Discussion
The number of natural frequencies a discrete system possesses equals
Answer: A
Correct answer: its number of degrees of freedom
The eigenvalue problem \( (K - \omega^2 M)\phi = 0 \) is of the order of the model, so an \( n \)-degree system has \( n \) natural frequencies and \( n \) mode shapes.
The count is a property of the model, not of the building: the same structure idealised with one lumped mass per floor and with a full finite element mesh yields quite different numbers. A real structure is continuous and has infinitely many; discretisation is what makes the count finite.
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