Natural modes and frequencies of in-plane vibrations of a fixed elastic ring

Abstract

The natural modes and frequencies of an elastic ring fixed at a point are determined by numerically solving a boundary-value problem for a differential operator of the sixth order. The ring models a large circular antenna that slowly expands under zero gravity. The in-plane flexural vibrations of the ring are analyzed. Numerical results are presented Keywords: natural frequency, natural mode, flexible ring, vibrations, differential operator Introduction. The problem of determining the natural frequencies and modes of a ring arose in studying the dynamics of a spacecraft that carries an antenna of variable geometry (a compact roll of an elastic ribbon slowly deployed, in a programmed manner, into a ring). Assuming that the deployment duration is much longer than the period of vibration of the ring in the first mode, we can neglect the perturbations of natural modes of the ring caused by the variation in its radius with time.Structures as a circular antenna can be modeled by thin curved bars. If the elongation is small, the question whether such a model is applicable can be answered only after solving the appropriate problem of elasticity. If it is, then it remains to be seen whether the model involves factors that disturb the vibrations. The linear analysis of the free and forced vibrations of curved bars is based on the assumption that the displacements and angles of rotation are small. In this case, the Bernoulli hypothesis applies well and vibrations are described by a system of differential equations for the vector of displacements and angular rates of rotation [1,3,7]. The solution can only be found if the coefficients of this system are constant. There are three such cases [3]: a straight bar , ring, and helix, the first two being special cases of the third. For the coefficients of the equations of motion to be constant, it is particularly necessary that the bar have constant cross-section. In the case of a closed ring, the vector system breaks up into two scalar subsystems describing in-plane and out-of-plane vibrations of the ring. There are Hoppe's approximate solutions for the natural frequencies of in-plane flexural vibrations of a closed ring [7] and Michell's solutions for the natural frequencies of out-of-plane vibrations of a closed ring [7]. There are also two more axisymmetric approximate solutions obtained by Hoppe [7] for the torsional in-plane vibrations of a ring and by Basset [7] for the torsional out-of-plane vibrations of a ring.The analytic solution for an open ring appears so complex that most authors preferred approximate analytic and numerical methods to find its natural frequencies and modes [5].Though the natural modes and frequencies of a ring were already studied hundred years ago, modern computers allow us to approach this problem from a qualitatively different level. There are published numerical data on the several first natural frequencies and modes of in-plane and out-of-plane vibrations of closed rings with certain boundary conditions. However, the accuracy of these data is l...

Natural frequencies and modes of out-of-plane vibrations a fixed ring

“…An extensive bibliography on the subject can be found in [2,3,11]. We failed to find any discussion of the case of an elastic ring with one fixed point in the literature.The present paper continues the studies of the modes and frequencies of flexural in-plane vibrations of a circular ring fixed at one point [13,14]. In the present paper, we will discuss the natural modes and frequencies of such a ring under zero gravity that were obtained by solving a boundary-value problem for the equations of motion of an elastic ring with direct numerical methods easily implementable on modern computers.…”

“…Hence zero values of its determinant. This explanation, however, does not exclude the possibility for the determinant to vanish on the solutions satisfying (14) at these points. Therefore, the points should be examined individually.…”

Total positive curvature of circular DNA

“…Fluctuations in DNA miniplasmids have been modeled as a Kirchhoff-Clebsch rod with no strain [31]. It would be interesting to study excitations and fluctuations, e.g., breathing modes, and to include strain and stretching in detailed studies of the different kinds of modes and their stability in closed loops of chiral filaments [32][33][34]. It would also be interesting to include higherorder curvature terms in the twist neutrality condition.…”

Effect of the Boundary Conditions and Influence of the Rotational Inertia on the Vibrational Modes of an Elastic Ring

“…for a free ring, and several other authors have since complemented and enriched these results. The case of a ring clamped at one point, however, has received little, if any, attention (to the best of our knowledge there is only one reference, by Zakrzhevskii et al [2], that addresses the subject). The lack of attention may stem in part from the fact that, for such boundary conditions, the relevant equations need to be solved numerically.…”

“…The lack of attention may stem in part from the fact that, for such boundary conditions, the relevant equations need to be solved numerically. The vibration of a ring clamped at one point is, nevertheless, a problem that arises in various contexts, such as the design of antennae [2], the study of fluctuations in circular biopolymers [3–6], and the dynamics of nano-structures [7,8]. …”