To reproduce this result with matlab using analytical method

To reproduce this result with matlab using analytical method


A Compact Matrix Formulation Using the Impedance and Mobility Approach for the Analysis of Structural-Acoustic Systems 0022±460X/99/210097 ‡ 17 $30.00/0 # 1999 Academic Press A COMPACT MATRIX FORMULATION USING THE IMPEDANCE AND MOBILITY APPROACH FOR THE ANALYSIS OF STRUCTURAL^ ACOUSTIC SYSTEMS S. M. KIM AND M. J. BRENNAN Institute of Sound and Vibration Research, University of Southampton, Southampton, Hampshire, SO17 1BJ, U.K. (Received 20 March 1997, and in ®nal form 16 December 1998) This paper describes a compact matrix formulation for the steady-state analysis of structural±acoustic systems. A new approach to the problem is adopted that uses impedance and mobility methods commonly found in the analysis of purely structural or purely acoustic systems. The advantages of the approach are that an investigation into the coupling between the structural and acoustic systems is made easier, and it facilitates improved physical insight into the behaviour of structural±acoustic systems. In addition, because the equations describing the complete system are in matrix form, they can be solved easily using a computer. Due to the mismatch of dimensions between structural mobility and acoustic impedance, new terms are introduced for the coupled system analysis; the coupled acoustic impedance and the coupled structural mobility. F±u (force±velocity) and p±Q (pressure±source strength) diagrams are also introduced for impedance and mobility representations of a complete coupled system. Experimental work is presented, in which a simple rectangular acoustic enclosure with ®ve rigid and one ¯exible side was used, to validate the analytical model and to investigate structural±acoustic coupling. # 1999 Academic Press 1. INTRODUCTION The interaction between an acoustic space and its ¯exible boundaries is an important problem in the ®eld of acoustics. Analysis of this interaction has been of interest to many researchers during the last half a century, as reviewed by Pan et al. [1, 2] and Hong and Kim [3]. A comprehensive theoretical model for coupled responses in a structural±acoustic coupled system has been presented by Dowell et al. [4]. They provided solutions for coupled responses in terms of the modal characteristics of the uncoupled structural and acoustic systems. This paper considers the analysis of a similar coupled system, but uses the impedance- mobility approach, which results in a compact matrix formulation. The impedance-mobility approach is well known to electrical engineers and physicists and is particularly applicable to the analysis of coupled systems, which are Journal of Sound and Vibration (1999) 223(1), 97±113 Article No. jsvi.1998.2096, available online at http://www.idealibrary.com on 98 S. M. KIM AND M. J. BRENNAN composed of several individual linear systems. Each system at the connection can be characterised by impedance or mobility, and the dynamics of a complete coupled system can be described at some or all of the points of interest. They are particularly useful concepts to judge the degree of coupling when two or more systems are connected, and are often used for the analysis of electrical systems [5]. In the 1950s, the method was adapted by mechanical engineers, who applied it to mechanical vibration problems [6]. A general theory of the approach and application examples to mechanical systems can be found in reference [7]. Furthermore, the approach has been successfully applied to various sound and vibration problems, such as the coupling between actuators and substructures, sound radiation from a plate to the acoustic free ®eld, and wave propagation through media with different physical properties, as can be found in many textbooks, for example references [8±10]. In this paper, the classical theory by Dowell et al. [4] is re-examined from the impedance-mobility point of view, and a general method of structural±acoustic coupling analysis is presented. The basic theory of the impedance±mobility approach is considered in Section 2 with a simple conceptual structural±acoustic coupled system. In fact this represents the coupling between a single structural mode and a single acoustic mode. The approach is extended in Section 3 to u ZCA ZS FS F FA YCS QS YA p (b) (a) Q Figure 1. Impedance and mobility representation of a conceptual structural±acoustic system excited by (a) structural excitation, force F, and (b) acoustic excitation, source strength Q. (a) F±u representation for structural excitation; (b) p±Q representation for acoustic excitation. STRUCTURAL±ACOUSTIC COUPLED SYSTEM 99 analyse general structural±acoustic coupled systems in modal co-ordinates. The methodology can also be applied to structural±acoustic systems described by their physical co-ordinates, and this has been described in detail by Kim [11]. A criterion to establish whether or not a structural±acoustic system is strongly or weakly coupled is proposed in Section 3, and this criterion is presented in terms of acoustic impedance and structural mobility. In Section 4, some experimental results are presented that validate the analytical model developed, and illustrate the effects of structural±acoustic coupling. Finally, some conclusions are drawn in Section 5. There is also an Appendix to this paper that gives the relevant equations for the model problem used in the simulations and the experimental work. 2. BASIC THEORY OF THE IMPEDANCE-MOBILITY APPROACH In this section a simple model of a conceptual structural±acoustic system is described, which forms the basis of the comprehensive model of a general structural±acoustic system discussed in Section 3. The conceptual model could, in fact, be used to describe the behaviour of a single structural mode coupled with a single acoustic mode. In a single input structural system, the frequency domain quantities of mobility YS and impedance ZS are de®ned as [7]: YS ˆ u F , ZS ˆ F u , …1a, b† where the subscript S denotes the structural system and F and u are applied force and resulting velocity, respectively. In a single input acoustic system, the impedance and mobility are de®ned as [9]: ZA ˆ p Q , YA ˆ Q p , …2a, b† where the subscript A denotes the acoustic system and Q and p are the source strength and acoustic pressure, respectively. It is important to note that the dimensions of impedance and mobility in structural and acoustic systems are different; the dimension of structural impedance being [Ns/m] and the dimension of acoustic impedance being [Ns/m5]. This dimension difference makes the theoretical description for structural±acoustic coupled systems different from that for general mechanical systems considered in textbooks, for example reference [7]. Consider the conceptual structural±acoustic coupled system consisting of impedances ZS and ZCA excited by a single known structural force F as shown in Figure 1(a). The impedance ZS is de®ned as the uncoupled structural impedance and is the ratio of the effective force applied to the structure FS to the velocity u. The impedance ZCA represents the acoustic reaction force FA to the structural input velocity u and may be de®ned as the coupled acoustic impedance. Thus, ZS ˆ FS u , ZCA ˆ FA u : …3a, b† 100 S. M. KIM AND M. J. BRENNAN Using the force equilibrium condition, F=FS+FA, one gets an expression for the velocity of the structure u in terms of the structural mobility YS and the coupled acoustic impedance ZCA. u ˆ YS 1‡ YSZCA F, …4† where YS=1/ZS. When a single acoustic source of strength Q excites the conceptual structural±acoustic coupled system, it can be represented by the series combination of mobilities YA and YCS as shown in Figure 1(b). Hereafter it is called the p±Q representation since the physical parameters are pressure and source strength, while the diagram in Figure 1(a) is called the F±u representation, i.e., the force±velocity representation. The mobility YA is de®ned as the uncoupled acoustic mobility and is the ratio of the effective source strength QA acting on the acoustic system to the acoustic pressure p. The mobility YCS represents the induced structural source strength QS to the acoustic pressure p and is de®ned as the coupled structural mobility. thus, YA ˆ QA p , YCS ˆ ÿQS p : …5a, b† Note the minus sign of YCS because the direction of QS is de®ned opposite to the u ZCAYS SZAQF QS ZAYCS SYSF p Q (b) (a) Figure 2. F±u and p±Q representation for the conceptual structural±acoustic system with both structural and acoustic excitation. (a) F±u representation; (b) p±Q representation. Administrador Resaltado Administrador Resaltado Administrador Resaltado Administrador Resaltado STRUCTURAL±ACOUSTIC COUPLED SYSTEM 101 acoustic pressure. Since both source strengths are acting toward the acoustic system, the effective source strength acting on this system is QA=Q+QS. Thus, p ˆ ZA 1‡ ZAYCS Q, …6† where ZA=1/YA. When there is both force and acoustic excitation, a coupling factor, which connects the F±u and the p±Q representations is required. For the conceptual system studied in this section an area S may be simply used to match the dimensions. Thus, the relationship between the coupled and uncoupled acoustic impedances and the coupled and uncoupled structural mobilities are given by: ZCA ˆ S2ZA, YCS ˆ S2YS: …7a, b† Conversions between the F±u representation and the p±Q representation can be achieved by using Thevenin and Norton's theorems [7]. The F±u and the p±Q representations for the conceptual structural±acoustic system subject to both structural and acoustic excitation are given in Figure 2(a) and (b), respectively. The equations relating the structural velocity and the acoustic pressure to the applied force and acoustic strength are given by: u ˆ 1 1‡ YSZCA YS…Fÿ SZAQ†, p ˆ 1 1‡ ZAYCS ZA…Q‡ SYSF †: …8a, b† These are the key equations for the analysis of general coupled systems, and can be extended to vector and matrix forms to deal with multi-degree-of-freedom systems with several excitation points. In Section 3 these equations are expanded to model a general structural±acoustic system. If the system is excited by a structural source and the structure responds predominantly as though it was in vacuo then the coupled acoustic impedance has a negligible effect on the structure. In this case the system is said to be weakly coupled. Moreover, if the system is excited acoustically and the cavity responds predominantly as though the structure were in®nitely rigid it is also said to be weakly coupled. These conditions can be examined mathematically using equations (8a) and (8b). If one sets Q=0 in equation (8a) then u=YSF provided that |YSZCA|5 1 so that one can set YSZCA=0. If one sets F=0 in equation (8b) then p=ZAQ provided that |ZAYCS |5 1 so that one can set ZAYCS=0. Noting the relationship between the coupled and the uncoupled acoustic impedance and structural mobility in equations (7a) and (7b), one can see that YSZCA=ZAYCS, and thus the condition for weak coupling is independent of the type of excitation. If there is both structural and acoustic excitation in a weakly coupled conceptual system then the equations for the structural velocity and acoustic pressure are given by u ˆ YS…Fÿ SZAQ†, p ˆ ZA…Q‡ SYSF†: …8c, d† The F±u and the p±Q representations for a weakly coupled conceptual structural±acoustic system are shown in Figure 3(a) and (b), respectively. Administrador Resaltado Administrador Resaltado Administrador Resaltado Administrador Resaltado Administrador Resaltado 102 S. M. KIM AND M. J. BRENNAN 3. STRUCTURAL±ACOUSTIC COUPLING THEORY IN MODAL COORDINATES In this section the impedance and mobility approach described in Section 2 is used to analyse the dynamic behaviour of an arbitrary shaped enclosure surrounded by a ¯exible structure and an acoustically rigid wall such as that shown in Figure 4. The acoustic source strength density function s(x, o) and the force distribution function f(y, o) excite the cavity and the ¯exible structure, respectively. Co-ordinate x is used for the acoustic ®eld in the cavity, and co- ordinate y is used for vibration on the structure. It is assumed that coupled responses can be described by ®nite sets of uncoupled acoustic and structural modes. The uncoupled modes are the rigid- walled acoustic modes of the cavity and the in vacuo structural modes of the structure. Full coupling is considered between the ¯exible structure and the acoustic cavity system. However, weak coupling is assumed between the ¯exible structure and the acoustic ®eld outside the cavity. This is because it is assumed that the vibration of the structure is not in¯uenced by the radiated acoustic ®eld outside the cavity. The acoustic pressure and the structural vibration are described by the summation of N and M modes, respectively. Hence, both the acoustic pressure p at x inside the enclosure and the structural vibration velocity u at y are given by [4]: u YS SZAQ F QS ZA SYSF p Q (b) (a) Figure 3. F±u and p±Q representations for a weakly coupled conceptual structural±acoustic system with both structural and acoustic excitation. (a) F±u representation; (b) p±Q represen- tation. Administrador Resaltado Administrador Resaltado STRUCTURAL±ACOUSTIC COUPLED SYSTEM 103 p…x, o† ˆ XN nˆ1 cn…x†an…o† ˆ CCCTa, …9a† u…y, o† ˆ XM mˆ1 fm…y†bm…o† ˆ FFFTb, …9b† where, the N length column vectors CCC and a consist of the array of uncoupled acoustic mode shape functions cn(x) and the complex amplitude of the acoustic pressure modes an(o), respectively. Likewise the M length column vectors FFF and b consist of the array of uncoupled vibration mode shape functions fm(y) and the complex amplitude of the vibration velocity modes bm(o), respectively. The superscript T denotes the transpose. The mode shape functions cn(x) and fm(y) satisfy the orthogonal property in each uncoupled system, and are normalised as follows: V ˆ … V c2n…x† dV, Sf ˆ … Sf f2m…y† dS, …10a, b† where V and Sf are the volume of the cavity and the surface area of the ¯exible structure, respectively. The complex amplitude of the nth acoustic mode under structural and acoustic excitation is given by [4, 12]: an…o† ˆ roc 2 o V An…o† … V cn…x†s…x, o† dV‡ … Sf cn…y†u…y, o† dS ! …11† where r0 and co denote the density and the speed of sound in air, respectively. Volume V Acoustically rigid