Comments on “Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function”

Abstract

In Ref. [1], Kang and Lee presented a non-dimensional dynamic influence function (NDIF) method for plate vibration. This paper extended the NDIF method from membrane vibration [2] to plate vibration problems. Kang and Lee [3] also applied this method to solve the membrane vibration by using domain partition for multiply connected and concave problems. Although Laura and Bambill [4] commented that the considered problem is very simple, it really proposed an easy method for engineers. Since only boundary node is required, the approach is meshless. Many successful examples of the clamped boundary conditions were demonstrated. It seems that this method is very attractive. However, this method can be treated as one kind of the Trefftz method [5], [6], [7] or boundary collocation method [8]. Based on the dual formulation developed by Chen and Hong [9], [10], [11], the interpolation function is nothing but the imaginary-part of the fundamental solution (W(s,x)=(i/(8λ2)){H0(2)(λr)+H0(1)(iλr)), where Kang and Lee chose J0(λr) and I0(λr) as radial basis functions. The method proposed by Kang and Lee [1] can be treated as a special case of the imaginary-part dual BEM, and its occurrence of spurious eigenvalues has been verified in [12], [13]. In addition, Chen et al. [14] and Kuo et al. [15] employed the theory of circulants to prove that spurious eigensolutions and ill-posed problems may occur in case of circular membrane. For general shape problems, the two drawbacks are also inherent. To overcome the problem of spurious eigensolutions, a net approach was proposed by Kang and Lee [16]. Another alternative to avoid the occurence of spurious eigensolution was also proposed by Chen et al. [12] using the double-layer approach. This singularity-free method also results in the ill-posed behavior when the number of degrees of freedom becomes large [14]. Kuo et al. [17] employed the generalized singular value decomposition (GSVD) method in conjunction with the Tikhonov regularization to deal with the ill-posed problem for the incomplete boundary element formulation. Until now, research of ill-posed problem is an active area. However, why the spurious eigenvalues occur in Ref. [1] for a circular plate was not studied analytically. In this letter to editor, we will prove it.