# Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations

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## Abstract

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## 1. Introduction

_{2}and H

^{∞}optimization techniques [25,26].

## 2. Positive Position Feedback (PPF) Controller

#### 2.1. Mathematical Analysis

#### 2.2. Steady-State Vibration and Stability Investigations

#### 2.3. Response Curves and Numerical Validations

## 3. Adaptive Positive Position Feedback (APPF) Controller

#### Frequency-Response Equation of APPF Controller

## 4. Comparison between the PPF and APPF Controllers

## 5. Conclusions

- The conventional PPF controller can eliminate the primary resonance vibrations of the considered system in the presence of 1:1 internal resonance.
- Once the resonance conditions between the main system and the PPF controller are lost, the controller adds excessive vibrational energy to the main system rather than suppressing it.
- At the large excitation force amplitudes, the main system may lose its stability to respond with a quasiperiodic motion when the resonance condition between the main system and the PPF controller is lost.
- Regardless of the main system natural frequency, once the controller natural frequency is properly tuned to be the same value as the excitation frequency (${\omega}_{2}=\mathsf{\Omega}$), the controller can suppress the main system vibrations when subjected to any excitation force amplitude and/or any excitation frequency.
- According to point (4) of the conclusion, the adaptive positive position feedback controller is the best control strategy that can eliminate the main system vibrations regardless of the excitation frequency and excitation force amplitude.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$x,\dot{x},\ddot{x}$ | Dimensionless displacement, velocity, and acceleration of the main system. |

$y,\dot{y},\ddot{y}$ | Dimensionless displacement, velocity, and acceleration of the controller. |

${\mu}_{1},{\mu}_{2}$ | Dimensionless linear damping coefficients of the main system and controller, respectively. |

${\omega}_{1},{\omega}_{2}$ | Dimensionless linear natural frequencies of the main system and controller, respectively. |

$\alpha $ | Dimensionless cubic nonlinear stiffness coefficient. |

$f$ | Dimensionless excitation force amplitude. |

$\mathsf{\Omega}$ | Excitation force angular frequency. |

$\gamma $ | Dimensionless control signal gain. |

$\lambda $ | Dimensionless feedback signal gain. |

${\sigma}_{1}$ | The detuning parameter ${\sigma}_{1}=\mathsf{\Omega}-{\omega}_{1}$ |

${a}_{1},{a}_{2}$ | Dimensionless steady-state oscillation amplitudes of the main system and controller, respectively. |

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**Figure 2.**The main system response curves before control: (

**a**) frequency-response curve, (

**b**) force-response curve.

**Figure 3.**Frequency-response curves of the main system and controller when $f=0.03$: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 4.**The effect of varying ${\sigma}_{2}$ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 5.**The effect of increasing the control signal gain $\gamma $ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 6.**The effect of increasing the feedback signal gain $\lambda $ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 7.**The effect of increasing the controller linear damping coefficient ${\mu}_{2}$ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 8.**The effect of increasing the excitation force amplitude $f$ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 9.**Force response-curves of the controlled system at five different values of the external detuning parameter ${\sigma}_{2}$: (

**a**) the main system, (

**b**) the PPF controller.

**Figure 10.**The controlled system time-response, Poincaré map, and frequency spectrum according to Figure 9 when $f=0.5$ and ${\sigma}_{1}=0.0$: (

**a**,

**b**) the temporal oscillation of the main system and PPF controller, (

**c**,

**d**) the Poincaré map of the main system and PPF controller, and (

**e**,

**f**) the frequency spectrum of the main system and PPF controller.

**Figure 11.**The controlled system time-response, Poincaré map and frequency spectrum according to Figure 9 when $f=0.5$ and ${\sigma}_{1}=0.05$: (

**a**,

**b**) the temporal oscillation of the main system and PPF controller, (

**c**,

**d**) the Poincaré map of the main system and PPF controller, and (

**e**,

**f**) the frequency spectrum of the main system and PPF controller.

**Figure 12.**The controlled system time-response, Poincaré map, and frequency spectrum according to Figure 9 when $f=0.5$ and ${\sigma}_{1}=0.14$: (

**a**,

**b**) the temporal oscillation of the main system and PPF controller, (

**c**,

**d**) the Poincaré map of the main system and PPF controller, and (

**e**,

**f**) the frequency spectrum of the main system and PPF controller.

**Figure 13.**The schematic diagram describing the connection between the main system and the APPF controller.

**Figure 14.**The effect of increasing the excitation force amplitude $f$ on the frequency-response curve of the controlled system: (

**a**) the main system, (

**b**) the APPF controller.

**Figure 15.**Force response-curves of the controlled system at five different values of the external detuning parameter ${\sigma}_{2}$: (

**a**) the main system, (

**b**) the APPF controller.

**Figure 16.**A comparison between the frequency-response curves of the main system without the control, with the PPF controller, and with the APPF controller: (

**a**) the main system, (

**b**) the PPF and APPF controllers.

**Figure 17.**The effect of online switching between the PPF and APPF control strategies on the main system nonlinear vibration when $\mathsf{\Omega}=0.9$ according to Figure 16: (

**a**) the main system, and (

**b**) the controllers.

**Figure 18.**The effect of online switching between the PPF and APPF control strategies on the main system nonlinear vibration when $\mathsf{\Omega}=1.0$ according to Figure 16: (

**a**) the main system, and (

**b**) the controllers.

**Figure 19.**The effect of online switching between the PPF and APPF control strategies on the main system nonlinear vibration when $\mathsf{\Omega}=1.1$ according to Figure 16: (

**a**) the main system, and (

**b**) the controllers.

**Figure 20.**A comparison between the force-response curves of the main system without the control, with the PPF controller, and with the APPF controller: (

**a**) the main system, (

**b**) the PPF and APPF controllers.

**Figure 21.**The controlled system bifurcation diagram according to Figure 20 in the case of the PPF controller: (

**a**) the main system, and (

**b**) the PPF controller.

**Figure 22.**The controlled system bifurcation diagram according to Figure 20 in the case of the APPF controller: (

**a**) the main system, and (

**b**) the APPF controller.

**Figure 23.**The time histories of the main system when the excitation force $f$ is swept from 0.01 to 0.45 according to Figure 20 in the case of both the PPF and APPF controllers: (

**a**) the main system, (

**b**) the PPF and APPF controllers.

The Main System Parameters | PPF Controller Parameters | ||
---|---|---|---|

${\omega}_{1}$ | $1.0$ | ${\omega}_{2}$ | $1.0$ |

${\mu}_{1}$ | $0.01$ | ${\mu}_{2}$ | $0.001$ |

$\alpha $ | $0.05$ | $\gamma $ | $0.2$ |

$f$ | $0.03$ | $\lambda $ | $0.2$ |

${\sigma}_{1}$ | $0.0$ | ${\sigma}_{2}$ | $0.0$ |

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**MDPI and ACS Style**

Saeed, N.A.; Awwad, E.M.; Abdelhamid, T.; El-Meligy, M.A.; Sharaf, M.
Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations. *Symmetry* **2021**, *13*, 255.
https://doi.org/10.3390/sym13020255

**AMA Style**

Saeed NA, Awwad EM, Abdelhamid T, El-Meligy MA, Sharaf M.
Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations. *Symmetry*. 2021; 13(2):255.
https://doi.org/10.3390/sym13020255

**Chicago/Turabian Style**

Saeed, N. A., Emad Mahrous Awwad, Talaat Abdelhamid, Mohammed A. El-Meligy, and Mohamed Sharaf.
2021. "Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations" *Symmetry* 13, no. 2: 255.
https://doi.org/10.3390/sym13020255