On the stability of linear circulatory systems

Abstract

The paper investigates the stability of a linear mechanical system subjected to potential and circulatory forces. Three theorems which provide stability conditions directly in terms of the coefficient matrices are established.Mathematics Subject Classification (1991). 70J25.

A note on the complete instability of linear non-conservative undamped systems

“…The next proposition supplements a result given in [9]. Also, it shows that the introduction sufficiently large non-degenerate circulatory forces (det ̸ = 0) in a stable conservative system of even degree of freedom destroys stability and makes the system completely unstable.…”

“…For many years, it has been well known that circulatory forces − can destabilize a stable equilibrium of purely potential (conservative) system, and that they can stabilize an unstable potential system [2,4]. Various results concerning the stability problem for circulatory systems can be found in [2][3][4][5][6][7][8][9][10][11][12].…”

Stability of Gyroscopic Circulatory Systems

“…Proof: We set G sk ¼ 0 in Eq. (8). Since the zero matrix commutes with all matrices, the conditions of the theorem then simply require the matrices K s and N sk to commute.…”

“…The third is dealt with by the celebrated Merkin's theorem [7], which was generalized in Refs. [8] and [9]. Reference [8] does not deal with the quintessential problem of the severe restriction of Merkin's theorem, which is only applicable to systems whose vibrational frequencies are all identical, thereby making the result very narrow in its applicability to many real-life systems.…”

“…(3) may be thought of as a generalization of the one considered in Ref. [8] in which only circulatory forces (N sk ) are included and no gyroscopic forces (G sk ¼ 0).…”