Flutter Stability Maps

Panovski and Kielb (2000) showed, using flitter stability maps, how the modeshape and the reduced frequency were the basic parameters that con­trolled the stability of a two-dimensional LPT section. In practice only the mode-shape is relevant from a design perspective since the possible range of variation of the reduced frequency is very limited. We have extended such analysis to pairs of airfoils moving as a rigid body. The aim is to mimic the mode shapes obtained when pairs of blades are welded to increase the aero­dynamic damping of the bladed-disk assembly. The edgewise and flap modes are defined as bending modes along and perpendicular to the line that joins the leading and trailing edges, respectively. The center of torsion of the third fundamental mode is located at the l. e. of the airfoil, when pairs of blades are considered the pair is formed adding a new airfoil adjacent to the pressure side of the reference airfoil and the center of torsion of the fundamental node is kept at the l. e. of the reference section. The airfoil used in all the simulations

corresponds to the mid-section of a representative rotor blade (ainlet = 37°, aexit = 64°, MiS = 0.76).

Figure 3 displays the damping coefficient as a function of the IBFA for the different fundamental modes previously described. For both configurations it may be seen the stabilizing effect of the reduced frequency although for the single blade configuration there always exists a region of unstable IBPA for the computed range of reduced frequencies. The stabilizing effect of the welded – pair configuration may be clearly seen at the bottom of the same figure. In this case all the fundamental modes are stable for k = 0.4 being the fhp mode the most critical one. The torsion mode is highly stabilised for the welded-pair configuration and becomes neutrally stable for k = 0.1.

The damping curves of the fundamental modes have been fitted to a sine curve and the methodology described in the previous section used to construct the stability maps for both configurations to conduct a complete study of mode shape in a practical and systematic manner.

Figure 4 shows the flitter stability maps for the single blade configuration, the middle section represents the reference section and the shadow regions the locus of the stable torsion centres. It may be appreciated firstly how the airfoil is intrinsically unstable in torsion and secondly how increasing the reduced

frequency the stable region is enlarged. It is worth noting as well that while the axial mode (bending in the x direction) is stable the fhx mode (bending in the y direction) is unstable, this may inferred by realizing that a torsion axis at infinity (y ^ ж for instance, which is a stable region) generates a pure axial bending stable mode.

Figure 5 shows the equivalent map for a pair of airfoils moving as a rigid body. The upper airfoil of the pair corresponds to the upper section of the figure. The increase of the aerodynamic damping with respect the single blade configuration is clearly seen and for k = 0.4 the airfoil is stable in torsion modes whose centre of torsion is in the vicinity of the blade and in a wide range of bending directions, the only unstable mode is the fhx mode.

Only qualitative comparisons are possible with the results obtained by the research efforts of Panovski & Kielb (2000) since neither the geometry nor all the aerodynamic conditions are available, still it may be concluded that the basic steady aerodynamic conditions are comparable in first approximation and the stability map of both cases is similar as well confirming the idea that the sensitivity to the geometry and aerodynamic conditions is low.