# Improved Modelling of a Nonlinear Parametrically Excited System with Electromagnetic Excitation

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Apparatus

#### 2.2. Mathematical Model

#### 2.3. Parameter Identification

## 3. Free Response of the NPE System

#### 3.1. Experimental Results

#### 3.2. Effects of Cubic Parametric Stiffness and Nonlinear Damping

#### 3.3. The Effects of Cubic Stiffness

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Cantilever beam with a pair of identical magnets and a pair of identical coils on a wooden support. (

**b**) Schematic model of the experimental set-up. A programmable power supply is used to generate controllable current from controlling input voltage. The vertical movement of the beam (in the direction perpendicular to both the x and z axes is minimized by precisely locating the beam between the axis of the two coils. This position is adjusted by entering the attractive mode, and then returning to the repulsion mode once the centered state is reached.

**Figure 2.**(

**a**) Schematic of the NPE oscillator. (

**b**) Circuit diagram. Controllable power supply to generate current ${I}_{\mathrm{c}}$ is connected to a resistor R and the coils.

**Figure 4.**(

**a**) The analytical transition curve for a LPE system. (

**b**) Analytical and experimental amplitude-frequency plot for a NPE system. Stable branches are indicated by solid lines, and unstable branches by dashed lines. The unstable trivial branch is shown by dotted line. The distance between the coils $h=0.03\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$ limits the maximum displacement of the beam to $z=0.021\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$, shown by the grey line.

**Figure 5.**Experimental results for point A in Figure 4b. (

**a**) Measured displacement at the cantilever beam tip. (

**b**) Current measured across the coils in series connection. (

**c**) Power spectrum density (PSD) of the displacement signal. $\mathsf{\Omega}$ is at $78.5\phantom{\rule{0.166667em}{0ex}}\mathrm{rad}{\mathrm{s}}^{-1}$. (

**d**) Power spectrum density of the current. (

**e**) Phase portrait plot. (

**f**) Poincaré map.

**Figure 6.**Analytical and experimental amplitude-frequency plots. (

**a**) Comparing solutions when $\gamma =313.05\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$ and $\gamma =0$. (

**b**) Comparing solutions of Equation (1) when ${\zeta}_{\mathrm{es},\mathrm{app}}=430.26$, ${\delta}_{\mathrm{es},\mathrm{app}}=56.32$, and ${\gamma}_{\mathrm{es},\mathrm{app}}=83508.48$ and when they are zero (indicated in grey). System parameters are presented in Table 2.

**Figure 7.**The ratio between the cubic stiffness $\alpha $, and time-varying stiffness $\delta $ for different positions and input currents. (

**a**) with high DC current, and high AC current. (

**b**) with low DC current and low AC current. (

**c**) with high DC current but low AC current. The ⨂ labels show positions, which are chosen to compare two cases in each individual graph with the current specified. The comparison is based on the amplitude-frequency plots shown in Figure 8. The “8a–f” annotations in the plots correspond to Figure 8a to f.

**Figure 8.**Experimental and analytical amplitude-frequency plot. The system parameters are shown in Table 3. Stable branches are indicated by solid lines, and unstable branches by dashed lines. The unstable trivial solutions are shown by dotted lines.

Property | Value | Units |
---|---|---|

Radius of the Neodymium (N42) disc magnets | $0.015$ | $\mathrm{m}$ |

Residual magnetic flux density of the permanent magnet (${B}_{\mathrm{r}}$) | $1.1$ | T |

Permeability (${\mu}_{0}$) | $4\pi {10}^{-7}$ | $\mathrm{N}{\mathrm{A}}^{-2}$ |

Inner radius of the coil type L71-3,30 from Mundorf (${r}_{1}$) | $0.0085$ | $\mathrm{m}$ |

Outer radius of the coil (${r}_{2}$) | $0.0225$ | $\mathrm{m}$ |

Mean radius of the coil (${r}_{\mathrm{c}}$) | $0.0135$ | $\mathrm{m}$ |

Number of turns of in coil (N) | 485 | - |

Length of wire in one rotation (${l}_{\mathrm{w}}$) | 0.078 | $\mathrm{m}$ |

Diameter of the coil (${D}_{\mathrm{w}}$) | 0.00071 | $\mathrm{m}$ |

Height of the coil with shield (${h}_{\mathrm{coil}}$) | 0.02 | $\mathrm{m}$ |

Coordinate for coil (${z}_{1}$) | 0.007 | $\mathrm{m}$ |

Coordinate for coil (${z}_{2}$) | −0.007 | $\mathrm{m}$ |

Measured electrical resistance of the coil and extra wiring (${R}_{\mathrm{coil}}$) | 1.91 | Ohm |

Resistor (R) | 0.1 | Ohm |

Width of the beam (${b}_{\mathrm{b}}$) | $0.01$ | $\mathrm{m}$ |

Thickness of the beam (${t}_{\mathrm{b}}$) | $0.002$ | $\mathrm{m}$ |

Total physical mass (the effective mass of the beam and magnets) (${m}_{\mathrm{t}}$) | $0.104$ | $\mathrm{kg}$ |

Static stiffness of the beam with magnets and coils when ${I}_{\mathrm{c}}=0$ (${k}_{\mathrm{b}}$) | 32.84 | ${\mathrm{Nm}}^{-1}$ |

Measured first natural frequency of the beam with magnets and coils when ${I}_{\mathrm{c}}=0$ (${\omega}_{\mathrm{n}},\mathrm{exp}$) | 17.76 | $\mathrm{rad}{\mathrm{s}}^{-1}$ |

Measured second natural frequency of the beam with magnets and coils when ${I}_{\mathrm{c}}=0$ | 202 | $\mathrm{rad}{\mathrm{s}}^{-1}$ |

Mechanical damping coefficient of the beam with magnets and coils when ${I}_{\mathrm{c}}=0$ (${c}_{\mathrm{m}}$) | 0.011 | ${\mathrm{Nsm}}^{-1}$ |

**Table 2.**System parameters for Figure 6a,b.

Tests | Measured Parameters | Calculated Parameters from Equations (23)–(28) | |||||||
---|---|---|---|---|---|---|---|---|---|

h (m) | ${\zeta}_{\mathrm{m}}$ | ${\omega}_{\mathrm{n}}$ ($\mathrm{rad}{\mathrm{s}}^{-1}$) | ${\zeta}_{\mathrm{es},\mathrm{app}}$ | ${\delta}_{\mathrm{es},\mathrm{app}}$ | ${\gamma}_{\mathrm{es},\mathrm{app}}$ | $\delta $ | $\alpha $$({\mathrm{m}}^{-2})$ | $\gamma $$({\mathrm{m}}^{-2})$ | |

Figure 6a | 0.035 | 0.001 | 30.77 | 78.46 | 14.18 | 17762.4 | 0.25 | 782.63 | Refer to Figure 6a |

Figure 6b | 0.025 | 0.001 | 50.3 | Refer to Figure 6b | 0.093 | 1262.17 | 138.83 |

Tests | Measured Parameters | Calculated Parameters from Equations (23)–(28) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

h (m) | ${I}_{\mathrm{DC}}$ (A) | ${I}_{\mathrm{AC}}$ (A) | ${\zeta}_{\mathrm{m}}$ | ${\omega}_{\mathrm{n}}$ ($\mathrm{rad}{\mathrm{s}}^{-1}$) | ${\zeta}_{\mathrm{es},\mathrm{app}}$ | ${\delta}_{\mathrm{es},\mathrm{app}}$ | ${\gamma}_{\mathrm{es},\mathrm{app}}$ | $\delta $ | $\alpha $$({\mathrm{m}}^{-2})$ | $\gamma $$({\mathrm{m}}^{-2})$ | |

Figure 8a | 0.03 | 0.92 | 0.14 | 0.001 | 48.60 | 146 | 17.44 | 24,761.7 | 0.129 | 1203.19 | 183.1 |

Figure 8b | 0.035 | 0.97 | 0.155 | 0.001 | 40.12 | 60.16 | 8.93 | 11,188.82 | 0.122 | 956.40 | 152.82 |

Figure 8c | 0.025 | 0.50 | 0.055 | 0.001 | 50.3 | 430.26 | 56.33 | 83,508.47 | 0.093 | 1262.17 | 138.83 |

Figure 8d | 0.03 | 0.48 | 0.06 | 0.001 | 37.11 | 191.26 | 28.82 | 40,897.67 | 0.092 | 1055.86 | 131.98 |

Figure 8e | 0.03 | 0.98 | 0.08 | 0.001 | 49.23 | 144.17 | 16.55 | 23,497.5 | 0.069 | 1214.5 | 99.14 |

Figure 8f | 0.035 | 0.96 | 0.06 | 0.001 | 39.81 | 60.64 | 9 | 11,277.63 | 0.047 | 954.05 | 59.62 |

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**MDPI and ACS Style**

Zaghari, B.; Rustighi, E.; Ghandchi Tehrani, M.
Improved Modelling of a Nonlinear Parametrically Excited System with Electromagnetic Excitation. *Vibration* **2018**, *1*, 157-171.
https://doi.org/10.3390/vibration1010012

**AMA Style**

Zaghari B, Rustighi E, Ghandchi Tehrani M.
Improved Modelling of a Nonlinear Parametrically Excited System with Electromagnetic Excitation. *Vibration*. 2018; 1(1):157-171.
https://doi.org/10.3390/vibration1010012

**Chicago/Turabian Style**

Zaghari, Bahareh, Emiliano Rustighi, and Maryam Ghandchi Tehrani.
2018. "Improved Modelling of a Nonlinear Parametrically Excited System with Electromagnetic Excitation" *Vibration* 1, no. 1: 157-171.
https://doi.org/10.3390/vibration1010012