Structure Model
The structure rotor blades model is based on a modal approach of the coupled fbidstructure problem (Bathe and Wilson 1976, Rzadkowski 1998). The first step of the modal approach consists of solving the problem of the natural mode shapes and eigenvalues without damping and in a vacuum. Then the displacement of each blade can be written as a linear combination of the first N modes shapes with the modal coefficients depending on time. Taking into account the orthogonality property of the mode shapes the equation of motion reduces to the set of independent differential equations relatively to modal coefficients of natural modes. The modal forces are calculated for each iteration with the use of the instantaneous pressure field calculated form the fbw code (Gnesin and Rzadkowski 2000).
2. Numerical Results
The numerical calculations presented below were carried out for the stage of the turbine with rotor blades length of 0.765 m. The number of stator blades is equal to 56, the number of rotor blades is equal to 96. The stator to rotor blade number ratio of 56:96 (7:12). All geometrical parameters of the blade are presented in Rzadkowski 1998.
It was assumed that the pressure behind the rotor blades is changing in the circumferential direction (measured by the angle around the axis of rotation of the turbine). For circumferential angle а Є (0, 90°) p2=6000 Pa, а Є (90°, 180°) p2=7500 Pa, а Є(180°, 270°) p2=9000 Pa, а Є (270°, 360°) p2=7500 Pa (see Figures 2 pressure = p2/(pka2),pka2 =9467 Pa). The unsteady forces acting on the ith rotor blades, in axial, tangential and radial directions were found.
Figure 2. The pressure distribution behind the rotor wheel 
The numerical and experimental verification of the numerical code is presented in Rzadkowski and Gnesin 2000.
The numerical calculations have been made using the computational Hgrid of 11*24*60 grid points for each stator passage and 11*14*60 grid points for each rotor passage.
One of the important aspects of statorrotor interaction is the effect of the blade response with taking into account the excitation caused by the flow uniformity and excitation due to blades oscillations.
The blade vibrations are defined with taking into account the first ten natural modes shapes of rotating blade. The values of natural frequencies and the mechanical damping coefficients hi = 2ші fi, are given in Table 1. The modal damping coefficients were assumed (Rzadkowski 1998): = 0.00075, =
0.00094, ^3 = 0.0011, £з = ^4 = £io.
Table 1. Natural frequencies and mechanical damping coefficients of the rotating rotor blade L=0.765 m

The unsteady force is the unperiodic function in time. The forces acting on the various blades differ one from another. We are using here the term the unsteady modal force which is equal along the blade length and correspond to the particular mode shape. This is disadvantage of the modal superposition calculations, where modal force averaged along the length of the blades is calculated. After the start regime, there began the coupled vibrations where unsteady forces in the turbine stage are the result of continuous interaction between gas fbw, rotation of the rotor wheel and blades vibration. So, it is impossible to separate the unsteady effects caused by the external excitation and the unsteady effects due to blades vibration.
Figures 3 – 4 shown the unsteady modal forces corresponding to the 1st, 2nd, 4th and 8th modes for the 1st blades. Generally the low frequency excitation is predominant.
Figures 5 a, b present the modal components of the unsteady modal force corresponding to the first mode. The high frequency excitation appeared for 2800 Hz and is equal to 1 % of the steady force Ao=27.5 [N]. The low frequency excitation caused by nonuniform pressure distribution is 158 % of Ao for frequency 50 Hz (see Figure 5b).
Figures 6a, b present the modal components of the unsteady modal force corresponding to the second mode. The high frequency excitations appeared for 2800 Hz and is equal to 2% of the steady force Ao=35.5 [N]. The low frequency excitation is 38 % of Ao for frequency 50 Hz (see Figure 6b).
Figures 7a, b present the modal components of the unsteady modal force corresponding to the fourth mode. The high frequency excitations appeared for 2800 Hz and is equal to 2 % of the steady force Ao=27.0 [N]. The low frequency excitation is 78 % of Ao for frequency 50 Hz (see Figure 7b).
Figure 3. The unsteady modal forces of the 1st, 2nd modes 
Figures 8a, b present the modal components of the unsteady modal force corresponding to the 8th mode. The high frequency excitations appeared for 2800 Hz and is equal to 5 % of the steady force Ao = 6.8 [N]. The low frequency excitation is 600 % of Ao for frequency 50 Hz (see Figure 8b).
It should be noted that only first four modes bring their contribution to the blade motion. The low frequency unsteady forces caused by nonuniform pressure distribution are higher in comparison to the high frequency excitations.
The modal coefficients of the 1st blade motion corresponding to the 1st, 2nd, 4th and 8th modes shape have been shown in Figures 9 12.
The unsteady amplitude of the first mode (see Figure 9) has frequency 73 Hz (99 Hz the natural frequency) and the frequency closes to 100 Hz. The
Figure 4. The unsteady modal forces of the 4th, 8th modes 
unsteady amplitude of the second mode (see Figure 10) has frequency 70 Hz and 157 Hz (160 Hz the natural frequency).
The unsteady amplitude of the fourth mode (see Figure 11) has frequencies 77 Hz and 280 Hz (297 Hz the natural frequency). The unsteady amplitude of the 8th mode (see Figure 12) has frequencies 77 Hz (862 Hz the natural frequency).
The spectrum includes mainly the blade oscillation frequencies closed to their natural ones (not multiple to the rotation frequency).
3. Conclusions
A partially – integrated method based on the solution of the coupled aerodynamic and structure problem is used for calculation of the unsteady 3D fbw through a turbine stage with taking into account the rotor blades oscillations. The paper has investigated the mutual influence of both outer nonuniform distribution of the pressure behind the rotor blade and rotor blades rotation and oscillations. The interblade phase angle of blades oscillations depends not
Figure 6. The amplitudefrequency spectrum for the modal force of the 2nd mode
only on unsteady forces lag but on the blade natural frequencies, as well. The low frequency unsteady forces caused by nonuniform pressure distribution are higher in comparison to the high frequency excitations. It has shown that
Figure 9. Amplitudefrequency spectrum of the blade oscillations by 1st mode 
Figure 10. Amplitudefrequency spectrum of the blade oscillations by 2nd mode 
References
Bakhle, M. A., Reddy, T. S.R., and Keith T. G. (1992). Time Domain Flutter Analysis of Cascades Using a FullPotential Solver, AIAA J. vol.30, No 1, p.163.
Bathe K., Wilson E. (1976). Numerical Methods in Finite Element Analysis, PrenticeHall, Inc., Englewood Cliffs, New Jersey.
Bendiksen O. (1998). Nonlinear blade vibration and flitter in transonic rotors, Proc. of IS – ROMAC – 7, The 7th Intern. Symp. on Transport Phenomena and Dynamics of Rotating Machinery, 2226 February, Honolulu, Hawaii, USA, 664.
Carstens V., Belz J. (2000). Numerical investigation of nonlinear fliidstructure interaction in vibrating compressor blades, ASME paper 2000GT0381, 2000.
Amplitudefrequency spectrum of the blade oscillations by 4th mode
Amplitudefrequency spectrum of the blade oscillations by 8th mode
Chew J. W., Marshall J. G., Vahdati M. and Imregun M. (1998). PartSpeed Flutter Analysis of a WideChord Fan Blade, T. H. Fransson(ed.), Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Kluwer Academic Publishers, Printed in the Netherlands. 707724.
Gnesin V., Rzadkowski R. and Kolofyazhnaya, L., V. (2000). A coupled fludstructure analysis for 3D flitter in turbomachines, ASME paper 2000GT0380.
Gnesin V., and Rzadkowski R. (2000). The theoretical model of 3D flitter in subsonic, transonic and supersonic inviscid fbw, Transactions of the Institute of FluidFlow Machinery, No. 106, 4568.
Gnesin V., Rzadkowski R. and Kolodyazhnaya, L., V. (2000). A coupled fliidstructure analysis for 3D fitter in turbomachines, ASME paper 2000GT0380.
Hall, K. C. and Silkowski, P. D. (1997). The Infhence of Neighbouring Blade Rows on the Unsteady Aerodynamic Response of Cascades, ASME Journal of Turbomachinery, 119,8593.
He L. (1994). Integration of 2D fliid/structure coupled systems for calculation of turbomachinery aerodynamic, aeroelastic instabilities, Journal of Computational Fluid Dynamics. 3, 217.
He L. and Ning W. (1998). Nonlinear harmonic analysis of unsteady transonic inviscid and viscous fbws, unsteady aerodynamics and aeroelasticity of turbomachines, Proceedings of the 8th International Symposium held in Stockholm, Sweden, 1418 September, 183189.
Moyroud F., JacquetRichardet G., and Fransson T. H. (1996). A modal coupling for fliid and structure analysis of turbomachine flitter application to a fan stage, ASME Paper 96GT – 335, 119.
Namba, M. and Ishikawa, A. (1983). Threedimensional Aerodynamic Characteristics of Oscillating Supersonic and Transonic ennular Cascades, ASME J. of Engineering for Power 105,138146.
Rzadkowski R., Gnesin V. (2000). The numerical and experimental verification of the 3D inviscid code, Transactions of the Institute of FluidFlow Machinery, No. 106, 2000, 6995.
Rzadkowski R., Gnesin V. (2002). 3D Unsteady Forces of the Transonic Flow Through a Turbine Stage with Vibrating Blades, ASME Paper GT2002300311.
Rzadkowski R. (1998). Dynamics of steam turbine blading. Part two: Bladed discs, Ossolineum, WrocSawWarszawa.
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