# Vibration Isolation and Alignment of Multiple Platforms on a Non-Rigid Supporting Structure

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## Abstract

**:**

## 1. Introduction

## 2. General AVI Framework

#### 2.1. Transmissibility Model for the Isolation System

#### 2.2. Supporting Structure Model

#### 2.3. Formulation of the VI and Alignment Problem

## 3. Interaction Problem Using a Feedback AVI Control Law

#### AVI Control Law

## 4. Application Example

#### 4.1. System Dynamics

#### 4.2. Design Criterion

#### 4.3. Numerical Results

#### 4.3.1. Symmetrical Case: ${L}_{12}={L}_{23}={L}_{b}/4$

#### 4.3.2. Non-Symmetrical Case: ${L}_{12}={L}_{b}/4$ and ${L}_{23}={L}_{b}/3$

#### 4.3.3. Robustness Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

VI | Vibration Isolation |

PVI | Passive Vibration Isolation |

SAVI | Semi-Active Vibration Isolation |

AVI | Active Vibration Isolation |

SISO | Single-Input–Single-Output |

MIMO | Multiple-Input–Multiple-Output |

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**Figure 5.**Ratio ${f}_{fv}\left({\widehat{\mathit{K}}}_{V}\right)/{f}_{fv}\left({\widehat{\mathit{K}}}_{FV}\right)$ for: (

**a**) a single disturbance force; and (

**b**) three disturbance forces for the symmetrical case.

**Figure 6.**Ratio ${\widehat{k}}_{{v}_{1,3}}/{\widehat{k}}_{f{v}_{1,3}}$ for: (

**a**) a single disturbance force; and (

**b**) three disturbance forces for the symmetrical case.

**Figure 7.**Impulse responses for three particular cases: (r

_{m}= 0.01, r

_{ω}= 0.3) (

**a**,

**d**); (r

_{m}= 0.001, r

_{ω}= 0.5) (

**b**,

**e**); and (r

_{m}= 0.0001, r

_{ω}= 0.9) (

**c**,

**f**), for a single disturbance input (

**top row**) and three disturbance inputs for the symmetrical case (

**bottom row**).

**Figure 8.**Ratio ${f}_{fv}\left({\widehat{\mathit{K}}}_{V}\right)/{f}_{fv}\left({\widehat{\mathit{K}}}_{FV}\right)$ for: (

**a**) a single disturbance force; and (

**b**) three disturbance forces for the non-symmetrical case.

**Figure 9.**Rations ${\widehat{k}}_{{v}_{1}}/{\widehat{k}}_{f{v}_{1}}$ and ${\widehat{k}}_{{v}_{3}}/{\widehat{k}}_{f{v}_{3}}$ for: (

**a**) a single disturbance force; and (

**b**) for three disturbance forces for the non-symmetrical case.

**Figure 10.**Impulse responses for three particular cases: (r

_{m}= 0.01, r

_{ω}= 0.3) (

**a**,

**d**); (r

_{m}= 0.001, r

_{ω}= 0.5) (

**b**,

**e**); and (r

_{m}= 0.0001, r

_{ω}= 0.9) (

**c**,

**f**), for a single disturbance input (

**top row**) and three disturbance inputs for the non-symmetrical case (

**bottom row**).

**Figure 11.**Effect on the VI and alignment performance due to 5% (a–c) and 10% variation (d–f) in system parameters ${\zeta}_{{p}_{2}}$, ${\omega}_{{p}_{2}}$ of the mid-span isolator—symmetrical case.

**Figure 12.**Effect on the VI and alignment performance of 5% variation (a–c) and 10% variation (d–f) in the dynamic parameters ${\zeta}_{{p}_{2}}$, ${\omega}_{{p}_{2}}$ of the mid-span isolator—non-symmetrical case.

**Table 1.**Mass ratios ${r}_{m}$ and frequency ratios ${r}_{\omega}$ used in the numerical experiment.

${\mathit{r}}_{\mathit{m}}$ | ${\mathit{r}}_{\mathit{\omega}}$ |
---|---|

$[1\xb7{10}^{-4},1.5]$ | $[0.1,1.5]$ |

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**MDPI and ACS Style**

Pérez-Aracil, J.; Pereira, E.; Aphale, S.S.; Reynolds, P. Vibration Isolation and Alignment of Multiple Platforms on a Non-Rigid Supporting Structure. *Actuators* **2020**, *9*, 108.
https://doi.org/10.3390/act9040108

**AMA Style**

Pérez-Aracil J, Pereira E, Aphale SS, Reynolds P. Vibration Isolation and Alignment of Multiple Platforms on a Non-Rigid Supporting Structure. *Actuators*. 2020; 9(4):108.
https://doi.org/10.3390/act9040108

**Chicago/Turabian Style**

Pérez-Aracil, Jorge, Emiliano Pereira, Sumeet S. Aphale, and Paul Reynolds. 2020. "Vibration Isolation and Alignment of Multiple Platforms on a Non-Rigid Supporting Structure" *Actuators* 9, no. 4: 108.
https://doi.org/10.3390/act9040108