Vibration reduction by the use of beam-type metastructures with nonlinear attachments

Özceylan, Bekir Mert
Metamaterials are artificial materials engineered to have special properties which are not possessed by natural materials. Metastructures are metamaterial-based finite structures with specified boundary conditions coupled with an array of resonators. Due to their resonant nature, metastructures can have bandgaps at very long wavelengths compared to their lattice size. Previous studies have shown that this phenomenon arises as dynamic effective mass becomes negative around a local resonance frequency interval which is called as stopband region. If excitation frequency is within this range, due to the negative dynamic effective mass of the structure, local resonators accelerate in the reverse direction of the forcing and start to create opposing shear forces to straighten the metastructure beam. Therefore, locally resonant metastructures can be utilized to attenuate vibrations at low frequencies. In the literature, several types of base structures such as bars, beams, membranes, plates, etc. were examined from the vibration suppression perspective by using linear models. However, beam-type metastructure with nonlinear attachments is investigated very little. In this dissertation, vibration absorption capability of a metastructure beam with nonlinear inserts is investigated. First, bandgap analysis is performed on the single cell of an infinitely long structure by using Bloch-Floquet Theory. Linear and nonlinear cells endowed with one and two resonators are studied separately. Once the behavior of the single cell is analyzed, then studies are extended on the finite metastructure beam with nonlinear attachments. Next, examinations are focused on the fundamental mode vibration mitigation performance of a mass-conserved metastructure beam. Conserved-mass approach implies that mass dedicated to resonators are reduced from the mass of the base beam. Thus, improvement in vibration reduction capability is studied without adding any additional mass. Mathematical model of the structure is constructed by using finite element discretization, and structural matrices are derived based on assumptions of Euler-Bernoulli Beam Theory. Initial studies are performed on mass-conserved linear metastructure to identify the influence of various parameters on the vibration mitigation performance quantified by H2 and H∞ norms. Then, optimization studies are carried on linear metastructure to obtain maximum attainable performance later to compare with nonlinear metastructure performance. Next, vibration absorption performance of the nonlinear metastructure is investigated for two types of nonlinear elements -cubic stiffness and dry friction-. Describing Function Method (DFM) is utilized to represent the nonlinear differential equations of motion as a set of nonlinear algebraic equations which is solved by using arc-length continuation method. Modal Superposition Method (MSM) in which the nonlinear response is written in terms of the mode shapes of the linear system is employed to decrease the number of nonlinear equations to be solved. Finally, parameter and optimization studies are performed on nonlinear metastructure, and results are compared with linear metastructure and equally weighted base beam without any attachments.


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Citation Formats
B. M. Özceylan, “Vibration reduction by the use of beam-type metastructures with nonlinear attachments,” M.S. - Master of Science, Middle East Technical University, 2022.