# Optimal Voltage Distribution on PZT Actuator Pairs for Vibration Damping in Beams with Different Boundary Conditions

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## Abstract

**:**

## 1. Introduction

## 2. Optimal Voltage Distribution on Piezoelectric Actuators Coupled with Beams under Different Constraints

## 3. Numerical Model and Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

L | beam’s length |

${d}_{31}$ | piezoelectric coefficient |

$\rho $ | beam’s density |

${E}_{a}$ | Young’s modulus of the piezoelectric actuator |

${E}_{b}$ | Young’s modulus of the beam |

${M}_{p}$ | piezoelectric bending moment |

${T}_{a}$ | piezoelectric actuator thickness |

${T}_{b}$ | beam’s thickness |

b | beam’s width |

A | beam’s cross-section area |

w | vertical displacement |

$\stackrel{\sim}{w}$ | virtual vertical displacement |

${w}_{i}\left(x\right)$ | i-th flexural mode of the cantilever beam |

$\overline{x}$ | dimensionless length of the beam: $\frac{x}{{L}_{b}}$ |

${x}_{i}$ | points where the potential changes its sign |

$\overline{x}=\frac{x}{L}$ | dimensionless length |

${}^{\prime}$ | derivative with respect to the $x-$axis |

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**Figure 1.**Beams coupled with PZT actuator pairs (PPs) for different boundary conditions: free-free, fixed-fixed, and pinned-pinned.

**Figure 4.**Example of the optimal voltage distribution to achieve the maximum work $\delta {L}_{p}$ for the third mode and free-free boundary condition. $\delta {L}_{p}$ is only dependent on the first derivative of the eigenmode (${w}_{3}^{\prime}\left(\overline{x}\right)$): (

**a**) no voltage’s sign switch is provided, (

**b**) optimal voltage distribution (two sign switches entailed). In the case (

**a**), the total work $\delta {L}_{p}=\delta {L}_{p1}+\delta {L}_{p2}+\delta {L}_{p3}$ is not the highest since the work $\delta {L}_{p2}$ is negative and reduces the total work. Case (

**b**) shows that the optimal voltage distribution entails two voltage sign switches, respectively, at $\overline{{x}_{1}}=0.36$ and $\overline{{x}_{2}}=0.64$. In this way, the resulting total work $\delta {L}_{p}^{max}=\delta {L}_{p1}+\delta {L}_{p2}+\delta {L}_{p3}$ is the highest since $\delta {L}_{p2}$ is actually positive. The solid red line represents the function ${w}_{3}^{\prime}\left(\overline{x}\right)$ when the voltage’s sign is switched for $0.36\le \overline{x}\le 0.64$.

**Figure 5.**First derivatives of the first three modes for free-free boundary condition: (

**a**) first mode, (

**b**) second mode, and (

**c**) third mode. The FEM results highlight the optimal location of the voltage sign switch (illustrated with blue stars).

**Figure 6.**Example of discretization errors when evaluating the damping effectiveness in the case of free-free boundary conditions and third mode excitation. The total analytical virtual work $\delta {L}_{p,an}=|\delta {L}_{p1,an}|+|\delta {L}_{p2,an}|+|\delta {L}_{p3,an}|$ is the maximum theoretically achievable. Since 10 PZT actuator pairs are considered in the FEM simulations, the total virtual work, obtained via FEM, will be always lower than $\delta {L}_{p,an}$ and it can be written as: $\delta {L}_{p,fem}=|\delta {L}_{p1,fem}|+|\delta {L}_{p2,fem}|+|\delta {L}_{p3,fem}|\le \delta {L}_{p,an}$. ${e}_{1}$ and ${e}_{2}$ can be obtained as: ${e}_{1}=\delta {L}_{p1,an}-\delta {L}_{p1,fem}$ and ${e}_{2}=\delta {L}_{p3,an}-\delta {L}_{p3,fem}$. It is worth noting that $\delta {L}_{p2,an}-\delta {L}_{p2,fem}={e}_{1}+{e}_{2}$. The total percentage loss of damping due to the discretization error (Equation (8)) is $e(\%)=\frac{\delta {L}_{p,an}-\delta {L}_{p,fem}}{\delta {L}_{p,an}}=\frac{2{e}_{1}+2{e}_{2}}{\delta {L}_{p,an}}=8\%$.

**Figure 7.**First derivatives of the first three modes for fixed-fixed boundary condition: (

**a**) first mode, (

**b**) second mode, and (

**c**) third mode. The FEM results highlight the optimal location of the voltage sign switch (illustrated with blue stars).

**Figure 8.**First derivatives of the first three modes for pinned-pinned boundary condition: (

**a**) first mode, (

**b**) second mode, and (

**c**) third mode. The FEM results highlight the optimal location of the voltage sign switch (illustrated with blue stars).

**Figure 9.**Example of the damping efficacy achieved activating the voltage distributions listed in Table 4. ${w}_{\overline{x}=0.2}$ vs. time plots in the case of: (

**a**) activation of the voltage distribution A, (

**b**) activation of the voltage distribution B, (

**c**) activation of the voltage distribution OVD${}_{2}$, and (

**d**) activation of the voltage distribution OVD${}_{1}$. The first eigenmode of the fixed-fixed beam is excited by the external load $F\left(t\right)$, evenly distributed on the beam length, with $F\left(t\right)={F}_{0}cos\left({\omega}_{1}t\right)$, ${F}_{0}=15$ N and ${\omega}_{1}$ being the angular frequency of the first mode (1174.4 rad/s). The vertical displacement of a point located at $\overline{x}=0.2$ is used as the reference point. The dashed red line shows the instant of voltage distribution activation.

**Figure 10.**Virtual works of the piezoelectric forces ($\delta {L}_{p}$) in the case of the first eigenmode excitation in a fixed-fixed beam. The FEM results highlight the optimal location of the voltage sign switch (illustrated with blue stars).

beam | L (mm) | b (mm) | ${h}_{b}$ (mm) |

300 | 30 | 3 | |

PZT | ${L}_{p}$ (mm) | ${b}_{p}$ (mm) | ${h}_{p}$ (mm) |

30 | 30 | 0.3 |

Distribution\PP | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 |

2 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 |

3 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | - | +1 |

4 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | +1 |

5 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 | +1 | +1 |

… | … | … | … | … | … | … | … | … | … | … |

k | +1 | −1 | +1 | −1 | −1 | +1 | −1 | +1 | +1 | −1 |

… | … | … | … | … | … | … | … | … | … | … |

${2}^{10}=1024$ | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 |

Label | Material | Density (kg/m${}^{3}$) | Young’s Modulus (GPa) | Poisson’s Ratio | ${\mathit{d}}_{31}({10}^{-12}\mathit{C}/\mathit{N})$ |
---|---|---|---|---|---|

Beam | Aluminium | 2700 | 70 | 0.3 | – |

Actuator | PZT-5A | 7750 | 39 | – | 374 |

**Table 4.**Selected voltage distributions for the damping effectiveness comparison illustrated in Figure 9 and considering the first eigenmode excitation of the fixed-fixed beam. OVD${}_{1}$ and OVD${}_{2}$ are the optimal voltage distributions when the mode is excited, respectively, for the first and the second mode.

Distribution\PP | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

A | +1 | +1 | 0 | 0 | 0 | 0 | 0 | 0 | +1 | +1 |

B | +1 | +1 | −1 | −1 | −1 | +1 | +1 | −1 | −1 | +1 |

OVD${}_{1}$ | +1 | +1 | −1 | −1 | −1 | −1 | −1 | −1 | +1 | +1 |

OVD${}_{2}$ | +1 | +1 | −1 | −1 | −1 | +1 | +1 | +1 | +1 | −1 |

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**MDPI and ACS Style**

Rossi, A.; Botta, F.
Optimal Voltage Distribution on PZT Actuator Pairs for Vibration Damping in Beams with Different Boundary Conditions. *Actuators* **2023**, *12*, 85.
https://doi.org/10.3390/act12020085

**AMA Style**

Rossi A, Botta F.
Optimal Voltage Distribution on PZT Actuator Pairs for Vibration Damping in Beams with Different Boundary Conditions. *Actuators*. 2023; 12(2):85.
https://doi.org/10.3390/act12020085

**Chicago/Turabian Style**

Rossi, Andrea, and Fabio Botta.
2023. "Optimal Voltage Distribution on PZT Actuator Pairs for Vibration Damping in Beams with Different Boundary Conditions" *Actuators* 12, no. 2: 85.
https://doi.org/10.3390/act12020085