Some results on the frequencies of nonhomogeneous rods

Abstract

In this paper we consider the transverse vibrations of a thin cylindrical rod of variable density and constant f lexural r igidity which is clamped at both ends. In Section I we prove two results on the principal frequency of these rods. These theorems are analogous to theorems on the principal frequency of nonhomogeneous strings. Theorem 1 corresponds to a result of Beesack [1] and Theorem 2 to a result of Krein [2]. In Section I I we consider nonhomogeneous clamped rods with equimeasurable density and we prove an inequality for the min imum of the nth frequency (n =2~). This Theorem 3 should be compared with a much stronger statement on the extrema of the frequencies of equimeasurable strings. We rely heavily upon results of Beesack [1, 3]. Notation and methods are similar to those used in our paper on nonhomogeneous strings [4].