Complex Mode Shapes

The mode shapes examined in the Test Cases Section, are simple mode shapes based on classical flutter analysis for turbomachinery, which assumes that the blades are not structurally coupled. However, real turbomachinery blades are structurally coupled, as they are attached to the same disk. The level of structural coupling is increased if the blades are shrouded. The structural coupling causes different sections of the same blade to oscillate at different phases for some mode shapes. These mode shapes are called complex mode shapes as complex numbers can be used to describe the magnitude and phase of the modal displacements.

Complex Mode Shape Example

The example shown below, is for the subsonic flow case of the Standard Configuration 10. The aerodynamic damping versus interblade phase-angle for four mode shapes are shown in the Figure below. It is assumed that the mode shapes do not vary with interblade phase-angle. The first two mode shapes are simple torsion and flap. The last two mode shapes are combinations of a torsion and a flap mode shape with a 90 degree phase difference. The flap mode leads the torsion mode for the first complex mode, and the torsion mode leads for the second complex mode. The reduced frequency for each mode shape is 0.5. Note that the two simple mode shapes and the second complex mode shape are stable for all interblade phase-angles. However, the first complex mode is unstable for almost all interblade phase-angles.

The aerodynamic damping for all mode shapes shown above has been normalized by the maximum modal displacement.