Structural Dynamics / module 11
Natural frequencies and mode shapes
The eigenproblem in general — and why a mode-shape ordinate printed by software means nothing on its own, while the ratio between two of them means everything.
What you will understand
- 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
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.
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.
Module summary
- (K − ω²M)φ = 0 — n degrees of freedom give exactly n frequencies and n shapes
- Mode shape magnitude is arbitrary; only the ratios between its entries mean anything
- φᵢᵀMφⱼ = 0 and φᵢᵀKφⱼ = 0 for i ≠ j — orthogonality, and it follows from symmetry alone
- Mass normalisation makes φᵀMφ = 1, so φᵀKφ = ω² and modal equations have unit mass
- A rigid-body mode has ω = 0 and stores no strain energy — legitimate only if unrestrained
- Repeated frequencies are normal in symmetric structures and any combination is also a mode
- A local mode has a large ordinate at one point and almost no effective mass
- Mode n of a shear building has n − 1 sign changes — a free credibility check
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A uniform four-storey shear building has a first natural frequency of 12 rad/s. Using the uniform-shear-building ratios (0.3473, 1.0000, 1.5321, 1.8794), what is its fourth natural frequency in rad/s?
Practice
A mode shape {0.25, 0.55, 0.80, 1.00} is on a building with 350 tonnes per floor. What is its modal mass, in tonnes?
Check yourself
A modal analysis of a symmetric square building returns modes 1 and 2 at exactly the same frequency. Is this an error?
Practice
A mass-normalised mode has a period of 0.8 s. What is its modal stiffness φᵀKφ?
Check yourself
Which of these quantities is UNCHANGED when a set of mode shapes is renormalised?