Vibration Prediction of Space Large-Scale Membranes Using Energy Flow Analysis

Abstract

In this work, vibration prediction of space large-scale membranes from the energy point of view is investigated. Based on the Green kernel of vibrating membranes, a new analytical representation of energy response of infinite membranes is derived. Averaged energy is used as the main variable so that the response fluctuation can be smoothed. Then membranes of various shapes can be taken into account by introducing the mean free path into the formulation to describe travel distances of energy waves. The energy response of finite membranes is obtained with the superposition of energy waves subsequently. Considering uncertainties usually becomes significant in large-scale structures, the formulation expressed with random variables is obtained for membranes with uncertain properties. The mathematical expectation and variance of energy response are derived subsequently. And the confidence interval of random response is obtained. Finally, numerical simulations are performed to validate the proposed formulations and characteristics of the random energy responses are analyzed by taking a space large-scale membrane structure as a model. The developed formulations make the analysis of membranes with uncertainties more convenient than Finite Element Method (FEM) since they are expressed in analytical forms. Compared with existing formulations of energy flow derived from deterministic travel distances of waves that only apply to regular shapes of structures, the proposed formulations are suitable for membranes of various shapes. This work provides an alternative analytical approach to vibration prediction for space large-scale membranes with uncertainties. And the approach is thought helpful for the vibration analysis of other two-dimensional structures.

1. Introduction

With the development of aerospace technologies, the large-scale spacecraft is playing an increasingly important role. Large-scale space membrane structures, which is lightweight and can be folded in launching vehicles, are being considered in many applications of spacecrafts, such as large-scale antennas of satellites, large solar arrays of space solar powers, solar sails, and sun shields of space telescopes [1,2,3,4,5,6,7]. These structures with high flexibility and poor stiffness are sensitive to external excitations, such as orbital maneuver, attitude adjustment, and unsteady solar radiation pressure. Undesirable vibrations will occur to affect the surface accuracy of membranes and the attitude of spacecrafts. Dynamic modeling and vibration response prediction of large-scale membranes are necessary to ensure spacecrafts to operate normally [8,9,10].

At present, one of the most popular methods for structural vibration prediction is finite element method (FEM). Kukathasan and Pellegrino presented a finite-element modelling technique to simulate the air-membrane interaction and obtained the results with ABAQUS [11]. To simulate the ground tests, Tessler et al. developed finite element models for a 10-m four-quadrant solar sail system and a 20-m four-quadrant solar sail system. The results from analysis showed reasonable agreement with the test data [12]. Shen et al. obtained mode shapes and natural frequencies of a membrane antenna under varying pre-tension loads with ABAQUS [13]. Houmat applied the trigonometric p-version of FEM to vibration prediction of membranes for higher accuracy [14].

In FEM, structures are divided into finite elements. To ensure the precision of results, the element number will increase with increasing vibration frequency and structure size. Thus, for high frequency vibrations or large-scale structures, FEM will lead to a huge amount of computation which may become impracticable. A potential solution is using averaged variables to describe vibration behaviors of structures which leads to statistical energy analysis (SEA) [15,16]. In SEA, structures are divided into subsystems and one single statistical energy value is provided for every subsystem. The computation cost decreases dramatically and the applications were studied in many fields on structural vibrations [17,18,19,20]. However, one of the most obvious shortages should be paid attention to is unavailability of the energy distribution which is unacceptable in many applications. To provide more precious results, researchers investigated several alternative tools among which energy flow analysis (EFA) is a representative method. EFA predicts vibration response by governing differential equations whose main variable is space- and frequency-averaged energy of far-field vibrating waves. Vibrational energy distributions can be provided for structures. FEM was introduced into EFA by Nefske and Sung when they solved the governing equation of the vibrating Euler-Bernoulli beam [21]. The energy response of the rod was studied with EFA by Wohlever and Bernhard [22]. Bouthier et al. derived the energy governing equation for the Kirchhoff plate. The response is obtained by solving the governing differential equation with FEM [23,24]. Cho derived coupling relationships of beams and plates with power transmission and reflection coefficients for built-up structures [25]. The vibration of membranes was not studied before Bouthier et al. who developed the governing energy equation of the membrane and solved the equation with a numerical method [26]. With the effort of many researchers, EFA has made great progress and been applied in various fields [27,28,29,30,31].

EFA, which requires much less computation cost than FEM and provides more detailed results than SEA, is potentially a preferable approach to the vibration prediction. However, there is another problem to apply it to large-scale membranes. Most works of EFA focus on deterministic structures which means the parameters are known precisely and unchanged. There exist natural uncertainties in practical parameters due to assemblage and manufacturing errors. Uncertainties of vibration responses resulted from uncertainties of parameters will increase with increasing size of structures. For the vibration prediction of large-scale membranes, it is meaningful to take uncertainties into account. There are many methods to deal with uncertainties in structures [32,33,34,35,36]. And one of the most popular methods is using probabilistic distribution. Parameters with uncertainties are described with random variables that satisfy certain probabilistic distributions. Then the vibration responses are obtained in probabilistic form which is thought to be more feasible than deterministic results.

Analytical formulations are usually preferred since they enable to obtain the vibration energy directly without solving differential equations with numerical methods as FEM. And the analytical expressions facilitate the analysis of membranes with uncertainties. However, existing analytical expressions of energy flow only apply to structures of regular shapes since they are usually derived with exact travel distances of waves. In this work, a new analytical formulation of averaged energy for the space large-scale membrane with deterministic parameters is proposed based on the Green kernel of vibrating membranes. The formulation, using mean free path to describe travel distances of energy waves, is applicable to membranes of various shapes. Then the vibration energy formulation for the large-scale membrane with random parameters is developed. A validation is presented with the comparison of the averaged energy response from the proposed formulation and the exact energy response from traditional displacement solution. Considering the length of the membrane as an uncertain parameter, results from both random and deterministic analysis are provided to illustrate the characteristics of random energy responses. The derived analytical formulation of energy response provides more reasonable results for the space large-scale membrane with random parameters and is an instructive representation for membranes with uncertainties of other types. Besides, the work gives a reference for research on the vibration of other large-scale structures from the energy point of view.

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]

$$T\left(1+j\eta \right){\nabla }^{2}u+\rho {\omega }^{2}u=0$$

(1)

where $T$ is the tension in the membrane, $u=U{e}^{j\omega t}$ 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

$$\frac{1}{r}\frac{d}{dr}\left(r\frac{dU}{dr}\right)+k^{2}U=-\frac{F}{T}\delta \left(r\right)$$

(2)

where $F$ indicates the magnitude of the exciting force, and $k$ is a complex wave number

$$k^2=\frac{\rho {\omega }^2}{T\left(1+j\eta \right)}$$

(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

$$U=\frac{j}{4T}{H}_0^{\left(2\right)}\left(kr\right)$$

(4)

Then the displacement solution for membranes under an exciting force with magnitude $F$ is

$$U=\frac{jF}{4T}{H}_0^{\left(2\right)}\left(kr\right)$$

(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

$$U=\frac{F}{4T}\left(1+j\right)\sqrt{\frac{2}{\pi kr}}{e}^{-jkr}$$

(6)

The averaged kinetic energy can be derived in terms of displacement [25]

$$E_k=\frac{\rho }{2}{\omega }^{2}U{U}^{*}$$

(7)

where the star * indicates the conjunction. And the averaged potential energy is

$$E_p=\frac{T}{2}\left(\frac{{\partial }^{2}U}{\partial r^2}+\frac{1}{r}\frac{\partial U}{\partial r}\right){\left(\frac{{\partial }^{2}U}{\partial r^2}+\frac{1}{r}\frac{\partial U}{\partial r}\right)}^{*}$$

(8)

Substituting Equation (6), into Equations (7) and (8), the averaged potential and kinetic energy are written as

$$E_k=\frac{\rho {\omega }^2{F}^2}{32T^2}\frac{1}{\pi k_{1}r}{e}^{2k_{2}r}$$

(9)

$$E_p=\frac{T{F}^2{k}_1^3}{32\pi r}{e}^{2k_{2}r}$$

(10)

with

$$k=k_1+j{k}_2$$

(11)

The total vibrational energy of the membrane is the sum of the potential and kinetic energy and can be obtained as

$$E=E_k+E_p=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}{e}^{2k_{2}r}$$

(12)

by noticing

$$k_1=\sqrt{\frac{\rho {\omega }^2}{T}}$$

(13)

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.

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

$$n_x=\frac{c\cos \theta }{a}$$

(14)

and the number of reflections between two boundaries $y=0$ and $y=b$ is

$$n_y=\frac{c\sin \theta }{b}$$

(15)

Then the reflection number of an energy ray in 1 s is

$$n_r=n_x+n_y=c\left(\frac{\cos \theta }{a}+\frac{\sin \theta }{b}\right)$$

(16)

Since the energy rays travel with equal probability in every direction, the probability in $d\theta$ is

$$p=\frac{d\theta }{2\pi }$$

(17)

Assuming $2\pi n$ energy rays are emitted by the excitation in 1 s, then the reflection number of total energy rays is

$$N=4\underset{0}{\overset{\frac{\pi }{2}}{{\displaystyle \int }}}nc\left(-\frac{\cos \theta }{a}+\frac{\sin \theta }{b}\right)d\theta =4nc\left(\frac{1}{a}+\frac{1}{b}\right)=\frac{2nc{L}_c}{S}$$

(18)

where $n$ is the ray number in per unit angle, $L_c=2\left(a+b\right)$ 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

$$L_m=\frac{L}{N}=2\pi c\frac{S}{2nc{L}_c}=\frac{\pi S}{L_c}$$

(19)

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

$$E=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}{\displaystyle \sum }_{i=0}^N{e}^{2k_2\left(r+i{L}_m\right)}$$

(20)

where $i$ indicates the ith reflection

The averaged energy response for the membrane of finite size is obtained when the number of reflections approaches infinity.

$$E=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{1}{1-e^{2k_2{L}_m}}{e}^{2k_{2}r}$$

(21)

Then we have the energy kernel for finite membranes

$$G_e=\frac{\rho {\omega }^2}{16\pi k_{1}r{T}^2}\frac{1}{1-e^{2k_2{L}_m}}{e}^{2k_{2}r}$$

(22)

For a vibrating membrane with random parameters, the energy representation can be written as

$$\tilde{E}=\frac{\tilde{\rho }{\omega }^2{F}^2}{16\pi k_1\tilde{r}{T}^2}\frac{1}{1-e^{2k_2{\tilde{L}}_m}}{e}^{2k_2\tilde{r}}$$

(23)

where $\tilde{x}$ indicates an independent random parameter whose mathematic expectation is $x$ and the standard deviation is ${\sigma }_x$.

The expectation of the random energy is

$$\overline{E}=E+EXP\left({\displaystyle \sum }_{x=\rho ,L_m,r}\frac{\partial E}{\partial x}\delta x\right)$$

(24)

in which $E$ is the energy response from deterministic parameters,

$$\delta x=\tilde{x}-x$$

(25)

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

$${\sigma }_E^2={\displaystyle \sum }_{a=\rho ,L_m,r}{\left(\frac{\partial E}{\partial a}\delta a\right)}^2$$

(26)

$${\sigma }_E=\sqrt{{\displaystyle \sum }_{a=\rho ,L_m,r}{\left(\frac{\partial E}{\partial a}\delta a\right)}^2}$$

(27)

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

$$\tilde{E}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{1}{1-e^{2k_2{\tilde{L}}_m}}{e}^{2k_{2}r}$$

(28)

The mathematic expectation and the variance of the random mean free path ${\tilde{L}}_m$ are

$$EXP\left({\tilde{L}}_m\right)=L_m$$

(29)

$$VAR\left({\tilde{L}}_m\right)={\sigma }_{L_m}^2$$

(30)

Using Taylor’s series expansion at $L_m$ for Equation (28), the first two terms are retained to obtain the approximate formulation of the energy response. Then formulation becomes

$$\tilde{E}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{1}{1-e^{2k_2{L}_m}}{e}^{2k_{2}r}\left(1+\frac{e^{2k_2{L}_m}}{1-e^{2k_2{L}_m}}\left({\tilde{L}}_m-L_m\right)\right)$$

(31)

Assuming the mean free path as a Gaussian random parameter, the expectation and the standard deviation of the energy response can be written as

$$\overline{E}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{1}{1-e^{2k_2{L}_m}}{e}^{2k_{2}r}$$

(32)

$${\sigma }_E=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{1}{{\left(1-e^{2k_2{L}_m}\right)}^2}{e}^{2k_{2}r}$$

(33)

Within $n$ standard deviations, the resultant energy response interval is

$$E^I=\left[\overline{E}-n{\sigma }_E,\overline{E}+n{\sigma }_E\right]$$

(34)

in which at least $1-1/n^2$ of the values of response are covered. The maximum and minimum response in the interval are

$$E_{max}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{e^{2k_{2}r}}{1-e^{2k_2{L}_m}}\left(1+\frac{n{\sigma }_{L_m}{e}^{2k_{2}r}}{1-e^{2k_2{L}_m}}\right)$$

(35)

$$E_{max}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\frac{e^{2k_{2}r}}{1-e^{2k_2{L}_m}}\left(1-\frac{n{\sigma }_{L_m}{e}^{2k_{2}r}}{1-e^{2k_2{L}_m}}\right)$$

(36)

For large-scale membranes, we have $L_m\gg 1$. Then the results can be approximated by

$$E_{max}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\left(1+n{\sigma }_{L_m}{e}^{2k_{2}r}\right)e^{2k_{2}r}$$

(37)

$$E_{min}=\frac{\rho {\omega }^2{F}^2}{16\pi k_{1}r{T}^2}\left(1-n{\sigma }_{L_m}{e}^{2k_{2}r}\right)e^{2k_{2}r}$$

(38)

3. Results

For the purpose of method validation, a large-scale tensioned membrane in space is chosen as the model. The edges of the membrane are connected to a support frame with tension $T=10 \mathrm{N}$ (see Figure 2). It is assumed that the membrane structure is excited by a transverse harmonic point force at a corner and the membrane is made of polyimide film with length $a=100 \mathrm{m}$, width $b=100 \mathrm{m}$, thickness $h=0.025 \mathrm{mm}$, elastic modulus $E=3 \mathrm{GPa}$, material damping $\eta =0.005$, and mass density $\rho =1400 \mathrm{kg}/{\mathrm{m}}^3$.

The averaged energy response from the proposed formulation and the exact energy response from traditional displacement solution at a point $\left(x=20 \mathrm{m},y=20 \mathrm{m}\right)$ are provided in frequency field in Figure 3. And the energy distributions in space at $\omega =50 \mathrm{rad}/\mathrm{s}$ from the two methods are shown in Figure 4 and Figure 5. It can be observed that the exact solution is of wave form. And the averaged solution is smoothed. The proposed solution represents well the global variation of the exact solution in frequency field which is the result of the averaging process to reduce computation cost. The difference between the two solution decreases with increasing frequency. However, the energy distributions from the two methods act in a similar way in space. The energy response decreases smoothly with increasing distance from the excitation.

Figure 6 shows the energy response intervals in frequencies at the point by assuming the length $a$ as a Gaussian random variable and its relative standard deviation $\sigma =0.05, 0.1, 0.15$. The relative standard deviations of the energy responses are shown Figure 7. It is observed that, due to the random effect of the length, the energy responses are no longer deterministic. Potential energy responses form confidence intervals and the practical responses can be any value in the corresponding interval. However, the mean responses are equal to the results from deterministic parameters without deviations. The width of intervals and the relative standard deviation of energy responses decrease with increasing excitation frequency and decreasing standard deviation which indicates that the uncertainty of responses becomes more significant for lower frequencies and greater uncertainty of structures.

Figure 8 shows the standard deviation of the energy responses in space at $\omega =50 \mathrm{rad}/\mathrm{s}$ with the relative deviation of the length is 0.05. It can be observed that the standard deviation decreases with increasing distance from the excitation. However, Figure 9 shows the relative standard deviation remains unchanged in space. It is indicated that, although the absolute uncertainty of the energy response changes at different locations, the relative uncertainty stays at a steady level at a fixed frequency regardless of the coordination.

4. Discussion

Vibration prediction of large-scale membrane structures demands more feasible methods considering computation cost and structural uncertainty. As a widely applied method, EFA is potentially a reasonable method. Existing analytical formulations of energy flow derived from deterministic representations only apply to structures of regular shapes. FEM can be introduced in solving energy governing equations to extend the applications. However, numerical expressions raise the bar in dealing with structures with uncertainties. And extra computation is brought in since a set of equations needs to be solved to find even only one single response value. In this work, a general method was proposed to predict the energy responses for space large-scale membranes. Based on the Green kernel of vibrating membranes, a new formulation of averaged energy response for membranes was obtained with the summation of reflected waves. With the proposed method, formulations of energy response were derived for membranes with random parameters. Simulations were performed for a membrane with the length being a Gaussian random variable to validate the formulations. The developed formulation is of analytical form with which uncertainties can be dealt with directly and demands much less computation than numerical methods. Besides, the formulations can be applied to various shapes of membranes since it describes travel distances of energy waves with the mean free path derived from the statistical point of view. This work provides a systematic way to analyze energy responses of two-dimensional large-scale structures of various shapes with structural uncertainties and promotes the analysis of vibration in energy point of view. The extension of the method to large-scale structures of other types is the subject of future research.