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: