A consistent-mode indicator for the eigensystem realization algorithm

A new method is described for assessing the consistency of structural modal parameters identified with the Eigensystem Realization Algorithm. Identification results show varying consistency in practice due to many sources including high modal density, nonlinearity, and inadequate excitation. Consistency is considered to be a reliable indicator of accuracy. The new method is the culmination of many years of experience in developing a practical implementation of the Eigensystem Realization Algorithm. The effectiveness of the method is illustrated using data from NASA Langley's Controls-StructuresInteraction Evolutionary Model. Introduction The dynamic behavior of most aerospace structures is adequately described using modal parameters (natural frequencies, mode shapes, damping factors, and modal masses). The objective of structural modal identification is to obtain a valid modal representation over a specified frequency range for all spatial degrees-of-freedom. This objective is considerably different than identifying an input-output map only at particular degrees-of-freedom where control actuators and sensors are located. 1 A full spatial modal representation permits several tasks to be performed which cannot be performed using an inputoutput map derived for control purposes. These tasks include validation of structural modeling procedures and assumptions, prediction of system dynamics using modal parameters of individual components, investigation of more-effective actuator and sensor locations for control purposes, and improved characterization of disturbances occurring at unexpected locations on a spacecraft during operation. It is relatively straightforward to estimate structural modal parameters experimentally using a variety of available approaches.2, 3 However, it is generally much more difficult to establish reliable confidence values for each result. Confidence criteria based on noise characteristics are available 4 but are of limited usefulness in practical applications. In modal-survey tests, identification difficulties arise primarily from high modal density, nonlinearity, weakly excited modes, local modes, nonstationarities, rattling, etc., not from instrumentation noise. The simultaneous effects of these conditions are in general impossible to include explicitly in confidence calculations. The Consistent-Mode Indicator (CMI) described in this paper provides a reliable, relative measure of accuracy for structural modal parameters identified with the Eigensystem Realization Algorithm (ERA). 5"7 A single value ranging from 0 to 100 percent is obtained for each identified mode. Furthermore, the results can be decomposed into constituent components associated with each input (initial condition) and output (response measurement), or input-output pair. Both temporal and spatial consistency calculations are included in the formulation. Modes with CMI values greater than approximately 80 percent are identified with high confidence. Modes with values ranging from 80 to 1 percent display moderate to large uncertainty. Fictitious "computational modes" have CMI values of zero. The first part of this paper contains a brief summary of ERA followed by a complete description of CMI. The second part illustrates the concepts using recent laboratory data from NASA Langley's Controls-Structures-Interaction (CSI) Evolutionary Model (CEM). 8 The CEM is a large flexible research structure being used to experimentally assess the level of confidence with which CSI technolegy can be applied to future spacecraft. The Eigensystem Realization Algorithm A finite-dimensional, linear, time-invariant dynamic system can be represented by the state-variable equations: i(t) = Acx(t) + Bu(t) y(t) = Cx(t) where x is an n-dimensional state vector, u is a pdimensional excitation vector, and y is a q-dimensional response vector. A special solution to these equations is the impulse response function: