The statistical errors in the estimated correlation function matrix for operational modal analysis

Abstract

Given the random vibration of a linear and time-invariant system, the correlation function matrix is equivalent to free decays when the system is excited by Gaussian white noise. Correlation-driven Operational Modal Analysis utilises these properties to identify modal parameters from systems in operation based on the response only. Due to the finite length of the system response, the correlation function matrix must be estimated and this introduces statistical errors. This article focuses on the statistical errors due to this estimation process and the effect it has on the envelope and zero crossings of the estimated correlation function matrix. It is proven that the estimated correlation function matrix is a Gaussian stochastic process. Furthermore, it is proven that the envelope of the modal correlation function matrix is Rice distributed. This causes the tail region of the correlation function to become erroneous -called the noise tail. The zero crossings are unbiassed, but the random error related to the crossings increases fast in the noise tail. The theory is tested on a simulated case and there is a high agreement between theory and simulation. A new expression for the minimal time length is introduced based on the bias error on the envelope.

Experimental determination of structural damping of a full‐scale building with and without tuned liquid dampers

“…In reality, the correlation function matrix is estimated with finite data, and this introduces statistical errors; thus, the estimated correlation function matrix is a stochastic process 28,29 . In general, the estimated correlation function matrix has a physical part, which has persistent features in the beginning and a noise dominated part, which behaves erratically, in the tail region.…”

“…In general, the estimated correlation function matrix has a physical part, which has persistent features in the beginning and a noise dominated part, which behaves erratically, in the tail region. The width of the persistent part is called the correlation time, and the noise part is called the noise tail 29,30 . The noise tail has a biassed envelope, 29 which must be excluded from an identification process to minimize bias errors on the damping estimates.…”

“…The width of the persistent part is called the correlation time, and the noise part is called the noise tail 29,30 . The noise tail has a biassed envelope, 29 which must be excluded from an identification process to minimize bias errors on the damping estimates. Generally, increasing the time length of the measured response decreases the statistical errors, thus, the time length of the measurement series is an important parameter in OMA 29 .…”

“…The noise tail has a biassed envelope, 29 which must be excluded from an identification process to minimize bias errors on the damping estimates. Generally, increasing the time length of the measured response decreases the statistical errors, thus, the time length of the measurement series is an important parameter in OMA 29 . Brincker and Ventura 26 recommend as the minimum time length where T min is the minimum time length, f 0 is lowest natural frequency, and ζ 0 is the damping ratio.…”

Automated reduction of statistical errors in the estimated correlation function matrix for operational modal analysis

“…The statistical errors are system dependent and they cause the estimated correlation function matrix to become a stochastic process that depends on the modal parameters and the time length. These statistical errors create random errors in the correlation functions [5][6][7][8] that increase with the number of time lags.…”