PREDICTION OF ACOUSTIC NATURAL FREQUENCIES FOR TWO DIMENSIONAL SIMPLIFIED AIRCRAFT CABIN BY IMPEDANCE MOBILITY COMPACT MATRIX (IMCM) APPROACH

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1 PREDICION OF ACOUSIC NAURAL FREQUENCIES FOR WO DIMENSIONAL SIMPLIFIED AIRCRAF CABIN BY IMPEDANCE MOBILIY COMPAC MARIX (IMCM) APPROACH Veerabhadra REDDY 1 ; Venkatesham B 2 1 Department of Mechanical and Aerospace Engineering, II Hyderabad, India 2 Department of Mechanical and Aerospace Engineering, II Hyderabad, India ABSRAC Prediction of modal characteristics of acoustic cavities with irregular shapes is an important topi c in vibro-acoustic analysis of systems. In this paper, proposed method of impedance and mobility compact matrix (IMCM) provided the approach of Integro Modal Method (IMM) in the matrix form for prediction of modal characteristics of acoustical cavities with irregular shape. his method, consists of discretizing the whole cavity into a series of sub cavities either regular or irregular cavities. Acoustic pressure in regular sub cavities has been decomposed over a modal basis and for irregular sub cavities over that of bounding surface. Continuity of both pressure and velocity between adjacent sub cavities is ensured using a membrane with zero mass and stiffness. he objective of this method is to develop a physical basis rather than a numerical approach. Mathematical formulation of the method for acoustic cavities with irregular shape has been explained in detail. Predicted natural frequencies of the simplified aircraft cabin based on the proposed method have been compared with the available results in the literature. Keywords: Impedance, mobility, vibro-acoustic analysis I-INCE Classification of Subjects Number: INRODUCION Prediction of modal characteristics of acoustic cavities with irregular shapes is an important issue in vibro-acoustic analysis of systems. Modal characteristics include natural frequencies, modal pressure and velocity of the particles of the medium. Analytical expressions available to calculate natural frequencies and mode shapes of cavities are limited to regular and simple geometries, such as rectangular and cylindrical cavities. In literature methods available for prediction of modal characteristics of cavities with arbitrary shape, are limited. hese methods include Finite element method (FEM), (1, 2), Acousto Elastic Method (AEM) (3), Green s function method (GFM) (4), and Integro Modal Method (IMM) (5). FEM is very popular and widely used for prediction of acoustic natural frequencies of arbitrary geometries; however it needs large number of degrees of freedom and computational time. AEM depends on the irregularity of the cavity shape. GFM is limited to a cavity slightly distorted from a regular one. IMM provides a combined approach of AEM and GFM retaining the advantages of both the methods. his method consists of discretising the whole cavity into a series of subcavities, whose acoustic pressure is decomposed either over a modal basis of regular subcavities or over that of bounding cavities in the case of irregular-shaped boundaries. An integral formulation is then established to ensure continuity of both pressure and velocity between adjacent subcavities using a membrane with zero mass and stiffness (5). E. Anyunzoghe and L. Cheng (6, 7) improved the integro-modal approach developed previously, and introduced the technique of overlapped cavities for better convergence of pressure gradient in the vicinity of the junctions between subcavities. S. M Kim and M. J Brennan (8) presented a compact matrix formulation for the steady-state analysis of vibroacoustic system. It is based on the Impedance and mobility approach using the uncoupled mobility of the structure and uncoupled acoustic impedance, both in modal coordinate systems. his method is very effective for investigating coupling between structural and acoustic systems. B. Venkatesham et al. (9) presented free vibration analysis of coupled acoustic-structural systems. 1 me13p1005@iith.ac.in 2 venkatesham@iith.ac.in 5526

2 In this paper an alternative approach; impedance and mobility compact matrix (IMCM) method is presented for prediction of acoustic modal characteristics of irregular cavities. Mathematical formulation and generalization of the method for acoustic cavities with irregular shape has been explained in detail. A standard Eigen value problem is then established. Numerical results are then presented and discussed. 2. IMPEDANCE AND MOBILIY COMPAC MARIX MEHOD (IMCM) he mathematical formulation consists in treating an irregular shaped cavity as a combination of connected sub-cavities separated by elastic panels. In each sub-cavity, sound pressure P can be calculated using Kirchhoff s Helmholtz integral (5). G( 2 P + λ 2 P)dv = P( 2 G + λ 2 G)dv + (G P G P n n ) ds (1) V V where λ is the wave number; n the outward normal vector of the boundary surface S b of the enclosure with volume V; G is the Green s function corresponding to a transfer function obtained between an observation point (r) and the source point (r 0 ). Green s function G, for interior cavities in terms of mode shapes can be written as (10) G(r, r 0 ) = c2 φ n (r)φ n (r 0 ) (ω 2 n ω 2 (2) )Λ n n Where c is the speed of the sound in the internal medium, ω n is the angular resonance frequency of the cavity, φ n the corresponding mode shape and S b Λ n is the acoustic mass of the cavity. 2 Λ n = φ n (r)dv (3) v Figure 1. Shows a typical irregular cavity and for the illustration purpose, the formulation is first developed for the cavity shown in Figure 1(a). As shown in Figure 1. he cavity investigated can be divided into a regular (Figure 1(b).) and an irregular (Figure 1(c).) subcavity. he junction between the two subcavities is replaced by a virtual vibrating membrane with zero mass and stiffness. a) Junction n n b) c) Figure -1. Discretization procedure a) Real Cavity b) Regular Subcavity c) Irregular Subcavity 2.1 Regular Subcavity In the case of a regular-shaped cavity, analytical expressions are available for the mode shapes and the natural frequencies. ypical examples are rectangular, cylindrical, or semi-cylindrical enclosures. he acoustic pressure P(r, ω) and the vibration velocity of the panel u(r, ω) are 5527

3 described by (8) N P(r, ω) = φ n (r) a n (ω) = Φ a (4) n=1 M u(r, ω) = ψ m (r)b m (ω) = Ψ b (5) m=1 where, n ( consisting of n x, n y ) is a acoustic and m is a structural index, the N length column vectors and a consists of array of uncoupled acoustic mode shape function for the rigid wall cavities amplitude of the acoustic pressure mode a n (ω) Ψ Φ φ n (r), and respectively. Similarly the M length column vectors and b consists of array of uncoupled vibration mode shape functions of the membraneψ m (r), and the uncoupled amplitude of the vibration velocity modes b m (ω) respectively. he amplitude of the nth acoustic mode in regular cavities is given by (8); M a n (ω) = ρ 0 c 2 V A n (ω) (q n + C n,m. b m (ω)) (6) where q n is the generalised acoustic source strength, ρ 0 is density of air, V is volume of the cavity and C n,m represents the geometrical coupling relationship between the uncoupled structural and acoustic mode shape functions on the surface of the vibrating membrane m=1 S f and is given by C n,m = φ n (r) ψ m (r)ds (7) S f Assuming damping is negligible A n (ω) = jω ω n 2 ω 2 (8) he modal acoustic pressure vector a can be expressed as a = Z a (q + q s ) (9) where q is the N length modal source strength vector and q s = Cb is the modal source strength vector due to vibration of the structure and Z a is the uncoupled acoustic modal impedance matrix. he vibration amplitude of the mth mode is expressed as (8) b m (ω) = N 1 B ρ s hs m (ω) c n,m a n (ω) (10) f In the above equation external force is assumed to be zero. ρ s is the density, h is the thickness and S f is the area of the of the vibration membrane. B m (ω) is the structural mode resonance term for the imaginary junction vibration membrane and assumed to be unity. he modal vibration amplitude vector b can be expressed as b = Y s g a (11) where g a = C a is the modal force vector acting on the acoustic system. Y s is the (M M) n=1 diagonal matrix defined as the uncoupled structural modal mobility matrix and is given by Y s = 1 Λ s Λ s = ψ m (r) 2 ds (12) S f 2.2 Irregular Subcavity An irregular shaped subcavity is deviation from regular shape as shown in Figure 1(c). In this the procedure consists of enclosing the irregular cavity by a regular one, called the enveloping or bounding cavity for which modal information is available. Since the natural modes of the irregular 5528

4 shaped cavity are not known analytically, the modes of the bounding cavity are used to obtain the pressure. he amplitude of the nth acoustic mode in irregular cavities is given by a n (ω) = ρ 2 0 c 0 jω V ω 2 n ω 2 (q n + C n,m. b m (ω) M m=1 1 an jω Λ n,n ) (13) n V, ω n, and acoustic mass Λ n belong to the bounding cavity. n,n is the spatial coupling between the t nth and n th acoustic modes of the bounding cavity S d and is given by φ n n,n = φ n ds (14) S d n d In the Eq. (14). Integration is performed over the surface of the irregular cavity, either analytically or numerically. Any irregularity of the boundary shape has the effect of coupling the acoustic modes of its envelope. 2.3 Generalization of the formulation In order to generalize the procedure, four subcavity systems is taken as shown in Figure 2. Middle cavities have two membranes one each on its left (L) and right (R) sides. End cavities have one membrane either left or right side. L R Figure -2. Four subcavity system Applying Eqs. (6, 10) to each subcavity k (k =1, 2, 3, 4) and assuming q = 0 a k = Z ak (C k b k ) k=1, 4 (15) a k = Z ak (C k L b K L + C k R b K R ) k=2, 3 (16) b k = Y sk C k a k k=1, 2, 3,4 (17) o ensure continuity of pressure on both sides of the membranes the following equations are imposed R b k = b k+1 k=1 (18) b L R k = b k+1 k=2, 3 (19) L b k 1 = b k k=4 (20) he above Eqs. (15 to 20) on simplification and assuming = λ, can be rearranged in to the following form (λ 2 [M] + λ[l] + [S])X = 0 (21) where = {a n (ω)}. n n MatricesM, L, and S are given below. o convert Eq. (21) in to standard eigen value problem, Y = λx is assumed and the same is rearranged as 0 I [ M 1 S M 1 L ] {X Y } = λ {X Y } (22) where AZ = λz (23) 0 I A = [ M 1 S M 1 L ], and Z = {X Y } 5529

5 Λ sk Λ nk 0 ω 2 nk Λ sk Λ nk 0 M = [ ], L = [ ] and 0 Λ sk Λ nk 0 ω 2 nk Λ sk Λ nk C k C k 0 S = [ C k C k 1 C k C k ] 0 C k C k 1 0 On solving Eq. (23). Imaginary values of λ 2π are the natural frequencies of the cavity 3. Results and Discussions 3.1 Regular Subcavity A simplified rectangular cavity is used to test the method. he dimensions of the rectangular cavity considered are Lx X Ly=2.0 X1.1 m; hree subcavities with five acoustic and structural modes (n x = n y =5, m=5) is used to validate the results. he dimensions of the three cavities Lx 1 =0.5, Lx 2 =1.05, Lx 3 =0.45 and Ly 1 = Ly 2 = Ly 3 =1.1. he connecting zero mass and stiffness membrane is located at x 1 =0.5 and x 2 =1.55. he mode shapes for membrane and bounding cavities are taken as ψ m (y) = sin ( mπy L y ) (24) φ n (x, y) = cos ( n xπx (L x ) cos ( n yπy L y ) (25) able 1. Natural frequencies of the rectangular cavity calculated with ICMM S. No Mode order Exact Solution (Hz) IMM (Hz) ICMM (Hz) % of Deviation of IMCM results from the exact solution 1 (1,0) (0,1) (2,0) (1,1) (2,1) (3,0) (3,1) (0,2) (1,2) (4,0)

6 able 1 shows the calculated natural frequencies of rectangular cavity using IMCM method. Results obtained from IMCM method are compared with exact solution and IMM results. he calculated results match well with exact solution and IMM results and the deviation is less than 3% at all modes. he convergence of the results to the exact solution improves with increasing number of cavities, number of acoustic and membrane modes. 3.2 Irregular Sub-cavity A simplified two-dimensional aircraft cabin with floor is considered as shown in Figure 3. he dimensions of the simplified cabin θ=49 0 and radius =1 m; wo subcavities with five acoustic and structural modes (n x = n y =5, m=5) is used to validate the results. he dimensions of the two cavities Lx 1 =0.66 m, Lx 2 =1 m and Ly 1 = Ly 2 = 2 m. he connecting zero mass and stiffness membrane is located at x 1 =0.66 m. he mode shapes for membrane and bounding cavities are taken same as shown in Eqs. (23 to 24) θ Figure- 3. Irregular cavity (Simplified two-dimensional aircraft cabin) able 2 below shows the calculated natural frequencies of simplified aircraft cabin with floor using IMCM method. Results obtained from IMCM method are compared with IMM results. he calculated results match well with IMM results and the deviation is less than 5% at all modes. he convergence of the results to the exact solution improves with increasing number of cavities, number of acoustic and membrane modes. 5531

7 able 2. Natural frequencies of the irregular cavity calculated with ICMM Mode order FE Analysis Frequency (Hz) IMM (Hz) ICMM (Hz) % of Deviation of IMCM results from the IMM CONCLUSIONS A new approach has been proposed for the computation of acoustic modes of irregular shaped cavities. he method approximates the solution via impedance and mobility formulation using multi-connected subcavities. he formulation is general and flexible enough to handle different cavity configurations. his method calculates the natural frequencies accurately even when few acoustic modes of the bounding cavity are considered. Future work is required to extend this approach to the prediction of interior noise inside irregular shaped cavities with vibrating structures. REFERENCES 1. Petyt M, Lea J, Koopmann GH. A finite element method for determining the acoustic modes of irregular shaped cavities. Journal of Sound and Vibration Apr 22; 45(4): Joppa PD, Fyfe IM. A finite element analysis of the impedance properties of irregular shaped cavities with absorptive boundaries. Journal of Sound and Vibration Jan 8; 56(1): Dowell EH, Gorman GF, Smith DA. Acoustoelasticity: general theory, acoustic natural modes and forced response to sinusoidal excitation, including comparisons with experiment. Journal of Sound and vibration Jun 22; 52(4): Morse PM, Feshbach H. Methods of theoretical physics, Vol-II, New York; Mc-GrawHill. 5. Missaoui J, Cheng L. A combined integro-modal approach for predicting acoustic properties of irregular-shaped cavities. he Journal of the Acoustical Society of America Jun 1;101(6): Anyunzoghe E, Cheng L. Improved integro-modal approach with pressure distribution assessment and the use of overlapped cavities. Applied Acoustics Nov 30; 63(11): Anyunzoghe E, Cheng L. On the extension of the integro-modal approach. Journal of sound and vibration Aug 8; 255(2): Kim SM, Brennan MJ. A compact matrix formulation using the impedance and mobility approach for the analysis of structural-acoustic systems. Journal of Sound and Vibration May 27; 223(1): Venkatesham B, iwari M, Munjal ML. Free vibration analysis of coupled acoustic structural systems. IISC Centenary-International Conference on Advances in Mechanical Engineering (IC-ICAME); July 2-4; Bangalore, India. 10. Morse PM, Ingard KU. heoretical acoustics. Princeton university press;