Chaotic vibrations of carbon nanotubes subjected to a traversing force considering nonlocal elasticity theory

The existence of chaos in the lateral vibration of the carbon nanotube (CNT) can contribute to source of instability and inaccuracy within the nano mechanical systems. So, chaotic vibrations of a simply supported CNT which is subjected to a traversing harmonic force are studied in this paper. The model of the system is formulated by using nonlocal Euler–Bernoulli beam theory. The equation of motion is solved using the Rung–Kutta method. The effects of the nonlocal parameter, velocity and amplitude of the traversing harmonic force on the nonlinear dynamic response of the system are analyzed by the bifurcation diagrams, phase plane portrait, power spectra analysis, Poincaré map and the maximum Lyapunov exponent. The results indicate that the nonlocal parameter, velocity and amplitude of the traversing harmonic force have considerable effects on the bifurcation behavior and can be used as effective control parameters for avoiding chaos.

Nonlinear and Dual-Parameter Chaotic Vibrations of Lumped Parameter Model in Blisk under Combined Aerodynamic Force and Varying Rotating Speed

In this paper, the nonlinear and dual-parameter chaotic vibrations are investigated for the blisk structure with the lumped parameter model under combined the aerodynamic force and varying rotating speed. The varying rotating speed and aerodynamic force are respectively simplified to the parametric and external excitations. The nonlinear governing equations of motion for the rotating blisk are established by using Hamilton’s principle. The free vibration and mode localization phenomena are studied for the tuning and mistuning blisks. Due to the mistuning, the periodic characteristics of the blisk structure are destroyed and uniform distribution of the energy is broken. It is found that there is a positive correlation between the mistuning variable and mode localization factor to exhibit the large vibration of the blisk in a certain region. The method of multiple scales is applied to derive four-dimensional averaged equations of the blisk under 1:1 internal and principal parametric resonances. The amplitude-frequency response curves of the blisk are studied, which illustrate the influence of different parameters on the bandwidth and vibration amplitudes of the blisk. Lyapunov exponent, bifurcation diagrams, phase portraits, waveforms and Poincare maps are depicted. The dual-parameter Lyapunov exponents and bifurcation diagrams of the blisk reveal the paths leading to the chaos. The influences of different parameters on the bifurcation and chaotic vibrations are analyzed. The numerical results demonstrate that the parametric and external excitations, rotating speed and damping determine the occurrence of the chaotic vibrations and paths leading to the chaotic vibrations in the blisk.

Polyharmonic Vibrations of Human Middle Ear Implanted by Means of Nonlinear Coupler

This paper presents a possibility of quasi-periodic and chaotic vibrations in the human middle ear stimulated by an implant, which is fixed to the incus by means of a nonlinear coupler. The coupler represents a classical element made of titanium and shape memory alloy. A five-degrees-of-freedom model of lumped masses is used to represent the implanted middle ear for both normal and pathological ears. The model is engaged to numerically find the influence of the nonlinear coupler on stapes and implant dynamics. As a result, regions of parameters regarding the quasi-periodic, polyharmonic and irregular motion are identified as new contributions in ear bio-mechanics. The nonlinear coupler causes irregular motion, which is undesired for the middle ear. However, the use of the stiff coupler also ensures regular vibrations of the stapes for higher frequencies. As a consequence, the utility of the nonlinear coupler is proven.

Suppression of chaotic vibrations in nonlinear systems by the example of a mechanical tachometer

A method for controlling dynamic chaos is proposed by introducing state feedback and changing the spectrum of Lyapunov characteristic parameters of a closed system to achieve the desired result - the transition from chaotic mode to regular motion. The solution of this problem is considered on the example of stabilization of a mechanical tachometer. The parameters of the controller in the feed-back circuit are determined by the method of modal con-trol synthesis.

Controlling Chaotic Vibrations of a Quarter-Car Model Excited by Road Surface Profile using Parametric Excitation.

Chaotic Vibrations are considered for a quarter-car model excited by the road surface profile. The equation of motion is obtained in the form of a classical Duffing equation and it is modeled with deliberate introduction of parametric excitation force term to enable us manipulate the behavior of the system. The equation of motion is solved using the Method of Multiple Scales. The steady-state solutions with and without the parametric excitation force term is investigated using NDSolve MathematicaTM Code and the nonlinear dynamical system’s analysis is by a study of the Bifurcations that are observed from the analysis of the trajectories, and the calculation of the Lyapunov. In making the system more strongly nonlinear the excitation amplitude value is artificially increased to various multiples of the actual value. Results show that the system’s response can be extremely sensitive to changes in the amplitude and the that chaos is evident as the system is made more nonlinear and that with the introduction of parametric excitation force term the system’s motion becomes periodic resulting in the elimination of chaos and the reduction in amplitude of vibration.