The effect of a linear constraint on the small oscillations of a dynamical system with three degrees of freedom
Abstract
En
We consider the effect of a linear constraint on the small oscillations about an equilibrium position of a conservative dynamical system which has just three degrees of freedom. Explicit expressions are presented for the eigenfrequencies and eigenvectors of the constrained system in terms of the eigenfrequencies and eigenvectors of the unconstrained system. The unconstrained system is described in terms of two positive definite 3×3 matrices with which two concentric ellipsoids may be associated. A ‘plane’ corresponds to the linear constraint. It is seen that in general it is possible to choose two constraints such that the constrained motion has a double eigenfrequency,or equivalently, two central planes may be chosen which cut the two concentric ellipsoids in a pair of similar and similarly situated ellipses.
We consider the effect of a linear constraint on the small oscillations about an equilibrium position of a conservative dynamical system which has just three degrees of freedom. Explicit expressions are presented for the eigenfrequencies and eigenvectors of the constrained system in terms of the eigenfrequencies and eigenvectors of the unconstrained system. The unconstrained system is described in terms of two positive definite 3×3 matrices with which two concentric ellipsoids may be associated. A ‘plane’ corresponds to the linear constraint. It is seen that in general it is possible to choose two constraints such that the constrained motion has a double eigenfrequency,or equivalently, two central planes may be chosen which cut the two concentric ellipsoids in a pair of similar and similarly situated ellipses.
DOI Code:
10.1285/i15900932v27n2p13
Keywords:
Linear contraints; Small oscillations; Conservative dynamical systems
Linear contraints; Small oscillations; Conservative dynamical systems
Full Text: PDF
References
- Ph. Boulanger, M. Hayes, "The common conjugate directions of plane sections of two concentric ellipsoids", Z. angew. Math. Phys., 46, Special Issue (Theoretical, experimental, and numerical contributions to mechanics of fluis and solids, ed. by J. Casey & M.J. Crochet), (1995), pp. 356-371
MR1359328 | Zbl 0829.73026 - J. L. Synge, "Classical Dynamics", Handbuch der Physik, Bd III, 1, Springer-Verlag, Berlin, 1960
MR0117945 | Zbl 0118.39901
