On the Generalization of Merkin’s Theorem for Circulatory Systems: A Rank Criterion

Abstract

This paper provides a generalization of the celebrated Merkin theorem. It provides new results on the destabilizing effect of circulatory forces on stable potential systems. Previous results are described and discussed, and the paper uncovers a deeper understanding of the fundamental reason for the destabilization. Instability results in terms of rank conditions that deal with the potential and circulatory matrices that describe the system are obtained, thereby generalizing this remarkable theorem. These new results are compared to those obtained earlier.

On the stability and instability criteria for circulatory systems: A review

“…This generalization was obtained in [15], and independently and, more recently, in [20], where it was also pointed out that in this case the potential matrix has at least one repeated eigenvalue (i.e., the corresponding conservative system has at least two equal natural frequencies). Although there is an uncountable infinity of skew-symmetric matrices that commute with the given potential matrix having multiple eigenvalues, as shown in [20], the commutation condition is very restrictive and some attempts have been made recently to weaken this restriction [21][22][23][24]. This criterion is a special case of a result that is related to general positional perturbations [21], and it was also obtained, in the case = , in [22].…”

“…However, for the given matrix with multiple eigenvalues, the orthogonal matrix that diagonalizes is not unique, making it difficult to apply this result. An alternative rank criterion which gives the same instability result as Theorem 3.8, but which avoids the non-uniqueness of the matrix , as well as the calculations of its eigenvalues and eigenvectors, was developed in [23].…”

“…In condition (3.5) the matrices Φ and Ω can be replaced by the matrices and , respectively, since rank Φ = rank and rank Ω = rank . The application of Theorem 3.8 and Theorem 3.9 is illustrated by the following example [23].…”

On the stability and instability of linear potential systems subjected to infinitesimal circulatory forces

“…The removal of the necessity of delving into the eigen structure of potential matrices has only recently been accomplished for MDOF systems subjected to circulatory forces. An instability criterion that uses instead the ‘gross' property of the rank of the products of potential and perturbatory matrices has been obtained, thereby providing a further generalization of Merkin's classical result [5].…”

“…This result contains, as a special case, the famous Merkin's theorem [1], which assumes the commutativity of the matrices K and N [2,3,12]. It should be noted that in applications, the condition given in (1.6) relies on an analysis of the eigen structure of the matrix K , and an appropriate choice of the orthonormal eigenvectors that are to be included in the submatrix T p [5]. Recently, a rank condition that is equivalent to (1.6) but obviates the need to examine the eigen structure and to pick the proper orthonormal vectors in T p has been developed [5].…”