# Rational Design Approach for Enhancing Higher-Mode Response of a Microcantilever in Vibro-Impacting Mode

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Enhanced VEH Configuration

_{i}is cross-sections of structure components, i = 1, 2, m. If one employs the method of nonlinear programming (e.g., gradient projection), the following inequality-shaped constraints should be incorporated:

^{+}), 2nd (OPT II

^{+}), and 3rd (OPT III

^{+}) natural frequencies. These optimal microcantilever structures would attain increased natural frequencies if compared to their counterparts with constant cross-sections. Should one like to use microcantilevers with reduced natural frequencies (with respect to the constant cross-section versions), symmetrically inverted optimal structures would be obtained, as depicted in Figure 1b for the operation in the 1st (OPT I

^{−}), 2nd (OPT II

^{−}), and 3rd (OPT III

^{−}) natural frequencies.

^{+}), the minimum cross-section is always located at the distance of 0.24 l from the clamping site, whereas for the structure that is optimal for operation at the increased 3rd frequency (OPT III

^{+}), the distance is 0.15 l and 0.5 l, respectively (l is the microcantilever length). A more detailed analysis of optimal microcantilever structures is presented in Figure 1, which reveals that the recurrence of maximum and minimum cross-sections corresponds to the positions of particular (maximum amplitude and nodal) points of the respective transverse vibration modes.

^{−}). This structural simplification was performed with the objective to develop an easily manufacturable VEH configuration, which retains the prescribed dynamic characteristics of the optimal structure. Thus, the proposed rational configuration of the VEH substrate can be considered as a near-optimal version. More precisely, the intricate optimal substrate shape was replaced by a simplified design with the hump located at 0.24 l. Contrary to the optimal substrate structure, the rational structural configuration is amenable to conventional (micro) fabrication methods used for piezoelectric material deposition or bonding via manual assembly, and also ensures straightforward segmentation of the piezoelectric layer at the strain node of the 2nd vibration mode.

## 3. Frequency Response Measurement

#### 3.1. Experimental Setup

#### 3.2. Experimental Results

_{1 OPT0}(ω—excitation frequency, ω

_{1 OPT0}—the 1st natural frequency of optimal microcantilever OPT 0) and amplitudes of transverse vibrations were registered, enabling the determination of the first three natural frequencies of the microcantilever. Measurements were performed with the freely-vibrating microcantilevers (non-impacting) as well as for vibro-impacting ones, i.e., microcantilevers impacting against a stopper located at points approximately coinciding with the nodes of the 2nd (x/l = 0.8) and 3rd (x/l = 0.9) modes of transverse vibrations of a cantilever with constant cross-sections (Figure 5).

_{1 OPT0}) accompanied by the lowest amplitude peak. Optimal microcantilever OPT III was characterized by the lowest 1st natural frequency of 0.77 ω/ω

_{1 OPT0}, which constituted a 23% reduction with respect to OPT 0, while the amplitude peak was higher by more than six times. Microcantilever OPT II was somewhat in the middle between the other configurations in terms of 1st natural frequency and modal amplitude. Analysis of the shift of the 2nd natural frequency revealed that the OPT II configuration possessed a similar 2nd natural frequency (6.53 ω/ω

_{1 OPT0}) compared to its respective constant cross-section counterpart, OPT 0 (6.27 ω/ω

_{1 OPT0}). The same trend was observed for the 3rd natural frequency of microcantilever OPT III (18.24 ω/ω

_{1 OPT0}), which was very close to the corresponding natural frequency of OPT 0 (17.8 ω/ω

_{1 OPT0}).

## 4. Study of Piezoelectric Vibro-Impacting VEH Prototype Based on Rationally-Shaped Microcantilevers

^{−}(Figure 2), intended for operation at the reduced 2nd natural frequency.

_{31}) mode. A rational design approach that is adopted here implies that an optimally-shaped zone of increased cross-section (with center located at x/l = 0.24) in the optimal microcantilever OPT II

^{−}(Figure 1b) is replaced by a hump-like zone, the length of which varies from 0.01 l to 0.07 l, while sections of the structure outside the hump retain constant cross-sections (overall length and width of the microcantilever do not change with respect to the initial microcantilever of constant cross-sections).

^{2}. The mathematical expression of the approximated random signal with the coefficient of determination of R

^{2}= 0.7869 is presented below:

_{1 OPT0}, 2nd—4.52 ω/ω

_{1 OPT0}, 3rd—13.17 ω/ω

_{1 OPT0}and 1st—1.16 ω/ω

_{1 OPT0}, 2nd—7.15 ω/ω

_{1 OPT0}, 3rd—19.9 ω/ω

_{1 OPT0}, respectively.

_{1 OPT0}, 2nd—3.61 ω/ω

_{1 OPT0}, 3rd—9.95 ω/ω

_{1 OPT0}. The discrepancy between the measured and simulated frequency values was attributed to non-ideal clamping of the fabricated device, which contributed to the decrease of the measured natural frequencies. A VEH prototype OPT 0 was also fabricated following the same procedure but using the nickel microcantilever of constant cross-sections as a substrate.

_{1 OPT0}). A comparison of voltage responses generated by both prototypes operating in vibro-impact mode (Figure 15b and Figure 16b) indicated a markedly larger content of higher-order harmonics in the case of OPT RAT response, though its vibration amplitudes were comparable or larger with respect to the OPT0 case. This demonstrated that the OPT RAT prototype undergoes self-excitation at higher vibration modes with the dominant 2nd mode of transverse vibrations. Thus, base excitation of the proposed rationally-shaped vibro-impacting VEH at low 1st natural frequency (0.64 ω/ω

_{1 OPT0}) leads to self-excitation of the vigorous vibrations at much higher frequencies (>3.6 ω/ω

_{1 OPT0}), which translates into higher deflection velocities and strain rates. As a result, the observed amplification of the higher-order mode response in the OPT RAT device would lead to higher power output since it is strongly dependent on the strain rate in the piezoelectric material. In addition, it would also improve the efficiency of mechanical-to-electrical energy conversion, which can be explained by means of the following relationships [21,22]:

_{n}and ω

_{op}are the natural and operational (excitation) frequencies, respectively; C

_{p}and ${k}_{e}^{2}$ are the capacitance and alternative electromechanical coupling coefficient of the piezoelectric transducer, respectively; ζ

_{T}, ζ

_{m}, ζ

_{e}are total, mechanical and electrical damping ratios, respectively.

_{e}), which in turn improves its efficiency. This means that, with higher vibration frequencies, more mechanical energy is removed from VEH during energy harvesting process. Higher vibration frequencies of the piezoelectric transducer result in lower matched load resistance (R

_{ML}= 1/ω

_{n}C

_{p}), which enhances the average power generated by VEH (P

_{av}= (V

_{rms})

^{2}/R

_{ML}).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Optimal cantilever structures for the operation at increased (

**a**) and reduced (

**b**) natural frequencies of transverse vibrations (from top to bottom: the 1st (OPT I), the 2nd (OPT II) and the 3rd (OPT III) natural frequencies).

**Figure 2.**Schematics of the optimal OPT II

^{−}(

**top**) and the proposed rational (

**bottom**) vibration energy harvester (VEH) substrate configurations.

**Figure 3.**Photos of experimentally tested microcantilevers of three different configurations: OPT 0; OPT II and OPT III.

**Figure 4.**Photo of experiment setup: 1—Polytec OFV 072 Microscope adapter with Polytec OFV 073 Microscope scanner unit and Polytec OFV 071 Microscope manual positioner; 2—Nikon microscope Eclipse LV100 with digital video camera Pixelink PL-A662; 3—piezoelectric actuator PSt 150/4/20VS9 (Piezomechanik GmbH, Munich, Germany) with a researched object; 4—Polytec OFV 512 fiber-optic interferometer; 5—Polytec MSV-Z-40 Scanner controller; 6—PC oscilloscope PicoScope 3424 (PicoTechnology Ltd.,GB); 7—Function waveform generator Agilent 33220A; 8—Linear amplifier EPA-104 (Piezo Systems Inc., Woburn, MA, USA); 9—Polytec OFV 5000 Vibrometer controller; 10—Polytec Vibrascan DAQ PC.

**Figure 5.**Stopper location at points approximately coinciding with the nodes of the 2nd (x/l = 0.8), 3rd (x/l = 0.9) modes of transverse vibrations and at the end of cantilever (x/l = 1).

**Figure 6.**Shift of the natural frequencies for different microcantilever configurations: (

**a**) shift of the 1st natural frequency; (

**b**) shift of the 2nd natural frequency.

**Figure 7.**Frequency responses of different microcantilever configurations when the rigid stopper is located at x/l = 0.78 ≈ 0.8.

**Figure 8.**Frequency responses of different microcantilever configurations when the rigid stopper is located at x/l = 0.87 ≈ 0.9.

**Figure 9.**Finite element model of the proposed rationally-shaped substrate (implemented in Comsol Multiphysics).

**Figure 10.**Schematics of the fabricated piezoelectric VEH prototypes: OPT 0 (

**top**), rational microcantilever structure (OPT RAT) (

**bottom**).

**Figure 11.**Comparison or real (

**blue**) and approximated (

**green**) random excitation signals (R

^{2}= 0.7869).

**Figure 12.**Response of the tip of randomly excited (

**a**) microcantilever OPT 0 and (

**b**) rational microcantilever OPT RAT.

**Figure 13.**Phase diagram of the tip of randomly excited (

**a**) microcantilever OPT 0 and (

**b**) rational microcantilever OPT RAT.

**Figure 14.**Microfabrication of microcantilever OPT RAT: (

**a**) deposition of sacrificial layer; (

**b**) deposition and patterning of photoresist; (

**c**) patterning of sacrificial layer and removal of photoresist; (

**d**) deposition and patterning of photoresist; (

**e**) patterning of sacrificial layer and removal of photoresist; (

**f**) deposition of nickel; (

**g**) deposition and patterning of photoresist; (

**h**) patterning of nickel and removal of photoresist; (

**i**) screen-printing of piezocomposite and removal of sacrificial layer.

**Figure 15.**Measured open-circuit voltages for the VEH prototype OPT 0 operating in (

**a**) freely vibrating mode; (

**b**) vibro-impacting mode with a stopper at x/l = 1 (1st piezocomposite segment (near the clamped part of the microcantilever)—blue line (1), 2nd segment (in the middle)—red line (2), 3rd (at the free end)—green line (3)).

**Figure 16.**Measured open-circuit voltages for the VEH prototype OPT RAT operating in (

**a**) freely vibrating mode; (

**b**) vibro-impacting mode with a stopper at x/l = 1 (1st piezocomposite segment (near the clamped part of the microcantilever)—blue line (1), 2nd segment (in the middle)—red line (2), 3rd (at the free end)—green line (3)).

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**MDPI and ACS Style**

Migliniene, I.; Ostasevicius, V.; Gaidys, R.; Dauksevicius, R.; Janusas, G.; Jurenas, V.; Krasauskas, P.
Rational Design Approach for Enhancing Higher-Mode Response of a Microcantilever in Vibro-Impacting Mode. *Sensors* **2017**, *17*, 2884.
https://doi.org/10.3390/s17122884

**AMA Style**

Migliniene I, Ostasevicius V, Gaidys R, Dauksevicius R, Janusas G, Jurenas V, Krasauskas P.
Rational Design Approach for Enhancing Higher-Mode Response of a Microcantilever in Vibro-Impacting Mode. *Sensors*. 2017; 17(12):2884.
https://doi.org/10.3390/s17122884

**Chicago/Turabian Style**

Migliniene, Ieva, Vytautas Ostasevicius, Rimvydas Gaidys, Rolanas Dauksevicius, Giedrius Janusas, Vytautas Jurenas, and Povilas Krasauskas.
2017. "Rational Design Approach for Enhancing Higher-Mode Response of a Microcantilever in Vibro-Impacting Mode" *Sensors* 17, no. 12: 2884.
https://doi.org/10.3390/s17122884