5.1 Expected modes of interactions
The components interactions are important sources of pressure
fluctuations and vibrations in turbomachinery. It is therefore useful to
know the modal frequencies in respect of the resonance leading to
operation instability and fatigue of blades. The \(n^{\text{th}}\) blade
passing in front of the \(m^{\text{th}}\) vane receives the same fluid
force as passing in front of the next \({(m+1)}^{\text{th}}\) vane,
and after an entire revolution the\(\ \) \(n^{\text{th}}\) blade
receives \(N_{s}\) times the same force [31]. Also, for a number of
blades \(N_{r}\) and vanes\(\text{\ N}_{s}\), after an entire revolution
each blade receives \(N_{s}\) times the same fluid force. The blade
takes \(\frac{2\pi}{\mathrm{\Omega}}\ \)for completing an entire
revolution and receives a repetitive force with a
period\(\ T_{s}=\frac{2\pi}{\mathrm{\Omega}N_{s}}\), and thus the
resulting force over the blades related to RSI could be described by the
Fourier series [32], as follows: