Co-existing complexity-induced traveling wave transmission and vibration localization in Euler-Bernoulli beams

We analytically and numerically examine the dynamic behavior of an undamped, linear, uniform and homogeneous Euler-Bernoulli beam of finite length, partially supported in its interior by local, grounded, linear spring-dashpot pairs and subjected to a harmonic displacement at its pinned left boundary or at both its ends. The Euler-Bernoulli beam is known to be dispersive and, thus, to exhibit a non-constant relationship between frequency and wave number. Local dissipation due to the interior support results in a non-classically damped system and, consequently, mode complexity. An analytical framework is developed to examine the coexistence of propagating and standing harmonic waves in complementary regions of the beam for four distinct boundary conditions at the right end: pinned, fixed, free and linear elastic. We show that the system can be designed so that, for a particular input frequency and interior support location, nearly perfect spatial separation of traveling and standing waves can be achieved; the imperfection is shown to be caused by the non-oscillatory evanescent components in the solution. We further demonstrate that vibration localization is achieved by satisfying necessary and sufficient wave separation conditions, which correspond to frequency- and position-dependent support stiffness and damping values, and that linear viscous damping in the interior support, but not necessarily linear stiffness, is required to achieve the separation phenomenon and vibration localization.