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This article presents methods for modeling, analysis, and design of metamaterial beams for broadband vibration absorption/isolation. The proposed metamaterial beam consists of a uniform isotropic beam with many small spring-mass-damper subsystems integrated at separated locations along the beam to act as vibration absorbers. For a unit cell of an infinite metamaterial beam, governing equations are derived using the extended Hamilton principle. The existence of stopband is demonstrated using a model based on averaging material properties over a cell length and a model based on finite element modeling and the BlochFloquet theory for periodic structures. However, these two idealized models cannot be used for finite beams and/or elastic waves having short wavelengths. For finite metamaterial beams, a linear finite element method is used for detailed modeling and analysis. Both translational and rotational absorbers are considered. Because results show that rotational absorbers are not efficient, only translational absorbers are recommended for practical designs. The concepts of negative effective mass and stiffness and how the spring-massdamper subsystems create a stopband (i.e., no elastic waves in this frequency range can propagate forward) are explained in detail. Numerical simulations reveal that the actual working mechanism of the proposed metamaterial beam is based on the concept of conventional mechanical vibration absorbers. It uses the incoming elastic wave in the beam to resonate the integrated spring-mass-damper absorbers to vibrate in their optical mode at frequencies close to but above their local resonance frequencies to create shear forces and bending moments to straighten the beam and stop the wave propagation. This concept can be easily extended to design a broadband absorber that works for elastic waves of short and long wavelengths. Numerical examples validate the concept and show that, for high-frequency waves, the structure's boundary conditions do not have significant influence on the absorbers' function. However, for absorption of low-frequency waves, the boundary conditions and resonant modes of the structure need to be considered in the design. With appropriate design calculations, finite discrete spring-mass-damper absorbers can be used, and hence expensive micro-or nanomanufacturing techniques are not needed for design and manufacturing of such metamaterial beams for broadband vibration absorption/isolation.

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a b s t r a c tA new Modified Displacement Component (MDC) method is proposed to accurately predict wrinkling characteristics in the membrane by eliminating the singularity of the displacement solution. In MDC method, a singular displacement component is primarily obtained at the wrinkling point by introducing the first-order characteristic vector multiplied by a positive intermediate parameter in the singular stiffness matrix. The non-singularity displacement solution is then obtained by modifying the singular displacement component based on three equality relationships at the wrinkling point. Where, the accurate introduction and the timely removal of the critical wrinkling mode are two key steps. In our simulation, we use a direct perturbed method to accurately consider these two key steps. In the direct perturbed method, some small, quantitative, out-of-plane forces are applied onto the membrane surface directly based on the first wrinkling mode, and then removed immediately after wrinkling starts. Several effective strategies are then used to advance the convergence. A wrinkling test using photogrammetry is used to verify the validity of our method. In addition, we also studied the secondary wrinkling characteristics occurred in the post-wrinkling phase in the end. The secondary wrinkling characteristics are the basic explanations of the wrinkling expansion and evolution in the post-wrinkling phase.

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