Periodic Solutions of Gravity Oriented Axisymmetric Systems

TLDR: In this article, the Rayleigh quotient for the transverse vibration of a massless string under constant tension with five equal masses equally spaced along the string was derived from the simpler (and more traditional) point of view of the dynamicist.

Abstract: quadrature schemes may be used in effecting the numerical integrations. In this sense, specific discrete element "condensation" techniques represent only one of many possible ways for carrying out the required process of integration and summation in the Rayleigh-Ritz method. To further illustrate the alternative ways in which Rayleigh's principle may be applied and the fact that the use of a vibration mode determined from a first iteration does not necessarily lead to best results, the numerical example given by Fried 3 will be considered from the simpler (and more traditional) point of view of the dynamicist. The Rayleigh quotient may be formed from Fried' s Eq. (6) by multiplying the first row of the matrix equation by xl9 the second by x2, etc., and summing the results to give Ml Apart from physical constants which are absorbed in the definition of A, this may be viewed as Rayleigh's quotient for the transverse vibration of a massless string under constant tension with five equal masses equally spaced along the string. In this case the numerator represents the strain energy and the denominator represents the kinetic energy divided by frequency squared. In the traditional approach of the dynamicist, a reasonable (and typical) assumption for the shape of the lowest symmetric mode of this system would 5e a parabola (as in Appendix 1 of Ref. 4). Thus Xj_ = 5, x2 = 8, x3 = 9, x4 — 8, x5 = 5 Substituting these values into Eq. (1) gives A! - 70/259 - 0.2703 This compares to Fried's result for the singly iterated mode, A! - 0.2860, for the doubly iterated mode, ^ = 0.2680, and the exact value, Aj_ = 0.2679. The accuracy of the result obtained here by application of the primitive Rayleigh method is thus much better than the result obtained by Fried using a singly iterated mode and differs from the exact solution by less than i°/ For the an ti symmetrical case, applying a similar assumed parabola to each half of the system leads to A 2 = 1.000 which corresponds to the exact solution. By contrast, Fried obtained /2 = 1.200 for his singly iterated mode and A2 = 1.024 for his doubly iterated mode. Fried's results are a consequence of the fact that his assumed antisymmetric loading for the first iteration destroyed the subsymmetry of the first antisymmetric vibration mode in each half of the system. This example is not, of course, typical of all problems to which Rayleigh's principle might be applied. (For instance, in the bending vibration of cantilever beams of variable properties use of the first iteration for the mode shape in application of Rayleigh's principle to determination of the lowest natural frequency is often a most satisfactory procedure.) It does, however, emphasize that to obtain best results with minimum effort from Rayleigh's principle, it is necessary for the analyst to exercise a degree of judgment based on the geometrical, structural, and dynamic properties of the system under consideration and to choose analytical techniques accordingly.