# Dynamic Modeling and Vibration Suppression of a Rotating Flexible Beam with Segmented Active Constrained Layer Damping Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Modeling

#### 2.1. Basic Assumptions

- (1)
- The SACLD beam consists of three layers, a piezoelectric constraining layer, a viscoelastic damping layer, and a base layer, without considering the sliding between the layers;
- (2)
- The shear displacement of the piezoelectric and base layers is neglected; only the shear displacement of the viscoelastic damping layer is considered;
- (3)
- The piezoelectric layer is polarized along the thickness direction;
- (4)
- The longitudinal contraction caused by transversal displacement is considered for each beam;
- (5)
- The beam rotates on a horizontal plane, without considering the effect of gravity;
- (6)
- The transverse displacements of the three layers are considered the same.

#### 2.2. Displacement Description

#### 2.3. Kinetic Energy and Potential Energy

#### 2.4. Equations of the System

## 3. Results and Discussion

#### 3.1. Validations of the Hub–SACLD Beam System

#### 3.2. Dynamic Response Analysis

_{p}= 1, K

_{d}= −0.005. It can be seen that the amplitude of the vibration of the SACLD beam becomes smaller and the vibration suppression of the beam is better when the control gains are applied to the SACLD beam. Therefore, the control for the vibration suppression is pronouncedly effective. Figure 8 shows the transverse displacement of the SACLD rotating beam for different proportional control coefficients K

_{p}, while K

_{d}= 0. It can be seen that the vibration-suppression effect of the SACLD rotating beam is better as the proportional control coefficient K

_{p}increases. The transverse displacement of the SACLD rotating beam decreases the most when K

_{p}= 5. K

_{p}increases further on this basis; the transverse displacement of the structure also decreases, but the reduction is no longer obvious.

_{p}= 1, K

_{d}= −0.005. It can be seen that, applying control to the beam, the vibration amplitude of tip transverse displacement and longitudinal displacement of the beam are significantly reduced. Both the longitudinal displacement and transverse displacement attenuate compared with the case without the control. Figure 10 compares the effect of K

_{d}on the dynamic response of the SACLD rotating beam. It can be seen that the tip displacement of the SACLD rotating beam decreases during the derivative control gain. K

_{d}is set to be −0.01 and −0.005, respectively, while the decreasing trend is faster when the gain is −0.01. Figure 11 compares the transverse and longitudinal displacement of the rotating SACLD beam when the VEM layer thickness is h

_{2}= 0.25 mm and 0.5 mm, respectively. It can be seen that, with the increase of the VEM layer thickness, the tip transverse displacement and tip longitudinal displacement of the SACLD beam are reduced, while the displacement amplitude reduction is more pronounced when the VEM layer thickness is larger. Figure 12 shows the dynamic response of the rotating SACLD beam with the central hub radius R = 0 and 0.025 m, respectively. The results show that the tip transverse displacement and tip longitudinal displacement of the rotating SACLD beam increase after the radius of the central hub increases. And the displacement increase is due to the increase of the hub radius resulting in an increase in centrifugal force during the SACLD beam rotation. The radius of the central hub has less influence on the displacement of the end of the SACLD rotating beam. Figure 13 shows the effect of beam width b on the dynamic response of the rotating SACLD beam. It can be seen that the tip transverse displacement and the tip longitudinal displacement of the SACLD rotating beam increase when the width of the beam decreases. And the width b impacts the flexural rigidity and the tension rigidity of the beam; the decrease of the width makes the beam more flexible. The width of the beam has a more significant effect on the tip displacement of the SACLD rotating beam.

#### 3.3. Vibration Analysis

_{b}, H

_{v}, and H

_{c}are the relative thicknesses of the base layer, VEM layer, and PZT layer, respectively. X

_{k}is the location of the cutout.

_{b}= H

_{c}of 0.001, 0.005, and 0.012 when the angular velocity $\omega $ = 0, K

_{p}= 0, and K

_{d}= 0. According to the study of Tian et al. [55], the notch position of one notch is set as X

_{k}= 0.7. Set the notch positions for two cutouts to X

_{k}

_{1}= 0.3 and X

_{k}

_{2}= 0.7, and for the case of three notches, set the notch positions to X

_{k}

_{1}= 0.3, X

_{k}

_{2}= 0.4, and X

_{k}

_{3}= 0.7 according to Mohammed’s optimal design of the notch positions [52]. It can be seen in Figure 14 that the segmentation method consistently reduces the first three frequencies of the beam when H

_{b}= H

_{c}= 0.001. The reduction effect on the first three frequencies is always greater for three cuts than for two cuts, and one cut has the relatively smallest reduction on the first three frequencies. The segmentation method can always increase the first three damping ratios when H

_{b}= H

_{c}= 0.001. The enhancement effect on the first three damping ratios is always the best with three cuts, followed by two cuts, and the worst with one cut. It can be shown that the SACLD beam has a better damping effect than the ACLD beam when H

_{b}= H

_{c}= 0.001. With the increased number of cuts, the vibration-damping effect of the segmental method is better when H

_{b}= H

_{c}= 0.001. It can be seen from Figure 15 that the segmentation method consistently reduces the first three frequencies when H

_{b}= H

_{c}= 0.005. As the number of cuts increases, the segmentation method is more effective in reducing the first three frequencies of the beam. However, the segmentation method does not always reduce the first three damping ratios, with the base layer’s relative thickness and the PZT layer’s relative thickness becoming thicker. The segmentation method can improve the first-order damping ratio when H

_{b}= H

_{c}= 0.005. The best vibration-suppression effect is achieved at the two cuts. However, as the VEM layer’s relative thickness increases, the damping effect of three cuts changes from better than that of one cut to worse than that of one cut. With the increase of the VEM layer’s relative thickness, the second-order damping ratio is smaller than that of the ACLD beam. At this time, the damping effect of one cut is the best, followed by two cuts, and that of three cuts is the worst. For the third-order damping ratio, with the increase of the VEM layer’s relative thickness, the third-order damping ratio of the SACLD beam is smaller than that of the ACLD beam. It can be seen from Figure 16 that the segmentation method always reduces the first three frequencies when H

_{b}= H

_{c}= 0.012. Applying one cut to the ACLD beam can always increase the first-order damping ratio. However, only when the VEM layer’s relative thickness is small, applying two notches and three notches can improve the first-order damping ratio. In this case, the segmentation method cannot increase the second-order and third-order damping ratios. A comparative analysis of Figure 14, Figure 15 and Figure 16 shows that the segmentation method can always reduce the first three frequencies when H

_{b}= H

_{c}= 0.001–0.012. However, the vibration-suppression effect of the segmentation method gradually becomes smaller as the relative thickness of the base layer and the PZT layer increases. It can be shown that, with the increase of the base layer’s relative thickness and the PZT layer’s thickness, the shear-strain level within the VEM layer increases, and the segmentation method changes from effective to ineffective.

_{b}= H

_{c}= 0.012, H

_{v}= 0.2. It can be seen that the first three frequencies increase gradually with the increase of angular velocity. The segmentation method can reduce the first three natural frequencies. Only by applying one cut to the beam can the first damping ratio be increased. In this case, applying two and three notches to the beam reduces the first damping ratio. For the second-order and third-order damping ratios, the damping effect of three notches is the worst, followed by two notches, and the damping effect of one notch is relatively the best.

_{b}= H

_{c}= 0.012 and H

_{v}= 0.2. It can be seen that the first three natural frequencies and damping ratios increase with the increase of $\eta $. While the natural frequencies and the damping ratios decrease with the increase of the cuts number, and the effect is obvious to the first and the third natural frequencies and damping ratio while the cut number is one and three, while the cut number is two, the vibration suppression is obvious to the second natural frequency and damping ratio.

_{p}of the PD control law with angular velocity in a closed-loop case when n = 1 and X

_{k}= 0.7. It can be seen that the first natural frequency of the rotating SACLD beam decreases, and the first damping ratio increases as K

_{p}increases, which leads to the conclusion that the vibration-suppression effect of the rotating SACLD beam is better with K

_{p}.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Matrices in Equation (30)

## Appendix B. The Matrices in Equation (33)

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**Figure 1.**Schematic diagram of the rotating flexible beam with SACLD treatment attached to a rigid hub.

**Figure 6.**Tip transverse displacement of the ACLD–SACLD rotating beam in an open-loop case. (

**a**) 0 < t < 3.0 s; (

**b**) 2.00 s < t < 2.10 s.

**Figure 8.**Transverse displacement of the rotating SACLD beam with different K

_{p}. (

**a**) 0 < t < 3.0 s; (

**b**) 2.00 s < t < 2.10 s.

**Figure 9.**Effect of control on the dynamic response of the rotating SACLD beams: (

**a**) tip transverse displacement of the beam; (

**b**) tip longitudinal displacement of the beam.

**Figure 10.**Effect of different control gains on the dynamic response of the rotating SACLD beams: (

**a**) tip transverse displacement of the beam; (

**b**) tip longitudinal displacement of the beam.

**Figure 11.**Effect of VEM layer thickness h

_{2}on the dynamic response of the rotating SACLD beams: (

**a**) tip transverse displacement of the beam; (

**b**) tip longitudinal displacement of the beam.

**Figure 12.**Effect of central rigid body radius R on the dynamic response of the rotating SACLD beams: (

**a**) tip transverse displacement of the beam; (

**b**) tip longitudinal displacement of the beam.

**Figure 13.**Effect of beam width b on the dynamic response of the rotating SACLD beams: (

**a**) tip transverse displacement of the beam; (

**b**) tip longitudinal displacement of the beam.

**Figure 14.**Variations of the first three natural frequencies and damping ratios with the VEM layer’s relative thickness when H

_{b}= 0.001.

**Figure 15.**Variations of the first three natural frequencies and damping ratios with the VEM layer’s relative thickness when H

_{b}= 0.005.

**Figure 16.**Variations of the first three natural frequencies and damping ratios with the VEM layer’s relative thickness when H

_{b}= 0.012.

**Table 1.**Comparison between the model of Ref. [12] and the present model at different angular velocities (unit: Hz).

$\mathit{\omega}$ (rpm) | Mode No. | Ref. [12] | Present | Err (%) |
---|---|---|---|---|

200 | 1 | 20.21 | 20.21412 | 0.0253 |

2 | 104.384 | 104.4093 | 0.0242 | |

3 | 277.427 | 277.4587 | 0.0114 | |

600 | 1 | 20.5604 | 20.55981 | −0.0028 |

2 | 106.685 | 106.7338 | 0.0457 | |

3 | 280.155 | 280.1561 | 0.11 | |

1000 | 1 | 21.1927 | 21.19537 | 0.0126 |

2 | 111.178 | 111.2351 | 0.0514 | |

3 | 285.44 | 285.466 | 0.0091 |

**Table 2.**The comparison of the cantilever SACLD beam model at different relative thicknesses of the base layer (unit: Hz).

H_{b} | Mode No. | Ref. [55] | Present | Err (%) |
---|---|---|---|---|

0.01 | 1 | 18.9 | 19.4 | 2.6% |

2 | 101 | 101.2 | 0.19% | |

3 | 283 | 283.8 | 0.28% | |

0.012 | 1 | 21.5 | 21.9 | 1.8% |

2 | 119 | 119.4 | 0.34% | |

3 | 334 | 334.8 | 0.24% | |

0.015 | 1 | 25.6 | 25.9 | 1.17% |

2 | 147 | 147.0 | 0.00% | |

3 | 412 | 412.4 | 0.97% |

**Table 3.**The comparison between the FEM model by ANSYS and the present model with ACLD–SACLD treatment at different angular velocities (unit: Hz).

$\mathit{\omega}$ (rpm) | Mode No. | ACLD | SACLD | ||
---|---|---|---|---|---|

Present | ANSYS | Present | ANSYS | ||

0 | 1 | 20.16942 | 20.08 | 19.4 | 19.1 |

2 | 104.1151 | 104.09 | 101.2 | 101.068 | |

3 | 277.1195 | 277.1 | 283.8 | 283.26 | |

200 | 1 | 20.21412 | 20.261 | 19.47999 | 19.25 |

2 | 104.4093 | 105.01 | 101.5351 | 101.451 | |

3 | 277.4587 | 277.96 | 284.176 | 283.76 | |

600 | 1 | 20.55981 | 20.321 | 19.95035 | 20.047 |

2 | 106.7338 | 106.3 | 103.8929 | 102.732 | |

3 | 280.1561 | 279.27 | 286.812 | 285.63 | |

1000 | 1 | 21.19537 | 21.179 | 20.78848 | 20.665 |

2 | 111.2351 | 111.201 | 108.4595 | 107.457 | |

3 | 285.466 | 286.2 | 292.0034 | 291.32 |

Parameter | Value | Parameter | Value |
---|---|---|---|

L | 0.3 m | ρ_{1} | 2700 kg/m^{3} |

L_{e} | 0.03 m | ρ_{2} | 1250 kg/m^{3} |

b | 0.0127 m | ρ_{3} | 7600 kg/m^{3} |

h_{1} | 1.8 × 10^{−3} m | G_{2} | 2 × 10^{6} Pa |

h_{2} | 0.25 × 10^{−3} m | η | 0.38 |

h_{3} | 0.762 × 10^{−3} m | d_{31} | 23 × 10^{−12} m/V |

E_{1} | 64.9 GPa | g_{31} | 0.216 V·m/N |

E_{2} | 2(1 + η)G* | k_{31} | 0.12 |

E_{3} | 64.9 Gpa | k_{3t} | 12 |

X_{k} | 0.7 | R | 0 |

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## Share and Cite

**MDPI and ACS Style**

Wang, Y.; Fang, Y.; Li, L.; Zhang, D.; Liao, W.-H.; Fang, J.
Dynamic Modeling and Vibration Suppression of a Rotating Flexible Beam with Segmented Active Constrained Layer Damping Treatment. *Aerospace* **2023**, *10*, 1010.
https://doi.org/10.3390/aerospace10121010

**AMA Style**

Wang Y, Fang Y, Li L, Zhang D, Liao W-H, Fang J.
Dynamic Modeling and Vibration Suppression of a Rotating Flexible Beam with Segmented Active Constrained Layer Damping Treatment. *Aerospace*. 2023; 10(12):1010.
https://doi.org/10.3390/aerospace10121010

**Chicago/Turabian Style**

Wang, Yue, Yiming Fang, Liang Li, Dingguo Zhang, Wei-Hsin Liao, and Jianshi Fang.
2023. "Dynamic Modeling and Vibration Suppression of a Rotating Flexible Beam with Segmented Active Constrained Layer Damping Treatment" *Aerospace* 10, no. 12: 1010.
https://doi.org/10.3390/aerospace10121010