The internal moment can be obtained by integrating the first moment of the stress distribution at a cross section over the cross-sectional area. The piezoelectric constitutive relations (13) give the stress-strain (and electric field) relations and they are expressed for the substructure and the PZT layers asDisplay Formula

Display Formula$T1p=Yp(S1p\u2212d31E3)$

(4)

respectively. Here,

$T$ is the stress,

$S$ is the strain,

$Y$ is Young’s modulus;

$d$ is the piezoelectric constant, and

$E$ is the electric field. Equation

4 is obtained from the piezoelectric constitutive relation

$S1=s11ET1+d31E3$, where

$s11E$ is the elastic compliance at constant electric field and therefore

$Yp$ is simply the reciprocal of

$s11E$. Furthermore, subscript/superscript

$p$ and

$s$ stand for PZT and substructure layers, respectively; 1 and 3 directions are coincident with

$x$ and

$y$ directions, respectively (where 1 is the direction of axial strain and 3 is the direction of polarization). Then, the internal moment can be written as

Display Formula$M(x,t)=\u2212\u222bhahbT1sbydy\u2212\u222bhbhcT1pbydy$

(5)

where

$b$ is the width of the beam,

$ha$ is the position of the bottom of the substructure layer from the neutral axis,

$hb$ is the position of the bottom of the PZT layer (therefore, top of the substructure layer) from the neutral axis, and

$hc$ is the position of the top of the PZT layer from the neutral axis (see the Appendix). Expressing the bending strain in terms of radius of curvature (

18) and employing Eqs.

3,

4 in Eq.

5 give

Display Formula$M(x,t)=\u222bhahbYsb\u22022wrel(x,t)\u2202x2y2dy+\u222bhbhcYpb\u22022wrel(x,t)\u2202x2y2dy\u2212\u222bhbhcv(t)Ypbd31hpydy$

(6)

where the uniform electric field is written in terms of the voltage

$v(t)$ across the PZT and the thickness

$hp$ of the PZT

$(E3(t)=\u2212v(t)\u2215hp)$. Equation

6 can be reduced to

Display Formula$M(x,t)=YI\u22022wrel(x,t)\u2202x2+\u03d1v(t)$

(7)

where

$YI$ is the bending stiffness of the composite cross section given by

Display Formula$YI=b[Ys(hb3\u2212ha3)+Yp(hc3\u2212hb3)3]$

(8)

and the coupling term

$\u03d1$ can be written as

Display Formula$\u03d1=\u2212Ypd31b2hp(hc2\u2212hb2)$

(9)

If the PZT layer and/or the electrodes do not cover the entire length of the beam but the region

$x1\u2a7dx\u2a7dx2$, then the second term in Eq.

7 should be multiplied by

$H(x\u2212x1)\u2212H(x\u2212x2)$, where

$H(x)$ is the Heaviside function. Note that, in energy harvesting from higher vibration modes, it becomes necessary to use segmented electrode pairs in order to avoid cancellation of the charge collected by continuous electrode pairs (

15). In such a case, one needs to define separate voltage terms, which should appear in Eq.

7 as separate terms multiplied by Heaviside functions. However, in our case, we assume that the electrodes and the PZT layer cover the entire length of the beam displayed in Fig.

1 and it is convenient to rewrite Eq.

7 as

Display Formula$M(x,t)=YI\u22022wrel(x,t)\u2202x2+\u03d1v(t)[H(x)\u2212H(x\u2212L)]$

(10)

where

$L$ is the length of the beam. In Eq.

10, although the PZT layer and the electrodes cover the entire beam length, Heaviside functions are associated with the second term in Eq.

10 in order to ensure the survival of this term when the internal moment expression

$M(x,t)$ is used in the differential equation of motion given by Eq.

2. Then, employing Eq.

10 in Eq.

2 yields

Display Formula$YI\u22024wrel(x,t)\u2202x4+csI\u22025wrel(x,t)\u2202x4\u2202t+ca\u2202wrel(x,t)\u2202t+m\u22022wrel(x,t)\u2202t2+\u03d1v(t)[d\delta (x)dx\u2212d\delta (x\u2212L)dx]=\u2212m\u22022wb(x,t)\u2202t2\u2212ca\u2202wb(x,t)\u2202t$

(11)

where

$\delta (x)$ is the Dirac delta function and it satisfies (

19)

Display Formula$\u222b\u2212\u221e\u221ed(n)\delta (x\u2212x0)dx(n)f(x)dx=(\u22121)ndf(n)(x0)dx(n)$

(12)