This paper investigates the Nonlinear-coupled radial-axial vibration of single-walled carbon nanotubes (SWCNTs) based on numerical methods. Two coupled partial differential equations that govern the Nonlinearcoupled radial-axial vibration for such nanotube are derived using nonlocal doublet mechanics (DM) theory. To obtain the Nonlinear natural frequencies in coupled radial-axial vibration mode, these equations are solved using Homotopic perturbation method (HPM). It is found that the coupled radial-axial vibrational frequencies are complicated due to coupling between two vibration modes. The infl uences of some commonly used boundary conditions, changes in vibration modes and variations of the nanotubes geometrical parameters on the Nonlinear-coupled radial-axial vibration characteristics of SWCNTs are discussed. It was shown that boundary conditions and maximum vibration velocity play significant roles in the Nonlinear-coupled radial-axial vibration response of SWCNTs. It was shown that unlike the linear one, the Nonlinear natural frequencies are dependent to maximum vibration velocity. Increasing the maximum vibration velocity increases the natural frequency of vibration compared to the prediction of the linear model. However, with increase in tube length, the effect of the maximum vibration velocity on the natural frequencies decreases. It was also shown that the amount and variation of Nonlinear natural frequencies are more apparent in higher vibration modes and two clamped boundary conditions. To show the accuracy and capability of this method, the results obtained herein are compared with the fourth order Runge-Kuta numerical results and also with the other available results and good agreement is observed. It is notable that the results generated herein are new and can be served as a benchmark for future works.

This paper investigates the coupled axial-radial (CAR) vibration of single-walled carbon nanotubes (SWCNTs) based on doublet mechanics (DM) with a scale parameter. Two coupled forth order partial differential equations that govern the CAR vibration of SWCNTs are derived. It is the first time that DM is used to model the CAR vibration of SWCNTs. To obtain the natural frequency and dynamic response of the CAR vibration, the equations of motion are solved and the relation between natural frequencies and scale parameter is derived. It is found that there are two frequencies in the frequency spectrum and these CAR vibrational frequencies are complicated due to coupling between two vibration modes. The advantage of these analytical formulas is that they are explicitly dependent to scale parameter and chirality effect. The influence of changing some geometrical and mechanical parameters of SWCNT on its CAR frequencies has been investigated, too. It is shown that the chirality and scale parameter play significant role in the CAR vibration response of SWCNTs. The scale parameter decreases the higher band CAR frequency compared to the predictions of the classical continuum models. However, with increase in tube radius and length, the effect of the scale parameter on the natural frequencies decreases. The lower band CAR frequency is nearly independent to scale effect and tube diameter. The CAR frequencies of SWCNTs decrease as the length of the tube increases. This decreasing is higher for higher band CAR frequency. To show the accuracy and ability of this method, the results obtained herein are compared with the existing theoretical and experimental results and good agreement is observed.

This paper presents a fixed-end two-mass system (TMS) with end constraints that permits uncoupled solutions for different masses. The coupled Nonlinear models for the present fixed-end TMS were solved using the continuous piecewise linearization method (CPLM) and detailed investigation on the effect of mass-ratio on the TMS response was conducted. The investigations showed that increased mass-ratio leads to decreased oscillation frequency and an asymptotic response was obtained at very large mass-ratios. Theoretical solutions to determine the asymptotic response were derived. Also, it was observed that distinct responses can be obtained for the same mass-ratio depending on the mass combination in the TMS. The present fixed-end TMS and the analyses presented give a broader understanding of fixed-end TMS.

An analytical solution for a sandwich circular FGM plate coupled with piezoelectric layers under one-dimension heat conduction is presented in this paper. A Nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations. By adding an incremental dynamic state to the pre-vibration state, the differential equations are derived. The role of thermal environment and control effects on Nonlinear static deflections and natural frequencies imposed by the piezoelectric actuators using high input voltages are investigated. The good agreement between the results of this paper and those of the finite element (FE) analyses validated the presented approach. The emphasis is placed on investigating the effect of varying the applied actuator voltage and thermal environment as well as gradient index of FG plate on the dynamics and control characteristics of the structure.

In this paper, Homotopy Analysis Method is used to analyze free non-linear vibrations of a beam simply supported by pinned ends under axial force. Mid-plane stretching is also considered for dynamic equation extracted for the beam. Galerkin decomposition technique is used to transform a partial dimensionless Nonlinear differential equation of dynamic motion into an ordinary Nonlinear differential equation. Then Homotopy Analysis Method is employed to obtain an analytic expression for Nonlinear natural frequencies. Effects of design parameters including axial force and slenderness ratio on Nonlinear natural frequencies are studied. Moreover, effects of factors of Nonlinear terms on the general shape of the time response are taken into account. Combined Homotopy-Pade technique is used to reduce the number of approximation orders without affecting final accuracy. The results indicate increased speed of convergence as Homotopy and Pade are combined. The obtained analytic expressions can be used for a vast range of data. Comparison of the results with numerical data indicated a good conformance. Having compared accuracy of this method with that of the Homotopy perturbation analytic method, it is concluded that Homotopy Analysis Method is a very strong method for analytic and vibration analysis of structures.

The aim of this article is to analyze Nonlinear electro-magneto vibration of a double-piezoelectric composite microplate-system (DPCMPS) pursuant to the nonlocal piezoelasticity theory. The two microplates are assumed to be connected by an enclosing elastic medium, which is simulated by the Pasternak foundation. Both of piezoelectric composite microplates are made of poly-vinylidene fluoride (PVDF) reinforced by agglomerated carbon nanotubes (CNTs). The Mori-Tanaka model is employed to compute the mechanical properties of composite. Applying Nonlinear strain-displacement relations and contemplating charge equation for coupling between electrical and mechanical fields, the motion equations are derived in consonance to the energy method and Hamilton's principle. These equations can't be solved analytically as a result of their Nonlinear terms. Hence, the differential quadrature method (DQM) is employed to solve the governing differential equations for the case when all four ends are clamped supported and free electrical boundary conditions. The frequency ratio of DPCMPS is inspected for three typical vibrational states, namely, out-of-phase, in-phase and the case when one microplate is fixed in the DPCMPS. A detailed parametric study is conducted to scrutinize the influences of the small scale coefficient, stiffness of the internal elastic medium, the volume fraction of the CNTs, agglomeration and magnetic field. The results reveal that with increasing volume fraction of the CNTs, the frequency of the structure increases. This study might be beneficial for the design and smart control of nano/micro devices such as MEMS and NEMS.

We investigate the existence of non-radial positive normalized solutions to coupled fractional Nonlinear Schrödinger systems characterized by competing Nonlinearities and subject to multiple \( L^2 \) norm constraints. Considering a local minimization strategy within specially constructed symmetric function spaces and applying the concentration-compactness principle, we demonstrate the existence of multiple non-radial solutions that exhibit symmetry breaking relative to the radial symmetry of the external potential. Additionally, we conduct an asymptotic analysis as the semiclassical parameter \( \varepsilon \) approaches zero, revealing that the solutions localize around multiple distinct points where the potential attains its maximum values. These concentration points are arranged according to the symmetry imposed by a finite group of orthogonal transformations, leading to the formation of multi-bump profiles.