Stability of Potential Systems to General Positional Perturbations

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].…”

“…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]. It is obvious that under condition (3.2) equations (1.2) may be decoupled by using a coordinate transformation determined by the orthogonal matrix into two subsystems, one of which assures instability.…”

“…[21]. Let the potential matrix have a single eigenvalue 0 with multiplicity 2,and let = [ | ] be an orthogonal matrix, where the × submatrix contains any 2 eigenvectors (their order is immaterial) of corresponding to the multiple eigenvalue, and the × submatrix contains the remainder (i.e., = − ) of the eigenvectors of .…”

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

“…Though the subject has been investigated for several decades, our knowledge of the conditions for stability and instability of linear potential systems subjected to positional perturbative forces is apparently still far from complete. This is highlighted by the recent observation that such instabilities can be induced even when the potential and perturbatory matrices do not commute and are not necessarily circulatory [4]. Routes to instability caused by such positional forces, both finite and infinitesimal, are shown to be dependent on the interaction of the positional perturbative matrices (forces) and the potential matrix through the potential matrix's eigen structure.…”

“…r = n − p ) of the eigenvectors of K . Then, if the following conditions hold TpTNTp≠0,1emTpTNTr=0, the system (1.4) is unstable by flutter [4]. 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].…”