**5. The vertical displacement and electric potential approach**

Understanding physical problems can be accomplished when the numerical simulation of equations within the variables that describe them are represented. In the case of the electric energy prediction from the vehicle suspension system, the variables are vertical displacement *w* and electric potential *V* of different points, or nodes, and the domain within the model is considered valid.

Some numerical methods are often used to understand physical problems, and among them, the most widely used in engineering is the finite element method. Within this method, the variables in the equations that describe the physical

*Energy Harvesting Prediction from Piezoelectric Materials with a Dynamic System Model DOI: http://dx.doi.org/10.5772/intechopen.96626*

problem must be approximated to the polynomial functions of these variables. For the application of this method in solving problems, all elements that compose it must be represented by matrices, called elementary matrices, which are the result of the adopted polynomial functions. For further details on this method, the following references [9, 10] are recommended.

If the linear approximation function of vertical displacement *w* and electric potential *V* over thickness *hp* in direction z of the piezoelectric disk are considered, stiffness elementary matrices of the piezoelectric finite element can be obtained by applying the Lagrange equations [11] over the energy expressions presented by Eq. (2).

$$\begin{aligned} \left[K\_m\right] &= \frac{c\_{zx}^E S\_p}{h\_p} \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} = k\_m \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} \\ \left[K\_{m,el}\right] &= -\frac{e\_{zx} S\_p}{2h\_p} \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} = -k\_{m,el} \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} \\ \left[K\_{el}\right] &= \frac{c\_z^S S\_p}{h\_p} \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} = k\_{el} \begin{bmatrix} \mathbf{1} & -\mathbf{1} \\ -\mathbf{1} & \mathbf{1} \end{bmatrix} \end{aligned} \tag{5}$$

Where ½ � *Km* , *Km*\_*el* ½ � and *Kel* ½ � are mechanical elementary stiffness, electromechanical coupling elementary stiffness and electric elementary stiffness matrices, respectively, and *Sp* is the cross-sectional area of the piezoelectric disk.

Moreover, if the same linear approximation function of the vertical displacement *w* over stiffness *ks* and viscous damping *cs* of the shock absorber in direction z are considered, the suspension system's differential equation of motion, illustrated in **Figure 2**, can be obtained by applying the Lagrange equations [11] over the energy expressions presented by Eq. (4).

$$[\mathbf{M}]\{\ddot{w}\} + [\mathbf{C}]\{\dot{w}\} + [\mathbf{K}]\{w\} = \{F(t)\} \tag{6}$$

Where the elementary matrices and the vectors are:

$$\begin{aligned} \begin{bmatrix} M \\ \end{bmatrix} &= \begin{bmatrix} m\_s & 0 \\ 0 & m\_u \\ \end{bmatrix} \\ \begin{bmatrix} K \\ \end{bmatrix} &= \begin{bmatrix} k\_i & -k\_i \\ -k\_i & k\_i \\ \end{bmatrix} \\ \begin{bmatrix} C \\ \end{bmatrix} &= \begin{bmatrix} c\_i & 0 \\ 0 & c\_i \\ \end{bmatrix} \\ \begin{Bmatrix} w \\ \end{Bmatrix} &= \begin{Bmatrix} w\_i(t) \\ w\_u(t) \\ w\_u(t) \\ \end{Bmatrix} \\ \begin{Bmatrix} \dot{w} \\ \end{Bmatrix} &= \begin{Bmatrix} \dot{w}\_i(t) \\ \dot{w}\_u(t) \\ \end{Bmatrix} \\ \begin{Bmatrix} F(t) \\ \end{Bmatrix} &= \begin{Bmatrix} 0 \\ 0 \\ k\_{\text{if}} \end{Bmatrix} \end{aligned} (7)$$

Using the technique of assembling the elementary matrices of the finite element method, the suspension system's differential equation of motion and piezoelectric disk coupled model, as illustrated in **Figure 3**, is as shown in Eq. (8):

$$[\mathcal{M}\_m]\{\ddot{w}\} + [\mathcal{C}\_m]\{\dot{w}\} + [\mathcal{K}\_m]\{w\} + [\mathcal{K}\_{m,el}]\{V\} = \{F(t)\} \tag{8}$$

$$\left[K\_{m\\_el}\right]^t \{w\} + \left[K\_{el}\right] \{V\} = \{\mathbf{0}\} \tag{9}$$

Eq. (9) can be manipulated and substituted for Eq. (8). Thus, the final equation of motion as a function of only mechanical variables is:

$$\left[\left[\boldsymbol{M}\_{m}\right]\{\ddot{\boldsymbol{w}}\}+\left[\boldsymbol{C}\_{m}\right]\{\dot{\boldsymbol{w}}\}+\left[\left[\boldsymbol{K}\_{m}\right]-\left[\boldsymbol{K}\_{m\_{-}\epsilon l}\right]\left[\boldsymbol{K}\_{\epsilon l}\right]^{-1}\left[\boldsymbol{K}\_{m\_{-}\epsilon l}\right]^{t}\right]\{\boldsymbol{w}\} = \left\{\boldsymbol{F}(t)\right\}\tag{10}$$

Eq. (10) can be solved in the time domain with an integration method, such as the Newmark Method [12]. The mechanical force is due to road roughness as well as the tire characteristics of the wheel as shown in Eq. (7). These data are used to apply the condition in each time step in solving Eq. (10), in which all variables *ws* ¼ *w*2, *w*1, and *wu* and their time derivatives are obtained.

The response in the time domain in respect to vertical displacements is obtained by using Eq. (10) and subsequently, the electric potential is obtained as:

$$\{V(t)\} = -[K\_{el}]^{-1}[K\_{mel}]^{t} \{w(t)\} \tag{11}$$

The electrical energy can be calculated as presented by Eq. (2). The development of this equation follows:

$$U\_{\epsilon} = \frac{1}{2} \int\_{0}^{h\_p} \int\_{S\_p} \left[ c\_{\text{xx}} \left( \frac{\partial w}{\partial \mathbf{z}} \right)^{\circ} \frac{\partial V}{\partial \mathbf{z}} \right. \\ \left. + c\_x^{\mathcal{S}} \left( \frac{\partial V}{\partial \mathbf{z}} \right)^2 \right] d\mathcal{S} \text{ } dz \tag{12}$$

Thus, the expression of the electrical energy due to the piezoelectric disk on this suspension system is as:

$$U\_{\epsilon} = \frac{1}{2} \frac{e\_{\rm xr} S\_p}{h\_p} \ (w\_2 - w\_1) \ (V\_2 - V\_1) + \frac{e\_x^S S\_p}{h\_p} \ (V\_2 - V\_1)^2 \tag{13}$$

In addition, the electrical power can be calculated as presented by Eq. (3) and its development is as follows:

$$P\_{\epsilon} = \frac{1}{2} \int\_{0}^{h\_p} \int\_{S\_p} \left[ e\_{\text{zx}} \left( \frac{\partial}{\partial t} \left( \frac{\partial w}{\partial \mathbf{z}} \right) \cdot \frac{\partial V}{\partial \mathbf{z}} + \frac{\partial w}{\partial \mathbf{z}} \cdot \frac{\partial}{\partial t} \left( \frac{\partial V}{\partial \mathbf{z}} \right) \right) + e\_x^S \cdot \frac{\partial}{\partial t} \left( \frac{\partial V}{\partial \mathbf{z}} \right)^2 \right] dS \,\, d\mathbf{z} \tag{14}$$

Thus, the expression of the electrical power due to the piezoelectric disk on this suspension system is as:

$$\begin{split} P\_{\varepsilon} &= \frac{1}{2} \frac{e\_{xx} S\_p}{h\_p} \quad \left[ (\dot{w}\_2 - \dot{w}\_1) \ (V\_2 - V\_1) + (w\_2 - w\_1) \ (\dot{V}\_2 - \dot{V}\_1) \right] \\ &+ \frac{e\_x^{\mathcal{S}} S\_p}{h\_p} \ (V\_2 - V\_1) \ (\dot{V}\_2 - \dot{V}\_1) \end{split} \tag{15}$$

Where *w*\_ 1, *w*\_ 2, *V*\_ <sup>1</sup> and *V*\_ <sup>2</sup> are time derivatives of the displacements and electric potential of nodes 1 and 2 of the piezoelectric disk.

*Energy Harvesting Prediction from Piezoelectric Materials with a Dynamic System Model DOI: http://dx.doi.org/10.5772/intechopen.96626*
