diff --git a/lrftubes.bib b/lrftubes.bib index 9669a45..946a177 100644 --- a/lrftubes.bib +++ b/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} -} \ No newline at end of file +} + +@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} +} diff --git a/lrftubes.lyx b/lrftubes.lyx index e80c2dc..f20fccc 100644 --- a/lrftubes.lyx +++ b/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