Multiplication of Blade Loading Frequency

Assume that the blades of rotor 1 are vibrating with a single frequency и 10 and an inter-blade phase angle 2na10 /Nв 1, so that the displacement normal to
the blade chord of the m-th blade is given by

ai(r, z )вІШ10 t+i2™10m/NB 1 : m = 0,1 ,…,Nb i – 1. (2)

Here ст10 is an integer between —Nb 1/2 and Nb 1/2.

Then as described in details in references [6, 7], aeroacoustic coupling be­tween the rotors in mutual motion produces ft>w disturbances of multiple fre­quencies, resulting in blade loading of multiple frequencies. Thus we can de­scribe the unsteady blade loading (pressure difference between upper and lower surfaces of blades) on the m-th blade of each rotor as summations of multiple frequency components:

Ap1-1>(v)(r, zi)еіШ1"t+i2naivm/NB 1

V = — Ж

and

Y AP2-1,(p)(r, Z2 )eiU2 t+i2™2^ m/NB 2 , (3)

^,= — Ж

where

W1v = W10 — vNB 2 (^1 — ^2), ^1v = vNB 2 + V10, (4)

W2^ = W10 + (^NB 1 + C10 )(^1 — ^2), 02^ = ^NB1 + ^10. (5)

Further subscripts 1-1 and 2-1 imply the loading on rotor 1 blades due to vi­bration of rotor 1 blades themselves and the loading on rotor 2 blades due to vibration of rotor 1 blades respectively. It is worth emphasizing that all fre­quency components are coupled with each other and can not be determined independently.

In the case of blade vibration of rotor 2 with displacement normal to blade surface given by

a2(r, Z2)eiW20t+i2™20m/NB2 : m = 0,1,…, Nb2 — 1, (6)

similar formulations can be made, and the blade loadings may be written as

Y Ap2-2,(v) (r, Z2)еіШ2^t+i2™2"m/NB2

V = — ^

and

Y Ap1-2)(^)(r, Z1 )еіШ1^t+i2n^m/NB 1, (7)

^,= — Ж

where

W2v = W20 — vNb 1 (^2 — ^1) W1^ = W20 + (^NB 2 + ^20 )(^2 — ^1)