**1. Introduction**

Energy use is widely discussed nowadays, as energy conversion and management. In this way, new sources of energy are required to be investigated. Thus, one of the energy sources that can be used is from vibration systems, which can be subject to different excitations. This wasted energy to the environment can be recovered through vibration energy harvesting devices, which convert mechanical vibration into useful electrical energy in a way that low power devices may utilize.

Piezoelectric materials are known to have the electro/mechanical coupling effect. This property has a large range of applications in engineering. Currently, they are extensively used as sensors and actuators in vibration control systems. As a sensor, it can monitor the vibrations when bonded to a flexible structure. As an actuator, it can control the vibration level by introducing a restored force or by adding damping to the system.

In the context of recovered energy from mechanical vibrations based on the conversion of piezoelectric harvesting devices and its application on powering electronic devices, this subject has received the attention from various researchers [1–3]. An example of this type of recovered energy is the suspension system vibration for use of the vehicle itself, such as an energetic source for an active and semiactive suspension.

On a typical road, vehicles suffer accelerations due to its roughness, which excite undesired vibration. Some recently conducted reviews mentioned the potential of recovering a few hundred watts for a passenger car driven in experimental tests as well as some mathematical models [4, 5]. One of the ways to convert the mechanical energy from the vehicle suspension to electric energy is through piezoelectric materials [6]. Therefore, the objective of this chapter is to present the coupling between a piezoelectric element and a dynamic system in the context of predicting the recovered electric power and energy density for piezoelectric materials, especially the PZT (Lead Zirconate Titanate).

## **2. Piezoelectric material modelling**

Mathematical models for predicting the harvesting energy in piezoelectric materials submitted to axial loads consider its geometric properties, diameter *Dp* and thickness *hp*, and its mechanical and electrical properties, Young's modulus is *cE zz*, the piezoelectric constant is *ezz* and the dielectric constant is *ϵ<sup>S</sup> zz*, as shown in **Figure 1**. Points 1 and 2 represent the two faces of the piezoelectric disk, and *w* and *V* are the mechanical displacement and electric potential, respectively, at these two points.

The electromechanical coupling effect of the piezoelectric material can be described using a set of basic equations as given in the IEEE Standard on Piezoelectricity [7]:

$$\begin{aligned} \sigma\_x &= \mathfrak{e}\_{xx}^E \cdot \frac{\partial w}{\partial \mathbf{z}} - \mathfrak{e}\_{xx} \cdot \frac{\partial V}{\partial \mathbf{z}}\\ D\_x &= \mathfrak{e}\_{xx} \cdot \frac{\partial w}{\partial \mathbf{z}} + \mathfrak{e}\_{xx}^S \cdot \frac{\partial V}{\partial \mathbf{z}} \end{aligned} \tag{1}$$

Where *σ<sup>z</sup>* is the normal stress, *Dz* is the electric flux density, both in direction z. As seen in the above equations, the electro/mechanical coupling occurs due to the piezoelectric constant *ezz*.

The mechanical strain energy *Um* and the electric energy *Ue* of the piezoelectric material are written as [7]:

$$\begin{aligned} U\_m &= \frac{1}{2} \int\_V \sigma\_x \cdot \frac{\partial w}{\partial \mathbf{z}} dV = \frac{1}{2} \int\_V \left( c\_{xx}^E \cdot \frac{\partial w}{\partial \mathbf{z}} - e\_{xx} \cdot \frac{\partial V}{\partial \mathbf{z}} \right) \frac{\partial w}{\partial \mathbf{z}} dV \\\ U\_\epsilon &= \frac{1}{2} \int\_V D\_x \cdot \frac{\partial V}{\partial \mathbf{z}} dV = \frac{1}{2} \int\_V \left( e\_{xx} \cdot \frac{\partial w}{\partial \mathbf{z}} + c\_x^S \cdot \frac{\partial V}{\partial \mathbf{z}} \right) \frac{\partial V}{\partial \mathbf{z}} dV \end{aligned} \tag{2}$$

**Figure 1.** *The piezoelectric material model.* *Energy Harvesting Prediction from Piezoelectric Materials with a Dynamic System Model DOI: http://dx.doi.org/10.5772/intechopen.96626*

The electrical power can be calculated as the partial derivative of the electric energy, presented by Eq. (2), in respect to time:

$$P\_{\varepsilon} = \frac{\partial U\_{\varepsilon}}{\partial t} = \frac{1}{2} \left[ \frac{\partial}{\partial t} \left( D\_{x} \frac{\partial V}{\partial x} \right) dV = \frac{1}{2} \left[ \frac{\partial}{\partial t} \left( e\_{xx} \frac{\partial w}{\partial x} + e\_{x}^{S} \frac{\partial V}{\partial x} \right) \frac{\partial V}{\partial x} dV \right] \tag{3}$$

### **3. Dynamic system modelling**

This chapter aims to predict the wasted energy from vibration systems that can be further transformed into electrical energy. A typical vibration system can be described as mass, spring and damper elements, and that can represent a suspension system assembly.

The suspension system is an assembly of suspension arms or linkages, springs and shock absorbers that connect the wheels to the vehicle's chassis in order to isolate passengers from vibrations due to bumps and roughness of the road. Furthermore, it must maintain the contact of the wheels with the road to ensure drivability. Thus, the suspension system is the mechanical system where the stability and handling of the vehicle, besides energy harvesting, must be equilibrated.

Mathematical models were initially developed for vertical vehicle performance, and the one-dimensional quarter car model is the simplest from the frequently used suspension system [8]. It is composed of the sprung mass *ms*, which represents ¼ of the vehicle's body and the unsprung mass *mu*, which represents the wheel assembling mass. Both are considered rigid bodies. Its displacements are *ws* and *wu*, respectively and both are vertically aligned. There are some studies that include a third degree in the system to describe road roughness excitation *wr*. In this case, only the bounce input mode, or the vertical displacement can be implemented. Other elements of the suspension system are included, such as tire stiffness *kt*, suspension stiffness *ks*, and viscous damping *cs*. All vertical displacements are a function of the independent variable *t* that represents the time. **Figure 2** illustrates this 1D quarter car model, which could represent both the front and rear of the vehicle.

The expressions of kinetical energy from the masses, strain energy and dissipation function from the shock absorber, and the virtual work from the road roughness excitation are written as:

$$\begin{aligned} T &= \frac{1}{2} m\_i \dot{w}\_i^2 + \frac{1}{2} m\_u \dot{w}\_u^2\\ U &= \frac{1}{2} k\_s (w\_s - w\_u)^2\\ R &= \frac{1}{2} c\_s \left(\dot{w}\_s^2 - \dot{w}\_u^2\right)\\ \delta W &= F\_r(t) \delta w\_u = k\_t w\_r(t) \delta w\_u \end{aligned} \tag{4}$$

### **4. The suspension system and piezoelectric disk coupling**

The conversion of the mechanical energy from the vehicle suspension to electric energy through piezoelectric materials can be predicted by piezoelectric disk coupling illustrated in **Figure 1** and the suspension system illustrated in **Figure 2**. Since the conversion of mechanical energy to electrical energy in this system is produced by compressive efforts, the piezoelectric disk is located between the shock absorber *Piezoelectric Actuators - Principles, Design, Experiments and Applications*

**Figure 2.** *1D quarter car model.*

**Figure 3.** *The suspension system and piezoelectric coupling model.*

system composed by stiffness *ks* and viscous damping *cs* on the bottom side, along with the sprung mass *ms* on the upper side, as illustrated in **Figure 3**. As stated previously, vertical displacements *w* and now the electric potential *V*, are all a function of the independent variable *t*.
