# Dynamics Analysis of Rotating Cantilever Beams with Free End Mass
## 4 Frequency of Free Vibration
As shown in Figure 6a, the first order frequencies of chordwise vibration from the proposed and previous models are almost overlapped. The frequencies increase with the increasing dimensionless rotating angular speed \(\gamma\). However, the frequency of the model without stiffening decreases with the increasing dimensionless speed \(\gamma\). The phenomenon in which the frequencies increase with the angular speed is called the dynamic stiffening effect. The proposed model is an effective model for exhibiting the dynamic stiffening effect as the same as the previous model in chordwise vibration. Generally, the influence of rotating motion on chordwise motion is the primary focus of different modeling methods, but it is still necessary to investigate the influence on the axial motion. A pretty different result is exhibited in Figure 6b; that is, the axial vibration frequency of the proposed model increases with angular speed \(\gamma\), but the other two models decrease. The previous model shows the same rule of the axial vibration frequency as the model without stiffening. The difference means that the proposed model is dynamic stiffening in axial vibration, but the other models are dynamic softening. From the underlined terms in Equations (16) and (17), the reason is that the influence of equilibrium axial deformation is considered in axial motion in the proposed model, but those are not the consideration in the other models. Figure 7 shows the variation in different free end mass ratios on the vibration frequencies when \(\delta=0.1\), \(\alpha=70\), and the variable parameter mass ratio \(\varphi\)= 0, 0.3, and 0.5, respectively. It can be observed in Figure 7a,b that the attached lumped mass makes the fundamental frequencies of the no-rotation cantilever beam decrease both in chordwise and axial vibration. As the dimensionless rotating angular speed increases, the first three frequencies increase. Different-order frequencies have different sensitivities to free end mass and angular speed. As shown in Figure 8, it is the magnification of Figure 7. The first order frequencies with free end mass are always smaller than those without free end mass. A bigger free end mass leads to a smaller first frequency at the same speed, consistent with the response results observed in Figure 5. However, it shows differences in high-order frequencies. As shown in Figure 8a, the second-order frequencies with free end mass are smaller than those without free end mass in low angular speed range, but those with free end mass will transcend the frequencies without free end mass as the dimensionless angular speed continues increasing. The value of dimensionless angular speed where the transcendence occurs can be called the overtaking speed. In the comparison of the same order frequency, it is observed that a bigger free end mass corresponds to a smaller  
Figure 6: The first-order dimensionless frequency of rotating cantilever beams with free end mass: (**a**) Chordwise frequency; (**b**) Axial frequency. overtaking speed. Meanwhile, the overtaking speed of high-order frequencies is greater than the lower orders under the same free end mass. The same rules occur in axial motion, as presented in Figure 8b. In the previous model shown in Figure 9, although the chordwise frequencies show the same rule as the proposed model, the frequencies of axial motion exhibit differences. The high-order frequencies with free end mass are always smaller than those without free end mass. The performance of chordwise and axial motions for different free mass ratios is consistent in the proposed model, but it is not consistent in the previous model. The slope of each order of dimensionless frequency curve increases with the value of the free end mass ratio. The free end mass has more significant influence on high-order frequencies than on low-order frequencies. The increasing free end mass can strengthen the dynamic stiffening effect of the high-order frequencies. Figure 10 is the corresponding variation of free end axial deformation to beam length.