Positive Position Feedback Active Vibration Control of a Smart Cantilever Beam Using ANSYS

Abstract

In this work, the active vibration control of a uniform cantilever beam using piezoelectric materials subjected to transverse vibrations is studied. The equation of motion of a beam bonded with the piezoelectric actuator is realized based on the Euler Bernoulli beam theory and the Hamilton's principle. A linear time invariant state space model is derived. Numerical simulations of the equation of motion are performed. Moreover, a finite element model of the beam-piezo system is done using ANSYS APDL©. Two control algorithms were also implemented using ANSYS APDL code to reduce the flapwise bending vibrations of the beam. The two control techniques are: Linear quadratic regulator (LQR) and positive- position-feedback (PPF). Results of the PPF simulations were compared with that of the LQR control and the advantage of using PPF control over LQR control in the finite element simulations is presented. The area of vibration analysis and control is an interesting and important field of research. In fact, any structure that has certain mass and elasticity is said to vibrate and so it is important to study the frequencies at which it vibrates to avoid resonance which can lead to failure of the system. The need to control such vibrations is important in the industrial field to get better functioning machines and increase the quality of the products. The importance of vibration control is also clear in the field of aerospace where flexible light weight structures are subjected to vibrations. Moreover, tall buildings and long bridges need vibration control to avoid their failure. In the past, passive vibration control was used in many structures by adding damping and stiffness but there were drawbacks for this type of control like slow response and increased weight of the structure. Therefore, active control techniques using piezoelectric materials attracted many engineers to be used in vibration control for their low weight and fast response. The use of piezoelectric materials as sensors and actuators is increasingly growing, as it can easily convert mechanical energy to electrical energy which is useful in many applications such as vibration control, energy harvesting and aerospace industry. Aldraihem (1) studied the effect of the location of the piezoelectric actuator on the response of the beam under different boundary conditions and formulated an optimization criterion based on the modal controllability. Al-Ashtari (2) derived a mathematical model presenting the smart cantilever beam and studying its behavior under different applied loads either static or cyclic. Moreover, he studied the effect of changing the number of actuators and proved that as the number of actuators increases, better damping of the oscillations is achieved. For the field of vibration control using piezoelectric materials, Meyer et al. (3) studied the effect of PPF control and LQG control on flexible structures using piezoelectric materials. Then, Song et al. (4) proved that PPF control is robust to frequency variations and so it is an effective method for vibration suppression. Since ANSYS APDL © is more efficient in finite element modeling, researchers used to export the finite element matrices representing the model from finite element analysis software to Matlab™ and perform the control techniques. However, for complex structures it would be more efficient if the control algorithms can be implemented using the finite element software and withdrawing the need of exporting or importing the model. Malgaca and Karagulle (5) implemented a numerical model for a cantilever beam under free and forced response then integrated the control actions which are strain and displacement feedback with the finite element model using finite element program which is ANSYS APDL©. They also implemented the model experimentally and proved that numerical and experimental results are in good match. However, in all research mentioned there was no clear explanation of the method used to implement the controller using the finite element software until Takacs and Rohal'-Ilkiv (6) implemented a digital LQR controller for a cantilever beam using ANSYS APDL© and compared their results with the experimental results which were in a good match. This was the first implementation for a controller represented in state space form on finite element software, however only the LQR control was studied while other control techniques like PPF control are not yet implemented on the finite element software. The goal of this paper is to implement the positive position feedback (PPF) controller in ANSYS APDL© and compare ANSYS results with the PPF direct numerical integral of the closed loop system. Also, an optimal LQR controller is implemented and compared with that of PPF.