**1. Introduction**

Periodic structures can be defined as structures consisting of a series of repeating segments with the same physical properties and sizes. Theoretical investigations of such structures have usually been carried out by the assumption of infinite dimensions [1–5]. However, certain features of periodic structures may also manifest even in the case of structures of finite dimensions if they include a sufficient number of segments.

One of specific features of periodic structures are band gaps in their frequency spectra. Band gaps define frequency ranges within which signals cannot propagate within these structures. The locations and the widths of these gaps in the frequency spectra are strongly dependent on the size of the unit cell and such material properties as modulus of elasticity [6–8]. These special features of periodic structures can be employed for very efficient vibration damping. On the other hand, active vibration damping methods include techniques that use piezoelectric materials. Structures with active piezoelectric elements enable one the conversion of mechanical vibrations to electrical vibrations and thus to control damping properties of the system. Therefore, only the balance between passive and active damping allows one to maximise the effectiveness of the damping process.

Additionally, while modeling periodic structures, the influence of features resulting from the application of a particular numerical model, on the results of calculations, should be taken into account carefully. Almost every computational model of a discretised structure (finite element method (FEM) or time domain spectral finite element method (TDSFEM) models), has certain characteristics of a periodic structure. Therefore, it is worth to analyse if certain features of periodic structures may be utilised in a directed manner in order to make practical use of the unusual behaviour of such structures. This approach may potentially allow to reduce, or enhance, periodic properties of the computational model.

In this paper, the authors propose to combine all the aspects mentioned above. They propose a special numerical model of the beam with active piezoelectric elements, by means of which the dynamic characteristics of the beam can be analysed and the width of band gaps can be controlled.

## **2. Numerical Model**

The structure under consideration, presented in Figure 1, is a sequence of 50 unit cells, consisting of an aluminium beam with piezoelectric rectangular strips (from APC International, Ltd., Cat.No. 70-1000, item 721) attached on both sides. Each pair of piezoelements is connected to the RLC resonant circuit with a controlled inductance.

**Figure 1.** A concept of an electromechanical periodic structure (**a**), unit cell (**b**).

The material parameters taken into calculations were as follows: for aluminium E = 67.5 GPa, *ρ* = 2700 kg/m<sup>3</sup> and *ν* = 0.33 and for piezoelectric material E = 63 GPa, *ρ* = 7800 kg/m3, *ν* = 0.33 and piezoelectric electro-mechanical coupling coefficient *k*31 = 0.35. The geometry of the analysed beam was as follows: the length *L* = 1 m, the width *b* = 0.02 m, the height *h* = 0.01 m, the single RLC element length was 0.01 m.

Numerical modelling of the piezoelectric material properties was based on the approach proposed in the literature [9,10]. The authors presented there a formulae for calculation of the effective Young's modulus of the piezoelectric element being an element of a resonant circuit. The piezoelectric material has frequency-dependent stiffness and damping, and the frequency itself depends on the parameters of the resonance circuit. Therefore, the effective Young's modulus of the piezoelectric material in the resonant circuit can be described by the equation:

$$E\_p^{SII}(\omega) = E\_p^D \left( 1 - \frac{k\_{31}^2}{1 + i\omega \mathcal{C}\_p^c Z^{SII}(\omega)} \right),\tag{1}$$

where *ESUp* is the effective Young's modulus of the piezoelectric material in a closed circuit mode, *EDp* is the effective Young's modulus of the piezoelectric material in an open circuit mode, *k*31 is the electro-mechanical coupling coefficient of thee piezoelectric material, *Cp* is the capacitance of the piezoelectric element, and *ZSU* is the impedance of a resonant circuit. In the carried out numerical calculations, PZT impedance has been taken into account as an electrical circuit parameter. The impedance of the aluminium beam itself has been neglected because the aluminium beam is not a part of a controlled electrical circuit. In the presented paper, changes in the vibration characteristics of the aluminium beam with attached, actively controlled PZT elements have been analysed.

The displacement and deformation fields of the analysed beam structure have been assumed according to the Timoshenko theory. The mathematical formulae can be expressed by [11,12]:

$$\begin{cases} u(\mathbf{x}) = z\phi(\mathbf{x}) \\ w(\mathbf{x}) = w\_0(\mathbf{x}) \end{cases} \tag{2}$$

$$\begin{cases} \varepsilon\_{\mathbf{x}} = \frac{\partial u(\mathbf{x})}{\partial \mathbf{x}} = z \frac{d\phi(\mathbf{x})}{d\mathbf{x}} \\\\ \gamma\_{\mathbf{x}\mathbf{z}} = \frac{\partial w(\mathbf{x})}{\partial \mathbf{x}} + \frac{\partial u(\mathbf{x})}{\partial \mathbf{z}} = \frac{dw\_{0}(\mathbf{x})}{d\mathbf{x}} + \phi(\mathbf{x}), \end{cases} \tag{3}$$

where *u*(*x*) and *w*(*x*) are respectively the longitudinal and transverse components of the element displacements expressed in the global coordinate system, while the independent rotation *φ*(*x*) around the *y* axis and the lateral displacement *<sup>w</sup>*0(*x*) are nodal displacements, defined in neutral element axis.

Following the standard FEM procedures, the inertia matrix *M* and the stiffness matrix *K* were evaluated: 

$$\mathbf{M} = \rho \iiint\_{V} \mathbf{N}^{t} \mathbf{N}dV, \quad \mathbf{K} = \rho \iiint\_{V} \mathbf{B}^{t} \mathbf{D} \mathbf{B}dV,\tag{4}$$

where *ρ* is the density of the material, **D** is the matrix of elasticity coefficients, and **N** and **B** are the shape function and strain-displacement matrices, respectively.

The presented numerical simulations have been obtained by the use of the classical Finite Element Method (FEM), the Frequency Domain Spectral Element Method (FDSFEM, details have been widely presented by Doyle in [13], the interested Reader is encouraged to follow the source) or the Time Domain Spectral Finite element Method (TDSFEM) approach.

In the classical FEM approach, the unit cell has been divided into three finite elements while in the case of the spectral approach the unit cell has been represented by one finite element, as shown in Figure 2.

**Figure 2.** Modelling a unit cell of an electromechanical periodic structure: (**a**) by the finite element method (FEM), (**b**) by the spectral finite element method (SFEM).

The main difference of the TDSFEM in comparison to the FEM is that in the TDSFEM approach the element nodes are not equally distributed. Coordinates of the nodes are defined as roots of a certain orthogonal polynomial:

$$T\_p^c = (1 - \mathfrak{J}^2) \mathcal{U}\_{p-2}(\mathfrak{J}),\tag{5}$$

which in the analysed case has been *Up*−<sup>2</sup>(*ξ*)—the second order Chebyshev polynomial. The element nodes in the element coordinate system may be calculated as follows:

$$\xi\_i = -\cos\frac{\pi(j-1)}{p} \quad j = 1, \ldots, p+1. \tag{6}$$

Such a definition of node distribution allows one to use higher order shape functions without the risk of causing the Runge effect. The node distribution used in the calculations performed for this paper has been shown in Figure 3.

**Figure 3.** (**a**) A unit cell in the global coordinate system (**b**) A node distribution in element coordinate system.

The stiffness and inertia matrices corresponding to the piezoelectric element within the respective integration limits have been joined with the stiffness and inertia matrices of the aluminium element respectively, as shown in Figure 4. The procedure has been precisely described in [14] for the case of passive periodic structures being a beam and rod with a sequence of drilled holes.

**Figure 4.** Construction of inertia and stiffness matrices in the case of the SFEM.

The aforementioned mathematical operations ensures that the stiffness matrix of the piezoelectric element is dependent on the frequency, therefore it is possible to actively control the mechanical responses of the analysed element by frequency variation.

## **3. Numerical Analysis**

In order to examine whether the proposed numerical approach is appropriate, a series of numerical experiments were carried out to verify the impact of the resonance circuit parameters on the physical properties (the width and placement of band gaps) of the periodic beam. During calculations the periodic boundary conditions were assumed.

The graphs shown in Figure 5 represent frequency response functions in the ranges from 0 to 250 kHz for the periodic beam with a resonant circuit being: open, closed or tuned to the specific frequency. The left column of Figure 5 shows the results obtained by the use of the FEM, the right hand side column of this Figure—by the TDSFEM [12] respectively. It may be noticed that there appeared two natural band gaps in given frequency ranges for the passive structure. Tuning the PZT circuits to the resonant frequency introduced an artificial band gap in the range of that frequency. However tuning the circuits to the frequency in the range of the natural periodic beam band gap, with the inactive PZT element, significantly widened the band gap. It should be also mentioned that for a lower range of the frequency spectra both the FEM and the TDSFEM results were quite similar, but in a higher frequency range the FEM results were distorted. The reason for that has been widely discussed in [14], where several features of the numerical models have been addressed.

Figure 6 shows the influence of the PZT circuits resonance frequency on the width of beam band gaps. This example was calculated with the TDSFEM. In this case it has been demonstrated how changes in the resonant frequency of PZT circuits allows one to control the ranges of blocked frequencies in the case of forced vibrations. Red colour represents frequency ranges that will propagate freely in the structure, the other colours (blue, green and yellow) represent different levels of attenuation.

The graphs presented in Figure 7 illustrate the effect of changes in the electrical resistance on the active periodic structure frequency band gaps. Here the TDSFEM has been used. The figures shown on the left hand side present the results calculated for 1 Ω RLC circuit resistance, the right hand-side—5 Ω respectively. As it can be noticed higher values of the resistance in the RLC circuit increased the energy dissipation and, as a result, widens the band gap. This effect is more significant in case, when the resonant frequency of the RLC circuit is equal to the passive structure frequency band gap. It it may be noticed in the bottom right graph from Figure 7.

**Figure 5.** Frequency response functions of a passive and active periodic structure with the resonant circuit tuned to the frequency 50 kHz or 100 kHz (marked with a red line) and resistance 1 Ω.

**Figure 6.** Dependence of the vibration amplitude on the resonant frequency of the RLC circuits.

**Figure 7.** Frequency response functions of an active periodic structure with the resonant frequency of 50 kHz or 100 kHz (red line), and the resistance of 1 Ω (**left**) or 5 Ω (**right**).
