A. W. Phillips

Autonomous Modal Parameter Estimation: Methodology

Traditionally, the estimation of modal parameters from a set of measured data has required significant experience. However, as the technology has matured, increasingly, analysis is being performed by less experienced engineers or technicians. To address this development, frequently software solutions are focusing upon either wizard-based or autonomous/semiautonomous approaches. A number of autonomic approaches to estimating modal parameters from experimental data have been proposed in the past. In this paper, this history is revewed and a technique suitable for either approach is presented. By combining traditional modal parameter estimation algorithms with a-priori decision information, the process of identifying the modal parameters (frequency, damping, mode shape, and modal scaling) can be relatively simple and automated. Examples of the efficacy of this technique are shown for both laboratory and real-world applications in a related paper.

Autonomous Modal Parameter Estimation: Application Examples

Autonomous modal parameter estimation is an attractive approach when estimating modal parameters (frequency, damping, mode shape, and modal scaling) as long as the results are physically reasonable. Frequently, significant post processing is required to tune the autonomous estimates. A general autonomous method is demonstrated with no post processing of the modal parameters. Example case histories are given for simple measurement cases taken from the laboratory (circular plate) as well as realistic field measurement cases involving significant noise and difficulty (bridge). These application case histories explore the successes and failures of the autonomous modal parameter estimation method and demonstrate the limitations of practical application of automated methods.

Combined State Order and Model Order Formulations in the Unified Matrix Polynomial Method (UMPA)

The unified matrix polynomial (coefficient) method (UMPA) has been used by the authors to provide a single, educational framework that encompasses most commercial and research methods used to estimate modal parameters from measured input-output data (normally frequency response functions). In past publications of this methodology, the issue of state order has not clearly been identified in the formulation of the UMPA model. State order refers to the order of the base vector that is an elementary part of the basic UMPA model and has been a part of the modal parameter estimation development since the Ibrahim Time Domain methods in the mid 1970s. The UMPA model is restated to clearly identify the role of base vector order and the relationship between base vector (state) order and polynomial model order. This relationship provides a mechanism for explaining a number of modal parameter estimation methods that have not previously been identified and helps to explain the sensitivity of different modal parameter estimation methods to noise.

Modal Parameter Estimation Using Acoustic Modal Analysis

Acoustic modal analysis (AMA) is of interest in cases where accelerometer measurements are limited by mounting techniques and where the mass of sensors affects the system dynamics. Major problems in performing AMA are time delay adjustment and the inability of obtaining true driving point measurements. For an impact test, the former problem causes difficulties because each measured acoustic frequency response function (AFRF) will have its own time delay as a function of the position of the reference microphone with respect to the structure. Thus, obtaining consistent modal parameters conventional multi-input, multi-output (MIMO) modal parameter estimation methods utilizing several microphones (MIMO AFRFs) becomes rather difficult. The latter problem complicates the computation of modal scaling which is frequently required in model validation. As an example, both microphone measurements and accelerometer measurements are utilized in an impact test for a heavy ringdisc structure. The results from each method are compared to study the effectiveness of estimating modal parameters from AFRFs compared to conventional FRFs. While some conventional modal parameter estimation tools such as the consistency diagram and the complex mode indicator function (CMIF) look slightly different, the frequencies, damping and mode shapes estimated using AFRFs are consistent with those of standard modal analysis.

Normalization of Experimental Modal Vectors to Remove Modal Vector Contamination

When modal vectors are estimated from measured frequency response function (FRF) data, some amount of contamination in terms of random and bias errors is always present. The sources of these errors may be the experimental data acquisition process (calibration inconsistencies, averaging limitations, leakage errors, etc.) or due to limitations of the modal parameter estimation methods (mismatch between measured FRF data and the model form). These random and bias errors include uncertainty in complex magnitude about the central axis of the modal vector as well as rotation of the central axis. In a number of practical applications, particularly those involving close modal frequencies, the contamination of a modal vector will often have a significant influence from the modal vector that is near in frequency. In these situations, the numerical procedure for estimating the final, scaled modal vector, in terms of residue, generally involves a linear estimation method that, with MIMO FRF data, utilizes a weighted least squares solution procedure. This numerical solution process is reviewed and a real normalization of the weighting vectors used for estimating each modal vector in the MIMO FRF case is shown to reduce the contamination from nearby modal vectors. Theoretical evaluations for both proportional and non-proportional analytical cases are evaluated, as well as, results for a real application with pseudo-repeated modal frequencies and associated modal vectors that has historically demonstrated the problem.

Autonomous Modal Parameter Estimation: Statisical Considerations

Autonomous modal parameter estimations may involve sorting a large number of possible solutions to develop one consistent estimate of the modal parameters (frequency, damping, mode shape, and modal scaling). Once the final, consistent estimate of modal parameters is established, this estimate can be compared to related solutions from the larger set of solutions to develop statistical attributes for the final, consistent set of modal parameters. These attributes will include sample size, standard deviation and other familiar variance estimates. New variance estimates are introduced to categorize the modal vector solution. These modal vector statistics are based upon the residual contributions in a set of correlated modal vectors that are used to estimate a single modal vector. Examples of this statistical information is included for a number of realistic data cases.

Integrating Multiple Algorithms in Autonomous Modal Parameter Estimation

Recent work with autonomous modal parameter estimation has shown great promise in the quality of the modal parameter estimation results when compared to results from experienced user interaction using traditional methods. Current research with the Common Statistical Subspace Autonomous Mode Identification (CSSAMI) procedure involves the integration of multiple modal parameter estimation algorithms into the autonomous procedure. The current work uses possible solutions from different traditional methods like Polyreference Time Domain (PTD), Eigensystem Realization Algorithm (ERA) and Polyreference Frequency Domain (PFD) that are combined in the autonomous procedure to yield one consistent set of modal parameter solutions. This final, consistent set of modal parameters is identifiable due to the combination of temporal information (the complex modal frequency) and the spatial information (the modal vectors) in a Z domain state vector of relatively high order (5–10). Since this Z domain state vector has the complex modal frequency and the modal vector as embedded content, sorting consistent estimates from hundreds or thousands of possible solutions is now relatively trivial based upon the use of a state vector involving spatial information.