Vibration control of smart hull structure with optimally placed piezoelectric composite actuators

International Journal of Mechanical Sciences 53 (2011) 647–659

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Vibration control of smart hull structure with optimally placed piezoelectric composite actuators Jung Woo Sohn a, Seung-Bok Choi a,n, Heung Soo Kim b a b

Smart Structures and Systems Laboratory, Department of Mechanical Engineering, Inha University, 253 Young-Hyun Dong, Nam-Gu, Incheon 402-751, South Korea Department of Mechanical, Robotics and Energy Engineering, Dongguk University-Seoul, 26 Pil-dong 3-Ga Jung-Gu, Seoul 100-715, South Korea

a r t i c l e i n f o

abstract

Article history: Received 10 August 2010 Received in revised form 25 May 2011 Accepted 30 May 2011 Available online 6 June 2011

Active vibration control to suppress structural vibration of the smart hull structure was investigated based on optimized actuator configurations. Advanced anisotropic piezoelectric composite actuator, Macro-Fiber Composite (MFC), was used for the vibration control. Governing equations of motion of the smart hull structure including MFC actuators were obtained using the Donnell–Mushtari shell theory and Lagrange’s equation. The Rayleigh–Ritz method was used to obtain the dynamic characteristics of the smart hull structure. Experimental modal tests were conducted to verify the proposed mathematical model. In order to achieve high control performance, optimal locations and directions of the MFC actuators were determined by genetic algorithm. Optimal control algorithm was then synthesized to suppress structural vibration of the proposed smart hull structure and experimentally implemented to the system. Active vibration control performances were evaluated under various modes excitations. Vibration tests revealed that optimal configurations of MFC actuators improved the control performance of the smart hull structure in case of the limited number of actuators available. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Cylindrical shell structure Macro-fiber composite actuator Piezoelectric actuator Structural vibration control Genetic algorithm Optimal configuration

1. Introduction Significant progresses in the field of smart materials and smart structures have been made in the last decade. A smart structure has the capability to respond to a change of external environment as well as to a change of internal environment. It incorporates smart materials that allow the alteration of system characteristics such as stiffness or damping in a controlled manner. Many types of smart materials are being considered as actuators and sensors, such as piezoelectric materials, shape memory alloys, electrorheological and magnetorheological fluids [1–4]. Especially, piezoelectric materials are most commonly used as smart material owing to their quick response, wide bandwidth and easy implementation. Moreover, piezoelectric materials can be employed as both actuators and sensors by taking advantage of direct and converse piezoelectric effects. Numerous research works have been carried out on active vibration control of beam and plate structure with piezoelectric actuators and sensors [5–15]. Crawley and de Luis provided pioneering work in this area involving the development of the induced strain actuation mechanism [5]. Crawley and de Luis developed a model of the mechanical coupling of bonded piezoelectric actuators to the dynamics of the structural member using a shear lag analysis. n

Corresponding author. Tel.: þ82 32 860 7319; fax: þ82 32 868 1716. E-mail address: [email protected] (S.-B. Choi). URL: http://www.ssslab.com (S.-B. Choi).

0020-7403/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2011.05.011

Thereafter, many researches have been conducted to improve structural performance based on the induced strain actuators and sensors [6,7]. Hangud et al. established equations of motion of the cantilever beam with piezoelectric actuators and sensors using finite element method and evaluated vibration control performance [8]. Thakkar and Ganguli studied the dynamic behavior of rotating beams with piezoceramic actuation for application to structures such as helicopter and wind turbine rotor blades [9]. Hashemi et al. proposed an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates [10]. Kapuria et al. discussed current research issues in the development of efficient analysis models and their efficient numerical implementation for smart piezoelectric laminated structures [11]. Many of these researches were focused on beam and plate structure. Hull structure or cylindrical shell structure is widely used in many engineering structures such as submarines, aircraft fuselage, and pressure vessels. Thus, active vibration and noise control of hull structure is very important for structural stability and has a wide range of applications in engineering science and technology. However, research works on vibration control of hull structure or cylindrical shell structure with piezoelectric actuators is relative rare. Banks et al. proposed a new piezoelectric actuator model for active vibration and noise control in thin cylindrical shells [16]. Lester and Lefebvre made analytical models for piezoelectric actuators and vibrating circular cylinders [17]. Sonti and Jones derived a differential equations of motion for a bimorph configured thin curved uniformly polarized piezoactuator pair surface bonded to a cylindrical shell [18].

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However, piezoceramic patch has inherent brittle nature and cannot be applied to curved surface of the hull structure in practical application. NASA Langley Research Center developed Macro-fiber Composite (MFC) intended to mitigate many of the disadvantages associated with traditional piezoceramics and piezocomposites [19]. Nguyen and Kornmann conducted performance comparison between MFC and other piezoelectric actuators [20]. Wilkie et al. reported reliability evaluation of the device under externally applied mechanical loads at various operating temperature [21]. Since the MFC consist of piezoceramic fibers, it is flexible and applicable to the curved structures. In addition, in-plane poling with d33 property can be achieved by interdigitated electrode, which produces more induced actuating strain than monolithic piezoelectric ceramic patch. Since the MFC has directional actuating performance from the anisotropic characteristic, optimization of location and direction for MFC actuator is required to achieve high control performance. Many research works on optimal numbers, sizes and locations of

Hull Structure

MFC Actuator

piezoelectric actuators and sensors have been conducted through decades. In the optimization procedure, genetic algorithm was widely used since genetic algorithm has many advantages compared to gradient-based optimization. Sadri et al. reported on modeling and optimal placeement of piezoelectric actuator using genetic algorithm [22]. Han and Lee determined optimal placement of piezoelectric sensors and actuators for vibration control of composite plate using genetic algorithm [23]. Sun and Tong evaluated vibration control performance of optimized distributed piezoelectric actuators [24]. Rao et al. studied the multi-mode structural vibration control in the composite fin–tip of an aircraft with optimally placed piezoelectric actuators [25]. Gupta et al. reported the various optimization criteria used by researchers for optimal placement of piezoelectric sensors and actuators on a smart structure [26]. However, research on vibration control of hull or cylindrical shell structure associated with optimal placement of anisotropic MFC actuator is rare. The main contribution of the present work is to control the structural vibration of hull structure using optimally distributed piezoelectric composite actuators (MFC). The governing equations of the smart hull structure with MFC actuators are derived using the Donnell–Mushtari shell theory and Lagrange equation. the Rayleigh–Ritz method is used to obtain approximate solutions. Dynamic characteristics of the smart hull structure are analytically identified and compared to experimental results. In order to achieve high control performance, the location and direction of MFC actuators are optimized using genetic algorithm. Linear quadratic Gaussian (LQG) controller is then designed and experimentally implemented for the vibration control. Vibration control performances of the hull structure are presented and compared between with and without optimal placements of actuators. 2. System modeling

0.25 m

0.5 m

0.2075 m

0.057 m

0.085 m

0.005 m 0.002 m

0.0003 m

Fig. 1. Configuration of the proposed smart hull structure: (a) schematic diagram and (b) dimensional geometry.

The schematic diagram of the proposed smart hull structure for vibration control is shown in Fig. 1(a). Simple end-capped cylindrical shell structure is considered as a host hull structure, which can be used for underwater vehicles such as submarine and fuselage of aircrafts. The MFC actuators are bonded on the surface of the host structure and perfect bonding is assumed between the host structure and actuators. The structure is considered in the space with free–free boundary conditions. The dimensional geometry of the proposed smart hull structure is presented in Fig. 1(b). The length, radius and thickness of the host structure are 0.5, 0.125 and 0.002 m, respectively. The length, width and thickness of the actuator are 0.085, 0.057 and 0.0003 m, respectively. Perfect bonding is assumed and the thickness of the bonding layer is ignored. Aluminum is used for the hull and material properties of the aluminum and MFC are shown in Table 1. The coordinate configuration and variables of the hull structure and MFC are shown in Fig. 2. The nth mode equations of the motion of the smart hull structure with MFC actuator is obtained as follows:   Mq€ þKq ¼ ðrRLhpÞMS þ ðrMFC RLhMFC pÞMMFC q€    ERhp KS þ ðRhMFC pÞKMFC q ¼ F ð1Þ þ ð1n2 ÞL

Table 1 Material properties of the aluminum and MFC. Aluminum Young’s modulus (E) Poisson ratio (n) MFC (1: poling direction) Young’s modulus 1 direction (E1) Shear modulus (G12) Poisson ratio (n12) Piezoelectric constant (d11) Permittivity (e11/e0)

68 (GPa) 0.32 30.34 (GPa) 5.52 (GPa) 0.31 400 (pC/N) 830 (C/m2)

Density (r)

2698 (kg/m3)

Young’s modulus 2 direction (E2) Density (r) Poisson’s ratio (n21) Piezoelectric constant (d12) Permittivity (e22/e0)

15.86 (GPa) 7750 (kg/m3) 0.16  170 (pC/N) 916 (C/m2)

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

where2

3

Muu

6 MS ¼ 4

Mvv Mww

7 5,

2

Kuu 6 T KS ¼ 4 Kuv KTuw

3

Kuv

Kuw

Kvv

Kvw 7 5 Kww

KTvw

u

7 5DHx DHy

Mvv_MFC

2

Kuu_MFC 6 T K KMFC ¼ 6 4 uv_MFC KTuw_MFC

Kuv_MFC Kvv_MFC KTvw_MFC

Mww_MFC 3 Kuw_MFC 7 Kvw_MFC 7DHx DHy 5 Kww_MFC

ðQ xx e1,MFC þ Q xy e2,MFC Þ

R x2 R y2 x1

y1

cos nyF0u dy dx

3

h



r

6 MMFC ¼ 4

6 7 R x2 R y2 6 n ðQ e 7 6 R yx 1,MFC þ Q yy e2,MFC Þ x1 y1 cos nyFv dy dx 7 6 7 R x2 R y2 7 RhMFC 6 ^an 6 7 ðQ e þ Q e Þ cos n yF d y dx w yx 1,MFC yy 2,MFC x1 y1 F¼ R2 6 o7 2 6 n 7 6 þ 1 ðQ yx e1,MFC þ Q yy e2,MFC ÞaðQ ^ xx e1,MFC þ Q xy e2,MFC Þ 7 R 6 7 4R R 5 x2 y2 00 cos n yF d y dx w x1 y1

x w

3

Muu_MFC

2

L

v

2

649

R

2

1 

(x0,0)

x2

1



2

x1

x Fig. 2. Coordinate configurations: (a) hull structure and (b) MFC actuator.

The detailed derivation of Eq. (1) is presented in the appendix. The first six mode shapes of smart hull structure are shown in Fig. 3 and corresponding natural frequencies are shown in Table 2. For the cylindrical shell structure, the mode is defined as (m , n), where m is number of the waves in circumferential direction and n is the number of the waves in longitudinal direction. The natural frequencies with and without MFC actuators are compared in Table 2. The natural frequencies of the smart hull with MFC actuators are decreased because of the mass effect of actuators. Experimental test for dynamic characteristics of the hull structure is conducted to verify the proposed analytical model. Experimental apparatus for the modal test is shown in Fig. 4(a). PULSE 3560B and ¨ PULSE software of the Bruel&Kjær (FFT analyzer), accelerometer and impact hammer were used for the modal test. Dynamic characteristics of aluminum hull structure with and without MFC actuators are measured. Three MFC actuators are attached to the surface of the hull structure at the center of longitudinal direction with equal space

Fig. 3. Simulated mode shapes of the first six modes. (a) (3,1) mode, (b) (4,1) mode, (c) (2,1) mode, (d) (5,1) mode, (e) (4,2) mode and (f) (5,2) mode.

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J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

Table 2 First six modes and corresponding natural frequencies.

1

40

(3,1) (4,1) (2,1) (5,1) (4,2) (5,2)

Natural frequency w/o MFC

Natural frequency with 3 MFCs

Analytic (Hz)

Experiment (Hz)

Analytic (Hz)

Experiment (Hz)

481.45 553.58 767.88 799.31 998.94 1012.1

482.42 582.51 706.80 850.32 1010.90 1074.83

478.77 550.29 762.39 793.82 991.64 1004.76

474.63 578.19 696.50 848.57 1006.23 1074.03

25 49

20 Magnitude (dB)

Mode (m,n)

0

-20

-40

-60 0

60

400

600 800 Frequency (Hz)

1000

1200

with MFC without MFC

40

Magntude (dB)

200

20 0 -20 -40 -60 0

Fig. 4. Experimental setup for modal test: (a) experimental apparatus and (b) measurement points.

(1201) through circumferential direction as shown in Fig. 4(a). Measurement is carried out through 120 points selected by 5 points for longitudinal direction and 24 points for circumferential direction as shown in Fig. 4(b). The system response of the hull structure with MFC actuators along longitudinal direction is shown in Fig. 5(a). ‘Point 1’ is the left end point of the structure, ‘point 49’ is the center point of the structure and ‘point 25’ is the middle between ‘point 1’ and ‘point 49’. As shown in Fig. 5(a), natural frequencies are matched perfectly through the three points. Since natural frequencies under 200 Hz are related to end-caps or rigid modes and not concerned with vibration through the thickness direction of the shell structure, they are ignored in the vibration control. The fifth and sixth natural frequencies are not measured at the center point (point 49) because ‘point 49’ is geometrically symmetric point under the fifth and sixth mode shape as shown in Fig. 3. The system responses measured at the center of longitudinal direction (point 49) with and without MFC actuators are compared in Fig. 5(b). As shown in Fig. 5(b), natural frequencies with MFC actuators are shifted down. This means that natural frequencies are decreased just by attaching MFC actuators. This is due to mass effect of MFC actuators. The measured natural frequencies of the system with and without MFC actuators are shown in Table 2 and compared with analytical results. The natural frequencies of first mode obtained by analytical analysis and experimental measurements correlates well each other and the relative differences are 0.2% for without MFC and 0.9% for with MFC, respectively. Higher mode

200

400

600 800 Frequency (Hz)

1000

1200

Fig. 5. Measured frequency responses of the hull structure: (a) at three different measurement points and (b) with and without MFC actuators.

natural frequencies show 5–8% differences between analytical and experimental results. This is because of the effect of boundary conditions. Free–free boundary condition of the structure cannot be perfectly satisfied in the experiments and shear diaphragm boundary condition is also an approximated boundary condition for end-caps. In spite of the approximate boundary conditions, the proposed analytical model can predict accurate dynamic characteristics of the smart hull. Mode shapes of the smart hull structure give more proof of this analogy. Mode shapes of the hull structure obtained by experiment are shown in Fig. 6. Experimental mode shapes are exactly coincided with those of analytical analysis. The equations of motion given by Eq. (1) include infinite number of modes and thus are not suitable for controller design. Therefore, the reduced order dynamic model is constructed using modal characteristics. After solving the eigenvalue problem corresponding to the nth circumferential mode, the natural frequency, on and modal matrix, F can be obtained. Then, the modal matrix is used to transform the global displacement vector q to the modal displacement vector Z as follows: q ¼ FZ ð2Þ By substituting Eq. (2) into Eq. (1), modal equations of motion of the smart hull structure are obtained as follows: € þ LH ¼ F^ H

ð3Þ

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

651

3. Optimal configuration of actuators

Fig. 6. Experimental mode shapes of the first six modes. (a) (3,1) mode, (b) (4,1) mode, (c) (2,1) mode, (d) (5,1) mode, (e) (4,2) mode and (f) (5,2) mode.

where H, L and F^ is modal coordinate vector, square of natural frequency vector and modal force vector, respectively, and these are given by 2 h H ¼ Z1

Z2    Zn

iT

,

6 6

L¼6 6 4

3

o21 o22 &

o

2 n

7 7 7 7 5

In order to increase vibration control performance with limited numbers of actuators, optimal configurations of the actuators are investigated using genetic algorithm. The genetic algorithm is a probabilistic search technique that has its roots in the principles of genetics. The beginning of the genetic algorithm is credited to John Holland, who developed the basic ideas in the late 1960s and the early 1970s [27]. The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on the process that drives biological evolution. The genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm selects individuals at random from the current population to be parents and uses them to produce the children for the next generation. Over successive generations, the population ‘evolves’ toward an optimal solution. The genetic algorithm can be applied to solve a variety of optimization problems that are not well suited for standard optimization algorithms, including problems in which the objective function is discontinuous, nondifferentiable, stochastic or highly nonlinear. The information considering the interaction of individual modes with the actuators can be obtained from the modal forces. From the linear control theory, the controllability of a linear system can be determined by calculating the controllability grammian. This means that the overall control performance of the actuators can be predicted from the controllability grammian. If the controllability grammian is nonsingular, the system is said to be controllable with that set of actuators. Since this concept is of binary nature, it cannot be effectively applied in evaluating the performance of distributed actuators. For example, if a system is uncontrollable for a particular actuator location, it can be made controllable by shifting the location by a small amount. However, great amount of control energy may be exerted by the actuator for effective control in this case. Then, performance index, which is based on the eigenvalues of controllability grammian matrix, is proposed that can minimize the control energy requirement [28,29]. From the control state-space system for n modes of vibration control, the controllability grammian can be calculated. The problem of minimum control energy requirement to regulate the system from an initial state, x(0), to a final state, x(tf), can be defined as follows: Z tf Minimize JC ¼ uT ðtÞuðtÞ dt ð6Þ 0

The solution of Eq. (6) can be determined using Pontryagin’s minimum principle:

Augmenting a modal damping ratio, zn, modal equation is expressed as follows:

JC,min ¼ ½eAtf xð0Þxðtf ÞT WC ðtf Þ1 ½eAtf xð0Þxðtf Þ

ð7Þ

where

€ þ 2ZOH _ þ LH ¼ F^ H

ð4Þ

where Z and O is modal damping ratio vector and natural frequency vector, respectively: 2 3 2 3 z1 o1 6 7 6 7 z2 o2 6 7 7 7, O ¼ 6 Z¼6 6 7 6 7 & & 4 5 4 5

on

zn

 T By defining a state vector x ¼ Z1 Z_ 1    Zn Z_ n , the statespace control model can be derived as follows: _ ¼ AxðtÞ þ BuðtÞ þ wðtÞ, xðtÞ

yðtÞ ¼ CxðtÞ þ vðtÞ

ð5Þ

WC ðtf Þ ¼

Z

tf

T eAt BBT eA t dt

0

The matrix WC(tf) is called the controllability grammian matrix and it depends on the input matrix B. Maximizing a norm of controllability grammian matrix can lead to the minimum control energy requirement. In addition, a small eigenvalue of the controllability grammian matrix would lead to at least one mode requiring very high control effort. This implies that all the eigenvalues of the controllability grammian matrix should be as high as possible. It is also more desirable to have the condition of minimum energy requirement be independent of the final time, tf. A grammian matrix independent of tf is obtained using the following relation: T

where A is system matrix, B is input matrix and C is output matrix, w(t) is the external disturbance and v(t) is sensor noise. Since MFC actuators and sensors are physically collocated, output matrix is defined as C¼BT.

WC ðtf Þ ¼ WC ð1ÞeAtf WC ð1ÞeA

tf

ð8Þ

For a stable system, as tf increase, the effect of the second term in Eq. (8) decreases and hence it is appropriate to impose the minimization problem based on WC(N), which is independent of

652

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

the final time. The matrix WC(N) can be calculated by solving the Lyapunov stability equation: AWC ð1Þ þ WC ð1ÞAT þBBT ¼ 0

Table 3 Parameters for genetic algorithm.

ð9Þ

The eigenvalues of the matrix WC(N) play a crucial role in determining the performance of the actuators. Then the performance index of actuators can be defined as follows: ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2N X 2N 2N PIC ¼ ð10Þ li P ðli Þ i¼1

i¼1

where li are the eigenvalues of WC(N). The summation term in Eq. (10) represents the size of the grammian and must be large for good control performance. To ensure that all the eigenvalues of the grammian are high for good controllability of each mode, the performance index includes the product of all the eigenvalues. In this work, MFC sensor is collocated with actuator. Thus, the optimization of sensor location is not considered. In order to select the optimal location of three MFC actuators, the actuator location on the hull structure is divided into 35 points as shown in Fig. 7. Five points in longitudinal direction and seven points in circumferential direction are selected for the candidate of optimal location. 0, p/4, p/2 and 3p/4 are considered for the rotation angle of the MFC actuator. Rotation angle in chromosome structure 1, 2, 3 and 4 stands for 0, p/4, p/2 and 3p/4, respectively. The length, width and rotation of MFC actuator are considered and each point stands for the center point of the MFC actuator. The total number of possible combinations for the MFC actuator location is about 40C3(¼9880). The performance criterion in Eq. (10) is used as the objective function. The number of controlled modes is set to first three modes. The chromosome structure is shown in Fig. 8. Each 31

Parameter

Values

Population size Maximum number of generations Crossover probability Mutation probability

50 100 0.5 0.05

31

35

6

10



1

x

5

31

35

6

10

35





6

1

10

x

5

Fig. 7. Numbering of points for MFC actuator locations.

1

x

5

Fig. 9. Initial and optimal configuration of the actuators. (a) initial configuration and (b) optional configuration.

Fig. 8. Chromosome structure for actuator configurations.

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

algorithm, two-point crossover method is used. The optimization procedure using genetic algorithm is summarized as follows:

set of nine bits indicates the corresponding actuator location and direction, six bits for the location and three bits for the rotation angle. The several important parameters used in genetic algorithm are shown in Table 3. In order to improve the efficiency of the

(1) The initial chromosomes of three actuators are chosen randomly in population 100. (2) The fitness value is calculated for each location set. (3) Genetic operators, crossover and mutation, are applied to produce a new set of chromosomes. (4) Steps (2) and (3) are repeated 500 times. (5) The computation is terminated and the location set based on the minimizing control spillover and input energy from the current generation is selected as the optimal locations of three actuators.

Fig. 9 shows the initial configuration and optimal configuration of the MFC actuators. The initial configurations are determined in random. It is observed that MFC actuators are located in the center of longitudinal direction and the poling direction is parallel to

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

Sensor Output (V)

Sensor Output (V)

Fig. 10. Experimental setup for vibration control.

1

2 3 Time (sec)

4

400 300 200 100 0 -100 -200 -300 -400 0

1

2

3

4

0.4 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4

5

Input Voltage1 (V)

Input Voltage1 (V)

0

0

1

0

1

5

Input Voltage2 (V)

Input Voltage2 (V) 2

3

4

5

0

Input Voltage3 (V)

Input Voltage3 (V) 2

3

Time (sec)

2

3

4

5

1

2

3

4

5

4

5

Time (sec)

400 300 200 100 0 -100 -200 -300 -400 1

5

400 300 200 100 0 -100 -200 -300 -400

Time (sec)

0

4

Time (sec)

400 300 200 100 0 -100 -200 -300 -400 1

2 3 Time (sec)

400 300 200 100 0 -100 -200 -300 -400

Time (sec)

0

653

4

5

400 300 200 100 0 -100 -200 -300 -400 0

1

2

3

Time (sec)

Fig. 11. Experimental control response and control input under (3,1) mode excitation: (a) initial configuration and (b) optimal configuration.

654

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

In the above, Q is the state weighting semi-positive matrix and R is the input weighting positive matrix. Since the system (A, B) in Eq. (5) is controllable, we can obtain the following linear quadratic regulator (LQR):

circumferential direction after optimization. This can be explained by mode shapes of the first three modes. The center of the longitudinal direction has the largest deflection in the all three mode. Then the actuator should be located at the center of the longitudinal direction.

uðtÞ ¼ KG xðtÞ 4. Vibration control performance

In the above, KG is the state feedback gain matrix and can be obtained by

The control purpose is to regulate unwanted vibrations of the hull structure. Thus, the performance index to be minimized is chosen as follows [30]: Z 1 J¼ fxðtÞT Q xðtÞ þ uðtÞT RuðtÞg dt ð11Þ

KG ¼ ðR þ BT PBÞBT PA

AT PAPAT PBðR þ BT PBÞ1 BT PA þ Q ¼ 0

0.3

0.2

0.2

Sensor Output (V)

Sensor Output (V)

0.3

0.1 0.0 -0.1 -0.2

0.1 0.0 -0.1 -0.2 -0.3

-0.3 0

1

2

3

4

0

5

1

Input Voltage1 (V)

Input Voltage1 (V)

400 300 200 100 0 -100 -200 -300 -400 0

1

2 3 Time (sec)

4

Input Voltage 2 (V) 1

2 3 Time (sec)

4

Input Voltage3 (v) 1

2 3 Time (sec)

4

5

4

5

0

1

2 3 Time (sec)

4

5

0

1

2 3 Time (sec)

4

5

0

1

2 3 Time (sec)

4

5

400 300 200 100 0 -100 -200 -300 -400

5

400 300 200 100 0 -100 -200 -300 -400 0

3

400 300 200 100 0 -100 -200 -300 -400

5

400 300 200 100 0 -100 -200 -300 -400 0

2

Time (sec)

Time (sec)

Input Voltage2 (V)

ð13Þ

where P is the solution of the following algebraic Riccati equation:

0

Input Voltage3 (v)

ð12Þ

400 300 200 100 0 -100 -200 -300 -400

Fig. 12. Experimental control response and control input under (4,1) mode excitation: (a) initial configuration and (b) optimal configuration.

ð14Þ

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

Covðv,wT Þ ¼ 0

Since the state variables of LQR are not available from direct measurement, the Kalman–Bucy Filter (KBF) is formulated. The KBF is a state estimator which is considered optimal in the statistical sense. The state space model, considering observation modes, can be given by

ð17Þ

^ The estimated state, xðtÞ, can be obtained from _^ ¼ AxðtÞ ^ þ BuðtÞ þ L½yðtÞCxðtÞ ^ xðtÞ T

_ ¼ AxðtÞ þ BuðtÞ þ wðtÞ xðtÞ

ð15Þ

yðtÞ ¼ CxðtÞ þvðtÞ

ð16Þ

655

T

ð18Þ

1

where L ¼ ASC ðCSC þVÞ . In the above, L is the observer gain matrix, and S is the solution of the following observer Riccati equation: ASAT SASCT ðCSCT þ VÞ1 CSCT þ W ¼ 0

In the above equations, w and v are uncorrelated white noise characterized by covariance matrices W and V as follows:

ð19Þ

Using the estimated states, the control input is obtained as follows:

Covðv,vT Þ ¼ V

^ uðtÞ ¼ KG xðtÞ

0.03

0.03

0.02

0.02

Sensor Output (V)

Sensor Output (V)

Covðw,wT Þ ¼ W

0.01 0.00 -0.01 -0.02

ð20Þ

0.01 0.00 -0.01 -0.02 -0.03

-0.03 0

1

2

3

4

0

5

1

400 300 200 100 0 -100 -200 -300 -400 1

2 3 Time (sec)

4

400 300 200 100 0 -100 -200 -300 -400 0

1

2

3

4

0

1

5

Input Voltage3 (V)

Input Voltage3 (V) 2 3 Time (sec)

5

0

1

2 3 Time (sec)

4

5

2

3

4

5

4

5

Time (sec)

400 300 200 100 0 -100 -200 -300 -400 1

4

400 300 200 100 0 -100 -200 -300 -400

Time (sec)

0

3

400 300 200 100 0 -100 -200 -300 -400

5

Input Voltage2 (V)

Input Voltage2 (V)

0

2

Time (sec)

Input Voltage1 (V)

Input Voltage1 (V)

Time (sec)

4

5

400 300 200 100 0 -100 -200 -300 -400 0

1

2 3 Time (sec)

Fig. 13. Experimental control response and control input under (2,1) mode excitation: (a) initial configuration and (b) optimal configuration.

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J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

Evaluation of vibration control performance is conducted by experimental investigation and experimental setup is established as shown in Fig. 10. Aluminum hull is prepared and three MFC actuators are attached on the surface of the structure. Collocated MFC sensors are used to measure the vibration of the structure. In order to reduce the bonding effect between MFC and structure, 3M’s DP 460 Epoxy adhesive, which is recommended by Smart Material Corp., is used and kept as thin as possible. One MFC is attached inside of the hull structure and used as an exciter. Measured signal of the MFC sensor is sent to computer through the AD converter. Control input is determined by control algorithm, which is implemented in DS1104 control board of dSPACE, and sent to high voltage amplifier through DA converter. Amplified input voltage is applied to the each MFC actuator. Experimental results of structural vibration control with and without optimal configuration of MFC actuator is compared. The control response and applied control input voltages under (3,1) mode excitation are shown in Fig. 11. It is observed that the structural vibration was reduced effectively by applying control input voltages. After optimization of MFC configuration, the vibration suppression is increased by 17% and control input voltages are decreased almost 20% compared against results without optimization. The control response and control input voltages under (4,1) mode excitation and (2,1) mode excitation are shown in Figs. 12 and 13, respectively. Similarly to Fig. 11, when the configurations of actuators are optimized, structural vibration was suppressed more effectively with smaller control inputs.

5. Conclusion The mathematical model of smart hull structure with MFC actuators was proposed and vibration control performance with optimized actuator configurations was evaluated. Dynamic model of smart hull structure was derived from Lagrange equation based on the simple Donnell–Mushtari shell theory. The Rayleigh–Ritz method was used to analyze modal characteristics of smart hull structures. The state-space model was established using approximate solutions. Modal characteristics, such as natural frequencies and mode shapes were analytically obtained and compared to experimental modal test results. Because of directional performance of MFC actuator, location and direction of MFC actuators were optimized using genetic algorithm based on minimum energy problem. LQG controller was then developed and implemented into the system for the vibration control of the smart hull structure. Vibration control performance of the proposed system was evaluated via experimental investigation. It has been demonstrated that structural vibration was effectively controlled by applying proper control input voltages to the MFC actuators whose location and directions are optimally determined by genetic algorithm. The vibration control of the hull structure in underwater is also going to be studied as a second phase of this research. In addition, the vibration control performance with considering system time delay and parameter uncertainties will be studied in the next phase of this research using robust control algorithms.

Acknowledgment This work was supported by research grant from the Underwater Vehicle Research Center of Agency for Defense Development and Defense Acquisition Program Administration, Korea. This financial support is gratefully acknowledged.

Appendix As a first step, kinetic and potential energy of the hull structure is defined in terms of the displacement of the structure based on the Donnell–Mushtari shell model. In this work, a thin hull structure of radius R, thickness h and length L is considered. x and y are the coordinates of the axial and circumferential direction, respectively. The coordinate configuration and variables of the hull structure is shown in Fig. 2(a). The kinetic energy of the hull structure can be expressed as follows: Z L Z 2p Z h=2 ( 2  2  2 ) 1 @u @v @w TS ¼ r þ þ Rdz dy dx 2 @t @t @t 0 0 h=2 ðA  1Þ where u, v and w is the axial, tangential and radial displacement, respectively. The potential energy of the hull structure can be obtained in terms of stress and strain of the structure: Z Z Z 1 L 2p h=2 ðsx ex þ sy ey þ sxy gxy Þ R dz dy dx UE,S ¼ 2 0 0 h=2  Z L Z 2p Z h=2  E 1n 2 2 2 e þ e þ2 n e e þ g ¼ R dz dy dx x y x y 2 xy 2ð1n2 Þ 0 0 h=2 ðA  2Þ Eq. (A-2) can be expressed in terms of displacements using the strain–displacement relationship [31]: !   Z L Z 2p (  2 ERh @u h2 @2 w 1 @v 2 þ þ UE,S ¼ @x 12 @x2 2ð1n2 Þ 0 0 R2 @y !      2 2 2 w h @ w 2 @v 2n @u @v þ 2 þ wþ þ 2 @y R @x R @y R 12R4 @y2 ! !     2n @u nh2 @2 w @2 w ð1nÞ @v 2 wþ 2 þ þ 2 2 2 @x R @x @x 6R @y 9 !2   =  2 ð1nÞ @u ð1nÞh2 @2 w ð1nÞ @v @u þ þ þ dy dx @x@y R @x @y ; 2R2 @y 6R2 ðA  3Þ By considering the coordinate configuration and variables of the MFC which is shown in Fig. 2(b), the kinetic energy of the MFC actuator is obtained as follows: 1 TMFC ¼ rMFC 2 Z L Z 2p Z h=2 þ hMFC ( 2  2  2 ) @u @v @w þ þ DHx DHy Rdz dy dx @t @t @t 0 h=2 0 ðA  4Þ where

DHx ¼ ½Hx1 Hx2 , Hxj ¼ Hðxxj Þ, j ¼ 1,2 DHy ¼ ½Hy1 Hy2 , Hyj ¼ Hðyyj Þ, j ¼ 1,2 In the above, H is a Heaviside unit step function, rMFC is the density and hMFC is the thickness of the MFC actuator, respectively. The potential energy of the MFC actuator can be expressed in terms of the structural strain energy, piezoelectric induced stain energy and electric energy as follows: Z Z Z 1 L 2p h=2 þ hMFC UE,MFC ¼ ðsx,MFC_s ex þ sy,MFC_s ey 2 0 0 h=2 Z Z Z 1 L 2p h=2 þ hMFC þ ss,MFC_s gs ÞDHx DHy R dz dy dx þ 2 0 0 h=2 ðsx,MFC_p ex þ sy,MFC_p ey ÞDHx DHy R dz dy dx Z Z Z 1 L 2p h=2 þ hMFC ðb11 D21 ÞDHx DHy R dz dy dx þ 2 0 0 h=2

ðA  5Þ

J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

where D1 ¼ e11E1 ¼ e11(Vin/hMFC) and b11 ¼(1/e11). Here, e11 is the permittivity of MFC actuator in 11 direction. The structural strain energy term of Eq. (A-5) can be expressed in terms of the axial, tangential and radial displacements, u, v and w as follows: Z Z Z 1 L 2p h=2 þ hMFC ðsx,MFC_s ex þ sy,MFC_s ey 2 0 0 h=2 (  Z Z Qxx RhMFC L 2p @u 2 þ ss,MFC_s gs ÞDHx DHy R dzdy dx ¼ @x 2 0 0 9 ! ! 2   2 @u @ w @2 w = DHx DHy dy dx 2a^ þ b^ 2 ; @x @x @x2 (     Z Z Qyy RhMFC L 2p 1 @v 2 2 @v w2 þ þ wþ 2 2 R @y 2 R @y R 0 0 ! !2 9   2 ! b^ @2 w = 2a^ @v @ w 2a^ @2 w DHx DHy dy dx  3  3 wþ 4 2 2 2 ; R R @y R @y @y @y (       Z Z Qss RhMFC L 2p @v 2 2 @v @u 1 @u 2 þ þ 2 @x R @x @y 2 R @y 0 0   2 !   2 ! ^ ^ 4a @v @ w 4a @u @ w  2  R @x @x@y @x@y R @y

 ðcos3 a sin1 acos1 a sin3 aÞ Qss ¼ Q11 cos2 a sin2 a þ Q22 cos2 a sin2 a 2Q12 cos2 a sin2 a þ Q66 ðcos2 asin2 aÞ2 E1 E2 n21 E1 n12 E2 Q11 ¼ , Q22 ¼ ,Q12 ¼ ¼ , 1n12 n21 1n12 n21 1n12 n21 1n12 n21 Q66 ¼ G12 The piezoelectric induced stain energy term of Eq. (A-5) can be also expressed in terms of the axial, tangential and radial displacements, u, v and w as follows: Z 1 ðsx,MFC_p ex þ sy,MFC_p ey ÞDH dVMFC 2 VMFC Z Z Z 1 x2 y2 h=2 þ hMFC ¼ fðQ xx e1,MFC þQ xy e2,MFC Þex þ ðQ yx e1,MFC 2 x1 y1 h=2

þ Q yy e2,MFC Þey R dzdy dx Z Z Z RðQ xx e1,MFC þ Q xy e2,MFC Þ x2 y2 h=2 þ hMFC ¼ ðex þ zkx Þ dz dy dx 2 x1 y1 h=2

þ

þ

þ

!2 9 @2 w =

4b^ R2 @x@y

þ Qxy RhMFC

¼

DHx DHy dy dx

;

Z

b^ R2

@2 w @x2

!

þ Qxs RhMFC

@2 w @y

Z

!)

2

L

Z 2p (   @u @v

Q 11 ¼ 

@x @x 0   2 !   2 !  1 @u @u 2a^ @u @ w @v @ w þ  a^ R @x @y R @x @x@y @x @x2 ! !)   2 ! a^ @u @ w 2b^ @2 w @2 w DHx DHy dy dx þ  R @y R @x2 @x@y @x2      Z L Z 2p     1 @v @v 1 @v @u w @v þ 2 þQys RhMFC þ R @y @x @y R @x R @y 0 0 !     2 ! 2 w @u 2a^ @v @ w 2a^ @ w  2 w þ 2  2 @x@y R @y R @y R @x@y   2 !   2 ! a^ @v a^ @u @ w @ w  2  3 2 2 R @x R @y @y @y ! !) 2b^ @2 w @2 w DHx DHy dy dx ðA  6Þ þ 3 2 @x@ y R @y 

where a^ ¼

  1 1 3 3 ðhþ hMFC Þ, b^ ¼ h2MFC þ hMFC h þ h2 2 3 2 4

Qxx ¼ Q11 cos4 a þ Q22 sin4 a þ2Q12 cos2 a sin2 a þ4Q66 cos2 a sin2 a 4

2

2

Qyy ¼ Q11 sin a þQ22 cos4 a þ 2Q12 cos2 a sin a þ 4Q66 cos2 asin a Qxy ¼ Q11 cos2 a sin2 a þQ22 cos2 a sin2 a þðsin4 a þ cos4 aÞQ12 4Q66 cos2 a sin2 a 1

3

1

3

Qxs ¼ Q11 cos a sin aQ22 cos a sin a þðcos1 a sin3 acos3 a sin1 aÞQ12 þ 2Q66  ðcos1 a sin3 acos3 a sin1 aÞ Qys ¼ Q11 cos1 a sin3 aQ22 cos3 a sin1 a 3

1

1

3

Z

Z

x2

x1 x2

x1

y2

Z

y1

Z

y2

y1

h=2 þ hMFC

ðey þzky Þdz dy dx

h=2

!) (  @u ðh þhMFC Þ @2 w dy dx þ  2 @x 2 @x

where

DHx DHy dy dx

0



RhMFC ðQ xx e1,MFC þ Q xy e2,MFC Þ 2

Z

þ

R @x @y 0 0 !     2 !   2 ! a^ @u a^ @v a^ @2 w w @u @ w @ w  2 þ   w 2 R @y R @x2 R @x R @x @x2 @y

þ

RðQ yx e1,MFC þQ yy e2,MFC Þ 2

  Z Z RhMFC ðQ yx e1,MFC þQ yy e2,MFC Þ x2 y2 1 @v w  þ 2 R @y R x1 y1 !) ðhþ hMFC Þ 1 @2 w  2 2 dy dx þ ðA  7Þ 2 R @y

Z 2p    1 @u @v

L

657

þðcos a sin acos a sin aÞQ12 þ 2Q66

E1 , 1n21 n12

Q 22 ¼ 

e1,MFC ¼ ðex þzkx Þ ¼ d11

E2 1n12 n21

Vin 1 V ðe þ zky Þ ¼ d12 in , e2,MFC ¼ 1 þz=R y hMFC hMFC 2

2

Q xx ¼ Q 11 cos2 a, Q yy ¼ Q 22 cos2 a, Q xy ¼ Q 22 sin a, Q yx ¼ Q 11 sin a

An approximate solution is obtained by the Rayleigh–Ritz method. The displacements u, v and w are assumed as follows: u¼ v¼ w¼

1 X n¼1 1 X

1 X

un ðx, y,tÞ ¼ vn ðx, y,tÞ ¼

n¼1 1 X n¼1

n¼1 1 X

cos nyFu ðxÞqun ðtÞ sin nyFv ðxÞqvn ðtÞ

n¼1 1 X

wn ðx, y,tÞ ¼

cos nyFw ðxÞqwn ðtÞ

ðA  8Þ

n¼1

where Fu(x), Fv(x) and Fw(x) are mode shape function vector of each direction. qun(t), qvn(t), qwn(t) are generalized displacement vector of each direction for circumferential nth mode. Considering m mode shape functions, the mode shape function vector and generalized displacement vector can be written as follows:   Fu ðxÞ ¼ Fu1 Fu2    Fum   Fv ðxÞ ¼ Fv1 Fv2    Fvm   Fw ðxÞ ¼ Fw1 Fw2    Fwm ðA  9Þ   qun ðtÞ ¼ qun1 qun2    qunm   qvn ðtÞ ¼ qvn1 qvn2    qvnm   qwn ðtÞ ¼ qwn1 qwn2    qwnm

ðA  10Þ

The kinetic energy and potential energy for the circumferential nth mode of the hull structure can be expressed in terms of mode shape function vector and generalized displacement vector

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J.W. Sohn et al. / International Journal of Mechanical Sciences 53 (2011) 647–659

as follows:  TS ¼ 12rRhLp q_ Tu Muu q_ u þ q_ Tv Mvv q_ v þ q_ Tw Mww q_ w Þ

ðA  11Þ

 ERhp 1 T 1 1 q Kuun qu þ qTv Kvvn qv þ qTw Kwwn qw UE, S ¼ 2 2 ð1n2 ÞL 2 u þ qTu Kuvn qv þ qTv Kvwn qw þ qTu Kuwn qw

ðA  12Þ

 1 T 1 1 UE,MFC ¼ RhMFC p q Kuu_MFC qu þ qTv Kvv_MFC qv þ qTw Kww_MFC qw 2 u 2 2 þqTu Kuv_MFC qv þ qTv Kvw_MFC qw þ qTu Kuw_MFC qw DHx DHy " Z x2 Z y2 RhMFC ðQ xx e1,MFC þ Q xy e2,MFC Þ þ cos nyF0u dy dxqu 2 x1 y1 ! Z Z x2

where

x1

Muu ¼ Fuu , Mvv ¼ Fvv , Mww ¼ Fww ð1n2 Þa2 n2 ð1nÞ Kuu ¼ Fuu þ Fuu , Kvv ¼ a2 n2 Fvv þ Fvv 2 2 ! 2 2 a2 b n4 ð1nÞb n2 Kww ¼ a2 þ Fww þ Fww 12 6 6 12a ð1n2 Þan ^ ~ uv , Kvw ¼ a2 nFvw , Kuw ¼ naF ~ uw Kuv ¼ Fuv þ nanF 2 Z 1 Z 1 Z 1 Fuu ¼ FTu Fu dx, Fvv ¼ FTv Fv dx, Fww ¼ FTw Fw dx Z

0

0

1 0 F0T u Fu dx, Fvv ¼

0

~ uv ¼ F

Z

1

0

Fvw ¼

Z

1 0

^ F0T u Fv dx, Fuv ¼

0

Z

1 0 F0T v Fv dx, Fww ¼

0

Z

^ ww ¼ FTv Fw dx, F

1 0

Z

0 ~ F0T u Fv dx, F uw ¼ 1

0

Z

1

0 Z 1 0

0 ~ F0T w Fw dx, F ww ¼

0 F0T w Fw dx

F0T u Fw dx

Z 0

1

F00T w Fw dx

x ¼ x=L, a ¼ L=R, b ¼ h=R

cosnyF00w dy dxqw þ ðQ yx e1,MFC þ Q yy e2,MFC Þ

the equations of motion of the circumferential nth mode of the hull structure is derived in matrix form as follows:   ERhp KS ¼ 0 ðA  14Þ ðrRLhpÞMS q€ þ 2 ð1n ÞL

6 MS ¼ 4

3

Muu Mvv Mww

7 5,

2

Kuu 6 T KS ¼ 4 Kuv KTuw

3

Kuv

Kuw

Kvv

Kvw 7 5 Kww

KTvw

x2

Z

x1

y2

y1

cos nyFv dy dxqv þ

1 R

Z

x2

Z

x1

y2

y1

cos nyF00w dy dxqw

ðA  19Þ

0

0

where Muu_MFC ¼ Fuu_MFC , Mvv_MFC ¼ Fvv_MFC , Mww_MFC ¼ Fww_MFC Ln2 1 Kuu_MFC ¼ 2 Qss Fuu_MFC þ Qxx Fuu_MFC L R Ln2 1 Kvv_MFC ¼ 2 Qyy Fvv_MFC þ Qss Fvv_MFC L R ! ^ 4 bLn 2a^ Ln2 L Kww_MFC ¼ Qyy þ 2 Qyy þ 4 Qyy R R3 R ^ 2 4bn Qss Fww_MFC R2 L ! 2 ^ ^ 2a^ 2bn ~ ww_MFC þ b Qxx F ^ ww_MFC Qxy þ 2 Qxy F  RL L3 R L n ~ uv_MFC  n Qss Fuv_MFC Kuv_MFC ¼ Qxy F R R   ^ 3 ^ an n 2an Qss Fvw_MFC Kvw_MFC ¼ LQ þ LQ yy Fvw_MFC þ RL R2 yy R3 ^ an ~ vw_MFC  Qxy F RL   a^ n2 1 ~ uw_MFC a^ 1 Qxx Fuw_MFC Qxy þ 2 Qxy F Kuw_MFC ¼ R L2 R 2 ^ 2an ^ uw_MFC  2 Qss F R

ð1n2 ÞrL2 2 o E

at x ¼ 0,L

Z

1

0

ðA  15Þ

The proposed smart hull structure satisfies shear diaphragm boundary conditions as follows: v ¼ w ¼ Mx ¼ Nx ¼ 0

Z

!# Z Z y2 a^ n x2 þ 2 cos nyFw dy dxqw R x1 y1 Z 1 Z 2p b11 D21 hMFC L dy dxDHx DHy þ

Fuu_MFC ¼

Then, eigenvalue problem can be defined as following equation: 9KS O2 MS 9 ¼ 0, O2 ¼

y1

Fww_MFC þ

By substituting Eqs. (A-11) and (A-12) into the following Lagrange’s equation:   d @L @L  ¼ 0, L ¼ TV ðA  13Þ dt @q_ @q

where 2

n R



b2 ^ nb2 n2 ~ Fww  F ww 2

þ

Fuu ¼

y2

a^

ðA  16Þ

These conditions can be closely approximated in physical application simply by means of rigidly attaching a thin, flat, circular cover plate at each end. Mode shape functions which satisfy shear diaphragm boundary condition are expressed as follows: pffiffiffi pffiffiffi Fum ðxÞ ¼ 2 cos mpx, Fvm ðxÞ ¼ Fwm ðxÞ ¼ 2 sin mpx ðA  17Þ The kinetic and potential energy of the MFC actuator also can be expressed in terms of mode shape function vector and generalized displacement vector as follows: TMFC ¼ 12rMFC RhMFC Lp  T q_ u Muu_MFC q_ u þ q_ Tv Mvv_MFC q_ v þ q_ Tw Mww_MFC q_ w ÞDHx DHy ðA  18Þ

Fuu_MFC ¼

Z

0 F0T u Fu dx, Fvv_MFC ¼

Z

~ ww_MFC ¼ F

Fvw_MFC ¼

Z

1 0 1

0

Z

1 0

~ uw_MFC ¼ F

Z

1

0

~ uv_MFC ¼ F

FTu Fu dx, Fvv_MFC ¼

Z 0

1

0

Z

FTv Fv dx, Fww_MFC ¼

1 0 F0T v Fv dx, Fww_MFC ¼

0

^ F00T w Fw dx, Fww_MFC ¼ Z

F0T u Fv dx, Fuv_MFC ¼ FTv Fw dx, Fvw_MFC ¼

1

F0T u Fw dx, Fuw_MFC ¼

Z 0

Z

1 0

1

0

Z

FTw Fw dx,

1 0 F0T w Fw dx

0

1 0

1 0

1

Z

Z

00 F00T w Fw dx

FTu F0v dx

0 ~ F0T v Fw dx, F vw_MFC ¼

00 ^ F0T u Fw dx, Fuw_MFC ¼

Z

1

0

Z 0

1

FTv F00w dx FTu F0w dx

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