Natural frequencies and mode shapes

What this module covers

• State and solve the general free-vibration eigenproblem for any number of degrees of freedom
• Explain why mode-shape magnitude is arbitrary and normalisation is a choice
• Derive orthogonality through both the mass and the stiffness matrix
• Compute modal mass and modal stiffness and say what mass normalisation buys
• Recognise rigid-body modes, repeated frequencies and local modes
• Judge whether a computed set of modes is credible

Lessons

1. 11.1

   The general eigenproblem

   From two degrees of freedom to any number: what changes is only that the determinant can no longer be expanded by hand.

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2. 11.2

   Orthogonality and normalisation

   The property that makes modal analysis possible, derived from the symmetry of M and K alone — and the scaling choices that change every printed number without changing any physics.

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