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J Mech Phys Solids. 2018 Mar;112:187-208. doi: 10.1016/j.jmps.2017.12.002. Epub 2017 Dec 8.

Vibration of Mechanically-Assembled 3D Microstructures Formed by Compressive Buckling.

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Departments of Civil and Environmental Engineering, Mechanical Engineering, and Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA.
Department of Materials Science and Engineering, Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China.
Center for Mechanics and Materials and Center for Flexible Electronics Technology, Tsinghua University, Beijing 100084, China.
Man-machine-Environment Engineering Institute, Department of Aeronautics & Astronautics Engineering, Tsinghua University, Beijing 100084, China.
Departments of Materials Science and Engineering, Biomedical Engineering, Chemistry, Mechanical Engineering, Electrical Engineering and Computer Science, Neurological Surgery, Center for Bio-Integrated Electronics, Simpson Querrey Institute for BioNanotechnology, McCormick School of Engineering and Feinberg School of Medicine, Northwestern University, Evanston, Illinois 60208, USA.


Micro-electromechanical systems (MEMS) that rely on structural vibrations have many important applications, ranging from oscillators and actuators, to energy harvesters and vehicles for measurement of mechanical properties. Conventional MEMS, however, mostly utilize two-dimensional (2D) vibrational modes, thereby imposing certain limitations that are not present in 3D designs (e.g., multi-directional energy harvesting). 3D vibrational microplatforms assembled through the techniques of controlled compressive buckling are promising because of their complex 3D architectures and the ability to tune their vibrational behaviour (e.g., natural frequencies and modes) by reversibly changing their dimensions by deforming their soft, elastomeric substrates. A clear understanding of such strain-dependent vibration behaviour is essential for their practical applications. Here, we present a study on the linear and nonlinear vibration of such 3D mesostructures through analytical modeling, finite element analysis (FEA) and experiment. An analytical solution is obtained for the vibration mode and linear natural frequency of a buckled ribbon, indicating a mode change as the static deflection amplitude increases. The model also yields a scaling law for linear natural frequency that can be extended to general, complex 3D geometries, as validated by FEA and experiment. In the regime of nonlinear vibration, FEA suggests that an increase of amplitude of external loading represents an effective means to enhance the bandwidth. The results also uncover a reduced nonlinearity of vibration as the static deflection amplitude of the 3D structures increases. The developed analytical model can be used in the development of new 3D vibrational microplatforms, for example, to enable simultaneous measurement of diverse mechanical properties (density, modulus, viscosity etc.) of thin films and biomaterials.

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