Nonlinear vibrations and control of PFG-GR laminated composite cantilever rectangular variable cross-section plate with NPPF controller

Abstract

This paper studies the nonlinear vibrations and control of the piezoelectric functionally graded graphene-reinforced laminated composite cantilever rectangular (PFG-GRLCCR) variable cross-section plate with the NPPF controller under the simultaneous resonant cases. The nonlinear dynamic governing equations of motion for the PFG-GRLCCR variable cross-section plate are obtained by using the extended Halpin-Tsai mode, linear constitutive equations of the piezoelectric materials, classical laminated plate theory, von Karman large deformation theory, and Hamilton principle. Considering Galerkin method, the nonlinear ordinary differential governing equations are obtained. The nonlinear positive position feedback (NPPF) controllers are designed according to the similarity principle. The multiple scale method is used to investigate the nonlinear vibration characteristics of the stable motion for the PFG-GRLCCR variable cross-section plate with the NPPF controllers. The amplitude-frequency response curves, force-amplitude response curves, time histories, phase portraits, and bifurcation diagrams are given to investigate the nonlinear dynamic characteristics of the PFG-GRLCCR variable cross-section plate with the NPPF controller. The results demonstrate that the NPPF controller can suppress the nonlinear vibrations of the structure well, and the effectiveness of the controllers is best when the NPPF controller has a similar nonlinear characteristic with the structure. The complex nonlinear vibration characteristics appear for the PFG-GRLCCR variable cross-section plate with the NPPF controller.