# Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{31}and d

_{33}simultaneously. In this design, proof mass can be mounted on the duralumin beam in between piezoelectric patches and the screw side. This variation in fixing the proof mass endows flexibility onto the natural frequencies of the PEGs. As per the mechanical vibration input, the first natural frequency can be adjected within the limit for high power output. This design provides flexibility and enhances the output power options compared to the other previous designs. The mechanical vibrations are used as an input parameter. This study concerns the new design of a generator, in which proof mass plays a key role in achieving higher power output. The analysis is also carried out on the electrical load dependency.

## 2. Description of Model Parameters and Electric Scheme

#### Working Principle of the Model

## 3. Theoretical Description of the Model of Composite Elastic, Electroelastic and Acoustic Media by FE Simulation

- ${u}_{i}$are the components of the displacement vector;
- ${\sigma}_{ij}$are the components of the stress tensor;
- $f$ are the components of the vector of the density of mass forces;
- ${D}_{i}\text{}$are the components of the electric induction vector;
- ${c}_{ijkl}$are the components of the fourth rank tensor of the elastic moduli;
- ${e}_{ikl}$are the components of the third rank tensor of piezoelectric coefficients;
- ${\epsilon}_{kl}$are the components of strain tensor;
- ${E}_{k}$are the components of the electric field vector;
- ${\phi}_{k}\text{}$is the electric potential;
- ${\u03f5}_{ik}\text{}$are the components of the dielectric constants tensor;
- $\alpha ,\beta ,\mathsf{\zeta}\text{}$are non-negative damping coefficients (the value of ζ
_{d}is used in ANSYS software).

## 4. Modeling

_{1}= 0.7 10

^{11}, Pa, ρ

_{1}= 2600 kg/m

^{3}, ν

^{1}= 0.33; and fiberglass: E

_{2}= 0.06 10

^{11}, Pa, ρ

_{2}= 1600 kg/m

^{3}, ν

_{2}= 0.33) and corresponding thickness options h = {1; 2} mm base plate. In this case, the attached mass was fixed in the position L

_{m}= 150 mm and its value was M = 0.5 gr, 30 gr. As an active load in the form of resistance, R for each piezoelectric element was taken, equal to 1000 Ω.

^{2}for all structural units at the corresponding harmonics. For the first and second vibration modes, the dependences of their value on the mass and location of the load were calculated. The corresponding graphs (Figure 5) show that the lowest frequency values are in the range of 212–220 Hz for the first vibration mode and values of 501–510 Hz can be achieved with the central location of the load and its maximum mass of 9 g. This, at the point of attachment of the load, will be maximum, which is shown in Figure 6. Figure 7 shows the calculated parameters of the dependence of the output voltage at the electrodes of piezocylinders (U

_{1}) and piezoelectric elements in the form of plates (U

_{2}and U

_{3}), respectively, with their calculated arrangement from left to right (see Figure 1). In the calculations, the value of the active load of the corresponding PE was taken to be 1000 Ω for one vibration mode.

_{i}of the base 7 (Figure 1) and its thickness h

_{i}. The proof mass varied within M = 0.5 gr, 30 gr. The place of fixation-proof mass L

_{m}= 150 mm is the point in the middle of the base. Accordingly, at fixed dimensions of the load, the specific density of the material was calculated. It was assumed that the most sensitive characteristic for changing the natural frequency of PEG vibrations is the value of the mass-proof mass. The modulus of elasticity of the volume-proof mass was taken equal to the properties of the base. The base modeling was considered, using the following properties of duralumin for calculations: E

_{1}= 0.7 10

^{11}, Pa, ρ

_{1}= 2600 kg/m

^{3}, ν

_{1}= 0.33; and using the properties of fiberglass: E

_{2}= 0.06 10

^{11}, Pa, ρ

_{2}= 1600 kg/m

^{3}, ν

_{2}= 0.33. The thickness of the base (7) was assumed to be h

_{1}= 2 mm and h

_{2}= 1 mm for calculations. Thus, four options for the layout of parameters for modeling were used: 1—E

_{1}, h

_{1}; 2—E

_{1}, h

_{2}; 3—E

_{2}, h

_{1}; 4—E

_{2}, h

_{2}. The results of numerical calculations obtained on the basis of the modal analysis carried out are presented on Figure 8. An analysis of the obtained frequency dependences shows the following: with an increase in mass for all design simulation options, the first natural frequency decreases. Therefore, with a conditionally small mass of 0.5 gr, the first natural frequency for options was 1—287.4 Hz; 2—154 Hz; 3—122.8 Hz; 55.3 Hz. In this case, the first natural frequency with a mass of 30 gr was, respectively, 1—143.4 Hz; 2—59.7 Hz; 3—50 Hz; 18.5 Hz. In a comparative analysis for all calculation options, the first frequency changed, respectively, for variations, more than 1–2 times, 2–2.57 times, 3–2.45 times and 4–2.97 times. Thus, with various initial parameters of the properties of the base model, the use of this PEG is possible in various loading ranges, both in the low-frequency region up to 50 Hz and in the region of higher frequencies, using only the first oscillation mode up to 287 Hz. The loading mode in the region of more than 50 Hz involves the use of devices for mechanical excitation of oscillations, for example, rotary motors with magnetic media. In addition, these modes of operation of the PEG can be used as the use of the PEG in the form of vibration sensors of the impulse action on the structure in a certain fixed frequency range.

## 5. Experimental Probe

#### 5.1. The Principle of Operation of the Laboratory Test Setup LTS -01

- (i)
- a range of measurable lateral displacements of the PEG substrate from 0 to 5 mm;
- (ii)
- frequency of forced oscillations from 1 to 1000 Hz;
- (iii)
- linear range of forced vibration amplitudes from 20 to 1000 Hz;
- (iv)
- the sensitivity limit of the optical displacement sensor is not less than 5 μm;
- (v)
- electric voltage at the input of the electromagnetic exciter of oscillations from 0.1 to 10 V.

#### 5.2. PEG Results Validation

#### 5.3. Method of Natural Modeling of the Oscillatory Processes of PEG

_{Exp}− ω

_{Sol})/ω

_{Sol}] × 100%.

## 6. Conclusions

_{1}2138.9 μW. For the PE plate type for this frequency 39 Hz, the maximum peak power was P

_{2}446.9 μW and P

_{3}423.2 μW. When conducting a comparative analysis with literature data, the analysis shows that the output parameters of the generator are at an average level and may require further modification of the design.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Structure scheme of PEG with proof mass: 1—piezocylinder; 2,3—plate piezoelectric elements; 4—rigid base of the generator; 5—PEG fixing supports; 6—L-clamping bar; 7—base; 8—proof mass; 9—screw.

**Figure 2.**Electric scheme of axial-type PEG under active electric load and structure scheme of PEG with proof mass: 1—piezoelectric cylindrical element; 2—piezoelectric bimorph element; 3—substrate; 4—proof mass; 5—clamping bar of PEs (1) fixing; 6—place of PEG fixing. (B) is the movable base; p is the PE’s polarization direction.

**Figure 5.**Dependence of 1 (

**a**) and 2 (

**b**) natural frequencies of PEG on the magnitude and location of proof mass at the lowest frequency.

**Figure 6.**Dependence of 1 (

**a**) and 2 (

**b**) natural frequencies of PEG on the magnitude and location of proof mass.

**Figure 7.**Dependence of the output voltage U on the electrodes of the corresponding PE at an active load of 1000 Ω for 1 vibration mode. Respectively, for (

**a**) piezocylinders (U

_{1}) and (

**b**,

**c**) piezoelectric elements in the form of plates (U

_{2}and U

_{3}).

**Figure 8.**Dependence 1 of the first natural frequency of the PEG on various variations in the parameters of the base plank (E

_{i}—the modulus of elasticity and h

_{i}—the height of the base).

**Figure 9.**Block scheme of the LTS -01 laboratory test setup for studying the PEG output characteristics: 1—computer with monitor; 2—AFG 3022B Tektronix signal generator; 3—power amplifier LV-102; 4—electromagnetic vibrator VEB Robotron 11077; 5—vibrator working table for fixing the model; 6—RF603 optical linear displacement transducer; 7—optical sensor switching unit (6); 8—optical linear displacement transducer optoNCDT; 9—optical sensor switching unit (8); 10—external ADC/DAC E14-440D module; 11—bank of electric load resistances Rl; 12—the base of the investigated object; 13—matching path; 14—piezoelectric generator of axial type; 15—grounding path.

**Figure 10.**General view of laboratory test setup LTS -01 for studying the output parameters of PEG. 1—computer with monitor; 2—AFG 3022B Tektronix signal generator; 3—power amplifier LV-102; 4—electromagnetic vibrator VEB Robotron 11077; 5—vibrator working table for fixing the model; 6—RF603 optical linear displacement transducer; 7—optical sensor switching unit (6); 8—optical linear displacement transducer optoNCDT; 9—optical sensor switching unit (8); 10—external ADC/DAC E14-440D module; 11—bank of the active electric load resistance R1–R6 of the corresponding PEG elements; 12—the base of the investigated object; 13—matching path; 14—piezoelectric generator of axial type; 15—grounding path; 16—piezoelectric cylindrical type; 17—piezoelectric elements of the plate type; 18—tripods for mounting optical sensors.

**Figure 12.**Visualization of the process of measuring the PEG output characteristics during sweeping.

**Figure 13.**Dependence of the output voltage on the magnitude of the load. (

**a**) Piezocylinders and (

**b**) piezoplates.

**Figure 14.**Dependence of the output power P on the value of the electric load, (

**a**) Piezocylinders and (

**b**) piezoplates.

No. | Name | Geometric Parameters, mm | |||||
---|---|---|---|---|---|---|---|

1 | Piezocylinder | R_{pc} | 14 | h_{pc} | 12.5 | ||

2, 3 | Piezoelements | L_{p} | 20 | b_{p} | 15 | h_{p} | 0.5 |

4 | Rigid base | L | 300 | B | 50 | H | 25 |

t | 2 | ||||||

5 | Fixing supports | L_{1} | 50 | b_{1} | 25 | h_{1} | 25 |

t | 2 | ||||||

6 | L-clamping bar | L_{L} | 74 | b_{L1} | 25 | b_{L2} | 24 |

t_{1} | 15 | h_{L} | 2 | ||||

d_{p} | 10 | ||||||

7 | Base | L_{b} | 260 | b_{m} | 14 | h_{m} | 1.5 |

8 | Mass | L_{m} | 16 | b_{m} | 10 | h_{m} | 4 |

9 | Screw | L_{d} | 44 | R_{d} | 13 |

№ | PEG Element | Material | ρ, kg/m^{3} | E × 10^{11}, Pa | ν |
---|---|---|---|---|---|

1 | Piezocylinders | CTS-19 | 7280 | - | 0.33 |

2, 3 | Piezoelements | CTS-19 | 7280 | - | 0.33 |

4 | Rigid base | duralumin | 2600 | 0.7 | 0.33 |

5 | Fixing supports | duralumin | 2600 | 0.7 | 0.33 |

6 | L-clamping bar | steel | 7700 | 2.1 | 0.33 |

7 | Base | duralumin | 2600 | 0.7 | 0.33 |

8.1 | Proof mass | plastic | 1600 | 0,06 | 0.33 |

8.2 | Proof mass | duralumin | 2600 | 0.7 | 0.33 |

8.3 | Proof mass | steel | 7700 | 2.1 | 0.33 |

9 | Screw | steel | 7700 | 2.1 | 0.33 |

R | Active electric load | resistor | $\text{}1\text{}\mathrm{k}\mathsf{\Omega}-2\text{}\mathrm{M}\mathsf{\Omega}$ | ||

damping value | $\xi =0.031$ |

**Table 3.**Elastic moduli C

^{E}

_{pq}(×10

^{10}Pa), piezoelectric coefficients e

_{kl}(C/m

^{2}) and relative permittivity ε

^{ξ}

_{kk/ε0}of piezoceramics (based on measurements at room temperature).

Piezoelement Type | C^{E}_{11} | C^{E}_{12} | C^{E}_{13} | C^{E}_{33} | C^{E}_{44} | e_{31} | e_{33} | e_{15} | $\frac{{\mathit{\epsilon}}_{11}^{\mathit{\zeta}}}{{\mathit{\epsilon}}_{0}}$ | $\frac{{\mathit{\epsilon}}_{33}^{\mathit{\zeta}}}{{\mathit{\epsilon}}_{0}}$ |
---|---|---|---|---|---|---|---|---|---|---|

CTS-19 | 10.9 | 6.1 | 5.4 | 9.3 | 2.4 | −4.9 | 14.9 | 10.6 | 820 | 840 |

**Table 4.**Elastic compliance of S

^{E}

_{pq}(×10

^{−12}PA), piezoelectric d

_{fp}moduli (pC/N) and relative permittivity ε

^{σ}

_{kk}/ε

_{0}of piezoceramics (based on measurements at room temperature).

Pizoelement Type | S^{E}_{11} | S^{E}_{12} | S^{E}_{13} | S^{E}_{33} | S^{E}_{44} | d_{31} | d_{33} | d_{15} | $\frac{{\mathit{\epsilon}}_{\mathbf{11}}^{\mathit{\sigma}}}{{\mathit{\epsilon}}_{\mathbf{0}}}$ | $\frac{{\mathit{\epsilon}}_{\mathbf{33}}^{\mathit{\sigma}}}{{\mathit{\epsilon}}_{\mathbf{0}}}$ |
---|---|---|---|---|---|---|---|---|---|---|

CTS-19 | 15.1 | −5.76 | −5.41 | 17.0 | 41.7 | −126 | 307 | 442 | 1350 | 1500 |

Mode | Stimulation | Experimental | Error % |
---|---|---|---|

1 Natural Frequency (NF) | 283 Hz | 302 Hz | 6.29 |

2 Natural Frequency (NF) | 581 Hz | 587 Hz | 1.1 |

Deflection at 1 NF | 1.59 mm | 1.5 mm | 5.6 |

Deflection at 2 NF | 0.72 mm | 0.7 mm | 2.7 |

**Table 6.**Comparison of the experimental and numerical values 1 and 2 of the natural frequencies of PEG oscillations at an active load of PE of 10 kΩ and a load value of 3.71 g at different positions.

Location of Proof Mass L_{m}, mm | |||||
---|---|---|---|---|---|

150 | 170 | 190 | 210 | 230 | |

1 natural frequency, Hz | |||||

Experiment ω_{Exp} | 255.0 | 258.0 | 267.0 | 288.0 | 302.0 |

Calculation ω_{Sol} | 252.2 | 253.0 | 261.2 | 274.8 | 292.2 |

∆, % | 1.1 | 2.0 | 2.2 | 4.8 | 4.0 |

2 natural frequency, Hz | |||||

Experiment ω_{Exp} | 581.0 | 576.0 | 560.0 | 563.0 | 587.0 |

Calculation ω_{Sol} | 564.4 | 547.5 | 538.1 | 546.0 | 574.2 |

∆, % | 2.9 | 5.2 | 4.1 | 3.1 | 2.2 |

**Table 7.**Experimental measurements of the voltage amplitude on the PE electrodes at the corresponding vibration frequencies and the corresponding active electric load.

PE N | $\mathbf{Voltage}\text{}\mathbf{Amplitude}\text{}\left(\mathbf{V}\right)\text{}\mathbf{across}\text{}\mathbf{the}\text{}\mathbf{Electrodes}\text{}\mathbf{at}\text{}\mathbf{Oscillation}\text{}\mathbf{Frequency}\text{}\mathbf{of}\text{}\mathbf{39}\text{}\mathbf{Hz}\text{}\mathbf{and}\text{}\mathbf{the}\text{}\mathbf{Corresponding}\text{}\mathbf{Active}\text{}\mathbf{Electric}\text{}\mathbf{Load}\text{}\mathbf{R}\text{}\left(\mathbf{k}\mathbf{\Omega}\right)$ | |||||
---|---|---|---|---|---|---|

$10\text{}\mathrm{k}\mathsf{\Omega}$ | $51\text{}\mathrm{k}\mathsf{\Omega}$ | $75\text{}\mathrm{k}\mathsf{\Omega}$ | $150\text{}\mathrm{k}\mathsf{\Omega}$ | $300\text{}\mathrm{k}\mathsf{\Omega}$ | $2000\text{}\mathrm{k}\mathsf{\Omega}$ | |

1 | 0.14625 | 0.294872 | 0.355079 | 0.392063 | 0.560649 | 1.317638 |

2 | 0.06685 | 0.096381 | 0.100432 | 0.222399 | 0.207643 | 0.187802 |

3 | 0.06505 | 0.085808 | 0.077982 | 0.1905 | 0.183093 | 0.180334 |

$\mathbf{Voltage}\text{}\mathbf{Amplitude}\text{}\left(\mathbf{V}\right)\text{}\mathbf{across}\text{}\mathbf{the}\text{}\mathbf{Electrodes}\text{}\mathbf{at}\text{}\mathbf{an}\text{}\mathbf{Oscillation}\text{}\mathbf{Frequency}\text{}\mathbf{of}\text{}\mathbf{107}\text{}\mathbf{Hz}\text{}\mathbf{and}\text{}\mathbf{the}\text{}\mathbf{Corresponding}\text{}\mathbf{Active}\text{}\mathbf{Electric}\text{}\mathbf{Load}\text{}\mathbf{R}\text{}\mathbf{\left(}\mathbf{k}\mathbf{\Omega}\mathbf{\right)}$ | ||||||

$10\text{}\mathrm{k}\mathsf{\Omega}$ | $51\text{}\mathrm{k}\mathsf{\Omega}$ | $75\text{}\mathrm{k}\mathsf{\Omega}$ | $150\text{}\mathrm{k}\mathsf{\Omega}$ | $300\text{}\mathrm{k}\mathsf{\Omega}$ | $2000\text{}\mathrm{k}\mathsf{\Omega}$ | |

1 | 0.043576 | 0.094412 | 0.061982 | 0.117833 | 0.127969 | 0.467393 |

2 | 0.032606 | 0.043102 | 0.06355 | 0.136766 | 0.111208 | 0.081129 |

3 | 0.02439 | 0.082987 | 0.078731 | 0.109417 | 0.106021 | 0.106581 |

**Table 8.**Experimental measurements of the output power peak on the PE electrodes at the corresponding vibration frequencies and the corresponding active load.

PE N | Peak Power (μW) at the PE Electrodes at an Oscillation Frequency of 39 Hz and the Corresponding Active Load R (kΩ) | |||||
---|---|---|---|---|---|---|

$10\text{}\mathrm{k}\mathsf{\Omega}$ | $51\text{}\mathrm{k}\mathsf{\Omega}$ | $75\text{}\mathrm{k}\mathsf{\Omega}$ | $150\text{}\mathrm{k}\mathsf{\Omega}$ | $300\text{}\mathrm{k}\mathsf{\Omega}$ | $2000\text{}\mathrm{k}\mathsf{\Omega}$ | |

1 | 2138.9 | 1704.9 | 1681.1 | 1024.8 | 1047.8 | 868.1 |

2 | 446.9 | 182.1 | 134.5 | 329.7 | 143.7 | 17.6 |

3 | 423.2 | 144.4 | 81.1 | 241.9 | 111.7 | 16.3 |

Peak power (μW) at the PE Electrodes at an Oscillation Frequency of 107 Hz and the Corresponding Active Load R (kΩ) | ||||||

$10\text{}\mathrm{k}\mathsf{\Omega}$ | $51\text{}\mathrm{k}\mathsf{\Omega}$ | $75\text{}\mathrm{k}\mathsf{\Omega}$ | $150\text{}\mathrm{k}\mathsf{\Omega}$ | $300\text{}\mathrm{k}\mathsf{\Omega}$ | $2000\text{}\mathrm{k}\mathsf{\Omega}$ | |

1 | 189.89 | 174.78 | 51.22 | 92.56 | 54.59 | 109.23 |

2 | 106.31 | 36,43 | 53.85 | 124.7 | 41.22 | 3.29 |

3 | 59.49 | 135.04 | 82.65 | 79.81 | 37.47 | 5.68 |

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**MDPI and ACS Style**

Cherpakov, A.V.; Parinov, I.A.; Haldkar, R.K. Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base. *Appl. Sci.* **2022**, *12*, 1700.
https://doi.org/10.3390/app12031700

**AMA Style**

Cherpakov AV, Parinov IA, Haldkar RK. Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base. *Applied Sciences*. 2022; 12(3):1700.
https://doi.org/10.3390/app12031700

**Chicago/Turabian Style**

Cherpakov, Alexander V., Ivan A. Parinov, and Rakesh Kumar Haldkar. 2022. "Parametric and Experimental Modeling of Axial-Type Piezoelectric Energy Generator with Active Base" *Applied Sciences* 12, no. 3: 1700.
https://doi.org/10.3390/app12031700