Letter to the Editor
Preservation of the fundamental natural frequencies of rectangular plates with mass and spring modifications

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Introduction

For most cases where plates are used in engineering structures, mass and stiffness modifications become necessary. Ingber et al. [1] investigated experimentally, vibrations of clamped plates with concentrated mass and spring attachments by using a modal analysis technique and mixed boundary-element method. Boay [2] analyzed the natural frequencies of plates with and without a concentrated mass. The Rayleigh-energy method was used in the theoretical formulation. Lin and Lim [3] derived the receptances based on mode superposition and then used it to calculate the receptances of the plate with arbitrary mass and stiffness modification. McMillan and Keane [4] developed the direct modal sum method for shifting resonances from a frequency band by applying concentrated masses to a thin rectangular plate. Cha [5] applied the hybrid approach to analyze the free vibration of a simply supported rectangular plate carrying a concentrated mass. Wu and Luo [6] determined the natural frequencies and mode shapes of a rectangular plate carrying any number of point masses and springs by means of the analytical-and-numerical combined method. Dowell and Tang [7] studied the high-frequency response of a plate carrying a concentrated mass/spring system. Ref. [8] was concerned with satisfying a design aim such that the fundamental frequency of a cantilever beam remains the same in spite of the addition of a mass at some point on the beam. The present study represents to some extent the counterpart of the publication [8] for plate vibrations. Within this framework, the present study aims to investigate the possibility of using springs to preserve the fundamental frequency of a thin rectangular plate carrying any number of point masses. The numerical results obtained in this study are not only related to the fundamental frequency of the plate, but the formulation can also be adopted when any one of the natural frequencies of the plate is desired to be kept constant. The problems on plates carrying concentrated masses are encountered, e.g., in the design of electronic systems. The printed circuit boards and plate-like chassis can be approximated as flat rectangular plates carrying concentrated masses and subjected to vibration [2]. By calculating the receptance matrix of the unmodified plate and the receptance matrix corresponding to the modification, the receptance matrix of the modified plate are obtained based on substructuring analysis. Then the natural frequencies of the modified plate are calculated by analyzing the receptance data. Finally, the required coefficients of the springs to be placed at certain locations such that the fundamental frequencies will remain the same although there are added point masses that can be calculated.

Section snippets

Receptances of a rectangular plate

According to the classical thin-plate theory, the governing equation in terms of the lateral displacement w(x,y,t) is given byD∇4w+ρh∂2w/∂t2=P,where ∇4 is the two-dimensional biharmonic operator, h is the thickness of the plate, D is the flexural rigidity, ρ is the mass density, and P is the lateral load per unit area. The flexural rigidity is given byD=Eh3/12(1−ν2),where E is the modulus of elasticity, and ν is the Poisson ratio. The receptance of a plate αij(ω) is the response at the location

Frequency response function (FRF) method of coupled structure analysis

This method is often referred to as the ‘impedance coupling method’ or the ‘dynamic stiffness method’ [9]. The two components A and B shown in Fig. 2 are to be connected by the coupling co-ordinates to form the connected system C. The substructures, A and B, the 'input’ to coupling process will comprise two square FRF matrices, one of nA×nA and the other nB×nB which will then be combined to yield a corresponding FRF matrix for the coupled structure, C, which is of nC×nC. Both components A and B

Numerical results

This section is devoted to the testing of the expressions obtained. The rectangular plates shown in Fig. 1 are taken as examples. One of them is simply supported on its four edges denoted as (S–S–S–S) and two edges of the second plate are simply supported whereas the remaining two are clamped, denoted as (S–C–S–C). Before following the design aim that the fundamental frequency remains unchanged despite mass attachment, the two natural frequencies of the simply supported (S–S–S–S) rectangular

Conclusions

The present study is concerned essentially with the derivation of the receptance matrix of a rectangular thin plate to which several point-masses and springs are attached, by the so-called ‘impedance coupling method’. The study enables one to obtain the eigenfrequencies of the combined system described above. Further, an examination was carried out of the problem of determining the stiffness coefficient of the spring to be placed at a specified position so that the fundamental frequency of the

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