Solitons and other solutions to a new coupled nonlinear Schrodinger type equation

In this paper, the first integral method combined with Liu's theorem is applied to integrate a new coupled nonlinear Schrodinger type equation. Using this combination, more new exact traveling wave solutions are obtained for the considered equation using ideas from the theory of commutative algebra. In addition, more solutions are also obtained via the application of semi-inverse variational principle due to Ji-Huan He. The used approaches with the help of symbolic computations via Mathematica 9, may provide a straightforward effective and powerful mathematical tools for solving nonlinear partial differential equations in mathematical physics.     

Vibration suppression of a dynamical system to multi-parametric excitations via time-delay absorber

The aim of this work is to control the dynamic system behavior represented by a beam at simultaneous primary and sub-harmonic resonance condition, where the system damage is probable. Control is conducted via time delay absorber to suppress chaotic vibrations. A comprehensive investigation of the effect of the time delay on the control of a beam when subjected to multi- parametric excitation forces is presented. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. Stability of the steady state solution for the selected resonance case is investigated applying Rung-Kutta fourth order method and frequency response equations via Matlab 7.0 and Maple11. Time delay absorber is effective like ordinary one within a specified range of time delay. The delay time is an important factor in selecting the absorber. The effects of the different parameters of the absorber on the system behavior are studied numerically. The reported results are compared with the available published work.     

Vibration and stability of an aircraft tail under simultaneous primary-combined and internal resonance

This paper adds a negative velocity feedback to the dynamical system of twin-tail aircraft to suppress the vibration. The system is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to both multi-harmonic and multi-tuned excitations. The method of multiple time scale perturbation is adopted to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the proposed analytic solution near the simultaneous primary, combined and internal resonance is studied and its conditions are determined. The effect of different parameters on the steady state response of the vibrating system is studied and discussed by using frequency response equations. Some different resonance cases are investigated numerically     

Vibration suppression in multi-tool ultrasonic machining to multi-external and parametric excitations

Ultrasonic machining （USM） is of particular interest for the machining of non-conductive, brittle materials such as engineering ceramics. In this paper, a multi-tool technique is used in USM to reduce the vibration in the tool holder and have reasonable amplitude for the tools. This can be done via dynamic absorbers. The coupling of four nonlinear oscillators of the tool holder and tools representing ultrasonic cutting process are investigated. This leads to a four-degree-of-freedom system subjected to multi-external and multi-parametric excitation forces. The aim of this work is to control the tool holder behavior at simultaneous primary, sub-harmonic and internal resonance condition. Multiple scale perturbation method is used to obtain the solution up to the second order approximations. The different resonance cases are reported and studied numerically. The stability of the system is investigated by using both phase-plane and frequency response techniques. The effects of the different parameters of the tools on the system behavior are studied numerically. Comparison with the available published work is reported.     

Vibration reduction in ultrasonic machine to external and tuned excitation forces

The dynamic response of mechanical and civil structures subject to high-amplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in ultrasonic machining (USM). Ultrasonic machining (USM) is the removal of material by the abrading action of grit-loaded liquid slurry circulating between the workpiece and a tool vibrating perpendicular to the workface at a frequency above the audible range. A high-frequency power source activates a stack of magnetostrictive material, which produces a low-amplitude vibration of the toolholder. This motion is transmitted under light pressure to the slurry, which abrades the workpiece into a conjugate image of the tool form. This can be achieved via passive and active control methods. In this paper, multi-tool techniques are used in the ultrasonic machining via reducing the vibration in the tool holder and providing reasonable amplitudes for the tools represented by the absorbers. The coupling of the tool holder and absorbers simulating ultrasonic cutting process are investigated. This leads to a multi-degree-of-freedom system subject to external and tuned excitation forces. Multiple scale perturbation method is applied to obtain the solution up to the second order approximation. Different resonance cases are reported and studied numerically. The stability of the system is investigated applying both phase-plane and frequency response techniques. The effects of the different parameters of the absorbers on the system behavior are studied numerically. Comparison with the available published work is reported.     

Active vibration control of a nonlinear magnetic levitation system via Nonlinear Saturation Controller (NSC)

In this paper, a nonlinear saturation controller (NSC) is proposed to reduce the horizontal vibration of a magnetically levitated body described by a nonlinear differential equation. An approximate solution is obtained applying the multiple scales perturbation technique to analyze the nonlinear behavior of this model. Based on the quadratic coupling, the energy transfer between the main system and the controller is achieved by designing the natural frequency of the NSC to be close to one half of the natural frequency of the main system. Time histories of the main system and the controller are included to show the response with and without control. Validation curves are included to show the closeness between the perturbation solution and the numerical one. A comparison with previously published work is included.     

Positive position feedback (PPF) controller for suppression of nonlinear system vibration

In this paper, a study for positive position feedback controller is presented that is used to suppress the vibration amplitude of a nonlinear dynamic model at primary resonance and the presence of 1:1 internal resonance. We obtained an approximate solution by applying the multiple scales method. Then we conducted bifurcation analyses for open and closed loop systems. The stability of the system is investigated by applying the frequency-response equations. The effects of the different controller parameters on the behavior of the main system have been studied. Optimum working conditions of the system were extracted to be used in the design of such systems. Finally, numerical simulations are performed to demonstrate and validate the control law. We found that all predictions from analytical solutions are in good agreement with the numerical simulation. A comparison with the available published work is included at the end of the work.