STEM is taught by Dr. C. It is a fully project-based class, with a focus on both individual and group projects. Currently, I am working on my STEM I project, which involves resonant frequencies and energy generator optimization. Later in the year, we will work on our STEM II project, which involves creating assistive technologies for people living in an elder care facility.

**Overview:**
In essence, cantilever-based piezoelectric energy harvesters (PCEHs) operate optimally at the resonant frequency of the cantilever. However, the optimal resonant frequency has not been heavily researched.

** Abstract: **Cantilever-based piezoelectric energy generators (PCEHs) are most efficient at the resonant frequency of the cantilever, however, different length cantilevers have different resonant frequencies. A thorough understanding of how efficiency is correlated to cantilever properties can help to optimize PCEHs. The engineering need was to increase the efficiency of PCEHs, and the project objective was to employ frequency optimization for an increase in both output and efficiency of these generators. The determined frequency implies different cantilever lengths depending on the material, thickness, and other factors. A piezoelectric element was attached at the bend of a thin metal cantilever, and then force was applied to the cantilever at its resonant frequency. The power used to vibrate the cantilever and the power produced by the element was measured. Efficiency may be determined by dividing power out by power in and can be graphed as a function of frequency. The optimal frequency found was 46Hz, with an efficiency of 5.44%. This means that this optimized generator may be incorporated at any location with a frequency an integer number of octaves below 46Hz. Presently, these findings provide a basis for all cantilever-based piezoelectric research, to optimize efficiency and output. They also corroborate past research conducted by Xiong et. al. (2019), which indicated that optimal frequencies hover around 48 Hz.

**Phrase 1, Engineering need:** To increase the efficiency of piezoelectric energy generators

**Phrase 2, Engineering objective:** To employ frequency optimization for the increase in output of a piezoelectric energy generator.

**Background:** The piezoelectric effect is a material’s ability to produce an electric charge upon deformation (Carter & Kensley, n.d.). This deformation may be physical compression, torque, or stress. Some common natural piezoelectric materials include bone, amber, and wood. Although all piezoelectric materials share this property, some are more effective in producing a greater electrical charge with constant deformation. Generally, artificial materials such as Lead Zirconium Titanate (PZT) or Polyvinylidene Difluoride (PVDF) are the most effective (Zhao & You, 2014).

Every object has a natural frequency of oscillation, also called its resonant frequency. Applying a force to an object at its resonant frequency is the most efficient method of increasing oscillatory amplitude. At their resonant frequencies, the power output of the piezoelectric generators is optimal (Kim et al., 2013). When vibrated at that frequency or any number of octaves below its resonant Frequency, this optimal output can be attained.

Although it is known that for any configuration, its resonant frequency is optimal, but when comparing the optimized frequencies for multiple configurations, which is most efficient?

**Procedure: **First, multiple length rulers were precisely cut to ensure multiple data points. Secondly, once installed with the PZT clamped at the bend, the resonant frequency was determined using Audacity audio analysis software tools, Then, a stepper motor was programmed to rotate at a number of octaves below the determined frequency. To determine the power used to vibrate the cantilever, the power used to rotate the motor without the cantilever was subtracted from the power used when the cantilever was disturbed. Both of these quantities were determined using a MAX471 chip connected to an Arduino. Each measurement was the result of n=125,000 rapid measurements over the course of 5 seconds, which were averaged out to output one resulting power. To determine the power produced by the generator, the output was connected to a capacitor with a voltmeter attached. The generator ran for 15 seconds at a time, at which point Logger Pro would produce a line of best fit (which always had a correlation of above 0.99). This line, which describes voltage across the capacitor as a function of time, was then manipulated into a function describing energy as a function of time, which was used to determine the power of the generator over 15 seconds. Dividing the power out of the system by the power into the system, efficiency was determined. This technique provided valuable data, as well as reassurance that the output measurement methodology was solid, and that all devices were working as intended. However, it had two major flaws. First, due to the physical limitations of the stepper motor, it had to rotate a number of octaves below the resonant frequency. This introduced a major confounding variable, because the number of octaves could affect outcome. Secondly, duty cycle, or the percentage of time in a cycle when the impulse is active, was not constant. This confounding variable likely had the greatest impact on results, because a lower duty cycle may result in significantly lowered outputs, due to muffling of cantilever vibrations.
Electromagnet-Based Prototype
The second and final attempt to establishing a clear relationship between resonant frequency and efficiency successfully eliminated the two confounders present in initial testing. To accomplish this, the stepper motor was replaced with an electromagnet module, which had numerous advantages over the motor. First and foremost, the duty cycle of the electromagnet was set to 20% by the Arduino. Secondly, the electromagnet was able to operate at far higher frequencies that the motor, easily covering the necessary range to reach the resonant frequencies of the cantilevers. All other methodology is the same as was used in with the first prototype.

** Analysis: **To evaluate the statistical significance of the data shown in Table 1, a Hoeffding dependence coefficient test was applied to the data using the Wolfram Mathematica software. This is because the Hoeffding D test is well suited for non-linear non-monotonic data. Few tests exist for such data, leaving Hoeffding as a reliable option.

The data was found to be very significant, with a p<0.0001, indicating an extremely low chance of the data following the trend found by chance. The Hoeffding D coefficient was 0.32599, indicating that the null hypothesis that efficiency is independent from frequency can be rejected. Thus, the alternate hypothesis is accepted, meaning that the efficiency is strongly dependent on frequency.

** Discussion/Conclusions/Applications: **
**Overview: **
The optimal frequency was found to be 46Hz. With an efficiency of 5.44%, a PCEH with a resonant frequency of 46Hz is over 290 times more efficient than a PCEH with a resonant frequency of 9Hz, and about 7 times more efficient than a PCEH with a resonant frequency of 76Hz. Overall, little research has been conducted in the field of frequency optimization, with most studies focusing on materials and configuration optimization. One study, conducted by Xiong et. al. in 2019, concluded than the optimal frequency is 48Hz, which is withing a reasonable range of the results obtained in this paper. That study, however, focused primarily on structural changes to a cantilever beam and their effect on output, rather than exploring areas of application for current PCEH systems.
**Statistical Tests: **
A Hoeffding dependence coefficient test was used on the data to determine its significance. This is because the Hoeffding D test is well suited for non-linear non-monotonic data. Few tests exist for such data, leaving Hoeffding as a reliable option. The data was found to be overwhelmingly significant, with a p value below 0.0001, and a positive D coefficient.
**Limitations and Confounding Variables: **
The initial data-gathering prototype created, which employed a stepper motor, had major limitations and confounding variables. Firstly, the stepper motor, although precise, could not spin at over 5rpm, and with a dual attachment, this was converted into 10Hz. Due to this, resonant frequencies had to be divided by an integer value so that the motor could handle them. Although generally speaking, resonant properties remain nearly constant across octaves, at low levels, a difference in amplitudes may be examined. A second limitation was the duty cycle, which could not be controlled with a stepper motor. Both of these confounders were removed through the use of an electromagnet module in replacement of the motor. The module was able to generate frequencies of thousands of hertz, and the duty cycle was able to be set to a constant value.
**Applications: **
With the newly found quadratic regression for efficiency as a function of frequency, a vibration source may be more optimally fitted with a PCEH. For example, if a piece of machinery is losing energy to vibrations of x Hz, a PCEH with a resonant frequency of x Hz may not be optimal. In reality, any PCEH with a resonant frequency of x·n Hz, where n is a positive integer, would experience the resonant effect. However, only one value of n would be optimal. To find this value, the quadratic regression formula may be used.
**Future Research: **
Future research conducted in the field should focus on improving testing prototypes to increase accuracy and precision, as well as simply gathering more data. In addition, similar tests can be run for other PEG configurations.

** References (APA): **

Akopyan, V. A., Parinov, I. A., Rozhkov, E. V., Zakharov, Y. N., & Shevtsova, M. S. (2014). Energetic efficiency of cantilever type piezoelectric generators. *Springer Proceedings in Physics, 355*–363. https://doi.org/10.1007/978-3-319-03749-3_29

Brennan, E. (2021, December 12). *RC Circuit Formula Derivation Using Calculus. *Owlcation. Retrieved December 12, 2021, from https://owlcation.com/stem/RC-Circuit-Time-Constant-Analysis

Carter, R., & Kensley, R. (n.d.). *An introduction to piezoelectric transducers.* Piezo.com.
https://blog.piezo.com/hubfs/Manuals/Introduction-to-Piezoelectric-Transducers-Carter-Kensley.pdf

*Canva.* (2013). [Graphic design platform]. Canva. https://www.canva.com/

Circuit Diagram. (2021). [Circuit diagram software]. Circuit Diagram.https://www.circuit-diagram.org/

Ewald, W. (2021, February 13). *MAX471 Current Sensor.* Wolles Elektronikkiste. https://wolles-elektronikkiste.de/en/max471-current-sensor

Feenstra, J., Granstrom, J., & Sodano, H. (2008). Energy harvesting through a backpack employing a mechanically amplified piezoelectric stack. *Mechanical Systems and Signal Processing, 22*(3), 721–734. https://doi.org/10.1016/j.ymssp.2007.09.015

Kang, M. G., Jung, W. S., Kang, C. Y., & Yoon, S. J. (2016). Recent Progress on PZT Based Piezoelectric Energy Harvesting Technologies. *Actuators, 5*(1), 5. https://doi.org/10.3390/act5010005

*KS0320 Keyestudio Electromagnet Module (Black and Environmental-friendly) - Keyestudio Wiki. *(n.d.). Keystudio. https://wiki.keyestudio.com/KS0320_Keyestudio_Electromagnet_Module_(Black_and_Environmental-friendly)

Kim, D., Hong, S., Li, D., Roh, H. S., Ahn, G., Kim, J., Park, M., Hong, J., Sung, T.-hyun, & No, K. A spring-type Piezoelectric Energy Harvester. *RSC Advances, 3*(10), 3194. https://doi.org/10.1039/c2ra22554a

Odarno, L. (2021). *1.2 Billion People Lack Electricity. Increasing Supply Alone Won’t Fix the Problem. * World Resources Institute. https://www.wri.org/insights/12-billion-people-lack-electricity-increasing-supply-alone-wont-fix-problem

Libretexts. (2020, August 13). 3.1.1: Resonance Examples. Physics LibreTexts. https://phys.libretexts.org/Bookshelves/Waves_and_Acoustics/Book%3A_Sound_-_An_Interactive_eBook_(Forinash_and_Christian)/03%3A_Resonance/3.01%3A_Resonance/3.1.01%3A_Resonance_Examples

Minazara, E., Vasic, D., & Costa, F. (2008). Piezoelectric generator harvesting bike vibrations energy to supply portable devices. *Renewable Energy and Power Quality Journal, 1*(06), 508-513. doi:10.24084/repqj06.344

Murray, J. ( 2019). *Is the lithium-ion battery having a positive impact on the environment? *(2019, October 14). NS Energy. Retrieved October 30, 2021, from https://www.nsenergybusiness.com/features/lithium-ion-battery-environmental-impact/

StackPath. (n.d.). ElectronicDesign. https://www.electronicdesign.com/power-management/article/21801833/what-is-the-piezoelectric-effect

Xiong, Y., Song, F., & Leng, X. (2019). A piezoelectric cantilever-beam energy harvester (PCEH) with a rectangular hole in the metal substrate. *Microsystem Technologies, 26*(3), 801–810. https://doi.org/10.1007/s00542-019-04608-8

Yamashita, S. (1978). Portable piezoelectric electric generating device. US4091302A. United States Patent 4,091,302. Patent office unknown.

Yang, Z., Zhou, S., Zu, J., & Inman, D. (2018). High-Performance Piezoelectric Energy Harvesters and Their Applications. *Joule, 2*(4), 642–697. https://doi.org/10.1016/j.joule.2018.03.011

Yusran, A., Madia, A., & Zainuddin, Z. (2020). Prototype of Electrical Generator device from piezoelectric with serial configuration - based on mechanical pressure. IOP Conference Series: *Materials Science and Engineering, 875,* 012033. https://doi.org/10.1088/1757-899x/875/1/012033

Zhao, J., & You, Z. (2014). A Shoe-Embedded Piezoelectric Energy Harvester for Wearable Sensors. *Sensors, 14*(7), 12497–12510. https://doi.org/10.3390/s140712497