**4**, 2, pp. 175–225, 1956

### Stateczność dynamiczna płaskiej postaci zginania przy różnych warunkach brzegowych

**Dynalmical stalbillty of plane form of bending with various boundary conditions**

The paper is composed of two parts. In the first, the boundary problem for Eq. (0.1) is set forth and solved. Using a method analogous to that used for the equa,uon of Mathieu, [4], the eigenvalues *λ* = *λ* (*γ*_{1}, *γ*_{2}) are found as well as some first coefficients of the corresponding eigenfunctions. Next, using Haupt theorem, [2], the obtained „body of dynamical stability” is divided into stable and unstable regions. It is found that the boundary surfaces *λ _{C}*

_{1}, and

*λ*

_{S}_{1}intersect along a certain curve, which is of practical importance. With a suitable choice of the parameters

*γ*

_{1}and

*γ*

_{2}, the second resonance region does not appear. The results obtained cotnstitute a complement of the tables of K. Klote r and G. Kotowski, [9], Eq. (0.1) is often met in physical and engineering problems and the knowledge of the form of the function

*λ*=

*λ*(

*γ*

_{1},

*γ*

_{2}) will make the discussion of these problems more easy. The expressions obtained can also be used when one of the parameters

*γ*

_{1},

*γ*

_{2}is a known function of the other.

The investigation of the stability of a bar of narrow rectangular crosssection loaded at the extremities with a moment *M*(*t*) and an axial force *P*(*t*), of the same angular frequency *ω*, forms the subject of the second part of the paper. Boundary conditions of simple suport arnd other types are considered. It is found that in the case of free support, *M*_{0}, *M*_{l}, *P*_{0} and *C* can be chosen in such a way as to obtain an incomplete sequence of resonance regions. In pa,rticula,r the basic resonance region is lacking. With other boundary conditions the method of K. Klotter is used in a modified form, reducing the problem to the investigation of two independent equations of the type (0.1). The analytical representation of resonance regions obtained in the case of simply supported extremities is formally the same as in the case of other boundary conditions, with suitably calculated roots of the characteristic equations of free vibration. The resonance regions for various boundary conditions are sums of resonance regions for each equation of the system (2.3.4.13).

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#### References

A. Grzędzielski i J. Nowiński, Obliczanie ostrosłupów trójprętowych, Sprawozd. I.T.L. 4,26 (1938), s. 5-66.

O. Haupt, über lineare homogene Differentialgleichungen zweiter Ordnung mit periodischen Koeffizienten, Math. Annalen, t. 79, 278 (1919).

K. K1otter, Die Biegeschwingungen eines Stabes unter pulsierender Achsialkraft bel beliebigen Randbedingungen, Ing. Arch. 18 (1950), s. 363.

N.W. McLach1an, Tieorja i priłożenja funkcji Mathieu, Moskwa 1953 (tłum. z jęz. ang.).

J. Naleszkiewicz, Zagadnienia stateczności sprężystej, Warszawa 1953.

J. A. Pratusemricz, Wariacjonnyje mietody w stroitielnoj miechanikie, Moskwa-Leningrad 1948.

W. E. Sałion, Dinamiczeskaja ustojczitvost' płoskoj formy izgiba, Dokł. AN ZSSR, t. 78, 5 (1951).

N.K. Snitko, Mietody rasczota soorużenij na wibracju i udar, Leningrad 1953.

K. Klotter i G. Kotowski. Zeitsehr. Angew. Math. Mech., t. 23, 3 (1943).