Damping-induced Localization Of Traveling And Standing Waves In Elastic Continua

Dynamic problems involving elastic continua coupled to local spring-damper attachments are often of importance for understanding the passive vibration confinement in structures.Mode complexity due to the presence of non-classical damping can reach a maximum via tuning of the stiffness and damping of the local attachment,such that localization of traveling and standing waves can be achieved in elastic continua.Therefore,the vibration energy can be effectively transferred through traveling wave transmission and localized into a specific structural domain of the elastic continua.The responses to harmonic excitation that exhibit complete separation of traveling and standing waves can be found in non-dispersive structures such as a linear elastic string with a local viscoelastic support.Generalization enables one to apply the wave localization strategy to transfer vibration energy preferentially in one-dimensional dispersive distributed parameter systems and typical two-dimensional elastic structures.Firstly,the free and forced vibration are examined in a finite linear string resting on a partial elastic foundation and attached to a local spring-damper,as well as a linear string resting on a viscoelastic foundation.The presence of the foundation renders the system dispersive.Analytical expressions are obtained for the free and forced response due to a harmonic excitation at one end,and are validated by numerical simulations.Free vibration of the string-foundation system shows eigenvalue loci veering and normal mode localization for weakly coupled subsystems under both symmetric and asymmetric placements.The forced vibration results show that co-existing regions of purely traveling and standing waves are still achievable by exploitation of mode complexity induced by non-classical damping.This wave transition phenomenon is influenced by the relative locations between the discrete spring-damper and the partial foundation and the difference between the excitation frequency and the cutoff frequency.When the former frequency is lower than the latter,wave transition cannot be attained for a string on an elastic foundation,but is possible if the string is on a viscoelastic foundation.The present example may have useful applications to the vibration control of structures interacting with soil foundations.Secondly,the wave separation mechanism in an Euler-Bernoulli beam is realized by a local spring-viscous damper pair connected in parallel through parameter tuning of the local attachment.An analytical framework is developed to examine the wave separation phenomena for four distinct boundary conditions at the right end: pinned,fixed,free and linear elastic.The sufficient and necessary conditions for wave separation are derived analytically.The results 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 caused by the non-oscillatory evanescent components in the solution.Vibration localization is achieved by satisfying necessary and sufficient wave separation conditions,which correspond to frequency-and position-dependent support stiffness and damping values.It is found that linear viscous damping in the interior support,but not necessarily linear stiffness,is required to achieve the separation phenomena and vibration localization.Thirdly,the transient finite element approach is employed to verify the phenomena of traveling wave transmission and vibration localization in one-dimensional,dispersive structural systems.The strong and weak forms for these problems are developed.The element stiffness,mass and damping matrices are derived from the variational approach,after assembly leading to a finite number of discrete equations.Along with the welldefined finite element equations and the appropriately constructed mesh,the equations are numerically solved and the corresponding results are visualized after post-processing.When the system reaches steady state,good correspondence can be observed between the analytical and FEM results.Finally,vibrations of a uniformly stretched circular membrane coupled to an internal concentric viscoelastic support are investigated for understanding the localization of traveling and standing waves in two-dimensional axisymmetric structures.Parameter values of the viscoelastic ring support are designed to separate the traveling and standing waves in the circular membrane under asymmetric excitation.At steady state,the analytical results are verified through finite element simulation using a commercial code.This work presents possibilities for new passive vibration control strategies in elastic continua via the mechanisms of separation of traveling and standing waves(one-way energy propagation)and vibration localization.The ability to channel input energy through traveling waves to a specific region of the structure may prove useful as a protective strategy.It also lays the foundation for effective designs of passive energy transfer and vibration energy harvesting method.