Vibroacoustic Simulation - Alexander Peiffer

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Peiffer.

      2.7.3.1 Impedance Concept

      The radiation impedance of the piston is calculated from the pressure averaged over the surface related to the piston velocity vz. As shown by Lerch and Landes (2012) the mechanical impedance of the piston due to radiation is given by

      According to equation (2.141) assuming a constant velocity vz over the surface A the pressure is

       bold-italic p left-parenthesis r right-parenthesis equals StartFraction j omega rho 0 bold-italic v Subscript z Baseline Over 2 pi EndFraction integral Underscript upper A Endscripts StartFraction e Superscript minus j k s Baseline Over s EndFraction d upper A period (2.147)

      Thus, we get the pressure at r from integrating the contribution from the rest of the piston in circles of radius s. The angle integration over φ0 runs from 0 to 2π. From every angle φ0 follows the integration limits smax of the second integral.

       s Subscript m a x Baseline equals r cosine phi plus StartRoot upper R squared minus r squared sine squared phi EndRoot (2.148)

      Using those limits gives

      Inserting equation (2.149) into (2.146) leads to the expression

       bold-italic upper Z equals rho 0 c 0 pi upper R squared left-parenthesis 1 minus StartFraction 1 Over pi upper R squared EndFraction integral Subscript phi equals 0 Superscript 2 pi Baseline integral Subscript r equals 0 Superscript upper R Baseline e Superscript minus j k r cosine phi minus j k StartRoot upper R squared minus r squared sine squared phi EndRoot Baseline r d r d phi right-parenthesis period (2.150)

      Running through quite a lot of algebraic modifications we get the expression for the impedance of a piston

       bold-italic upper Z equals rho 0 c 0 pi upper R squared left-parenthesis 1 minus StartFraction upper J 1 left-parenthesis 2 k upper R right-parenthesis Over k upper R EndFraction plus j StartFraction upper H 1 left-parenthesis 2 k upper R right-parenthesis Over k upper R EndFraction right-parenthesis (2.151)

      or

      H1(z) is the Hankel function of first order. In Figure 2.14 the real and imaginary parts of the acoustic radiation impedance are compared to those of the pulsating sphere. Both sources have a similar shape except some waviness for the piston resulting from interference effects from the integration over the piston surface. For large kR the impedance is real for both radiators and approaches the acoustic impedance of a plane wave z0=ρ0c0.

      Figure 2.14 Acoustic radiation impedance of the piston. Source: Alexander Peiffer.

      The main use of Equation (2.153) is that the required velocity to achieve (or prevent) a certain sound power can be calculated from it, for example if one must define the boundary condition for a radiating piston in simulation software and only the radiated power is known.

      2.7.3.2 Inertia Effects

      The Bessel functions can be approximated by a series in 2kR taking the first series term of both functions (Jacobsen, 2011)

       bold-italic upper Z equals rho 0 c 0 pi upper R squared left-parenthesis one-half left-parenthesis 2 k upper R right-parenthesis squared plus j StartFraction 8 Over 3 pi EndFraction k upper R right-parenthesis (2.154)

      This expression is valid for ka<0.5. From the imaginary part we get for the mass

       m equals StartFraction upper I m left-parenthesis bold-italic upper Z right-parenthesis Over omega EndFraction equals StartFraction 8 upper R cubed rho 0 Over 3 EndFraction (2.155)

      Assuming a cylindrical volume V=πR2lc of the fluid above the piston we can calculate the length of the moving mass cylinder to be

       l Subscript c Baseline equals StartFraction 8 upper R Over 3 pi EndFraction almost-equals 0.85 upper R (2.156)

      meaning that at low frequencies the piston is moving a fluid layer of 0.85 times the radius acting as an inertia without radiation.

      2.7.4 Power Radiation

      For

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