On the mismatch between predicted and measured frequencies in model testing

Abstract

Mathematical models verified against ground test data are often necessary in the aerospace industry. They are used for predicting system performance in the intended as well as in any altered operating environments. They also provide means for evaluating corrective measures when warranted. However, in aerospace model or prototype testing, predictions from analysis seldom agree with measurements. The disagreement or mismatch is essentially due to uncertainties and errors in the way system descriptions or parameters are realized in mathematical models and prototypes and in the way analysis and measurements are carried out. The uncertainties, often called random and systematic errors, can be ascertained a priori in terms of their statistics from the modeling methods, fabrication processes, and test procedures. From these statistics, those of the system characteristics (which are more often than not the system natural frequencies) could be deduced. Hence, for a chosen level of confidence, the acceptance zones for various characteristics can be obtained, both for the measurements on the model and for the prediction from analysis. However, these computations are possible only for very simple spring-mass systems. For more complex ones and for real-life aerospace systems, neither an ideal analysis is feasible nor an ideal model obtainable. Therefore, in practice, an available mathematical model is modified or updated to make it agree as well as possible with measurements. However, any such updating is based on an a priori assumption that both the physical model and the analytical representation contain no inadvertent errors, which cannot be avoided completely. The designer so far has been assuring himself about the absence of such inadvertent errors by ensuring that the difference between the values from predictions and measurements is within acceptable ranges, which he arrives at heuristically. The validity of this procedure is investigated in this thesis. For the investigation, to start with, a simple single degree of freedom system is considered. Both uncertainties and errors are introduced using appropriate models into the system parameters. To accommodate the current practice followed by the designers, a new statistic corresponding to the percentage difference between two observations—one from analysis and the other from model testing—is introduced. Further, available theory of hypothesis testing is introduced as a basis for decision-making for the designer. The procedure developed allows the designer to rule out with a high degree of confidence the presence of large errors when the percentage difference is within an acceptable range, under certain favorable conditions. The scope of the procedure is also extended to include two degrees of freedom systems by using an extension of hypothesis testing to vector-valued variables.