Transition from normal to local modes in an elastic beam supported by nonlinear springs

The transient mode localization phenomenon is considered in a mechanical model combining a simply supported beam and transverse nonlinear springs with hardening characteristics. Two different approaches to the model reduction, such as normal and local mode representations for the beamʹs center line, are discussed. It is concluded that the local mode discretization brings advantages for the transient localization analysis. Based on the specific coordinate transformations and the idea of averaging, explicit equations describing the energy exchange between the local modes and the corresponding localization conditions are obtained. It was shown that when the energy is slowly pumped into the system then, at some point, the energy equipartition around the system suddenly breaks and one of the local modes becomes the dominant energy receiver. The phenomenon is interpreted in terms of the related phase-plane diagram which shows qualitative changes near the image of the out-of-phase mode as the total energy of the system has reached its critical level. A simple closed form expression is obtained for the corresponding critical time estimate.