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Analysis of the Vibration Suppression of Double-Beam System via Nonlinear Switching Piezoelectric Network^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Single-Beam System with SSDI Shunt

## 3. Double-Beam System with SSDI Network

#### 3.1. Modeling of the Double-Beam System with SSDI Network

#### 3.2. The Lumped Electromechanical Coupling Model

#### 3.3. Nonlinear Complex Modal Analysis

#### 3.4. Forced Response Analysis

## 4. Experiment Design

#### 4.1. Implementation of the Low-Power SSDI Circuit

#### 4.2. Experiment Setup

#### 4.2.1. Single-Beam Experiment with SSDI Shunt

#### 4.2.2. Double-Beam Experiment with SSDI Network

## 5. Experimental Results and Discussions

#### 5.1. Single-Beam System with SSDI Shunt

#### 5.1.1. Vibration Control Performance with Single-Frequency Excitation

#### 5.1.2. Multi-Mode Vibration Control Performance

#### 5.2. Double-Beam System with SSDI Network

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Configuration of the single beam with the SSDI shunt: (

**a**) model of the single beam with the SSDI shunt; (

**b**) the equivalent circuit of the SSDI shunt.

**Figure 3.**Configuration of the double-beam system with SSDI network: (

**a**) model of the double-beam system with SSDI network; (

**b**) the equivalent circuit of an SSDI network.

**Figure 5.**Frequency vs. modal amplitude: (

**a**) frequencies of the lumped model; (

**b**) frequency change in different ${\gamma}_{net}$s.

**Figure 8.**Modal damping ratio vs. modal amplitude: (

**a**) modal damping ratios of the lumped model; (

**b**) modal damping ratio changing curves in different ${\gamma}_{net}$s.

**Figure 21.**Displacement response (piezo-patch 2) and voltage response (piezo-patch 1) with SSDI shunt (50 Hz).

**Figure 26.**Experimental design for the vibration reduction of a bladed disk system excited by a traveling wave.

**Figure 27.**Displacement comparison curves in three cases (50 Hz): (

**a**) displacement comparison curves; (

**b**) displacement comparison curves (zoom-in).

**Figure 28.**Damping comparison in three cases (white noise): (

**a**) displacement response curves in the time domain; (

**b**) frequency response curves.

Variable | ${\mathit{\delta}}_{\mathit{m}}$ | ${\mathit{\gamma}}_{\mathit{k}}$ | ${\mathit{\gamma}}_{\mathit{m}\mathit{e}}$ | ${\mathit{\gamma}}_{\mathit{e}}$ | ${\mathit{\xi}}_{1}$ | ${\mathit{\xi}}_{2}$ |
---|---|---|---|---|---|---|

Value | 1 | 1 | 0.1 | 0.05 | 0.02 | 0.02 |

Variable | Length (mm) | Width (mm) | Thickness (mm) | Elastic Modulus (Gpa) | Poisson’s Ratio | Density (kg/m ^{3}) |
---|---|---|---|---|---|---|

Value | 200 | 50 | 2 | 70 | 0.33 | 2700 |

Variable | Length (mm) | Width (mm) | Thickness (mm) | Elastic Modulus (Gpa) | Density (kg/m ^{3}) | Force Factor (N/V) | Inherent Capacitance (nF) |
---|---|---|---|---|---|---|---|

value | 43 | 35 | 0.5 | 66 | 7650 | 1.27 × 10^{−4} | 20 |

1st-Order Frequency (Hz) | Max Displacement (mV) | 2nd-Order Frequency (Hz) | Max Displacement (mV) | |
---|---|---|---|---|

Uncontrolled system | 50.05 | 164.6 | 280 | 3.891 |

SSDI shunt | 50.3 | 119.3 | 280.4 | 3.03 |

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**MDPI and ACS Style**

Zhang, F.; Liu, J.; Tian, J.
Analysis of the Vibration Suppression of Double-Beam System via Nonlinear Switching Piezoelectric Network. *Machines* **2021**, *9*, 115.
https://doi.org/10.3390/machines9060115

**AMA Style**

Zhang F, Liu J, Tian J.
Analysis of the Vibration Suppression of Double-Beam System via Nonlinear Switching Piezoelectric Network. *Machines*. 2021; 9(6):115.
https://doi.org/10.3390/machines9060115

**Chicago/Turabian Style**

Zhang, Fengling, Jiuzhou Liu, and Jing Tian.
2021. "Analysis of the Vibration Suppression of Double-Beam System via Nonlinear Switching Piezoelectric Network" *Machines* 9, no. 6: 115.
https://doi.org/10.3390/machines9060115