Solution Philosophy

The lifting surface theory represents the disturbance flow field as summation of disturbances induced by unsteady blade loadings of both rotors. The distur­bance flow quantities are mathematically expressed in integral forms involving blade loading functions Ap1-1,(v) (r, z), etc. Once the blade loading functions are determined, the disturbance flow quantities are obtained by straightforward computation of the integrals.

In the present problem the blade loading functions are not prescribed but are unknown functions to be determined. The ft>w tangency condition at the blade surfaces gives a set of simultaneous integral equations for the unsteady blade loading functions Ap1-1,(v)(r, z) : v = 0,±1,±2,… and Ap2-i,0)(r, z) : ^ = 0, ±1, ±2,… (the case of rotor 1 vibration), or Ap1-2j(M)(r, z) : ^ = 0, ±1, ±2,… and Ap2-2,(v)(r, z) : v = 0, ±1, ±2,… (the case of rotor 2 vibration) [6, 7]. There are available various methods to solve the equations numerically. A method based on combination of Galerkin formulation and expansion of blade loading functions in terms of double mode function series is applied to the present study. Details are omitted to save space.