Piezoelectric Energy Harvesting from a Non-ideal Portal Frame System Including Shape Memory Alloy Effect

Abstract

In this work, the investigation of energy harvesting for a U-frame-type (portal frame) structure is presented. The structure is considered as a composite made out of a Shape Memory Alloy (SMA), accounting for a non-ideal DC motor of limited power supply attached to its rigid body. The energy harvesting is carried out using a piezoelectric material (PZT), accounting for a nonlinear electromechanical coupling model. For the behavior of the SMA, a polynomial constitutive model is adopted, which relates the voltage variation to the temperature. Numerical results demonstrate that both PZT and SMA material has a significant influence on energy harvesting. In addition, it is highlighted that the use of SMA makes controlling the vibrations of the structure possible, increasing the harvested energy.

Appendices

Appendix

The 0-1 Test Method

The 0-1 test consists of estimating a single parameter $\kappa$ by [28]:

$$\kappa = \frac{{{\text{cov}} \left( {Y,M\left( {\overline{c}} \right)} \right)}}{{\sqrt {{\text{var}} (Y){\text{var}} (M\left( {\overline{c}} \right))} }}$$

(10)

where: $\overline{c} \in \left( {0,\pi } \right)$, $M\left( {\overline{c}} \right) = \left[ {M\left( {1,\overline{c}} \right), M\left( {2,\overline{c}} \right), \ldots ,M\left( {n_{max} \overline{c}} \right)} \right]$ and $Y = \left[ {1,2, \ldots ,n_{max} } \right]$.

If $\kappa$ is close to 0 the system is periodic. On the other hand, if $\kappa$ is close to 1 the system is chaotic. The test utilizes a system variable $x(j)$, where two new coordinates (p, q) are defined as follows [29]:

$$p\left( {n,\overline{c}} \right) = \sum\limits_{j = 0}^{n} {x(j)\cos \left( {j\overline{c}} \right)}$$

(11)

$$q\left( {n,\overline{c}} \right) = \sum\limits_{j = 0}^{n} {x(j)\sin \left( {j\overline{c}} \right)}$$

(12)

The mean square displacement of the new variables $p\left( {n,\overline{c}} \right)$ and $q\left( {n,\overline{c}} \right)$ is given by [29]:

$$M(n,c) = \mathop {\lim }\limits_{n \to \infty } \frac{1}{N}\sum\limits_{j = 1}^{N} {\left[ {\left( {p(j + n,\overline{c}) - p(j,\overline{c})} \right)^{2} \ldots + \left( {q(j + n,\overline{c}) - q(j,\overline{c})} \right)^{2} } \right]}$$

(13)

where $n = 1,2, \ldots ,N$.