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The spectrum of a selfadjoint quadratic operator pencil of the form$\lambda^2M-\lambda G-A$ is investigated where $M\geq 0$, $G\geq 0$ are boundedoperators and $A$ is selfadjoint bounded below is investigated. It is shownthat in the case of rank one operator $G$ the eigenvalues of such a pencil areof two types. The eigenvalues of one of these types are independent of theoperator $G$. Location of the eigenvalues of both types is described. Examplesfor the case of the Sturm-Liouville operators $A$ are given.

On spectra of quadratic operator pencils with rank one gyroscopic linear part