D. L. Brown

Complex Mode Indication Function and its Applications to Spatial Domain Parameter Estimation

Abstract This paper introduces the concept of the Complex Mode Indication Function (CMIF) and its application in spatial domain parameter estimation. The concept of CMIF is developed by performing singular value decomposition (SVD) of the Frequency Response Function (FRF) matrix at each spectral line. The CMIF is defined as the eigenvalues, which are the square of the singular values, solved from the normal matrix formed from the FRF matrix, [H(jω)]H[H(jω)], at each spectral line. The CMIF appears to be a simple and efficient method for identifying the modes of the complex system. The CMIF identifies modes by showing the physical magnitude of each mode and the damped natural frequency for each root. Since multiple reference data is applied in CMIF, repeated roots can be detected. The CMIF also gives global modal parameters, such as damped natural frequencies, mode shapes and modal participation vectors. Since CMIF works in the spatial domain, uneven frequency spacing data such as data from spatial sine testing can be used. A second-stage procedure for accurate damped natural frequency and damping estimation as well as mode shape scaling is also discussed in this paper.

A frequency domain global parameter estimation method for multiple reference frequency response measurements

Abstract A method of using the matrix Auto-Regressive Moving Average (ARMA) model in the Laplace domain for multiple-reference global parameter identification is presented. This method is particularly applicable to the area of modal analysis where high modal density exists. The method is also applicable when multiple reference frequency response functions are used to characterise linear systems. In order to facilitate the mathematical solution, the Forsythe orthogonal polynomial is used to reduce the ill-conditioning of the formulated equations and to decouple the normal matrix into two reduced matrix blocks. A Complex Mode Indicator Function (CMIF) is introduced, which can be used to determine the proper order of the rational polynomials.

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.

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.

Practical Experence with Identification of Large Flexible Structures

This paper summarizes work done to experimentally characterize the combined actuator/structure/sensor system, and the control system hardware, of the ACES facility at the NASA Marshall Space Flight Center. The input-output frequency response functions of the actuator/structure/sensor system were measured for all analog actuator inputs and sensor outputs. System nonlinearity and time variance, measurements delays, and control hardware sampling delays and alias protection were investigated. Modal models of the system were generated from frequency response function data. State space models are formed from these modal parameters, for use in controller design.

Practical Trouble Shooting Test Methodologies

Trouble shooting acoustic, vibration and controls problems has been a historical application where modal analysis has been employed to characterize the troubled system or systems. For many of the troubled systems no existing analytical models or previous test results are available. Many problems involve systems which have very high overhead or loss production cost, therefore the time required to obtain a solution is important. This paper will examine test methods and characterization techniques which are well suited to trouble shooting applications including the use of semi-autonomous and autonomous modal parameter estimation and modal modeling methods which can be used to simplify and speed up the process.