Active shape and vibration control of functionally graded thin plate using functionally graded piezoelectric material

Abstract

This article deals with the geometric nonlinear static and dynamic analyses of thin functionally graded structure sandwiched between functionally graded piezoelectric materials. The properties of functionally graded material are graded in thickness direction according to power law distribution, and the variation in electric potential is assumed to be quadratic across the thickness of functionally graded piezoelectric material layers. The functionally graded material structure is modeled using finite element modeling considering complete Green–Lagrangian strains. The finite element formulation is derived using Hamilton’s principle based on first-order shear deformation theory. The ensued nonlinear algebraic equations are then solved using the modified Newton–Raphson method. Shape and vibration control of functionally graded plate is presented using functionally graded piezoelectric material. Fuzzy logic controller is used to control the vibrations. The numerical results predict that functionally graded piezoelectric material can control the shape and vibration of functionally graded plate.

Keywords

Functionally graded piezoelectric materials shape control active vibration control geometric nonlinear analysis fuzzy logic controller

Introduction

Piezolaminated functionally graded (FG) structures are widely used in the field of shape and vibration control. The piezoelectric ceramics are bonded on the surface of FG structure and are used as actuators, transducers, or sometimes even as micro positioners. However, its poor surface bonding capabilities and low reliability have been limiting their use in such applications. Moreover, the structural dissimilarities may cause debonding due to cyclic strains. The reliability of the structure can be increased by optimizing the stress distribution across the thickness. To overcome these shortcomings, new type of piezoelectric materials commonly known as functionally graded piezoelectric material (FGPM) has attained more attention in recent years. Using these materials, the problems at layer interfaces, such as high stress concentrations, creep at high temperature and failures from interfacial debonding are diminished to greater extent. Keeping this in mind, Zhu and Meng (1995) designed FG piezoelectric actuator where the electrostatic properties of laminae are graded through the thickness direction. The top layer had high dielectric constant and low piezoelectric constant, the mid layer was graded smoothly across the thickness, and the bottom layer had low dielectric constant and high piezoelectric constant. Takagi et al. (2002) fabricated an FG bimorph actuator using a mixture of lead zirconate titanate (PZT) and Pt which was used to predict its deflection property and an electrically induced stress in the actuator. Analytical and numerical methods based on modified classical plate theory and the finite element methods were used for calculations. The sensing and actuating mechanisms of an FG piezoelectric shell were investigated by Wu et al. (2002) using higher order theory. The authors claimed the superiority of FGPMs compared to homogeneous piezoelectric materials. Shi and Chen (2004) studied the direct and converse behavior of FGPM cantilever beam analytically under different loading conditions. Lee (2005) investigated analytically the displacement and stress response of an FG piezoelectric bimorph actuator, the formulation of which was based on the principles of linear thermo-piezoelectricity and accounts for the coupled mechanical, electrical, and thermal responses of piezoelectric materials. Yang and Xiang (2007) have done a parametric study on the static bending, free vibration, and dynamic response of monomorph, bimorph, and multimorph actuators made of FGPMs under a combined thermal-electro-mechanical load, and Hamilton’s principle was employed in theoretical formulation and the governing differential equations were then solved using differential quadrature method to determine deflection, reaction force, natural frequencies, and dynamic response of various FGPM actuators. The analytical static solution of an FGPM sandwich cantilever under thermal load was predicted using Airy stress function method by Xiang and Shi (2009). They demonstrated that the electromechanical coupling cannot be neglected when the piezoelectric coefficient of the material is large and the distribution of electric field or normal stress in FGPM actuators is continuous along the thickness. The equilibrium conditions and continuity conditions are satisfied for the stress, displacement, electric displacement, and electric potential on the interfaces. Free and transient response of FG piezoelectric cylindrical panel was studied analytically by Bodaghi and Shakeri (2012). Sharma et al. (2016) presented active vibration control of PZT–Pt-based FGPM considering linear conditions. First-order shear deformation theory was used to predict static and dynamic responses of the vibrating structure using finite element modeling.

In addition to linear analysis of FGPM, few research groups have included nonlinearities in their studies. The nonlinear stability analysis of FGPM beams was investigated by Komijani et al. (2012) considering von Karman nonlinearity. Timoshenko beam theory and trigonometric distribution of electric potential were used to formulate the governing equations which were solved based on Newton–Raphson and Picard iterative techniques. Arbind and Reddy (2013) used a modified couple stress theory with one length scale parameter to predict accurate deflections of micro- and nano-beam structures considering the von Karman geometric nonlinearity, power law distribution of material gradient, and microstructure length scale parameter. Bodaghi et al. (2014) monitored the static and dynamic behaviors of the structure using integrated FG piezoelectric sensors. A geometrically nonlinear thermo-elastic analysis of FG beams under thermo-mechanical loads was performed using the Hamilton principle. The equations of motion of smart FG beams were derived based on the first-order shear deformation theory and the geometrical nonlinearity in von Karman sense. Komijani et al. (2014) studied the effect of various types of in-plane and out-of-plane loads and boundary conditions on the thermo-electrical stability and electromechanical bending response of the actuators using finite element method. The effects of applied actuator voltage, power law index, and microstructural length were also presented through a number of numerical studies considering Timoshenko beam theory and von Karman nonlinearity.

Although there are considerable articles dealing with linear analysis of FGPM beams and plates, most of them neglect the implications of nonlinearities. Very few studies have been performed for nonlinear behavior of FGPM presenting analysis of plate and shells. However, none has shown their application to shape and vibration control. The smart structure deployed in space applications has to work in extremely harsh environments, and therefore, it becomes important to consider nonlinearities while studying them. In this article, capabilities of FGPM as sensors and actuators are studied and compared with that of PZT actuators and sensors. An FG plate with complete Green-Lagrangian nonlinear strains are considered, the properties of which are graded based on a power law form across the thickness. Thereafter, the performance of FGPM has been investigated for static bending, free vibration analysis, and dynamic response of the actuators and sensors under different boundary conditions. A fuzzy logic controller is used to attenuate the vibration developed in the plate.

Theoretical and numerical formulation

An FG substructure layer (plate) is sandwiched between two FGPM layers. The top FGPM layer acts as the sensor and the bottom FGPM layer acts as the actuator. The properties of both substructure and FGPM layers are graded through the thickness according to a power law distribution in terms of the volume fractions. Therefore, the effective material properties in each layer can be written as (Amini et al., 2015)

\begin{array}{l} P_{e f f}^{s} (z) = (P_{T}^{s} - P_{B}^{s}) {(\frac{1}{2} + \frac{z}{h_{s}})}^{n_{1}} + P_{B}^{s} \\ - \frac{h_{s}}{2} \leq z \leq \frac{h_{s}}{2} \\ P_{e f f}^{P T} (z) = (P_{T}^{P T} - P_{B}^{P T}) {(\frac{2 z - h_{s}}{2 h_{P T}})}^{n_{2}} + P_{B}^{P T} \\ \frac{h_{s}}{2} \leq z \leq \frac{h_{s}}{2} + h_{p} \\ P_{e f f}^{P B} (z) = (P_{T}^{P B} - P_{B}^{P B}) {(1 + \frac{2 z + h_{s}}{2 h_{P B}})}^{n_{3}} + P_{B}^{P B} \\ - \frac{h_{s}}{2} - h_{p} \leq z \leq - \frac{h_{s}}{2} \end{array}

(1)

where n is the non-negative volume fraction index which varies from 0 to $\infty$ , and h is the thickness of the layer. The superscripts s, PT, and PB stand for the structure, top piezo layer, and bottom piezo layer, respectively. P_T and P_B are the properties of the top and bottom layers, respectively, and P_eff is the effective property at any point across the thickness.

Displacement fields and strains

The displacement of any point within the element (shown in Figure 1) is given as

u_{i} (r, s, t) = \sum_{k = 1}^{4} h_{k} u_{i}^{k} + \sum_{k = 1}^{4} ((- \frac{a_{k}}{2} + \sum_{i = 1}^{n} l_{k}^{i} - \frac{l_{k}^{n}}{2}) + \frac{t^{n} l_{k}^{n}}{2}) (- {{}^{0}V}_{2 i}^{k} α_{k} + {{}^{0}V}_{1 i}^{k} β_{k})

(2)

where $u_{i}^{k}$ are the nodal mid-point displacements in the Cartesian coordinate directions; $h_{k}$ represents the two-dimensional interpolation functions corresponding to node k. $a_{k}$ is the thickness of the shell in t-direction at nodal point k, measured along the vector ${{}^{l}V}_{ni}^{k}$ ; $l_{k}^{i}$ is the thickness of ith layer at node k; $t^{n}$ is the natural coordinate of nth layer through the thickness; $m_{k}$ is the distance at node k between the element neutral surface and the mid plane of layer n. $α_{k}$ and $β_{k}$ are the rotations of the director vector ${{}^{0}V}_{n}^{k}$ about ${{}^{0}V}_{1}^{k}$ and ${{}^{0}V}_{2}^{k}$ axes, respectively. Left superscript l has value equal to 0 for initial geometry of the element and equal to 1 for the deformed element geometry.

Figure 1.

Piezolaminated degenerated shell element.

The Green–Lagrangian strain and second Piola–Kirchhoff stress tensor are considered as work conjugate pair. Green–Lagrangian strain is given in Appendix 1. Strain–displacement relationship can be written in compact form as

{_{0} ε}_{e} = [{}_{0}{\bar{B}}_{L}]_{e} {{}_{0}q'}_{e} + [_{0} {\bar{B}}_{NL}]_{e} {{}_{0}q'}_{e}

(3)

where $[{}_{0}{\bar{B}}_{L}]_{e} = [{}_{0}{\bar{B}}_{L 0}]_{e} + [{}_{0}{\bar{B}}_{L 1}]_{e}$ .

The description of these matrices can be followed in Appendix 1. Equation (3) can be further modified to include mixed interpolation of displacement and transverse shear strains as Dvorkin and Bathe (1984), to overcome the effect of shear locking.

Electric potential distribution

To model realistic electric potential, a quadratic distribution of the electric potential can be considered as (Balamurugan and Narayanan, 2008)

φ = (1 - t^{2}) \sum_{n = 1}^{4} h_{k} φ^{k} + \frac{1}{2} (t + t^{2}) Δ φ^{E}

(4)

The electric field inside the nth piezoelectric layer within the element can be given as

{_{0} E}_{n} = - {_{0} B_{φ}}_{e} {{{}_{0}φ}_{p_{n}}}

(5)

$B_{φ}$ is defined in Appendix 1.

Temperature field distribution

The temperature distribution is assumed to be linear within the element along the thickness direction. The temperature of any point within the element can be given as

{{}_{0}Θ}_{e} = [N_{Θ}]_{e} {{}_{0}θ}_{e} + [N_{φ}]_{e} {{}_{0}θ}_{e}

(6)

Finite element equation

The finite element equations are derived using Hamilton’s principle

\int_{t_{1}}^{t_{2}} (δ T - δ U + δ W_{ext}) dt = 0

(7)

where U, T, and $W_{ext}$ are the potential energy, kinetic energy, and external work, respectively. In a piezolaminated shell element, the potential energy (Tanveer and Singh, 2010) can be written as

U = \int_{{}^{0}V} (\frac{1}{2} {{}_{0}^{t + Δ t}{\vec{ε}}^{T}} {{}_{0}^{t + Δ t}{\vec{S}}} - \frac{1}{2} {{}_{0}^{t + Δ t}{\vec{E}}^{T}} {{}_{0}^{t + Δ t}{\vec{D}}}) d {}^{0}V

(8)

where the left subscript represents the configuration with respect to which the different variables are defined and the left superscript represents the configuration at which the different variables are to be calculated. Similarly, the converse and direct constitutive equations of piezoelectric material with respect to configuration at time $t + Δ t$ can be written as (Behjat and Khoshravan, 2012)

{{}_{0}^{t + Δ t}S} = [Q] {{}_{0}^{t + Δ t}ε} - [e^{T}] {{}_{0}^{t + Δ t}E} - {λ} {}_{0}^{t + Δ t}Θ

(9)

{{}_{0}^{t + Δ t}D} = [e] {{}_{0}^{t + Δ t}ε} + [b] {{}_{0}^{t + Δ t}E} + {p} {}_{0}^{t + Δ t}Θ

(10)

where ${D}$ , ${S}$ , ${ε}$ , and ${E}$ are the electric displacement, stress, strain, and electric field, respectively. [Q], [e], and [b] are the plane-stress reduced elastic stiffness coefficients, the piezoelectric constant matrix, and permittivity matrix, respectively. ${λ}$ and ${P}$ are the stress–temperature coefficient and pyro-electric coefficient vectors, respectively. $Θ$ is the temperature rise from the stress free reference temperature.

Substitute equations (9) and (10) in equation (8) and apply the linearization procedure as discussed by Bathe (1996). Thereafter, using the Hamilton’s principle, the equation of motion for the FG shell element for the piezoelectric continuum on the elemental level can be obtained as (see Appendix 2 for detailed formulation)

[M_{uu}] {\overset{\cdot\cdot}{q}} + ([K_{L}] + [K_{NL}] - [H_{NL}]) {q} + [K_{u ϕ}] {ϕ_{p_{n}}} + [K_{u θ}] {θ} = {F_{m}} - {F_{int}}

(11)

[K_{ϕ u}] {q} + [K_{ϕ ϕ}] {ϕ_{p_{n}}} + [K_{ϕ θ}] {θ} = {F_{q}} - {q_{int}}

(12)

The details of the matrices used are given in Appendix 2.

Nonlinear analysis

If the method of solving linear equations is used in the nonlinear case without any improvements, the errors due to linearization may introduce solution shift or instability. Therefore, an incremental-iterative scheme such as modified Newton–Raphson method used for solving the first-order differential equation is employed to trace the static behavior of thin-walled structures. Newton–Raphson method is an incremental load method in which equilibrium is checked at every load step. One or more iterations are required to remove the imbalance in the internal and external forces. The stiffness matrix is reformed only once during the iterative procedure in the modified Newton–Raphson method, which is used in this work. The iterative procedure is repeated until the imbalance in the forces reached the required tolerance level of 1e−3. This is checked using two different convergence criteria as follows:

The displacement convergence criteria

\frac{{‖ Δ u^{(i)} ‖}_{2}}{{‖ {{}^{t + Δ t}u}^{(i)} ‖}_{2}} \leq DTOL

(13)

The energy convergence criteria

\frac{Δ {u^{(i)}}^{T} ({}^{t + Δ t}F - {{}^{t + Δ t}F}_{int}^{(i - 1)})}{u^{T} ({}^{t + Δ t}F - {{}^{t}F}_{int})} \leq ETOL

(14)

In case of dynamic analysis, the second-order differential equation is solved using the Newmark method. The parameters β and γ usually have the values of 0.25 and 0.5, respectively. If the value of γ ≥ 0.5 and β ≥ (2γ + 1)²/16, the method is unconditionally stable. This means the size of time step does not have any affect; however, the accuracy may be affected.

Control algorithm

To control vibration of the host structure, FGPM sensor voltage is coupled with FGPM actuator voltage via fuzzy logic controller. Fuzzy logic controller includes four basic steps, fuzzification, fuzzy rule, fuzzy interface engine, and defuzzification (Zhang, 2010) as shown in Figure 2. Fuzzification is a process which interprets the interaction between the input and output variables. First of all, the input and output variables for the fuzzy logic controller are decided. Thereafter, the range of these variables is decided such that they remain within the safe operating conditions. The most frequently used input variables for fuzzy logic controller are error and change in error. In this study, displacements and rate of change in displacements of the structure are used as the inputs for the fuzzy logic controller and the output is the control force. Here, “displacement” stands for the difference between the existent displacement and the desired displacement which is 0, and “velocity” means the difference between the existent velocity and the desired velocity which is 0. Then the crisp input values are converted into the fuzzy value within the fuzzy range of 0–1. The fuzzification step converts the crisp (real) inputs into fuzzy inputs: the second process of controller (inference mechanism) uses the fuzzy rules in the rule base to produce fuzzy conclusions, and finally, the defuzzification block transforms these fuzzy conclusions into crisp outputs. Seven membership functions for fuzzy input and output variables are used and classified into linguistic values of the displacement and velocity as follows: negative large (N++), negative medium (N+), negative small (N), zero (Ze), positive small (P), positive medium (P+), and positive large (P++). The combination of trapezoidal and triangular membership functions is used to fuzzify the variables as shown in Figure 3. The linguistic quantification described is thereafter used to define the relationship between the input and output variables using simple IF–THEN rules which are shown in Table 1. The form of the rules can be given as

IF the displacement is D “AND” the change of the displacement is “V”

THEN the voltage is “v”

where D, V, and v belong to their own fuzzy sets. The total number of rules for the fuzzy logic controller is 49. The center of gravity method is used for defuzzification.

Figure 2.

Different stages of fuzzy logic controller.

Figure 3.

Membership functions for fuzzy input and output variables.

Table 1.

Rule base for fuzzy logic controller.

Fuzzy If–Then rules
	N++	N+	N	Ze	P	P+	P++
N++	P++	P++	P++	P++	P+	Ze	Ze
N+	P++	P++	P++	P++	P+	Ze	Ze
N	P+	P	P+	P+	Ze	N	N
Ze	P+	P+	P	Ze	N	N+	N+
P	P	P	Ze	N+	N+	N+	N+
P+	Ze	Ze	N+	N++	N++	N++	N++
P++	Ze	Ze	N+	N++	N++	N++	N++

N++: negative large; N+: negative medium; N: negative small; Ze: zero; P: positive small; P+: positive medium; P++: positive large.

Validation and numerical results

Validation of finite element formulation

Based on the finite element formulation presented in section “Theoretical and numerical formulation,” a code is developed and validated by comparing with the results existing in the literature. A square cantilever functionally graded material (FGM) plate consisting of combined zirconia and aluminum as material constituents with continuously varying mix ratios is considered. The bottom surface of the FGM plate is assumed to be metal rich and the top surface to be ceramic rich. The FGM plate is integrated with piezoelectric sensor and actuator layers. The G-1195N piezoelectric patches are bonded on both the surfaces of the square plate having length of 200 mm and thickness of 30 mm. The thickness of the piezoelectric layers is 0.1 mm, and the relevant material properties are the same as reported by Liew et al. (2001). A thermal gradient of 100°C/m is applied to the top surface of the cantilevered square plate. Figure 4 represents the comparison of centerline deflection using the present formulation with Liew et al. (2001). It can be observed that the numerical results are in excellent agreement with the published results.

Figure 4.

Centerline deflection along x-axis for FGM plate under thermal gradient of 100°C/m.

Thereafter, a piezolaminated FG plate is considered to validate the natural frequency as a function of the volume fraction for different boundary conditions. Table 2 shows the comparison of natural frequency for FGM plate with He et al. (2001) for clamped free (CFFF)- and simply supported (SSSS)-type boundary conditions at various material indexes, where, C represents Clamped, F represents Free and S represents Simply supported. Table 3 shows the first five natural frequencies for CFFF-type boundary condition at material index of 0, 1, and 100. The same FG plate is further used to validate the transient response. The vibrations in the plate are introduced by applying unit force as an impact force. All the initial and boundary conditions are the same as taken by He et al. (2001). Figure 5 shows the variation in the displacements at the tip of the cantilever plate for two volume fractions (0 and 1000). It is clear from Figure 5 that the obtained displacement time response shows good agreement with the published results.

Table 2.

Natural frequency (Hz) for FGM plates with actuator layers bonded on the top and bottom surfaces.

	Volume fraction
	0	0.2	0.5	1	5	15	100
He et al. CFFF	25.58	29.87	32.84	35.33	40.97	43.97	46.12
Present nonlinear analysis	25.52	29.20	32.51	35.95	44.52	47.60	47.89
He et al. SSSS	144.25	168.74	185.45	198.92	230.46	247.30	259.35
Present nonlinear analysis	147.89	167.73	185.67	204.27	250.51	267.05	268.59

CFFF: Clamped Free; SSSS: Simply supported.

Table 3.

First five natural frequencies (Hz) for FGM plates (CFFF) with actuator layers bonded on the top and bottom surfaces.

Mode no.	Volume fraction
	He et al.	Present	He et al.	present	He et al.	present
	0		1		100
1	25.58	25.52	35.33	35.95	46.12	47.89
2	62.75	62.63	87.52	88.80	114.93	119.40
3	157.20	158.21	218.04	223.39	284.96	298.56
4	200.19	201.35	276.89	283.80	361.66	378.52
5	228.22	229.68	317.43	324.95	415.69	435.53

Figure 5.

A plot of decay envelopes showing the variation in the displacements at the tip of the cantilever plate.

Numerical study

For numerical study, a square plate integrated with PZT–Pt-based FG piezoelectric sensor and actuator layers is considered for active shape and vibration control application. The host layer is also made of FGMs (zirconia and aluminum), the properties of which are given in Table 4. For both FGPM sensor and actuator, the top layer is enriched with PZT–0% Pt and the bottom layer is enriched with PZT–20% Pt. The material properties are uniformly varied through the thickness of an FGPM layer with different material power law indexes.

Table 4.

Material properties.

Material properties	PZT–0% Pt	PZT–20% Pt	Zirconia	Aluminum
Young’s modulus, Y (GPa)	63	84.8	151	70
Density, ρ (kg/m³)	7600	10,290	3000	2707
Poisson’s ratio, υ	0.3	0.342	0.3	0.3
Piezoelectric constant, d₃₁ (10⁻¹⁰ C/m²)	−1.64	−0.98	0	0
Relative dielectric constant, ε₃₃/ε_o	1653	2927	0	0

PZT: lead zirconate titanate.

Shape control of FG square plate using FGPM

An FG square plate of length 0.4 m and thickness 2 mm sandwiched between two FGPM layers is considered. The plate is functionally graded in the thickness direction from aluminum at bottom to zirconia at the top. A uniformly distributed load of 50 N/m² is applied on simply supported FG plate as shown in Figure 6. Due to symmetry, only quarter part of the plate is considered and is divided into a mesh of 10 × 10 elements. Due to external applied load, the plate structure will undergo deflection. Figure 7 shows the stress distribution along the cross section at the center point of the plate. The strain in the x-direction is developed in the plate due to the bending moment. The piezoelectric patch is assumed to be placed in order to extract d₃₁ mode. Therefore, only the stress in the x-direction is of interest and hence presented in Figure 7. It can be observed from Figure 7 that the jump in stresses at the lower interface (A) is higher compared to upper interface (B). This happens due to the fact that the lower interface is in between aluminum (metal) and FGPM ceramic, whereas the upper interface is between zirconia (ceramic) and FGPM ceramic. Considering this, it is recommended that in order to decrease stress variation at both the interfaces, the middle layer should be functionally graded in such a way that the top and bottom faces of the host FGM structure are enriched with pure ceramic while the metal should be enriched at the middle of the structure. One of the most suitable and effective way to do the same is by considering two layers of FGM host structures. The resulting host structure will consist of two layers, that is, one layer is functionally graded from ceramic at the bottom to metal at the top and the second layer is graded from metal at the bottom to ceramic at the top. Figure 8 shows the stress distribution across the thickness considering two layers of FGM host structures, and it can be seen that stresses developed at both the interfaces are almost the same and no jump in the stress level from one layer to another is observed. This decrease in the stresses at the interface will help in decreasing stress concentrations, creep at high temperature and failure from interfacial debonding, which increases the reliability and life of the structure. The same host structure is further used to compare the deformation induced in FGM plate using PZT and FGPM layers. In the first case, the FGM plate is sandwiched between two PZT layers and 30 V voltage is applied to bring back the structure to its original position. In the second case, the FGM plate is sandwiched between FGPM layers and 30 V voltage is applied. The material index for FGPM layers is 1, and for host structure it varies from 0 to 10. The centerline displacement of the simply supported plate in both cases is shown in Figure 9. It can be seen that the centerline deflection due to the applied voltage on PZT and FGPM actuators is almost the same. Therefore, FGPMs can be used as an alternative to PZT as actuators. The centerline displacement of the cantilever plate with different material gradient indexes is shown in Figure 10. It can be seen from Figure 10 that centerline deflection increases with a decrease in the material gradient index as expected. This happens due to the change in the material properties which changes from ceramic to metallic. Thereafter, the above-mentioned FGPM is used in the shape control application of simply supported square FGM plate under the application of uniformly distributed loading of 50 N/m². Basically, the deformed shape of FGM plate due to external loading is regained to its original shape (undeformed shape) after applying external voltage on FGPM layers. Figures 11 to 14 shows shape control at material index 0, 1, 5, and 10 at the applied voltages of 0–40 V. It can be seen from the results that the shape of the FG plate can be controlled effectively using FGPM actuators.

Figure 6.

Functionally graded plate laminated with FGPM.

Figure 7.

Comparison of stresses developed at the interfaces using PZT and FGPM in case I.

Figure 8.

Comparison of stresses developed at the interfaces using PZT and FGPM in case II.

Figure 9.

The comparison of central deflection of plate under applied voltage of 30 V for PZT and FGPM at n = 1.

Figure 10.

Centerline deflection of simply supported plate with FGPM at various material indexes.

Figure 11.

Shape control of simply supported plate using FGPM at gradient index of n = 0.

Figure 12.

Shape control of simply supported plate using FGPM at gradient index of n = 1.

Figure 13.

Shape control of simply supported plate using FGPM at gradient index of n = 5.

Figure 14.

Shape control of simply supported plate using FGPM at gradient index of n = 10.

Active vibration control of FG square plate using FGPM

To actively control the vibration induced in the host structure, two FGPM layers are used. One layer acts as the sensor while the other acts as the actuator. The FG host layer is sandwiched between these FGPM layers. The same geometry and boundary conditions of FG plate are used as discussed in the previous section. To induce vibrations in the structure, a uniformly distributed load of 50 N/m² is applied. Figure 15 shows the comparison between linear and nonlinear vibrations. Figure 16 shows the variation in the vibration of plate at various material gradient indexes. Thereafter, the application of FGPM in the active vibration control of the simply supported plate under suddenly applied loading of 50 N/m² is considered. The induced vibrations are attenuated using a fuzzy logic controller and the results obtained are compared with the proportional and derivative feedback controllers as shown in Figure 17. It is assumed that there is no internal material damping in the system. To compare the effect of various controllers on the vibration amplitude of the structure, logarithmic decay is calculated for the first five peaks. It has been seen that the value of logarithmic decay increases by 75.58%, 81.83%, and 90.56% for proportional, derivative, and fuzzy logic controller, respectively, when the first five peaks are considered. Therefore, nonlinear control using fuzzy logic controller shows better attenuation capabilities than constant gain proportional and derivative controller for active vibration control application.

Figure 15.

Comparison of linear and nonlinear free vibrations under constant force in FGPM-laminated FG plate.

Figure 16.

Nonlinear free vibrations under impact force in FG plate at various gradient indexes.

Figure 17.

Comparison of active vibrations under impact force in simply supported plate.

Conclusion

In this article, nonlinear finite element modeling of FG structures laminated with FG piezoelectric ceramic is presented. The developed formulation is validated with those reported in the literature. It is shown that FGPMs can be effectively used to control the shape and vibration of the FG plate. Moreover, the application of FGPM can increase the reliability of the structure by decreasing the interface stress levels. The numerical results also predict that fuzzy logic controller damped out the vibrations faster than proportional and derivative controllers.

Footnotes

Appendix 1

Appendix 2 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Fuzzy If–Then rules
	N++	N+	N	Ze	P	P+	P++
N++	P++	P++	P++	P++	P+	Ze	Ze
N+	P++	P++	P++	P++	P+	Ze	Ze
N	P+	P	P+	P+	Ze	N	N
Ze	P+	P+	P	Ze	N	N+	N+
P	P	P	Ze	N+	N+	N+	N+
P+	Ze	Ze	N+	N++	N++	N++	N++
P++	Ze	Ze	N+	N++	N++	N++	N++

Fuzzy If–Then rules
	N++	N+	N	Ze	P	P+	P++
N++	P++	P++	P++	P++	P+	Ze	Ze
N+	P++	P++	P++	P++	P+	Ze	Ze
N	P+	P	P+	P+	Ze	N	N
Ze	P+	P+	P	Ze	N	N+	N+
P	P	P	Ze	N+	N+	N+	N+
P+	Ze	Ze	N+	N++	N++	N++	N++
P++	Ze	Ze	N+	N++	N++	N++	N++

Fuzzy If–Then rules
	N++	N+	N	Ze	P	P+	P++
N++	P++	P++	P++	P++	P+	Ze	Ze
N+	P++	P++	P++	P++	P+	Ze	Ze
N	P+	P	P+	P+	Ze	N	N
Ze	P+	P+	P	Ze	N	N+	N+
P	P	P	Ze	N+	N+	N+	N+
P+	Ze	Ze	N+	N++	N++	N++	N++
P++	Ze	Ze	N+	N++	N++	N++	N++