Procedures to compute the statistical accuracy achievable in estimating the natural frequency and damping parameters of randomly excited structural systems are demonstrated. Particular worked examples yield quantitative and qualitative results.
A regularly sampled stationary randomly excited n degree-of-freedom differential equation model and the corresponding mixed autoregressive-moving average (AR-MA) time series of order 2n that is known to represent the sample sequence are assumed. The elements of the Fisher information matrix, the mixed second partial derivatives of the log-likelihood functional of the maximum likelihood estimates of the AR-MA parameters, are computed from those AR-MA parameters. The matrix inverse of the information matrix, asymptotically the Cram&-Rao lower bound on the covariance matrix of the errors in the AR-MA parameter estimates, and the result that the n damping and n natural frequency parameters can be expressed in terms of the 2n AR parameters yield the achievable accuracy results on the parameters of the assumed differential equation model system.
It is demonstrated that cj. and cr respectively the coefficient of variation (the ratio of the standard deviation to the mean) of natural frequency and damping parameters, can be of the order of 0-01 and 0.2 for N = 1000 regularly sampled vibration observations and that the theoretically achievable values of ct and c~ have the following properties: (a) c~, and c~ are inversely proportional to V"~'; (b) cs and cr are essentially identical for displacement velocity and acceleration data taken at the same observation point; (c) cs and cr are quite insensitive to the presence of additive noise in the observations when the maximum likelihood parameter estimation procedure is used; (d) ct and cr are essentially independent of the intensity and correlation structure of the random excitation; (e) cs and cr are inversely proportional to T,, the data sampling interval; (f) ct and c~ for a fixed T,, for any particular mode in a system, are quite insensitive to the number of modes in that system; (g) cs varies directly and cr varies inversely with an increase in ~.