# Vibration Prediction of Space Large-Scale Membranes Using Energy Flow Analysis
## 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). [PERSON] and [PERSON] presented a finite-element modelling technique to simulate the air-membrane interaction and obtained the results with ABAQUS [11]. To simulate the ground tests, [PERSON] et al. developed finite element models for a 10-m four-quadrant solar sail system and a 20-m four-quadrant solar sailsystem. The results from analysis showed reasonable agreement with the test data [12]. [PERSON] et al. obtained mode shapes and natural frequencies of a membrane antenna under varying pre-tension loads with ABAQUS [13]. [PERSON] 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 [PERSON] and [PERSON] when they solved the governing equation of the vibrating Euler-Bernoulli beam [21]. The energy response of the rod was studied with EFA by [PERSON] and [PERSON] [22]. [PERSON] 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]. [PERSON] 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 [PERSON] 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.