Damping and Natural Frequency Estimation Using the Least Pth Optimization Technique
Authors: Lu Xinsen , Vandiver J. Kim 
Published: October 2008

Abstract
The Least Pth Optimization method is developed for estimation of structural dynamic parameter from estimated response spectra. In this method both one-degree-of-freedom and mutli-degree-of-freedom vibration modules may be evaluated. Ordinates of the response spectrum with high signal to noise ration may be selected in fitting the model.
The method is particularly appropriate for evaluation of structures with closely spaced natural frequencies. As an example, the structural dynamic parameters of an offshore platform, equipped with a dynamic absorber, are estimated. In this example, two peak frequencies are so close that the methods based on one degree-of-freedom models, such as the half-power band- width method or the spectral moment method, fail to estimate the individual dynamic parameters. However, a two-degree-of-freedom model combined with the method of Least Pth Optimization is shown to work well.
The combination of the Maximum Entropy Method with Least Pth Optimization is suggested for estimating structural dynamic parameters from ambient response measurements.

Nomenclature
List of nomenclatures (Available in full paper)

Introduction
Estimation of structural dynamic parameters from ambient measurement in a random environment is important in collection of design and performance data and has been suggested for detection of structural damage[1], An unknown white noise input is usually assumed in the resonant frequency range since in most cases the real input force is very complex and not easily measured.
Response measurements alone are used in the estimation. The estimate can be processed in time domain or frequency domain. In the time domain the damping ratio may be estimated from the decay rate of the auto-correlation function or from the random dec signature [2, 3]. These methods are limited to estimation of the damping ratios of well separated modes. Another approach in time domain is to identify the stochastic linear system by using an autoregressive moving average (ARMA) model and the maximum likelihood method with some information criterion [4]. It must be used to estimate all system parameters simultaneously and is not suitable for estimation of only a part of the parameters; for instance, the parameters of some specific mode in a multi-degree of freedom system.
In the frequency domain, the dynamic properties of different modes of interest can be obtained from different portions of the spectral density function. There are two steps in estimating natural frequencies and damping ratios. The first step is from measurement records to the power spectral density function: The second step is from the power spectrum to system parameter estimates. In the first step, a measurement record is typically discretized and processed into an auto-correlation function. The power spectrum is then obtained by using the Fourier transform with some smoothing operation. This is often called the Blackman- Tukey method [5] or the conventional method. The fast Fourier transform, operating directly on the data is also commonly used to obtain the response spectrum. The basic problem with these methods is the conflict between high resolution and small statistical uncertainty. The only way to solve this problem is to increase the record length, which is sometimes not feasible.