Energy harvesting from nonlinear vibrations of an L-shaped beam using piezoelectric patches

Abstract

In the present study, the energy harvesting from nonlinear vibrations of an L-shaped beam with the use of piezoelectric patches is evaluated. The beam was assumed to be a Euler–Bernoulli type in which the effects of rotational inertia and shear forces are neglected. Also, the effective mass of the beam is assumed to be at the end of the beam. The axial displacement of the beam is included with the assumption of small excitation amplitude. The effect of the piezoelectric patch on the equations of motion is also considered. First, the nonlinear motion equations of an L-shaped beam are extracted by the use of the Lagrange method, and then, the obtained equations of motion are solved by multiple scales technique. Also, the mode shapes of the system are calculated for the linear undamped system. The dimensions of the L-shaped beam are chosen in such a way that the ratio of the first-to-second natural frequency is 1 to 2, in order to increase the bandwidth of the energy harvesting. Finally, the results of the analytical solutions are compared and validated with the experimental results. Good agreement was observed between the theoretical values and the measured data.

Appendix 1

See Fig. 11

Fig. 11

A magnified view of the piezoelectric material layers

• (a)

  $$\left( {{\text{EI}}} \right)_{{\text{k}}} = \left( {\text{Y}} \right)_{{\text{k}}} \frac{{\left( {\text{b}} \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{top}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{bottom}}}} } \right)_{{\text{k}}}^{3} } \right]$$

  (60a)
  $$\begin{aligned} \left( {{\text{EI}}_{{\text{p}}} } \right)_{{\text{k}}} = & \left( {{\text{Y}}_{{\text{p}}} } \right)_{{\text{k}}} \frac{{\left( {{\text{n}}_{{\text{p}}} {\text{b}}_{{\text{p}}} } \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{tp}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{ts}}}} } \right)_{{\text{k}}}^{3} } \right] + \left( {{\text{Y}}_{{\text{l}}} } \right)_{{\text{k}}} \frac{{\left( {{\text{n}}_{{\text{l}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{tl}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{top}}}} } \right)_{{\text{k}}}^{3} } \right] \\ & \; + \left( {{\text{Y}}_{{\text{s}}} } \right)_{{\text{k}}} \frac{{\left( {{\text{n}}_{{\text{s}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{ts}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{tl}}}} } \right)_{{\text{k}}}^{3} } \right] + \left( {{\text{Y}}_{{\text{c}}} } \right)_{{\text{k}}} \frac{{\left( {{\text{n}}_{{\text{c}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{tc}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{tp}}}} } \right)_{{\text{k}}}^{3} } \right] + \left( {{\text{Y}}_{{\text{o}}} } \right)_{{\text{k}}} \frac{{\left( {{\text{n}}_{{\text{o}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} }}{3}\left[ {\left( {{\text{h}}_{{{\text{to}}}} } \right)_{{\text{k}}}^{3} - \left( {{\text{h}}_{{{\text{tc}}}} } \right)_{{\text{k}}}^{3} } \right]. \\ \end{aligned}$$

  (60b)

  Here,

  $$\left( {{\text{h}}_{{\text{t}}} } \right)_{{\text{k}}} = \frac{{\left( {{\Xi}_{1} } \right)_{{\text{k}}} + 2\left( {{\Xi}_{2} } \right)_{{\text{k}}} }}{{2\left( {{\text{bh}} + {\text{n}}_{{\text{l}}} {\text{b}}_{{\text{c}}} {\text{h}}_{{\text{l}}} + {\text{n}}_{{\text{s}}} {\text{b}}_{{\text{c}}} {\text{h}}_{{\text{s}}} + {\text{n}}_{{\text{p}}} {\text{b}}_{{\text{p}}} {\text{h}}_{{\text{p}}} + {\text{n}}_{{\text{c}}} {\text{b}}_{{\text{c}}} {\text{h}}_{{\text{c}}} + {\text{n}}_{{\text{o}}} {\text{b}}_{{\text{c}}} {\text{h}}_{{\text{o}}} } \right)_{{\text{k}}} }},$$

  (61a)

  where

  $$\left( {\Xi_{1} } \right)_{{\text{k}}} = \left( {\text{b}} \right)_{{\text{k}}} \left( {\text{h}} \right)_{{\text{k}}}^{2} + \left( {{\text{n}}_{{\text{l}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} \left( {{\text{h}}_{{\text{l}}} } \right)_{{\text{k}}}^{2} + \left( {{\text{n}}_{{\text{s}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} \left( {{\text{h}}_{{\text{s}}} } \right)_{{\text{k}}}^{2} + \left( {{\text{n}}_{{\text{p}}} {\text{b}}_{{\text{p}}} } \right)_{{\text{k}}} \left( {{\text{h}}_{{\text{p}}} } \right)_{{\text{k}}}^{2} + \left( {{\text{n}}_{{\text{c}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} \left( {{\text{h}}_{{\text{c}}} } \right)_{{\text{k}}}^{2} + \left( {{\text{n}}_{{\text{o}}} {\text{b}}_{{\text{c}}} } \right)_{{\text{k}}} \left( {{\text{h}}_{{\text{o}}} } \right)_{{\text{k}}}^{2}$$

  (62a)
  $$\left( {{\Xi}_{2} } \right)_{{\text{k}}} = \left( {{\text{n}}_{{\text{l}}} {\text{h}}_{{\text{l}}} {\text{h}}_{{\text{s}}} } \right)_{{\text{k}}} + \left( {{\text{n}}_{{\text{s}}} {\text{h}}_{{\text{s}}} \left( {{\text{h}} + {\text{h}}_{{\text{l}}} } \right)} \right)_{{\text{k}}} + \left( {{\text{n}}_{{\text{p}}} {\text{h}}_{{\text{p}}} \left( {{\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} } \right)} \right)_{{\text{k}}} + \left( {{\text{n}}_{{\text{c}}} {\text{h}}_{{\text{c}}} \left( {{\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} + {\text{h}}_{{\text{p}}} } \right)} \right)_{{\text{k}}} + \left( {{\text{n}}_{{\text{o}}} {\text{h}}_{{\text{o}}} \left( {{\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} + {\text{h}}_{{\text{p}}} + {\text{h}}_{{\text{c}}} } \right)} \right)_{{\text{k}}}$$

  (62b)

  and

  $$\left( {{\text{n}}_{{\text{i}}} } \right)_{{\text{k}}} = \left( {{\text{Y}}_{{\text{i}}} } \right)_{{\text{k}}} /\left( {\text{Y}} \right)_{{\text{k}}} ,\,\left( {{\text{h}}_{{{\text{bottom}}}} } \right)_{{\text{k}}} = - {\text{h}}_{{\text{t}}} ,\left( {{\text{h}}_{{{\text{top}}}} } \right)_{{\text{k}}} = {\text{h}} - {\text{h}}_{{\text{t}}}$$

  (63a,b,c)
  $$\left( {{\text{h}}_{{{\text{tl}}}} } \right)_{{\text{k}}} = {\text{h}} + {\text{h}}_{{\text{l}}} - {\text{h}}_{{\text{t}}} { },\;\left( {{\text{h}}_{{{\text{ts}}}} } \right)_{{\text{k}}} = {\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} - {\text{h}}_{{\text{t}}} { },$$

  (63d,e)
  $$\left( {{\text{h}}_{{{\text{tp}}}} } \right)_{{\text{k}}} = {\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} + {\text{h}}_{{\text{p}}} - {\text{h}}_{{\text{t}}} { },{ }\left( {{\text{h}}_{{{\text{tc}}}} } \right)_{{\text{k}}} = {\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} + {\text{h}}_{{\text{p}}} + {\text{h}}_{{\text{c}}} - {\text{h}}_{{\text{t}}}$$

  (63f,g)
  $$\left( {{\text{h}}_{{{\text{to}}}} } \right)_{{\text{k}}} = {\text{h}} + {\text{h}}_{{\text{l}}} + {\text{h}}_{{\text{s}}} + {\text{h}}_{{\text{p}}} + {\text{h}}_{{\text{c}}} + {\text{h}}_{{\text{o}}} - {\text{h}}_{{\text{t}}}$$

  (63h)

• (b)

  $${\text{C}}_{11} = \frac{9}{5} + 3{\text{e}}_{1} + \frac{{4{\text{e}}_{1}^{2} }}{3}{ },{\text{C}}_{13} = \frac{9}{5} + 3{\text{e}}_{2} + \frac{{4{\text{e}}_{2}^{2} }}{3}$$

  (64a,b)
  $${\text{C}}_{12} = \frac{9}{5} + \frac{3}{2}\left( {{\text{e}}_{1} + {\text{e}}_{2} } \right) + \frac{4}{3}{\text{e}}_{1} {\text{e}}_{2} { }$$

  (64c)
  $${\text{C}}_{21} = {\uplambda }_{1} \left( {3 + 2{\text{e}}_{1} } \right)^{2} + \left\{ {4{\uplambda }_{2} \left( {3 + 2{\text{e}}_{1} } \right)\left( {3 + {\text{e}}_{1} } \right)/3} \right\} + \left\{ {8{\uplambda }_{1} {\uplambda }_{2} \left( {3 + {\text{e}}_{1} } \right)^{2} /15} \right\}$$

  (64d)
  $${\text{C}}_{23} = {\uplambda }_{1} \left( {3 + 2{\text{e}}_{2} } \right)^{2} + \left\{ {4{\uplambda }_{2} \left( {3 + 2{\text{e}}_{2} } \right)\left( {3 + {\text{e}}_{2} } \right)/3} \right\} + \left\{ {8{\uplambda }_{2} {\uplambda }_{3} \left( {3 + {\text{e}}_{2} } \right)^{2} /15} \right\}$$

  (64e)
  $$\begin{aligned} {\text{C}}_{{22}} = \lambda _{1} \left( {3 + 2{\text{e}}_{1} } \right)\left( {3 + 2{\text{e}}_{2} } \right) + \left\{ {8\lambda _{2} \lambda _{3} \left( {3 + {\text{e}}_{1} } \right)\left( {3 + {\text{e}}_{2} } \right)/15} \right\} \hfill \\ + \left( {2\lambda _{2} /3} \right)\left[ {\left( {3 + 2{\text{e}}_{1} } \right)\left( {3 + {\text{e}}_{2} } \right) + \left( {3 + 2{\text{e}}_{2} } \right)\left( {3 + {\text{e}}_{1} } \right)} \right] \hfill \\ \end{aligned}$$

  (64f)

• (c)

  $${\text{Y}}_{11} = \frac{{2{\text{R}}\left( {{\text{A}}_{21} {\text{C}}_{11} - {\text{A}}_{11} {\text{C}}_{21} } \right)}}{{{\text{N}}_{1} }}{ },\;{\text{Y}}_{12} = \frac{{{\text{R}}\left( {{\text{A}}_{22} {\text{C}}_{11} + {\text{A}}_{21} {\text{C}}_{12} - {\text{A}}_{12} {\text{C}}_{21} - {\text{A}}_{11} {\text{C}}_{22} } \right)}}{{{\text{N}}_{1} }}$$

  (65a,b)
  $${\text{Y}}_{13} = \frac{{2{\text{R}}\left( {{\text{A}}_{21} {\text{C}}_{12} - {\text{A}}_{11} {\text{C}}_{22} } \right)}}{{{\text{N}}_{1} }}{ },{\text{ Y}}_{14} = \frac{{{\text{R}}\left( {{\text{A}}_{22} {\text{C}}_{12} + {\text{A}}_{21} {\text{C}}_{13} - {\text{A}}_{12} {\text{C}}_{22} - {\text{A}}_{11} {\text{C}}_{23} } \right)}}{{{\text{N}}_{1} }}$$

  (65c,d)
  $${\text{Y}}_{21} = \frac{{{\text{N}}_{1} {\text{Y}}_{12} }}{{{\text{N}}_{2} }},{\text{ Y}}_{22} = \frac{{2{\text{R}}\left( {{\text{A}}_{22} {\text{C}}_{12} - {\text{A}}_{12} {\text{C}}_{22} } \right)}}{{{\text{N}}_{2} }}$$

  (65e,f)
  $${\text{Y}}_{23} = \frac{{{\text{N}}_{1} {\text{Y}}_{14} }}{{{\text{N}}_{2} }},{\text{Y}}_{24} = \frac{{2{\text{R}}\left( {{\text{A}}_{22} {\text{C}}_{13} - {\text{A}}_{12} {\text{C}}_{23} } \right)}}{{{\text{N}}_{2} }}$$

  (65 g,h)
  $${\text{A}}_{{1{\text{i}}}} = 1 + {\text{e}}_{{\text{i}}} { },{\text{A}}_{{2{\text{i}}}} = {\uplambda }_{1} \left( {3 + 2{\text{e}}_{{\text{i}}} } \right) + \frac{2}{3}{\uplambda }_{2} \left( {3 + {\text{e}}_{{\text{i}}} } \right)$$

  (65j,k)
  $${\text{N}}_{{\text{i}}} = \left( {1 + {\text{R}}} \right)\left( {1 + {\text{e}}_{{\text{i}}} } \right)^{2} + {\text{R}}\left[ {{\uplambda }_{1} \left( {3 + 2{\text{e}}_{{\text{i}}} } \right) + \frac{2}{3}{\uplambda }_{2} \left( {3 + {\text{e}}_{{\text{i}}} } \right)} \right]^{2} for\,\,{\text{i}} = 1,2$$

  (65l)

• (d)

  $${\text{Z}}_{{11}} = \frac{{{\text{R}}\Omega {\text{C}}_{{21}} }}{{2{\text{N}}_{1} }},{\mkern 1mu} Z_{{12}} = \frac{{{\text{R}}\Omega {\text{C}}_{{22}} }}{{2{\text{N}}_{1} }},{\mkern 1mu} Z_{{21}} = \frac{{{\text{R}}\Omega {\text{C}}_{{22}} }}{{2{\text{N}}_{2} }},{\mkern 1mu} Z_{{22}} = \frac{{{\text{R}}\Omega {\text{C}}_{{23}} }}{{2{\text{N}}_{2} }}$$

  (66a,b,c,d)

• (e)

  $${\text{X}}_{11} = {\text{Y}}_{11} /2,{\text{ X}}_{12} = {\text{Y}}_{13} { },{\text{X}}_{13} = \frac{{{\text{R}}\left( {{\text{A}}_{21} {\text{C}}_{13} - {\text{A}}_{11} {\text{C}}_{23} } \right)}}{{{\text{N}}_{1} }}{ }$$

  (67a,b,c)
  $${\text{X}}_{{21}} = \frac{{{\text{R}}\left( {{\text{A}}_{{22}} {\text{C}}_{{11}} - {\text{A}}_{{12}} {\text{C}}_{{21}} } \right)}}{{{\text{N}}_{2} }},{\mkern 1mu} {\text{X}}_{{22}} = {\text{Y}}_{{22}} ,{\mkern 1mu} {\text{X}}_{{23}} = {\text{Y}}_{{24}} /2.$$

  (67d,f,g)

  and

  $${\text{K}}_{{\text{i}}} = \frac{{\Omega ^{2} \left( {1 + {\text{R}}} \right){\text{A}}_{{1{\text{i}}}} }}{{2{\text{N}}_{{\text{i}}} }}$$

  (67h)

• (f)

  $$J_{1i} = \vartheta_{1} \left( {3 + 2e_{i} } \right)/N_{1} ,\,J_{2i} = \left\{ {\vartheta_{2} \left( {3 + e_{i} } \right)\frac{{\lambda_{2} }}{{\lambda_{1} }}} \right\}/N_{2}$$

  (68a,b)

• (g)

  $$\zeta_{11} = \frac{{i\frac{{\omega_{1} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{11} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{21} }}{{T_{c1} }}} \right) - l_{1} \varphi ^{\prime}_{11} \omega_{1}^{2} }}{{i\omega_{1} \lambda_{t} - \omega_{1}^{2} }} , \,\zeta_{12} = \frac{{i\frac{{\omega_{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{12} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{22} }}{{T_{c1} }}} \right) - l_{1} \varphi ^{\prime}_{12} \omega_{2}^{2} }}{{i\omega_{2} \lambda_{t} - \omega_{2}^{2} }}$$

  (69a,b)
  $$\zeta_{21} = \frac{{i\frac{{\omega_{1} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{21} }}{{T_{c1} }} - \frac{{\varphi ^{\prime}_{11} }}{{T_{c2} }}} \right) - l_{1} \varphi ^{\prime}_{21} \omega_{1}^{2} }}{{i\omega_{1} \lambda_{t} - \omega_{1}^{2} }} , \;\zeta_{22} = \frac{{i\frac{{\omega_{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{22} }}{{T_{c1} }} - \frac{{\varphi ^{\prime}_{12} }}{{T_{c2} }}} \right) - l_{1} \varphi ^{\prime}_{22} \omega_{2}^{2} }}{{i\omega_{2} \lambda_{t} - \omega_{2}^{2} }}$$

  (69c,d)

  where

  $$\varphi ^{\prime}_{ij} = \varphi_{ij} /C_{pi} ,\;T_{ci} = R_{1} C_{pi} ,\;\lambda_{t} = \frac{1}{{\beta T_{c1} }} + \frac{1}{{\beta T_{c2} }}$$

  (70a,b)

• (h)

  $$G_{r1} = \frac{{\frac{{\omega_{1}^{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{11} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{21} }}{{T_{c1} }}} \right)\lambda_{t} + l_{1} \varphi ^{\prime}_{11} \omega_{1}^{4} }}{{\left( {\omega_{1} } \right)^{4} + \left( {\lambda_{t} \omega_{1} } \right)^{2} }} ,\;G_{m1} = \frac{{\frac{{\omega_{1}^{3} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{11} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{21} }}{{T_{c1} }}} \right) + l_{1} \varphi ^{\prime}_{11} \omega_{1}^{3} \lambda_{t} }}{{\left( {\omega_{1} } \right)^{4} + \left( {\lambda_{t} \omega_{1} } \right)^{2} }}$$

  (71a,b)
  $$G_{r2} = \frac{{\frac{{\omega_{2}^{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{12} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{22} }}{{T_{c1} }}} \right)\lambda_{t} + l_{1} \varphi ^{\prime}_{12} \omega_{2}^{4} }}{{\left( {\omega_{2} } \right)^{4} + \left( {\lambda_{t} \omega_{2} } \right)^{2} }} ,\;G_{m2} = \frac{{\frac{{\omega_{2}^{3} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{12} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{22} }}{{T_{c1} }}} \right) + l_{1} \varphi ^{\prime}_{11} \omega_{2}^{3} \lambda_{t} }}{{\left( {\omega_{2} } \right)^{4} + \left( {\lambda_{t} \omega_{2} } \right)^{2} }}$$

  (71c,d)
  $$G_{r3} = \frac{{\frac{{\omega_{1}^{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{12} }}{{T_{c1} }} - \frac{{\varphi ^{\prime}_{11} }}{{T_{c2} }}} \right)\lambda_{t} + l_{1} \varphi ^{\prime}_{11} \omega_{1}^{4} }}{{\left( {\omega_{1} } \right)^{4} + \left( {\lambda_{t} \omega_{1} } \right)^{2} }} ,\;G_{m3} = \frac{{\frac{{\omega_{1}^{3} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{21} }}{{T_{c1} }} - \frac{{\varphi ^{\prime}_{11} }}{{T_{c2} }}} \right) + l_{1} \varphi ^{\prime}_{21} \omega_{1}^{3} \lambda_{t} }}{{\left( {\omega_{1} } \right)^{4} + \left( {\lambda_{t} \omega_{1} } \right)^{2} }}$$

  (71e,f)
  $$G_{r4} = \frac{{\frac{{\omega_{2}^{2} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{22} }}{{T_{c2} }} - \frac{{\varphi ^{\prime}_{21} }}{{T_{c1} }}} \right)\lambda_{t} + l_{1} \varphi ^{\prime}_{22} \omega_{2}^{4} }}{{\left( {\omega_{2} } \right)^{4} + \left( {\lambda_{t} \omega_{2} } \right)^{2} }} , \;G_{m4} = \frac{{\frac{{\omega_{2}^{3} l_{1} }}{\beta }\left( {\frac{{\varphi^{\prime}_{22} }}{{T_{c1} }} - \frac{{\varphi ^{\prime}_{12} }}{{T_{c2} }}} \right) + l_{1} \varphi ^{\prime}_{22} \omega_{2}^{3} \lambda_{t} }}{{\left( {\omega_{2} } \right)^{4} + \left( {\lambda_{t} \omega_{2} } \right)^{2} }}.$$

  (71g,h)