directivity-1

1. Directionality of sound radiation from rectangular panels Pattabhi R. Budarapu a,⇑ , T.S.S. Narayana b , B. Rammohan c , Timon Rabczuk a,d a Institute of Structural Mechanics, Bauhaus University of Weimar, 99423 Weimar, Germany b Assystem India Private Limited, Bangalore, India c IFB Automotive Private Limited, Bangalore, India d School of Civil, Environmental and Architectural Engineering, Korea University, Republic of Korea a r t i c l e i n f o Article history: Received 31 December 2013 Received in revised form 17 August 2014 Accepted 5 September 2014 Keywords: Receptance Acoustic radiation Directivity a b s t r a c t In this paper, the directionality of sound radiated from a rectangular panel, attached with masses/springs, set in a baffle, is studied. The attachment of masses/springs is done based on the receptance method. The receptance method is used to generate new mode shapes and natural frequencies of the coupled system, in terms of the old mode shapes and natural frequencies. The Rayleigh integral is then used to compute the sound field. The point mass/spring locations are arbitrary, but chosen with the objective of attaining a unique directionality. The excitation frequency to a large degree decides the sound field variations. How- ever, the size of the masses and the locations of the masses/springs do influence the new mode shapes and hence the sound field. The problem is more complex when the number of masses/springs are increased and/or their values are made different. The technique of receptance method is demonstrated through a steel plate with attached point masses in the first example. In the second and third examples, the present method is applied to estimate the sound field from a composite panel with attached springs and masses, respectively. The layup sequence of the composite panel considered in the examples corresponds to the multifunctional structure battery material system, used in the micro air vehicle (MAV) (Thomas and Qidwai, 2005). The demonstrated receptance method does give a reasonable estimate of the new modes. Ó 2014 Elsevier Ltd. All rights reserved. 1. Introduction The rectangular plate is one of the most widely used structures in the industrial world. The sound radiation from vibrating unbaffled panels [2–4], baffled panels [4–9] and submerged panels [10–12] has been the subject of active research for many years. Particularly, the sound radiation from vibrating plates is a common problem in automobiles, airplanes, industrial machinery, buildings and electro-acoustical devices, to name a few. Understanding the sound radiation characteristics of these structures is important for the researchers to maintain the noise levels within the specified limits. Regular or periodic excitation forces are likely to be experi- enced by the plates when they form part of a structure. The driving force spectrum may be composed of a single frequency alone or of a large number of frequencies. There is usually little that can be done to change the nature of the driving forces. Therefore, researchers are studying various techniques related to acoustic radiation for making engineering systems quieter. One of the methods is to arrange the design so that the forces act on a nodal line for the mode shape about to be excited. This method is useful when the applied forces act on a concentrated area only. Naghshineh et al. [13] proposed material tailoring of structures for designing structures that radiate sound inefficiently in light fluids. They solved the problem in two steps. In the first step, given a frequency and overall geometry of the structure, a surface velocity distribution for minimum radiation condition was found. In the second step, a distribution of Young’s modulus and density distribution was found for the structure such that it exhibits the weak radiator velocity profile as one of its mode shapes. Wodtke and Lamancusa [14] discussed the use of damping layers in sound power minimization. They mainly concentrated on the minimization of sound power radiated from plates under broad band excitation by redistribution of unconstrained damping layers, by assuming the total radiated sound power is represented by the power radiated at structural resonances. Apart from the above methods, Jog [15] proposed the reduction of dynamic compliance. St Pierre and Koopmann [16] worked on the point mass attachments to the structures to control the sound. Sonti [9] studied the variation of the sound power from a baffled point-force-driven simply supported rectangular plate subjected to a line constraint, http://dx.doi.org/10.1016/j.apacoust.2014.09.006 0003-682X/Ó 2014 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: pattabhib@gmail.com (P.R. Budarapu). Applied Acoustics 89 (2015) 128–140 Contents lists available at ScienceDirect Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

2. as a function of the constraint angle. Sastry et al. [17] have studied the sound radiation from the baffled rectangular panels attached with point masses and Ramaiah et al. [18,19] have estimated the sound radiation from the baffled composite panels. Xuefeng and Li [20,21] tried to reduce the sound radiation from a plate by modifying the boundary conditions. Fahy [22] proposed the vibro-acoustic noise control based on the reciprocity principle. These methods are grouped as passive methods of noise attenua- tion and works well at high frequencies. Complimentary to the above technique is active noise control, which covers the low frequency range. In active noise control, global control can be achieved for enclosed sound fields at low frequencies, by appropriate placement of sensors and actuators [23,24]. In contrast, global control in the unbounded domains, such as external radiation is still a challenge [25]. Sonti and Jones [26] have developed a curved piezo-actuator model for active vibration control of cylindrical shells. Guo and Pan [27] have demonstrated the active noise control in free field environments. It requires appreciable hardware and achieves reasonable broadband control when the microphones and speakers are optimally located in the sound field. In the free fields, exact cancellation of sound occurs only when the secondary source is a replica of the primary and placed at the same location, which cannot happen in practice. In applications such as stealth in submarines and ships, one alternative might be to achieve the control over the directivity of the external radiated sound, rather than attenuating sound totally. Even in industrial applications it is useful to direct the sound away from the work place and make the environment acceptable. The main objective of the current work is to achieve a change in the directivity of a point driven rectangular plate set in a baffle by attaching point masses/springs to it. The strategy here involves deliberate changes in the mode shapes of the radiator in order to achieve the stated objective. The analysis of the coupled system i.e., determination of new resonances, modes and the response, is performed based on the receptance method [28–37]. The sound field is estimated through the Rayleigh integral [38,39]. The developed methodology has been applied to three different coupled systems as presented in the examples. In the first example, we demonstrate the receptance method through a steel plate attached with point masses. In the second and third examples, the methodology is applied to estimate the sound directivity from a composite panel with attached point springs and masses. The layup sequence of the composite panel considered in examples 2 and 3 corresponds to the multifunctional structure battery material system, used in the MAV [1]. The arrangement of the article is as follows: the receptance method is introduced in Section 1. Details of the receptance method are explained in Section 2. Section 3 explains the estimation of sound field using the Rayleigh integral. Numerical examples are presented in Section 4. In the first example, acoustic directivity of the plate-mass system is studied. The size(s), location(s) and the excitation frequencies are varied to arrive at a particular configuration where the directionality is significant. In the second example, directionality of the composite plate-spring system is studied. The third example is on estimating the acoustic directivity of a composite panel with five attached point masses along a line, at a particular orientation. Section 5 concludes the article. 2. The receptance method The receptance method is well developed and a detailed description of the method can be found in [28–37]. With the receptance method, vibrational characteristics of a combined system can be estimated from the characteristics of the component systems. A feature of the receptance method is that the receptances of the component systems may be determined by any method that is suf- ficiently accurate. In this paper, the receptances are written in terms of the natural frequencies and modes, which can be obtained from any finite element programs or through the experiments. The advantage of the receptance method is that it is a pure analytical method and the new mode shapes and the natural frequencies of the combined system are determined in terms of the old mode shapes and natural frequencies. Receptance is defined as the ratio of response at a certain point (location i) to the harmonic force or moment input at the same or different point (location j), as given below: aij ¼ Response of system A at location i Harmonic force or moment input to system A at location j : ð1Þ The response may be either a line displacement or a rotation. The notation adopted in this paper is as follows: capital letters such as A, B, C refer to subsystems and the Greek letters a; b; c will denote the receptances of the subsystems. The material coordinates of a point in domain X are denoted by X, whose spatial coordinates are denoted by x. ‘M’ indicates the external mass attached to the plate and ma is the mass per unit area of the plate. Note that from the reciprocity theorem aij ¼ aji [22]. Consider two systems, a plate (system A) and mass(es)/ spring(s) (system B), connected in the domain X at two points 1 and 2 respectively, shown in Fig. 1(a). Let the combined system be subjected to harmonic excitation and a’s denote the receptance Fig. 1. Two systems connected at two points. (a) Two masses (system B) m1 and m2 connected at points 1 and 2 on a rectangular plate (system A). (b) Equilibrium displacements and force distribution on systems A and B, subjected to harmonic excitation. P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 129

3. of system A. The force displacement relationship of system A at equilibrium (refer Fig. 1(b)) are given by [37] XA1 XA2 ¼ a11 a12 a21 a22 FA1 FA2 ð2Þ where XA1 and XA2 are the displacements and FA1 and FA2 are the amplitude of forces of system A, at locations 1 and 2, respectively. The matrix notation of Eq. (2) is given below fXAg ¼ ½aŠfFAg: ð3Þ Similarly, the equilibrium equations of the system B are given by [37] XB1 XB2 ¼ b11 b12 b21 b22 FB1 FB2 ð4Þ and the matrix notation of Eq. (4) is fXBg ¼ b½ ŠfFBg: ð5Þ Thus, a11; a22; b11 and b22 denote the drive point receptances and a12; a21; b12, and b21 denote the cross receptances. According to Eq. (1) a11 is the displacement at point 1 due to a unit force applied at point 1 and a12 is the displacement at point 1 due to a unit force applied at point 2. When two such systems are joined together, forces FA and FB become internal forces and they have to add to zero. Therefore fFAg ¼ ÀfFBg: ð6Þ and the continuity of the displacements at the point of contact yields fXAg ¼ fXBg ð7Þ Substituting Eqs. (3) and (5) in Eq. (7) and after simplifying using Eq. (6), we have ða11 þ b11ÞFA1 þ ða12 þ b12ÞFA2 ¼ 0 ð8Þ ða21 þ b21ÞFA1 þ ða22 þ b22ÞFA2 ¼ 0 Eq. (8) can be expressed in the matrix notation as ½aŠ þ ½bŠ½ ŠfFAg ¼ 0: ð9Þ A non-trivial solution of Eq. (9) can be found by setting the determinant of the receptance matrix to zero, i.e., ½aŠ þ ½bŠj j ¼ 0: ð10Þ 2.1. Receptance of coupled system In this section, the receptance of the plate-mass(es)/spring(s) is estimated. Based on Eq. (1) inorder to compute the receptance of a system, the forcing and the response functions are required to be estimated. In the present work, the panel is excited with a harmonic point force. The harmonic point force response of a rectangular panel can be expressed in terms of mode shapes and natural frequencies [40]. The natural frequency is a function of the system characteristics and the mode shape depends on the boundary conditions. Therefore, Section 2.1.1 explains the calculation of natural frequencies and the mode shapes of the clamped rectangular metal panel. Estimation of the natural frequencies and the mode shapes of a composite panel for the simply supported boundary conditions are explained in Section 2.1.2. Section 2.1.3 is focussed on estimating the receptance of the plate-mass system and the receptance of the plate-spring system is calculated in Section 2.1.4. The damped harmonic response of a rectangular panel subjected to a harmonic point force excitation at point (xp; yp) is given by [41], wðx; y; tÞ ¼ 1 maCmn X1 m¼1 X1 n¼1 Umnðxp; ypÞFejxt ðx2 mn À x2 þ 2jfmnxmnxÞ Umnðx; yÞ ð11Þ where ma is the mass per unit area of the plate, F is the force amplitude, xmn is the ðm; nÞth natural frequency, x is the driving frequency and Umn represent the mode shapes. The mass per unit area (ma ) is estimated as ma ¼ qh for the metal plate q1h1 þ q2h2 þ Á Á Á þ qnhn for the composite plate ð12Þ where q and h are the density and thickness of the metal plate, respectively and qn and hn are the density and thickness of the nth layer of the composite plate, respectively. The mode shapes of a rectangular panel can be expressed as the product of the mode shape functions along the x and y directions [40], as given below Umnðx; yÞ ¼ XðxÞYðyÞ ð13Þ where X and Y are chosen as the fundamental mode shapes of the beam. Estimation of the modes for the clamped rectangular metal panel and the rectangular simply supported composite panel are explained in Sections 2.1.1 and 2.1.2, respectively. The constant Cmn can be evaluated as, Cmn ¼ Z a 0 Z b 0 U2 mndxdy: ð14Þ If k is the equivalent viscous damping factor, the modal damping coefﬁcient fmn is given by fmn ¼ k 2maxmn ð15Þ 2.1.1. Modes of the rectangular metal plate The expressions for the mode shapes (X) of a beam having the clamped boundary conditions are given below [40], XðxÞ ¼ cosc1 x a À 1 2 þ sinðc1 2 Þ sinhðc1 2 Þ coshc1 x a À 1 2 for m ¼ 2;4;6;... ð16Þ where the values of c1 are obtained as roots of tan c1 2 þ tanh c1 2 ¼ 0 ð17Þ and XðxÞ ¼ sinc2 x a À 1 2 À sinðc2 2 Þ sinhðc2 2 Þ sinhc2 x a À 1 2 for m ¼ 1;3;5;... ð18Þ where the values of c2 are obtained as roots of tan c2 2 À tanh c2 2 ¼ 0 ð19Þ The functions Y in Eq. (13) are obtained by replacing x by y, a by b and m by n in Eqs. (16)–(19). The natural frequencies xmn of the clamped plate are given by [40], x2 mn ¼ p4 D a4q G4 x þ G4 y a b 4 þ 2 a b 2 ½lHxHy þ ð1 À lÞJxJyŠ ð20Þ where D is expressed as D ¼ Eh 2 12ð1 À l2Þ : ð21Þ The expressions for Gx; Gy; Hx; Hy; Jx and Jy are given in Table 1. 130 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

4. 2.1.2. Modes of the rectangular composite plate The damped point force response of a rectangular composite plate due to a harmonic point force at ðxp; ypÞ is given by Eq. (11), where ma is estimated based on Eq. (12). We consider the simply supported boundary conditions in the analysis of composite panels, where the mode shapes are given by [42] Umn ¼ sin mpx a sin npy b ð22Þ The natural frequencies xc mn of the simply supported rectangular composite plate are given by [42] x2 mn ¼ p4 q D11 m a 4 þ 2ðD12 þ 2D66Þ m a 2 n b 2 þ D22 n b 4 ð23Þ the constants D11; D12; D66 and D22 can be calculated as explained in [42]. The constant Cc mn is estimated by using Eq. (22) in Eq. (14). And the constant fc mn is calculated from Eq. (15), using Eq. (12). 2.1.3. Receptance of the plate-mass system Consider a rectangular plate attached with two point masses, at points 1 and 2 as shown in Fig. 1(a). The cross receptances of the masses are zero, since the force on one mass does not cause the other mass to respond. Let a’s represent the receptances of the plate and b’s represent the receptances of the masses. Elements of the receptance matrix for the plate given in Eq. (2) can be estimated in the following steps. First, the drive point receptance aii is given by, aii ¼ xi Fiejxt ¼ 1 maCmn X1 m¼1 X1 n¼1 Umnðxi; yiÞ ðx2 mn À x2 þ 2jfmnxmnxÞ Umnðxi; yiÞ ð24Þ and secondly, the cross receptance aij; i – j, is given by aij ¼ xi Fjejxt ¼ 1 maCmn X1 m¼1 X1 n¼1 Umnðxj; yjÞ ðx2 mn À x2 þ 2jfmnxmnxÞ Umnðxi; yiÞ: ð25Þ The receptance of a mass can be estimated from the steady- state response of a mass subjected to a harmonic force input. From Newton’s third law M€xBi ¼ FBiekxt ð26Þ since xBi ¼ XBiekxt , the drive point receptance of the mass bii is given by, bii ¼ XBi FBi ¼ À 1 Mx2 : ð27Þ The cross receptances of the mass are zero, since the force on one mass does not cause the other mass to respond. Therefore, as explained in Section 2, the new natural frequencies (xk) of the plate attached with two point masses are obtained by solving the below equation a11 À 1 mB1x2 a12 a21 a22 À 1 mB2x2

14. ¼ 0: ð28Þ In the current work, Eq. (28) is solved for the new natural frequencies of the plate-mass system by a numerical procedure. It can also be solved graphically, by plotting the determinant as a function of the excitation frequency x and capturing the points where the determinant is zero. The above procedure can be extended to the case where N masses are attached to a plate. New natural frequencies of the plate-mass system can be obtained by setting the determinant of the N Â N receptance matrix to zero. The new mode shapes of the plate-mass system can be determined from the point force response expression of the plate without masses [37]. For the case of a plate attached with single mass, Eq. (11) gives the new mode shape, when the excitation frequency x, is set to the new natural frequency xk and ðx; yÞ becomes the location of the mass. For a plate attached with N point masses there will be N new resonances. Since the plate is constrained at N points, it will experience point forces at those N locations. The magnitudes of these point forces are given by the elements of the eigenvector corresponding to the zero eigenvalue of the receptance matrix evaluated at the new natural frequency, xk. Thus, the new kth mode shape is given by substituting xk for x in Eq. (11) with an additional summation term as shown below Ukðx; yÞ ¼ 1 maCmn X1 m¼1 X1 n¼1 PN i¼1Umnðxmi; ymiÞFik ðx2 mn À x2 k þ 2jfkxkxÞ Umnðx; yÞ ð29Þ where Fik is the ith element of the eigenvector of zero eigenvalue, corresponding to the kth new natural frequency and ðxmi; ymiÞ is the location of the ith mass. The response of the plate-mass system subjected to a point force can now be calculated using the new mode shapes obtained from Eq. (29) as wkðx; y; tÞ ¼ 1 ma XN k¼1 1 Ck Ukðxp; ypÞFejxt ðx2 k À x2 þ 2jfkxkxÞ Ukðx; yÞ ð30Þ where F is the amplitude of the force at location ðxp; ypÞ, xk the kth natural frequency, x is the driving frequency, fk the modal damping coefﬁcient, given by fk ¼ k 2maxk ð31Þ and the constant Ck is evaluated as, Ck ¼ Z a 0 Z b 0 U2 kdxdy: ð32Þ 2.1.4. Receptance of the plate-spring system Consider a rectangular plate attached with two linear springs of stiffness k1 and k2, respectively, as shown in Fig. 2. The displacement and force relationships for the coupled system can be derived on the similar lines of the plate-mass system shown in Fig. 1. Let Table 1 Variables to estimate the natural frequencies of the clamped rectangular metal plate. Variable For m = 1 or n = 1 For all the other modes Gx 1.506 ðm þ 1Þ À 0:5 Gy 1.506 ðn þ 1Þ À 0:5 Hx 1.248 ððm þ 1Þ À 0:5Þ2 1 À 2 ððmþ1ÞÀ0:5Þp Hy 1.248 ððn þ 1Þ À 0:5Þ2 1 À 2 ððnþ1ÞÀ0:5Þp Jx 1.248 ððm þ 1Þ À 0:5Þ2 1 À 2 ððmþ1ÞÀ0:5Þp Jy 1.248 ððn þ 1Þ À 0:5Þ2 1 À 2 ððnþ1ÞÀ0:5Þp Fig. 2. Two springs of stiffness k1 and k2 connected at two points on a rectangular plate. P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 131

15. a’s represent the receptances of the plate and b’s represent the receptances of the springs. The drive point and the cross receptances of the plate aii and aij are given by Eqs. (24) and (25), respectively. The receptance of a spring can be estimated from the steady-state response of a spring subjected to a harmonic force input. From Newton’s third law kxB1 ¼ FB1ejxt ð33Þ since xB1 ¼ XB1ejxt b11 ¼ XB1 FB1 ¼ 1 k ð34Þ Therefore, for the plate attached with two spring system, the new natural frequencies (xc k) are estimated by solving the equation, a11 þ 1 kB1 a12 a21 a22 þ 1 kB2

25. ¼ 0: ð35Þ The structure of the matrix can now be extended to the case where N springs are attached, as explained in the previous section. Hence, the new natural frequencies and the new mode shapes of the plate- spring system can be obtained from the Eqs. (35) and (29), respectively. Knowing the new natural frequencies and the mode shapes, the point force response of the plate-spring system can be estimated from Eq. (30). 3. Sound power calculation During the normal vibrations of a rectangular plate, the normal velocity of the acoustic medium on the surface of the plate loaded at r1 must be equal to the normal velocity of the plate v(r1), refer Fig. 3. Due to the acoustic perturbation on the plate surface, the generated acoustic pressure p(r2) at r2 can be estimated by the Rayleigh’s integral [43,38] as given below pðr2; tÞ ¼ jq0x 2p ejxt Z S1 vnðr1; tÞeÀjkR R dS1 ð36Þ where r2 is the position vector of the observation point, r1 the position vector of the elemental surface dS1 having the normal velocity vnðr1Þ; R ¼j r2 À r1 j; q0 is the density of air, k is the acoustic wave number and S1 is the area of the plate. Considering a hemispherical measurement surface in the far field as shown in Fig. 3, when r2 is much larger than the source size as defined by the larger edge of the two panel dimensions a and b, i.e r2 ) a and a b, R and r2 are related by the approximate relationship [43] R ’ r2 À x sin h cos / À y sin h sin /: ð37Þ The sound intensity is given by the time-averaged product of the sound pressure and the particle velocity vector. In the far field, the component of acoustic particle velocity in phase with the pressure is radially directed. As a result, the sound intensity vector is also radially directed and given by the product of the sound pressure and the radial component of the particle velocity. Therefore, for harmonic motion the time averaged sound intensity in the far field is given by [43] I ¼ j pðr; h; /; xÞj2 2q0c : ð38Þ In otherwords, I is the time-averaged rate of energy transmission through a unit area normal to the direction of propagation. Hence, the sound power Wp radiated into the semi infinite space above the plate is the integral of the sound intensity over the panel surface, which is given by Wp ¼ Z S1 Iðr2ÞdS2; ð39Þ where S2 is an arbitrary surface which covers area S1, see Fig. 3. Eq. (39) is evaluated by using Eqs. (36)–(38). When the two surfaces S1; S2 becomes equal i.e, S2 ¼ S1, then r1 and r2 would represent any two arbitrary position vectors on the plate surface. Therefore, the radiated sound power from the plate can be expressed as Wp ¼ q0x 4p Z S2 Z S1 Re vðr1Þ je ÀjkR R ! vÃ ðr2Þ # dS1dS2 ð40Þ where S1 and S2 are the areas on the xy plane with 0 x a and 0 y b. Using the Maxwell’s reciprocity relation between source at r1 and receiver at r2 Eq. (40) can be simplified as Wp ¼ q0x 4p Z b 0 Z a 0 Z b 0 Z a 0 vðr1Þ sinðkRÞ R vÃ ðr2Þdx1dy1 # dx1dy2: ð41Þ 4. Numerical examples In this section three examples are presented. In the first example, the sound directivity of the steel plate attached with point masses is achieved for a particular size(s) and location(s) of the mass(es), when the system is excited at a specific frequency. The Fig. 3. A rectangular panel set in a baffle showing the details of the parameters required to estimate the radiated sound power in the far field, along with highlighting the integration areas S1 and S2. 132 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

26. second and third examples focussed on directing the sound from a composite rectangular panel by attaching point springs and masses, respectively. The considered composite panel forms part of the structure battery material used in the fabrication of MAV [1]. 4.1. Example 1: Rectangular plate-mass system In this example, we study the sound directivity of a steel plate- mass system. The plate dimensions are 380 mm Â 300 mm along the x and y directions with a thickness of 1 mm. The density, Young’s modulus and Poisson’s ratio of the plate are 7815 kg/m3 , 205 GPa and 0.285, respectively. Clamped boundary conditions are considered for all the edges of the plate. The plate is excited by a point force of unit amplitude located at (100 mm, 75 mm) from the bottom left corner of the plate, refer to Fig. 1(a). Natural frequencies and the point force response of the coupled system are estimated from the analytical model developed in Section 2. The radiated sound field is estimated based on the Rayleigh Fig. 4. Schematic of the experimental setup. (a) Connections of various equipment to the plate to measure the plate response. (b) Plate set in a baffle along with the microphone in the anechoic chamber to measure the sound radiation from the plate. Fig. 5. Comparison of natural frequencies from the analytical, numerical and the experimental models. Fig. 6. Comparison of the point force response from analytical and numerical models of the plate along a particular line in the x (left) and the y (right) directions, at 500 Hz. Table 2 Comparison of natural frequencies of the plate attached with three and five-mass system from analytical and numerical models. Mode Three masses Five masses Anal. (Hz) Num. (Hz) % Error Anal. (Hz) Num. (Hz) % Error 1 77.1 76.7 0.519 72.3 71.8 0.692 2 135.1 134.6 0.370 117.3 116.8 0.426 3 180.0 177.5 1.389 173.5 170.4 1.789 4 216.8 215.5 0.600 195.8 192.4 1.736 5 233.0 230.0 1.288 211.5 209.9 0.757 Table 3 Mass locations. Case Locations (mm) from the bottom corner of the plate Single mass (76, 60) Three masses (76, 60), (76, 180) and (150, 190) Five masses (76, 60), (76, 180), (150, 190), (250, 120) and (310, 200) P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 133

27. integral as discussed in Section 3. The analytical results are validated with the numerical model and the experimental model. A schematic of the experimental model is shown in Fig. 4. As shown in Fig. 4(a) the baffled plate is excited with a piezoelectric exciter and the amplitude of the excitation is controlled through a piezoelectric controller. The response of the plate is captured through the response of the piezoelectric accelerometers attached at several locations of the plate. The total weight of the accelerometers is small compared to the weight of the plate, hence neglected in the present work. The output signal from the accelerometer is further analyzed in the spectrum analyser to estimate the natural frequencies and the modes. The experiment is carried out in an anechoic chamber in a reverberating room, as shown in Fig. 4(b). The sound power from the plate is estimated by measuring the sound pressure levels at several identified locations in the free field. The measured sound pressure levels are first converted to sound pressure. Secondly, the sound intensity is estimated using Eq. (38). Finally, the sound power is calculated based on Eq. (39). Fig. 5 compares the natural frequencies from the analytical, numerical and experimental models for the bare plate. The analytical and numerical models agree with each other. The experimental model deviates for the higher order modes. The amplitudes of the displacement between the analytical and numerical models, displacements along two particular lines in the x and y directions are captured and compared in Fig. 6. Table 2 lists the first five natural frequencies of the plate with attached masses, from the analytical and the numerical models. Locations of masses used in the Fig. 7. Schematic of the locations of masses (a) listed in Table 3 and (b) five infinite masses arranged along a 30° line. Table 4 Comparison of natural frequencies from analytical and numerical models, of the bare plate and the plate with five infinite masses aligned at different orientations. Mode Without masses (Hz) Five infinite masses along 15° line (Hz) 30° line (Hz) 45° line (Hz) 60° line (Hz) 1 87.6 120.59 161.78 146.02 112.12 2 152.6 180.56 233.24 233.08 210.80 3 202.1 275.57 241.67 252.34 239.28 4 258.1 289.26 308.04 315.36 326.03 5 262.1 354.83 369.48 388.57 382.84 Fig. 8. Point force response of the plate with five infinite masses oriented along 15°, 30°, 45° and 60° lines at different frequencies. The first row corresponds to the point force responses close to the first natural frequency and the second row corresponds to responses close to the second natural frequency. The mode bifurcation along the orientation angle can be observed. 134 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

28. calculation are given in Table 3, where a schematic with approximate locations of the masses listed in Table 3 along with the point of excitation on the plate is shown in Fig. 7(a) and (b) shows five infinite masses arranged along the 30° line. Size(s) of the mass(es) is taken as 50 g each. From Table 2, we observed that the natural frequencies are decreasing with the addition of masses, as the natural frequencies are inversely proportional to the mass. The per- centage error is estimated as the ratio of difference between the analytical and the numerical results multiplied by 100, as given below %error ¼ xanal À xnum xanal Â 100: ð42Þ where xanal and xnum are the natural frequencies estimated from the analytical and numerical models, respectively. The minimum and maximum errors are observed to be 0.370% and 1.789%, occured while estimating the second natural frequency of the plate attached with 3 masses and the third natural frequency of the plate attached with 5 masses, respectively. When the magnitude of the attached mass(es) reaches a large value, in the limit of infinity, the point of mass attachment will be completely arrested in all degrees of freedom. Therefore, infinite masses can be used to restrain and stiffen a particular point. In otherwords, infinite masses increases the stiffness of the system and hence the natural frequencies. Table 4 summarises the natural frequencies of the plate with five infinite masses aligned along the 15°, 30°, 45° and 60° lines to the x axis. We considered 10000 kg as the infinite mass in the present study. Since the mass variable is in the denominator of Eq. (27), the drive point receptances of the mass (bii) in Eq. (27) will be significantly small compared to the drive point receptances of the plate (aii) in Eq. (28). Hence, the value of bii with 10,000 kg mass is very small and its contribu- tion to the receptance matrix in Eq. (28) can be neglected. Alter- nately, bii goes to zero as ‘M’ approaches infinity in Eq. (27). Because of the infinite stiffness(es)/mass(es) at a particular point, the response of the structure at that particular point will be zero. Therefore, infinite point mass(es)/spring(s) will act as restraints. From the values of natural frequencies it can be observed that Fig. 9. Hemispherical distribution of sound intensity of the plate with; for the case of plate with single point mass excited at 1300 Hz (a) projected on to two dimensional plane and (b) in three dimensional plane and for the case plate with three point masses excited at 2500 Hz (c) projected on to two dimensional plane and (d) in three dimensional plane; with permission from [17]. P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 135

29. the natural frequencies are significantly raised by attaching the infinite masses. Therefore, infinite masses can be used as stiffeners. Since the infinite masses acts like restrainers, when they are aligned along a particular line, the line acts like a rigid boundary. Hence the mode bifurcation across the line should be observed. The point force response of the plate at different frequencies with five infinite masses oriented along 15°, 30°, 45° and 60° lines at various frequencies are plotted in Fig. 8. Each row corresponds to the deformed configuration at a particular orientation. The responses are captured close to the natural frequencies, so that the mode bifurcation can be observed. At the natural frequencies the response pattern matches with the mode shape. For example, the point force responses in the first row are captured close to the first natural frequency, so that the shape of the response will be close to the first mode. From Fig. 8(a)–(c), it can be observed that the first mode is clearly restrained along the constrained line. Similarly, the second row pictures correspond to the point force responses of the above plate close to the second natural frequency. From Fig. 8(d)–(f), the second mode is restrained along the constrained line. The goal of the present work is to achieve the sound directionality. To achieve the objective a Guess and check method is employed by varying the number, locations of the masses and the frequency of excitation. A 50 g mass(es) are considered in this paper. Therefore, several analytical simulations are carried out at different combinations of number, locations of the masses and the frequency of excitation, to achieve the directivity. Finally, clear directivity was observed in 2 cases; (1) single mass excited at 1300 Hz and (2) three masses at 2500 Hz. The locations of the masses are given in Table 3. The sound intensity contours corresponding to cases 1 and 2 are plotted in Fig. 9. The acoustic intensity is calculated and plotted for different in-plane angles (h) and out of plane angles (/) values based on Eq. (38). Fig. 9(a) and (b) shows the intensity distribution of the plate with single mass, excited at 1300 Hz in two and three dimensions, respectively. Fig. 9(c) and (d) plots the distribution of sound intensity when the plate is attached with three point masses and excited at 2500 Hz. From the intensity plots the sound field is clearly directed by attaching the masses. Furthermore, to validate the estimated analytical sound power calculations, sound power levels are calculated from a numerical model of the completely clamped plate model without attaching any masses, developed in Sysnoise [44]. On top of that, the analytical results are verified with the experiment as explained in Fig. 4. Fig. 10 compares the sound power levels estimated from the analytical, numerical and the experimental models. The sound power levels from the three models are observed to deviate by around 10 dB. Fig. 10. Comparison of the sound power level from the analytical, numerical and the experimental models. Fig. 11. Layup sequence of different layers of the composite plate used in example 2. Table 5 Properties of different layers of the composite plate. Material Young’s modulus E (GPa) Density (kg/m3 ) Thickness (mm) Orientation (Deg) (No. of Layers) Carbon Epoxy 72.40 1100 0.15 0/45/45/0 (4 Layers) Packaging 4.60 1290 0.20 0/0 (2 Layers) PLI 1.02 2540 0.50 0/0 (2 Layers) Table 6 Comparison of natural frequencies of the composite plate with zero, two, four and five-spring system with large spring stiffness from analytical and numerical models. Mode Without springs Two springs along 15° line Four springs along 30° line Five springs along 60° line Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error % Anal. (Hz) Num. (Hz) Error % 1 62.3 62.05 0.40 95.89 93.97 2.00 139.97 137.79 1.55 95.25 92.68 2.69 2 140.7 139.87 0.59 160.98 160.63 0.21 236.43 229.91 2.75 197.27 197.66 À0.19 3 173.8 174.11 À0.17 246.76 242.96 1.54 269.37 270.14 À0.28 238.17 229.23 3.75 4 249.4 246.37 1.21 283.37 279.97 1.20 355.63 353.07 0.72 334.78 337.23 À0.73 5 272.9 274.59 À0.61 312.97 314.58 À0.51 395.89 393.34 0.64 400.67 392.52 2.03 136 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

30. 4.2. Example 2: Rectangular composite plate-spring system Composite structures are extensively used in all the engineering disciplines due to their high strength to weight ratios, ease of han- dling and transportation. Metallic structures do not offer the flex- ibility to achieve the required response. Replacement of the metallic structure with composite structures can be aeroelastically tailored [45,46]. In this example, we estimate the sound radiation from the composite rectangular panels. Fig. 11 shows the lay up sequence along with the orientation angles of different layers of a panel structure used in a typical MAV [1]. Plastic Lithium Ion (PLI) is the main battery of the structure, offering additional structural support and carbon epoxy layers resists the mechanical the loads on to the structure. Packaging material is used to couple the carbon epoxy layer with PLI. Combining structure and battery (power) functions in a single material entity permits improve- ments in system performance, which is not possible through inde- pendent subsystem optimization. Consider a 0.4 m Â 0.3 m panel, with the material properties, thicknesses and the orientations of the individual laminas listed in Table 5. The simply supported boundary conditions are considered on all the edges. The natural frequencies of the composite panel attached with linear springs along a particular line, aligned in different orientations are listed in Table 6. Table 6 also compares the natural frequencies estimated from the analytical and the numerical models and the %error estimated from Eq. (42) is also listed. Each spring of infinite stiffness is considered in all the calculations. A spring of infinite stiffness acts as a restraint, hence the point of attachment acts as a node. Therefore, infinite stiffness springs can be used to restrain and stiffen a particular point. In otherwords, infinite stiffness springs increases the stiffness of the system and hence the natural frequencies. Table 6 summarises the natural frequencies of the plate with five infinite stiffness springs aligned along lines at 15°, 30°, and 60°. Ref. [47] for the acoustic radiation of a rectangular plate reinforced by springs at arbitrary locations. From the values of natural frequencies it can be observed that the natural frequencies are raised by attaching large stiffness springs. The minimum and maximum errors are observed to be À0.73% and 3.75%, occured while estimating the fourth and third natural frequencies of the plate attached with five springs of infinite stiffness along the 60° line, respectively. Fig. 12 plots the mode shapes and the point force response of the composite rectangular panel. The first mode at 95.89 Hz when the springs are aligned along the 15° line is plotted in Fig. 12(a) and the second mode at Fig. 12. Mode shapes and response of the composite rectangular panel. (a) First mode, springs aligned along the 15° line at 95.89 Hz. (b) Second mode, springs aligned along the 30° line at 236.43 Hz. (c) First mode, springs aligned along the 60° line at 95.25 Hz. (d) Point force response of the panel at 150 Hz with springs aligned along the 60° angle. P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 137

31. 236.43 Hz when the springs are aligned along the 30° line is shown in Fig. 12(b). Fig. 12(c) shows the first mode at 95.25 Hz when the springs are aligned along 60° line and the point force response of the panel at 150 Hz with springs aligned along 60° angle is plotted in Fig. 12(d). As explained the first example, the mode bifurcation is observed in Fig. 12. Fig. 13(a) shows the acoustic intensity distribution of the panel constrained with five springs of infinite stiffness along the 60° line at 200 Hz. The acoustic directivity is observed at much lower frequencies in the composite plates, compared to the steel plate. Variation of the sound power with the constraint angle is plotted in Fig. 13(b). 4.3. Example 3: Rectangular composite plate-mass system In this example, we want to study the acoustic directivity of the composite rectangular panel attached with five masses along the 60° line. The layers of the composite panel and their sequence is the same as in example 2. Consider five point masses of 50 g each attached along the 60° line. The natural frequencies of the bare composite plate with simply supported boundary conditions on all the edges are listed in the second column of Table 6. Table 7 compares the first five natural frequencies from the analytical and the numerical models of the composite plate attached with five masses along the 60° line. The point force response contours of the panel at 200 Hz from the analytical and the numerical models are plotted in Figs. 13 and 14(b), respectively. The corresponding hemispherical distribution Fig. 13. (a) Hemispherical distribution of the sound Intensity of the composite plate with five springs of infinite stiffness along the 60° line at 200 Hz. (b) Variation of sound power with the constraint angle; with permission from [18]. Table 7 Comparison of the natural frequencies from the analytical and the numerical models of the composite plate attached with five masses along the 60° line. Mode Analytical (Hz) Numerical (Hz) 1 54.51 54.21 2 118.01 116.20 3 149.52 150.18 4 218.44 218.70 5 245.81 241.28 Fig. 14. Point force response of the composite panel at 200 Hz, attached with five masses attached along the 60° line from (a) the analytical model and (b) the numerical model. The response plot in (a) is generated through the current method and the response plot in (b) is produced through MSC Nastran, for comparison. The pattern and the amplitudes of the response at several locations from both the models are observed to be in agreement. 138 P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140

32. of the sound intensity in two and three dimensions is plotted in Figs. 14 and 15(b), respectively. 5. Conclusions We developed a methodology to analyze a coupled plate-mass/ spring system based on the receptance method. In the first example, we applied the developed methodology for a completely clamped steel plate-mass system set in a baffle, to estimate the new natural frequencies and the structural response of the coupled system. The estimated structural response is validated with com- mercial software. The hemispherical acoustic field is estimated from the structural response based on the Rayleigh integral. Sound powers are compared with the results from the Sysnoise [44] for the uncoupled system. The methodology is further extended for a system of a plate attached with very large (infinite) masses. Infinite masses will act as restraints. When such infinite masses are attached along a line at particular orientation, they form a line constraint. The developed methodology has been validated for line constraints along different orientations. Two particular combinations of the mass(es) location(s), number and frequency of excitation are identified where the sound is significantly directed. In the second example, the methodology is extended for a composite rectangular panel set in a baffle with attached springs. We considered springs of infinite stiffness, so that each spring acts as a constraint. Hemispherical acoustic field has been estimated along with the sound power for different orientations of the line constraints. The acoustic directivity of the composite panel attached with five masses along the 60° line is studied in the third example. The directionality of the sound from the composite panels is observed at significantly lower frequencies compared to the steel panel. A platform is developed to quickly perform the acoustic analysis of coupled rectangular metal and composite panels set in a baffle with attached point masses and/or springs. In future we would like to extend the developed technique 1. To study the acoustic field from a baffled plate attached with an active damper like a Magneto-Rheological fluid. 2. To estimate the sound field of a rotating system such as motor where a drift in the predicted natural frequencies is expected due to the coriolis effect. Acknowledgements The first two authors acknowledge the support received from the Facility for Research in Technical Acoustics (FRITA), Indian Institute of Science, Bangalore, for carrying out majority of the present work. The first two authors gratefully acknowledge the technical interac- tions with Prof. V.R. Sonti, Department of Mechanical Engineering, Indian Institute of Science. PRB acknowledges the financial support from the International Research Staff Exchange Scheme (IRSES), FP7-PEOPLE-2010-IRSES, through the project ‘MultiFrac’. References [1] Thomas JP, Qidwai MA. The design and application of multifunctional structure-battery materials systems. JOM: J Miner, Met Mater Soc (TMS) 2005;57(3):18–24. [2] Williams EG. Numerical evaluation of the radiation from unbaffled finite panels using the FFT. J Acoust Soc Am 1983;74:343–7. [3] Oppenheimer CH, Dubowsky S. A radiation efficiency of unbaffled plates with experimental validation. J Sound Vib 1997;199(3):473–89. [4] Putra A, Thompson DJ. Sound radiation from rectangular baffled and unbaffled plates. Appl Acoust 2010;71:1113–25. [5] Gomperts MC. Sound radiation from baffled, thin, rectangular plates. Acustica 1977;37:93–102. [6] Berry Alain, Guyader Jean-Louis, Nicolas Jean. A general formulation for the sound radiation from rectangular, baffled plates with arbitrary boundary conditions. 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Material tailoring of structures to achieve a minimum radiation condition. J Acoust Soc Am 1992;92(2):841–55. [14] Wodtke HW, Lamancusa JS. Sound power minimization of circular plates through damping layer placement. J Sound Vib 1998;215(5):1145–63. [15] Jog CS. Reducing radiated sound power by minimizing the dynamic compliance. In: International union of theoretical and applied mechanics (IUTAM). Bangalore, India: Indian Institute of Science; 2000. [16] St Pierre RL, Koopmann GH. Minimization of radiated sound power from plates using distributed masses. In: ASME winter annual meeting, paper no. 93-WA/ NCA-11. New Orleans, LA; 1993. Fig. 15. Hemispherical distribution of the sound intensity with five masses attached along the 60° line excited at 200 Hz (a) when projected on to a two dimensional plane and (b) in three dimensions; with permission from [19]. P.R. Budarapu et al. / Applied Acoustics 89 (2015) 128–140 139

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