Higher mode elliptical integral solution for the large amplitude free vibration of a rectangularplate backed by a cavity

Lee, Y. Y.

Published in:Results in Physics

Published: 01/09/2020

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Published version (DOI):10.1016/j.rinp.2020.103239

Publication details:Lee, Y. Y. (2020). Higher mode elliptical integral solution for the large amplitude free vibration of a rectangularplate backed by a cavity. Results in Physics, 18, [103239]. https://doi.org/10.1016/j.rinp.2020.103239

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Results in Physics

journal homepage: www.elsevier.com/locate/rinp

Microarticle

Higher mode elliptical integral solution for the large amplitude freevibration of a rectangular plate backed by a cavityY.Y. LeeDepartment of Architecture and Civil Engineering, City University of Hong Kong, Kowloon Tong, Kowloon, Hong Kong

A R T I C L E I N F O

Keywords:Large amplitude vibrationElliptical integralStructural-acoustic interaction

A B S T R A C T

In this article, the elliptic integral solution form is adopted to obtain the higher mode natural frequencies of thelarge amplitude rectangular plate backed by a cavity. The main advantage of the proposed method is the conciseacoustic structural modal formulation which do not require heavy computation effort. A set of numerical resultsare presented for studying the higher mode natural frequencies.

Introduction

Numerous nonlinear vibration/oscillation or structural-acousticproblems have been studied (e.g. [1–3]), but there are limited worksabout nonlinear structural-acoustics. One of the two main difficulties isthat the nonlinear structural modes and acoustic modes are fully cou-pled, and thus result into tedious nonlinear multi-modal formulationprocedures. The other difficulty in these problems is the complicatedsolution procedures and heavy computational effort (e.g. setting upresidual equation or global matrix equation to obtain the eigenvaluesolutions). Some remarks about the fundamental natural frequency ofthe present nonlinear problem were given by Hui et al. [4], which alsoadopted the same elliptic integral approach. There are many nonlinearproblems solved by the classic harmonic balance method (e.g. [5–9]).This most common method keeps all the nonlinear terms to produce thepossible multiple solutions in a set of nonlinear algebraic equations.The higher harmonic solutions to any desired accuracy are obtained bysolving more nonlinear algebraic equations. In other words, if moreaccurate solutions are needed, more nonlinear algebraic equationswould be generated. It is very time consuming. The main advantage ofthe proposed method is that the elliptic integral solution form includesthe fundamental and all higher harmonic characters, and thus there isonly one nonlinear equation generated. Thus, the study adopts the el-liptic integral approach and develops the multi-acoustic and singlehigher structural mode formulation for investigating the higher modenatural frequencies.

Theoretical formulation

According to Hui et al. [4] and Lee [5], the homogeneous waveequation and the acoustic force within the cavity (see Fig. 1) are given

by

= =PC

Pt

P t hµ c

µA h t1 0; ( ) ϼ ( )

coth( ) ( ) cos( )h

a

hch

au

U

v

Vuvh

uvh

mnh

uvmn

cos sin

22 2

22

(1,2)

where = ( ) ( ) dxdycos coscosb a u x

av y

b0 02 2

; = ssinb a

0 0

( ) ( ) dxdyin sinm xa

n yb

2 2; P t( )c

h is the hth-order harmonic modal acoustic

force; =uvmn ( ) ( ) ( ) ( )dxdysin sin cos cosb a m x

an y

bu x

av y

b0 0 ; =µuvh

+ ( )( ) ( )ua

vb

hC

2 2 2

a; ρa = air density; a and b are the panel length

and width; u and v are the acoustic mode numbers; m and n are thepanel mode numbers; U and V are the numbers of the acoustic modesused; ω is the natural frequency to be determined

The governing equation of the (m,n) mode nonlinear panel vibrationis given by

+ + + =d Adt

A A F 0mno mn mn c

2

22 3

(3)

where Amn(t) is the modal displacement of the double sinepanel mode(i.e. sin(mπx/a) sin(nπy/b)); = ( )E

vm

a4(1 )

42

+ +( )( )( ) ( )v1 nm

v nm

4 34 4

22is the nonlinear stiffness coefficient;

= =F t P t( ) ( )c hH

ch

1,3,5 is the total modal acoustic force within the cavity;ωo is the fundamental linear natural frequency; γ is the aspect ratio; τ isthe panel thickness; ρ is the panel surface density; t is time; E is Young’smodulus; ν is Poisson’s ratio.

Introduce a dummy term K Aa mn into Eq. (3) and consider the fol-lowing Duffing equation

https://doi.org/10.1016/j.rinp.2020.103239Received 28 May 2020; Received in revised form 6 July 2020; Accepted 9 July 2020

E-mail address: bcraylee@cityu.edu.hk.

Results in Physics 18 (2020) 103239

Available online 14 July 20202211-3797/ © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

T

+ + + +

= + + + =

d Adt

K A A F K A

d Adt

K A A

( )

0; ( ) 0

mno a mn mn c a mn

o a

2

22 3

2

22 3

(4,5)

where U=A A cn ( ( ))o ;U is the elliptic integral; =+ +

AK A2( )

o

o a o

22 2 is

the modulus of U ; cn is the elliptic cosine; Ao is the initial modal dis-placement; Ka is a constant to be determined.

Then replace Amn by A in equation (4) and consider a residual termR on the right side

=F K A Rc a (6)

The value of Ka is determined by minimization of R in Eq. (6). Theperiod and frequency of the nonlinear vibration is given by

=+

=+

T KK

dT K

( ) 4 11 ( ) sin( )

; 2( )a

oK A a a2 0

/2

2 2a o2

(7,8)

Results

This case study considers a simply supported square aluminum plateof 0.3048 m× 0.3048 m× 1.2192 mm backed by a 0.0508 m cavity atvarious vibration amplitude ratios. The material properties are: Young’smodulus E = 7 × 1010 N/m2, Poisson’s ratio ν = 0.3, and mass densityρ= 2700 kg/m3. The first 16 symmetrical acoustic modes and the firstthree harmonic terms are used. The frequency ratio is defined as ω/ωL,where ωL is the linear natural frequency. In Fig. 2, vibration amplituderatio is plotted against frequency ratio. The (1,1) mode frequency ratioobtained by the elliptical integral method agrees reasonably well withthat obtained from the harmonic balance finite element method in Lee[6]. The comparison between the 3 curves indicates that the (1,1) mode

frequency ratio shows a lower degree of sensitivity to the vibrationamplitude (in other words, the slopes of the curves are deeper). Thecurves of the 1st anti-symmetric mode and 2nd symmetric mode (i.e.the (2,1) mode and (3,1) mode) are generally close each other. Whenthe amplitude ratio changes from 0 to 0.5, 1, and 1.4, the two frequencyratios increase by about 15%, 50% and 85%, respectively. In Fig. 3,aspect ratio is plotted against frequency ratio. On contrary to that inFig. 2, the (1,1) mode frequency ratio shows a much higher degree ofsensitivity to the aspect ratio. The (2,1) and (3,1) mode curves are al-most vertical or slightly inclined to the left side. It is implied that thefrequency ratios are almost indifferent to the aspect ratio change.

Conclusions

The nonlinear structural-acoustic formulation has been introducedfor the large amplitude free vibrations of a flexible panel backed by acavity, and the elliptical integral method is applied to solve this non-linear problem. The present solution agrees reasonably well with theresults obtained from the harmonic balance method. The higher modefrequency ratios shows a high degree of sensitivity to the vibrationamplitude; and low degree of sensitivity to the aspect ratio.

CRediT authorship contribution statement

Y.Y. Lee: Conceptualization, Data curation, Formal analysis,Investigation, Methodology, Validation, Writing - original draft,Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.

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[3] Xu L. Application of He’s parameter-expansion method to an oscillation of a massattached to a stretched elastic wire. Phys Lett A 2007;368:259–62.

[4] Hui CK, Lee YY, Reddy JN. Approximate elliptical integral solution for the largeamplitude free vibration of a rectangular single mode plate backed by a multi-acoustic mode cavity. Thin-walled Struct 2011;49(9):1191–4.

[5] Lee YY. The effect of large amplitude vibration on the pressure-dependent absorptionof a structure multiple cavity system. PLoS One 2019; 14(7): Article Number:e0219257.

Fig. 1. Rectangular plate backed by a cavity.

Fig. 2. Vibration amplitude ratio versus frequency ratio.

Fig. 3. Aspect ratio versus frequency ratio.

Y.Y. Lee Results in Physics 18 (2020) 103239

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[6] Lee YY. Structural-acoustic coupling effect on the nonlinear natural frequency of arectangular box with one flexible plate. Appl Acoust 2002;63(11):1157–75.

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