Longitudinal vibration analysis of nanorods with multiple discontinuities based on nonlocal elasticity theory using wave approach
Discontinuities such as cracks and steps on the length, and the mass of attached buckyball on the tip, of nanoresonators have large effects on nanoresonators natural frequencies. In this paper, free longitudinal vibration of multiple cracked, stepped nanorods with a buckyball at tip is studied using the Eringen nonlocal elasticity theory. From wave viewpoint, vibrations can be considered as travelling waves along structures. Waves propagate in a waveguide and transmit and reflect at discontinuities. The propagation, transmission and reflection functions are derived for nanorods. Cracks and steps as discontinuities, and buckyball at the end, clamped end and free end for boundary conditions, are considered. Cracks are modelled by an infinitesimal length, massless linear spring. Steps are formed by connecting two nanorods with different cross section areas and mechanical properties and buckyballs are considered as lumped masses. These propagation, transmission and reflection functions are combined to provide a short comprehensive systematic analytical approach to analyze the free longitudinal vibration of nanorods. Also, explicit expressions for natural frequencies are derived. The effects of crack intensity, sectional change as step, mass of buckyball, location of crack and step and also small-scale effect on natural frequencies are discussed. The approach is explained using several examples and the closed-form solutions are derived for some cases. The results are compared with the existing methods and can be used as benchmark for other future works.
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