Hindawi Publishing CorporationAdvances in Acoustics and VibrationVolume 2013, Article ID 972409, 6 pageshttp://dx.doi.org/10.1155/2013/972409

Research ArticleDetermination of the Fundamental Frequency ofPerforated Rectangular Plates: Concentrated Negative MassApproach for the Perforation

Kiran D. Mali and Pravin M. Singru

Department of Mechanical Engineering, Birla Institute of Technology and Science, Pilani, K.K. Birla Goa Campus,Zuarinagar, Goa 403726, India

Correspondence should be addressed to Kiran D. Mali; malikirand@gmail.com

Received 28 February 2013; Revised 1 May 2013; Accepted 12 May 2013

Academic Editor: Joseph CS Lai

Copyright © 2013 K. D. Mali and P. M. Singru. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

This paper is concerned with a vibration analysis of perforated rectangular plates with rectangular perforation pattern of circularholes. The study is particularly useful in the understanding of the vibration of sound absorbing screens, head plates, end covers,or supports for tube bundles typically including tube sheets and support plates used in the mechanical devices. An energy methodis developed to obtain analytical frequencies of the perforated plates with clamped edge, support conditions. Perforated plate isconsidered as platewith uniformly distributedmass.Holes are considered as concentrated negativemasses.The analytical procedureusing the Galerkin method is adopted. The deflected surface of the plate is approximated by the cosine series which satisfies theboundary conditions. Finite element method (FEM) results have been used to illustrate the validity of the analytical model. Thecomparisons show that the analytical model predicts natural frequencies reasonably well for holes of small size.

1. Introduction

Perforated plates are widely used in nuclear power equip-ments, heat exchangers, and pressure vessels. The holes inthe plate are arranged in various regular penetration patterns.Industrial applications include both square and triangulararray perforation patterns. Cutouts are found in mechanical,civil, marine, and aerospace structures commonly to accessports for mechanical and electrical systems, or simply toreduce weight. Cutouts are also made to provide ventilationand modify the resonant frequency of the structures. Perfo-rated plates are often utilized as head plates, end covers, orsupports for tube bundles typically including tube sheets andsupport plates.

Many studies have been done on perforated plates havingrectangular/square and triangular array of holes, especially[1–3] dealing with equivalent properties of material for per-forated plate. These equivalent material properties are usedin vibration analysis to consider perforated plate as full solidplate. Burgemeister and Hansen [4] showed that, to predictaccurately the resonance frequencies of simply supported

perforated panel, effective material constants cannot be usedin classical equations. They used cubic function fitted fromANSYS results to determine the effective resonance fre-quency ratio for large range of panel geometries with an errorof less than 3%. Mali and Singru [5] introduced concept ofconcentrated negativemasses for perforation holes and deter-mined fundamental frequency of rectangular plate carryingfour circular perforations in rectangular pattern. Mali andSingru formulated an analytical model by using unit stepfunctions [6] and the greatest integer functions [7] to expressnonhomogeneity in Young’s modulus and density and deter-mined fundamental frequency of free vibration of perforatedplate.

Low [8, 9] determined the vibration frequencies of rect-angular plates with weights mounted at various locations.He also developed improved model to determine the fre-quency of vibrating plates carryingmultiplemasses at variouspositions. Low et al. [10, 11] obtained natural frequencies ofrectangular plates carrying a single and multiple concen-trated masses by using Rayleigh-Ritz method for differentboundary conditions. They compared results obtained from

2 Advances in Acoustics and Vibration

the analytical study using the energy method with thosemeasured experimentally. Boay [12] analyzed free vibrationanalysis of rectangular isotropic plates carrying a concen-trated mass. The Ritz approach is applied to rectangularplates with various edge support combinations of clamp andsimple support conditions. The effect of different locationsof the concentrated mass on the fundamental frequency ofthe plate is presented in detail. Wu and Luo [13] determinedthe natural frequencies and the corresponding mode shapesof a uniform rectangular flat plate carrying any numberof point masses and translational springs by means of theanalytical and numerical combinedmethod (ANCM). Avaloset al. [14] studied transverse vibrations of simply supportedrectangular plates with rectangular cutouts carrying an elas-tically mounted concentrated mass. Ostachowicz et al. [15]presented new results for the identification of the locationof a concentrated mass on isotropic plates by means of agenetic algorithm search technique based on changes innatural frequencies. Li [16] presented an exact approach forfree vibration of an isotropic rectangular plate carrying aline-concentrated mass and with a line-translational springsupport or carrying a line-spring-mass system. Altintas [17]investigated the special behaviors of linear vibrating plateswith special parameters near degenerate modes. The specialparameters considered in the study are the location and quan-tity of an additional mass, which have an effect on removingthe system symmetry. Zhang [18] presented and compareddifferent methods on the eigenfrequency computation of abeam and a plate carrying arbitrary number of concentratedmass/spring. The advantages and disadvantages of thesemethods are analyzed and discussed. Amabili [19] studiednonlinear forced vibrations of rectangular plates carryinga central concentrated mass. Amabili et al. [20, 21] studiedeffect of concentratedmasseswith rotary inertia on vibrationsof rectangular plates and large-amplitude vibrations of rect-angular plates carrying concentrated masses.

From the literature on vibration of perforated plates,authors found that, there is no evidence on the analyticalformulation by considering negative mass approach for holesexcept for reference [5] where authors have studied onlyfour specimens of rectangular plates with four circular per-forations. They have not discussed the limiting condition ofthe approach for obtaining the accuracy in predicted valueof fundamental frequency. In this paper, free vibration ofperforated plate with rectangular perforation pattern of ninecircular perforations is studied. Boundary condition used forthe plate is clamped on all four edges. In the current approachGalerkin method is used for determining the fundamentalfrequency of rectangular perforated plate with rectangularperforation pattern of circular holes. Perforated plate is con-sidered as plate with uniformly distributed mass, and holesare considered as concentrated negativemasses.Thedeflectedsurface of the plate is approximated by the cosine serieswhich satisfies the boundary conditions. Ten specimens areanalyzed numerically with different perforation diametersto obtain fundamental frequency. To validate the proposedmodel FEM results have been used. Percentage error infundamental frequency obtained by comparing numericaland FEM results is plotted against ratio of plate area (𝐴) to

perforation hole area (𝐴𝑝). For the variation of the percenterror in the fundamental frequency, trendline is establishedby regression analysis. Further from the equation of thetrendline limiting condition in terms of the ratio of platearea (𝐴) to perforation hole area (𝐴𝑝), to obtain accuracy infundamental frequency is obtained. Present analytical modelis thus more useful for predicting accurately the fundamentalfrequencies of wide range of small size perforation geome-tries, for rectangular plates with rectangular penetration pat-tern for all edges clamped support condition.

2. Analytical Formulations

2.1. Equation of Motion for Isotropic Rectangular Plates. TheGalerkin method is applied to the plate vibration problems[5, 22] to get the equation of motion. The partial differentialequation governing the free transverse vibration of an iso-tropic thin platewith dimensions 𝑎, 𝑏 shown in Figure 1 is [22]

𝐷(𝜕4𝑤

𝜕𝑥4+ 2

𝜕2𝑤

𝜕𝑥2

𝜕2𝑤

𝜕𝑦2+𝜕4𝑤

𝜕𝑦4) + 𝜌ℎ

𝜕4𝑤

𝜕𝑡2= 0, (1)

𝐷∇2∇2𝑤 (𝑥, 𝑦, 𝑡) + 𝜌ℎ

𝜕4𝑤

𝜕𝑡2(𝑥, 𝑦, 𝑡) = 0, (2)

where ℎ is the uniform plate thickness, 𝜌 is the density, 𝑤 istransverse displacement, 𝐷 is the flexural rigidity, and ∇

2 istwo-dimensional Laplacian operator. 𝐷 and ∇

2 are given as[22, 23]

𝐷 =𝐸ℎ3

12 (1 − 𝜐2), ∇

2=

𝜕2

𝜕𝑥2+

𝜕2

𝜕𝑦2, (3)

where 𝐸 is modulus of elasticity and ] is Poisson’s ratio.To solve (2) and obtain𝑤(𝑥, 𝑦, 𝑡) in general, the following

solution can be assumed [22]:

𝑤 (𝑥, 𝑦, 𝑡) = (𝐴 cos𝜔𝑡 + 𝐵 cos𝜔𝑡)𝑊 (𝑥, 𝑦) . (4)

This is separable solution of the shape function 𝑊(𝑥, 𝑦)

describing the modes of the vibration and some harmonicfunction of time;𝜔 is the natural frequency of the plate vibra-tion which is related to vibration period 𝑇 by the relationship

𝜔 =2𝜋

𝑇. (5)

Introducing (4) into (2) we get [22]

𝐷∇2∇2𝑤 (𝑥, 𝑦) − 𝜔

2𝜌ℎ𝑊 = 0. (6)

The shape function satisfying the boundary conditions forrectangular plate with dimensions 𝑎 and 𝑏 is assumed in theform of series as [22]

𝑊(𝑥, 𝑦) =

𝑛

∑

𝑖=1

𝐶𝑖𝑊𝑖(𝑥, 𝑦) , (7)

where 𝐶𝑖are unknown coefficients.

Now following the general procedure of the Galerkinmethod [22] the unknown coefficients 𝐶

𝑖can be determined

from orthogonality conditions.

Advances in Acoustics and Vibration 3

x

y

b

aO

Figure 1: Rectangular plate coordinates.

For the plate problem vibration given by (6), the orthogo-nality conditions together with (7) result in (8) as given below[22].

∬𝐴

(𝐷

𝑛

∑

𝑖=1

𝐶𝑖∇2∇2𝑊𝑖− 𝜔2𝜌ℎ

𝑛

∑

𝑖=1

𝐶𝑖𝑊𝑖)𝑊𝑘𝑑𝑥 𝑑𝑦 = 0;

𝑘 = 1, 2, . . . , 𝑛.

(8)

The above conditions when implemented numericallyleads to the Galerkin system of linear algebraic equations ofthe form

𝑎11𝐶1+ 𝑎12𝐶2+ ⋅ ⋅ ⋅ = 0

...

𝑎𝑛1𝐶1+ 𝑎𝑛2𝐶2+ ⋅ ⋅ ⋅ = 0,

(9)

where

𝑎𝑖𝑘= 𝑎𝑘𝑖= ∬𝐴

[𝐷∇2∇2𝑊𝑖− 𝜔2𝜌ℎ𝑊𝑖]𝑊𝑘𝑑𝑥 𝑑𝑦. (10)

This system of homogeneous equations has a nontrivialsolution if its determinantΔ(𝜔)made up of the coefficients 𝑎

𝑖𝑘

is equal to zero. Therefore we obtain 𝑛th order characteristicequation for determining the natural frequencies [22].

Δ (𝜔) = 0. (11)

This equation will have an infinite number of solutionswhich constitute the frequency spectrum for the given plate.In general, the frequencies will depend on two parameters:𝑚 and 𝑛 (𝑚 = 1, 2, . . .; 𝑛 = 1, 2, . . .). The lowest frequencyis called the frequency of the fundamental mode or thefundamental natural frequency, and all other frequenciesare called the frequencies of higher harmonics. For eachfrequency 𝜔

𝑚𝑛, there is a corresponding shape function

𝑊(𝑥, 𝑦), which on the basis of the homogeneous equations, isdetermined by a constant multiplier (which can be assumedas being equal to unity).

2.2. Determination of the Fundamental Frequency of Plateswith Perforations. A rectangular plate with coordinate sys-tem (𝑂; 𝑥, 𝑦, 𝑧), having the origin 𝑂 at one corner, is consid-ered as shown in Figure 2. Coordinates of the plate clampedon all edges carrying circular holes. The displacement of anarbitrary point of coordinates (𝑥, 𝑦) on the middle surface ofthe plate is denoted by W, in out-of-plane (𝑧) direction. Theboundary conditions considered here are all edges clamped.Geometric parameters of the rectangular plate are 𝑎 and 𝑏

for sides, ℎ for thickness and 𝑑 is hole diameter which foruniform for all perforations. The assumptions made in thefollowing formulation are that transverse defections are smallso that the dynamic behavior of the plate is governed byclassical thin plate theory. Mass of the plate (𝑚) is consideredas uniformly distributed with nine concentrated negativemasses denoted by “−𝑀.”Thenegative sign indicates that theconcentrated mass “−𝑀” cancels out the effect of the sameamount ofmass of the homogenous plate at the position of thecutouts. This is an equivalent approach to apply the Galerkinmethod for the given perforated plate vibration problem.Analytical model in the present work does not consider anyrotary inertia of the plate.

The middle surface displacement is approximated byusing shape function 𝑊(𝑥, 𝑦) in the form of a series, whichsatisfies the boundary conditions on the edges 𝑥 = 0, 𝑥 = 𝑎

and 𝑦 = 0, 𝑦 = 𝑏. Let us represent the shape function𝑊(𝑥, 𝑦)

for a rectangular plate with dimensions 𝑎and 𝑏in the form[22]

𝑊(𝑥, 𝑦) = ∑

𝑖

∑

𝑘

𝐶𝑖𝑘𝑊𝑖𝑘(𝑥, 𝑦) , (12)

where 𝐶𝑖𝑘are unknown coefficients representing the ampli-

tudes of the free vibrationmodes and𝑊𝑖𝑘(𝑥, 𝑦) is the product

of the pertinent eigenfunctions of lateral beam vibrations.Consider

𝑊𝑖𝑘(𝑥, 𝑦) = 𝐹

𝑖(𝑥) 𝐹𝑘(𝑦) (13)

which satisfy the prescribed boundary conditions on theedges 𝑥 = 0, 𝑥 = 𝑎 and 𝑦 = 0, 𝑦 = 𝑏. In (13) 𝐹

𝑖(𝑥) and 𝐹

𝑘(𝑦)

represent the 𝑖th and 𝑘th modes of freely vibrating beams ofspans 𝑎and 𝑏, respectively.

By applying Galerkin method for determining funda-mental frequency, (10) can be modified for the coefficients 𝑎

𝑖𝑘

of Galerkin’s system of equations, to the present problem, asfollows [22]:

𝑎𝑖𝑘= 𝑎𝑘𝑖= ∬𝐴

[𝐷∇2∇2𝑊𝑖− 𝜔2𝜌ℎ𝑊𝑖)𝑊𝑘𝑑𝑥 𝑑𝑦

−∑𝑀𝜔2𝑊𝑖𝑊𝑘.

(14)

The deflected surface of the vibrating plate is approxi-mated by the series

𝑊(𝑥, 𝑦) =

∞

∑

𝑖=1

∞

∑

𝑘=1

𝐶𝑖𝑘(1 − cos 𝑖2𝜋𝑥

𝑎)(1 − cos

𝑘2𝜋𝑦

𝑏) .

(15)

This satisfies the boundary condition clamped on all edges.

4 Advances in Acoustics and Vibration

x

y

b

−M −M−M

2b/4

b/4

a/4

2a/4

3a/4

−M −M−M

−M −M−M

a

3b/4

O(0,0)

Figure 2: Coordinates of the plate clamped on all edges carryingcircular holes.

For the first approximation retaining only first term in thein the expansion of (15) we obtain from (14)

𝑎11

= ∫

𝑎

0

∫

𝑏

0

[𝐷(𝜕4𝑊1

𝜕𝑥4+ 2

𝜕2𝑊1

𝜕𝑥2

𝜕2𝑊1

𝜕𝑦2+𝜕4𝑊1

𝜕𝑦4)

−𝑚𝜔2𝑊1]𝑊1𝑑𝑥 𝑑𝑦 −∑𝑀𝜔

2𝑊2

1.

(16)

Introducing in (16)𝑊1,

𝑊1= (1 − cos 2𝜋𝑥

𝑎) (1 − cos

2𝜋𝑦

𝑏) . (17)

After simplification of (16) the following expressions areobtained:

𝐷(12𝑏𝜋4

𝑎3+12𝑎𝜋4

𝑏3+8𝑎𝜋4

𝑎𝑏) − 𝑚𝜔

2(9𝑎𝑏

4) + 32𝑀𝜔

2= 0,

𝜔 =

−4𝜋2√−𝑎𝑏 (−9𝑚𝑎𝑏 + 128𝑀)𝐷 (3𝑏4 +3𝑎

4+ 2𝑎2𝑏)

𝑎2𝑏2 (−9𝑚𝑎𝑏 + 128𝑀).

(18)

3. Numerical Simulation

Analytical model developed in Section 2.2 is applicable torectangular perforated plates with different side dimensions,provided that the perforations are of the same size. By virtueof the symbolic forms presented in this work, the methodcan be applied to analytical studies of perforated plates withdifferent boundary conditions. To estimate the sensitivityof the method for various cases of hole sizes and platesizes, numerical analysis is carried out for ten different plate

Table 1: Specimen parameters.

Specimen no. 𝑎 (mm) 𝑏 (mm) ℎ (mm) 𝑑 (mm) (𝐴/𝐴𝑝)

1 138 216 2 5 1518.1542 138 216 2 10 379.53843 138 216 2 15 168.68374 138 216 2 25 60.726155 138 216 2 30 42.170946 276 432 2 10 1518.1547 276 432 2 20 379.53848 276 432 2 25 242.90469 276 432 2 50 60.7261510 276 432 2 65 35.93263

specimens listed in Table 1. Last column of Table 1 lists theratio of the plate area (𝐴) to the area of perforation hole (𝐴

𝑝)

𝐴 = 𝑎 × 𝑏,

𝐴𝑝=𝜋𝑑2

4.

(19)

All the specimens considered have thickness 2mm. Coordi-nates of the perforation centers (xmm, ymm) are given inTable 2. The following are the material properties [24] con-sidered for all specimen plates analyzed:

𝐸 = 2.1 × 1011N/m2,

𝜐 = 0.3,

𝜌 = 7850 kg/m3.

(20)

4. Finite Element Method (FEM) Analysis

The proposed analytical model is validated by comparing thenumerical analysis results with FEM modal analysis results.FEM modal analysis is carried out by ANSYS 11 using Shell63 element. Parameters of the plate specimen considered inthis study are shown in Table 1. Analysis is carried out forclamped steel plates having 2mm thickness and carrying nineholes at positions shown in Figure 2. Meshing is done by freemeshing with smart size option, and quadrilateral elementsare used. Mesh convergence for FEM results is checkedfor every specimen. This is checked by running differentsimulations. Final solution is chosen based on the meshquality as well as mesh size. Thus converged solution is givenin Table 3. It is assumed that structure is formed of isotropichomogeneous elastic material, that is, structural steel withmaterial properties [24] same as that used in numericalanalysis.

5. Results and Discussions

Comparison of numerical and FEMnatural frequencies of theplate specimens for firstmode is given in Table 3. Last columnof Table 3 shows percentage error in fundamental frequencyvalues obtained by numerical analysis and by FEM analysis.The agreement between the analytical approach and the finite

Advances in Acoustics and Vibration 5

Table 2: Coordinates of the perforation centers.

Specimen size (mm ×mm) Coordinates of the perforation centers (𝑥mm, 𝑦mm)𝑎 × 𝑏 𝑎/4, 𝑏/4 𝑎/4, 2𝑏/4 𝑎/4, 3𝑏/4 2𝑎/4, 𝑏/4 2𝑎/4, 2𝑏/4 2𝑎/4, 3𝑏/4 3𝑎/4, 𝑏/4 3𝑎/4, 2𝑏/4 3𝑎/4, 3𝑏/4138 × 216 34.5, 54 34.5, 108 34.5, 162 69, 54 69, 108 69, 162 103.5, 54 103.5, 108 103.5, 162276 × 432 69, 108 69, 216 69, 324 138, 108 138, 216 138, 324 207, 108 207, 216 207, 324

Table 3: Fundamental frequency results of numerical and FEM sim-ulations.

Specimen 𝑑 (𝐴/𝐴𝑝) 𝜔, Numerical 𝜔, FEM % Error

no. (mm) (Hz) (Hz)Specimens with dimensions (138mm × 216mm)

1 5 1518.154 718.63 693.38 3.642 10 379.5384 729.04 693.85 5.073 15 168.6837 747.46 695.97 7.404 25 60.72615 817.34 710.17 15.095 30 42.17094 878.60 724.94 21.20

Specimens with dimensions (276mm × 432mm)6 10 1518.154 179.65 173.37 3.627 20 379.5384 182.26 173.46 5.078 25 242.9046 184.29 173.65 6.139 50 60.72615 204.33 177.53 15.1010 65 35.93263 230.04 183.55 25.33

element results is excellent when perforation size is small.The maximum difference is of the order 25.33% especiallyfor the specimen having 𝐴/𝐴

𝑝= 35.93263. Figure 3 shows

the variation of the percent error in fundamental frequencywith respect to 𝐴/𝐴

𝑝ratio. Dotted line shows the trendline

for this variation. The power trendline clearly demonstratesthe decrease in percentage error with increase in 𝐴/𝐴

𝑝ratio.

Equation used to calculate the least squares fit through pointsfor the trendline is

% Error = 𝑓(𝐴

𝐴𝑝

)

𝑔

. (21)

In the above expression 𝑓and𝑔 are constants. For thepercentage error, 𝐴/𝐴

𝑝ratio data given in Table 3 values of

constants 𝑓 and 𝑔 are found to be 123.97 and −0.51, respec-tively. Thus equation of the trendline shown in Figure 3becomes

𝑦 = 123.97𝑥−0.51

, (22)

where 𝑦 and 𝑥 are values of percentage error and𝐴/𝐴𝑝ratio,

respectively.𝑅-squared value shown in Figure 3 is a correlation coef-

ficient which reveals how closely the estimated values for thetrendline correspond to actual data

𝑅2= 0.9371. (23)

It is observed that percent error increases above 10%,when𝐴/𝐴

𝑝ratio decreases below the value of 139.2380.Thus

present approach has limitation in predicting the value of

272421181512

9630

0 200 400 600 800 1000 1200 1400 1600 1800A/Ap ratio

Variation of % errorTrendline for variation of % error

R2 = 0.9371

y = 123.97×−0.51

% E

rror

Figure 3: Variation of the percent error in fundamental frequencywith 𝐴/𝐴

𝑝ratio.

fundamental frequency, as it is based on the basic assumptionof concentrated negative mass. Further from Table 3 it isobserved that for specimens of different configurations buthaving the same 𝐴/𝐴

𝑝ratio analytical model gives results

with the same accuracy (specimens 1, 6 and 2, 7).When perforation size increases, results obtained by

proposed method deviate more. This occurs because, as per-foration size increases, it changes stiffness also significantly,but proposed model considers only change in/reduction inthe mass or kinetic energy but not strain energy. Due to thisunaccounted change in strain energy, fundamental frequen-cies obtained by this method are higher than FEM results.Thus deviation between numerical and FEM fundamentalfrequencies becomes more as perforation size increases.

The reasonably good performance of the proposedmodel,for small holes is demonstrated in Table 3 and Figure 3.Variation of the percent error in fundamental frequency with𝐴/𝐴𝑝ratio is due to the fact that the effects of both the

different holes and their locations on the frequency have beenaccounted.

6. Conclusions

This work presents an analytical model to estimate the fun-damental frequency of uniform thickness plates carryingcircular perforations in rectangular pattern.The effect of per-foration on the natural frequency of plate has been modeledusing concept of concentrated negative mass in the Galerkinmethod.The proposedmodel has been verified by comparingthe numerical results with FEM results. It is found that theerror in the fundamental frequency is of the order of 5%whenratio of the area of plate (𝐴) to the area of single perforation

6 Advances in Acoustics and Vibration

(𝐴𝑝) is 542.0170. It is found that the error in the fundamental

frequency is more when perforation size increases. Thus,fundamental frequency of perforated plate can be obtainedfor small size of the perforation by a proper choice of the plateparameters and shape function depending on the boundarycondition.

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