Vibrational excitations in graded elastic chains

We study vibrational excitations in linearly graded materials and/or systems. Graded systems demonstrate a unique spectrum and mode profiles, resulting in a new type of localization-delocalization transition. The nature of gradients can confine certain vibrational excitations, and redistribute them spatially. These features are contrasted to the two extreme cases of inhomogeneous system, i.e., periodically modulated system and randomly disordered system, We show in detail vibrational normal modes sustained by one dimensional graded force constant and graded mass networks, in particular, a unusual kind of modes called gradons. We propose an approach to study vibrational modes in a grades elastic system with the help of a series of homogeneous systems. Using this approach, we elaborate the features of the elastic gradons and the phonon-gradon transition.