Added derivation of circular plate membrane impedance to the documentation
This commit is contained in:
parent
b743b9da3e
commit
635003673d
19
lrftubes.bib
19
lrftubes.bib
@ -381,4 +381,21 @@ The full solution of the problem has been obtained by Kirchhoff (1868) in the fo
|
||||
date = {2009-04},
|
||||
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}
|
||||
}
|
||||
|
||||
399
lrftubes.lyx
399
lrftubes.lyx
@ -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
|
||||
|
||||
Loading…
Reference in New Issue
Block a user