Wave Analysis of Mode Localization and Delocalization in Elastically Constrained Strings and Beams

The free vibration response of both a string and a Euler-Bernoulli beam supported by intermediate elastic constraints is studied and analyzed. For both the string and beam systems, curve veering and mode localization are observed in the lower modes when the distance between the elastic constraints is varied. As the mode number increases, the modes of the system become extended indicating that the coupling springs have little effect on the systems at higher modes. A wave analysis is employed to show the effects of the constraints on the coupling of the subsystems and high frequency behavior. The beam may exhibit a delocalization phenomenon where a particular mode experiences no localization while other neighboring modes may be localized. The frequency (termed the delocalization frequency) at which this occurs corresponds to a transmission resonance. The delocalization frequency is predicted well by the vibration ratio (Langley, 1995). The existence and behavior of the delocalization are explained analytically by the wave approach.