# Fractional Dynamical Behavior of an Elastic Magneto Piezo Oscillator Including Non-Ideal Motor Excitation

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Magneto Piezo Oscillator Device

## 3. Mathematical Modeling

_{0}= 0 we have no action of the non-ideal motor. Figure 2 depicts the first five of these functions, which aids in understanding the effect of the nonideal excitation.

_{1}> 0 and a purely hysteretic, D > 0 is the constant flow displacement and k

_{1}∈ (0,1) is the post/precedence stiffness ratio. A, β, γ, and n are parameters of shape and the size of the hysteresis loop [1,13,14,15,42,43,44].

## 4. Numerical Results

_{avg}considering the parameters of Table 1 to observe the behavior of the parameter related to non-ideal motor rotation and the parameter of the fractional derivative. The values a0 close to zero with b

_{0}= 0.5 and values of q

_{4}close to 0.85 have a low average power value (Figure 2a). However, for values of b

_{0}with values between 0 and 1 and q

_{4}close to 0.85, there is a decrease in average power with a

_{0}= 0.5.

_{avg}regions with the variation of the parameters (a

_{0}, b

_{0}, and q

_{4}), and changes in the behavior in the hysteresis of the piezoceramic materials caused by the system of Equation (8) reaches orbits with larger beam displacement amplitudes. This larger displacement causes vibrations in the piezoceramic material, allowing for a higher average power, considering the parameters adopted (See Table 1). Another aspect associated with the piezoceramic material is defined by [40,41,42,43,44,45,46,47,48,49,50,51,52,53], which establishes that changes observed in the hysteresis curves can characterize hard-type piezoceramic materials, which have a piezoelectric constant with low values, or soft-type piezoceramic materials with high piezoelectric constants.

_{4}. For values of q

_{4}= 0.8586 and q

_{4}= 0.9495, we can observe the results in Figure 4a,b obtained by numerical simulations using the algorithm described in Appendix A and Equations (8) and (10), where it can be seen that there is a maximum of P

_{avg}with maximum values close to a

_{0}= 0.5 and values above b

_{0}= 0.5 (Figure 3a). For the case of q

_{4}= 0.9495, we will have a minimum region for the values of a a

_{0}= 1.0 and b

_{0}= 1.0, as seen in Figure 3b.

_{4}

`→`1.0 there is a larger region for P

_{avg}values, as we can see in Figure 5a,b.

_{1}) and velocity (x

_{2}).

_{c}, considering the time series described by the integration of the system of fractional differential equations. For a better description of the 0–1 test see Refs. [54,55,56]. Thus, if K

_{c}is close to 0, the system has a periodic behavior, however, if K

_{c}is close to 1 the behavior will be chaotic. The parameter K

_{c}confirms these intervals obtained from the bifurcation diagrams. For the other intervals, the behavior was chaotic according to the bifurcation diagrams and the K

_{c}. In Figure 5a we show the bifurcation diagram of displacement ${x}_{1}$ and Figure 6b shows (${K}_{c}$) referring to the ${x}_{1}$ is displacement of cantiliver. This diagram makes it possible to observe the dynamic behavior of the displacement of the bar containing the piezoceramic patches, thus observing the intervals in which the system has a periodic and chaotic regime when considering the parameter of the fractional derivative operator.

**a**)${q}_{4}=0.8586$ and (

**b**)${q}_{4}=0.9495$, (

**c**) ${q}_{4}=0.9999,$ and (

**d**)${q}_{4}=1.0$.

## 5. Conclusions

_{0}and 𝑏

_{0}according to the order of the RL derivative operator (𝑞

_{4}). We show that in the limit 𝑞

_{4}

`→`1.0, the system tends to the high-power region.

_{4}= 0.8486, 0.9495, 0.9999, and 1.0 with the parameters of the external force 𝑎

_{0}and 𝑏

_{0}, and we observed that the power goes through the significant changes in its regions. In the next step, we analyzed the behavior of the fractional system dynamics based on 𝑞

_{4}for the values of 𝑎

_{0}= 0.2 and 𝑏

_{0}= 0.5. Therefore, we noted that values of 𝑞

_{4}in the intervals ${q}_{4}$ and in the intervals $\left. [0.85,0.8911\right]$, [0.9272, 0.9439], [0.945, 0.9459], [0.9538,0.9548], [0.9606,0.9619], and [0.9801, 0.9833] have periodic behavior for displacement (x

_{1}) and velocity (x

_{2}). Thus, we used K

_{c}to confirm these intervals obtained from the bifurcation diagrams. For other intervals, the behavior of the equations of motion was chaotic. The intervals were confirmed with the 0–1 Test and with the phase maps and the Poincaré maps with values of 𝑞

_{4}pertaining to those intervals.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Energy Harvesting model subjected to the kinematic excitation ${f}_{ext}\left(t\right)$ and magnetic poles symmetrically arranged. R is load resistance and $\phi $ is angular displacement.

**Figure 3.**${P}_{avg}$ behavior. (

**a**) ${a}_{0}\in \left. [0,1\right]\times {q}_{4}\in \left. [0.85,1.0\right]$ and (

**b**)${b}_{0}\in \left. [0,1\right]\times {q}_{4}\in \left. [0.85,1.0\right]$.

**Figure 4.**${P}_{avg}$ behavior. (

**a**) ${a}_{0}\in \left. [0,1\right]\times {b}_{0}\in \left. [0,1\right]$ with ${q}_{4}=0.8586$ and (

**b**) ${a}_{0}\in \left. [0,1\right]\times {b}_{0}\in \left. [0,1\right]$ with ${q}_{4}=0.9495$.

**Figure 5.**(

**a**) ${a}_{0}\in \left. [0,1\right]\times {b}_{0}\in \left. [0,1\right]$ with ${q}_{4}=0.9999$ and (

**b**) ${a}_{0}\in \left. [0,1\right]\times {b}_{0}\in \left. [0,1\right]$ with${q}_{4}=1.0$.

**Figure 6.**(

**a**) Bifurcation diagram for displacement ${x}_{1}$ and (

**b**) ${K}_{c}$ of displacement (x

_{1}).

**Figure 8.**Representation of $\left({x}_{1}\times {x}_{2}\right)$ systems of the Equation (8), the black line is portrait phase and the red dot is the Poincaré Map. (

**a**) ${q}_{4}=0.8586$ and (

**b**) ${q}_{4}=0.9495$.

**Figure 9.**Representation of $\left({x}_{1}\times {x}_{2}\right)$ systems of the Equation (8), the black line is the portrait phase and the red dot is the Poincaré Map. (

**a**) ${q}_{4}=0.9999$ and (

**b**)${q}_{4}=1.0$.

**Figure 10.**Representation of $\left({x}_{1}\times {x}_{4}\right)$ is displacement x of the beam and ${x}_{4}$ represents z for the Bouc–Wen damping. The hysteresis curves of the Equation (8). (

**a**) ${q}_{4}=0.8586$ and (

**b**) ${q}_{4}=0.9495$, (

**c**) ${q}_{4}=0.9999$ and (

**d**) ${q}_{4}=1.0$.

**Table 1.**Parameters used for numerical analysis of fractional differential equations described in Equation (6). Adapted from [8].

Parameter | Values | Parameter | Values |
---|---|---|---|

$\eta $ | 0.01 | $\lambda $ | 0.01 |

${k}_{1}$ | 0.25 | $\kappa $ | 0.5 |

$\chi $ | 0.05 | $A$ | 1.0 |

$D$ | 1.0 | $\beta $ | 0.55 |

$\omega $ | 1.1 | $\gamma $ | 0.45 |

$n$ | 3 | ${f}_{0}$ | 0.2 |

a | 0.5 |

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**MDPI and ACS Style**

Ribeiro, M.A.; Balthazar, J.M.; Lenz, W.B.; Felix, J.L.P.; Litak, G.; Tusset, A.M. Fractional Dynamical Behavior of an Elastic Magneto Piezo Oscillator Including Non-Ideal Motor Excitation. *Axioms* **2022**, *11*, 667.
https://doi.org/10.3390/axioms11120667

**AMA Style**

Ribeiro MA, Balthazar JM, Lenz WB, Felix JLP, Litak G, Tusset AM. Fractional Dynamical Behavior of an Elastic Magneto Piezo Oscillator Including Non-Ideal Motor Excitation. *Axioms*. 2022; 11(12):667.
https://doi.org/10.3390/axioms11120667

**Chicago/Turabian Style**

Ribeiro, Mauricio A., Jose M. Balthazar, Wagner B. Lenz, Jorge L. P. Felix, Grzegorz Litak, and Angelo M. Tusset. 2022. "Fractional Dynamical Behavior of an Elastic Magneto Piezo Oscillator Including Non-Ideal Motor Excitation" *Axioms* 11, no. 12: 667.
https://doi.org/10.3390/axioms11120667