^{1}

^{2}

^{3}

^{4}

^{1}

^{2}

^{3}

^{4}

In this paper, the maximum obtainable energy from a galloping cantilever beam is found. The system consists of a bluff body in front of wind which was mounted on a cantilever beam and supported by piezoelectric sheets. Wind energy caused the transverse vibration of the beam and the mechanical energy of vibration is transferred to electrical charge by use of piezoelectric transducer. The nonlinear motion of the Euler–Bernoulli beam and conservation of electrical energy is modeled by lumped ordinary differential equations. The wind forces on the bluff body are modeled by quasisteady aeroelasticity approximation where the fluid and solid corresponding dynamics are disconnected in time scales. The linearized motion of beam is limited by its yield stress which causes to find a limit on energy harvesting of the system. The theory founded is used to check the validity of previous results of researchers for the effect of wind speed, tip cross-section geometry, and electrical load resistance on onset speed to galloping, tip displacement, and harvested power. Finally, maximum obtainable average power in a standard RC circuit as a function of deflection limit and synchronized charge extraction is obtained.

Piezoelectric energy harvester uses the ambient energy and transfers it into electric charge [

The schematic of the system is shown in Figure

Schematic of the galloping piezoelectric energy harvester for the case of wind direction normal to the cantilever beam.

By assuming the following functions for motion and voltage,

The integration of equations (

Before going further to derive the optimal values of the system parameters, same as of Tan and Yan [

The numerical (solving equations (

Analytical solutions (dash lines), numerical solutions (symbols), and corrected solutions (lines) of the amplitudes of the tip displacement versus the electrical impedance and free stream velocity.

The variation in the harvested power with the electrical impedances at different free stream velocities from the analytical and numerical solutions is revealed in Figure ^{−1} watts, and for other cases, the order of the harvested power before failure is milliwatts. The results are in a good agreement with the experimental results [

Analytical solutions (dash lines), numerical solutions (symbols), and corrected solutions (lines) of the amplitudes of the harvested power versus the electrical impedance and free stream velocity.

By differentiating equation (_{max}/∂

Analytical solutions, numerical solutions, and corrected solutions of the amplitudes of the harvested power versus the parameter

Analytical solutions (dash lines), numerical solutions (symbol), and corrected solutions (lines) of the amplitudes of the harvested power versus the parameter

When the value of expression under the square root in eqution (_{o} is negative (the velocities higher than 16.1 m/s), the optimal values of the

As for the current parameters, this value of the

Analytical solutions for galloping-based piezoelectric energy harvesters with various interfacing circuits are summarized in Table

Regular circuits.

Name | Output | Schematic |
---|---|---|

Standard RC | ||

Synchronized charge extraction | ||

For the case of wind direction parallel to the cantilever beam (see Figure

Schematic of the galloping piezoelectric energy harvester for the case of wind direction parallel to the cantilever beam.

Various circuit interfaces are shown in Table

New circuit interfaces.

Name | Output | Schematic |
---|---|---|

Parallel-synchronized | ||

Series-synchronized | ||

Maximum obtainable various galloping piezoelectric energy harvester for the case of wind direction parallel to the cantilever beam.

Nomenclature.

Symbol | Description |
---|---|

_{b} | 2_{n}/^{2} (_{t}) |

1/^{2}_{t}) | |

_{n}^{2}/^{2} (_{t}) | |

Θ | _{t}) |

Air density | |

Length of the beam | |

_{tip} | |

_{tip} | |

_{base} | |

_{tip} | Width of the tip body |

_{tip} | Length of the tip body |

_{1}, _{3} | Aerodynamic force coefficients |

Load resistance | |

Quality factor | |

Damping ratio of the structure | |

Mode shape of the structure | |

Electromechanical coefficient of piezoelectric material | |

Yield strength of piezoelectric material | |

_{p} | Capacity of piezoelectric layer |

Wind velocity | |

Piezoelectric voltage | |

Ω | Angular velocity of the motion |

First natural angular velocity of the cantilever beam | |

Cantilever-beam displacement |

In this study, the nonlinear model of the galloping cantilever beam used for piezoelectric energy harvesting is simulated numerically with respect to the failure criteria as a limit of the maximum obtainable power.. The ideal case of such system is compared with the case of maximum stress limited due to the yielding stress of piezoelectric material. The results show that the mechanical limits of the system do not allow us to obtain the anticipated values in theory, and the feasible values are 2-3 orders of magnitude lower than prediction values. Hence, the fracture limitation should be considered in the process of the design of galloping-based energy harvesters with piezoelectric materials. Furthermore, the current research proposes for engineering applications, and designing the control system for the amplitude of galloping is necessary as well. Finally, maximum obtainable average power in a standard RC circuit as a function of deflection limit and synchronized charge extraction is obtained. In addition, four electrical interfaces in galloping-based energy harvesters are assessed. The results are for a feeble coupling SCE circuit, which is reasonable at higher wind while SSHI suits low wind speed. The standard circuit is suggested for strong electromechanical pairing, and the SCE has the best strength against the wind and can produce the highest value of power.

No data were used to support this study.

The authors declare that they have no conflicts of interest.