# Enhancing Vibration Control in Cable–Tip–Mass Systems Using Asymmetric Peak Detector Boundary Control

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## Abstract

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## 1. Introduction

- Introduction of a boundary strategy centered on detecting peak vibration values and subsequent proof of its bounded-input bounded-output (BIBO) stability.
- Proposal of two design parameters aimed at enhancing the flexibility of peak detection. Their values can be determined using specific performance indices.
- Conducting simulations that compare the performance of this approach with a classic boundary controller, demonstrating its efficacy.

## 2. Materials and Methods

#### 2.1. Cable–Tip–Mass System

- (i)
- The amplitude of $u(x,t)$ is very small.
- (ii)
- ${T}_{o}$ is constant all along the cable.

#### 2.2. Asymmetric Peak Detector Model

#### 2.2.1. Definition and Characterization

#### 2.2.2. Bounded-Input Bounded-Output Analysis

- If ${v}_{i}\left(t\right)\ge y\left(t\right)$, then $\left|y\left(t\right)\right|\le |{v}_{i}\left(t\right)|\le K$. In fact, we can obtain:$$\left|y\left(t\right)\right|\le K{e}^{-{\alpha}_{1}{t}_{0}}.$$
- If ${v}_{i}\left(t\right)<y\left(t\right)$, then $\left|y\left(t\right)\right|\le |{y}_{0}|{e}^{-{\alpha}_{2}(t-{t}_{0})}$, for $t\ge {t}_{0}$ with $y\left({t}_{0}\right)={y}_{0}$.

#### 2.3. Asymmetric Peak Detector Boundary Controller

- (a)
- If ${f}_{d}\left(t\right)\ge y\left(t\right)$, then $\dot{V}\left(t\right)\le -{k}_{d}{u}_{t}(L,t){u}_{t}(L,t)=-{k}_{d}{u}_{t}^{2}(L,t)\le 0$.
- (b)
- If ${f}_{d}\left(t\right)<y\left(t\right)$, then $-y\left(t\right)<-{f}_{d}\left(t\right)$. Moreover, from (7) we induce that ${u}_{t}(L,t)=-\frac{1}{{k}_{d}}{f}_{d}\left(t\right)$. So, $\dot{V}\left(t\right)\le y\left(t\right){u}_{t}(L,t)=-y\left(t\right)\phantom{\rule{0.166667em}{0ex}}\frac{1}{{k}_{d}}{f}_{d}\left(t\right)<-\frac{1}{{k}_{d}}{f}_{d}^{2}\left(t\right)\le 0$.

## 3. Results

- (i)
- Standard boundary controller: $f\left(t\right)={f}_{d}\left(t\right)$ (7);
- (ii)

#### 3.1. Unperturbed Case Experiments

#### 3.2. External Disturbance Case Experiments

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

PDE | Partial Differenial Equation |

BIBO | Bounded-Input Bounded-Output |

D | Diode |

C | Capacitor |

R | Resistor |

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**Figure 1.**Floating platform for offshore wind turbines (cleantechnica.com); oil-drilling actuators (EnggCyclopedia.com).

**Figure 3.**A simplified cable–tip–mass system, with local coordinates and boundary conditions. One end of the string is pinned while the other end is linked to an actuator $f\left(t\right)$. Our assumption accounts for uniform tension along the entire length of the string.

**Figure 4.**Simplified electronic circuit of the peak detector system. $y\left(t\right)$ displays the peak information on the input signal ${v}_{i}\left(t\right)$. The memory time to keep the peak value of ${v}_{i}\left(t\right)$ stored in the capacitor is controlled through the values of R and C.

**Figure 5.**Simulation results of the asymmetric peak detector system (5), when different classes of input ${v}_{i}\left(t\right)$ are considered; in blue: input ${v}_{i}\left(t\right)$; in red: output y(t). Three cases of parameters ${\alpha}_{1}$ and ${\alpha}_{2}$ values in (5) are considered for each input function, to expose the performance of the proposal. We obtain a positive response, with different input values detected. The designer needs to determine the preferred scenario for their system.

**Figure 10.**Simulation results $u(x,t)$ by using the standard boundary damper (7), with ${k}_{d}=1000$.

**Figure 11.**Numerical result of $u(L,t)$ versus t, by using the standard boundary damper (7). The vibration attenuation is reached in fifty seconds.

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**MDPI and ACS Style**

Acho, L.; Pujol-Vázquez, G.
Enhancing Vibration Control in Cable–Tip–Mass Systems Using Asymmetric Peak Detector Boundary Control. *Actuators* **2023**, *12*, 463.
https://doi.org/10.3390/act12120463

**AMA Style**

Acho L, Pujol-Vázquez G.
Enhancing Vibration Control in Cable–Tip–Mass Systems Using Asymmetric Peak Detector Boundary Control. *Actuators*. 2023; 12(12):463.
https://doi.org/10.3390/act12120463

**Chicago/Turabian Style**

Acho, Leonardo, and Gisela Pujol-Vázquez.
2023. "Enhancing Vibration Control in Cable–Tip–Mass Systems Using Asymmetric Peak Detector Boundary Control" *Actuators* 12, no. 12: 463.
https://doi.org/10.3390/act12120463