## 1 Introduction  
Modern spacecraft are usually equipped with large flexible appendages, such as deployable antennas, robot arms, and solar panels, which may cause the spacecraft to have extremely high flexibility and low-frequency vibration modes [1]. The solar flux shock is one of the most important excitations that induce those low-frequency vibration modes which usually occur during the diurnal or diurnal shifts in orbit. The sudden changes in the thermal load can cause temperature gradients in the appendage and further lead to thermal stress, deformation, and vibrations [2, 3]. In addition, vibration can also interact with the attitude motion and attitude maneuver of spacecraft [4], significantly reducing the accuracy of the spacecraft attitude, and even leading to complete failure of the mission [2]. For example, the pointing disruptions were caused by thermal gradients of the Hubble Space Telescope's solar array [2], and the solar arrays caused the attitude disturbances on the Upper Atmosphere Research Satellite [4]. Thus, the research on the dynamical behaviors and effective control strategies for the coupled attitude and vibration motion is essential and indispensable for spacecraft equipped with large flexible appendages [5, 6]. There are many investigations about the modeling methods for rigid spacecraft with flexible appendages. The Hamilton principle was generally used to establish the governing equations for rigid-flexible coupled continuum systems [7, 8]. Most investigations focused on appendages with regular shapes such as a beam, plate, or shell. [PERSON] and [PERSON] [9] derived the coupled control partial differential equation of motion by usingthe Hamilton principle for the spacecraft with flexible sandwich panels. [PERSON] et al. [10] investigated the coupled model of a spacecraft composed of a rigid platform and two flexible solar arrays. The nonlinear dynamic equations, which remain the second-order coupling terms of axial displacement caused by the transverse motion of solar arrays, were obtained by using the Hamilton principle. The dynamic model of a flexible hub-beam-tip mass system is established and researched by [PERSON] et al. [11]. [PERSON] and [PERSON] [12] presented a problem concerning the free vibrations of a set of \(n\) axially loaded stretched Bernoulli-Euler beams connected by elastic layers. A normal-mode solution is applied to the governing partial differential equations to derive a set of coupled ordinary differential equations which are used to determine the natural frequencies and mode shapes. Another type of method to model the rigid-flexible system is based on multi-body dynamics. The absolute node coordinate formula (ANCF) and [PERSON]'s method can accurately describe rigid-flexible systems, especially those with large deformation. [PERSON] et al. [13] used [PERSON]'s method to derive the coupled dynamics equation for a rigid body with multiple flexible bodies. [PERSON] et al. [14] used ANCF to derive the dynamics of a single composite laminated plate. [PERSON] [15] utilized natural coordinates and the ANCF for rigid-flexible multibody systems with flexible beam elements. For flexible appendages with irregular shapes, such as the hoop-truss antennas and inflatable structures, one of the major problems that researchers are faced with is solving the dynamic equation of motion in linear and nonlinear behaviors [16]; the finite element model has advantages in obtaining the frequency and modes analytically [17]. [PERSON] and [PERSON] [18] developed the so-called free formulation (FF) for the construction of displacement-based incompatible finite elements. [PERSON] [19] lifted strict invertibility conditions linking by recasting the free formulation within the framework of a mixed-hybrid functional that allows internal stresses. An efficient thermal-structural coupling finite element method (FEM) formulation for the dynamics of the beam-cable tensegrity structure was developed and validated by [PERSON] [20].