Abstract
We use the incremental harmonic balance (IHB) method to analyse the dynamic stability problem of a nonlinear multiple-nanobeam system (MNBS) within the framework of Eringen’s nonlocal elasticity theory. The nonlinear dynamic system under consideration includes MNBS embedded in a viscoelastic medium as clamped chain system, where every nanobeam in the system is subjected to time-dependent axial loads. By assuming the von Karman type of geometric nonlinearity, a system of m nonlinear partial differential equations of motion is derived based on the Euler–Bernoulli beam theory and D’ Alembert’s principle. All nanobeams in MNBS are considered with simply supported boundary conditions. Semi-analytical solutions for time response functions of the nonlinear MNBS are obtained by using the single-mode Galerkin discretization and IHB method, which are then validated by using the numerical integration method. Moreover, Floquet theory is employed to determine the stability of obtained periodic solutions for different configurations of the nonlinear MNBS. Using the IHB method, we obtain an incremental relationship with the frequency and amplitude of time-varying axial load, which defines stability boundaries. Numerical examples show the effects of different physical and material parameters such as the nonlocal parameter, stiffness of viscoelastic medium and number of nanobeams on Floquet multipliers, instability regions and nonlinear amplitude–frequency response curves of MNBS. The presented results can be useful as a first step in the study and design of complex micro/nanoelectromechanical systems.
Highlights
Most dynamic stability studies in the literature are considering single or double-micro/nanostructure-based systems [1,2,3,4]
We introduce the incremental harmonic balance (IHB) method to determine the periodic solutions of the presented system of m nonlinear ordinary differential equations, instability regions of the nonlinear multiple-nanobeam system (MNBS) and Floquet multipliers
The present analysis describes the dynamic stability problem of the nonlinear MNBS embedded in a viscoelastic medium, where each nanobeam in the system is supported and subjected to axial time-dependent loads
Summary
Most dynamic stability studies in the literature are considering single or double-micro/nanostructure-based systems [1,2,3,4]. Murmu and Adhikari conducted detailed vibration studies of a double-nanorod, nanobeam, and nanoplate systems [20,21,22,23], where partial differential equations of motion are obtained based on the D’ Alembert’s principle and nonlocal elasticity theory and solved by using analytical methods They investigated the influence of small-scale effects and other physical parameters on natural frequencies and critical buckling loads and compared analytical results with results obtained by molecular dynamics simulations. [28] presented a straightforward method to obtain analytical solutions for natural frequencies and critical buckling loads of multiple nanorods, nanobeams, and nanoplates systems based on the Eringen’s nonlocal elasticity theory and trigonometric method To this time, the mechanical behaviour of different nanostructures such as the longitudinal vibration of nanorods [29], transverse vibration.
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