## 1 Introduction  
A parallel mechanism (PM) is a closed-loop mechanism with a moving platform and a fixed base connected by at least two limbs. PMs have the advantage of higher stiffness, higher bearing capacity, and better dynamic response compared to serial mechanisms [1]. PMs have been used as machine tools [2], legged robots [3], pick-and-place applications [4], etc. In many applications, for example, machining processes, the end effector of the PM suffers high frequency vibration and a large load [5]. In other cases, the link of the PM is moving at a fast speed, so the elastic deformation of the mechanism needs to be considered to compensate for the error [6]. Therefore, the elastodynamic modeling of the PM is important to evaluate and optimize its performance. The dynamics of PMs can be classified into three classes. The first class is rigid body dynamics. When the links and joints of the PM are rigid enough under external force, all parts of the mechanism can be considered rigid bodies. Screw theory [7, 8] and the Newton-Euler method [9] have been widely used to establish rigid body dynamics. In the second class, the PM is moving at a low speed while subjected to external vibration force, for example, friction stir welding [10]. In this case, the rigid body inertia force is relatively low, whereas the structure vibration has a great impact on the performance. Usually, the mechanism in the workspace is considered a varying structure with vibration force on the end effector. Matrix structural analysis (MSA) [11] and the virtual joint method (VJM) [12, 13] have been developed to construct the model. In the third class, the PM goes through both large rigid body motion and large deformation, and neither of their impacts can be ignored, for example, the Delta robot in the fast pick-and-place process. The kineto-elastodynamics (KED) [14, 15] method has been proposed to construct the model in the last century. However, KED neglects the coupling terms of rigid body movement and flexible deformation. The floating frame of reference formulation (FFRF) [16] and the absolute nodal coordinates formulation (ANCF) [17] were later proposed to handle this problem. However, the process of modeling is extremely complicated and has a higher computational cost compared to the first two classes. In our work, we focused on the second class, the structural elastodynamics of the PM. There are two main methods for the structural elastodynamics of PMs--the VJM and the MSA. With finite element analysis (FEA) of separate links, the VJM can obtain the numerical stiffness matrix of links with complex shapes [18]. Then, the overall stiffness matrix is derived using screw theory [19]. However, the force analysis in this method becomes very complicated when dealing with multiloop PMs [20]. The MSA is based on computational structural mechanics and can uniformly acquire the overall stiffness and mass matrix by summing all the energy together. Although traditional MSA simplifies all links into beam elements with the analytic element matrix, it can also combine the FEA result to obtain the numerical stiffness and mass matrix of complex shape links [21]. The MSA was first introduced by [PERSON] [22]. Different kinds of kinematic joints between links are handled using Lagrange multipliers. Later, strain energy minimization was proposed by [PERSON] [23; 24] to eliminate the Lagrange multipliers, and only the differential vibration equation is presented in the final result. With this method, all kinds of kinematic joints, including prismatic joints, rotational joints, universal joints, etc., can be well assembled into the stiffness and mass matrices. However, the long prismatic joints composed of outer and inner tubes have not been modeled. Tube structure has been widely used for pneumatic cylinders or screwballs in PMs. Many researchers tie the outer and inner tubes together as a single beam [25] or only add a single prismatic constraint on the tube [26]. Therefore, the existing methods are still not accurate enough for the tube structure.