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New Methods of Calculation for the Determination of a Special Form of Instability of the Elastic Equilibrium of a Cantilever Wing1

Published online by Cambridge University Press:  28 July 2016

Tullio Viola
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Summary

The superposition of the flexural instability of the plane form of a cantilever wing in its own plane, over that of the torsional elastic equilibrium, gives rise to a special phenomenon of mixed instability which has been translated analytically by Prof. C. Minelli into the problem of determining the first positive characteristic value of a certain system of two ordinary, linear and homogeneous equations, in two unknown functions, with given limiting conditions. The methods of calculation, both analytical and numerical, for the solution of this problem are described. The results confirm the intuitive inferences according to which, other conditions being equal, the backward displacement of the elastic axis worsens conditions by lowering the critical speed.

Type
Research Article
Information
The Aeronautical Journal , Volume 44 , Issue 359 , November 1940 , pp. 816 - 826
Copyright
Copyright © Royal Aeronautical Society 1940

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Footnotes

*

Work carried out at the National Institute of Applied Mathematics, Rome.

References

* Published by permission of the Ministry of Aircraft Production (R.T.P.).

2 Bending of the elastic axis in the wing plane cannot occur, if (as is nearly always realised in practice) the wing is assumed as infinitely rigid against bending in its own plane.

3 The initial conditions (7) are immediately satisfied, when r> – 1. On the other hand, this inequality is necessary in order to satisfy the third of equations (7).

4 Formulae (10) were also used for the purpose of checking the numerical calculations of the integration by series described above.

5 In effect in para. 5a, the function B(x) was assumed as an infinitesimal of the third order, so that, together with the determination of the other functions (11), it was possible to obtain in this way: (1) a wider determination of the region about the origin, within which the developments in series would be valid; (2) the satisfaction a priori of the initial conditions (7). (cf. footnote 3.)