An extension of the Southwell-Dunkerley methods for synthesizing frequencies,part II: Applications

The series-type synthetic method of obtaining frequencies and the combined synthetic one are examined mainly from the practical point of view, for some typical vibrating models. In section 2, the fundamental frequency of a two-degree-of-freedom system with two springs and two masses, connected alternately in series, is first discussed, and the accuracy of the approximate solutions is investigated without the influence of the error associated with Rayleigh's principle, which is completely eliminatable for such discrete systems. The convergence of the approximate solutions as finite power-series expansions in the perturbation parameter is also studied. Secondly, for two kinds of fairly complicated models composed of a beam, springs and concentrated masses as examples, a study is presented of the degree of the error of approximation accumulated when synthesizing many isolated frequencies on the basis of the series-type synthetic method or the combined synthetic one. This study shows that in almost all cases the lower limit of the true frequency obtained by the present methods is very close to the true frequency and the error due to the procedure of synthesizing isolated frequencies is smaller than that associated with Rayleigh's principle. In sections 3 and 4, it is shown that in elastic continua such as bars, beams and plates the bending deformation and the transverse shear one, or the former and the torsional one, if related, are coupled in series, so that one can utilize the concept of the series-type synthetic method to obtain an approximate frequency of composite systems with such deformations.