Comment on "Applications of Bolotin's Method to Vibrations of Plates"

Abstract

N a recent interesting Technical Note,1 King and Lin applied Bolotin's asymptotic method to the determination of natural frequencies of rectangular plates. The writer agrees with their observation that little attention to this method has been given in Western literature. The main purpose of this Comment is to mention earlier work, where the method was used to determine the natural frequencies of plate systems.2 In Ref. 2, the writer applied this method to single- and multiplate systems and gave approximate frequency equations for 2 two-plate systems, where the two plates were in perpendicular planes and were joined along a common edge. The first ten natural frequencies for each of these systems were obtained and agreed to within 0.5% with a solution based on a series method; the latter has been described in more detail elsewhere3 and was based on the work of Dill and Pister.4 A few frequencies were obtained for a closed box using Bolotin's method; again good agreement was obtained with frequencies from the series method. However, Bolotin's method has limitations, as it predicts mode shapes which have nodal lines parallel to the sides of the plates. For the two-plate systems considered in Ref. 2 some of the modes could not easily be approximated in this way, but nevertheless the Bolotin method gave accurate natural frequencies. For boxtype structures many modes have nodal patterns which cannot be represented by lines parallel to the plate edges and here Bolotin's method failed to predict the natural frequencies. This is one disadvantage of the method. Its other disadvantage is that frequencies of increasing accuracy cannot be obtained by increased computational effort; i.e., there is no procedure similar to the use of additional terms or a finer mesh in the series, Rayleigh-Ritz, finite difference, and finite element methods. If the writer had to determine natural frequencies of plate systems today, he would use the finite element method because of its versatility and the availability of a general program; the natural frequencies of the box-type structures of Refs. 2 and 3 have been obtained satisfactorily by this method.5 However, for simpleplate systems, a useful check on the accuracy of frequencies by the finite element method could be provided by both the series method and Bolotin's method, particularly for higher modes. The second purpose of this comment is to remark upon some of the natural frequencies in Table 3 of Ref. 1. King and Lin state that they were unable to find in the literature a value of the natural frequency for the sixth mode of a square plate with two adjacent edges clamped and the other two free to compare with that obtained by Bolotin's method. From Leissa's recent paper,6 the value of the nondimensional frequency factor for this mode is 65.833. He used 36 terms in a Rayleigh-Ritz solution. Thus the frequency factors of Ref. 6 are more accurate than Young's values, which are quoted in Table 3 and for which a smaller number of terms was used. As we mentioned, Bolotin's method predicts mode shapes with nodal lines parallel to the edges of the plate. Thus it gives identical frequencies for the second and third modes of this plate and also for the fifth and sixth modes, as pointed out by King and Lin. In this way it is similar to the use of the Rayleigh-Ritz method with a single term representing