# MATHEMATICAL FORMULATION

To optimize the electric field and the average stress of the piezoelectric cantilever beam while minimizing the volume of the piezoelectric beam, a finite element model was developed by using the commercial package ANSYS® in conjunction with optimization method-based direct search. The initial geometry and finite element mesh of the model used is depicted in Figure 5.13.

Grid-independence results analysis was carried out in this investigation, and it was deduced that a total number of 13,104 elements and 60,961 nodes were sufficient to produce mesh-independence results. The cantilever beam that was used in this study is composed of two layers: the top layer is a piezoelectric material and the bottom layer is an elastic material. The polarization (P) of the piezoelectric material is assumed along the у-direction. The interface between the two layers is grounded (voltage equals zero). A tip load boundary condition of 6 N was applied in the negative у-direction at the free end of the cantilever. The initial dimensions of the piezoelectric and substrate layers are listed in Table 5.1.

The mechanical and electrical properties of the piezoelectric material and the elastic substrate are given as follows [42]:

Elastic Bottom Layer: The material properties of the bottom layer are:

£ = 90 GPa, and a = 0.3

FIGURE 5.13 Geometry and mesh of the model used in the present investigation.

TABLE 5.1

Dimensions of the Initial Design

 Parameters Initial Value (mm) Thickness of PZT material 1 Thickness of elastic substrate 4 Length of elastic material 100 Width of PZT material 0.5 mm < Lra < 10 mm 10

Piezoelectric Layer: Assuming anisotropic materials of the piezoelectric beam, the anisotropic elastic properties used in this investigation can be given in a matrix form as follows [42]:

The piezoelectric coefficients in the strain form used in this investigation are also given in a matrix form as follows [42]:

# FORMULATION OF THE OPTIMIZATION PROBLEM

In this investigation, the optimal geometry of the piezoelectric cantilever is determined by direct search based optimization, which is used in conjunction with the ANSYS finite element software. The design problem is therefore to maximize the output electric field and simultaneously to minimize the structural volume of the piezoelectric beam, subject to the design constraints. The formulation of the optimization problem can be outlined as follows:

where X is the design variable, Vm is the volume of the piezoelectric beam to be minimized, EnT is the electric field, crmax is the maximum stress (i.e., maximum Von Mises stress), and cry is the yield strength of 25 MPa [43]. The subscript “initial” indicates the initial value that corresponds to the initial iteration. x‘"'n and x,max are the lower and upper limits of the design variables of .^respectively. There was one geometrical design variable in the structural model, which represented the free end width of the piezoelectric beam. The lower and upper bounds are set to 0.5 mm and 10 mm, respectively. The optimization process starts with an initial geometry in Ansys. Each iteration, the geometry will be changed and results are obtained until maximum electric field and minimum volume are achieved.

# CODE VALIDATION

The initial model is validated against analytical results reported in the literature by Smits et al. [45]. Smits et al. [45] developed a relationship between the deflection of the beam and the applied voltage using the energy density as follows:

where V is the applied voltage, and tp is the thickness of piezoelectric beam. Figure 5.14 demonstrates an excellent comparison of the deflection along the beam between the present results and the analytical results of Smits et al. [45].

FIGURE 5.14 Comparison of the deflection along the beam between the present results and the analytical results of Smits et al. [45]. (V = 100 volt. tr = 1 mm, dM = 2.3 x 10~M C/N). (From Smits, J.G., et al.. Sen. Actuators A, 28, 781-784, 1991. With permission.)

# RESULTS AND DISCUSSION

The direct optimization of the piezoelectric beam length required 20 trials to optimize the electric field of the piezoelectric beam. Figure 5.15 illustrates the optimized geometry and mesh of the model. The width of the free end was reduced to very small value of 0.5 mm compared the initial value of 10 mm, the same as at the fixed end of the beam. Table 5.2 summarizes the initial and optimized volumes of the substrate and the piezoelectric layers. This table illustrated that the structural volume of both piezoelectric and the substrate decreased considerably from 5 x 10'' mm3 to 2.62 x 103 mm3, representing a 47.6% saving in materials.

The effect of varying the geometry on the area normalized quantities, voltage, electric field, and normal strain (ej distribution along the length of the piezoelectric beam is depicted in Figure 5.16. For the studied designs, the trapezoidal beam geometry was found to produce the improved results. Figure 5.16 clearly shows that in the case of the trapezoidal geometry, the deformation increases because of the beam stiffness reduction. Consequently, the voltage, electric field, and the normal strain (eA) distribution along the center line of the piezoelectric beam rise significantly compared with the rectangular geometry. Figure 5.16a illustrates a comparison of the voltage/area distribution along the centerline length of the piezoelectric layer between the optimum and initial designs. The maximum voltage for both designs was found to occur in the vicinity of the fixed end due to maximum bending moment location. Moreover, optimum design was found to exhibit higher voltage/area distribution compared wfith the initial design as depicted in Figure 5.16a. The maximum value of

FIGURE 5.15 Optimized geometry and Finite Element Mesh.

TABLE 5.2

Volumes of the Initial and Optimized Geometries of the Substrate and Piezoelectric Layers

 Parameters Initial Geometry (mm3) Optimized Geometry (mm3) Volume of the substrate 4 x 10' 2.09 x 103 Volume of the piezoelectric 1 x 10' 5.23 x 101 Total volume 5 x 103 2.62 x Ю3

FIGURE 5.16 Comparison of the area normalized quantities (a) voltage, (b) electric field, and (c) normal strain (t) distribution along the center length of the piezoelectric beam between the rectangular design and the trapezoidal design for the same tip force.

the voltage/area was found to increase by 96.4%. Figure 5.16b demonstrates that the maximum electric field per unit area was also found to increase from 5.87 x 108 V/m3 for the initial design to 1.14 x 109 V/m3 for the optimum design, which represents an increase of 94.2%. As can be seen from Figure 5.16c, the trapezoidal shape exhibits higher strain than the rectangular shape for identical load.

FIGURE 5.17 Comparison of the electric field distribution in у-direction between the rectangular design and the trapezoidal design of the piezoelectric beam (tip force of the beam = 6 N).

Figure 5.17 shows the contour plots for the comparison of the electric field distribution in у-direction between the initial rectangular design and the optimized geometry (trapezoidal design) of the piezoelectric beam (tip force of the beam = 6 N). The electric field is found maximum in the vicinity of the fixed end for both designs, and this is associated with maximum bending moment location. Furthermore, the trapezoidal design shows a more even distribution than the rectangular design.

A similar behavior is also observed, when the voltage distribution is considered, as depicted in Figure 5.18. The von Mises stress fields in both designs are compared, under the same tip load of 6 N, applied downward as shown in Figure 5.19. Although the structural volume of the optimized trapezoidal design reduced by 47.6% from its initial volume, the maximum values of the induced stresses in both designs are approximately of the same order (about 17 MPa), and they occur at the constrained

FIGURE 5.18 Comparison of the voltage distribution between the rectangular design and the trapezoidal design of the piezoelectric beam (tip force of the beam = 6 N).

end of each beam. The stress distribution in the trapezoidal beam shows less variation along the beam axis. The minimum stress value occurs in both cases at the free beam end, whereas a much higher value arises at the tip of the trapezoidal beam due to the very small cross-sectional area (0.25 MPa vs. 0.02 MPa).

Modal analysis was carried out on both beam designs to calculate the first four natural frequencies, given in Table 5.3. Figure 5.20 displays the resulting frequencies compared with the number of vibration mode for both beams. At all calculated modes, the trapezoidal cantilever design has higher natural frequency values than the rectangular design. The increased sensitivity of the trapezoidal design leads to higher power output.

FIGURE 5.19 Comparison of the von Mises stress between the rectangular and the trapezoidal designs.

TABLE 5.3

The First Four Natural Frequencies for Rectangular and Trapezoidal Cantilever Beam Designs

 Mode Rectangular Geometry f[Hz] Trapezoidal Geometry f[Hz] 1 331.07 619.12 2 2046.7 2669.6 3 5594.8 6388.5 4 9830.1 11509

FIGURE 5.20 Comparison of the natural frequency between the rectangular design and the trapezoidal design for different modes.

# CONCLUSIONS

Finite element method in conjunction with direct optimization was conducted in this review to maximize the output electric field of the piezoelectric beam while minimizing the structural volume. The present numerical method was validated against theoretical studies reported in the literature and excellent agreement was found between both results. The results of this review indicated that the structural volume of both piezoelectric and the substrate reduced significantly from 5 x 103 mm3 to 2.62 x 103 mm3, representing a 47.6% saving in materials. The maximum Von Mises stress results showed insignificant increase from 16.8 MPa to 17 MPa, respectively. These results are well within the strength constrains уШ = 24 MPa). The maximum electric field per unit area was found to increase from 5.87 x 10s V/m3 for the initial design to 1.14 x 109 V/m3 for the optimum design, which represents an increase of 94.2%. The modal analysis illustrated that the optimum design (trapezoidal cantilever) exhibited higher natural frequencies at all calculated modes, which may lead to higher power output. The preliminary results presented in this review revealed that varying the geometry of the piezoelectric layer may have a significant effect on the characteristics of piezoelectric beam. Therefore, the future work will focus on maximizing the output power by seeking optimum topology of the cantilever beam.

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