Acoustics Field and Active Structural Acoustic Control Modeling in ANSYS

This article attempts to examine acoustic finite element analysis coupled with structure, and provides the necessary information to apply ANSYS for a wide class of structural acoustics problem. First part of this article describes the acoustic field and hence the uses of acoustic point sources in ANSYS. Results are explained and compared with that of analytical solution. And second part deals with the active control of structural acoustic problems. Results are presented for global cancellation of a primary monopole's sound field by the use of multiple piezoelectric elements bonded to the surface of the elastic structure to provide control forces. Introduction: As we know, ANSYS can be applied to carry out the acoustic analysis, which includes the generation, propagation, scattering, diffraction, transmission, radiation, attenuation, and dispersion of sound pressure waves in a fluid medium [1, . In the ANSYS/Multiphysics and ANSYS/Mechanical programs, an acoustic analysis usually involves modeling the fluid medium and the surrounding structure. The ANSYS program supports a harmonic response analysis due to harmonic excitation, as well as modal and transient acoustic analyses. The normal procedure includes four major steps in a harmonic acoustic analysis: build the model, apply boundary conditions and loads, obtain the solution and review the results. One of the purposes of this paper is to address the acoustic field modeling during the model creation and the flow-type load application when considering the sound scattering and reflection problems. Active control of structural acoustic has been a great interest to the researchers in the recent years. Piezoelectric substance such as PZT has the ability to control acoustic field by the introduction of electric fields or voltage potentials. Few FEA packages are available where piezoelectric material strongly coupled with fluid medium can be included directly for the purpose of active control. This paper will also show the ability of ANSYS to examine the global reduction of the radiated sound pressure in a harmonically excited enclosed fluid medium, where piezoelectric material will be used as the control force. Acoustic Fluid-Structure Coupling: For the coupled fluid-structure interaction problem, the fluid pressure load acting at the interface is added to the structure equation of motion as follows, ) 1 ( 0 0 0 0 0 0         + =               +             +             L F F V u K K K K V u C V u M pr d T Z Z & & & & & & Where, matrix mass Structural = M matrix damping l Steructura = C matrix stifness Structural = K matrix ty conductivi dielectric = d K matrix coupling ric piezoelect = z K forces) body and forces suface forces, nodal of (vector vector load structural = F vector charge nodal applied vector, load electric = L vector load pressure fluid = pr F The fluid pressure load vector at the interface S is obtained by integrating the pressure over the area of the fluid/structure interface surface. { } { } (2) dS ∫∫ ′ = n P N F pr Where { are the shape functions employed to discretize the displacement components u, v, w (obtained from the structural element), {n} is the unit normal to fluid/structure boundary. Details of finite element formulation of fluid structure coupling along with the piezoelectric analysis can be found in reference [3]. } N ′ Acoustic Field: The theoretical model underlying all mathematical models of the acoustic propagation is the wave equation. The wave equation is derived from the more fundamental equation of state, continuity and motion. The assumptions made in acoustics and fluid-structure analyses are that the fluid behaves as an ideal acoustic medium. This implies that (i) the fluid is isotropic and homogeneous, (ii) thermodynamic processes are adiabatic, (iii) the fluid is inviscid (no viscous damping), and that (iv) acoustic pressure and displacement amplitudes are small relative to the fluid’s ambient state. The acoustic wave equation is given by ) 3 ( 1 2 2 2 2 t p c p ∂ ∂ = ∇ where, c is the acoustic wave speed. c = κ/ρ. ρ is the fluid density, and κ is the fluid bulk modulus. This paper will not dig into the finite element formulation of structural acoustics. Details of finite element formulation of the wave equation can be found in references [2,3]. Apply flow-type loads on the model: There is four load types in acoustic analysis of ANSYS: constraints (displacement, pressure); forces (force, moment, flow loading); surface loads (pressure impedance, fluid-structure interaction flag) and inertia loads (gravity, spinning, etc.). Generally, it is straightforward work to specify the load on the FEM model except for the flow loading when applying an acoustic loading at a node in the acoustic medium. How to interpret the physical concept of the flow-type source is crucial for us to apply it in a correct way. Sound pressure field in 2D acoustic problem: It is found that the flow-type acoustic source in the ANSYS 2-dimensional acoustic analysis behaves as a pulsating cylindrical source when the element behavior is set to be plane strain. Suppose that we have a long cylinder of radius a, which is expanding and contracting uniformly in such a manner that the velocity of the surface of the cylinder is v . The acoustic field close to the source is complicated, but a simple expression can be obtained in the far field approximation. t i Ve ω = The pressure and velocity at large distances from the pulsating cylindrical sound source are [4] ) 4 ( 4 π ct) -i (r k i e r cf V a ρ π p + = ) 5 ( 4 π ct) -i (r k i e cr f V a π ν + = where r is the distance from the source to the observing point. k is the wave number, which equals to ω/c. ω is the circular frequency. c is the sound speed and f is the frequency. ρ is the mass density of acoustic medium. The flowing load provided in the ANSYS is simulated using effective “fluid loads” [2] as ) 6 ( -A F 2 2 ρ ρ Q t u & & − = ∂ ∂ = where A is a representative area associated with the flowing source. u is the outward normal displacement of fluid particle to the surface of a fluid mesh. Q is the volume acceleration. & & From equation (6), it is seen that the flowing load is actually a product of the volume acceleration and medium density. Therefore, we have the volume acceleration expression as for a given cylindrical flow source with radius of a and length of l: ) 7 ( 2 t i Ve al i Q ω ω π = & & So the particle velocity close to the source will be, if the flow source F is given: ) 8 ( 2 t i e al i F v ω ωρ π = For plane strain problem, we may make t be one, Hence the velocity amplitude term is