# Vibration Prediction of Space Large-Scale Membranes Using Energy Flow Analysis
## 2 Materials and Methods
For a rectangular membrane whose length is \(a\) and width is \(b\) (see Figure 1), the number of reflections of an energy ray with propagating velocity \(c\) between two boundaries \(x=0\) and \(x=a\) in per unit time is  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]3  
and the number of reflections between two boundaries \(y=0\) and \(y=b\) is  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]4  
Then the reflection number of an energy ray in \(1\) s is  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]5  
Since the energy rays travel with equal probability in every direction, the probability in \(d\theta\) is  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]6  
Assuming \(2\pi n\) energy rays are emitted by the excitation in \(1\) s, then the reflection number of total energy rays is  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]7  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]8 An energy ray in the membrane. where \(n\) is the ray number in per unit angle, \(L_{c}=2(a+b)\) is the perimeter and \(S=ab\) is the area of the membrane. The travel distance of total rays in 1 s is \(L=2\pi nc\). Then the mean free path can be written as  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]9  
It can be observed that the mean free path is independent from the excitation location. And it should be noticed that, the mean free path formulation holds true for membranes of various shapes although it is derived with a rectangular one. Then the resultant energy can be obtained by the summation of a series of energy rays  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]0  
where \(i\) indicates the \(i\)th reflection  
The averaged energy response for the membrane of finite size is obtained when the number of reflections approaches infinity. \[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]1  
Then we have the energy kernel for finite membranes  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]2  
For a vibrating membrane with random parameters, the energy representation can be written as  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]3  
where \(\widetilde{x}\) indicates an independent random parameter whose mathematic expectation is \(x\) and the standard deviation is \(\sigma_{x}\). The expectation of the random energy is  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]4  
in which \(E\) is the energy response from deterministic parameters,  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]5  
is the deviation of a random parameter from its expectation, and \(EXP\) indicates the expectation of a mathematical expression. The variance \(\sigma_{E}^{2}\) and standard deviation \(\sigma_{E}\) of the random energy response can be obtained as  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]6  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]7  
As an example, for a vibrating membrane whose length \(a\) is a random variable, the mean free path becomes a random variable. The energy response is obtained as  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]8The mathematic expectation and the variance of the random mean free path \(\widetilde{L}_{m}\) are  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]9  
\[U=\frac{j}{4T}H_{0}^{(2)}(kr) \tag{4}\]0  
Using [PERSON]'s series expansion at \(L_{m}\) for Equation (28), the first two terms are retained to obtain the approximate formulation of the energy response.