advances in Vibration Engineering, The Scientific Journal of the Vibration Institute of India, Vol.8, No.2, 2009, 125-139.
9. Rzadkowski R., Drewczynski M.: Multistage Coupling of Eight Bladed Discs on a Solid Shaft, Proc. Of ASME Turbo Expo 2010: Power for Land, Sea and Air, June 14- 18, Glasgow, UK.
10. Rzadkowski R., Maurin A.: Multistage Coupling of Eight Mistuned Bladed Disk on a Solid Shaft, Part 1. Free Vibration Analysis, ASME paper GT-2012-68391, 2012.
MIDZYSTOPNIOWE SPRZENIE – DRGANIA SWOBODNE ROZSTROJONYCH UOPATKOWANYCH TARCZ
Summary:Artyku stanowi wprowadzenie do analizy dynamiki wirnika silnika odrzutowego zawierajcego osiem rozstrojonych, uopatkowanych tarcz. Ka da uopatkowana tarcza posiadaa ró n, rzeczywist liczb opatek. W celu wyjanienia wpywu wprowadzonego na poszczególnych stopniach turbiny rozstrojenia na powstawanie rezonansów w badanym ukadzie, obliczono drgania wasne wirujcych pojedynczych opatek, uopatkowanych tarcz oraz caego wirnika wykorzystujc metod elementów skoczonych. Zbadane zostay postacie drga wasnych poszczególnych elementów jak i caego modelu biorc pod uwag wpyw prdkoci obrotowej taki jak usztywnienie pod dziaaniem siy odrodkowej. Otrzymane wartoci czstotliwoci drga wasnych poszczególnych opatek, uopatkowanych tarcz, wau oraz caego ukadu zostay uwa nie przebadane pod ktem wykrycia warunków rezonansowych oraz dodatkowych sprz e. Stwierdzono, i rozstrojenie w ukadzie powoduje zdecydowanie intensywniejsze wystpowanie sprz e w porównaniu z ukadami nastrojonymi.
