Vibration confinement and localization patterns in weakly coupled oscillators with parameter gradients

Localization of vibration due to energy localized redistribution over a structure is defined by the presence of unusually large displacement at some specific locations of the structure. This article proposes a new understanding of the localization phenomena that occur in many different fields. These fields range from structural dynamics in lumped or continuous mechanical systems to the biomechanics of hearing. In this study, two kinds of turning point phenomena, are used as the fundamental blocs to explain different types of localization that were observed in various mechanical systems. A single system composed of coupled oscillators with parameter gradient is used to distinguish the influence of mass, stiffness and coupling on localization. It is shown that, depending on the parameters on which the gradient is applied as well as each parameter and gradient values, two kinds of turning points may occur. The turning points may combine to produce different patterns of localization in the system. These patterns are scrutinized in relation to the different localization phenomena with the literature. •A new understanding of normal mode localization.•A distinction between two kinds of turning points in lumped systems.•Tonotopic behavior in systems with parameter gradients.•Gradient amplitude over coupling ratio to monitor weak and strong localization.•Link between turning points and eigenvalues evolution convexity.