## 2 Model Design and Details  
For a kind of PDSR utilized in a cubic microsatellite, the main components are the central hub, petals, supporting bars, linkage butts, and drag springs as shown in Figure 1[13]. The parabolic surface is divided into a central hub and several identical parabolic petals around the central hub. These divided segments aim to maintain the parabolic profile as a reflector. Each petal is connected to the central hub by a supporting bar with two revolute joints on the two ends. The joint between the central hub and supporting bar is set along the circumferential direction of hub, while the joint between the petal and supporting bar is set along the radial direction of petal. These two single-direction joints are orthogonal to fulfill the petal deployment mechanism with the specific rotation angle combinations. The torsion spring is installed in each single-direction pin to provide the driving torque for petal deployment. The rotation limiter in each joint is designed to make a restriction of rotation angle and passively lock the structure in deployed state. The design parameters of reflector surface are defined in Cartesian coordinate system O-XYZ shown in Figure 1c. The vertex of the central hub is located at the coordinate origin O. A kind of 1200 mm aperture PDSR is employed here as a specific object to study its structural dynamic properties for the future application to microsatellites. The focal length of antenna is \(f=480\) mm and the focus-diameter-ratio is defined as \(\lambda=0.4\). The parabolic reflector surface equation is expressed as \(x^{2}+y^{2}=1920\ z\). The curved Surface CDFE shown in Figure 1 is the main reference petal that is symmetrical along the Plane XOZ. The projection radius of the central hub on the Plane XOY is r, while the projection radius of the reference petal on the Plane XOY is \([r,R]\), where \(r=200\) mm, \(R=600\) mm. The pins of two orthogonal revolute joints are, respectively, defined as pin A and B. The rotation angles are defined as \(\omega=-90^{\circ}\) and \(\theta=60^{\circ}\) along pin A and B, respectively. The horizontal distances from point A and point B to the vertex of central hub are defined as \(m=150\) mm, \(l=300\) mm. The vertical distances from pin A and pin B to the parabolic surface are both defined as \(a=b=15\) mm, respectively. Based on the models in Reference [13], two linkage butts are installed on the top of each petal as shown in Figure 1(a). For each of the two adjacent petals, the adjacent linkage butts are linked by the drag spring. The drag spring is tensioned with storing the elastic potential energy when the structure is folded. As soon as the folded structure is unlocked to deploy, the drag spring quickly shortens to drag the adjacent linkage butts together. Finally, in totally deployed state, the drag spring still has the tension stress to keep petals stably locked. The deploying processes of the two models were conducted by ADAMS software simulations in Reference [13]. In view of its simulation results, it was indicated that the drag springs set on linkage butts can improve the integrality and stability of antenna during the deploying. Nevertheless, the function of drag springs was re-examined to deeply explore the structural dynamic properties of this PDSR in deployed state. Figure 1: Schematic diagram of Petal-type Deployable Solid-surface Reflector (PDSR) model. The weakest parts of the deployed reflector should be the revolute joints in the structural dynamic analysis. The stiffness in each revolute joint of the PDSR is mainly provided by the torsion spring and the rotation limiter. The details of typical revolute joint are shown in Figure 2b [27]. The joint is vertically folded with the twisted torsion spring retaining elastic potential energy and then rotates counterclockwise around the pin axis during deploying. After rotating to horizontal configuration, the joint is restricted by the rotation limiter from rotating.