In this work, the nonlinear dynamics and active control of smart beams are investigated using the shear mode and extensional mode piezoelectric actuators. The numerical analysis is performed considering two smart beams for separate use of the two kinds of piezoelectric actuators. These smart beams are considered to operate under the transverse or axial compressive harmonic load while the actuators are activated by supplying the external electric field according to the velocity feedback control law. The dynamics of the smart beams is analysed numerically by deriving the geometrically nonlinear electro-elastic incremental equations of motion in the finite element (FE) framework. The FE equations of motion are solved using Bathe time integration method for evaluation of the transient responses while the frequency responses are evaluated by implementing the harmonic balance method in conjunction with a numerical continuation technique. The results for the transverse harmonic load reveal that both the actuators are capable of attenuating the large amplitude vibration of the smart beams; however, the application of the shear/extensional mode actuator is mainly limited to a certain extent of the load-amplitude. Under the axial compressive harmonic load, the dynamics of the smart beams in the pre-buckled state appears through the parametric instabilities. However, once the buckling occurs, the smart beams undergo complex nonlinear dynamics due to the appearance of the snap-through periodic motion and the chaotic motion through the symmetry breaking, period-doubling/demultiplying and saddle-node bifurcations. The presented results mainly reveal the suitability of the shear mode and extensional mode piezoelectric actuators in active control of such dynamic instabilities and the associated complex nonlinear dynamics of the smart beams.
Read full abstract