Resonance condition in Bladed disk (Cyclic structures)

[Domainpower]

i would highly appreciate if someone could clear my doubt regarding resonance condition of a cyclic structure (say Bladed disk). Most of the articles mention two criteria to be respected:
matching frequency ( excitation force and natural frequency)
matching nodal diameter of the structure with force shape (m=k).
My question is linked with the second condition
i have found in some papers this condition more completely mentioned in the form
k = Nn + m and k = Nn- m.
where N is the number of blades and n = 0,1,2,3...
which shows that a positive work can be done on a bladed disk even when (m not equal to k).
Is this complete form of shape matching applicable to a particular kind of bladed disk (say packeted bladed disk) or is applicable in general?

Regards
Rajesh Kachroo
Alstom Power

 

[amanuensis]

The most important thing to know is : modes are orthogonal, they form an orthogonal basis of infinite dimension.

So, if the structure is excited by a punctual force at frequency f, then all the modes (an infinity of modes) respond. But the amplitude of each mode is depending on the frequency range between the resonance and the excitation.

If the excitation is not punctual but has rather the same shape as one mode shape, then the orthogonality involves the cancellation of all the other modes.