% Template for submitting a manuscript to the Journal of Engineering of Catholic University of Petropolis. % version 2.0 dated 28 August 2012. % This file was adapted from cilamce 2011. Modifications may be freely made, % provided the edited file is saved under a different name %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% %% Important note on usage %% %% ----------------------- %% %% This file must be compiled with PDFLaTeX %% %% Using standard LaTeX will not work! %% %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[oneside,a4paper,english,links]{EngUCP} % \usepackage{graphicx} \usepackage{amsmath,amsfonts} \usepackage{epstopdf} \usepackage{subfigure} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\n{\noindent} \def\ba{\begin{eqnarray}} \def\ea{\end{eqnarray}} \def\bfi{\begin{figure}} \def\ef{\end{figure}} \def\sen{\rm sen} \def\nob{\nobreak} \def\n{\noindent} \def\D{\displaystyle} \def\bma{\begin{\bmatrix}} \def\ema{\end{\bmatrix}} \def\bvma{\begin{\vmatrix}} \def\evma{\end{\vmatrix}} \title{MAPPING OF RADIANT REGIONS IN VIBRATING RECTANGULAR PLATES WITH DIFFERENT BOUNDARY CONDITIONS THROUGH THE SUPERSONIC AND USEFUL INTENSITIES } \author[1]{Vera Lúcia D. Ferreira} \author[2]{Roberto A. Tenenbaum} \author[2]{Filipe O. Taminato} % \affil[1]{Universidade Federal do Pampa, 96413-170 - Bagé, RS, Brasil} % \affil[2]{Universidade do Estado do Rio de Janeiro, Instituto Polit\'{e}cnico, 28630-050 - Nova Friburgo, RJ, Brasil } %\affil[3]{Department of Civil Engineering, Pontifical Catholic %University of Rio de Janeiro, Rio de Janeiro, 25651-075, RJ, %Brazil } %% NOTE: IF ALL AUTHORS BELONG TO THE SAME AFFILIATION %% USE THE `\voidaffil' MACRO FOR THE AFFILIATION CODE. %% Example: %% \author[\voidaffil]{First A. Author} %% \author[\voidaffil]{Second B. Author} %% \author[\voidaffil]{Third C. Author} %% \author[\voidaffil]{Fourth D. Author} %% % %% \affil[\voidaffil]{CEC, Department of Engineering and Computation, %%Petrópolis Catholic University, Petrópolis, 25.685-070, RJ, Brazil} \begin{document} \vspace{3cm} \maketitle %E-mail Addresses: Put here! \let\oldthefootnote\thefootnote \renewcommand{\thefootnote}{\fnsymbol{footnote}} \footnotetext[1]{ E-mail addresses: \url{ veraferreira@unipampa.edu.br}, \url{ratenenbaum@gmail.com}, \url{filipeot@gmail.com} .} \let\thefootnote\oldthefootnote \begin{keywords} Supersonic intensity, Useful intensity, Sound radiation, Sound source identification. \end{keywords} \begin{abstract} This work investigates the acoustical intensity generated by the sound radiated from rectangular plates in four cases with distinct combinations of classical boundary conditions. The objective is to identify the regions of these plates that effectively contribute to the radiated sound power into the far-field, especially when the driven frequency occurs below the critical coincidence frequency. This identification is done by filtering the non-propagating waves, both using the (analytical) supersonic intensity method and the (numerical) useful intensity model. Brief theoretical formulations, both for the plates vibration and the resulting acoustical field, are discussed. The closed form solution of the normal velocity field for the four cases is given. Then, the supersonic intensity is estimated. In the numerical examples, the comparison of the supersonic intensity, the useful intensity and the classical acoustic intensity is shown. \end{abstract} \section{INTRODUCTION} Rectangular plates are structural elements of great importance, being present in several places and in many engineering applications, such as on mechanical and aeronautic ones. In this way, the discussion and analysis of the sound radiation due to plates vibration has received the special attention of researchers in the last decades, such as (Williams, 1995), (Magalhães and Tenenbaum, 2006), (Fernandez-Grande et al., 2012) and (Corrêa Jr. and Tenenbaum, 2013). A relevant concept was introduced by Williams~(1995), named by the author as \emph{supersonic intensity} (SI), to identify the regions on the source surface that radiates efficiently and contributes to the sound power. The calculus of the SI is originally based on the spatial Fourier transform, being formulated in the wavenumber space, where the sound field is processed. Its fundamental principle relies on the filtering of the non-propagating waves --- the evanescent ones (Williams, 1998). As a consequence, only the propagating components remains. Williams also concerns in distinguishing the SI from the classical acoustic intensity (AI), showing that in the near field there is energy recirculation and the AI is not as reliable as the SI to identify the actual sound sources. However, his work is focused in rectangular plates simply supported in the four edges. He shows that it is possible to identify the corner, border and surface modes, in conformity with the results found in the literature~(Maidanik, 1974). Fernandez-Grande et al.~(2012) examined the concept of supersonic intensity and reformulate it as a filtering operation directly in the space domain. The method allows the identification of the efficient radiation regions of a sound source and evaluates the actual contribution to the radiated sound power. A numerical example of a plate clamped in all edges is presented as well as an experimental study to illustrate the method and to examine its advantages and limitations. The method has the convenience of not requiring to move to the wavenumber space, but it is still restricted to separable geometries. Corrêa Jr. and Tenenbaum~(2013) present an innovative technique for the computation of a numerical equivalent to the SI, for sound sources with arbitrary geometry, which was named by the authors as \emph{useful intensity} (UI). As the SI, the technique filters the non-propagating components, extracting the propagating ones. The method is entirely formulated in the vibrating surface. The sound power is obtained through a matricial operator that is related to the normal velocity field, by using the boundary element method (BEM). This operator is Hermitian, guaranteeing that its eigenvalues are real and the associated eigenvectors form a basis for the velocity distribution. The use of a stop criteria allows the removing of the non-propagating components. The great advantage of the method is to be applied to non-separable geometries. Although there is a substantial number of works in the literature dealing with rectangular plates radiation (Hashemi, 2009; Liu, 2013; Yoo, 2010), the analytical determination of the \emph{hot spots} of such plates with other boundary conditions was not examined yet. In this work it is computed the supersonic intensity for rectangular plates with four distinct boundary conditions. Finally, the SI is compared with the classical acoustic and useful intensities. The four considered cases in this study are \mbox{S-S-C-S}, \mbox{C-C-C-C}, \mbox{S-C-C-C}, and \mbox{S-S-C-F}. The letters indicate the boundary conditions at the edges $x=0$, $x=a$, $y=0$ and $y=b$, as seen in Fig.~\ref{placasCC} and refers to {\em free} (F), {\em clamped} (C) and {\em simply supported} (S), as usual. \clearpage \begin{figure}[!htb] \begin{center} \includegraphics[width=10cm]{figures/placas1.eps} \caption{Four combinations of classical boundary conditions to be considered.} \label{placasCC} \end{center} \end{figure} \section{MATHEMATICAL BACKGROUND} \subsection{Equations of motion for a thin plate vibration}\label{2.1} Let us consider a rectangular plate in plane $xy$, with length $a$, width $b$ and thickness $h$. The classical Kirchhoff theory for thin plates will be considered. In the frequency domain, the differential equation that governs the transverse displacements $w(x,y)$, independently of the imposed boundary conditions, is \be \label{equalaplaci} \nabla^{4}w(x,y)+ k_b^{4}w(x,y)=0,\qquad {\rm with}\qquad k_b^4=\frac{\rho_s h \omega^2}{D}, \ee \n where $k_b$ is the flexural wavenumber, $\omega$ is the circular frequency, and $D$ is the flexural stiffness of the plate, given by \be\label{rigidez} D = \frac{Eh^{3}}{12\rho_{s}(1-\nu^{2})}, \ee \n where $E$ is the Young modulus, $\rho_s$ is the plate density and $\nu$ is the Poisson coefficient. The normal velocity distribution $\hat{v}(x,y)$, in the frequency domain, is \be \label{velomodal} \hat{v}(x,y)=i\omega w_{mn}(x,y), \ee \n where $i$ is the imaginary unit, $w_{mn}(x,y)$ is the modal shape function associated with $(m,n)$ mode, $m$ and $n$ being the mode counters, and $\omega_{mn}$ is the natural frequency for a given $(m,n)$ mode. The function $w_{mn}(x,y)$ can be written as a product of beam functions, in the form, see (Leissa, 1969) \be \label{solmodal} w_{mn}(x,y)=\psi_m(x)\phi_n(y), \ee \n so that $\phi_m(x)$ and $\psi_n(y)$ are chosen as fundamental modal shapes for beams with the same boundary conditions of the corresponding plate. The functions \be \label{eqmov1} \psi_{m}(x)=C_1 \sin(\beta_{1}x)+C_2\cos(\beta_{1} x)+C_3\sinh(\beta_{2} x)+C_4\cosh(\beta_{2} x); \ee \be \label{eqmov1} \phi_{n}(y)=B_1 \sin(\mu_{1}y)+B_2\cos(\mu_{2} y)+B_3\sinh(\mu_{2} y)+B_4\cosh(\mu_{2} y), \ee \n with a convenient choice of constants $C_i$ and $B_i$,\ $i=1,\ldots,4$, that must satisfy the boundary conditions listed in Table~\ref{tabela1}, and $\beta_{1,2}$ and $\mu_{1,2}$ are, respectively, eigenvalues in the coordinates directions $x$ and $y$. \begin{center} \begin{table}[htb!] \caption{Boundary conditions for a beam.} \label{tabela1} \begin{center} \begin{tabular}{ll| c} \hline Notation&Type of support & Boundary conditions\\ \hline a)~S&Supported & $w=\displaystyle{\frac{\partial^{2} w}{\partial x^{2}}=0}$\\ \hline b)~C&Clamped & $w$=$\displaystyle{\frac{\partial w}{\partial x}=0}$\\ \hline c)~F&Free & $\displaystyle{\frac{\partial^{2} w}{\partial x^{2}}}$=$\displaystyle{\frac{\partial^{3} w}{\partial x^{3}}=0}$\\ \hline \end{tabular} \end{center} \end{table} \end{center} It is worth noting that a closed form of the normal mode $w_{mn}$ is obtained, for each case, by using the symplectic dual method, shown in Xing and Liu~(2009). Those authors use the variable separation method to solve the Hamiltonian dual form of the eigenvalue problem in thin vibrating plates~(Eq.~(\ref{equalaplaci})). For the other two opposite edges $x=0$ e $x=a$, a characteristic equation and a set of eigenfunctions $\psi_m(x)$ can be obtained similarly. The natural mode of vibration, given in Eq.~(\ref{solmodal}) is obtained from the corresponding functions $\psi_m(x)$ and $\phi_n(y)$. \subsection{Analytical formulation of the supersonic acoustic intensity}\label{2.3} The mathematical formulation of the radiation problem is widely discussed in Corrêa Jr. and Tenenbaum~(2013). However, for the sake of completeness and better understanding, the mathematical development is very briefly outlined here. The supersonic acoustic intensity is a tool obtained using the inverse spatial Fourier transform to eliminate non-propagating (subsonic) waves, leaving the far-field radiating (supersonic) components. The supersonic acoustic pressure, $\hat{p}^{(s)}$, and the normal supersonic velocity, ${\hat{v}}^{(s)}$, both in the frequency domain, that are associated with the corresponding pressure $p$ and normal velocity $v$ inside the radiation circle $C_{r}$, are written as \begin{eqnarray} \hat{p}^{(s)}(x, y, 0, \omega) = \frac{1}{4\pi^{2}}\D\int\D\int_{C_r}\tilde{p}(k_x, k_y, 0, \omega)e^{i(k_xx+k_yy)}\;dk_x\;dk_y, \end{eqnarray} \begin{equation} \hat{v}^{(s)}(x, y, 0, \omega)=\frac{1}{4\pi^2}\D\int\D\int_{C_r}\tilde{v}(k_x, k_y, 0, \omega)e^{i(k_xx+k_yy)}\;dk_x\;dk_y, \end{equation} \n where $\tilde{p}$ and $\tilde{v}$ are, respectively, the pressure and normal component of velocity in the wavenumber domain, $k_{x}$ and $k_{y}$ are the wavenumbers in the plane directions, $\omega$ is the angular frequency and $i$ is the imaginary unit, as usual, see Williams~(1998). The flow of acoustic energy that is radiated effectively into the far-field, i.e., the supersonic (acoustic) intensity (SI), is then defined as \begin{equation} \hat{I}^{(s)} = \frac{1}{2}\Re[\hat{p}^{(s)}(x,y,0,\omega){\hat{v}}^{(s)}(x,y,0,\omega)^\textbf*], \end{equation} \noindent where the subscript ``*'' denotes the complex conjugate and $\Re$ stands for real part. Note that the supersonic intensity is essentially a spatial low-pass filtered version of the conventional active intensity, where the non-propagating waves are filtered out. An important result shown by Williams~(1995) is that the sound power, $\Pi$, calculated with the use of acoustic intensity (AI) is the same as that calculated with the SI. In other words, if $S$ is the vibrating surface, then \begin{equation}\label{conservapot} \Pi = \int_{S}\hat{I}(x, y, 0, \omega)\;dS = \int_{S}\hat{I}^{(s)}(x, y, 0, \omega)\;dS, \end{equation} \section{NUMERICAL FORMULATION}\label{3} It is well known that there is no closed form solution to compute the supersonic intensity for vibrating sound sources with arbitrary geometries. The alternative, in these cases, is to discretize the geometry and use numerical methods to model the radiation problem. These methods must be capable to identify, through computational tools, the regions of the source surface that effectively contributes to the radiated sound power. To obtain such numerical model, firstly the normal velocity field obtained in a closed form as presented in Subsection \ref{2.1}. The acoustical pressure field is then obtained with the \emph{boundary element method} (BEM). In this work, the useful intensity according to the definition given by \cite{Correa13} is also shown for each of the four cases %%%the useful intensity according to ~\cite{Correa13}, for each one of the four boundary condition cases is also presented. \subsection{Formulation of the boundary element method}\label{3.1} \medskip The boundary element method is based on Green's theorem to compute the fundamental solution of the Helmholtz equation to obtain an integral contour of the domain, called Kirchhoff-Helmholtz integral, given by \be c\hat{p}(X,\omega)=\int_{\Gamma}\left(i\omega\rho_{0} \hat{v}(X_{s},\omega) G(X_{s}|X)-\hat{p}(X_{s},\omega)\frac{\partial G(X_{s}|X)}{\partial n_{s}}\right)\,d\Gamma,\label{kirchoff} \ee \noindent where $c$ is a coefficient dependent on the position of point $X(x, y,z)$; $X_{s}(x, y, 0)$ is a point on the surface; $\Gamma$ is the surface contour and $G(X_{s}|X)$ is the free field Green function, which is given by, see \cite{Stakgold79}, \be G(X_{s}|X)=\frac{e^{{ik|X_{s}-X|}}}{4\pi|X_{s}-X|}. \ee A matrix relationship between pressure and normal velocity at the surface can then be obtained as \begin{equation} \hat{p}=\mathbf{R}\hat{v} \label{relacaopresevel} \end{equation} Knowing $\hat{v}$ and $\hat{p}$, then it can be obtained the normal component of the acoustic intensity (AI), as \be \hat{I}=\frac{1}{2}{\Re}[\hat{p}\hspace{0.05 cm}\hat{v}^{*}].\label{iacustica} \ee \noindent where $\mathbf{R}=H^{-1}G$ is called \textit{surface operator}. For details about $H$ and $G$ matrices, see \cite{Holmstrom01}. \subsection{Useful intensity}\label{UI}\label{3.2} Xu and Huang \cite{Xu10}, showed that the sound power can be expressed in terms of the normal velocity distribution. Such relationship is given as \begin{equation} \Pi=\hat{v}^{H}\bar{\bf Q}\hat{v}, \label{potcomovelocidade} \end{equation} \n where $H$ indicates the conjugate transpose and $\bar{\bf Q}$ is a Hermitian operator, i.e., \hbox{$\bar{\bf Q}=\bar{\bf Q}^H$}, called \emph{power operator} by~\cite{Correa13}. The characteristic of being Hermitian is a fundamental one, since it guarantees that the eigenvalues are all real and that any base of the invariant sub-spaces form an orthonormal set for the velocity distribution of the surface, see~Golub e Van Loan \cite{Golub96}. To formulate the model for the calculus of the sound power filtering the non radiating modes it is convenient to decompose the power operator in the form \begin{equation} {\bf \bar{Q}}={\bf V}{\bf D}{\bf V}^{-1}, \label{eq2} \end{equation} \n where {\bf V} is a matrix such that its columns are the eigenvalues of $\bar{\bf Q}$, ${\bf V}^{-1}$ is the inverse matrix of ${\bf V}$, and ${\bf D}$ is the diagonal matrix containing the eigenvalues of $\bar{\bf Q}$. Since the matrix ${\bf V}$ is unitary, one can write Eq.~(\ref{eq2}) as \begin{equation} \bar{\bf Q}={\bf VDV}^{H}, \label{potdecomposicaoespectralH} \end{equation} \n and, then, the sound power can be computed as \begin{equation} \Pi=\hat{v}^{H}{\bf VD}{\bf V}^{H}\hat{v}=\sum\limits_{ i=1}^{r}\lambda_{i}\langle{\hat{v}^{H},{\bf V}_{i}\rangle}\langle{{\bf V}_{{i}}^{H},\hat{v}\rangle} \label{potdecomposicaoespectralveloc} \end{equation} \n where $r$ is the global number of eigenvalues $\lambda_{i}$ of $\bar{\bf Q}$. Such eigenvalues are called eigenvalues of velocity, in analogy to the denomination given by Borgiotti~(1990) to the singular values. The~${\bf V}_{i}^{H}$ are the lines of ${\bf V}^{H}$ and ${\bf V}_{i}$ are the columns of ${\bf V}$. In other words, the eigenvectors of ${\bf \bar{Q}}$, are called own pattern of velocity, since they form a set of modes for the normal velocity distribution. The series presented in Eq.~(\ref{potdecomposicaoespectralveloc}) can be truncated for a limited number or radiation modes that effectively contribute to the sound power radiated. In this way, the modes associated to eigenvalues with negligible magnitude are discarded, obtaining an accurate approximation for the sound power. This means that \begin{equation} \label{eq6} \Pi=\sum\limits_{ i=1}^{r}\lambda_{i}\langle{\hat{v}^{H},V_{i}\rangle}\langle{V_{{i}}^{H},\hat{v}\rangle}\approx{\sum\limits_{ i=1}^{r_{c}}\lambda_{i}\langle{\hat{v}^{H},V_{i}\rangle}\langle{V_{{i}}^{H},\hat{v}\rangle}}, \end{equation} \n with $r_{c} < r$, where $r_{c}$ is a sufficient quantity of retained modes. Aiming at obtaining such a truncation procedure, the eigenvalues are organized --- without loss of generality --- in a decreasing order of absolute values, that means, $|\lambda_{1}|>|\lambda_{2}|>\cdots>|\lambda_{r}|$. To chose the best value $r_c