The Loading-Frequency Relationship in Multiple Eigenvalue Problems

Abstract

The free vibrations of a linear conservative system with multiple loading parameters are studied, attention being restricted to pure eigenvalue problems. It is shown that the smallest frequency and external loading parameters of such a system constitute a strictly convex (synclastic) surface which cannot have convexity toward the origin of the “parameter space.” It is further proved that in the case of systems with one degree of freedom only, the surface takes the form of a plane. The practical implications of these results regarding the estimation of the frequencies and/or the stability boundary of the system are discussed. Thus it is observed that, on the basis of the established theorems, lower bounds to the frequencies at any stage of external loading and/or upper bounds to the stability boundary are readily obtainable. A two-degree-of-freedom illustrative example is discussed.

Divergence instability of multiple-parameter circulatory systems

“…Thus, we have a theorem. This result is similar to that obtained in [1] for conservative systems. We now note that the stability boundary, being the intersection of F.C.S.…”

“…Keeping this in mind, combining Theorem 1 with the similar convexity theorem for the corresponding conservative system [1], and in view of the lemma just proved we reach the following obvious conclusions: Theorem 3. The fundamental characteristic surface of the non-conservative system is tangent to the fundamental characteristic surface of the corresponding conservative system at the first natural frequency point on the £2 axis.…”

“…Damping and gyroscopic forces are assumed to be absent. The paper essentially extends the concepts introduced in [1] to circulatory systems and establishes the convexity of both the stability boundary and the first characteristic surface. Similar investigations have previously led to several theorems regarding the shape of the stability boundary of linear and non-linear conservative systems [7,8,9].…”

“…(6) describes certain M-dimensional hypersurfaces in the (M + 1)-dimensional O -Pk space which was designated as the -parameter space in [1], the M-dimensional loading space being a subspace of it. The surfaces (6) are called characteristic surfaces, the critical surfaces (defined in [7]) being their intersections with the hyperplane ft = 0.…”

“…In a recent study [1] this relationship has been explored with regard to conservative multiple-eigenvalue problems, and two theorems concerning the convexity of the characteristic surfaces have been established. It has been observed that such general results have significant practical implications and can, in fact, be used to obtain upper and/or lower bounds for the critical loads and/or frequencies of the system.…”