Flutter Design of Low Pressure Turbine Blades with Cyclic Symmetric Modes 43 in the Panovsky-Kielb method, baseline unsteady aerodynamic analyses must be performed for the 3 fundamental motions, two translations and a rotation. Unlike the current method work matrices must be saved for a range of reduced frequencies and interblade phase angles. These work matrices are used to gen- erate the total work for the complex mode shape. Since it only requires knowl- edge of the reduced frequency and mode shape (complex), this new method is still very quick and easy to use. This paper describes the complete theory for extension of the Panovsky-Kielb method, gives example results, discusses the importance of the interaction effects of the cosine and sine modes, and discusses the contribution to work associated with the steady pressure. 2. Work for General Complex Mode Shapes Work per cycle can be calculated by integrating the dot product of the local velocity vector with the local force vector over one cycle of vibration and the entire airfoil surface. Figure 2. Coordinate Systems W cyc = A T 0 − ˙ Φ · NP dtdA where ˙ Φ=velocity N = normal vector P = surface pressure [...]... Propulsion and Power, Vol 5, No .2, 1989, pp 20 4 -21 1 [2] Nagai, K and Namba, M., "Effect of Acoustic Control on the Flutter Boundaries of Supersonic Cascade", Unsteady Aerodynamics and Aeroelasticity of Turbomachines, Fransson, T H ed., Kluwer Academic Publishers, 1998, pp.165-179 [3] Xiaofeng Sun, Xiaodong Jing, and Hongwu Zhao, "Control of Blade Flutter by SmartCasing Treatment", J .of Propulsion and Power,... V., Rzadkowski R (20 02) A Coupled Fluid-Structure Analysis for 3D Inviscid Flutter of IV Standard Configuration, Journal of Sound and Vibration, 25 1 (2) , 315- 327 Gnesin V., Rzadkowski R and Kolodyazhnaya, L., V (20 00) A coupled fl uid-structure analysis for 3D fl utter in turbomachines, ASME paper 20 00-GT-0380 Rzadkowski R., Gnesin V (20 00) The numerical and experimental verification of the 3D inviscid... cascade The comparison of the calculated and experimental results for different conditions of the cascade oscillations has shown the good quantitative and qualitative agreement Keywords: fl utter, inviscid, blades 53 K C Hall et al (eds.), Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 53–63 © 20 06 Springer Printed in the Netherlands 54 1 Introduction Cases of fl utter-type instability... respect of data aquisition and data transfer Steady fl conditions are ow measured by aerodynamic probes in the upstream and downstream sections and by pressure taps on the blades’ surfaces The aerodynamic probes were calibrated to obtain the total pressure p t1 and pt2 , the steady pressures p1 and p2 , the fl angles β 1 and 2 , and the Mach numbers Ma1 and Ma2 from the ow measured pressure data 1 .2 Cascade... displacement and the unsteady aerodynamic force of the No.1 and No .2 blades when δ = 45 (deg.) In Fig.13, dot- Figure 12 Amplitude of Trailing Edge Possibility of Active Cascade Flutter Control with Smart Structure 73 Figure 13 Time History of the Blade Displacement and the Unsteady Aerodynamic Force (δ = 45 degrees) ted line indicates the results of the case without control (trailing edges of the blades... self-oscillations of blades caused by cascade fl utter, Strength of Materials, 15, 1, 104-109 Kaminer A.A., Stel’makh A.L (1996) Effect of the aerodynamic connectedly between blades of the aerodynamic damping of their vibrations, and origin of cascade fl utter, Strength of Materials, 14, 12, 1667-16 72 Kaminer A.A., Chervonenko A.G., and Tsymbalyuk V.A (1988) Method for Studying Unsteady Aerodynamic Characteristics of. .. frequency and transonic fl conditions, some of the interblade phase angles appear to be ow aerodynamically unstable Keywords: Flutter, Reduced Frequency, Experiments, Annular Cascade 77 K C Hall et al (eds.), Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 77–91 © 20 06 Springer Printed in the Netherlands 78 Introduction The demand for a decrease in engine weight and a reduction... Transactions of the Institute of Fluid-Flow Machinery, No 106, 20 00, 69-95 Rzadkowski R (1998) Dynamics of steam turbine blading Part two: Bladed discs, Ossolineum, WrocŠaw-Warszawa Tsimbalyuk V.A (1996) Method of measuring transient aerodynamic forces and moments on vibrating cascade , Strength of Materials, 28 , 2, 150-157 Tsimbalyuk V.A., Zinkovski A.P and Rzadkowski R (20 02) Experimental Investigation of. .. 120 degrees The angle Ψ of No.1 and No.3 blades was impulsively changed to 143. 42 degrees from 120 degrees at the time when the displacement of the blades started to increase Figure 9 shows the time history of the displacements of No.1 and No .2 blades as well as the unsteady aerodynamic force on these blades The dotted line indicates the result when Ψ of all blades are kept 120 degrees for comparison... study, and two characteristic results of δ = 45 and δ = -135 (deg.) cases are reported here For the feasibility study of the method, the case in which all blades were forced to oscillate with 90 degrees of inter blade phase angle were analyzed first The trailing edges of No.1 and No.3 blades were oscillated and those of No .2 and No.4 blades were not oscillated Figure 13 shows the time history of the . quantitative and qualitative agree- ment. Keywords: flutter, inviscid, blades 53 Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines, 53–63. © 20 06 Springer. Printed in the Netherlands. (eds.), et. frequency is 0 .23 and the mode shapes for the forward traveling wave are {a 20 c } = −0 .26 0 −0.036 −0.638 and {a 20 s } = −0.0 72 −0.059 0.149 Since 92. 1% of the kinetic. Kielb, R. E., and Barter, J., 20 01, ”Comparative anal- ysis of blade mode shape influence on flutter of two-dimensional turbine blades”, ISABE- 20 01- 124 3, XV ISOABE conference, September 2- 7, Bangalore,