Abstract
A normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by Ince (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained.
Highlights
Normal vibration modes can be used to construct a general solution in the linear theory
The stability of the normal vibration modes is studied by deriving an approximation for the Poincare map by Month and Rand [8] and Pak [9]
The similar normal modes stability was investigated while the linearizing variational equations represent the Lame equation; criterions of a stability in the non-linear sense were obtained
Summary
Normal (principal) vibration modes can be used to construct a general solution in the linear theory. Since the equation will not change if z is replaced by - z, the following solutions will be associated with the singular point z = - 1: Y1(1 +z)=fl(l +z); y2 (1 +z)=Jl+zfz(1 +z), Here the analytical functions f 1 and f 2 converge within a circle 11 + z I :::::; 2. The origin of coordinates, z = 0, is, a regular point of the equation, no two independent even solutions can exist that are meaningful in the vicinity of the point z = 0 It follows that condition (1) must be rejected. Substituting z = [sn(t, k)] 2 , one obtains the following equation with three finite regular singular points: 1(l 1 -ddz2+ y2. Algebraic forms of the Lame equation are used in a number of problems considered in this article
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