On nonlinear frequency veering and mode localization of a beam with geometric imperfection resting on elastic foundation

This work presents an investigation on the effect of an initial geometric imperfection wavelength, amplitude and degree of localization on the in-plane nonlinear natural frequencies veering and mode localization of an elastic Euler–Bernoulli beam resting on a Winkler elastic foundation. The beam is assumed to be pinned–pinned with a linear torsional spring at one end. The effect of the axial force induced by mid-plane stretching is accounted for in the derivation of the mathematical model, due to its known importance and significant effect on the nonlinear dynamic behavior of the beam, as it was proved and presented in earlier investigations. The governing partial differential equation is discretized using the assumed mode method and the resulting nonlinear temporal equation was solved using the harmonic balance method to obtain results for the nonlinear natural frequencies and mode shapes. The results are presented in the form of characteristic curves which show the variations of the nonlinear natural frequencies of the first three modes of vibration, for a selected range of physical parameters like; torsional spring constant, elastic foundation stiffness and amplitude and wavelength of a localized and non-localized initial slack.