Exact Nonlinear Dynamic Analysis of a Beam With a Nonlinear Vibration Absorber and With Various Boundary Conditions

We study the nonlinear vibration of a beam with an attached grounded and ungrounded nonlinear vibration absorber (NVA) using the exact natural frequencies and mode shapes of the loaded beam. The nonlinearity in the beam is due to midplane stretching and that in the NVA is of cubic stiffness nonlinearity. We consider various boundary conditions and derive their closed-form characteristic equations and mode shapes. The method of multiple scales (MMS) is directly applied to the nonlinear partial differential equations of motion to obtain explicit expressions of the nonlinear frequency, modulation, and loci of the saddle-node bifurcation equations. Our analytical approach is validated using direct numerical simulation. Parametric studies demonstrate that the performance of the NVA does not only depend on its key design variables and location, but also on the boundary conditions, midplane stretching of the beam, and type of configuration (i.e., grounded NVA versus ungrounded NVA). Our analysis also indicates that the use of common approach such as employing approximate modes in estimating the nonlinear response of a loaded beam produces significant error (i.e., up to 1200% in some case). These observations suggest that the exact modes shape and natural frequencies are required for a precise investigation of the nonlinear dynamic of loaded beams. These findings could contribute to the design improvement of NVAs, microelectromechanical systems (MEMS), energy harvesters, and metastructures.