Added derivation of circular plate membrane impedance to the documentation
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lrftubes.bib
17
lrftubes.bib
@ -382,3 +382,20 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo
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langid = {english},
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file = {Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:/home/anne/.literature/storage/I9P5SZAE/Kino et al. - 2009 - Investigation of non-acoustical parameters of comp.pdf:application/pdf}
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}
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@article{leniowska_plate_resonance_1999,
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title = {Vibrations of circular plate interacting with an ideal compressible fluid},
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volume = {24},
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url = {https://acoustics.ippt.pan.pl/index.php/aa/article/viewFile/1117/952},
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pages = {427--441},
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number = {4},
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journaltitle = {Archives of acoustics},
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author = {Leniowska, L.},
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date = {1999}
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}
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@misc{calcdevice,
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title = {Natural frequency calculators (web page)},
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url = {https://calcdevice.com/natural-frequency-of-circular-plate-id224.html},
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urldate = {2022-05-25}
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}
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399
lrftubes.lyx
399
lrftubes.lyx
@ -7113,6 +7113,405 @@ Membrane
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A membrane is a mechanical
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\end_layout
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\begin_layout Section
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Circular plate membrane
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\end_layout
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\begin_layout Standard
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series_impedance/class CircPlateMembrane(SeriesImpedance)
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\end_layout
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\begin_layout Standard
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A thin circular plate can be modeled using CircPlateMembrane.
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It behaves like an acoustic compliance.
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A typical use is the attenuation of acoustic pressure by combining it with
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an enclosed volume.
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\end_layout
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\begin_layout Standard
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Two boundary condition cases can be applied: fixed/clamped edges and simply
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supported edges.
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The general equation for the static displacement of the plate is given
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by
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\begin_inset CommandInset citation
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LatexCommand cite
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after "p. 487"
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key "young_roarks_2002"
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literal "false"
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\end_inset
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:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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y\left(r\right)=y_{c}+\frac{M_{c}r^{2}}{2D\left(1+\nu\right)}+LT_{y}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $y_{c}$
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\end_inset
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and
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\strikeout off
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\xout off
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $M_{c}$
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\end_inset
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are
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\family default
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\series default
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\shape default
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\size default
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\emph default
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\bar default
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\strikeout default
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\xout default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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the displacement and moment at the center of the plate,
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\begin_inset Formula $LT_{y}$
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\end_inset
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is the load term in the y-direction,
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\begin_inset Formula $\nu$
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\end_inset
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is the Poisson's ratio of the plate material and
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\begin_inset Formula $D$
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\end_inset
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is the flexural stiffness of the plate, which is given by the equation:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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D=\frac{Et^{3}}{12\left(1-\nu^{2}\right)}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $E$
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\end_inset
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is the Young's modulus of the plate material and
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\begin_inset Formula $t$
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\end_inset
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is the plate thickness.
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Substituting
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\begin_inset Formula $D$
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\end_inset
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,
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\begin_inset Formula $y_{c}$
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\end_inset
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,
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\begin_inset Formula $M_{c}$
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\end_inset
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and
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\begin_inset Formula $LT_{y}$
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\end_inset
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for this specific load case (uniform load/pressure) and boundary conditions
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\begin_inset CommandInset citation
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LatexCommand cite
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after "p. 458 & p. 488"
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key "young_roarks_2002"
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literal "false"
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\end_inset
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and simplifying yields the following equations for the static plate deflection:
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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In these equations the distributed load
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\begin_inset Formula $q$
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\end_inset
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is replaced by
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\begin_inset Formula $-p$
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\end_inset
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.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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y_{ss}\left(r\right)=\frac{3p\left(1-\nu^{2}\right)}{16Et^{3}\left(1+\nu\right)}\left(a^{2}\left[a^{2}\left\{ 5+\nu\right\} -2r^{2}\left\{ 3+\nu\right\} \right]+r^{4}\left[1+\nu\right]\right)
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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y_{fix}\left(r\right)=\frac{3p\left(1-\nu^{2}\right)}{16Et^{3}}\left(a^{4}-2a^{2}r^{2}+r^{4}\right)
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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In which
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\begin_inset Formula $a$
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\end_inset
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is the radius of the plate and
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\begin_inset Formula $r$
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\end_inset
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is the radial coordinate.
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The static acoustic compliance of the plate is given by the equation:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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c_{stat}\left(r\right)=\frac{y\left(r\right)}{p}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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The static acoustic volume compliance for both cases can be calculated by
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integrating over the surface of the plate:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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C_{stat}=2\pi\int_{0}^{a}c_{stat}\left(r\right)rdr
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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Performing this integration for both boundary condition cases yields:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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C_{stat,ss}=\frac{\pi a^{6}}{16Et^{3}}\left(7-6\nu-\nu^{2}\right)
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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C_{stat,fix}=\frac{\pi a^{6}}{16Et^{3}}\left(1-\nu^{2}\right)
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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The dynamic acoustic volume compliance of the plate is given by the equation:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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C_{dyn}\left(f\right)=\frac{C_{stat}}{1-\left(\frac{f}{f_{r}}\right)^{2}}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $f$
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\end_inset
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is the frequency in Hz and
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\begin_inset Formula $f_{r}$
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\end_inset
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is the resonance frequency of the plate in Hz.
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The resonance frequency for the simply supported plate is given by the
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equation
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\begin_inset CommandInset citation
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LatexCommand citeyear
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key "calcdevice"
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literal "false"
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\end_inset
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:
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\begin_inset Note Note
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status collapsed
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\begin_layout Plain Layout
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This is an approximation from an online calculator.
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A more exact equation like the one for the fxed case should be found.
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\end_layout
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\end_inset
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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f_{r,ss}=\frac{0.8}{a^{2}}\sqrt{\frac{D}{\rho t}}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $\rho$
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\end_inset
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is the density of the plate material.
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The resonance frequency for the fixed plate is given by the equation
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\begin_inset CommandInset citation
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LatexCommand cite
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after "p. 430"
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key "leniowska_plate_resonance_1999"
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literal "false"
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\end_inset
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:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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f_{r,fix}=\frac{\gamma_{1}^{2}}{a^{2}}\sqrt{\frac{D}{\rho t}}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $\gamma_{1}$
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\end_inset
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is the first solution to the following equation:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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J_{0}\left(\gamma_{m}\right)I_{1}\left(\gamma_{m}\right)+J_{1}\left(\gamma_{m}\right)I_{0}\left(\gamma_{m}\right)=0\label{eq:gamma}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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in which
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\begin_inset Formula $J_{n}\left(\gamma_{m}\right)$
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\end_inset
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and
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\begin_inset Formula $I_{n}\left(\gamma_{m}\right)$
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\end_inset
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are the Bessel function of the first kind and modified Bessel functions
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of order
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\begin_inset Formula $n$
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\end_inset
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.
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Solving equation
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\begin_inset CommandInset ref
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LatexCommand ref
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reference "eq:gamma"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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yields
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\begin_inset Formula $\gamma_{1}=3.196$
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\end_inset
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.
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The impedance is given by the equation:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\begin{equation}
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Z_{cpm}\left(f\right)=\frac{1}{i2\pi fC_{dyn}\left(f\right)}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Section
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Holes in plate
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\end_layout
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