# Bandgap Calculation and Experimental Analysis of Piezoelectric Phononic Crystals Based on Partial Differential Equations

## TL;DR

This paper studies how piezoelectric phononic crystals reduce vibrations in plate-shell structures by calculating and validating bandgap properties using simulations and experiments.

## Contribution

The paper introduces a validated computational model for analyzing piezoelectric phononic crystals and their vibration-damping effects.

## Key findings

- Larger piezoelectric sheet areas lower the center band in the bandgap, while increased thickness widens the attenuation coefficient range.
- Negative capacitance significantly expands the low-frequency bandgap range.
- Experiments confirm a 4–5 dB vibration reduction in the 1710–1990 Hz range, validating the model's accuracy.

## Abstract

Focusing on the bending wave characteristic of plate–shell structures, this paper derives the complex band curve of piezoelectric phononic crystal based on the equilibrium differential equation in the plane stress state using COMSOL PDE 6.2. To ascertain the computational model’s accuracy, the computed complex band curve is then cross-validated against real band curves obtained through coupling simulations. Utilizing this model, this paper investigates the impact of structural and electrical parameters on the bandgap range and the attenuation coefficient in the bandgap. Results indicate that the larger surface areas of the piezoelectric sheet correspond to lower center bands in the bandgap, while increased thickness widens the attenuation coefficient range with increased peak values. Furthermore, the influence of inductance on the bandgap conforms to the variation law of the electrical LC resonance frequency, and increased resistance widens the attenuation coefficient range albeit with decreased peak values. The incorporation of negative capacitance significantly expands the low-frequency bandgap range. Visualized through vibration transfer simulations, the vibration-damping ability of the piezoelectric phononic crystal is demonstrated. Experimentally, this paper finds that two propagation modes of bending waves (symmetric and anti-symmetric) result in variable voltage amplitudes, and the average vibration of the system decreases by 4–5 dB within the range of 1710–1990 Hz. The comparison between experimental and model-generated data confirms the accuracy of the attenuation coefficient calculation model. This convergence between experimental and computational results emphasizes the validity and usefulness of the proposed model, and this paper provides theoretical support for the application of piezoelectric phononic crystals in the field of plate–shell vibration reduction.