Nonlinear dynamics of parametrically excited struts under harmonic axial loading

Nonlinear vibrations of beams and struts under parametric resonance are critically important phenomena for applications such as energy harvesting and vibration suppression systems. In such applications, the sudden changes in the mechanical response that are triggered by buckling instabilities can be harnessed to enhance performance. The present study investigates the nonlinear vibrational behaviour of simply-supported parametrically excited slender elastic struts under a harmonically-varying axial load. The strut is modelled as inextensible and by employing the extended Hamilton's principle, the governing differential equations of motion for lateral vibrations are derived. Galerkin’s method is subsequently used to simplify the analysis by focusing the discussion of the strut dynamics on its first vibration mode. The steady-state nonlinear dynamics are explored through numerical continuation of periodic solutions of the response amplitude. Frequency response functions are presented; bifurcations, including folds of cycles, and their loci are analysed through two- and three- parameter continuation. A parametric study reveals that these loci of bifurcations demarcate the parameter space into regions associated with distinct, classifiable responses. The results are verified numerically against dynamic solutions obtained from a finite element model developed in the commercial software ABAQUS and good agreement is found. The findings demonstrate that the proposed model offers a fast and accurate approach for analysing the nonlinear dynamics of parametrically excited buckled struts, providing valuable insights that would be useful for the aforementioned practical applications.