## 1 Introduction
[PERSON] in [24] developed four mathematical models to discuss the stiffening effect of rotating flexible links. These models are denoted as the Consistent Model (CM), Potential energy Model (PM), Kinetic energy Model (KM), as well as the Kinetic and Potential Model (KPM). A conclusion is obtained that CM, without considering the stiffening effect, leads to static instability at a critical angular velocity, while other three models are highly improved by considering the stiffening effect. Among PM, KM, and KPM, the effect of the rotating speed on the vibration behavior of all modes was discussed according to a series of simulation results. The main target of this paper is to develop the dynamic modeling of a Single Flexible-link Flexible-joint (SFF) manipulator by introducing the stiffening effect in the potential energy model, which was neglected in [10; 12]. Except for the link flexibility, there is another type of flexibility related to rotational joints which is also important in mechanical systems. A single flexible-link, flexible-joint manipulator investigated in the paper is illustrated in Figure 2. A rotational actuator connects the hub through a flexible joint, and a flexible beam is clamped at the hub with a tip payload attached at the distal end of the beam. As the counterpart of the SFF mechanism, a single flexible-link, rigid-joint mechanism and a single rigid-link, flexible-joint mechanism had also been utilized when performing the mode analysis [10], which showed several differences among these three mechanisms. In order to study the influence of the rotational  
Figure 1: Beam foreshortening effect. stiffness of the flexible joint on the vibration features, a sensitivity index is introduced from the closed-form characteristic equation of the flexible-link system. The results confirmed that the joint flexibility of the rotational actuator has a significant effect on the system frequencies [9]. In the process of modeling slender links, the position of the curvilinear points on flexible links is depicted in the relative coordinate systems. For the convenience of kinematic analysis, two typical coordinate systems illustrated in Figure 3 are defined as Virtual Link Coordinate System (VLCS) [28] and Tangential Coordinate System (TCS). Coordinate systems for flexible systems are also mentioned in [8] as pseudo-clamped, pseudo-pinned, and pseudo-pinned-pinned systems, where the pseudo-clamped reference frame is the same as TCS and pseudo-pinned-pinned to VLCS. As shown in Figure 3, the largest transversal deflection occurs at one end of a slender link for TCS, while it is represented near the middle point of the link for VLCS. Additionally, in VLCS, the deformation of the flexible link at both ends are assigned as zero. In comparison, in TCS, it is clearer to describe the transversal deformation of a slender link along with several boundary constraints. Therefore, TCS is employed in the present paper to facilitate the dynamic modeling of SFF systems in the following sections. In this paper, the relative coordinate system used for modeling the flexible link is defined as TCS. A rotational motor exerts a torque \(\tau\) to make the hub rotate. A flexible joint connects the motor and a hub as a pure torsional spring. A payload is exactly attached to the distal end of the flexible link without any deviation for simplicity. The proximal end of the flexible link is clamped by the hub. The governing equation of motion as PDE is  
Figure 3: Illustration of two coordinate systems: (**a**) tangential Coordinate System; (**b**) virtual Link Coordinate System. Figure 2: Sketch a single flexible-link, flexible-joint mechanism. deduced by the Hamilton principle in Section 2. The stiffening effect is considered in the dynamic modeling. Natural frequencies and mode shapes of an SFF system are discussed based on the dynamic modeling in Section 3. The stiffening effect of the flexible link system with constant rotational velocity is investigated in Section 4. Conclusions and remarks are included in Section 5.