# Dynamics Analysis of Rotating Cantilever Beams with Free End Mass
## 3 Dynamics Simulations  
In order to validate the proposed model, the dynamic responses of the rotating cantilever beam with free end mass are simulated in this section, and comparisons of the results are presented with other published works. The example is selected as cases considered in Refs. [31; 34]. The parameters of the system are listed as follows. The beam length is \(L=8\) m, the cross-section area is \(A=7.2968\times 10^{-5}\) m\({}^{2}\), the area rotary inertia is \(l=8.2189\times 10^{-9}\) m\({}^{4}\), the mass density is \(\rho=2.7667\times 10^{3}\) kg/m\({}^{3}\), and the modulus of elasticity is \(E=6.8952\times 10^{10}\) Pa. The lumped mass attached at the free end of the cantilever beam has the parameter \(m=0.085\) kg. The law of angular speed of the rotating cantilever beam system described in Refs. [31; 34] is:  
\[\Omega=\left\{ \begin{array}{ll}\frac{\Omega_{0}}{T}t-\frac{\Omega_{0}}{2\pi }\sin\Bigl{(}\frac{2\pi}{T}t\Bigr{)},&0\leq t\leq T\\ \Omega_{0},&t\geq T\end{array} \right. \tag{40}\]  
in which \(T=15\) s, and the simulation time is set as 20 s. The different values \(\Omega_{0}\) for the simulations are selected to be 0.6, 2, and 4 rad/s. When \(\Omega_{0}\) is set to be 0.6 rad/s, the response of the beam's free end when using the proposed model, the previous model from [PERSON] et al. [31], and the model without stiffening effect are presented in Figure 2. As shown in Figure 2a, since the rotating angular speed is low in this case, all three models can calculate the dynamic responses. The results of the response time history curves are similar, curves in the proposed model and the previous model fit well, and the model without a stiffening effect already exhibits a slight difference. Figure 2b is a larger image of Figure 2a for the time \(15\) s \(\leq t\leq 20\) s. It is observed that the vibration responses of the proposed model and the previous model are almost coincident, but the vibration of the model without the stiffening effect has a larger cycle. Considering the case when \(\Omega_{0}=2\) rad/s in Figure 3. The response time history curves in the first two models are still similar and stable. It only shows a slight difference in the uniform rotating stage. However, the response at the free end, which is calculated by the model without stiffening effect, shows a considerable difference compared to the others. The model without stiffening has a larger free end response in the acceleration stage, and the calculation result gives an even larger amplitude and vibration cycle at the uniform rotating stage. A larger vibration cycle means a smaller frequency. In the model without stiffening, the dynamic softening phenomenon occurs. A larger angular speed case is simulated where \(\Omega_{0}=4\) rad/s. It can be observed in Figure 4 that the result of the model without the stiffening effect is divergence. The model with the stiffening effect is unable to obtain an effective result. The proposed model in this study can still simulate the dynamic response of a rotating beam with free end mass at high speed condition as the previous model does. It proves that although no additional stiffening  
Figure 3: The response of the beam’s free end when \(\Omega_{0}=2\) rad/s: (**a**) Full time period; (**b**) Magnification of the uniform rotating stage. Figure 2: The response of the beam’s free end when \(\Omega_{0}=0.6\) rad/s: (**a**) Full time period; (**b**) Magnification of the uniform rotating stage. term is introduced in the process of deduction, the proposed model is an effective dynamic stiffening model by accounting for real-time deformation coordinates. The underlined terms in Equations (26) and (27) can provide the dynamic stiffening effect. Meanwhile, the proposed model exhibits larger vibration amplitudes in the uniform rotating stage compared to the previous model. It should be noted that the stiffness terms in the previous model are constant (time-independent) at this stage, but they are time-variant terms in the proposed model.