Vibroacoustic Simulation. Alexander PeifferЧитать онлайн книгу.

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the radiated power calculation of the piston we took the pressure at the piston surface and integrated the pressure–velocity product over the surface. Due to the fact that the velocity is constant the surface integral involves mainly the pressure as a space-dependent property. In case of vibrating structures with complex shapes of vibration the velocity distribution over the surface is not homogeneous, and we need a more detailed approach.

bold-italic p left-parenthesis bold r right-parenthesis equals double-integral Subscript negative normal infinity Superscript normal infinity Baseline StartFraction j omega rho 0 Over 2 pi l EndFraction e Superscript minus j k l Baseline bold-italic v Subscript z Baseline left-parenthesis bold r 0 right-parenthesis d bold r 0 with l equals StartAbsoluteValue bold r minus bold r 0 EndAbsoluteValue (2.157)

In the above equation a function with argument (r−r0) is multiplied by the velocity function for r0 and integrated over the two-dimensional space. Mathematically, this can be interpreted as a two-dimensional convolution in space

bold-italic p left-parenthesis bold r right-parenthesis equals StartFraction j omega rho 0 Over 2 pi StartAbsoluteValue bold r EndAbsoluteValue EndFraction e Superscript j k StartAbsoluteValue bold r EndAbsoluteValue Baseline asterisk bold-italic v Subscript z Baseline left-parenthesis bold r right-parenthesis (2.158)

Thus, when we apply the two-dimensional Fourier transform to the Rayleigh integral the result is the product of the Fourier transform of the vibration shape vz(r0) and the Green’s function in wavenumber space leading to

bold-italic p left-parenthesis bold k right-parenthesis equals StartFraction 1 Over 4 pi EndFraction StartFraction rho 0 omega Over StartRoot k Subscript a Superscript 2 Baseline minus k squared EndRoot EndFraction bold-italic v Subscript z Baseline left-parenthesis bold k right-parenthesis with k equals StartAbsoluteValue bold k EndAbsoluteValue (2.159)

So, we have replaced the expensive convolution operation by a multiplication. This simplification is at the cost of two-dimensional Fourier transforms that are required to get the expressions in wavenumber domain.

The time averaged intensity of a sound field is given by the product of pressure and velocity (2.45). As the velocity is not uniform over the surface we perform a surface integration over the vibrating area to get the total radiated power

(2.160)

Thus, for the determination of radiated power a double area integral is required that may become computationally expensive.

In the above expression we can also switch to the wavenumber domain. In this case the area integration is replaced by an integration over the two-dimensional wavenumber space.

(2.161)

The double integral is replaced by a single two-dimensional wavenumber integration. Thus, once the shape function is available the power calculation in wavenumber space is much faster than in real space (Graham, 1996).

2.7.4.1 Radiation Efficiency

The radiation efficiency is a quantity that relates the power of a plane wave to the radiated power of a vibrating surface with same surface averaged velocity. The definition of the radiation efficiency was motivated by experimental procedures because it allows the estimation of the radiated power from the measurements of the vibration velocity. The squared average velocity of a vibrating surface is

(2.162)

and the power radiated by a plane wave through the same area S is given by (2.47)

normal upper Pi 0 equals one-half upper S rho 0 c 0 mathematical left-angle ModifyingAbove v With caret Subscript z Superscript 2 Baseline mathematical right-angle Subscript upper S (2.163)

The radiation efficiency is defined as the ratio between the radiated power of a velocity profile vz(r) of a surface S and the standardized power of the plane wave:

sigma Subscript normal r normal a normal d Baseline equals StartFraction normal upper Pi Over normal upper Pi 0 EndFraction equals StartStartFraction normal upper Pi OverOver StartFraction upper S Over 2 EndFraction rho 0 c 0 mathematical left-angle ModifyingAbove v With caret Subscript z Superscript 2 Baseline mathematical right-angle Subscript upper S Baseline EndEndFraction (2.164)

The radiation efficiency is used to determine the radiated power of vibrating structures from calculated, estimated, or measured radiation efficiency of specific surfaces

(2.165)