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Description
The complex dynamical properties of particulate granular systems are yet to be fully understood by the scientific community. The need for a deeper understanding of granular systems is made evident by their commonplace status in both the natural and industrial environments. The formation of patterns within systems of particles subject to transversely driven vibrations is one focus of research in recent years that enhances an understanding of the connection between granular systems and their surroundings. The research suggests it is plausible that particulates of a granular assembly can be segregated and positioned via waves imposed on a medium's surface. As a study in the non-contact management of granular particles, we investegate a vibrating membrane with variable tension. The model is motivated on grounds that it may broaden an understanding of particle transport through the analysis of the non-uniform membrane's vibrational characteristics. The equations of motion of the membrane are developed via the constitutive relations of an elastic membrane as set forth by the theory of partial differential equations. Interesting eigenfunctions satisfying the Helmholtz equation are derived for which their eigenvalues are used to analyze and manipulate the membrane's motion through a range of tensile forces. The membrane is subjected to external homogenous periodic forcing of frequency _ and when _ is the same as a natural frequency of the membrane then resonance occurs. At resonance, standing waves similar to the membrane's analytical eigenfunctions give rise to quiescent nodal curves. Particles distributed across the membrane's surface are assumed to accumulate towards and rest along the nodal curves. Variable tension is an attempt to control node/particle location. By adjusting the tension gradient along the flow axis, node location is shifted to either the left or the right for each setting of the tension gradient. Transient behavior of the wave modes under changes to the tensile force are revealed by analyzing the resonant modes spectral decomposition. It is observed that while the forcing frequency corresponding to resonant wave modes agrees with predicted frequency response values the wave structure is sensitive to changing tensile conditions while under forced oscillations.
