Active control and forced vibrations of a smart sandwich viscoelastic micro beam resting on Vlasov’s foundation based on MCST

In this research, active control and forced vibrations of a smart sandwich micro beam with viscoelastic core and piezoelectric face sheets are investigated; in which the top and bottom layers are considered as actuator and sensor, respectively. Displacement fields and governing equations are obtained based on Timoshenko’s beam theory and modified couple stress theory (MCST). The micro sandwich is rested on Vlasov’s foundation and a harmonic force is applied to the structure. For optimal vibration control, the equations of motion are solved in state-space form using the linear quadratic regulator (LQR) approach. The effect of viscoelastic parameter, and LQR on vibration response is illustrated. Moreover, the influence of viscoelastic parameter, loss factor, Vlasov’s foundation, and length to thickness ratio on the frequency response, natural frequency, and three natural frequency modes of a smart sandwich micro beam is analyzed. The results are showed that the Vlasove’s foundation increases loss factor and natural frequency, especially for lower length to thickness ratios. For length to thickness ratios 2 and 10, the natural frequency of second mode increases by about 3.4 and 2.8 times relative to the natural frequency of first mode, respectively, while this value is 7.4 and 5 times for the natural frequency of third mode, but the length to thickness ratio has no significant effect on the loss factor. Also, LQR is significantly suppressed the amplitude of vibration for a smart sandwich micro beam.