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By H. Leipholz (eds.)
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Additional info for Stability of Elastic Structures
6) be defined. The coordinates of the "points" a are supposed to have the interpretation v x. ' q3 (3. 7) 1 If, for example, v were the variation of the deflection of the system and P1 were the time, then the coordinates q. of 1 a slopes, curvatures, and rates of deflection. were variations of deflection, Hence, each element abstract space represents a specific state of the system. a of the Approaches to the Solution 39 For the unperturbed state, the stability of which is to be investigated, v and all its derivatives are identically equal to zero.
48) Applying once more the inequality of Scharz , one obtains R. 3 R. v 2 < - f v 2 dx . E. H > K 2 . 9). 53) 0 v and curvatures v XX , one can ensure that (~i . 53). 37), H = 0. 55) Hence, H does not depend explicitely on t and (3:55) must hold true for any time. 58) is equivalent to lvl < £ for t > t . 23) and indicates that there is stability. SWIUilarizing, the following theorem can be fonnulated: For the rod shown in Figure 6, H= 0. 82a/i 3 , His positive definite. 59) for all time. That means, the lateral deflections of the rod remain small for all time.
E. Leipholz which means that the lateral deflections of the beam have become unstable. 3. Approaches to the Solution From the previous deliberations it can be concluded that under fairly general assumptions, (expandability of the Green function of the variational equation; sign definiteness, boundedness, finite duration of the perturbation), the stability problem of an elastic system can be reduced to the discussion of eigenvalues and eigensolutions of the fundamental problem. con5ists of the conditions.
Stability of Elastic Structures by H. Leipholz (eds.)