# Optimised Voltage Distribution on Piezoelectric Actuators for Modal Excitations Damping in Tapered Beams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimal Voltage Distribution on Piezoelectric Actuators Coupled with Cantilever Tapered Beams

## 3. Numerical Model and Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

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

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

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

L | beam length |

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

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

${T}_{b}\left(x\right)$ | beam thickness |

b | beam width |

c | tapering ratio |

w | vertical displacement |

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

${\varphi}_{i}\left(x\right)$ | 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 |

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

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**Figure 1.**Geometric characteristics of a tapered beam with linear tapering ratio c. (

**a**) top view and front view, (

**b**) 3D view of the tapered beam. $L,b,{T}_{{b}_{L}},{T}_{{b}_{R}}$ are respectively the length, width, left-end and right-end beam’s thickness.

**Figure 2.**Cantilever tapered beam coupled with n PZT actuator pairs (PPs). a denotes the axis of symmetry, ${x}_{i-1}$ and ${x}_{i}$ are, respectively, the left and right edges of the i-th PP. The step function ${\gamma}_{i}$ assumes $+1$ or $-1$ values and it is used to denote the two possible voltage distributions which can be provided to the generic i-th PP in order to induce two bending moments in the beam.

**Figure 3.**Comparison of the analytical (Equation (6)) and FEM bending moment exerted by the PPs considering $c=5$ and $\overline{x}=\frac{x}{L}$.

**Figure 4.**(

**a**) function ${f}_{1}\left(\overline{x}\right)$, (

**b**) graph of the integrand ${M}_{p}\left(\overline{x}\right){\varphi}_{1}^{\u2033}\left(\overline{x}\right)$. The first mode excitation and $c=5$ were considered. The O.V.D. provided no voltage sign change, since the minimum of ${f}_{1}\left(\overline{x}\right)$ coincided with $\overline{x}=0$, while the maximum was at $\overline{x}=1$. The same result could be obtained by observing the area subtended by ${M}_{p}\left(\overline{x}\right){\varphi}_{1}^{\u2033}\left(\overline{x}\right)$. The area of this function was always above the x-axis, so no voltage sign change was expected for the first mode.

**Figure 5.**(

**a**) function ${f}_{2}\left(\overline{x}\right)$, (

**b**) graph of the integrand ${M}_{p}\left(\overline{x}\right){\varphi}_{2}^{\u2033}\left(\overline{x}\right)$. The second mode excitation and $c=5$ were considered. The O.V.D. provided one voltage sign change, since ${f}_{2}\left(\overline{x}\right)$ included a minimum at $\overline{x}=0.315$, while the maximum was at $\overline{x}=1$. The same result could be obtained by observing the area subtended by ${M}_{p}\left(\overline{x}\right){\varphi}_{2}^{\u2033}\left(\overline{x}\right)$. In this case, the area of this function was negative (highlighted in red colour) within $0\le \overline{x}\le 0.315$, so by changing the voltage sign at $\overline{x}=0.315$ it was possible to enhance the system’s efficiency because the two areas would be on the same side of the graph.

**Figure 6.**(

**a**) function ${f}_{3}\left(\overline{x}\right)$, (

**b**) graph of the integrand ${M}_{p}\left(\overline{x}\right){\varphi}_{3}^{\u2033}\left(\overline{x}\right)$. The third mode excitation and $c=5$ were considered. The O.V.D. provided two voltage sign changes: ${f}_{3}\left(\overline{x}\right)$ included a local maximum at $\overline{x}=0.2$, a minimum at $\overline{x}=0.61$, and the absolute maximum was in $\overline{x}=1$. The same result could be obtained by observing the area subtended by ${M}_{p}\left(\overline{x}\right){\varphi}_{3}^{\u2033}\left(\overline{x}\right)$.The area of this function was negative (highlighted in red colour) within $0.2\le \overline{x}\le 0.61$, so, by changing the voltage sign both at $\overline{x}=0.2$ and $\overline{x}=0.61$, it was possible to enhance the system’s efficiency, because the three areas would be on the same side of the graph.

**Figure 7.**Parametric plot of ${f}_{1}\left(\overline{x}\right)$ considering the first flexural mode and several tapering ratios c.

**Figure 8.**Parametric plot of ${f}_{2}\left(\overline{x}\right)$ considering the second flexural mode and several tapering ratios c. The solid black line highlights the location of the minima of ${f}_{2}\left(\overline{x}\right)$ as c varies.

**Figure 9.**Parametric plot of ${f}_{3}\left(\overline{x}\right)$ considering the third flexural mode and several tapering ratios c. The solid black lines highlight the location of the extrema of ${f}_{3}\left(\overline{x}\right)$ as c varies.

**Figure 10.**Analytical and numerical optimal voltage distributions (O.V.D.) for the first eigenmode excitation versus tapering ratio. The stars represent the numerical optimal voltage distribution for each c value, calculated considering 13 PZT couples.

**Figure 11.**Analytical and numerical optimal voltage distributions (O.V.D.) for the second eigenmode excitation versus tapering ratio. Black lines highlight the analytical location of the points beyond which the voltage should be applied with the opposite phase with respect to the previous region. The dashed lines identify the PZT couples that have to be supplied with the opposite voltage sign. The stars represent the numerical optimal voltage distribution for each c value, calculated considering 13 PZT couples.

**Figure 12.**Analytical and numerical optimal voltage distributions (O.V.D.) for the third eigenmode excitation versus tapering ratio. Black lines highlight the analytical location of the points beyond which the voltage should be applied with the opposite phase with respect to the previous region. The dashed lines identify the PZT couples that have to be supplied with the opposite voltage sign. The stars represent the numerical optimal voltage distribution for each c value, calculated considering 13 PZT couples.

**Figure 13.**Example of numerical error in the assessment of the O.V.D. for the third mode excitation and $c=3$. The gray points highlight the position of each PP edge (13 PP were considered). The blue stars represent the FEM calculated O.V.D., the solid green line represents the function ${f}_{3}\left(\overline{x}\right)$ for $c=3$. The values ${e}_{1}$ and ${e}_{2}$ represent the discrepancies among the analytical and numerical $\delta {L}_{p}$. The total error was always less than 3%.

**Figure 14.**Example of damping efficacy of the O.V.D. when the first eigenmode was excited by a harmonic moment (amplitude = 1.8 Nm) concentrated at the tip. The dashed red line shows the instant of O.V.D. activation.

**Figure 15.**Example of damping efficacy of the O.V.D. when the second eigenmode was excited by a harmonic moment (amplitude = 1.2 Nm) concentrated at the tip. The dashed red line shows the instant of O.V.D. activation.

**Figure 16.**Example of damping efficacy of the O.V.D. when the third eigenmode was excited by a harmonic moment (amplitude = 0.9 Nm) concentrated at the tip. The dashed red line shows the instant of O.V.D. activation.

c | L (m) | b (m) | ${\mathit{T}}_{{\mathit{b}}_{\mathit{L}}}$ (m) |
---|---|---|---|

0.5 | 0.4 | 0.05 | 0.008 |

1 | 0.4 | 0.05 | 0.008 |

3 | 0.4 | 0.05 | 0.008 |

5 | 0.4 | 0.05 | 0.008 |

Configuration\PP | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

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

3 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | −1 | +1 |

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

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

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

${2}^{13}=8192$ | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 | −1 |

Component | Material | Density (kg/m^{3}) | Young’s Modulus (GPa) | Poisson’s Ratio | ${\mathit{d}}_{31}\phantom{\rule{3.33333pt}{0ex}}({10}^{-12}\phantom{\rule{3.33333pt}{0ex}}\mathit{C}/\mathit{N})$ |
---|---|---|---|---|---|

Beam | Steel | 7850 | 210 | 0.3 | – |

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

**Table 4.**Comparison between analytical and numerical eigenfrequencies for the first three modes and several tapering ratios. A = analytical result, N = FEM result.

$\mathit{c}=0.5$ | $\mathit{c}=1$ | $\mathit{c}=3$ | $\mathit{c}=5$ | |||||
---|---|---|---|---|---|---|---|---|

Mode | A (Hz) | N (Hz) | A (Hz) | N (Hz) | A (Hz) | N (Hz) | A (Hz) | N (Hz) |

1 | 42.79 | 45.23 | 44.36 | 46.47 | 48.45 | 50.39 | 50.88 | 51.91 |

2 | 227.71 | 238.31 | 212.50 | 221.66 | 187.75 | 201.83 | 179.28 | 193.49 |

3 | 607.25 | 642.64 | 548.32 | 590.23 | 449.83 | 495.8 | 412.71 | 456.31 |

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**MDPI and ACS Style**

Rossi, A.; Botta, F.
Optimised Voltage Distribution on Piezoelectric Actuators for Modal Excitations Damping in Tapered Beams. *Actuators* **2023**, *12*, 71.
https://doi.org/10.3390/act12020071

**AMA Style**

Rossi A, Botta F.
Optimised Voltage Distribution on Piezoelectric Actuators for Modal Excitations Damping in Tapered Beams. *Actuators*. 2023; 12(2):71.
https://doi.org/10.3390/act12020071

**Chicago/Turabian Style**

Rossi, Andrea, and Fabio Botta.
2023. "Optimised Voltage Distribution on Piezoelectric Actuators for Modal Excitations Damping in Tapered Beams" *Actuators* 12, no. 2: 71.
https://doi.org/10.3390/act12020071