Acoustic Modes and Frequencies

In the present problem disturbances are composed of multiple frequencies and multiple acoustic duct modes. The frequencies wv, ft viewed in the frame fixed to the duct, i. e., (r, в, z) system, and corresponding circumferential wave numbers nft, v of the duct modes are given in Table 1. Hereafter we denote the

Table 1. Frequencies and circumferential wave numbers of duct modes

Case Frequency Wavenumber n^,v

Vibration of rotor 1 wu + gNB 1П1 + vNb 202 + ffio^i gNB 1 + vNb 2 + &10

Vibration of rotor 2 ш2о + uNb2T12 + vNB 1П1 + (J20O2 nNB2 + vNB 1 + (J20

acoustic duct mode of (nft, v ,t) by (ц, v; t), where t denotes the radial order. A mode (ц, v; t) is cut-on if

(n)

where k denotes the radial eigenvalue [7].

Under this notation we can state that vibrating blades directly generate(>,0; t) modes. If all of these modes are cut-off and if the rotors are remotely sepa­rated, the inflience of the neighboring blade row will not be substantial. We should note, however, that vortical disturbances are convected without decay­ing. Therefore in the case of vibration of rotor 1, the vortical disturbances from rotor 1 always exert a finite inflience on rotor 2 even if all original modes (ц, 0; t) are cut-off and however large the rotor-to-rotor distance G may be. Further, any of the modes of v = 0 resulting from the interaction can be cut-on, giving backward reaction to rotor 1. The previous studies [6] indicate, how­ever, that the vortical disturbances play only a minor role in the aerodynamic interaction between the blade rows.

Note that ^i^2 < 0 in the case of contra-rotating cascades. Therefore duct modes of ц-v > 0 are of high frequency and low circumferential wave number, and are likely to satisfy the cut-on condition (10).