## 3 Natural Frequency Analysis of SFF
The physical parameters utilized in these two simulations follow the values in Table 1. In the first simulation, the flexible link rotates around the axis with a constant angular velocity of 1 rad/s during the period of 0-1.1 s and starts in a free vibration from 1.1 s. The fundamental frequency from the simulation is obtained as \(\omega_{1}^{\rm c}=2\pi/1.1562=5.4343\) rad/s, while the frequency from Equation (39) is \(\omega_{1}=5.4581\) rad/s--very close to the result of the first simulation. In the second simulation, the fundamental frequency of the SFF system is obtained from the simulation as \(\omega_{1}^{\rm s}=2\pi/1.21=5.19\) rad/s. The corresponding frequency obtained in the paper is 5. 1486 rad/s, with the relative error of 0.8%. For comparison, the displacements of the top mass in these two simulations are depicted in Figure 6, where the solid line denotes the displacement of SFF system and the dashed line denotes that of the cantilever beam with tip mass. Next, the sensitivity analysis of each parameter can be implemented to discuss the effect on the natural frequency. Four main variables are selected in a feasible range to study how the natural frequency changes. As depicted in Figure 7, the mass of the payload plays a highly important role in the fundamental natural frequency (denoted as diamond marks) of an SFF system. Furthermore, the fundamental natural frequency is less sensitive to the moment inertia of the motor. Regarding the second frequency (denoted as square marks), it mainly depends on the mass of the payload. These two conclusions are in agreement with the results reported in [10]. Additionally, the value of the third frequency is affected by the moment inertia of the tip payload, which is not discussed in [10]. Moreover, the fundamental and the second frequencies are also related to the stiffness of the flexible joint in the system. Although the numerical method can be applied to sensitivity analysis on natural frequency of the SFF, an analytical expression of natural frequency, together with the physical parameters of the SFF system, is more practical for sensitivity analysis. However, the analytical expression of natural frequency is deduced as an involved formula, which is omitted in the paper. Figure 6: Comparison of tip payload displacements from two simulation models. After determining the value of natural frequencies, the matrix \(\mathbf{\Pi}\) can be obtained by applying the frequencies. Then, the modal shape is able to be determined by employing the eigenvector of the matrix \(\mathbf{\Pi}\) with the corresponding eigenvalue equal to zero. Therefore,  
Figure 7: Sensitivity of parameters on natural frequencies of the SFF system: (**a**) effect of both the payload mass and motor inertia; (**b**) effect of both the payload mass and payload inertia; (**c**) effect of the mass of the tip payload; (**d**) effect of motor inertia; (**e**) effect of the moment inertia of the tip payload; (**f**) effect of the stiffness of the flexible joint. the exact dynamic equation of the points on the links is developed based on the modads of the SFF system. Using the frequencies in Table 2, the modal shapes (as illustrated in Figure 8) of the flexible link are denoted as  
\[ \begin{cases}\Psi_{1}(s)=27.1\sin(\beta_{1}s)-22.24\cos(\beta_{1}s)-25.92\sin \mathrm{h}(\beta_{1}s)+22.33\cos\mathrm{h}(\beta_{1}s)+s,\\ \Psi_{2}(s)=-0.0296\sin(\beta_{2}s)+0.63\cos(\beta_{2}s)+0.658\sin\mathrm{h}( \beta_{2}s)-0.63\cos\mathrm{h}(\beta_{2}s)+s,\\ \Psi_{3}(s)=-32.6\sin(\beta_{3}s)+31.82\cos(\beta_{3}s)+32.88\sin\mathrm{h}( \beta_{3}s)-31.82\cos\mathrm{h}(\beta_{3}s)+s,\\ \Psi_{4}(s)=-74.83\sin(\beta_{4}s)+75.74\cos(\beta_{4}s)+75.01\sin\mathrm{h}( \beta_{4}s)-75.84\cos\mathrm{h}(\beta_{4}s)+s,\\ \Psi_{5}(s)=-169.1\sin(\beta_{5}s)+169.1\cos(\beta_{5}s)+169.1\sin\mathrm{h}( \beta_{5}s)-169.1\cos\mathrm{h}(\beta_{5}s)+s,\end{cases}  \tag{33}\]  
where \(\beta_{1}=0.85,\ \beta_{2}=1.5917,\beta_{3}=3.55,\beta_{4}=5.508,\beta_{5}=8.061,\ \mathrm{and}\ s \in[0,L]\).