Abstract

Construction stiff features revision on results of experiment

V.A.Kostin, M.Yu.Toropov

When studying the mechanical systems, there is a need for designing their mathematical models. For this purpose drawings and other documentation are used. But the more complex a mathematical system is the harder it is to design an identical mathematical model. It is not always possible to increase accuracy of a model complicating it, and sometimes it is absolutely impossible. As the parameters going into equation are assumed with large assumptions, they may be considered as unknown for a specific problem of mathematical modelling. For their determining indirect data should be used: information about solution of the problem, which can be easier obtained experimentally. Such problems are mathematically inverse. The solution of the problem of identification of the dynamic parameters such as the mass and stiff features, shows that a simplified mathematical model with more precise equation factors can, in a number of cases, reflect real occurring processes better than a more complex model, but with the abundance of the approximate equation factors. In practice all identification methods are based on minimization of the divergence between the model calculations and experimental data.

This article deals with a common problem of recovering the stiff (rigid) features, on the basis of the results of the experiment. In other words, it is about revision of some parameters of the mathematical model by the method of identifications of the systems. The problem is solved for the stable process of stimulated oscillations; damping transients are not considered in this article. Its solution makes it possible to carry out experiments at conditions close to real.

Such a problem appears in the process of performing stand tests and diagnosing (of) real aircraft objects. In the process of using or testing, an obvious change of the design’s stiffness takes place, which is to be defined in the real scale of time, using the result of measurements as signal of feedback. Readings of tensor sensors are used as feedback signal-values, proportional to {y”}.

Examples of identification acerbity for the console beam of variable section and the flimsy design are given in the article. An individual mathematical model, reflecting unique features is used in every case. Now the choice of a mathematical model comes easy. There are a lot of them in the study of sturdiness, and they are all tried out. Therefore a designer must choose a proper model for the design.

In the example given, the {y”} values were calculated throughout the length of the design and then a casual unbalanced error within the ambit of ± 10% was entered. The error is the model of the result of experiment. After that rigidity in bending was identified.

The advantages of the integral methods are discussed, as in contrast with the differential methods. The differential methods have a significant inaccuracy of certainly-differential approximations. They are unstable to inaccuracy of raw data. Since raw data for identifications are the results of the experiment, the presence of inaccuracy in them is inevitable. Accuracy can be raised by reducing a step. However, step can be reduced to certain limit only, which is equal to the distance between the sensors.

The integral methods are used in this article. These methods can give you firmer solutions. What’s more, these methods are not very much sensitive to inaccuracies. In particular, the method of integrating matrices is used. It enables to present equations in the matrix type. Sign of integral is changed to a certain matrix presenting a numerical analogue of integral. Thereby the main complication in solution of inverse problems-obtaining correct solution-is overcome.

An experiment was conducted for checking the strategy. The results turned out to be a good coincidence with the theory. This shows a possibility of practical application of the stated method of identification. This method can be used for adaptive control of experiment, for mathematical modelling of design behavior in the models where making a special experiment is impossible or very expensive.

[Contents]