Generation of quasi-traveling waves in a finite rectangular membrane with two internal viscoelastic line supports

Abstract

In this work we study the forced vibration of a finite rectangular membrane driven by arbitrarily distributed boundary motion on two adjacent edges and internally coupled with multiple viscoelastic line supports, as an extension of the analysis of coexisting vibrations and waves in one-dimensional elastic strings and ducts. Using the static solution for the membrane displacement in the absence of interior supports, the response of the nonclassically damped system is obtained through a normal-mode expansion. A traveling-wave index based on complex orthogonal decomposition is utilized to identify and optimize the traveling waves in the rectangular membrane. A vibration localization condition in a semi-infinite membrane is analytically derived, from which the complex boundary excitations and constraints in a finite membrane are specified to effectively realize pure traveling waves and confine the vibration to a restricted area. Then the membrane is edge-driven by a simpler time-periodic, spatially sinusoidal excitation for practical realization, with the line-support parameters optimized using the traveling-wave index function and a derivative-free optimization routine with lower-bound constraints. Analytical results show that quasi-traveling waves are produced for ranges of line-support parameters (location, stiffness and damping) following optimization using the traveling-wave index. The developed wave patterns are examined with a power-flow analysis and numerically verified through finite element analysis. Wave localization is demonstrated to be valid in a finite membrane with specific boundary conditions and multiple viscoelastic line supports. The proposed method shows the predictive ability of the orthogonally stiffened membrane configuration as the basic unit cell for the design of periodic structures with directional wave propagation as well as vibration localization dynamics.