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This paper presents a dynamic study of sandwich functionally graded beam with piezoelectric layers that are used as sensors and actuators. This study is exploited later in the formulation of the active control laws, while using the optimal control Linear Quadratic Gaussian (LQG), accompanied by the Kalman filter. The mathematical formulation is based on Timoshenko’s assumptions and the finite element method, which is applied to a flexible beam divided into a finite number of elements. By applying the Hamilton principle, the equations of motion are obtained. The vibration frequencies are found by solving the eigenvalue problem. The structure is analytically then numerically modeled and the results of the simulations are presented in order to visualize the states of their dynamics without and with active control.

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An analytical method on the active vibration control of a functionally graded beam equipped with layers of piezoelectric sensors and actuators, in a thermal environment, is studied. The study based on Euler-Bernoulli theory and finite element method, applied to a flexible beam divided into a finite number of elements. The equations of motion are obtained by applying the principle of Hamilton. The structure is modeled analytically then numerically and the results of the simulations are presented to visualize the states of their dynamics.

This work presents the study of the dynamics and active control of a cantilever sigmoid FGM beam with porosities in a thermal environment. During this study, we considered the Timoshenko beam’s theory combined with the finite element method (FEM). This work also presents a comparative study with an experimental study for the vibration of a functionally graded piezoelectric beam (FGPM) to validate the numerical model. Linear quadratic Gaussian (LQG) optimal control with a Kalman filter was used for the vibration control using piezoelectric sensors and actuators as symmetrical layers to eliminate membrane effects. The controlled and uncontrolled responses are presented, considering the influence of thermal effect, the porosity of the FGM material, and the location of the sensor pair on the smart structure. The results indicate that the porosity effect of the FGM material, as well as the application of the thermal effect, involves an increase in vibration frequencies, in contrast to the increase in the power law index. The study also shows that the thermal and porosity effects result in an increase in vibration amplitudes.