Optimal unimorph and bimorph configurations of piezocomposite actuators for bending and twisting vibration control of plate structures

Abstract

Optimal configuration of anisotropic piezocomposite actuators is investigated for vibration control of flexible structures. Actuation effects of the actuators in unimorph and antisymmetric angle-ply bimorph configurations are analyzed. Dynamic equations of the piezo-actuated plate are established using finite element method. The locations and lead zirconate titanate fiber orientations of actuators are optimized simultaneously to improve vibration control performance of a plate. The optimization criterion is presented based on controllability Gramian matrix of the system. The properties of macro-fiber composite are used in the simulations of the actuators. The optimal unimorph and bimorph configurations of actuators are presented for three cases: bending control, twisting control, and coupled bending–twisting control. The results indicate that both optimal locations and fiber orientations are different for bending control and twisting control. A trade-off between control authority for bending and torsional modes can be found in coupled bending–twisting control. The actuators should be placed in the regions of high average strain of the interested modes with appropriate fiber orientations. Special piezocomposite actuators with optimal fiber orientations can be used to achieve improved coupled bending–twisting vibration control performance.

Keywords

Piezocomposite actuators vibration control plate structures optimization bending twisting

Introduction

In general, owing to increased requirement for weight savings, aerospace structures are becoming more flexible, whereas their performance requirements are becoming more stringent. Active vibration control using smart piezoelectric materials has drawn widespread interest in enhancing performance and control characteristics of the flexible structures (Kalaycioglu and Silva, 2000). Piezocomposite materials are advanced piezoelectric devices with effective material properties, good conformability, and anisotropic actuation effects (Bent et al., 1995; Ray and Reddy, 2013). Piezocomposite actuators, such as the active fiber composite (AFC) and the macro-fiber composite (MFC), have been widely used in active control of flexible wings, blades, and other aerospace structures (Arrieta et al., 2013; Bilgen et al., 2013; Monner et al., 2011; Moses et al., 2001).

A variety of researches toward active control of flexible structures using piezoelectric actuators have been carried out, including static, dynamic shape control, and vibration control (Gupta et al., 2011; Irschik, 2002). Active vibration control of piezo-actuated structures has been widely investigated in theoretical and experimental way (Forster and Yang, 1998; Gao et al., 2013; Giurgiutiu, 2000; Raja and Upadhya, 2007; Vasques and Rodrigues, 2006). Static and dynamic shape control of the structures, including bending and twisting shape, could be implemented using piezoelectric actuators (Adnan Elshafei et al., 2014; Gohari et al., 2016, 2017). Conventional piezoelectric actuators are typically capable of exerting high pressures, but with small strains and small flexibility, which has limited their applicability (Usher et al., 2013). In addition, these actuators with through-the-thickness poling possess a transverse isotropy in the plane and cannot supply the desired twisting actuation moment (Wetherhold and Aldraihem, 2001). Thus, the actuators should be inclined to achieve torsional actuation of the structures (Adnan Elshafei et al., 2014; Gohari et al., 2017). Piezocomposite materials are composed of piezoelectric fiber reinforcements and epoxy matrix so they can provide wide range of and strength integrity. Piezocomposite devices can utilize the d₃₃ piezoelectric effect to the direction of lead zirconate titanate (PZT) fibers, which is larger than the d₃₁ piezoelectric effect of the conventional piezoelectric actuator (Choi et al., 2007). Particularly, piezocomposite materials feature anisotropic actuation effects which can be used for twisting control and coupled bending–twisting control of structures (Bent et al., 1995; Wetherhold and Aldraihem, 2001).

With regard to the vibration control of structures using traditional piezoelectric or advanced piezocomposite devices, the configuration of the actuators has significant effect on the control performance (Frecker, 2003). For the purposes of flexible structures testing and control, it is useful to investigate the optimal configuration of the actuators and to evaluate their impact on the control performance. Extensive research on the optimal configuration of piezoelectric actuators for vibration control have been carried out (Biglar et al., 2014; Bruant et al., 2010; Bruant and Proslier, 2016; Crawley and De Luis, 1987; Han and Lee, 1999; Leleu et al., 2001; Liu et al., 2003; Qiu et al., 2007; Schulz et al., 2013; Wang and Wang, 2000). The actuators could also be optimized to enhance the shape control performance of smart structures (Gohari et al., 2017). A general optimization problem consists of three essential components: optimization criteria (objective function), design variables, and constraints. Optimization criteria is used to evaluate the configuration of the actuators. A detailed review on optimization criteria for optimal placement of piezoelectric actuators can be found in Gupta et al. (2010). In most papers, the design variables mainly include location, number and size of the actuator. The placement for traditional actuators can be determined by above variables. However, piezocomposite actuators are anisotropic whose mechanical properties and actuation characteristics are affected by PZT fiber orientation (Park and Kim, 2005; Williams et al., 2002). Therefore, the fiber orientation is also an essential and important parameter which affects the control authority of actuators, especially for twisting vibration control (Hu et al., 2004; Ray and Reddy, 2013). Smart Material Corp. (2017) provides two types of standard MFC actuator patches which are utilizing the d₃₃ effect: P1 types with 0° fiber orientation and F1 types with 45° fiber orientation. P1-type actuators are mainly used for bending control of structures, and F1-type actuators are used for twisting control. Dynamic response of many structures (such as flexible wings, blades, and solar panels) feature coupled bending–twisting vibration characteristics, so both bending and torsional modes should be controlled (Wetherhold and Aldraihem, 2001). As a result, the optimal fiber orientation of the actuator may be neither 0° nor 45°, and special piezocomposite actuators with customized fiber orientations would be designed and manufactured to achieve specific goals (Monner et al., 2011; Riemenschneider et al., 2009). Therefore, the fiber orientation can be optimized to obtain preferable bending and twisting vibration control performance. Of course, other design variables, such as locations, should also be optimized to take full advantage of the control authority of the actuator.

However, bimorph configuration of actuators is usually used to improve control performance (Wang and Wang, 2000). The two layers of piezocomposite material can be actuated in same or opposite polarity to enhance control authority. Additionally, positive and negative voltage directions of piezocomposite actuators may be asymmetrical, for example, the operational voltage range of MFCs is −500 ∼ 1500 V. The bimorph configuration can also remedy this asymmetry issue via current-direction-selective-voltage-divider circuit (Bilgen et al., 2010). Particularly, antisymmetric angle-ply bimorph configuration, which is composed of two actuators with opposite off-axis fiber orientations, can be adopted to realize pure twisting control and coupled bending–twisting control (Bent et al., 1995; Kwak and Yedavalli, 2001; Wang et al., 2016). For this specific type of actuator configuration, the optimization should also consider choosing favorable locations and fiber orientations of the actuators for coupled bending–twisting vibration control.

In this study, optimal location and fiber orientation of the piezocomposite actuators in unimorph and bimorph configurations are investigated for vibration control of a flexible plate. The mathematical model is established using the finite element method (FEM). The properties of MFCs are adopted in this study to model the piezocomposite actuators. The design variables include the placement locations and fiber orientations of the actuators. The primary purpose is to maximize the degree of controllability of the system, and the objective function is presented based upon controllability Gramian matrix. The optimal unimorph and bimorph configurations of the actuators are presented for three cases with different structural modes of interest. The explanations and discussions about the optimization results are also presented.

System modeling

The structure in this study is characterized by means of a cantilever plate with surface-bonded piezocomposite actuators, as depicted in Figure 1. The schematics of the detailed structure of the actuator are also depicted.

Unimorph and bimorph configurations

Figure 2 illustrates the strains and equivalent actuation moments of the actuator in unimorph laminate concept by applying a positive voltage. The anisotropic actuation characteristics are directly affected by fiber orientations. Figure 3 shows the strains in the 1–2 coordinate system and x-y coordinate system. $θ$ denotes the fiber orientation angle, that is, the angle from x-axis to 1-axis. It can be easily found that largest normal strain $ε_{x}$ can be obtained with 0° fiber orientation and corresponding shearing strain $γ_{x y}$ (and $γ_{y x}$ ) is zero, thus overall bending deformation of the structure can be generated. In contrast, the largest shearing strain $γ_{x y}$ (and $γ_{y x}$ ) may be obtained with 45° or −45° fiber orientations, but the normal strain $ε_{x}$ still exists. Then, both torsional and bending deformations would be produced using single actuator in unimorph configuration with off-axis fiber orientation. Multiple actuators with different fiber orientations may be required to cancel bending moments and achieve pure torsional deformation and twisting control. Another way to achieve pure torsional deformation is using bimorph concept.

An antisymmetric angle-ply bimorph concept is also considered in this study, as demonstrated in Figure 4. A collocated actuator pair is bonded to the upper and lower surface of the plate, respectively. Note that the two actuators are actually same, but their fiber orientations are opposite with respect to x-axis due to opposite bonded surfaces. This bimorph concept can convert the shearing strains into an overall twisting deformation by applying same voltage signals for the two actuators, as shown in Figure 4(a), whereas bending deformation could be generated using opposite voltages, as shown in Figure 4(b). As a result, bending and twisting vibrations can be controlled independently, and the control system may be designed separately.

For both the unimorph and bimorph configurations, the fiber orientation has significant effect on the bending and twisting vibration control performance. The optimal fiber orientation may be determined intuitionally in pure bending or twisting control. However, it is difficult to know the optimal fiber orientations intuitively for the coupled bending–twisting vibration control. Therefore, the fiber orientations of the actuators will also be optimized in this study, as well as their locations.

Finite element modeling

Under the linear elasticity assumption, the constitutive equation for the base structure is given as

σ_{b} = c_{b} ε_{b}

(1)

where $σ_{b}$ and $ε_{b}$ are stress and strain vectors of the substrate material, respectively. $c_{b}$ is the elastic stiffness matrix.

The electromechanical coupling effect of the piezocomposite device can be described by the following constitutive equations (Allik and Hughes, 1970)

σ_{p} = c_{p} ε_{p} - eE

(2)

D = e^{T} ε_{p} + ζ E

(3)

where $σ_{p}$ and $ε_{p}$ are stress and strain vectors of piezocomposite device, respectively. $c_{p}$ is the elastic matrix of piezocomposite material. $D$ and $E$ denote the electric displacement and electric field, respectively. $e$ and $ζ$ denote the piezoelectricity matrix and permittivity matrix, respectively.

It should be noted that stress–strain transformation from principal material coordinate $O' 123$ to global coordinate $O x y z$ is needed for piezocomposite laminate where fibers do not coincide with the global coordinate. The transformation matrix is given as (Williams et al., 2002)

T = [\begin{matrix} \cos^{2} θ & \sin^{2} θ & 2 \sin θ \cos θ \\ \sin^{2} θ & \cos^{2} θ & - 2 \sin θ \cos θ \\ - \sin θ \cos θ & \sin θ \cos θ & \cos^{2} θ - \sin^{2} θ \end{matrix}]

(4)

The structural characteristics of the piezocomposite actuated plate are formulated using the FEM. Quadrilateral plate elements are used in this study, as shown in Figure 5(a). There are two essential elements of the finite element model: passive elements without piezocomposite layer as shown in Figure 5(b) and the active elements with piezocomposite layers as shown in Figure 5(c) and (d). Composite laminate elements are used to specify the properties of the laminated composite materials. The equations of motions of elements can be derived from Hamilton’s principle (Allik and Hughes, 1970). For passive elements, the equations are written as

M_{u u}^{s} {\overset{\cdot\cdot}{u}}_{N} + K_{u u}^{s} u_{N} = f_{e}

(5)

For the active element which has active piezocomposite layers, the equations are written as

(M_{u u}^{s} + M_{u u}^{p}) {\overset{\cdot\cdot}{u}}_{N} + (K_{u u}^{s} + K_{u u}^{p}) u_{N} = f_{e} + f_{p}

(6)

In the above equations, $u_{N}$ denotes the node displacement vector. $M_{u u}^{s}$ and $K_{u u}^{s}$ are the element mass and stiffness matrix of the substrate plate layer, respectively. $M_{u u}^{p}$ and $K_{u u}^{p}$ denote the incremental element mass and stiffness matrices generated by piezocomposite layers, respectively. $f_{p}$ denotes the piezoelectric control force/moment vector. $f_{e}$ represents external load vector.

With the assembly of all elements and the elimination of fixed degree of freedom, the global dynamic equations can be obtained. The external loads would not be concerned in the subsequent modeling and optimization. Furthermore, internal damping matrix should be added to the model to approximate the true behavior of the structure, and classical damping is assumed in the present formulation. The governing equations for the finite element model are

M \overset{\cdot\cdot}{x} + C \overset{\cdot}{x} + Kx = F

(7)

where $x$ is the vector of nodal displacements. $M$ , $C$ , and $K$ are the global mass, damping, and stiffness matrices, respectively. The control forces of the actuators are denoted by $F$ , which can be written as (Chattopadhyay et al., 1999)

F = F_{p} u

(8)

where $F_{p}$ denotes piezoelectric control forces per unit voltage and $u$ is the vector of voltages.

State-space representation

The dynamic equation (7) has potentially many degrees of freedom. To facilitate the analysis and optimization, the modal decomposition is utilized in this study. The truncated modal decomposition is

x = Φ q

(9)

where $q$ is the modal participation factors and $Φ$ consists of structural mode shapes which are determined by solving an eigenvalue problem of structural finite element model. The modes used in the model reduction are chosen according to the control objective. Substituting equations (9) into (7) and left-multiplying with $Φ^{T}$ yields

\tilde{M} \overset{\cdot\cdot}{q} + \tilde{C} \overset{\cdot}{q} + \tilde{Kq} = \tilde{Bu}

(10)

where $\tilde{M} = Φ^{T} M Φ$ , $\tilde{C} = Φ^{T} C Φ$ , and $\tilde{K} = Φ^{T} K Φ$ denote the generalized mass, damping, and stiffness matrices, respectively. $\tilde{B} = Φ^{T} F_{p}$ is the generalized actuator distribution matrix. We would like to formulate the dynamic equations in state representation to facilitate analysis and control. By defining the state vector as $X = [\begin{matrix} q^{T} & {\overset{\cdot}{q}}^{T} \end{matrix}]^{T}$ , the state-space representation of equation (10) can be expressed as

\overset{\cdot}{X} = AX + Bu

(11)

with

A = [\begin{matrix} 0 & I \\ - {\tilde{M}}^{- 1} \tilde{K} & - {\tilde{M}}^{- 1} \tilde{C} \end{matrix}], B = [\begin{matrix} 0 \\ {\tilde{M}}^{- 1} \tilde{B} \end{matrix}]

(12)

If $n$ is the number of the mode shapes used in model reduction and $m$ is the number of the actuators, then the dimensions of the vectors and matrices in equation (11) are determined. $X$ is a $2 n \times 1$ state vector and $u$ is a $m \times 1$ control input vector. $A$ is a $2 n \times 2 n$ system matrix and $B$ is a $2 n \times m$ input matrix.

Optimization approach

Optimization criteria

The optimization criterion in this study is presented based upon maximization of degree of controllability (Gupta et al., 2010). The controllability Gramian matrix of the system in equation (11) is defined by (Han and Lee, 1999)

W (T) = \int_{0}^{T} e^{A τ} B B^{T} e^{A^{T} τ} d τ

(13)

Instead of using time-dependent $W (T)$ , a steady state $W_{c}$ can be considered to eliminate the dependency of the solution to time (Hac and Liu, 1993). The steady state controllability Gramian matrix is written as

W_{c} = \int_{0}^{\infty} e^{A τ} B B^{T} e^{A^{T} τ} d τ

(14)

where $W_{c}$ is a constant matrix that can be calculated by solving the Lyapunov equation

A W_{c} + W_{c} A^{T} + B B^{T} = 0

(15)

Eigenvalues of the Gramian matrix $W_{c}$ can be used to evaluate the degree of controllability (Gupta et al., 2010). Higher eigenvalues mean higher degree of controllability. If an eigenvalue is small, this indicates that the corresponding eigenmode is difficult to control. The primary purpose is to find optimal locations and fiber orientations of the actuators so as to provide the best Gramian properties for vibration control. In this study, a numerical criterion proposed in Leleu et al. (2001) is adopted as objective function which defined by

C r i t = trace (W_{c}) \sqrt[2 n]{det (W_{c})} / σ (λ_{i})

(16)

where $λ_{i}$ are eigenvalues of Gramian matrix $W_{c}$ and $σ (λ_{i})$ denotes the standard deviation of eigenvalues. The criterion contains three terms. $\sqrt[2 n]{det (W_{c})}$ represents the geometric mean of eigenvalues, that is, the dimensions of the ellipsoid. $σ (λ_{i})$ is used to penalize the configuration where there are both very large and very small eigenvalues. $trace (W_{c})$ represents the total energy transmitted from the actuators to the system. The objective function is set to obtain balanced, comprehensive controllability for the modes of interest via optimization of the actuators. Of course, if one mode should be particularly considered in vibration control, the objective function can be modified to strengthen the weight of this mode.

Design variables and constraints

As previously mentioned, the design variables include the locations and fiber orientations of the actuators. The dimensions of the actuators are held constant in the optimization. The number of actuator patches is $m$ , then the design variables are

V a r s = {[\begin{matrix} l_{1} & w_{1} & θ_{1} & \cdot \cdot \cdot & l_{m} & w_{m} & θ_{m} \end{matrix}]}^{T}

(17)

where $l_{i}$ , $w_{i}$ , and $θ_{i}$ denote the locations and the PZT fiber orientation of the ith actuator patch, respectively, as shown in Figure 6. The location of the actuator patch is defined in terms of its top-left corner node. It should be noted that the electrodes of the actuator always remain perpendicular to the fiber orientation.

Several geometric constraints must be implemented to ensure a physically meaningful design for the actuator configuration. A piezocomposite actuator patch must be within the structure surface. If $m > 1$ , additional constraints must be imposed to ensure that no overlap of the actuator patches shall occur. With regard to the fiber orientations, the full range of angle variation is $[- 90 °, 90 °]$ . With the above consideration, the constraints are expressed as

{\begin{matrix} 0 \leq l_{i} \leq L - l, 0 \leq w_{i} \leq W - w (i = 1, \dots, m) \\ ℜ_{i} \neq ℜ_{j} (i = 1, \dots, m, j = 1, \dots, m, i \neq j) \\ - 90 ° \leq θ_{i} \leq 90 ° (i = 1, \dots, m) \end{matrix}

(18)

where $L$ and $W$ denote the length and width of the plate, respectively. $l$ and $w$ denote the length and width of the actuator patch, respectively. The ith and jth regions $(ℜ_{i}, ℜ_{j})$ are defined by the ith and jth actuator boundaries, respectively. The boundaries of the actuators are assumed to correspond to the boundaries of the elements used in FEM. Obviously, the location parameters $l_{i}$ and $w_{i}$ are discrete variables. However, the fiber orientation $θ_{i}$ is a continuous variable which can vary from −90° to 90° continuously. In this study, the orientation variables are restricted as integers between −90 and 90, such that the optimization process can be more efficient.

Genetic algorithm

The optimization problem is solved using the genetic algorithm (GA) which is based on the mechanics of natural selection and genetics (Srinivas and Patnaik, 1994). The GA starts a search from initial population which consists of a series of various chromosomes. Each chromosome represents an available configuration of the piezocomposite actuators. Based on the evaluation by fitness function, the new population with updated chromosomes are generated by crossover and migration processes. Generally, more fit is a chromosome, greater chances are the chromosomes to be selected. After iterations, the best chromosome, that is, the optimal configuration of the actuators, will be determined.

A MATLAB (2014)-based program is developed to establish the mathematical model, implement the optimizations, and plot the results. The GA optimization is also implemented based on MATLAB Optimization Toolbox. Some important parameters used in GA approach are taken as: population size = 100, crossover fraction = 0.8, migration fraction = 0.2, and maximum number of generations: 300.

Results and discussion

The optimization results are presented for a cantilever plate whose parameters are shown in Table 1. The material properties of the plate and piezocomposite actuators are also listed in Table 1. The properties of commercially available MFCs (Smart Material Corp., 2017) are used in this study to simulate the electromechanical coupling effect of piezocomposite actuators. The dimensions of the standard M8557 P1-type and F1-type MFC actuators are used in the optimization. The operational voltage range of MFCs is −500 ∼ 1500 V. The plate structure is discretized by $40 \times 12$ elements and each actuator patch covers $4 \times 4$ elements.

Table 1.

The properties of the plate and actuators.

Parameters	Values
	Plate	Actuator
Length (mm)	L = 850	l = 85
Width (mm)	W = 171	w = 57
Thickness (mm)	H = 1	h = 0.3
Modulus of elasticity (GPa)	E = 70.3	E ₁ = 30.34, E₂ = 15.86
Poisson’s ratio	μ = 0.345	μ ₁₂ = 0.31, μ₂₁ = 0.16
Density (kg/m³)	ρ = 2700	ρ_p = 5400
Piezoelectric constants (m/V)		d ₁₁ = 400 × 10⁻¹², d₁₂ = −170 × 10⁻¹²

The optimal unimorph and bimorph configurations of the actuators will be presented for three cases:

Case I. Bending control (only the first two bending modes are considered, $n = 2$ );

Case II. Twisting control (only the first two torsional modes are considered, $n = 2$ );

Case III. Coupled bending–twisting control (both the first two bending and torsional modes are considered, $n = 4$ ).

Optimization results for unimorph configuration

The optimal unimorph configurations are presented in this subsection, and the actuators are required to be bonded to the upper surface of the plate. Figure 7 shows optimal configuration of a single actuator and Figure 8 shows optimal configurations of two actuators, respectively. The locations and fiber orientations are also listed in Table 2. For bending control in Case I, the optimal location of the actuator is the root region and the optimal fiber orientation is equal or close to 0°. For twisting control in Case II, the optimal location is at about 0.075 times length of the plate and the fiber orientations are ±43° (very close to ±45°). It can also be seen that the actuators are placed at elastic axis for bending control, while placed at the top or bottom edge for twisting control. The offset of the actuator to the elastic axis would also produce a certain impact on the torsional modes (Chattopadhyay et al., 1999), thus the optimal fiber orientations in Case II are not exactly equal to ±45°. As for coupled bending–twisting control in Case III, the optimal fiber orientations are ±15°. A trade-off between bending and twisting control authority can be found in both optimal locations and fiber orientations.

Table 2.

Optimal locations and fiber orientations of the actuators in unimorph configuration.

Cases	Optimal configuration ( $l_{i}$ , $w_{i}$ , $θ_{i}$ )
	Single actuator	Two actuators
Case I	(0, 0.33W, 0°)	(0, 0, −6°)(0, 0.33W, 0°)
Case II	(0.075L, 0.67W, −43°)	(0.075L, 0, 43°)(0.075L, 0.67W, −43°)
Case III	(0, 0.67W, −15°)	(0, 0, 15°)(0, 0.67W, −15°)

Optimization results for bimorph configuration

Figure 9 shows the optimal antisymmetric bimorph configuration for one actuator pair. The locations and fiber orientations are listed in Table 3. Note that the two actuators are identical, however, their fiber orientations are opposite with respect to x-axis due to opposite surfaces bonded to. The thick solid lines and thin dot dash lines denote the fiber orientations of the upper and lower surface actuators, respectively. The optimal location of the actuator is still the root area for bending control in Case I, and the optimal fiber orientation is $0 °$ . With regard to twisting control in Case II, the optimal location is at 0.1 times plate length. However, the actuator pair is placed at the elastic axis and the optimal fiber orientations are $\pm 45 °$ in bimorph configuration for twist control. Due to geometrical symmetry of the plate, there also exits another optimal configurations of the actuators with $\mp 45 °$ fiber orientations. The optimal bimorph configuration in Case III still indicates a trade-off between control authority for bending and torsional modes; however, the magnitude of the optimal fiber orientations $(\mp 20 °)$ are larger than the unimorph configuration $(\pm 15 °)$ . The optimization results show that the piezocomposite actuators with 0° and ±45° fiber orientations are appropriate for bending and twisting control, respectively. However, regarding coupled bending–twisting control, the optimal fiber orientation may be not 0° or ±45°, thus special piezocomposite actuators with optimal fiber orientations (such as 15° or 20°) can be designed and used to produce the improved coupled bending–twisting vibration control performance.

Table 3.

Optimal locations and fiber orientations of the actuator pair in bimorph configuration.

Cases	Optimal configuration ( $l_{i}$ , $w_{i}$ , $θ_{i}$ )
	Upper-side actuator	Lower-side actuator
Case I	(0, 0.33W, 0°)	(0, 0.33W, 0°)
Case II	(0.1L, 0.33W, 45°)	(0.1L, 0.33W, −45°)
Case III	(0, 0.67W, −20°)	(0, 0.67W, 20°)

Remarks and discussion

The above optimal configuration results could be explained from the perspective of strain distribution of the structural modes. Generally, the piezoelectric actuators which locally strain the substructure should be placed in regions of high average strain and away from areas of zero strain (Crawley and De Luis, 1987). Figure 10 shows the mode shapes for the first two bending and torsional modes. The color contour shows the corresponding normal or shearing strain distribution of the certain mode shape. As can be seen that the high normal strain regions for the first two bending modes exists at the root of the plate and there exists a certain distance from the high shearing strain region for the torsional modes to the root. By comparing the optimal configurations in bending, twisting control, and the strain distributions, it indicates that the optimal locations of the actuators depend on the high strain regions of the interested structural modes. Regarding coupled bending and twisting control, the configurations cannot be determined intuitively and can be optimized according to the certain criterion.

For the optimal unimorph configuration of two actuators in Case III as shown in Figure 8, same voltage values are required to generate pure bending deformation while opposite voltage values are required to produce pure torsional deformation. On the contrary, for the optimal bimorph configuration shown in Figure 9, bending deformation and torsional deformation can also be produced by applying opposite and same voltages, respectively. However, compared with unimorph configuration, the bimorph configuration has some unique advantages toward bending and twisting control. Two identical actuators in bimorph configuration are sufficient for both bending and twisting control, but two different actuators with opposite fiber orientations are required in unimorph configuration. Moreover, the width of the substrate may be a constraint for the actuators to be placed to in the unimorph configuration. Thus, the bimorph configuration could bring preferable actuation effect on the slender structures whose width is close to the actuator width. Of course, if the bimorph configuration is not available due to sensing or other constraints, the unimorph configuration can also be adopted according to the optimization results.

Furthermore, first two bending modes and torsional modes are considered in the optimizations to demonstrate the optimal configurations for different situations. In practice, the structural modes of interest that are required to be controlled may be chosen according to the inherent characteristics of the structures and external disturbances. The presented optimization approach can be applied to determine optimal configuration of anisotropic piezocomposite actuators on fixed wings, blades, and other flexible structures toward bending and twisting vibration control.

Conclusion

The configuration optimization of piezocomposite actuators on a plate structure is investigated for bending and twisting vibration control. Locations and PZT fiber orientations of the actuators in unimorph and bimorph configurations are optimized simultaneously to maximize the degree of controllability of the system. The results indicate that the optimal locations and fiber orientations are different for bending control and twisting control. For bending control, the actuators are placed in regions of high normal strain of bending modes with 0° fiber orientation. For twisting control, the actuators are placed in regions of high shearing strain of torsional modes with about ±45° fiber orientations. Regarding coupled bending–twisting control, special piezocomposite actuators with optimal fiber orientations (such as 15° or 20°) can be used to achieve improved vibration control performance. Antisymmetric angle-ply bimorph configuration is a preferable choice toward coupled bending–twisting control of flexible structures.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (11432010).

References

Adnan Elshafei

Farid

Omer

(2014) Modeling of torsion actuation of beams using inclined piezoelectric actuators. Archive of Applied Mechanics 85(2): 171–189.

Allik

Hughes

TJR

(1970) Finite element method for piezoelectric vibration. International Journal for Numerical Methods in Engineering 2(2): 151–157.

Arrieta

Bilgen

Friswell

et al . (2013) Modelling and configuration control of wing-shaped bi-stable piezoelectric composites under aerodynamic loads. Aerospace Science and Technology 29(1): 453–461.

Bent

Hagood

Rodgers

(1995) Anisotropic actuation with piezoelectric fiber composites. Journal of Intelligent Material Systems and Structures 6(3): 338–349.

Biglar

Mirdamadi

Danesh

(2014) Optimal locations and orientations of piezoelectric transducers on cylindrical shell based on Gramians of contributed and undesired Rayleigh–Ritz modes using genetic algorithm. Journal of Sound and Vibration 333(5): 1224–1244.

Bilgen

Butt

Day

et al . (2013) A novel unmanned aircraft with solid-state control surfaces: analysis and flight demonstration. Journal of Intelligent Material Systems and Structures 24(2): 147–167.

Bilgen

Kochersberger

Inman

et al . (2010) Lightweight high voltage electronic circuits for piezoelectric composite actuators. Journal of Intelligent Material Systems and Structures 21(14): 1417–1426.

Bruant

Proslier

(2016) Optimal location of piezoelectric actuators for active vibration control of thin axially functionally graded beams. International Journal of Mechanics and Materials in Design 12(2): 173–192.

Bruant

Gallimard

Nikoukar

(2010) Optimal piezoelectric actuator and sensor location for active vibration control, using genetic algorithm. Journal of Sound and Vibration 329(10): 1615–1635.

10.

Chattopadhyay

Seeley

Jha

(1999) Aeroelastic tailoring using piezoelectric actuation and hybrid optimization. Smart Materials and Structures 8(1): 83–91.

11.

Choi

Park

Kim

(2007) Vibration control of pre-twisted rotating composite thin-walled beams with piezoelectric fiber composites. Journal of Sound and Vibration 300(1–2): 176–196.

12.

Crawley

De Luis

(1987) Use of piezoelectric actuators as elements of intelligent structures. AIAA Journal 25(10): 1373–1385.

13.

Forster

Yang

HTY

(1998) Flutter control of wing boxes using piezoelectric actuators. Journal of Aircraft 35(6): 949–957.

14.

Frecker

(2003) Recent advances in optimization of smart structures and actuators. Journal of Intelligent Material Systems and Structures 14(4–5): 207–216.

15.

Gao

Fei

et al . (2013) Active vibration control based on piezoelectric smart composite. Smart Materials and Structures 22(12): 125032.

16.

Giurgiutiu

(2000) Review of smart-materials actuation solutions for aeroelastic and vibration control. Journal of Intelligent Material Systems and Structures 11(7): 525–544.

17.

Gohari

Sharifi

Vrcelj

(2016) New explicit solution for static shape control of smart laminated cantilever piezo-composite-hybrid plates/beams under thermo-electro-mechanical loads using piezoelectric actuators. Composite Structures 145: 89–112.

18.

Gohari

Sharifi

Vrcelj

(2017) A novel explicit solution for twisting control of smart laminated cantilever composite plates/beams using inclined piezoelectric actuators. Composite Structures 161: 477–504.

19.

Gupta

Sharma

Thakur

(2010) Optimization criteria for optimal placement of piezoelectric sensors and actuators on a smart structure: a technical review. Journal of Intelligent Material Systems and Structures 21(12): 1227–1243.

20.

Gupta

Sharma

Thakur

(2011) Mathematical modeling of actively controlled piezo smart structures: a review. Smart Structures and Systems 8(3): 275–302.

21.

Hac

Liu

(1993) Sensor and actuator location in motion control of flexible structures. Journal of Sound and Vibration 167(2): 239–261.

22.

Han

Lee

(1999) Optimal placement of piezoelectric sensors and actuators for vibration control of a composite plate using genetic algorithms. Smart Materials and Structures 8(2): 257–267.

23.

Jakiela

Pitt

et al . (2004) Reducing aerodynamic vibration with piezoelectric actuators: a genetic algorithm optimization. In: Proceedings of the smart structures and materials 2004: industrial and commercial applications of smart structures technologies, San Diego, CA, 14–18 March, pp. 276–287. Bellingham, WA: SPIE.

24.

Irschik

(2002) A review on static and dynamic shape control of structures by piezoelectric actuation. Engineering Structures 24(1): 5–11.

25.

Kalaycioglu

Silva

(2000) Minimization of vibration of spacecraft appendages during shape control using smart structures. Journal of Guidance Control and Dynamics 23(3): 558–561.

26.

Kwak

Yedavalli

(2001) New modeling and control design techniques for smart deformable aircraft structures. Journal of Guidance Control and Dynamics 24(4): 805–815.

27.

Leleu

Abou-Kandil

Bonnassieux

(2001) Piezoelectric actuators and sensors location for active control of flexible structures. IEEE Transactions on Instrumentation and Measurement 50(6): 1577–1582.

28.

Liu

Yang

(2003) Optimum positioning of actuators in tall buildings using genetic algorithm. Computers & Structures 81(32): 2823–2827.

29.

MATLAB (2014) Computer software, version R2014a. Available at: http://www.mathworks.com/

30.

Monner

Riemenschneider

Opitz

et al . (2011) Development of active twist rotors at the German aerospace center (DLR). In: Proceedings of the 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Denver, CO, 4–7 April. Reston, VA: AIAA.

31.

Moses

Wieseman

Bent

et al . (2001) Evaluation of new actuators in a buffet loads environment. In: Proceedings of the smart structures and materials 2001: industrial and commercial applications of smart structures technologies, Newport Beach, CA, 4–8 March, pp. 10–21. Bellingham, WA: SPIE.

32.

Park

Kim

(2005) Analytical development of single crystal macro fiber composite actuators for active twist rotor blades. Smart Materials and Structures 14(4): 745–753.

33.

Qiu

Zhang

et al . (2007) Optimal placement and active vibration control for piezoelectric smart flexible cantilever plate. Journal of Sound and Vibration 301(3–5): 521–543.

34.

Raja

Upadhya

(2007) Active control of wing flutter using piezoactuated surface. Journal of Aircraft 44(1): 71–80.

35.

Ray

Reddy

(2013) Active damping of laminated cylindrical shells conveying fluid using 1–3 piezoelectric composites. Composite Structures 98: 261–271.

36.

Riemenschneider

Opitz

Wierach

et al . (2009) Structural design and testing of active twist blades—a comparison. In: Proceedings of the 35th European rotorcraft forum 2009 (ERF 2009), Hamburg, 22–25 September, pp. 564–572. Bonn: DGLR.

37.

Schulz

Gomes

Awruch

(2013) Optimal discrete piezoelectric patch allocation on composite structures for vibration control based on GA and modal LQR. Computers & Structures 128: 101–115.

38.

Smart Material Corp. (2017) Smart Material—home of the MFC. Available at: http://www.smart-material.com/MFC-product-main.html

39.

Srinivas

Patnaik

(1994) Genetic algorithms—a survey. Computer 27(6): 17–26.

40.

Usher

Ulibarri

Jr Camargo

(2013) Piezoelectric microfiber composite actuators for morphing wings. ISRN Materials Science 2013: 189659-1–189659-9.

41.

Vasques

CMA

Rodrigues

(2006) Active vibration control of smart piezoelectric beams: comparison of classical and optimal feedback control strategies. Computers & Structures 84(22–23): 1402–1414.

42.

Wang

(2000) Optimal placement and size of piezoelectric patches on beams from the controllability perspective. Smart Materials and Structures 9(4): 558–567.

43.

Wang

Zhou

et al . (2016) Tracking control system design for roll maneuver via active wings using macro fiber composites. In: Proceedings of the AIAA modeling and simulation technologies conference, Washington, DC, 13–17 June. Reston, VA: AIAA.

44.

Wetherhold

Aldraihem

(2001) Bending and twisting vibration control of flexible structures using piezoelectric materials. Shock and Vibration Digest 33(3): 187–197.

45.

Williams

Grimsley

Inman

et al . (2002) Manufacturing and mechanics-based characterization of macro fiber composite actuators. In: Proceedings of the ASME 2002 international mechanical engineering congress and exposition, New Orleans, LA, 17–22 November, pp. 79–89. New York: ASME.