# Vibration Prediction of Space Large-Scale Membranes Using Energy Flow Analysis
## 2 Materials and Methods  
For membranes in orbit, the gravity is neglectable. The differential governing equation that describes the vibration of a uniform damped membrane with small loss factor is [24]  
\[T(1+j\eta)\
abla^{2}u+\rho\omega^{2}u=0 \tag{1}\]  
where \(T\) is the tension in the membrane, \(u=Ue^{i\omega t}\) is the transversely vibrating displacement, \(\eta\) is the material damping, \(\rho\) is the mass density per unit area and \(\omega\) indicates the excitation frequency. Considering an infinite membrane excited by a point sinusoidal force, the governing equation in cylindrical coordinates becomes  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]  
where \(F\) indicates the magnitude of the exciting force, and \(k\) is a complex wave number  
\[k^{2}=\frac{\rho\omega^{2}}{T(1+j\eta)} \tag{3}\]  
The displacement solution of Equation (2) for a unit exciting force is given by the Green kernel. Considering the response of vibration is uniform in the circumferential direction, the Green kernel can be obtained through the Hankel function of second kind  
\[U=\frac{j}{4T}H_{0}^{(2)}(kr) \tag{4}\]  
Then the displacement solution for membranes under an exciting force with magnitude \(F\) is  
\[U=\frac{jF}{4T}H_{0}^{(2)}(kr) \tag{5}\]  
The asymptotic expressions for the Hankel function can be used for large-scale membranes with near-field waves being neglected. Then the solution becomes  
\[U=\frac{F}{4T}(1+j)\sqrt{\frac{2}{\pi kr}}e^{-jkr} \tag{6}\]  
The averaged kinetic energy can be derived in terms of displacement [25]  
\[E_{k}=\frac{\rho}{2}\omega^{2}UU^{*} \tag{7}\]  
where the star \({}^{*}\) indicates the conjunction. And the averaged potential energy is  
\[E_{p}=\frac{T}{2}\left(\frac{\partial^{2}U}{\partial r^{2}}+\frac{1}{r}\frac{ \partial U}{\partial r}\right)\left(\frac{\partial^{2}U}{\partial r^{2}}+ \frac{1}{r}\frac{\partial U}{\partial r}\right)^{*} \tag{8}\]  
Substituting Equation (6), into Equations (7) and (8), the averaged potential and kinetic energy are written as  
\[E_{k}=\frac{\rho\omega^{2}F^{2}}{32T^{2}}\frac{1}{\pi k_{1}r}e^{2k_{2}r} \tag{9}\]  
\[E_{p}=\frac{TF^{2}k_{1}^{3}}{32\pi r}e^{2k_{2}r} \tag{10}\]with  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]0  
The total vibrational energy of the membrane is the sum of the potential and kinetic energy and can be obtained as  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]1  
by noticing  
\[\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta(r) \tag{2}\]2  
The derived formulation only holds true for one dimensional analysis of far-field wave propagation in infinite membranes. For the finite membrane of reverberant field, considering the energy waves as rays, it is can be assumed that every energy ray emitted by the excitation is reflected as by a mirror when it impinges on the membrane boundaries with neglectable energy loss. Between two successive reflections, an energy ray travels on a straight line of variable distance. However, the statistical averaged distance can be described by a mean free path.