Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory

The nonlinear free vibration of double-walled carbon nanotubes based on the nonlocal elasticity theory is studied in this paper. The nonlinear equations of motion of the double-walled carbon nanotubes are derived by using Euler beam theory and Hamilton principle, with considering the von Kármán type geometric nonlinearity and the nonlinear van der Waals forces. The surrounding elastic medium is formulated as the Winkler model. The harmonic balance method and Davidon–Fletcher–Powell method are utilized for the analysis and simulation of the nonlinear vibration. The simulation results show that the nonlocal parameter, aspect ratio and surrounding elastic medium play more important roles in the nonlinear noncoaxial vibration than those in the coaxial vibration of the double-walled carbon nanotubes. The noncoaxial vibration amplitudes of only considering nonlinear van der Waals forces are larger than those of considering both geometric nonlinearity and nonlinear van der Waals forces.     

Free vibration of embedded double-walled carbon nanotubes considering nonlinear interlayer van der Waals forces

In this article, nonlinear free vibration of embedded double-walled carbon nanotubes (DWCNTs) duo to the nonlinear interlayer van der Waals (vdW) force is studied based on the nonlocal Euler-Bernoulli beam theory. The interlayer vdW force is modeled as a nonlinear function of inner and outer tubes deflections considering the variation of the interlayer distance along the circumference of DWCNTs. The harmonic balance method is applied to analyze the relationship between the deflection amplitudes and the frequencies of in-phase and out-of-phase free vibrations for DWCNTs. Finally, the influences of the nonlocal parameter, surrounding elastic medium, nanotube length, end condition and vibrational mode on the nonlinear free vibration properties of DWCNTs are discussed in detail.     

Asymptotic behavior of solutions to a system of viscous conservation laws related to a phase transition problem

We study the asymptotic behavior of solutions for a system of viscous conservation laws with discontinuous initial data. We discuss mainly the case where the system without the viscosity term is of hyperbolic elliptic mixed type. This problem is related to a phase transition problem. We study the initial value problem and show the decay rates of solutions to piecewise constant states where two phases coexist. The modification necessary for the hyperbolic case is also discussed.     

Rigorous van der Waals effect on vibration characteristics of multi-walled carbon nanotubes under a transverse magnetic field

An analytical method is presented to investigate rigorous van der Waals interaction effect on vibration characteristics of multi-walled carbon nanotubes embedded in matrix under a transverse magnetic field. Each of the concentric tubes of multiwall carbon nanotubes is considered as an individual elastic shell and coupled with any two walls through a rigorous van der Waals interaction being dependent on the change of interlayer spacing and the radii of tubes. Results show that the rigorous van der Waals interaction effect makes the lowest magneto-vibration frequency of multi-walled carbon nanotubes decrease and the highest magneto-vibration frequency increase. The effect of rigorous van der Waals interaction on magneto-elastic vibrations of multi-walled carbon nanotubes is dependent on the transverse magnetic strength and the matrix constrained stiffness.     

Analysis of Mindlin micro plates with a modified couple stress theory and a meshless method

A modified couple stress theory and a meshless method is used to study the bending of simply supported micro isotropic plates according to the first-order shear deformation plate theory, also known as the Mindlin plate theory. The modified couples tress theory involves only one length scale parameter and thus simplifies the theory, since experimentally it is easier to determine the single scale parameter. The equations governing bending of the first-order shear deformation theory are implemented using a meshless method based on collocation with radial basis functions. The numerical method is easy to implement, and it provides accurate results that are in excellent agreement with the analytical solutions.     

Nonlinear dynamic analysis of a hybrid squeeze-film damper-mounted rigid rotor lubricated with couple stress fluid and active control

The hybrid squeeze-film damper bearing with active control is proposed in this paper and the lubricating with couple stress fluid is also taken into consideration. The pressure distribution and the dynamics of a rigid rotor supported by such bearing are studied. A PD (proportional-plus-derivative) controller is used to stabilize the rotor-bearing system. Numerical results show that, due to the nonlinear factors of oil film force, the trajectory of the rotor demonstrates a complex dynamics with rotational speed ratio s. Poincaré maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the rotor trajectory in the horizontal and vertical directions under different operating conditions. The maximum Lyapunov exponent and fractal dimension concepts are used to determine if the system is in a state of chaotic motion. Numerical results show that the maximum Lyapunov exponent of this system is positive and the dimension of the rotor trajectory is fractal at the non-dimensional speed ratio s = 3.0, which indicate that the rotor trajectory is chaotic under such operation condition. In order to avoid the nonsynchronous chaotic vibrations, an increased proportional gain is applied to control this system. It is shown that the rotor trajectory will leave chaotic motion to periodic motion in the steady state under control action. Besides, the rotor dynamic responses of the system will be more stable by using couple stress fluid.