Transverse Dynamic Responses and Optimization of a Flexibly Constrained Beam with Multiple Nonlinear Supports that Present Cubic Stiffness

Abstract

This paper establishes a simplified model of a flexible beam with multiple nonlinear supports that present cubic stiffness to study the potential application of cubic nonlinearities in the vibration control of beams. The Lagrangian method (LM) is used to predict transverse dynamic responses of the flexible beam with multiple nonlinear supports that present cubic stiffness, whereas the harmonic balance method (HBM) and Galerkin truncation method (GTM) are utilized to study the accuracy of the LM. Against the background, the effect of nonlinear supports that present cubic stiffness on the nonlinear transverse dynamic responses of the beam is studied. For this study, the accuracy of the LM is guaranteed under the 4-term truncation number. Nonlinear transverse dynamic responses of the flexibly constrained beam with multiple nonlinear supports that present cubic stiffness are sensitively influenced by their initial calculation values. For different boundary conditions or working statuses, the maximum restoring forces at both ends of the flexible beam can be suppressed effectively by optimizing the parameters of nonlinear supports that present cubic stiffness. Appropriate parameters of nonlinear supports that present cubic stiffness are good at vibration control of the flexibly constrained beam. In addition, complicated dynamic responses of the flexible beam with multiple nonlinear supports that present cubic stiffness are motivated under some inappropriate parameters of nonlinear supports.