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On The Stability Of Rotating Systems

Krešimir Veselić
Published 1995 · Mathematics

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We consider small vibrations of rotating systems with a finite number of degrees of freedom. We give an almost complete characterisation of the stability of these systems for large angular velocities. The criterion is the positive definiteness or indefiniteness of the ”effective stiffness” matrix, obtained by averaging the original, asymetric stiffness around the rotation axis. The effective stiffness may be positive definite even if the original stiffness is not. Its eigenfrequencies are shown to determine the behaviour of the real eigenfrequencies in the large angular velocity limit. The threshold of the upper stability region is determined by the highest eigenfrequency of the (possibly unstable) non-rotating system. Moreover, the coincidence of the rotation frequency with any of the frequencies of the non-rotating system almost always produces instability. We consider special gyroscopic systems (e.g. rotating shafts), described by the linear differential equation ([3], 5.6) [ M 0 0 M ] [ ẍ y ] + 2ξ [ 0 −M M 0 ] [ ẋ ẏ ] +W [ x y ] − ξ [ M 0 0 M ] [ x y ] = 0 . (1) Here M (the mass matrix) is symmetric and positive definite of order n, ξ > 0 is the angular velocity of the rotating system and W = [ W11 W12 W T 12 W22 ] is the stiffness matrix. The stiffness matrix describes the strain as well as the axial load and is symmetric but not necessarily positive definite. Some results concerning the stability of such systems were obtained in [5], [6], [4], [1] and [2]. In particular, [6], [1] and [2] are applicable to the systems of the form (1) with W − ξ [ M 0 0 M ] (2) negative definite. In this paper we consider our gyroscopic system as a function of the parameter ξ and ask under which conditions on M and W the system will be Ljapunov stable for ξ large enough. ∗Fernuniversitat Hagen, Lehrgebiet Math. Physik, Postfach 940, 5800 Hagen 1, Germany. e-mail: MA704@DHAFEU11.BITNET



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