Structures with surface-mounted or embedded sensors and actuators are referred to as smart structures. Piezoelectric materials as smart materials exhibit both direct and converse piezoelectric effects. The direct effect (electric field generation as a response to mechanical strains) is used in piezoelectric sensors and the converse effect (mechanical strain is produced as a result of an electric field) is used in piezoelectric actuators [2].

In this paper, a smart functionally graded beam is investigated. The smart beam consists of three layers: one layer of functionally graded material (FGM) and two layers of piezoelectric material used as sensor and actuator. The properties of the FGM layer are functionally graded in the thickness direction according to the volume fraction power law distribution (Reddy's model) [3]. For piezoelectric materials, the constitutive relationships describing the electrical and mechanical interactions are explained. Governing equations of the composite beam based on first order shear deformation theory (FSDT) are presented. Stress and stress fields, stress resultant , stress couple and transverse shear resultant are determined using these equations.

The potential field is assumed to be linearly varying across the thickness, in accordance with the same variation of the stress and strain. So, the electric field is uniform across the thickness of piezoelectric layers.

Using Hamilton's principle, carrying out the various variational operations yields the equation of motion of smart beam based on first order shear deformation theory (Timoshenko Theory) [4].

For deflection control, a constant velocity control algorithm is applied in a closed loop system to provide active feedback control of the FGM beam.

For solution of the governing equations of the beam, a displacement potential function was introduced to obtain the displacement components [5]. Introducing some operators and doing some simplifications using complex transformation, results in a differential equation for determining the displacement potential function. After obtaining the displacement potential function (), , and , mid-plane displacements in the and directions and rotation in the plane, respectively, can be determined using the governing equations.

The effect of feedback control gain on the tip transient response (deflection) is investigated. It is clear from the graphs in the paper that the amplitude of the beam deflection is damped quickly when the higher order feedback control gains are applied.

The effect of the volume fraction index of FGM layer on the vibration amplitude of the beam is also studied. It can be inferred that the vibration amplitude of the FGM beam increases as the power law exponent increases. It can be concluded that the present control algorithm is effective for the deflection control of FGM beams.

1

S.H. Chi, Y.L. Chung, "Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis", Int. J. Solids Struct., Article in press. doi:10.1016/j.ijsolstr.2005.04.011

2

J. Mackerle, "Smart materials and structures: a finite-element approach: a bibliography (1986-1997)", Modelling Simul. Mater. Sci. Eng., 6, 293-334, 1998. doi:10.1088/0965-0393/6/3/007

3

J.N. Reddy, C.D. Chin, "Thermoelastic analysis of functionally graded cylinders and plate", J. Thermal Stress, 21, 593-620, 1998. doi:10.1080/01495739808956165

4

J.N. Reddy, "Mechanics of laminated composite plates and shells: theory and and analysis. 2nd ed", CRC Press, 2004.

5

B. Sun, D. Huang, "Analytical vibration suppression analysis of composite beams with piezoelectric laminae", Smart Mater. Struct., 9, 751-760, 2000. doi:10.1088/0964-1726/9/6/303

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