Stochastic response of a piezoelectric ribbon-substrate structure under Gaussian white noise

Due to the excellent piezoelectric and ferroelectric properties of lead zirconate titanate (PZT), the buckled PZT ribbon-substrate structure has been widely used in the design of wearable electronic devices. However, wearable electronic devices must work in a complex vibration environment and are subjected to random excitations such as irregular human body motion. Hence, the reliability of the buckled piezoelectric ribbon-substrate structure in a dynamic context needs to be ensured. In this paper, Gaussian white noise is introduced to describe the broadband random environment. A voltage is applied to the PZT ribbon to realize the desired wavy configuration of the ribbon to influence the dynamic responses of this structure. Based on the Euler-Bernoulli beam theory and the Lagrange equation, the governing equation of the buckled piezoelectric ribbon-substrate structure is derived. By using a stochastic averaging method, the stationary probability density of stochastic responses of the buckled piezoelectric ribbon-substrate structure is obtained. Several numerical examples are analysed to reveal the effects of the intensity of the Gaussian white noise and the voltage applied to the PZT ribbon on the stochastic responses of this buckled structure. Through these numerical results, it can be found that when the applied voltage is above the critical voltage value, the piezoelectric ribbon would wrinkle into multiple small waves on top of the soft substrate. By increasing the applied voltage while keeping the intensity of noise excitation constant, the static buckling amplitude increases, and the number of transitions between two equilibrium positions decreases, which implies that the stretchability and stability of the ribbon-substrate structure would be improved. The results of this paper can be helpful for the design of robust piezoelectric ribbon-based stretchable electronics.