On One Approach to the Wing Rigidity Determining by Specified Deformations

Аuthors

Kostin V. A.^*, Valitova N. L.^**, Filyasova V. I.^***

Kazan National Research Technical University named after A.N. Tupolev, Kazan, Russia

*e-mail: VAKostin@kai.ru
**e-mail: NLValitova@kai.ru
***e-mail: FilyasovaVI@stud.kai.ru

Abstract

The main problem in structures diagnostics and quality control is that the overwhelming majority of physical quantities cannot be measured directly. The task emerges herefrom on the physical quantities determining by the results of their manifestation. The aircraft structures rigidity characteristics determining by the specified deformations and loads relates to this problem as well.
Problems in which coefficients of the equations are unknown, and the initial, boundary and other additional conditions are specified, form a wide class of so-called coefficient inverse problems. Inverse problems are mainly ill-posed. Their solution is often ambiguous, unstable, and errors in numerical methods lead to the resultant error increase. The up-to-date theory of their solution is largely associated with the name of O.M. Alifanov, Professor at Moscow State Technical University (MAI), and the scientific school he created.
The works cited in the review indicate a high level of research and the tools employed. However, according to the opinion of the authors of the article, the integrating matrices method is the most suitable one for the purpose of the listed problems solution in the framework of test laboratories and factory design bureaus, the more so, as as it gained further development through the approximations improving of the obtained relationships and modern software. Thus, the article considers this method, which naturally (due to integration) includes the necessary function of smoothing the experimental results as well.
The method consists of two stages. At the first stage, the initial differential equations are being reduced to a form convenient for the integrating matrices application. Integration of the problem differential equations is being performed herewith. Integration is being accomplished with account for the boundary conditions and continued so many times that the highest derivative in the equation becomes the one that determines the stress state. As the result, we obtain an integral-differential equation, which at the second stage is being reduced to a system of linear algebraic equations by replacing the integrals with their matrix analogs, and then this system is solved.
The article considers in detail the two examples of the proposed approach applying to the rigidity determining of a beam operating in both bending and torsion, with inaccurately specified initial information resulting from the experimental errors.
The results revealed that the integrating matrix method application for solving the inverse problem allows finding rigidities with almost the accuracy of the measurement error. Thus, this approach may be recommended as an effective tool for finding bending and torsional rigidities when the experimental results smoothing is practically not required.

Keywords:

numerical methods, identification, wing rigidity, oscillations, finite sums method, integrating matrices

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