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.

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.

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.

Analyzing the nonlinear vibration of beams is one of the important issues in structural engineering. According to this, an impressive analytical method which is called modified iteration perturbation method (MIPM) is used to obtain the behavior and frequency of a cantilever beam with geometric nonlinear. This new method is combined by the Mickens and iteration methods. Moreover, this method does not require small parameter in the equation which is difficult to be found for nonlinear oscillation. The accuracy of the solution that is obtained by the use of MIPM has been shown graphically and compared with exact solution. Comparison shows that good adaptation is obtained and MIPM as a powerful method for solving the vibrational behavior of structures analytically.

The current article is concerned with a comprehensive investigation in achieving approximate solutions of Coupled nonlinear oscillations with high nonlinearity. These equations are highly nonlinear second-order ordinary differential equations. Via a coupling of the Homotopy perturbation method and Laplace transforms, which is so-called the He-Laplace method, traditional approximate solutions involving the secular terms are accomplished. On the other hand, in order to cancel the secular terms, an expanded frequency technique is adapted to accomplish periodic approximate solutions. Therefore, a nonlinear frequency, for each differential equation, is achieved. Furthermore, for more convenience, these solutions are pictured to indicate their behavior. The multiple time-scales with the aid of the Homotopy concept are utilized to judge the stability criteria. The analyses reveal the resonance as well as the non-resonant cases. Additionally, numerical calculations are carried out, graphically, to address the regions that guaranteed the bounded solutions. It is found that the latter method, is the most powerful mathematical tool in extracting the stability analysis of the considered system.