Added derivation of circular plate membrane impedance to the documentation

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Thijs Hekman 2022-05-25 12:12:06 +02:00
parent b743b9da3e
commit 635003673d
2 changed files with 417 additions and 1 deletions

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@ -382,3 +382,20 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo
langid = {english},
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}
}
@article{leniowska_plate_resonance_1999,
title = {Vibrations of circular plate interacting with an ideal compressible fluid},
volume = {24},
url = {https://acoustics.ippt.pan.pl/index.php/aa/article/viewFile/1117/952},
pages = {427--441},
number = {4},
journaltitle = {Archives of acoustics},
author = {Leniowska, L.},
date = {1999}
}
@misc{calcdevice,
title = {Natural frequency calculators (web page)},
url = {https://calcdevice.com/natural-frequency-of-circular-plate-id224.html},
urldate = {2022-05-25}
}

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@ -7113,6 +7113,405 @@ Membrane
A membrane is a mechanical
\end_layout
\begin_layout Section
Circular plate membrane
\end_layout
\begin_layout Standard
series_impedance/class CircPlateMembrane(SeriesImpedance)
\end_layout
\begin_layout Standard
A thin circular plate can be modeled using CircPlateMembrane.
It behaves like an acoustic compliance.
A typical use is the attenuation of acoustic pressure by combining it with
an enclosed volume.
\end_layout
\begin_layout Standard
Two boundary condition cases can be applied: fixed/clamped edges and simply
supported edges.
The general equation for the static displacement of the plate is given
by
\begin_inset CommandInset citation
LatexCommand cite
after "p. 487"
key "young_roarks_2002"
literal "false"
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
y\left(r\right)=y_{c}+\frac{M_{c}r^{2}}{2D\left(1+\nu\right)}+LT_{y}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $y_{c}$
\end_inset
and
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $M_{c}$
\end_inset
are
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
the displacement and moment at the center of the plate,
\begin_inset Formula $LT_{y}$
\end_inset
is the load term in the y-direction,
\begin_inset Formula $\nu$
\end_inset
is the Poisson's ratio of the plate material and
\begin_inset Formula $D$
\end_inset
is the flexural stiffness of the plate, which is given by the equation:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
D=\frac{Et^{3}}{12\left(1-\nu^{2}\right)}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $E$
\end_inset
is the Young's modulus of the plate material and
\begin_inset Formula $t$
\end_inset
is the plate thickness.
Substituting
\begin_inset Formula $D$
\end_inset
,
\begin_inset Formula $y_{c}$
\end_inset
,
\begin_inset Formula $M_{c}$
\end_inset
and
\begin_inset Formula $LT_{y}$
\end_inset
for this specific load case (uniform load/pressure) and boundary conditions
\begin_inset CommandInset citation
LatexCommand cite
after "p. 458 & p. 488"
key "young_roarks_2002"
literal "false"
\end_inset
and simplifying yields the following equations for the static plate deflection:
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
In these equations the distributed load
\begin_inset Formula $q$
\end_inset
is replaced by
\begin_inset Formula $-p$
\end_inset
.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
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)
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
y_{fix}\left(r\right)=\frac{3p\left(1-\nu^{2}\right)}{16Et^{3}}\left(a^{4}-2a^{2}r^{2}+r^{4}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
In which
\begin_inset Formula $a$
\end_inset
is the radius of the plate and
\begin_inset Formula $r$
\end_inset
is the radial coordinate.
The static acoustic compliance of the plate is given by the equation:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
c_{stat}\left(r\right)=\frac{y\left(r\right)}{p}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
The static acoustic volume compliance for both cases can be calculated by
integrating over the surface of the plate:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
C_{stat}=2\pi\int_{0}^{a}c_{stat}\left(r\right)rdr
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
Performing this integration for both boundary condition cases yields:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
C_{stat,ss}=\frac{\pi a^{6}}{16Et^{3}}\left(7-6\nu-\nu^{2}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
C_{stat,fix}=\frac{\pi a^{6}}{16Et^{3}}\left(1-\nu^{2}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
The dynamic acoustic volume compliance of the plate is given by the equation:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
C_{dyn}\left(f\right)=\frac{C_{stat}}{1-\left(\frac{f}{f_{r}}\right)^{2}}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $f$
\end_inset
is the frequency in Hz and
\begin_inset Formula $f_{r}$
\end_inset
is the resonance frequency of the plate in Hz.
The resonance frequency for the simply supported plate is given by the
equation
\begin_inset CommandInset citation
LatexCommand citeyear
key "calcdevice"
literal "false"
\end_inset
:
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
This is an approximation from an online calculator.
A more exact equation like the one for the fxed case should be found.
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
f_{r,ss}=\frac{0.8}{a^{2}}\sqrt{\frac{D}{\rho t}}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $\rho$
\end_inset
is the density of the plate material.
The resonance frequency for the fixed plate is given by the equation
\begin_inset CommandInset citation
LatexCommand cite
after "p. 430"
key "leniowska_plate_resonance_1999"
literal "false"
\end_inset
:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
f_{r,fix}=\frac{\gamma_{1}^{2}}{a^{2}}\sqrt{\frac{D}{\rho t}}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $\gamma_{1}$
\end_inset
is the first solution to the following equation:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
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}
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
in which
\begin_inset Formula $J_{n}\left(\gamma_{m}\right)$
\end_inset
and
\begin_inset Formula $I_{n}\left(\gamma_{m}\right)$
\end_inset
are the Bessel function of the first kind and modified Bessel functions
of order
\begin_inset Formula $n$
\end_inset
.
Solving equation
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:gamma"
plural "false"
caps "false"
noprefix "false"
\end_inset
yields
\begin_inset Formula $\gamma_{1}=3.196$
\end_inset
.
The impedance is given by the equation:
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
Z_{cpm}\left(f\right)=\frac{1}{i2\pi fC_{dyn}\left(f\right)}
\end{equation}
\end_inset
\end_layout
\begin_layout Section
Holes in plate
\end_layout