Acoustic radiation by means of an acoustic dynamic stiffness matrix in spherical coordinates Kauê Werner and Júlio A. Cordioli. Department of Mechanical Engineering Federal University of Santa Catarina Florianópolis Brazil ABSTRACT In general numerical methods used to obtain the acoustic radiation of vibrating structures through a fully coupled analysis display large computational costs especially when the frequency range of interest involves several wavelengths. The Boundary Element Method (BEM) and the Finite Element Method (FEM) are examples of such methods and are therefore limited to the analysis at low frequencies. An alternative approach is given by the Rayleigh Integral and the calculation of an acoustic dynamic stiffness matrix (ASDM). However in its classical form such approach is restricted to planar structures. The purpose of this work is to extend the approach based on the acoustic dynamic stiffness matrix to include spherical geometries so that complex structures that are sphere-like can be analyzed. The surface displacement of the sphere is expressed in terms of pistons centered at nodes of a uniform mesh. Two validation cases are presented in the work including the radiated power of a breathing sphere (monopole) compared with its analytical solution; and the radiated power of a dipole compared with a FEM-BEM coupled model and the analytical solution. An average simulation time comparison between the FEM-BEM and the matrix method is also presented. Keywords: Sphere Vibration Radiated Power Acoustic Dynamic Stiffness Matrix 1. INTRODUCTION Sound radiation from vibrating structures has a great practical importance in engineering applications and research has been conducted since the nineteenth century to understand the mechanisms of the phenomenon. The theoretical modeling and estimation of the sound field from practical structures is generally complex and numerical methods are the usual option when analyzing such systems. In many cases it is also important to have a fluid-structure coupling considering the sound radiated into the fluid and the effect of the fluid loading on the vibrations of the structure. This effect must be considered when the body is immersed in a high density fluid for example in underwater acoustics. A way to solve this coupled analysis is having an acoustic dynamic stiffness matrix (ADSM) [12] that is added to the equation of motion so that (1) where and are vectors with displacement and force complex amplitudes at each degree of freedom (DOF) is a structural dynamic stiffness matrix given by " (2) 1
with being the stiffness matrix the inertia matrix and the damping matrix. The matrix is called the acoustic dynamic stiffness matrix (ADSM) and provides the coupling between all DOF troughs the fluid. The ADSM can be also used to calculate the radiated acoustic field. With the recent advances in computation modeling two numerical methods are widely applied for commercial use the Boundary Element Method (BEM) and the Finite Element Method (FEM). But due to the diversity of geometry properties and construction forms of some engineering systems some vibroacoustic models required large amounts of CPU storage and time when considering short wavelengths (high frequencies). In some cases is not possible to achieve a reliable solution for high frequency analysis. So it is relevant to consider the efficiency of the analysis method its limitations and potential to provide physical insight and suitable practical results. One of the purposes of this work is to find an alternative method to solve the problem of sound radiation from vibrating spheres in a more efficient way when compared with FEM/BEM methods. Many previous studies have presented alternative ways to solve numerically the problem of the radiated acoustic field due to vibrating structures with both planar and spherical geometries being assumed. In the case of planar geometries the Rayleigh integral is generally assumed. In [3] William and Maynard evaluated the radiated pressure via a wave number representation of the Rayleigh Integral so that the FFT algorithm could be applied to the problem. A non-modal method based on elementary radiator was presented by Mollo and Bernhard [4] and later used by Cunefare [5] where a baffled plate is divided in rectangular radiators whose vibrations are described in terms of the velocity at the center of each element. The relationship between pressure and velocity of each element is given by the Rayleigh integral so that one can build an impedance matrix correlating all elements and calculate the total radiated sound power. A similar approach was used by Langley [1] where an acoustic dynamic stiffness matrix (ADSM) was derived for a baffled planar structure represented by a finite number of DOF and using wavelets as shape functions. Regarding spherical geometries Sherman [6] proposes a model to calculate the mutual radiation impedance between two circular pistons located on the surface of a vibrating sphere immersed in an unbounded fluid. The work presented in this article establishes a connection between the approach of the ADSM used by Langley [1] and the model proposed by Sherman [6]. 2. MATHEMATICAL MODELING This section will lead to the derivation of the ADSM in spherical coordinates. Considering a spherical surface the displacement can be written in terms of generalized degrees of freedom located at so that with q being the number of degrees of freedom and being a prescribed shape function for the degree of freedom. A Rayleigh-like integral for a vibrating sphere is presented by William [7] so that the pressure field generated by the motion of the surface is given by where is the fluid density is the sphere radius is the wave number and is the Green s function in spherical coordinates that can be written in the form (3) (4) " " (5) where " is the order spherical Hankel function of first kind and are the spherical harmonics [7]. 2
A generalized force acting on the degree of freedom due to the pressure field generated by the surface harmonic displacement can be written as " where " are the elements of acoustic dynamic stiffness matrix in spherical coordinates given by " " " with " being the entries of the acoustic impedance matrix. The term " represents the force generated on the degree of freedom due to the displacement of the. In order to solve Eq. (7) the shape function was assumed as polar cap piston [5] so that (8) (6) (7) with being the polar cap angle. In this case the spherical surface was divided into a homogenous mesh of circular pistons with the same area so that the sum of all piston areas must be equal to the sum of the total surface area (9) where "#. (10) Substituting Eq. (5) and Eq. (8) into Eq. (7) and following a similar procedure to that used by Sherman [4] it is possible to arrive at an expression for each component of the ADSM given by " "# "# " "# " " (11) where " is the angular distance between the centre of pistons p and q is the sound velocity in the given fluid medium and is the Legendre polynomial of degree. It can be seen that the ADSM depends only on " so that only a few elements of the matrix actually need to be calculated due to the highly symmetric conditions presented by a sphere. The total acoustic radiated power can then be calculated as [1] "# " ".. (12) 3. VALIDATION CASES Two validation cases considering spheres with different surface velocity patterns are presented below. First the radiated power of a breathing sphere or monopole is calculated using the ADSM and compared with the existing analytical solution. The second case is a oscillating sphere or a dipole where its radiated power is calculated also using the ADSM and compared both with the existing analytical solution and a FEM/BEM analysis. The sphere dimensions and some properties can be seen in Table 1. 3
Table 1 Parameters of the problem Sphere radius 0.359m Fluid density 1.21kg/m Sound velocity 343m/s Order of functions 5 All simulations of the ADSM method were carried out using MATLAB. The FEM/BEM simulations and all FEM mesh generations were executed using the software VA-One from ESI Group [10]. To generate the ADSM mesh the FEM mesh nodes were extracted and defined as the piston centers. The angle size was defined using equations (7) and (8). Figure 1 presents the mesh and nodes used in the FEM/BEM analysis and to calculate the ADSM. Figure 1 (a) Spherical mesh from Va-One (b) Points of the spherical mesh used in Matlab. 3.1 Monopole For the monopole case all pistons were set with a constant velocity ( ") for the whole the frequency range. The analytic solution for the acoustic radiated power of a monopole is given by Mechel [6] as "# " (13) The comparison between analytical and the ADSM method results is shown in Figure 2. A very good agreement can be observed between numerical (ADSM approach) and analytical results. The over simplistic velocity field of the monopole may not be the best case to evaluate the proposed approach and the dipole case is considered in the next section. 4
Figure 2 Results for the radiated power of the monopole case. 3.2 Dipole The velocity field for the dipole case was obtained following two different approaches. At first the velocity field was extracted from a previous FEM simulation. On this simulation the sphere was submitted to a nodal point force ( ) with all nodes constrained in all directions except for the one in which the force was applied which resulted in a single rigid-body mode (other higher modes were not considered). That represents a dipole source as an oscillating sphere. A FEM/BEM simulation was also carried out for this case in order to calculate the radiated power. This approach can be later used to compare the results for a more complex velocity field (not included in this paper). In the second approach the velocity field was obtained by projecting a defined velocity vector on the radial direction at each mesh points. The analytic solution for the acoustic radiated power of an oscillating sphere is given by Blackstock [7] as "# (14) The oscillating velocity used was the one taken from the node where the point force was applied in the FEM/BEM simulation. A comparison between analytical FEM/ADSM and FEM/BEM results is shown in Figure 2. 5
Figure 3 Results for the radiated power of the dipole case. It can be seen in Figure 2 that all methods display very similar results for the dipole but the agreement is not as good as that observed for the monopole. In fact the FEM/ADSM and the FEM/BEM results display a small deviation from the analytical solution of around 10% over the whole frequency range. An analysis of the results has shown that this discrepancy is due to the fact that the velocity used in the analytical model was taken from the node where the force was applied which was not exactly located on the z-axis and therefore had a small angular position. Recalculating the results using the ADSM approach but with the velocities obtained analytically (by projecting a constant velocity " in the z direction onto the radial direction at each mesh point) a much better agreement with the analytical results was obtained as can be seen in Figure 3. The results for this case have an average discrepancy of 1% over the frequency range. It may be noted that the shape of curve obtained in Figure 3 for the radiated power is considerably different than the one given in Figure 2 since the velocity is constant over the frequency range of analysis in Figure 3 while in Figure 2 the results were obtained by assuming a constant force which result in a velocity decreasing with frequency. It may also be observed in Figure 2 some larger discrepancies between the FEM/BEM model and the other approaches at around 300 and 900 Hz which are believed to be due to irregular frequencies of the BEM solution. The analysis with the FEM/ADSM and FEM/BEM approaches where also carried on with different mesh resolutions in order to compare the average simulation time for one frequency step of the BEM and the ADSM method. The results are shown in Table 2. It can be seen that the average simulation time (one frequency step) for the BEM method has an exponential behavior with the number of DOF while for the ADSM method the simulation time grows more linearly. It is expected that the ADSM approach can be considerably faster that the BEM solution for large models. It is also important to note that the ADSM calculations where implemented in MatLab and the code was not optimized to reduce its computational cost. Even so the ADSM approach was 4 times faster than the BEM approach for the case with the largest number of DOF. 6
Figure 4 Results for the radiated power of the dipole case. Table 2 Average simulation time for one frequency step Mesh Number of pistons ADSM BEM 1 629 039s 1s 2 800 072s 16s 3 2551 487s 13s 4 4845 126s 498s 4. CONCLUSIONS A numerical procedure for the calculation of the radiated acoustic field of a spherical vibrating structure based on the Acoustic Dynamic Stiffness Matrix (ADSM) has been proposed. The ADSM was derived using a similar procedure used for planar structures that included the representation of the sphere velocity field using circular caps and a Rayleigh-like integral in spherical coordinates. The approach was validated through two cases: a monopole and a dipole and very good agreement between the ADSM method a FEM/BEM model and analytical results were obtained. The comparison of simulation times for different mesh resolutions has shown that the ADSM method can be much faster than conventional BEM. This work is still underway and the next steps include the validation of the radiated power for a more complex velocity field and considering the fluid loading effects. The final goal of the project is to apply the method to a real sphere-like structure (for example a refrigerator hermetic compressor) and compare the results with those obtained using conventional numerical methods. 7
ACKNOWLEDGEMENTS The authors would like to acknowledge the financial support from CNPQ (Brazilian research funding agency) through a Master Degree scholarship. REFERENCES [1] Langley R. S. Numerical evaluation of the acoustic radiation from planar structures with general baffle conditions using wavelets. J. Acoustical Society of America v. 121 n. 2 p. 766-777 2007. [2] Langley R. S. and Cordioli J. A. Hybrid deterministic-statistical analysis of vibro-acoustic systems with domain couplings on statistical components Journal of Sound and Vibration v. 321(3) p. 893 912 2009. [3] Williams E. G. and Maynard J. D. Numerical evaluation of the Rayleigh integral for planar radiators using the FFT The Journal of the Acoustical Society of America vol. 72 pp. 2020 1982. [4] Mollo C. G.; Bernhard R. Generalized method of predicting optimal performance of active controllers. AIAA v. 27 p. 1473 1989. [5] Cunefare K. A. Global optimum active noise control: surface and farfield effects. J. Acoustical Society of America v. 90 p. 365 1991. [6] Sherman C. H. Mutual Radiation impedance between pistons o spheres and cylinders. U.S. Navy Report 1958. [7] Williams E. G. Fourier Acoustics Sound Radiation and Nearfield Acoustical Holography. Academic Press 1999. [8] Mechel F. P. Formulas of Acoustics. Springer 2001. [9] Blackstock D. T. Fundamentals of Physical Acoustics. Wiley-Interscience 2000. [10] VA One 2012 User s Guide ESI Group Paris France 2012. 8