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Advances in Acoustics and Vibration
Volume 2009 (2009), Article ID 853407, 14 pages
http://dx.doi.org/10.1155/2009/853407
Research Article

Acoustic Pressure Waves in Vibrating 3-D Laminated Beam-Plate Enclosures

1Department of Systems Engineering, Faculty of Engineering, University of Lagos, Akoka-Yaba, Lagos 23401, Nigeria
2Department of Mechanical Engineering, Faculty of Engineering, University of Lagos, Nigeria

Received 22 September 2009; Accepted 5 December 2009

Academic Editor: Abul Azad

Copyright © 2009 Charles A. Osheku et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3-D laminated beam-plate enclosure is investigated analytically. For this problem, a set of well-posed partial differential equations governing the vibroacoustic wave interaction phenomenon are formulated and matched for the various vibrating boundary surfaces. By employing integral transforms, a closed form analytical expression is computed suitable for vibroacoustic modeling, design analysis, and general aerospace defensive applications. The closed-form expression takes the form of a kernel of polynomials for acoustic pressure waves showing the influence of linear interface pressure variation across the axes of vibrating boundary surfaces. Simulated results demonstrate how the mode shapes and the associated natural frequencies can be easily computed. It is shown in this paper that acoustic pressure waves propagation are dynamically stable through laminated enclosures with progressive decrement in interfacial pressure distribution under the influence of high excitation frequencies irrespective of whether the induced flow is subsonic, sonic , supersonic, or hypersonic. Hence, in practice, dynamic stability of hypersonic aircrafts or jet airplanes can be further enhanced by replacing their noise transmission systems with laminated enclosures.

1. Introduction

The control of vibration and noise propagation from industrial plants, aircraft engines and noise generating machines has remained an active research area for several decades. Nonetheless, limited literature exist in the area of noise-structure dynamic interaction modeling. However, within the context of analytical and experimental studies in acoustic -structure dynamics, a number of investigations have been reported in [19]. For these problems, analytical techniques were employed to study active control of acoustic-structure interaction in 2-D and 3-D enclosures.

In particular, Fang et al. [10], investigated acoustic-structure interaction through a 3-D rectangular enclosure to improve control design analysis. Nevertheless, reliable low-order finite model of 3-D acoustic-structure interaction has posed a formidable challenge in the field, given the fact that model density and order increase from 1-D to 3-D configurations, especially for severe multiple input and output systems. In the field of acoustic-structure interaction, the scarcity of literature in analytical modeling of 3-D enclosures having complex boundary conditions has been attributed mainly to the intractability of dynamic equations which preclude closed form solutions.

For some cases, with simplified assumptions, closed-form solutions are possible usually with rigorous mathematical intrigues and manipulations. One of such methods involves the computation of a transfer function from the frequency response data. Even at that, the limit of these identification techniques is that, the system identification algorithms work satisfactorily only for low-order systems and systems with separated nodes.

However, for 3-D acoustic enclosures, with vibrating boundary surfaces, these identification techniques are fronted with difficulty, especially with two-input one-output systems. Nevertheless, investigations of vibroacoustic pressure waves propagation through a 3-D acoustic enclosure having vibrating laminated boundary surfaces has not been widely reported in literature. Within the context of transverse vibrations of laminated structures, with non-uniform interfacial pressure distribution, some interesting results have been reported recently in [1115]. The effect of structural vibration on the propagation of acoustic pressure waves through a cantilevered 3-D laminated beam-plate enclosure is investigated. For this problem, a set of well posed partial differential equations governing the vibroacoustic pressure waves interaction phenomenon are formulated and matched for the various vibrating boundary surfaces. This paper is organized as follows. Section 1 introduces the problem under investigation within a general context. In the next section, the essential analytical mechanics is briefly reviewed. In Section 3, these relationships are incorporated into a simplified analytical model for the mathematical analysis of the vibroacoustic pressure waves problem. Section 4 is concerned with the analysis of the transmission intensity as modulated by interfacial pressure variation and excitation frequency.

In Section 5, simulated results for the vibroacoustic pressure waves, as influenced by the linear interface pressure variations across the axes of the vibrating boundary surfaces are analyzed whilst in Section 6, the paper ends with the summary and conclusion.

2. Problem Formulation

The problem here is to examine analytically, the effect of the nature of load, frequency variation and the pressure gradient on the acoustic pressure waves propagation.

A general theory of the energy dissipation properties of press-fit joints in the presence of Coulomb friction as originally developed by Goodman and Klumpp provides the basis for the physics of the problem. As illustrated in Figure 1(a) below the proposed 3-D enclosures is constructed with laminated elastic structure for each boundary surface. The contact conditions between the mating layers as itemised in Damisa et al. [11] hold, namely,(i)there is continuity of stress distributions at the interface to sufficiently hold the equivalent layers together both in the pre- and post-slip conditions,(ii)a stick elastic slip with presence of Coulomb friction occurs at the interface of the sandwich vibrating boundary to dissipate energy and does not remain constant as a function of some other variables such as spatial distance, time or velocity.

fig1
Figure 1

The formulated vibroacoustic wave equation is given by the relation2𝑃1𝜕2𝑃𝑐21𝜕𝑡22𝑢𝑥𝜕2𝑃𝑐21𝑢𝜕𝑡𝜕𝑥𝑥2𝑐21𝜕2𝑃𝜕𝑥2=1𝜕2𝑊1𝜌𝜕𝑡2(1) while the governing equations for each vibrating boundaries Ω1,Ω2, Ω3, Ω4 are derived in the appendix. In the meantime, the 2-D views of the structural loadings through Ω3, Ω4 are illustrated in Figures (1(b) and 1(c)) for brevity.

3. Mathematical Analysis of the Vibroacoustic Pressure Waves Equation

In this investigation we will simplify the solution of (1) by assuming that the prompting vibrating membrane is through domain Ω1 while the boundary effects on the pressure waves propagation are restricted to the following domains (Ω2, Ω3, Ω4).

Under this circumstance, (1) takes the form, viz, 21𝑃𝑐02𝜕2𝑃𝜕𝑡22𝑀𝑐0𝜕2𝑃𝜕𝑡𝜕𝑥𝑀2𝜕2𝑃𝜕𝑥21=𝜌𝜕2𝑊1𝜕𝑡2,𝑢𝑀=𝑥𝑐0.(2) Equation (2) can be further expressed as1𝑀2𝜕2𝑃𝜕𝑥2+𝜕2𝑃𝜕𝑦2+𝜕2𝑃𝜕𝑧21+2𝑀𝑐02𝜕2𝑃𝜕𝑡21=𝜌𝜕2𝑊1𝜕𝑡2.(3) By introducing the triple Fourier Finite Sine Transform, namely𝐹𝐹𝐹𝑠{[]}==𝑏0𝑎0𝑐0[][]=8𝔛𝑑𝑥𝑑𝑦𝑑𝑧,𝑎𝑏𝑐𝑘=1𝑚=1𝑛=1𝔛,(4) where 𝔛 denotes sin(𝑛𝜋𝑥/𝑎)sin(𝑚𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐), in conjunction with the Laplace transform, viz, (̃)=0()𝑒𝑠𝑡1𝑑𝑡,()=2𝜋𝑖𝜂+𝑖𝜂𝑖(̃)𝑒𝑠𝑡𝑑𝑠.(5) Equation (3) in the transform plane, subject to zero initial conditions takes the following form, viz, 1𝑀2𝑛2𝜋2𝑎2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘+,𝑠𝑛𝜋𝑎(1)𝑛+1𝑃𝐹𝑦𝐹𝑧𝑎,𝜆𝑚,𝜆𝑘+,𝑠𝑛𝜋𝑎𝑃𝐹𝑦𝐹𝑧0,𝜆𝑚,𝜆𝑘+𝑚,𝑠2𝜋2𝑏2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘+,𝑠𝑚𝜋𝑏(1)𝑚𝑃𝐹𝑥𝐹𝑧𝜆𝑚,𝑏,𝜆𝑘+,𝑠𝑚𝜋𝑏𝑃𝐹𝑥𝐹𝑧𝜆𝑚,0,𝜆𝑘+𝑘,𝑠2𝜋2𝑐2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘+,𝑠𝑘𝜋𝑐(1)𝑘+1𝑃𝐹𝑥𝐹𝑦𝜆𝑛,𝜆𝑚+,𝑐,𝑠𝑘𝜋𝑐𝑃𝐹𝑥𝐹𝑦𝜆𝑛,𝜆𝑚,0,𝑠1+2𝑀𝑐02𝑠2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘=1,𝑠𝜌𝑠2𝑊1𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘,,𝑠(6) where the following have been defined 𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘,𝑠=𝑏0𝑎0𝑐0𝑃sin𝑛𝜋𝑥𝑎sin𝑚𝜋𝑦𝑏sin𝑘𝜋𝑧𝑐𝑃𝑑𝑥𝑑𝑦𝑑𝑧;𝐹𝑦𝐹𝑧𝑎,𝜆𝑚,𝜆𝑘,𝑠=𝑏0𝑐0𝑃(𝑎,𝑦,𝑧,𝑠)sin𝑚𝜋𝑦𝑏sin𝑘𝜋𝑧𝑐𝑃𝑑𝑦𝑑𝑧;𝐹𝑦𝐹𝑧0,𝜆𝑚,𝜆𝑘,𝑠=𝑏0𝑐0𝑃(0,𝑦,𝑧,𝑠)sin𝑚𝜋𝑦𝑏sin𝑘𝜋𝑧𝑐𝑃𝑑𝑦𝑑𝑧;𝐹𝑥𝐹𝑧𝜆𝑛,0,𝜆𝑘,𝑠=𝑎0𝑐0𝑃(𝑥,0,𝑧,𝑠)sin𝑛𝜋𝑥𝑎sin𝑘𝜋𝑧𝑐𝑃𝑑𝑦𝑑𝑧;𝐹𝑥𝐹𝑧𝜆𝑛,𝑏,𝜆𝑘,𝑠=𝑏0𝑐0𝑃(𝑥,𝑏,𝑧,𝑠)sin𝑛𝜋𝑥𝑎sin𝑘𝜋𝑧𝑐𝑑𝑦𝑑𝑧.(7)Next we can evaluate the boundary stresses through the domains Ω2, Ω3, Ω4 from the following relations, viz, 𝑃𝐹𝑦𝐹𝑧0,𝜆𝑚,𝜆𝑘=,𝑠𝑏0𝑐0𝜇𝑃avsin𝑚𝜋𝑦𝑏sin𝑘𝜋𝑧𝑐𝑃𝑑𝑦𝑑𝑧,𝐹𝑦𝐹𝑧𝑎,𝜆𝑚,𝜆𝑘=,𝑠𝑏0𝑐0𝜇𝑃avsin𝑚𝜋𝑦𝑏sin𝑘𝜋𝑧𝑐𝑃𝑑𝑦𝑑𝑧,𝐹𝑥𝐹𝑦𝜆𝑛,𝜆𝑚=,0,𝑠𝑏0𝑎0𝑃av𝑠sin𝑛𝜋𝑥𝑎sin𝑚𝜋𝑦𝑏𝑃𝑑𝑥𝑑𝑦,𝐹𝑥𝐹𝑦𝜆𝑛,𝜆𝑚,𝑐,𝑠=𝑏0𝑎0𝑃av𝑠sin𝑚𝜋𝑥𝑎sin𝑘𝜋𝑦𝑏𝑃𝑑𝑥𝑑𝑦,𝐹𝑥𝐹𝑧𝜆𝑛,𝑏,𝜆𝑘=,𝑠𝑎0𝑐0𝑃av𝑠sin𝑛𝜋𝑥𝑎sin𝑘𝜋𝑧𝑐𝑃𝑑𝑥𝑑𝑧,𝐹𝑥𝐹𝑧𝜆𝑛,0,𝜆𝑘,𝑠=𝑎0𝑐0𝑃av𝑠sin𝑛𝜋𝑥𝑎sin𝑘𝜋𝑧𝑐𝑑𝑥𝑑𝑧.(8) So that on appropriate substitution, (6) is now expressed in Fourier-Laplace transform plane as 1𝑀2𝑛2𝜋2𝑎2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘,𝑠+𝔨1+(1)𝑚+1𝑚𝜋1+(1)𝑘+1𝑚𝑘𝜋2𝜋2𝑏2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘+,𝑠𝑚𝜋𝑏(1)𝑚+1+𝑚𝜋𝑏𝜇𝑃av(0,𝑏)𝑠𝑘(𝑏)2𝜋2𝑐2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘+,𝑠𝑘𝜋𝑐(1)𝑘+1+𝑘𝜋𝑐𝑃av(𝑎,𝑏)𝑠(𝒩)1+2𝑀𝑐02𝑠2𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘=1,𝑠𝜌𝑠2𝑊1𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘,,𝑠(9) where 𝔨 denotes (𝑛𝜋/𝑎(1)𝑛+1+𝑛𝜋/𝑎)𝜇𝑃av(𝑎,0)/𝑠, 𝑏 denotes ((1+(1)𝑚+1)/𝑚𝜋)((1+(1)𝑘+1)/𝑘𝜋), 𝒩 denotes ((1+(1)𝑛+1)/𝑛𝜋)((1+(1)𝑚+1)/𝑚𝜋). Subject to the following pressure profiles prescribed in Olunloyo et al. [12], viz, 𝑝(𝑥,0)=𝑝01+𝜓1𝑥𝑎,𝑝(0,𝑦)=𝑝01+𝜓2𝑦𝑏,(10) This now allows us to obtain the vibroacoustic pressure in the transform plane as 𝑃𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘,𝑠=𝑐021𝑀2𝑛𝜋𝑎(1)𝑛+1+𝑛𝜋𝑎×𝜇𝑃av(𝑎,0)𝑠×𝑏𝑐1+(1)𝑚+1𝑚𝜋1+(1)𝑘+1+𝑘𝜋𝑚𝜋𝑏(1)𝑚+1+𝑚𝜋𝑏𝜇𝑃av(0,𝑏)𝑠×𝑏𝑐1+(1)𝑚+1𝑚𝜋1+(1)𝑘+1+𝑘𝜋𝑘𝜋𝑐(1)𝑘𝑃+1av(𝑎,𝑏)𝑠×𝑎𝑏1+(1)𝑛+1𝑛𝜋1+(1)𝑚+1+1𝑚𝜋𝜌𝑠2𝑊1𝐹𝑥𝐹𝑦𝐹𝑧𝜆𝑛,𝜆𝑚,𝜆𝑘×𝑠,𝑠/(1+2𝑀)2+𝑐021+2𝑀1𝑀2𝑛2𝜋2𝑎2+𝑚2𝜋2𝑏2+𝑘2𝜋2𝑐2,(11)𝑃av(𝑎,0)=𝑃0𝜓1+12𝑃𝑎,av(0,𝑏)=𝑃0𝜓1+22𝑃𝑏,av(𝑎,𝑏)=𝑃0𝑎𝜓1+1/221+𝑟2+𝑟𝜓1+1/221+𝑟2,𝑏𝑟=𝑎.(12)To complete the solution of (10), we have employed the following double Fourier sine transform relations, namely,𝐹𝐹𝑆𝑊𝑥𝑥𝑥𝑥=𝑚4𝜋4𝑎4𝑊𝐹𝑥𝐹𝑦𝑠,𝜆𝑚,𝜆𝑛𝑚3𝜋3𝑎3𝑊𝐹𝑦𝑠,0,𝜆𝑛+(1)𝑚+1𝑊𝑠,0,𝜆𝑛+𝑚𝜋𝑎𝑊𝐹𝑥𝑥𝑦𝑠,0,𝜆𝑛+(1)𝑚+1𝑊𝑥𝑥𝑠,𝑎,𝜆𝑛,(13)𝐹𝐹𝑆𝑊𝑥𝑥𝑦𝑦=𝑚(𝑠,𝑥,𝑦)2𝑛2𝜋4𝑎2𝑏2𝑊𝐹𝑥𝐹𝑦𝑠,𝜆𝑚,𝜆𝑛𝑚2𝑛𝜋3𝑎2𝑏𝑊𝐹𝑦𝑠,𝜆𝑚,0+(1)𝑛+1𝑊𝑠,𝜆𝑚𝑏𝑚𝑛2𝜋3𝑎𝑏2𝑊𝐹𝑦𝑠,0,𝜆𝑛+(1)𝑚+1𝑊𝑠,𝑎,𝜆𝑛+𝑚𝑛𝜋2𝑎𝑏(1)𝑛+1𝑊(𝑠,0,0)+(1)𝑚+1𝑊(𝑠,0,𝑏)+(1)𝑚𝑚𝑛𝜋2𝑎𝑏𝑊(𝑠,0,0)+(1)𝑚+1,𝑊(𝑠,0,𝑏)(14)𝐹𝐹𝑆𝑊𝑦𝑦𝑦𝑦=𝑛(𝑠,𝑥,𝑦)4𝜋4𝑏4𝑊𝐹𝑦𝐹𝑥𝑠,𝜆𝑛,𝜆𝑚𝑛3𝜋3𝑏3𝑊𝐹𝑥𝑠,𝜆𝑚,0+(1)𝑛+1𝑊𝑠,𝜆𝑚+,0𝑛𝜋𝑏𝑊𝐹𝑦𝑦𝑥𝑠,𝜆𝑚,0+(1)𝑛+1𝑊𝑦𝑦𝑠,𝜆𝑚.,𝑏(15) By employing (11), (13), and (14) subject to (A.5), (A.6), (A.7), (A.8), (A.9), and (A.10), 𝑊1𝐹𝑥𝐹𝑦𝐹𝑧(𝜆𝑛,𝜆𝑚,𝜆𝑘,𝑠) can be evaluated from (A.1) as𝑊1𝐹𝜆𝑛,𝜆𝑚,𝜆𝑘=c,𝑠𝑚𝜋𝑎+𝑛𝜋𝑏𝑟2𝜐𝔴6𝜇𝑝0𝜓1+1/2𝑠𝐸𝐻𝑧2×𝑎+𝜇𝐻𝑧2𝑝0𝑎𝜓1+𝑝0𝑏𝜓2/𝜌𝐻𝑧𝑠2+𝐷𝜌𝐻𝑧×𝑛4𝜋4𝑎4𝑛+22𝑚2𝜋4𝑎2𝑏2+𝑚4𝜋4𝑏4(𝔖),(16) where 𝔴 denotes 6𝐹/𝐸𝑏𝐻𝑧3, and 𝔖 denotes (1+(1)𝑘+1)/𝑘𝜋. So that on appropriate substitution into (10), the vibroacoustic pressure can be evaluated in the transform plane as 𝑃𝐹𝑠,𝜆𝑛,𝜆𝑚,𝜆𝑘=𝑐201𝑀2𝜒1(𝑚,𝑛,𝑘)𝜇𝜌0𝜓1+12+𝜒2(𝑚,𝑛,𝑘)𝜇𝜌0𝜓1+12+𝜒3(𝑚,𝑛,𝑘)𝜇𝜌0Ψ𝜒+c4(𝑚,𝑛,𝑘)6𝐹01𝐸𝑏𝐻33(𝑠𝑖𝜔)6𝜇𝜌01+𝜓2/2𝑠𝐸𝐻2𝑧+𝜒5(𝑚,𝑛,𝑘)𝜇𝑟1𝑏𝐻𝑧𝜌0𝜓1+𝑟1𝜓2𝑠/𝔷/(1+2𝑀)2+Ω2𝑛1+2𝑚2𝑟2+𝑛𝑚4𝑟4𝜔02,(17) where 𝔷 denotes 𝜌2(𝑠2+𝜔20), where the following have been defined, viz, 𝜒1=(𝑚,𝑛,𝑘)(𝑛/𝑚)𝜌𝐻𝑧𝑏𝑐/𝑎4𝐷𝔄1+(1)𝑘+1/𝑘𝜋1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,𝜒2=𝑎(𝑚,𝑛,𝑘)4𝑏𝑐(𝑛/𝑚)1+(1)𝑚+1/𝑚3𝜋3𝔅1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,𝜒3=(𝑚,𝑛,𝑘)𝑎𝑏/𝑐(𝑘/𝑚)21+(1)𝑚+1/𝑚3𝜋31+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,𝜒4=𝑎(𝑚,𝑛,𝑘)4𝔇1+(1)𝑘+1/𝑘𝜋1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,𝜒5=𝑛(𝑚,𝑛,𝑘)𝑚(𝔈)1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,Ψ=𝔉21+𝑟2+(𝔊)21+𝑟2,(18) where 𝔄 denotes (1+(1)𝑚+1/𝑚3𝜋3)(1+(1)𝑛+1/𝑛𝜋), 𝔅 denotes (2/𝑛𝜋)(1+(1)𝑘+1/𝑘𝜋), (1+(1)𝑛+1/𝑛𝜋)(1+(1)𝑘+1/𝑘𝜋), 𝔇 denotes (1(𝑛/𝑚)𝑟3/𝜐2)(1/𝑚3𝜋3), 𝔈 denotes (1+(1)𝑘+1/𝑘𝜋)(𝑎4/𝑚3𝜋3)(1/𝑛𝜋), 𝔉 denotes (1+(𝜓1/2)), 𝔊 denotes 𝑟(1+(𝜓1/2)).

By invoking the Fourier inversion, the solution of (16) in the Laplace transform plane can be written as 8𝑃(𝑥,𝑦,𝑧)=𝑎𝑏𝑐𝑘=1𝑛=1𝑚=1𝑃𝐹𝜆𝑚,𝜆𝑛,𝜆𝑘,(19) where denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐).

The above equation can be simplified via the following relations, viz,𝑘=1𝑛=1𝑚=1𝜒1=(𝑚,𝑛,𝑘)𝑘=1𝑛=1𝑚=1(𝑛/𝑚)𝜌𝐻𝑧𝑏𝑐/𝑎4𝐷𝔏1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,(20) where  denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐), 𝔏 denotes ((1+(1)𝑚+1/𝑚3𝜋3)(1+(1)𝑛+1/𝑛𝜋)(1+(1)𝑘+1/𝑘𝜋)). 𝑘=1𝑛=1𝑚=1𝜒2=(𝑚,𝑛,𝑘)𝔣𝑘=1𝑛=1𝑚=1𝑎4𝑏𝑐(𝑛/𝑚)𝔥1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4𝔣,(21) where 𝔣 denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐), 𝔥 denotes (1+(1)𝑚+1/𝑚3𝜋3)(2/𝑛𝜋)(1+(1)𝑘+1/𝑘𝜋):𝑘=1𝑛=1𝑚=1𝜒3=(𝑚,𝑛,𝑘)𝒵𝑘=1𝑛=1𝑚=1𝑎𝑏/𝑐(𝑘/𝑚)21+(1)𝑚+1/𝑚3𝜋31+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4𝒵,(22) where 𝒵 denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐),  denotes (1+(1)𝑛+1/𝑛𝜋)(1+(1)𝑘+1/𝑘𝜋):𝑘=1𝑛=1𝑚=1𝜒4=(𝑚,𝑛,𝑘)𝑘=1𝑛=1𝑚=1𝜋𝑎4𝒬1+(1)𝑘+1/𝑘𝜋(1/𝑛𝜋)1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4,(23) where denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐), 𝒬 denotes (𝑛(𝑛2/𝑚)𝑟3/𝜐2)(1/𝑚3𝜋3).𝑘=1𝑛=1𝑚=1𝜒5=(𝑚,𝑛,𝑘)𝔰𝑘=1𝑛=1𝑚=1𝑛𝑚(𝒯)1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4𝔰,(24) where 𝔰 denotes sin(𝑚𝜋𝑥/𝑎)sin(𝑛𝜋𝑦/𝑏)sin(𝑘𝜋𝑧/𝑐), 𝒯 denotes (1+(1)𝑘+1/𝑘𝜋)(𝑎4/𝑚3𝜋3)(1/𝑛𝜋), in conjunction with the following Fourier closed form representations, namely, 1𝑞=𝜋𝜉=1(1)𝜉+1𝜉sin𝜉𝜋𝑞,0<𝑞<1,(25)𝜉=1sin𝜉𝜋𝑞𝜉3=𝜋2𝑞6𝜋𝑞24+𝑞3,120<𝑞<2,𝑥,𝑦,𝑧𝑞.(26) In view of the foregoing, (17) can now be written as 𝑃𝑠,𝑥,𝑦,𝑧=𝑐201𝑀2Λ1𝑥,𝑦,𝑧𝜇𝜌0𝜓1+1+/2Λ2𝑥,𝑦,𝑧𝜇𝜌0𝜓1+2+/2Λ3𝑥,𝑦,𝑧𝜇𝜌avΨ+cΛ4𝑥,𝑦,𝑧6𝐹01𝐸𝑏𝐻33(𝑠𝑖𝜔)6𝜇𝜌0𝜓1+2/2𝑠𝐸𝐻2𝑧×Λ5𝑥,𝑦,𝑧𝜇𝑟1𝑏𝐻𝑧𝜌0𝜓1+𝑟1𝜓2/𝜌2𝑠2+𝜔20𝑠/(1+2𝑀)2+Ω2×1+2(𝑛/𝑚)2𝑟2+(𝑛/𝑚)4𝑟4𝜔02,(27) where the following have been defined, namely,Λ1𝑥,𝑦,𝑧=𝑛𝑚1𝜐3𝑎5𝐻𝑧𝜌𝐸32𝑥84𝑥2+10𝑥3𝑦𝑧;Λ2𝑥,𝑦,𝑧=𝑎3𝐸𝜌𝑛𝑚32𝑥2𝑥𝑦𝑧;Λ3𝑥,𝑦,𝑧=1𝑐2𝐸𝜌𝑘𝑚294𝑥2𝑥𝑦𝑧;Λ4𝑥,𝑦,𝑧=𝑐𝜋𝑎3𝜌2𝑏𝐸𝐻𝑧2𝑛𝑛2𝑚𝑟3𝜐22𝑥+3𝑥223𝑥3𝑦𝑧;Λ5𝑥,𝑦,𝑧=𝑎3𝐻𝑧𝜌𝑛𝑚2𝑥+3𝑥223𝑥3𝑦𝑧,(28) while Ω2=𝑐021+2𝑀1𝑀2𝑚2𝜋2𝑎2+𝑛2𝜋2𝑏2+𝑘2𝜋2𝑐2,(29)𝜔02=𝐷𝜌𝐻𝑧𝑚4𝜋4𝑎4+𝑛4𝜋4𝑏4+𝑘4𝜋4𝑐4(30) are the natural frequencies of the vibroacoustic enclosures. For the limit case as Mach number, 𝑀0, the well known result in Laudau and Liftsitz [16] viz Ω2=𝑐02(𝑚2𝜋2/𝑎2+𝑛2𝜋2/𝑏2+𝑘2𝜋2/𝑐2), is correctly recovered. By employing the Laplace inversion, the closed form solution for the vibroacoustic pressure wave can be computed as 𝑃𝑥,𝑦,𝐹𝑧,𝜏=Π(𝑚,𝑛,𝑟,𝑀)1(𝜏)1𝑀2Λ1𝑥,𝑦,𝑧𝜇𝑝𝑜𝔯+Λ2𝑥,𝑦,𝑧𝜇𝑝𝑜𝜓1+22+Λ3𝑥,𝑦,𝑧𝜇𝑝0Ψ+Λ4𝑥,𝑦,𝑧6𝐹2(𝜏)6𝜇𝑝0𝐹3𝜓(𝜏)1+22+Λ5𝑥,𝑦,𝑧𝐹4(𝜏)𝑟1𝜇𝑝0𝜓1+𝑟1𝜓2,(31) where 𝔯 denotes (1+(𝜓1/2)), 𝐹1(𝜏)=𝜅sin2𝜋𝜂𝜏𝜂;𝑃𝑥,𝑦,=𝑃𝑧,𝜏𝑥,𝑦,𝑧,𝜏𝐹/𝑏𝐻𝑧,𝑝0=𝑝0𝐹/𝑏𝐻𝑧,𝑡=2𝜋𝜏𝜔0,𝐹2(𝜏)=𝜅2𝑒𝑖2𝜋𝜂𝜏1𝜂12𝜂2𝜂12+cos2𝜋𝜏+𝑖𝜂1sin2𝜋𝜏1𝜂21𝜂12cos2𝜂𝜋𝜏𝜂1𝜂2𝜂𝜂1,𝐹3(𝜏)=𝜅21𝜂2𝑒𝑖2𝜋𝜏21𝜂2𝑒𝑖2𝜋𝜏21𝜂2𝑒𝑖2𝜋𝜏2𝜂21𝜂2𝑒𝑖2𝜋𝜏2𝜂21𝜂2,𝐹4(𝜏)=𝜅sin2𝜋𝜂+𝜂sin2𝜋𝜏2𝜂31𝜂2,𝑐𝜅=𝑚𝜔0Ω,𝜂=𝜔0,𝜂1=𝜔𝜔0,𝑛Π(𝑚,𝑛,𝑟,𝑀)=(1+2𝑀)1+2𝑚2𝑟2+𝑛𝑚4𝑟4.(32)

4. Mathematical Analysis of Sound Intensity as Influenced by Interfacial Pressure Distribution

In acoustics, sound intensity 𝐼, is a vector quantity and can be defined as the time average of the net flow of sound energy per unit area in a direction normal to the area. By combining the fundamental equations governing sound field, mass continuity, the relation between sound pressure and density change in conjunction with Euler’s equation of motion, leads to the equation relating sound intensity with the net flow of energy, viz, 𝜕𝐸𝐼(𝑡)+𝑘𝑒+𝐸𝑝𝑒𝜕𝑡=0,(33) where (𝐸𝑘𝑒+𝐸𝑝𝑒) is total acoustic energy.

In this paper, we shall limit analysis to intensity variation along the spatial variable 𝑥 and for this purpose; the forgoing equation is modified to the following form, namely, =𝐼(𝑡)𝑥0𝜕𝑃𝐴𝜕𝑡𝑐𝜁+𝑃𝐴𝑐𝑐𝑚𝑑𝜁,(34) where 𝑃 is the vibroacoustic pressure; 𝐴𝑐 is the cross sectional area of the 3-D enclosure; 𝑐𝑚 is the velocity of sound through the 3-D enclosure; 𝜁 is a dummy variable of integration. Equation (31) can be normalized into to the form 𝐼=(𝜏)𝑥0𝜕𝑃𝜕𝜏𝜁+𝐼𝑃𝑀d𝜁,0𝑥0𝑒𝑐09𝐼(𝜏)=(𝑡)𝑏𝐻𝑧𝐴𝑐𝐹𝐿𝑐0,𝐿𝑡0𝑐0=1,𝑡=𝑡0𝜏,𝑃=𝑃𝜁,𝑦,z,𝜏(35) in conjunction with the ongoing mathematical representations, viz, 𝛾1𝑥,𝑦,𝑧=𝑛𝑚1𝜐3𝑎5𝐻𝑧𝜌𝐸16𝑥228𝑥3+52𝑥4𝑦𝛾𝑧,2𝑥,𝑦,𝑧=𝑎3𝐸𝜌𝑛𝑚3𝑥48𝑥32𝑦𝛾𝑧,3𝑥,𝑦,𝑧=1𝑐2𝐸𝜌𝑘𝑚29𝑥33164𝑥2𝑦𝛾𝑧,4𝑥,𝑦,𝑧=𝑐𝜋𝑎3𝜌2𝑏𝐸𝐻𝑧2𝑛𝑛2𝑚𝑟3𝜐22𝑥23+3𝑥342𝑥415𝑦𝛾𝑧,5𝑥,𝑦,𝑧=𝑎3𝐻𝑧𝜌𝑛𝑚2𝑥23+3𝑥342𝑥415𝑦𝛾𝑧,6𝑥,𝑦,𝑧=𝑛𝑚1𝜐3𝑎5𝐻𝑧𝜌𝐸16𝑥228𝑥3+52𝑥4𝑦𝛾𝑧,7𝑥,𝑦,𝑧=𝑎3𝐸𝜌𝑛𝑚𝑥323𝑥24𝑦𝛾𝑧,8𝑥,𝑦,𝑧=1𝑐2𝐸𝜌𝑘𝑚23𝑥3498𝑥2𝑦𝛾𝑧,9𝑥,𝑦,𝑧=𝑐𝜋𝑎3𝜌2𝑏𝐸𝐻𝑧2𝑛𝑛2𝑚𝑟3𝜐2𝑥2+𝑥316𝑥4𝑦𝛾𝑧,10𝑥,𝑦,𝑧=𝑎3𝐻𝑧𝜌𝑛𝑚𝑥2+𝑥316𝑥4𝑦H𝑧,1(𝜏)=2𝜅𝜋cos2𝜋𝜂𝜏,𝑡=2𝜋𝜏𝜔0,H2(𝜏)=𝜅22𝜋𝜂𝑖𝑒𝑖2𝜋𝜂𝜏1𝜂12𝜂2𝜂12+2𝜋sin2𝜋𝜏+𝑖2𝜋𝜂1cos2𝜋𝜏1𝜂21𝜂12+2𝜂𝜋sin2𝜂𝜋𝜏𝜂1𝜂2𝜂𝜂1,H3(𝜏)=𝜅22𝜋𝑖𝑒𝑖2𝜋𝜏21𝜂2+2𝜋𝑖𝑒𝑖2𝜋𝜏21𝜂22𝜋𝑖𝑒𝑖2𝜋𝜏2𝜂21𝜂2+2𝜋𝑖𝑒𝑖2𝜋𝜏2𝜂21𝜂2,H4(𝜏)=𝜅𝜋cos2𝜋𝜂+𝜋𝜂sin2𝜋𝜏𝜂21𝜂2.(36)

In view of the above relations, the magnitude of the sound intensity can be evaluated from (35) as×H𝐼(𝜏)=Π(𝑚,𝑛,𝑟,𝑀)1(𝜏)1𝑀2𝛾1𝑥,𝑦,𝑧𝜇𝑝𝑜𝜓1+12+𝛾2𝑥,𝑦,𝑧𝜇𝑝𝑜𝜓1+22+𝛾3𝑥,𝑦,𝑧𝜇𝑝0Ψ+𝛾4𝑥,𝑦,𝑧6H2(𝜏)6𝜇𝑝0H3𝜓(𝜏)1+22+𝛾5𝑥,𝑦,𝑧H4(𝜏)𝑟1𝜇𝑝0𝜓1+𝑟1𝜓2𝐹+𝑀1(𝜏)1𝑀2𝛾6𝑥,𝑦,𝑧𝜇𝑝𝑜𝜓1+12+𝛾7𝑥,𝑦,𝑧𝜇𝑝𝑜𝜓1+22+𝛾8𝑥,𝑦,𝑧𝜇𝑝0Ψ+𝛾8𝑥,𝑦,𝑧6𝐹2(𝜏)6𝜇𝑝0𝐹3𝜓(𝜏)1+22+𝛾10𝑥,𝑦,𝑧𝐹4(𝜏)𝑟1𝜇𝑝0𝜓1+𝑟1𝜓2.(37)

5. Analysis of Results

In this paper, vibration and noise propagation control emanating from complex engineering systems such as industrial power plants, aircraft engines, space propulsive devices and machine enclosures is investigated. The acoustic-structure configuration of interest is the one in which an acoustic disturbance is prompted by one of the vibrating boundaries of the enclosure such as in aircraft cabin noise transmission and systems for outer space exploration. The modeling techniques employed for this study derives from recent advances made in the mechanics of sandwich structures, with non-uniform interfacial pressure distribution. The acoustic-structure dynamic interaction problem is simplified by assuming zero initial conditions prior to the excitation of the upper boundary surface. In the formulated problem, the upper layer of the sandwich elastic plates is subjected to harmonic excitation force as illustrated in Figure 1. To demonstrate the practical significant of our model problem, we have utilized some characteristic values listed in [10] as shown in the Canadian Society for Mechanical Engineers (CSME), Transaction of Applied Mechanics for simulation. Figures 2, 3, 4, 5, and 6 illustrate the characteristic acoustic natural frequencies as couched in (29). In fact, our expression allows for the incorporation of Mach number and for the case where the Mach number is approaching zero, Landau and Lifshitz [16] characteristic frequencies equation is recovered as a special case. As can be seen, the ordering of the family of curves clearly displayed the existence of subsonic, sonic, supersonic and hypersonic zones.

853407.fig.002
Figure 2: Acoustic natural frequency profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3;𝑐=0.8.
853407.fig.003
Figure 3: Acoustic natural frequency profile for the case 𝑚=2;𝑛=2;𝑘=2;𝑏=0.3;𝑐=0.8.
853407.fig.004
Figure 4: Acoustic natural frequency profile for the case 𝑚=3;𝑛=3;𝑘=3;𝑏=0.3;𝑐=0.8.
853407.fig.005
Figure 5: Acoustic natural frequency profile for the case 𝑚=6;𝑛=6;𝑘=6;𝑏=0.3;𝑐=0.8.
853407.fig.006
Figure 6: Acoustic natural frequency profile for the case 𝑚=7;𝑛=7;𝑘=7;𝑏=0.3;𝑐=0.8.

In each zone, the general pattern of the frequency profiles showed that the results are dependent on the geometry of the 3-D enclosure and ambient sound velocity. With respect to the effect of modal parameters, Figure 2, showed the profiles for the principal modes as the axial length increases from 0.4 to 1.2 m whilst 𝑏 and 𝑐 are restricted to the values listed in Table 1.

tab1
Table 1: Parameters for simulation of results.

In the subsonic zone, the natural frequency profiles are inversely proportional to the axial length of the enclosure and decrease monotonically to a constant value irrespective of the axial length as we approach sonic flow. However, in the supersonic zone, the natural frequencies are further attenuated in reversed order to their respective critical values before increasing monotonically in the hypersonic zone.

With respect to the effect of higher modal parameters on the natural frequency profiles, as demonstrated in Figures 36, we observed similar pattern but with higher magnitude.

With respect to the profiles of vibroacoustic pressure, Figures 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16 displayed the various pictures for low and very high frequency excitations. In general the pressure wave profiles are strongly influenced by the values of the pressure gradients and the frequency of the vibrating boundary surface. Interestingly, several profiles can be simulated by playing with the values of 𝜓1 and 𝜓2 irrespective of the frequency of excitation, whereas the choice of any suitable combination can be arranged by progressively varying the tightening torques along the laminated surfaces. Such pressure gradients selection can have significant effect on the vibroacoustic pressure levels.

853407.fig.007
Figure 7: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=10Hz.
853407.fig.008
Figure 8: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3 m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1kHz.
853407.fig.009
Figure 9: Vibroacoustic pressure profile for the case𝑚=1;𝑛=1;𝑘=2;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1MHz.
853407.fig.0010
Figure 10: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1GHz.
853407.fig.0011
Figure 11: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=10Hz.
853407.fig.0012
Figure 12: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1kHz.
853407.fig.0013
Figure 13: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1MHz.
853407.fig.0014
Figure 14: Vibroacoustic pressure profile for the case 𝑚=2;𝑛=1;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1GHz.
853407.fig.0015
Figure 15: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=2;𝑘=1;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1GHz.
853407.fig.0016
Figure 16: Vibroacoustic pressure profile for the case 𝑚=1;𝑛=1;𝑘=2;𝑏=0.3m;𝑐=0.8m;𝑥=0.1270;𝑦=0.1905;𝑧=0.3112;𝜔=1GHz.

The pictures in Figures 710 illustrate the acoustic pressure profiles at the end of one cycle of vibration.

With respect to the effect of low excitation frequency through the boundary surface, the interfacial pressure gradients do not seem to have any strong effect on the acoustic pressure profiles. In fact we note in the lower range of Mach number, that the acoustic pressure waves rise initially before dropping at Mach 2 and increased marginally. However, as we move beyond Mach 16, the effects of the pressure gradients are noticeable.

On the other hand, Figures 811 displayed the pattern of the acoustic pressure waves from low to high excitation frequency in the vibrating boundary surface. We observed that the magnitudes of the pressure waves are significantly reduced with higher excitation frequency. Nonetheless, Figures 1116 illustrate the vibroacoustic pressure profile for the principal modes at Mach 3, as a function of the normalized cycles of vibration.

As can be observed, the pressure profiles are increasing with time. In particular, we note that pressure profile magnitude with lower boundary excitation frequency are significantly higher compare with the effect of higher boundary excitation frequency. On the other hand, Figure 13, showed the acoustic pressure profiles for an excitation frequency of 1 MHz. In respective of the pressure gradients combination, the profiles are monostatically stable. As the boundary excitation frequency increases to 1 GHz, the pressure profiles are significantly lower in the fundamental mode of vibration. However, this may not be the case as we vary the values of the modal parameters by comparison of Figures 1416. This suggests that in practice, dynamic stability of hypersonic aircrafts or jet airplanes can be further enhanced by replacing their noise transmission systems with laminated enclosures.

6. Summary and Conclusion

In this paper, explicit closed form solutions for the vibroacoustic characteristic frequencies, pressure waves and sound intensity or transmission quality through a 3-D enclosure with a vibrating laminated boundary surface is investigated. The vibroacoustic properties are shown to be dependent on, interfacial pressure gradients and boundary surfaces excitation frequencies. The results presented in this study can be positively exploited for the design of modern airplanes, aero-elastic structures and propulsive devices for the launching of space systems.

Appendix

For Case Ω1,𝐷𝜕4𝑊1𝜕𝑥4𝜕+24𝑊1𝜕𝑥2𝜕𝑦2+𝜕4𝑊1𝜕𝑦4+𝜌1𝜕2𝑊1𝜕𝑡2𝜇𝐻𝑧2𝜕𝑝(𝑥,0)+𝜕𝑥𝜇𝐻𝑧2𝜕𝑝(0,𝑦)𝜕𝑦=0.(A.1)

For Case Ω2,𝜕4𝑊2𝜕𝑥4+𝛽2𝜕2𝑊2𝜕𝑡2=𝛼2𝜕𝑃(𝑥,0)𝜕𝑥.(A.2)

For Case Ω3 (mirror reflection of Ω1), 𝐷𝜕4𝑊3𝜕𝑥4𝜕+24𝑊3𝜕𝑥2𝜕𝑦2+𝜕4𝑊3𝜕𝑦4+𝜌1𝜕2𝑊3𝜕𝑡2𝜇𝐻𝑧2𝜕𝑝(𝑥,0)+𝜕𝑥𝜇𝐻𝑧2𝜕𝑝(0,𝑦)𝜕𝑦=0.(A.3)

For Case Ω4 (mirror reflection of Ω2), 𝜕4𝑊4𝜕𝑥4+𝛽4𝜕2𝑊4𝜕𝑡2=𝛼4𝜕𝑃(𝑥,0),𝜕𝑥𝛼1=𝛼2=𝛼3=𝛼4=6𝜇𝐸2,𝛽1=𝛽2=𝛽3=𝛽4=𝜌𝑏.𝐸𝐼(A.4) Now for the 3-D cantilever enclosures under investigation, the following boundary conditions hold for each of the vibrating membranes, viz,

For Case Ω1,(i)𝑊1(0,0,𝑡)=𝑊1(0,𝑏,𝑡)=0,𝑧=0,(A.5)(ii)𝜕𝑊1(0,0,𝑡)=𝜕𝑥𝜕𝑊1(0,𝑏,𝑡)𝜕𝑥=0,𝑧=0,(A.6)(iii)𝜕𝑊21(𝑎,0,𝑡)𝜕𝑥2=𝜕𝑊21(𝑎,𝑏,𝑡)𝜕𝑥2=0,𝑧=0,(A.7)(iv)𝑊1(𝑎,0,𝑡)=𝑊1((𝑎,𝑏,𝑡)=0,𝑧=0,(A.8)v)𝜕𝑊1(0,0,𝑡)=𝜕𝑦𝜕𝑊1(0,𝑏,𝑡)𝜕𝑦=0,𝑧=0,(A.9)(vi)𝜕𝑊21(𝑎,0,𝑡)𝜕𝑦2=𝜕𝑊21(𝑎,𝑏,𝑡)𝜕𝑦2=0,𝑧=0.(A.10)

For Case Ω2, (i)𝑊2(0,0,𝑡)=𝜕𝑊2(0,0,𝑡)=𝜕𝑥𝜕𝑊22(𝑎,0,𝑡)𝜕𝑥2=0,𝑧=(A.11)

For Case Ω3, (i)𝑊3(0,0,𝑡)=𝑊3(0,𝑏,𝑡)=0,𝑧=0,(A.12)(ii)𝜕𝑊3(0,0,𝑡)=𝜕𝑥𝜕𝑊3(0,𝑏,𝑡)𝜕𝑥=0,𝑧=,(A.13)(iii)𝜕𝑊23(𝑎,0,𝑡)𝜕𝑥2=𝜕𝑊23(𝑎,𝑏,𝑡)𝜕𝑥2=0,𝑧=(A.14)(iv)𝑊3(𝑎,0,𝑡)=𝑊3(𝑎,𝑏,𝑡)=0,𝑧=(A.15)(v)𝜕𝑊3(0,0,𝑡)=𝜕𝑦𝜕𝑊3(0,𝑏,𝑡)𝜕𝑦=0,𝑧=(A.16)(vi)𝜕𝑊23(𝑎,0,𝑡)𝜕𝑦2=𝜕𝑊23(𝑎,𝑏,𝑡)𝜕𝑦2=0,𝑧=.(A.17)

For Case Ω4,𝑊4(0,𝑏,𝑡)=𝜕𝑊4(0,𝑏,𝑡)=𝜕𝑥𝜕𝑊24(𝑎,𝑏,𝑡)𝜕𝑥2=0,𝑧=(A.18) On the other hand the effects of the excitation forces at the free end of the 3-D cantilever enclosures can be captured via the following end conditions [11].

For case Ω1,(i)𝐻𝑦0𝜏(𝑥𝑧)1(1)𝐹(𝑥,𝑡)=1(𝑡)2𝑏,at𝑥=𝑎,(A.19)𝜏(𝑥𝑧)1(1)𝑥,𝑦,𝐻𝑧=𝐸𝑧2+𝑧𝐻𝑧21𝑣2𝜕3𝑊3𝜕𝑥3𝜕+𝜐3𝑊3𝜕𝑥𝜕𝑦2+𝐸𝑧2+𝑧𝐻𝑧(1𝜐)21𝜐2𝜕3𝑊3𝜕𝑥𝜕𝑦2+𝜇𝑝(𝑥,0)𝑧+𝐻𝑧𝐻𝑧,(A.20)(ii)𝐻𝑦0𝜏(𝑦𝑧)1(1)(𝑥,𝑡)=𝑇𝑅1𝐹1(𝑡)2𝑏,at𝑥=𝑎,(A.21)𝑇𝑅<1,𝜏(𝑦𝑧)1𝑥,𝑦,𝐻𝑧=𝐸𝑧2+𝑧𝐻𝑧21𝑣2𝜕3𝑊3𝜕𝑦3𝜕+𝜐3𝑊3𝜕𝑥2+𝐸𝑧𝜕𝑦2+𝑧𝐻𝑧(1𝜐)21𝜐2𝜕2𝑊3𝜕𝑦𝜕𝑥2+𝜇𝑝(𝑥,0)𝑧+𝐻𝑧𝐻𝑧(A.22)

For case Ω2,𝐻𝑦0𝜏(𝑥𝑦)2(1)(𝑥,𝑡)=𝐹2(𝑡)2,at𝑥=𝑎,(A.23)𝜏(𝑥𝑦)2(1)(=𝐸𝑥,𝑡)2𝑦2𝐻𝑦𝑦𝜕3𝑊2𝜕𝑥3𝜇𝑝𝑦𝐻𝑦𝐻𝑦.(A.24)

For case Ω3,(i)0𝐻𝑦𝜏(𝑥𝑧)3(1)𝐹(𝑥,𝑡)=3(𝑡)2𝑏,at𝑥=𝑎,(A.25)𝜏(𝑥𝑧)𝑥,𝑦,𝐻𝑧=𝐸𝑧2+𝑧𝐻𝑧21𝑣2𝜕3𝑊3𝜕𝑥3𝜕+𝜐3𝑊3𝜕𝑥𝜕𝑦2+𝐸𝑧2+𝑧𝐻𝑧(1𝜐)21𝜐2𝜕3𝑊3𝜕𝑥𝜕𝑦2+𝜇𝑝(𝑥,0)𝑧+𝐻𝑧𝐻𝑧,(A.26)(ii)0𝐻𝑦𝜏(𝑦𝑧)3(1)(𝑥,𝑡)=𝑇𝑅3𝐹3(𝑡)2𝑏,at𝑥=𝑎,𝑇𝑅<1,(A.27)𝜏(𝑦𝑧)𝑥,𝑦𝐻𝑧=𝐸𝑧2+𝑧𝐻𝑧21𝑣2𝜎3𝑊3𝜎𝑦3𝜎+𝜐3𝑊3𝜎𝑥2+𝐸𝑧𝜎𝑦2+𝑧𝐻𝑧(1𝜐)21𝜐2𝜎2𝑊3𝜎𝑦𝜎𝑥2+𝜇𝜌(𝑥,0)𝑧+𝐻𝑧𝐻𝑧.(A.28)Ω0𝐻𝑦𝜏(𝑥𝑦)4(1)𝐹(𝑥,𝑡)=4(𝑡)2,at𝑥=0,(A.29)𝜏(𝑥𝑦)4(1)=𝐸(𝑥,𝑡)2𝑦2+𝐻𝑦𝑦𝑊4𝑥𝑥𝑥(𝑥,𝑡)+𝜇𝑝𝑦+𝐻𝑦𝐻𝑦.(A.30)

For case 𝑐04,𝑐1

Nomenclature

a:Length of sandwich laminates
b:Width of sandwich laminates
c:Height of sandwich laminates
=𝑐:Ambient speed of sound
𝑑𝑑𝑥:Speed of sound through the 3-D enclosure
𝐼:
𝑊:Differential operator
E:Modulus of rigidity of the laminate
F:Applied end force amplitude
x:Space coordinate along the length of the laminate
y:Space coordinate along the width of the laminate
z:Space coordinate along the height of the laminate
𝑊𝐹:Sound Intensity
W:Dynamic response
𝑊𝐹:Dynamic response in Laplace transform plane
𝜇:Dynamic response in Fourier transform plane
𝜌0:Dynamic response in Fourier-Laplace transform plane
𝜌=𝜌01:Coefficient of friction at the interface of sandwich composite elastic beam
𝑝0:Ambient air density
𝜏(𝑥𝑧)1:
t:Time coordinate
𝜏(𝑥𝑧)2:Clamping pressure at the interface
H:Shear stress at the upper half of the laminates interface
H:Shear stress at the lower half of the laminates interface

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