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This article uses numerical methods to gain insight into the structural acoustic response of lightweight square aluminum panels. It takes into account the different positions of the main sound sources in the acoustic environment and how these positions affect the response of the structural panels. The numerical model is constructed by the finite element method and the first order deformation theory. Experimental measurements of the vibrating plate vibration shape and velocity frequency response were used to verify the results of the finite element model. In addition, the values of indicators of vibroacoustic emission, such as losses during sound transmission, sound pressure level and directionality of sound pressure in the far zone, were obtained. The results show that different positions of the main sound source significantly affect the noise reduction of the structural panel. Sound sources, usually located close to the structural plates, reduce the effectiveness of the vibrating plate to reduce noise. In addition, according to different positions of the sound source, the sound propagation curve of the emitting end of the vibrating plate is examined. The study shows that the obtained variations of the quiet zone, vibroacoustic emission parameters and sound propagation profiles can provide important information for optimal localization of structural sources for active control of vibration and noise.
Lightweight structural panels such as shells and sheets are becoming increasingly attractive for noise abatement/control, especially in the automotive, aerospace, marine and construction industries. Recently, researchers have found that these structural panels are particularly suitable for active noise and vibration control. In addition to their relatively good vibroacoustic response for low frequency noise suppression, their reduced weight allows them to be easily incorporated into windows and home appliances such as washing machines, refrigerators and many types of food processing equipment. In practice, these structural panels experience some unexpected operating conditions due to the location of the main sources of noise. As a result, the structural and acoustic performance of these panels for effective noise reduction is compromised. It is therefore very important to study the various locations where noise is generated and how these locations of noise sources affect the response of lightweight panels to noise control and/or reduction. A deep understanding of this study will provide researchers with important insights into the proper placement of secondary sound sources for noise reduction during active vibration and noise control.
Auxiliary sources for active structural and acoustic control, such as structural actuators or electrodynamic vibrators, as well as sensors, microphones or other noise suppression devices, were placed on or around the vibrating plate to suppress the noise generated by the main sound source. In recent years, various optimization methods have been carried out to find more efficient ways to place actuators and sensors on vibrating structures1,2. In addition, the control source 3 and the microphone 4 in the sound field, along with the main source, were used to provide information about the noise control results. However, a number of factors complicate the task of correctly placing these secondary sources on or around structural panels. The distance between secondary sources, the number of secondary sources, the distance from the secondary source to the vibrating plate, etc. are some of the factors that affect the noise reduction effect. These factors become more noticeable when dealing with low-frequency noise generated by outdoor vehicles and noise generated by industrial equipment.
For vibroacoustic problems in the low frequency region, lightweight structural panels (i.e., thin-walled structures) tend to give low transmission losses due to mass law effects. To meet this challenge, various lightweight structural panels made from composites6, smart materials7, functionally graded materials8 and reactive metallic materials9 have been developed and successfully used for active noise and vibration damping. The authors also demonstrate that vibroacoustic emission indicators, including sound transmission loss, emitted sound efficiency, sound power, far-field sound pressure, etc., are active indicators that are effective indicators for reliably displaying the response of structural panels to acoustic waves. . These panels allow incoming sound waves to pass through, pass those waves through their thickness, and then radiate them through the radiating surfaces. Several parameters have also been identified that affect these emission indices. For example, geometric parameters such as the thickness of the shell 11, the finite or infinite dimensions of the panels 12, and the boundary conditions (BC) of the vibrating structures 13 have been carefully studied to understand their effect on their vibroacoustic emissions. Other parameters related to material properties (eg stiffness, Mach number, etc.)14,15 and structural properties16 have also been investigated.
Several analytical solutions have been developed for solving vibroacoustic problems in noise barrier bushings17, as well as problems related to both limited and unlimited areas of building acoustics. However, numerical approximations such as the finite element method (FEM)18 have proven to be more effective tools for modeling structural acoustics problems. This method can be used in combination with other methods such as the boundary element method19 to provide a more robust solution, especially for mated structural-acoustic interfaces. However, this unified approach still requires significant computational resources. To this end, efforts are currently underway to address the limitations of computational efficiency. Cui et al. 20 combined edge smoothing FEM and gradient weighted FEM for more robust solutions to structural acoustic problems. In addition, Song and Wolf21 have developed FEM with scale limits implemented by Lehman et al. 22 and more recently by Lee et al. 23 to solve coupled bounded and unbounded vibroacoustic problems.
In most studies and analyzes of vibroacoustic problems of the above authors, noise sources located at fixed points of the structural-acoustic region are used to generate sound waves. However, little attention has been paid to where the generated primary sound waves come from. In this study, the authors sought to find an answer to how different positions of noise-producing loudspeakers located in a limited rigid enclosure affect the vibroacoustic response of a vibrating light square aluminum plate. One of the tasks of active control of vibration and noise is to obtain the optimal position of the secondary sound source in relation to the distribution of primary sound waves and the frequency spectrum. A good understanding of the various locations where sound waves are emitted will aid in the effective placement of actuators, sensors, microphones, or other active noise canceling devices on or around a vibrating plate, as described in this study. The study first uses finite element methods and shear deformation theory to consider the structural and acoustic responses of bounded-domain problems. Numerical results are obtained for various indicators of vibroacoustic emission, such as acoustic transmission loss \((STL)\), sound pressure distribution and sound pressure directivity in the far zone, as well as the influence of various positions of the sound source in the medium-low frequency range. range (i.e. 0-1000 Hz) to analyze these indicators. Finally, this study investigates the effect of source location on the far field directivity modes of a vibrating plate with fully clamped and freely maintained boundary conditions.
The model for this study consists of an isotropic thin-walled aluminum plate whose thickness (\(h\)) is much less than its length (\({a}_{x}\)) and width (\( {a} _ {y}\ ))24, size \({a}_{x}=420 \; \text{mm}\), \({a}_{y}=420 \; \text {mm} \ ) and \(h =1 \; \text{mm}\) as shown in Figure 1. The rationale for choosing these dimensions is to advance the research done by the authors4, especially the further study of active noise and vibration control. The fully clamped structural panel is placed in an infinite-dimensional spatial enclosure bounded by the acoustic region. The bounded region of the near zone consists of the structural region and the air region, defined by the Cartesian coordinates \((x,y,z)\), and the infinite-dimensional closed space is defined by the spherical coordinates in the far zone. field area. The sound source from the speaker is used to excite an aluminum plate, which in turn emits sound waves. For the study, the speakers were placed in nine different locations inside a rigid chassis, which was also fitted with lightweight square aluminum panels. For convenience, nine different sound source locations are named: Near Center (NCC), Mid Center (MCC), Far Center (FCC), Near Lower Center (NBC), Mid Lower Center (MBC), Far Lower Center (FBC). , near bottom edge (NBE), middle bottom edge (MBE), and far bottom edge (FBE). On fig. 2 schematically shows various designated locations where the loudspeaker emits sound waves. Distances along the \(x\), \(y\) and \(z\) axes from one given position point to another are expressed as \({l}_{x}\), \({l } _ {y} \) and \({l}_{z}\) respectively. The rigid shell is considered to be completely absorbing, so the vibroacoustic results are the same for any size of the rigid shell volume.
For aluminum square plates with dimensions \({a}_{x}\), \({a}_{y}\) and \(h\), as shown in Figure 1, on the bearing surface, the first order shear deformation theory used to get the offset as follows
where symbols \(\mathcal{U}\), \(\mathcal{V}\) and \(\mathcal{W}\) are along \(x\), \(y\) and \(z\ ) The direction is in the temporary variable \(\left(t\right)\) respectively. In addition, \(\mathcalligra{u}\), \(\mathcalligra{v}\) and \(\mathcalligra{w}\) are in \(x\), \(y\) and \(z\ ) direction. In addition, the symbols \({\alpha}_{x}=\partial \mathcalligra{w}/\partial x(x,y,t)\) and \({\alpha}_{y}=\partial \ mathcalligra {w}/\partial y(x,y,t)\), rotation of the transverse normal about the axes \(y\) and \(x\) respectively. , normal deformations (i.e. \({\varepsilon}_{x}\) and \({\varepsilon}_{y}\)) and shear deformations along the midplane (i.e. \({\gamma }_{xy}\}) can be expressed as
where \({\varepsilon}_{x}^{0}=\partial \mathcalligra{u}/\partial x\) and \({\varepsilon}_{y}^{0}=\partial \mathcalligra{ v}/\partial y\) represents a normal deformation, and \({\gamma}_{xy}^{0}=\partial \mathcalligra{u}/\partial y+\partial \mathcalligra{v}/\partial x \) – shear deformation on the supporting surface. In addition, the conditions \({\psi}_{x}\), \({\psi}_{y}\) and \({\psi}_{xy}\) represent changes in midsurface curvature. Due to the thinness of the aluminum plates, the transverse shear strain components \({\gamma}_{xz}\) and \({\gamma}_{yz}\) are assumed to be zero. The resulting stress \(\sigma (x,y) \) includes force (\(Q\)) and moment (\(M\)) resultant forces, which can be expressed in component form as
Where \({A}^{k}\), \({D}^{k}\) and \({F}^{k}\) are the stiffness coefficients of a thin-walled plate, respectively expressed as
The values of the material parameters \(E\) and \(\mu\) are given in Table 1. In addition, the total energy functional (\({\mathbf{E}}_{T}\)) can be calculated using the strain (\ ({{\varvec{S}}}_{\varvec{S}} }_{\varveclon}\}) and motion (\({{\varvec{T}}}_{E}\)) expressed like 25
where \(\dot{\mathcalligra{u}}\), \(\dot{\mathcalligra{v}}\) and \(\dot{\mathcalligra{w}}\) represent the speed on the mean surface and the symbols \ ({I}_{0}\), \({I}_{1}\) and \({I}_{2}\) respectively indicate that the moment of inertia is defined as
Applying the principle of virtual displacement, one can obtain the equation for controlling the movement of a square panel, which is expressed as
where the matrices \(\left[{{\varvec{M}}}_{{\varvec{p}}}\right]\) and \(\left[{{\varvec{K}}}_{{ \ varvec{p}}}\right]\) represent the mass and stiffness of the vibrating plate, respectively. Also, the vector \(\left\{\mathcalligra{u}\right\}\) represents the displacement field, and the symbol \(\omega \) represents the natural frequency. Substitute the function of time into the equation. (10a), the equation of motion takes the form
Apply the Rayleigh damping equation\({C}_{p}={a}_{M}\left[{M}_{p}\right]+{b}_{K}\left[{K}_ { p}\right]\) to the equation. (10b), where \({a}_{M}\) and \({b}_{k}\) are the damping coefficients of the mass and stiffness matrix, respectively. The values \({a}_{M}\) and \({b}_{K}\) used in this article are 3.8500347 and 0.0013203, respectively, calculated from the first natural frequency. plate in the time domain takes the form
In the midplane of a square thin-walled panel, you can use the node set \({x}_{k}\left\{k=1,\dots ,\zeta \right\}\) to get the final element part. The offset expression is as follows
where \({\mathbb{N}}_{k}\) represents the standard displacement shape function. Since the shear component of the lateral displacement of the square panel is ignored, the bending component in its region (\(\mathcalligra{w}\)) is
where \(\Delta\mathcalligra{w}\) is the vector of bending degrees of freedom. The stiffness matrix \({\varvec{K}}\) consists of two components \({{\varvec{K}}}^{e}\) and \({{\varvec{K}}}^{b} \) is the tensile and bending stiffness respectively. This is given as
where the notation \({{\varvec{B}}}^{e}\) is the tensile strain matrix and the notation \({{\varvec{B}}}^{b}\) is the flexural strain matrix. These terms are defined as
Using numerical procedures, the matrix terms \({\varvec{A}}\), \({\varvec{B}}\), \({\varvec{D}}\) and \(\mathcalligra { m}\ ) is easily solved, and the terms \({{\varvec{K}}}^{e}\), \({{\varvec{K}}}^{b}\) and \ ( { \varvec{M} }\) can be approximated using the Gauss integral method.
The acoustic fluid in the rigid shell of the cube is air. The discreteness of a closed air region is achieved through the use of regular hexagonal elements. The nodal pressure \({p}^{e}\) acting on each node of the element can be written as
Has a knot shape function, \({\mathbb{N}}_{f}=\left[{\mathbb{N}}_{1} {\mathbb{N}}_{2}\dots {\mathbb { N}}_{8}\right]\). Given the boundary of the main air region (\ ({\Omega} _{f}) \), the main mass matrix in a closed air region is defined as
where the symbol \({\rho }_{0}\) denotes the density of air, \({c}_{0}\) is the speed of sound in air, and \({J}_{f} \ ) is the Jacobi determinant .In addition, in a closed air domain, the main stiffness matrix can be defined as
The closed region of connection between the structure and air is the interface between the outer shell of the rigid shell and the inner wall of the rigid shell. An interface element with domain \({\Omega }_{sf}\) is limited to four nodes, each with two degrees of freedom. The displacement of a discrete element associated with a cubic shape function can be expressed as
where \({\mathbb{N}}_{ui}\) is the cubic function of the element’s normal displacement. The interface sound pressure acting on four element nodes is written as
where the notation \({\mathbb{N}}_{fi}\) denotes the linear shape function obtained at each element node. The fundamental connection matrix (\({\mathcal{K}}_{cf}^{e}\ }} bounded by \({\Omega}_{sf}\) is related to cubic and linear shape functions, which can now take the form
As already mentioned, the sound source and light aluminum panels are housed in a rigid cabinet, all of which are installed indoors. Sound waves from the speakers impinge on the surface of the panel, the waves are reflected, the sound waves pass through the thickness, and then the sound waves are emitted from the radiating surface of the panel. In this linear acoustic region, the equation governing the propagation of the wave according to the velocity potential (\(\phi \) ) is expressed as
where \(\phi \) is the main variable used in the acoustic formula. Also the gradient operator \(\nabla\) in the equation. (28) is given in the equation. (twenty one).Dynamic pressure
The relationship between the velocity potential and the velocity of an acoustic fluid particle is \({\varvec{\upnu}}=\nabla \phi\), and the Neumann velocity BC is also \({n}_{ 0}^ {T}\nabla \ phi = {{\varvec{\upnu))}_{n}\, where \({n}_{0}\) and \({{\varvec{\upnu }}} _{n}\) are outgoing the unit normal vector and the outgoing normal velocity of the sound region, respectively Excitation of the time harmonic by combining the equations (28) As a function of time (\({e}^{j\omega t}\)) the Helmholtz wave equation takes the form
where \({P}^{i}\) is the incident pressure wave, \({P}^{r}\) is the reflected pressure wave, and \({P}^{t}\) is the passage of the pressure wave. Note the corresponding characters \({A}_{0}^{i}\), \({A}_{0}^{r}\) and \({A}_{0}^{t} \ ) of these pressure waves, as described in the equation. (31) is the amplitude of each pressure wave. In addition, \({\eta}_{ix}\), \({\eta}_{iy}\) and \({\eta}_{iz}\) represent \(x\), \( axes y\) and \(z\).
In this section, acoustic far-field boundaries are discretized using finite elements. Consider the far field point \({p}_{0}\) on the discrete surface ABCD with nodes \(n\), as shown in Figure 3. The dimensionless axis \((\xi )\ ), namely the line segment \( O{p}_{0}\), provided that the Cartesian coordinates (\(x,y,z)\ at the origin\(O\) ) are connected to the point \({ p}_{0}\) on a surface with far-field coordinates (\(\xi ,\vartheta ,\varphi )\), arbitrary coordinates (\(\widehat{x} ,\widehat {y}, \widehat{z})\) and node coordinates (\({\varvec{x}},{\varvec{y}},{\varvec{z}})\).For \(n\) nodal discrete surface ABCD, the nodal coordinates of the point \({p}_{ 0}\) can be defined by a function of the form \({\mathbb{N}}_{a} \ left(\vartheta , \varphi \right)\) is described by the relation in far-field coordinates
Discrete model of an acoustic spherical limited area in the far zone with points in the far zone \({p}_{0}\).
where \({\mathbb{N}}_{a}=\left[{\mathbb{N}}_{1}, {\mathbb{N}}_{2},\dots , {\mathbb{N }}_{n}\right]\) is the shape function in the far field coordinates \(\vartheta\) and \(\varphi \) of the discrete surface ABCD. In the far field subdomain in Figure 3, the sound pressure amplitude can be discretized as
where \({P}_{a}\) is the pressure in the direction of the dimensionless axis. Similarly, the nodal velocity potentials in the far field subdomain of a finite element of the surface are obtained using the form function \({\mathbb{N}}_{a}\) in the far field subdomain. The ordinates \(\vartheta\) and \(\varphi \) given as
where \({\phi}_{a}\) is the potential of the velocities of all points in the direction \(\xi \). At the far point \({p}_{0}\) the gradient operator of the Helmholtz equation \(\nabla \) is equation (28), which can be rewritten as 26)
where the vectors \({{\varvec{b}}}_{a}^{1}\), \({{\varvec{b}}}_{a}^{2}\) and \({ { \varvec{b}}}_{a}^{3}\) is obtained from the coordinates of any point (\(\widehat{x},\widehat{y},\widehat{z})\)\ ( {p} _{0}\) on the surface ABCD and given in component form
where the term \({J}_{a}\) is obtained by computing the Jacobian at the point \({p}_{0}\). The surface domain coefficient matrix (\({\Omega}_{s})\), in far-field coordinates \(\vartheta\) and \(\varphi\), the constant \(\xi\) is given as
Their shape function relation is expressed as \({{\varvec{B}}}_{a}^{1}={{\varvec{b}}}_{a}^{1}{\mathbb{N} } _{a}\) and \({{\varvec{B}}}_{a}^{2}={{\varvec{b}}}_{a}^{2}{\mathbb{N } }_{a,\vartheta}+{{\varvec{b}}}_{a}^{3}{\mathbb{N}}_{a,\varphi}\).
In this section, one of the objectives is to obtain some emission parameters such as the sound pressure of the acoustic environment, the power of the emitted sound, and the loss in sound transmission. The effect of dynamic position on some of these emission indices is discussed in the Numerical Results section.
On fig. 4 shows a spherical closed sound region, the origin of which \(o\) of the Cartesian coordinate system \((x\),\(y,z)\) is transformed into the spherical coordinate system \(({R}_{0}\ ), \(\theta,\beta )\) along the far-field distance \({R}_{0}\).Sound pressure amplitude in the far-field of the receiving point\(({R}_{0}\),\( \theta ,\beta )\) Spherical coordinates can be expressed as
where \({{\varvec{\upnu))_{n}\) is due to the point \(o\) and the sign of the sound number \( \eta =\omega /{c}_{0}\ ). Let \ ({\lambda}_{a}=\eta cos\theta sin\beta\) and \({\lambda}_{b}=\eta sin\theta sin\beta \) lie along \(x \ ) and \(y\), then the speed characteristic of the vibrating structural plate is equal to
Given a certain point (\({r}_{0})\) located on the panel of the construction panel, up to the receiving point (\({r}_{f})\), the transmitted acoustic power is represented by the excitation result panels for 28
The symbol \(Re\) and the superscript * y \({\varvec{\upnu}}\) denote the real part and the complex conjugate of the sound intensity, respectively. It’s worth noting that speed is in the equation. (40) is obtained from equation (39) can also be obtained from the nodal displacements of the finite element solution described in the previous section. Let the reference power be equal to \({\Psi }_{ref}\), then the sound pressure level can be easily expressed how
The relationship between radiated sound power and far-field sound pressure along the far-field distance \({R}_{0}\) is approximately equal to
A very important acoustic emission parameter, often used to determine the ability of a vibrating plate to reduce noise, is the sound transmission loss. It is calculated as the ratio of the incident sound power to the transmitted sound power. To evaluate this parameter, the structural panel gain \((\tau )\) is first determined by dividing the transmitted power\(({\Psi }_{t})\) by the incident power\(\left ({\Psi } _{i}\right)\) is given by
with the transmission loss factor specified in the formula. (43) it is convenient to express the loss during sound transmission by a vibrating structural plate as
Experimental measurements were carried out and used to test the finite element solution. Both measurements and approximate solutions use the same parameters and fully clamped boundary conditions. This experiment is performed by the same authors and fully describes setup 8. In addition, finite element approximations are performed using ANSYS modeling tools. On fig. 5 shows the first 12 modal shapes of a square aluminum panel excited by acoustic sound from loudspeakers placed in a rigid enclosure. In the results of the experiment, it was observed that patterns numbered six (3.1), seven (2.3), ten (4.1) and twelve (2.4) were indistinguishable because they were located at the location of the FBC source. In the problem of vibroacoustics, the sound wave emitted by a square aluminum plate is proportional to the speed of vibrations of the square plate. On fig. 6 compares the speed response between numerical approximations and experimental measurements. The curves are plotted on a logarithmic scale and show similar trends to each other. However, a somewhat higher initial amplitude of the experimental results may be due to uneven excitation of the loudspeaker. The numerical results show a more responsive behavior of the square panels than the experimental results. For example, numerically fitted frequency response curves show the probability of excitation of an aluminum plate at the same natural frequency but with different amplitudes, as shown in Table 2.
The first twelve modal forms compare finite element approximation and measurements of an aluminum panel excited by a loudspeaker located at the location of the FBC source.
Post time: Sep-06-2022