Optimal locations of piezoelectric actuators and sensors for supersonic flutter control of composite laminated panels

Abstract

The optimal active flutter control of supersonic composite laminated panels is studied using the distributed piezoelectric actuators/sensors pairs. The supersonic piston theory is used to calculate the unsteady aerodynamic pressure, and Hamilton’s principle with the assumed mode method is employed to develop the equation of motion of the structural system. The controllers are designed by the proportional feedback control method and the linear quadratic Gauss (LQG) algorithm. The optimal locations of the actuator/sensor pairs are determined by the genetic algorithm (GA). The aeroelastic properties of the structural system are mainly analyzed using the frequency-domain method. The time-domain responses of the structure are also computed using the Runge–Kutta method. The influences of ply angle on the flutter bound of the laminated panel with different length–width ratios are analyzed. The optimal design for the locations for different numbers of piezoelectric patches used in the proportional feedback control is carried out through the GA. Meanwhile, the control effects using different numbers of actuator/sensor pairs are investigated. The flutter suppression by the LQG algorithm is also carried out. The control effects using the two different controllers are compared. Numerical simulations show that the optimal locations obtained by the GA can increase the critical flutter aerodynamic pressure significantly, and the LQG algorithm is more effective in flutter suppression for supersonic structures than the proportional feedback controller.

Keywords

Composite laminated panel supersonic air flow optimal active flutter control piezoelectric material linear quadratic Gauss genetic algorithm

1. Introduction

With the development of science and technology, the flight velocity of aircraft is being increased and the flight environment is being deteriorated. As a result, the aeroelasticity and servo aeroelasticity of the aircraft are more and more prominent, which limits the design of high-speed aircraft. The panel structures of aircraft are usually thin plates. When the aircraft fly at high speeds, self-oscillation for the aircraft panels will be generated under the effects of air flow. As a result, the flutter of the panel structures will occur. Panel flutter is a dynamic instability of the thin skin of flight vehicles occurring at a critical velocity under the coupling action of the elastic force, inertia force and aerodynamic pressure (Chattopadhyay et al., 2002; Na et al., 2006; Raja et al., 2006; Rao et al., 2006; Lee et al., 2007; Li et al., 2007; McNamara and Friedmann, 2007; Reddy et al., 2007; Carnahan and Richards, 2008; Oh and Kim, 2009). When flutter occurs, the stability, safety and reliability of the aircraft structures will be reduced seriously. So it is very important to investigate the aeroelastic flutter control of supersonic aircraft panels.

Active vibration control by piezoelectric patches and the optimal placements of piezoelectric actuators and sensors in the active vibration control are of great interest to researchers. Farhadi and Hashemi (2011) researched the vibration control of moderately thick rectangular plates using piezoelectric actuators. Trindade (2010) performed the experimental analysis of vibration control of a cantilever beam using three active–passive damping configurations. Fakhari and Ohadi (2010) studied the large amplitude vibration control of functionally graded material plates under thermal effects using integrated piezoelectric actuator/sensor layers.

Aeroelastic flutter analysis and active control using the piezoelectric materials have also been studied in some literatures. Matsumoto et al. (2010) developed a step-by-step analysis on the coupled flutter mechanism of plate and long span bridges. Eastep and Mcintosh (1971) analyzed the flutter and vibration response of a nonlinear panel under random excitation and aerodynamic load. Kouchakzadeh et al. (2010) studied the nonlinear aeroelastic flutter problems of the laminated composite plate based on the Von-Karman theory. Ganapathi and Touratier (1996) researched the influences of thermal effects on the aeroelastic flutter of supersonic laminated composite panels. McNamara and Friedmann (2007) investigated time-domain damping/frequency/flutter identification techniques. A flutter margin for discrete-time systems was evaluated. Ibrahim and Tawfik (2010) investigated the nonlinear flutter and thermal buckling of a functionally graded material plate using the finite element method.

Han et al. (2006) preformed a numerical and experimental investigation on the active flutter control of a sweptback cantilevered lifting surface using piezoelectric actuators. Onawola and Sinha (2011) studied panel flutter suppression using piezoelectric materials. The controller was designed by exact state transformations and feedback control. Shin et al. (2006, 2009) analyzed the aeroelastic and aerothermoelastic flutter characteristics of supersonic cylindrical composite panels and shells with viscoelastic damping treatments using the finite element method. Song and Li (2011) researched the aeroelastic flutter characteristics of the supersonic beam. The active flutter control was also investigated using piezoelectric material. Ibrahim et al. (2010) investigated the nonlinear flutter characteristic of shape memory alloy hybrid composite plates considering the thermal effect using a new nonlinear finite element model.

The genetic algorithm (GA) is an efficient optimization technique that is derived from the mechanism of natural selection and genetics. Bruant et al. (2010) studied the optimal locations of the actuators and sensors for active vibration control by the GA using two modified optimization criteria. Murat and Esref (2008) researched the optimal selection method for the actuators and sensors in flexible structures. Ramesh and Narayanan (2007) investigated the optimal placements of collocated piezoelectric actuator/sensor pairs on a thin plate using a model-based linear quadratic regulator (LQR) controller. The GA was used to solve the optimization problem.

Based on the analysis above, it is noted that very few literatures have studied the influences of the locations and coverage areas of the actuator/sensor pairs on the flutter control results of supersonic aircraft panels. In our recent work, we have studied the aeroelastic flutter analysis and active control of supersonic composite laminated panels using piezoelectric material (Song and Li, 2012). In this paper, the optimal design for the piezoelectric patch locations in active flutter suppression is further researched using the GA. The influences of ply angles on the aeroelastic properties of the laminated panel are also investigated. Apart from the proportional feedback controller (PFC), the LQR with the Kalman filter, which is known as the linear quadratic Gauss (LQG) controller, is also used in the active flutter control, and the control effects of the two controllers are compared.

2. Equation of motion of the aeroelastic structural system

The supersonic composite laminated panel with the piezoelectric actuator/sensor pairs is shown in Figure 1 in which the geometrical parameters of the structural system are also displayed. The length, width and thickness per layer of the base laminated panel are a, b and h_l, respectively. The total thickness of the laminated panel is h. The number of piezoelectric actuator/sensor pairs is n_p. The ith pair is positioned by the coordinates (x₁ _i , y₁ _i ) and (x₂ _i , y₂ _i ). The thicknesses of the piezoelectric actuators and sensors are both h_p.

Figure 1.

Schematic diagram of the supersonic laminated panel and piezoelectric actuator/sensor pairs.

It is known that, before the flutter of the aeroelastic structural system, the deflection is small, and the performances of the structure after the flutter are not considered in this study. So, in the modeling, the linear theory is used. Based on the Kirchhoff theory, the displacement fields can be expressed as

\begin{matrix} u = - z \frac{\partial w}{\partial x}, v = - z \frac{\partial w}{\partial y}, w = w, \end{matrix}

(1)

where u and v are the displacements in the x and y directions, respectively, w is the transverse displacement along the z direction and z is the coordinate in the transverse direction. The strain–displacement relations can be expressed as

\begin{matrix} ε = [- z \frac{\partial 2 w}{\partial x 2}, - z \frac{\partial 2 w}{\partial y 2}, - 2 z \frac{\partial 2 w}{\partial x \partial y}] T = z κ, \end{matrix}

(2)

where ε = [ɛ_x, ɛ_y, γ_xy]^T and κ = [κ_x, κ_y, κ_xy]^T are the strain and bending curvature vectors, respectively.

The bending moment resultant M ₀ of the laminated panel can be obtained by (Song and Li, 2012)

M_{0} = \sum_{k = 1}^{n_{l}} \int_{z_{k - 1}}^{z_{k}} σ_{k} z dz = D κ, D_{0} = \sum_{k = 1}^{n_{l}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{k} (z, z 2) dz,

(3)

where n_l is the number of the layers, σ _k is the stress vector of the kth lamina, z_k and z_k–₁ are the coordinates of the upper and lower surfaces, respectively, of the kth lamina in the z direction, D ₀ is the stiffness coefficient matrix and

{\bar{Q}}_{k}

is the transformed reduced stiffness matrix of the kth lamina, which can be obtained by

{\bar{Q}}_{k} = T_{r}^{- 1} {QT}_{r}^{- T},

(4)

where T _r is the transform matrix and Q is the lamina stiffness matrix. They can be written as

\begin{matrix} T_{r} = [\begin{matrix} \cos 2 ψ & \sin 2 ψ & 2 \sin ψ \cos ψ \\ \sin 2 ψ & \cos 2 ψ & - 2 \sin ψ \cos ψ \\ - \sin ψ \cos ψ & \sin ψ \cos ψ & \cos 2 ψ - \sin 2 ψ \end{matrix}], \\ Q = [\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}], \end{matrix}

(5)

in which ψ is the ply angle of the kth lamina and Q₁₁ = E₁/(1–υ₁₂υ₂₁), Q₂₂ = E₂/(1–υ₁₂υ₂₁), Q₁₂ = υ₂₁E₂/(1–υ₁₂υ₂₁) = υ₁₂E₁/(1–υ₁₂υ₂₁) and Q₆₆ = G₁₂ are the stiffness coefficients where E₁, E₂ and G₁₂ are the elastic and shear moduli and υ₁₂ and υ₂₁ are the Poisson’s ratios of the composite laminated panel.

The piezoelectric material is polarized in the z direction and is transversely isotropic in the x–y plane. The constitutive equations of the piezoelectric material can be expressed as

σ p = Q p ε p - e T E p,

(6a)

D p = e ε p + Ξ E p,

(6b)

where σ ^p and ε ^p are the stress and strain vectors of the piezoelectric material, respectively, E ^p = [0 0 E_z]^T is the electric field intensity vector in which E_z = V₀(t)/h_a where V₀(t) is the external voltage (Kim and Kim, 2005; Ray and Pradhan, 2006; Ray and Batra, 2007; Ramesh and Narayanan, 2008), D ^p = [0 0 D_z]^T is the electric displacement vector and e , Ξ and Q ^p are the piezoelectric constant, dielectric constant and stiffness matrices, respectively, which can be expressed as

\begin{matrix} e = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ e_{31} & e_{31} & 0 \end{matrix}], Ξ = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & Ξ_{33} \end{matrix}], \\ Q p = [\begin{matrix} C_{11} & C_{12} & 0 \\ C_{12} & C_{11} & 0 \\ 0 & 0 & C_{66} \end{matrix}], \end{matrix}

(7)

in which e₃₁ and Ξ ₃₃ are the piezoelectric constant and dielectric constant, respectively, and C₁₁ = E/(1–υ²), C₁₂ = υE/(1–υ²) and C₆₆ = E/[2(1 + υ)] are the elastic constants where E and υ are the elastic modulus and Poisson’s ratio of the piezoelectric material.

To formulate the equation of motion, Hamilton’s principle with the assumed mode method will be used. Hamilton’s principle is expressed as (Li et al., 2009; Song and Li, 2012)

\int_{t_{1}}^{t_{2}} [δ (T - U) + δ W] d t = 0,

(8)

where t₁ and t₂ are the integration time limits, and T, U and δW are the kinetic energy, potential energy and the virtual work done by the aerodynamic pressure, respectively. They are calculated by

\begin{matrix} T = \frac{1}{2} \int_{V} ρ (\overset{\cdot}{u} 2 + \overset{\cdot}{v} 2 + \overset{\cdot}{w} 2) d V \\ + \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{ai} + V_{si}} ρ_{p} (\overset{\cdot}{u} 2 + \overset{\cdot}{v} 2 + \overset{\cdot}{w} 2) d V, \end{matrix}

(9a)

\begin{matrix} U = \frac{1}{2} \int_{A} κ T M_{0} d A + \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{ai} + V_{si}} ε T σ p d V \\ - \sum_{i = 1}^{n_{p}} \frac{1}{2} \int_{V_{ai}} D_{z} E_{z} d V, \end{matrix}

(9b)

δ W = \int_{A} Δ p δ w d A,

(9c)

where ρ and ρ_p are the mass densities of the base panel and the piezoelectric material, A is the surface area of the base panel, V, V_ai and V_si are the volumes of the laminated panel and the ith piezoelectric actuator and sensor and Δp is the aerodynamic pressure, which is computed by the supersonic piston theory (Oh and Lee, 2006; Prakash and Ganapathi, 2006):

Δ p = - \frac{2 q_{\infty}}{β} (\frac{M_{\infty}^{2} - 2}{M_{\infty}^{2} - 1} \frac{1}{U_{\infty}} \frac{\partial w}{\partial t} + \frac{\partial w}{\partial x}),

(10)

where

q_{\infty} = \frac{1}{2} ρ_{\infty} U_{\infty}^{2}, β = \sqrt{M_{\infty}^{2} - 1},

(11)

where ρ_∞, U_∞ and M_∞ are the free stream air density, velocity and Mach number, respectively. In the analysis, a nondimensional aerodynamic pressure λ = 2q_∞a³/(βD) is introduced in which D = h³/[12(1–υ₁₂υ₁₂)]. In the case of M_∞ ≫ 1, [(

M_{\infty}^{2} - 2

)/(

M_{\infty}^{2} - 1

)]μ/β → μ/M_∞, where μ = ρ_∞a/ρh.

The assumed modes method is used to discretize the continuous structural system. The transverse displacement w is expressed in terms of the assumed modes and generalized coordinates as

w (x, y, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} W_{ij} (x, y) g_{ij} (t) = W T (x, y) g (t),

(12)

where m and n are the numbers of mode, W = [W₁₁, … , W₁ _n , W₂₁, … , W_m₁, … , W_mn]^T is the vector of the assumed modes and g = [g₁₁, … , g₁ _n , g₂₁, … , g_m₁, … , g_mn]^T is the vector of the generalized coordinates. For a fully simply supported laminated panel, the vibration modes are assumed as

\begin{matrix} W_{ij} (x, y) = \sin \frac{i π x}{a} \sin \frac{j π y}{b} i = 1, 2, \dots \dots, m; \\ j = 1, 2, \dots \dots, n . \end{matrix}

(13)

Substituting Equations (1)–(3), (6), (10) and (12) into Equation (9), by means of Equations (4), (5), (7), (11) and (13), and the results into Equation (8), after conducting the operation of variation, the equation of motion can be obtained as

M \overset{··}{g} (t) + C_{Δ p} \overset{\cdot}{g} (t) + (K + K_{Δ p}) g (t) + K_{a} U = 0,

(14)

where M is the mass matrix, C _Δ _p and K _Δ _p are the aerodynamic damping and stiffness matrices, respectively, K is the structural stiffness matrix, K _a is the electromechanical coupling matrix and U is the control voltage vector. They are all listed in Appendix A.

Equation (14) relates the applied voltage to the structural displacements and will be used to study the optimal active aeroelastic flutter control by different control methods.

3. The design of active controllers

In this section, the optimal aeroelastic flutter control is studied using the proportional feedback control and the LQG algorithm. When the external dynamic loads are applied on the piezoelectric sensor, the charge developed by the deformation of the ith piezoelectric sensor can be expressed as (Reddy, 1999; Gao and Liao, 2005)

Q_{si} (t) = \int_{A_{si} |_{(z = z_{si})}} D_{z} d A = K_{si} g (t), i = 1, 2, \dots n_{p},

(15)

where A_si is the surface area of the ith sensor, z_si is the transverse coordinate of the mid-plane of the ith sensor from the neutral plane of the base laminated panel and the matrix K _si is a coefficient vector, which is given in Appendix B.

The sensing voltage V_si of the ith sensor is obtained by (Park and Baz, 1999; Mukherjee et al., 2002; Koo and Hwang, 2004; Raja et al., 2006)

V_{si} = \frac{Q_{si} h_{p}}{Ξ_{33} A_{si}}, i = 1, 2, \dots n_{p},

(16)

Substituting Equation (15) into Equation (16), the sensor voltage can be obtained as

U_{s} = [V_{s 1}, V_{s 2}, \dots, V_{{sn}_{p}}] T = K_{s} g (t),

(17)

where K _s is the sensor voltage matrix.

3.1. Proportional feedback controller

If the proportional feedback control is applied on the aeroelastic structural system, the control voltage U in Equation (14) can be calculated by (Reddy, 1999)

U (t) = G_{v} U_{s} (t) = G_{v} K_{s} g (t) .

(18)

Substituting Equation (18) into Equation (14), the controlled equation of motion by the proportional feedback control can be obtained as

M \overset{··}{g} (t) + C_{Δ p} \overset{\cdot}{g} (t) + (K + K_{Δ p} + K_{a} G_{v} K_{s}) g (t) = 0 .

(19)

It is seen from Equation (19) that the proportional feedback control can provide an active stiffness for the aeroelastic structural system.

3.2. Linear quadratic Gauss controller

In using the LQR controller, all the states of the structural system should be known. Actually, it is difficult for the sensor to give accurate information of all the states. As a result, the LQG algorithm is used in this study to design the controller. Based on the equation of motion, Equation (14), the standard state-space equation of motion can be obtained as

\overset{\cdot}{X} = A_{0} X + B_{0} U, Y = C_{0} X,

(20)

where

\begin{matrix} X = [\begin{matrix} g \\ \overset{\cdot}{g} \end{matrix}], A_{0} = [\begin{matrix} 0 & I \\ - M - 1 (K + K_{Δ p}) & - M - 1 C_{Δ p} \end{matrix}], \\ B_{0} = [\begin{matrix} 0 \\ - M - 1 K_{a} \end{matrix}], C_{0} = [\begin{matrix} K_{s} & 0 \end{matrix}] . \end{matrix}

(21)

Firstly, the LQR is used to obtain the control voltage under the full-state condition, which is given as

U (t) = - GX (t),

(22)

which minimizes a quadratic performance index:

J = \int_{0}^{\infty} (X T QX + U T RU) dt .

(23)

By solving the functional extremum problem, the feedback control gain G in Equation (22) can be obtained as G = R ^–1 $B_{0}^{T}$ P , where P can be calculated through the Riccati equation:

\overset{\cdot}{P} + A_{0}^{T} P + {PA}_{0} - {PB}_{0} R - 1 B_{0}^{T} P + Q = 0 .

(24)

Secondly, to estimate the full state of the control system, a Kalman filter is introduced as

\overset{\cdot}{\overset{\land}{}} X = A_{0} \overset{\land}{X} + B_{0} U + K_{e} (Y - \overset{\land}{Y}), \overset{\land}{Y} = C_{0} \overset{\land}{X},

(25)

where K _e is the gain of Kalman filter, which can be calculated by K _e =

R_{e}^{- 1} C_{0}^{T}

P _e , where P _e is obtained by solving the following Riccati equation:

{\overset{\cdot}{P}}_{e} + A_{0}^{T} P_{e} + P_{e} A_{0} - P_{e} C_{0}^{T} R_{e}^{- 1} C_{0} P_{e} + Q_{e} = 0 .

(26)

Finally, the controlled state-space equation is obtained as

\overset{\land}{\overset{\cdot}{}} X = (A_{0} - B_{0} G - K_{e} C_{0}) \overset{\land}{X} + K_{e} Y,

(27a)

\overset{\land}{Y} = C_{0} \overset{\land}{X} .

(27b)

3.3. Solution of the equation of motion

The aeroelastic properties of the structural system are investigated mainly by the frequency-domain method. The general solution of the generalized coordinate g (t) is assumed as g (t) = g ₀exp(Ωt), where g ₀ and Ω are the eigenvector and eigenvalue. The imaginary part of Ω is the natural frequency ω of the structural system. It is well known that when certain two consecutive frequencies approach each other or the real part of a certain eigenvalue becomes positive from negative, the flutter will occur. The corresponding aerodynamic pressure is called the critical flutter aerodynamic pressure, which is denoted as λ_cr. The time-domain responses of the aeroelastic structural system are calculated using the Runge–Kutta method.

3.4. Genetic algorithm

To make full use of limited amount of piezoelectric material, optimal design for the locations of the piezoelectric actuator and sensor is necessary, and in this study, the GA is used. The GA is an efficient searching technique to find the global optimum solution. It is derived from the mechanism of natural selection and genetics (Peng et al., 2005).

To implement the GA, the base panel is divided into 64 square elements, which are numbered from 1 to 64, as shown in Figure 2. Each of the n_p actuator/sensor pairs is bonded on one element whose number is complied into a string of six binary codes, and then all the binary strings are combined into a chromosome, as shown in Figure 3. Parents are selected according to their fitness values via a select operator after initialization of a population of the first generation. Those individuals with relatively high fitness values are more likely to be selected as parents. If two strings of binary code in a chromosome represent the same value, the corresponding two actuator/sensor pairs are bonded on the same element of the base panel, which is not practical. So the fitness of such a chromosome is set to be a very small value to ensure that the chromosome can be discarded in the selection procedure. The child generation is produced with the recombination of the selected parents. After selecting a crossover point randomly, the crossover between two parent chromosomes is obtained. The mutation process is developed to change the bits of some of the chromosomes randomly.

Figure 2.

Numbering of the elements of the laminated panel.

Figure 3.

Chromosome structure of actuator/sensor pairs.

To obtain the highest flutter bound, the fitness function for the aeroelastic problem is defined as

f (x) = λ_{cr} .

(28)

4. Numerical simulations and discussion

4.1. Verifications

Prior to the investigation of the main topics, the numerical results are compared with the previous literatures and the results obtained by ANSYS to verify the formulations and the Matlab codes.

The nondimensional natural frequencies of the structural system are calculated. The nondimensional natural frequency is defined as ω* = ωa²[12(1–υ₁₂υ₂₁)ρ/(E₁h²)]^1/2. The material properties of the composite laminated panel used in the calculation are E₁/E₂ = 2.45, G₁₂ = 0.48E₂, υ₁₂ = 0.23, υ₂₁ = 0.0939, ρ = 8000 kg m^–3, h = 0.06 m. The Linear Layer 99 element is used in ANSYS, and the number of the finite element is 400. The results are shown in Table 1. It can be seen from Table 1 that the present results are in good agreement with those obtained by ANSYS and the results in the open literatures, which indicates that the formulations and the Matlab codes are correct.

Table 1.

Natural frequencies of simply supported square three-ply and four-ply laminated panels.

Modes
Various methods	1	2	3	4	5	6
[0°/0°/0°] a = 10 m, b = 10 m, h = 0.02 m
Exact (Whitney, 1987)	15.17	33.24	44.38	60.68	64.45	90.14
Dai et al. (2004)	15.22	33.76	44.79	61.11	66.76	91.69
Present results	15.17	33.25	44.38	60.68	64.45	90.13
[15°/–15°/15°] a = 10 m, b = 10 m, h = 0.02 m
Chow et al. (1992)	15.37	34.03	43.93	60.80	66.56	91.40
ANSYS	15.37	33.95	43.81	60.68	66.50	90.83
Present results	15.42	34.08	43.88	60.91	66.63	91.41
[30°/–30°/–30°/30°] a = 10 m, b = 10 m, h = 0.015 m
Leissa and Narita (1989)	16.02	36.30	42.62	62.57	71.68	85.81
ANSYS	15.96	36.32	42.56	62.45	71.57	85.72
Present results	16.08	36.38	42.73	62.84	72.02	86.37

To justify the correctness of the assumed mode method and to determine the number of the mode in solving the aeroelastic problem, the flutter bounds of a square simply supported laminated panel studied by Zhou et al. (1994) under different orders of mode are calculated and shown in Figure 4. It is noted by Zhou et al. (1994) that the flutter bound of the panel is λ_cr = 300, which is the same as that obtained under the mode numbers of m = 6 and n = 1. The results of Figure 4 also indicate that the mode is convergent at m = 6 and n = 1.

Figure 4.

Flutter bounds under different modes.

The mode shapes of the structure at λ = 0 and λ = 300 are calculated and shown in Figure 5. It can be observed from the figure that the modes are modified and the first two orders of the mode are the same at the flutter bound.

Figure 5.

The first two orders of vibration mode shapes at λ = 0 and λ = 300.

4.2. Aeroelastic analysis

In this section, the aeroelastic characteristics of simply supported laminated panels are analyzed. The influences of ply angles on the aeroelastic properties are also investigated. The geometrical sizes of the aeroelastic structural system are a = 0.2 m, b = 0.2 m, h_l = 0.0002 m. The material properties of the orthotropic laminated panel are ρ = 1600 kg m^–3, υ₁₂ = 0.3, E₁ = 150 × 10⁹N m^–2, E₂ = 9.0 × 10⁹N m^–2, G₁₂ =7.1 × 10⁹N m^–2, υ₂₁ = 0.018.

The flutter bounds of the laminated panel are firstly calculated. Figure 6 shows the variations of the first two orders of natural frequency of the panel with the aerodynamic pressure. It is seen from the figure that the flutter bounds of the panel under different ply angles are different. So it is necessary to analyze the influences of ply angle on the aeroelastic properties of the laminated panel.

Figure 6.

Variations of natural frequency with aerodynamic pressure.

Figure 7 shows the flutter bounds of square symmetric and anti-symmetric laminated panels with the ply angles of [θ₁/θ₂/θ₂/θ₁] and [θ₁/θ₂/θ₁/θ₂], respectively. It can be observed from the figure that for both the symmetric and anti-symmetric panels, with the increase of the ply angle, the flutter bound is decreased. It is also noted that, for the symmetric panel, the effects of outer laminas (corresponding to θ₁) on the flutter bound are more obvious. This is because if the ply angles of the outer laminas are fixed, the flutter bound of the panel changes little with the variation of the inner laminas. However, for the anti-symmetric ones, the effects of outer or inner laminas are the same.

Figure 7.

Variation of flutter bounds of symmetric and anti-symmetric square panels with ply angles. (a) [θ₁/θ₂/θ₂/θ₁] and (b) [θ₁/θ₂/θ₁/θ₂].

Figure 8 displays the variations of flutter bounds with the ply angles for a rectangular panel with a = 0.2 m and b = 0.1 m. The results obtained under this condition are quite different from those of the square panels. For both the symmetric and anti-symmetric panels, the highest critical flutter aerodynamic pressure occurs at [ ± 45°/ ± 45°/ ± 45°/ ± 45°]. The same conclusion is that, for the symmetric laminated panel, the effects of ply angles for outer laminas are more significant, and for the anti-symmetric panels, the effects of outer and inner laminas are the same.

Figure 8.

Variation of flutter bounds of symmetric and anti-symmetric rectangular panels with ply angles. (a) [θ₁/θ₂/θ₂/θ₁] and (b) [θ₁/θ₂/θ₁/θ₂].

4.3. Optimal flutter suppression

After analyzing the aeroelastic characteristics of the laminated panel, the optimal active flutter suppression is investigated in this section. In the numerical example, a symmetric square (a = b = 0.2 m) simply supported laminated panel with the same material properties as those given in Section 4.2 is considered. The ply angle of the panel is [45°/–45°/–45°/45°]. The material properties of the piezoelectric material are ρ_p = 7600 kg m^–3, E = 63 × 10⁹N m^–2, υ = 0.3, d₃₁ = d₃₂ = 245 × 10 ^− 12 m V^–1, Ξ ₃₃ = 1.5 × 10 ^− 8 F m^–1. The thicknesses of the base panel and the piezoelectric actuator and sensor are h = 0.0008 m and h_p = 0.0001 m, respectively.

Figure 9 shows the controlled flutter bounds for a laminated panel with fully covered actuator patch using the PFC under different control gains. It is observed from the figure that when the PFC is on, the critical flutter aerodynamic pressure is increased, which indicates that the PFC is effective on the flutter suppression. It is also noted that, with the increase of the control gain, the flutter bounds increased first and then decreased, and the coupling of the modes at the flutter bound changes from modes 1 and 2 to modes 2 and 3. Moreover, it is seen from Figure 9(b) that when the control gain is G_v = 12, the first order of frequency is equal to zero under low aerodynamic pressure, which means that the structure is buckling. However, at high aerodynamic pressure, the natural frequency is greater than zero, so the buckling disappears. That is to say that for the supersonic structural system, the aerodynamic pressure can eliminate the buckling of the structure. Based on the above analysis, it can be concluded that the control gain of PFC is very important. It can be neither too small nor too large. If it is too small, the control effect is not obvious; however, if it is too large, it will make the structure unstable. The variation of flutter bounds with the control gain is shown in Figure 10, from which one can observe that better control gains for the PFC are around G_v = 4.5.

Figure 9.

Controlled flutter bounds for an actuator patch fully covered on the base panel using the proportional feedback controller under different control gains.

Figure 10.

Variation of flutter bounds with the feedback control gain.

The above analysis is based on the patches of actuator/sensor pairs fully covered on the laminated panel. However, in practice, the number of piezoelectric patches is always limited. In this analysis, the GA is used to design the optimal locations of the actuator/sensor pair in order to make full use of the limited piezoelectric material.

Firstly, the optimal locations of six actuator/sensor pairs are calculated. The control gain for each actuator in the PFC is G_v = 50. The convergence curve of the GA is displayed in Figure 11. It can be seen from the figure that after the 24th generation, the flutter bound is convergent. The corresponding optimal locations of the six actuator/sensor pairs are shown in Figure 12(a) and the corresponding maximum critical flutter aerodynamic pressure is λ_cr = 1008.

Figure 11.

The convergence curve of the genetic algorithm for six pairs of actuator/sensor patches.

Figure 12.

The optimal and two arbitrary locations of the six pairs of actuator/sensor patches.

To justify the correctness of the optimal results, the flutter bound of the structural system with the optimally located actuator/sensor pairs is compared with those of two structures with arbitrary located piezoelectric patches shown in Figures 12(b) and (c) and plotted in Figure 13. It can be observed from the figure that the flutter bound of the optimally designed structural system is much higher than those of the other two structural systems.

Figure 13.

The flutter bounds under different locations of the six actuator/sensor pairs given in Figure 12.

In the same way, the laminated panel with eight piezoelectric patches is also studied. The optimal locations of the eight actuator/sensor pairs are also calculated using the GA. The convergence curve of the GA is plotted in Figure 14. It shows that the maximum flutter bound controlled using eight actuator/sensor pairs by the PFC is λ_cr = 1383. The corresponding optimal locations of the eight piezoelectric patches are shown in Figure 15. To investigate the optimal effect for the optimal design, the other two structural systems with arbitrary located piezoelectric patches are given and shown in Figure 15. The flutter bounds of the three types of structural systems in Figure 15 are computed and shown in Figure 16. It is obviously seen that the structural system with optimally located eight actuator/sensor pairs has the highest critical flutter aerodynamic pressure, which indicates the correctness of the GA.

Figure 14.

The convergence curve of the genetic algorithm for eight pairs of actuator/sensor patches.

Figure 15.

The optimal and two arbitrary locations of the eight pairs of actuator/sensor patches.

Figure 16.

The flutter bounds under different locations of the eight actuator/sensor pairs given in Figure 15.

To study the relations between the control effects and the number of piezoelectric patches, the flutter bounds of structural systems with different numbers of actuator/sensor pairs are calculated and shown in Figure 17. It can be seen from the figure that with the increase of the piezoelectric patch number, the critical flutter aerodynamic pressure is increasing. In other words, the larger the area of the piezoelectric material covered on the panel is, the better the control effect is. However, the quantity of the piezoelectric materials is limited in practice, so the optimal location design for the piezoelectric patches is necessary.

Figure 17.

The controlled flutter bounds using different numbers of actuator/sensor pairs.

The above analysis is based on the proportional feedback control method. Furthermore, the LQG controller is also designed in this paper to suppress the flutter. The parameters used in LQG controller are given as Q = 10 I , Q _e = 10 I , R = I and R _e = I , where I is the identical matrix.

Figure 18 shows the controlled flutter bounds under the LQG controller. It is noted from the figure when controlled by the LQG controller, the first two orders of natural frequency can no longer approach each other, which means that the flutter will not occur if the LQG controller is applied on the structural system.

Figure 18.

The controlled flutter bounds by the linear quadratic Gauss (LQG).

Furthermore, the two controllers of PFC and LQG are compared for their effects on the flutter suppression, and the results are shown in Figure 19. In the simulation, eight actuator/sensor pairs are used, and they are optimally located as shown in Figure 15(a). It is seen from Figure 19 that the flutter bound is enlarged to λ_cr = 1383 by the PFC, however, under the LQG controller, the flutter will not occur. In other words, in flutter suppression for the linear structural system studied in this paper, the LQG controller is more efficient.

Figure 19.

The controlled flutter bounds using different controllers. PFC: proportional feedback controller; LQG: linear quadratic Gauss.

To verify the above analysis, the time-domain responses at the position (x = 0.75a, y = 0.5b) of the panel controlled by the PFC and LQG controller are computed using the Runge–Kutta method realized by the Matlab program ode45. Figure 20 shows the time-response history of the uncontrolled structural system at λ = 290, which is larger than the critical flutter aerodynamic pressure. The response is divergent, which means that the flutter occurs. As shown in Figures 21 and 22, after being controlled by the PFC, the responses are convergent again. If the LQG controller is used, the time-domain response is convergent more quickly.

Figure 20.

Uncontrolled time-response history of the structural system.

Figure 21.

Controlled time-response history of the structural system by the proportional feedback controller (PFC).

Figure 22.

Controlled time-response history of the structural system by the linear quadratic Gauss (LQG) controller.

The control voltages applied on the eight actuators in the two kinds of controllers are displayed in Figures 23 and 24. It is seen from the figures that the voltages used in the LQG controller are much lower than those in the PFC. From Figures 25 and 26 one can observe that when the aerodynamic pressure reaches λ = 1800, although the PFC is on, the flutter will still happen. However, if the LQG controller is on, the time history is convergent. Both the frequency- and time-domain analyses indicate that the LQG controller is more efficient in flutter suppression.

Figure 23.

The control voltages applied on the eight actuators using in the proportional feedback controller.

Figure 24.

The control voltages applied on the eight actuators using the linear quadratic Gauss controller.

Figure 25.

Controlled time-response history of the structural system by proportional feedback controller (PFC) at λ = 1800.

Figure 26.

Controlled time-response history of the structural system by the linear quadratic Gauss (LQG) controller at λ = 1800.

5. Conclusions

In this paper, the optimal location design for the piezoelectric actuator/sensor patches in the active flutter suppression is researched using the GA. The influences of ply angles on the aeroelastic properties of the laminated panel are investigated. The PFC and the LQG controller are used in the active flutter control, and the control effects of the two kinds of controllers are compared. From the numerical simulations, the following conclusions can be drawn.

At the flutter bound, the modes of the aeroelastic structure are changed.

For both the square symmetric and anti-symmetric laminated panels, with the increase of the ply angle, the flutter bound is also decreased. However, for the rectangular ones with a/b = 2, the highest flutter bound occurs at [ ± 45°/ ± 45°/ ± 45°/ ± 45°].

For the symmetric square and rectangular laminated panels, the effects of outer laminas on the flutter bound are more significant than those of the inner laminas. However, for the anti-symmetric ones, the effects of outer and inner laminas on the critical flutter aerodynamic pressure are the same.

The PFC is effective in suppressing flutter. Nevertheless, the control gain can be neither too large nor too small. If it is small, the control effect is not obvious. However, if it is too large, the structural system will buckle.

The optimal locations of the actuator/sensor pairs obtained through the GA can increase the effect of flutter suppression significantly. Meanwhile, the larger the area of the piezoelectric patches covered on the base structure is, the higher the critical flutter aerodynamic pressure is.

The control effect of the LQG controller is more effective than that of the PFC in flutter suppression, and the control voltages used in the LQG controller are lower.

Footnotes

Funding

This work was supported by the National Basic Research Program of China (No. 2011CB711100) and the National Natural Science Foundation of China (No. 11172084, 10672017).

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