structures Prof. dr. Ines Lopez Arteaga Structural Acoustics - PowerPoint PPT Presentation

Sound radiation from structures Prof. dr. Ines Lopez Arteaga Structural Acoustics Department of Mechanical Engineering

What do they have in common?

What do they have in common? Source Transmission Receiver Soundboard

What do they have in common Source Transmission Receiver Soundboard Courtesy of Goodyear S.A.

Wave types • Longitudinal Acoustic / Structural C= constant • Transverse Structural • Bending C=C(f) Longitudinal Transverse Bending / Mechanical Engineering PAGE 18

Structural wave types - Pure / corrected bending wave - Mainly transverse vibrations Corrected bending wave includes effects of rotary inertia and shear deformation - most strongly excited (lowest mechanical impedance in audio Most important for freq.range) acoustic radiation - radiates most effectively (as compared to other wave types) / Mechanical Engineering PAGE 20

Transverse vibrations an infinite string      2 u         T dx T dx   L   2 x t  u    x   2 2 u 1 u T   where is the phase c  s   s 2 2 2 x c t velocity [m/s] L / Mechanical Engineering PAGE 22

Transverse bending of a beam Fourth order partial differential equation:   2 4 u EI u    0 where is the mass per unit length.    2 4 L t x L           i t kx 4 2 L u x , t Ae k Consider the solution EI    EI      2 L k c 4 4 B  B EI k B L  E   c L  c c fh 1 . 8 with , f and B L   2 h is the beam height. / Mechanical Engineering PAGE 23

Bending waves in plates Fourth order partial differential equation:       2 3 4 4 4 u Et u u u          2 0        s 2 2 4 2 2 4   t 12 1 x x y y  where is the mass per unit area. s Vectorial sum in x- and y-direction:    2   2  2  k k k k k k B x y B x y  E    c 1 . 8 c fh   with , f and c L B L    1  2 2 h is the plate thickness. / Mechanical Engineering PAGE 24

Dispersion Beam:  E   c L  c c fh 1 . 8 with , f and B L   2 h is the beam height. Plate:  E    c c fh 1 . 8   with c L , f and B L   1   2 2 h is the plate thickness.     ! In both cases: c ( f ) fh h / f B https://www.youtube.com/watch?v=dwMIaDg4Zeg / Mechanical Engineering PAGE 25

Phase velocity, group velocity Phase velocity:   c k Group velocity:    c g  k / Mechanical Engineering PAGE 27

Critical frequency plates At the critical frequency (coincidence frequency) the acoustic wave velocity and the bending wave velocity are equal:   E c c c f h  1 . 8 with   c L h : plate thickness. B L c   1  2 2 c  This leads to: f c c h 1 . 8 L The critical frequency of a plate only depends on the material properties and plate thickness / Mechanical Engineering PAGE 28

Sound radiation from infinite plates   2 f     k B k 2   sin  B . 1 8 c t k B L   2 2 f   k  c / Mechanical Engineering PAGE 29

Sound radiation from infinite plates p Solving for gives: max  c u    2 2   0 yp iy k k ik x i t ( , , ) p x y t max e e e B B 2 k  B 1 2 k   ik x With: u u e B yp yp max   2 2 Note that: k k k B y 2 k B  And that the pressure increases rapidly as 1 2 k / Mechanical Engineering PAGE 31

Sound radiation from infinite plates      air sin  Far field B f>f crit  (  B >  air )  B f<f crit Near field (  B <  air ) “hydrodynamic shortcut” 1 h    B  air f f / Mechanical Engineering PAGE 32

Radiation ratio (Radiation efficiency) Radiation ratio: Sound power radiated by the plate divided by the sound power radiated by a large rigid piston with the same surface area and same r.m.s. vibration velocity    plate Piston  plate piston / Mechanical Engineering PAGE 34

Sound power radiated by a vibrating piston Acoustic radiation of a large (compared to the acoustic wavelength) and rigid piston.    r 2 p rms u rms For plane waves the velocity and the pressure are related through the specific acoustic impedance . 1  u p  c 0 Therefore:    2 cSu 0 rms / Mechanical Engineering PAGE 35

Radiation ratio infinite plate Plate Piston 2 u       prms 2 cSu cS 0 0 prms 2 k  B 1 2 k  1    plate  plate 2 k B  piston 1 2 k / Mechanical Engineering PAGE 37

Radiation ratio infinite plate  1 1     plate  plate 2 f k   piston c 1 B 1 2 f k / Mechanical Engineering PAGE 38

Radiation from finite plates Below the critical frequency (subsonic modes) radiation efficiency depends on modeshape + + - - + - - + + Odd mode Even mode At low frequencies odd modes are better radiators (higher radiation efficiency) than even modes / Mechanical Engineering PAGE 41

Radiation from finite plates Below critical frequency dipole and quadrupole cancellations + + - Dipole + + - Dipole Dipole Dipole - + - Dipole Dipole - + + - + + Dipole Odd mode= Even mode= Edge radiation Corner radiation / Mechanical Engineering PAGE 42

Radiation from finite plates Above the critical frequency efficient radiation, each part of the plate radiates independently as a monopole + - + + - - - + - + - - - + + + + Odd mode Even mode / Mechanical Engineering PAGE 43

Radiation ratio finite plates Above the critical frequency approximately similar to infinite plates 1     1 f f for plate c f c  1 f / Mechanical Engineering PAGE 49

Radiation ratio finite plates Design curve for broadband mechanical excitation of flat plates / Mechanical Engineering PAGE 50

Conclusions • Infinite plates can only radiate sound above the critical frequency. • Real (finite) plates can radiate sound at all frequencies, but below the critical frequency they are inefficient radiators. / Mechanical Engineering PAGE 51

Recommended books • Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies (3rd Edition), L. Cremer, M. Heckl, and B. A. T. Petersson, Springer Berlin, 2005. • Fourier acoustics: Sound radiation and nearfield acoustic holography, E.G. Williams, Academic Press, London, 1999. • Fundamentals of noise and vibration analysis for engineers, M. Norton, d. Karckub, Cambridge University Press, 2003. / Mechanical Engineering PAGE 52

Sound radiation from structures Prof. dr. Ines Lopez Arteaga Structural Acoustics Department of Mechanical Engineering