Advanced Structural Dynamics and Acoustics

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1 Advanced Structural Dynamics and Acoustics Acoustics

2 Point-Driven Plate Radiation Cylindrical Coordinates z y Light Fluid Loading θ r h Fleural Wavenumber Particle Velocity

3 z Point-driven Plate Light Fluid Loading y θ r h I = 2 π i Σ Res Inverse Hankel Transform Comple Contour Integration Drive-point Impedance Fleural Wave Speed

4 Point Driven Plate Radiated Pressure Fleural Wave Speed Subsonic Evanescent Supersonic Radiating Coincidence Frequency Water

5 Point-Driven Plate Radiation Eact Formulation Plate Equation of Motion z y Fourier Transform θ r h Vertical Plate Displacement k r < k: Radiation Damping k r > k: Added Mass Vertical Particle Plate Velocity Velocity

6 z Point-driven Fluid-loaded Plate Eact Formulation y θ r h I = 2 π i Σ Res Comple Contour Integration Inverse Hankel Transform Fleural Wavenumber Equation Directivity Function

7 Point-driven Plate Evanescent Frequency Regime Light Fluid Loading 2 khz Eact

8 Point-driven Plate Radiation Frequency Regime 8 khz Eact Light Fluid Loading Radiation Damping

9 % % MATLAB script for plotting the directivity function for % a pointdriven elastic plate % % Parameters: % f Frequency % rho Density % c Speed of Sound % h Plate thickness % E Young's modulus % nu Poisson's ratio plate.m % rhos Plate density clear rhos=7700; cp=5600; h=0.05; E=rhos*cp^2; nu=0.33; D= E *h^3/(12*(1-nu^2)); rho=1000; c=1500; f=8000; omega=2*pi*f; k=omega/c ka=k; figure(1); hold off km=2*ka; nk=300; dk=2*km/(nk-1); =[-km:dk:km]; y=; o=ones(1,nk); k=' * o; ky=(y' *o)'; kr=abs(comple(k,ky)); kf= ((rhos*h+i*rho./sqrt(k^2-comple(kr,0.0).^2))*omega^2/d).^0.25; kfa=kf*a; ss=-rho*omega^2./(d*(2*pi)^2 *(comple(kr,0).^4 -kfa.^4)); wavei(dba(ss)',,y) shading('flat') ais('equal') b=label('k_') set(b,'fontsize',16); b=ylabel('k_y') set(b,'fontsize',16); tit=['point Driven Plate, h =' num2str(h) ', f = ' num2str(f)] b=title(tit); set(b,'fontsize',20); nphi=361; dphi=2*pi/(nphi-1); phi=[0:dphi:2*pi]; =k*a*cos(phi); yy=k*a*sin(phi); hold on ADVANCED STRUCTURAL DYNAMICS AND ACOUSTICS b=plot(,yy,'b'); figure(2) nphi=361. dphi=2*pi/(nphi-1) nth=181; dth=0.5*pi/(nth-0.5); phi=[0:dphi:(nphi-1)*dphi]' * ones(1,nth); th=([dth/2:dth:pi/2]'*ones(1,nphi))'; k=ka*sin(th).*cos(phi); ky=ka*sin(th).*sin(phi); kr=ka*sin(th); kf= ((rhos*h+i*rho./sqrt(k^2-comple(kr,0.0).^2))*omega^2/d).^0.25; kfa=kf*a; ss=-rho*omega^2./(d*(2*pi)^2 *(comple(kr,0).^4 -kfa.^4)); ss=dba(ss); sm=10.0*(ceil(0.1*ma((ma(ss))'))); for i=1:size(ss,1) for j=1:size(ss,2) ss(i,j)=ma(ss(i,j),sm-40.0)-(sm-40.0); end end =ss.*sin(th).*cos(phi); yy=ss.*sin(th).*sin(phi); zz=ss.*cos(th); surfl(,yy,zz); colormap('copper'); shading('flat'); ais('equal'); tit=['point Driven Plate, h =' num2str(h) ', f = ' num2str(omega/(2*pi)) ] b=title(tit); set(b,'fontsize',20); figure(3) b=polar([pi/2-fliplr(th(1,:)) pi/2+th((nphi-1)/2+1,:)], [fliplr(ss(1,:)) ss((nphi-1)/2+1,:)]); set(b,'linewidth',2) b=legend(['ref. ' num2str(sm-40.0) ' db']); set(b,'fontsize',14); tit=['point Driven Plate, h =' num2str(h) ', f = ' num2str(omega/(2*pi)) ] b=title(tit); set(b,'fontsize',20);

10 z Simply-supported Elastic Plate Homogeneous Equation of Motion L y L y Moments Boundary Conditions

11 Point-driven Rectangular Elastic Plate z Normal Modes Solutions for L y Orthogonality Relation (,y ) 0 0 F L y Dispersion Relation Moments Light Fluid Loading Homogeneous Equation of Motion Boundary Conditions Normal Mode Solution Transfer Mobility

12 Normal Modes of Simply-supported Elastic Plate Image removed due to copyright considerations. See Figure 2.26 in Williams, E. G. Fourier Acoustics. London: Academic Press, 1999

13 z Radiation from Point-driven Elastic Plate (,y ) 0 0 L y L y F Fourier Transforms Directivity Function

14 Radiation Efficiency z Low-order Modes Square Plate (,y ) 0 0 L y F L y Image removed due to copyright considerations. See Figure 2.27 in [Williams]. Radiation Efficiency RMS Velocity Mode m,n k/k f

15 Radiation Efficiency z High-order Modes Square Plate (,y ) 0 0 L y F L y Image removed due to copyright considerations. See Figure 2.28 in [Williams]. Radiation Efficiency RMS Velocity Mode m,n k/k f

16 Rectangular Elastic Plate Radiation Mode Types

17 Rectangular Elastic Plate Radiation Mode Ecitation Image removed due to copyright considerations. See Figure 2.34 in [Williams].

Evanescent Spectrum k Imaginary z Rectangular Elastic Plate Radiation Supersonic Intensity Radiating Spectrum k Real z M,n = 11,9 kl/2 = 1 k/k f =0.27 Image removed due to copyright considerations.

18 Evanescent Spectrum k Imaginary z Rectangular Elastic Plate Radiation Supersonic Intensity Radiating Spectrum k Real z M,n = 11,9 kl/2 = 1 k/k f =0.27 Image removed due to copyright considerations. See Figure 2.37 in [Williams]. Supersonic Intensity Image removed due to copyright considerations. See Figure 2.38 in [Williams]. Supersonic Intensity Normal Acoustic Intensity Image removed due to copyright considerations. See Figure 2.39 in [Williams].

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur

Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur LECTURE-13 WAVE PROPAGATION IN SOLIDS Longitudinal Vibrations In Thin Plates Unlike 3-D solids, thin plates have surfaces