Achievable precision of close modes in operational modal analysis: Wide band theory

This work takes up the challenge of deriving the ‘uncertainty law’ for close modes, i.e., closed form analytical expressions for the remaining uncertainty of modal parameters identified using (output-only) ambient vibration data. In principle the uncertainty law can be obtained from the inverse of the Fisher information matrix of modal parameters. The key mathematical challenges stem from analytical treatment of entangled stochastic dynamics with a large number of modal parameters of different nature and the quest for closed form expressions for identification uncertainty, whose possibility is questionable. Fortunately the problem still admits insightful closed form solution under long data, high signal-to-noise ratio and wide resonance band for identification. Up to modelling assumptions and the use of probability, the uncertainty law dictates the achievable precision of modal properties regardless of the identification algorithm used. A companion paper discusses the insights, verification, scientific implications and recommendation for ambient test planning.