Multiple Scale Perturbation Technique Research Articles

Bifurcation Analysis of a Transversely Cracked Nonlinear Jeffcott Rotor System at Different Resonance Cases

This work focuses on the dynamical behaviour and bifurcations of a vertically supported Jeffcott rotor system having a transverse crack and nonlinear stiffness characteristics at the primary, sub-harmonic, and super-harmonic resonance cases. The nonlinear restoring force due to the bearing-clearance, the crack breathing, the disc eccentricity, and the orientation angle between the crack and imbalance direction are considered in the system model. The equations governing the system motion are derived and solved analytically by applying the Multiple Scales Perturbation Technique (MSPT). The slow-flow modulating equations are obtained and the spinning speed response curve is plotted. The whirling orbit and amplitude spectrum are constructed in the three considered resonance cases. The acquired results provide a better understanding of the main reasons of the super- and sub-harmonic resonance excitations. In additions, we concluded that the suitable resonance case that can be used for early detections of the cracks in the rotating shafts is the sub-harmonic resonance case. Finally, the obtained results are confirmed numerically and compared with the work published in the literature

VIBRATIONS CONTROL OF THE HARMONICALLY EXCITED NONLINEAR SYSTEM VIA POSITIVE POSITION FEEDBACK CONTROLLER

In this paper, the vibration reducing of the harmonically excited nonlinear system is presented via applying positive position feedback controller (PPF). The analytical results are obtained by applying the multiple-scale perturbation techniques (MSPT) up to second-order approximations. The frequency response equation (FRE) is studied to test the be havior of the steady state solutions near the simultaneous resonances. The effects of the several parameters and the behavior of the system at resonance case are investigated to illustrate the optimum working conditions for the PPF controller. The stabilit y analysis for the uncontrolled system is investigated by applying the phase portrait technique. Moreover, numerical simulation is used to compare between time-history and the analytical solution. The analytical and numerical solutions are compared to show the validity of the results. Finally, a comparison with the available results in the literature is presented

Positive Position Feedback Controller for Nonlinear Beam Subject to Harmonically Excitation

In this paper, the vibration reduction of the harmonically excited nonlinear beam is introduced using positive position feedback controller (PPF). The multiple-scale perturbation techniques (MSPT) up is applied to second-order to obtain the analytic results. Numerical simulations are used to compare between time-history and the analytical solution. The frequency response equation (FRE) is studied to illustrate the steady state solutions near the simultaneous resonances. The influences of the different parameters and the system behavior at resonance case are studied to show the optimum conditions of decreasing the vibration. A comparison between the numerical and analytical solutions is presented to appear the validity of the results.

Active and Time Delay Controls on Vibrations of the Micro-Electro-Mechanical System (MEMS) Resonator

In this paper, the active control and time delay control are applied on a nonlinear mechanical system subjected to external force to reduce the resulted vibration. The system is modeled by a unique nonlinear differential equation. The multiple scale perturbation technique (MSPT) was applied to obtain an approximate solution and showing the response equation. The stability of the system at primary resonance case is investigated using both of phase plane and frequency response equation. Numerical solution is obtained using Runge – Kutta forth order method.Also, MATLAB 14.0 and Maple 18.0 programs were used to study the numerical solution and the effect of the different parameters for the response of the nonlinear dynamic mechanical system.

Multiple-scale analysis of open quantum systems

In this work, we present a multiple-scale perturbation technique suitable for the study of open quantum systems, which is easy to implement and in few iterative steps allows us to find excellent approximate solutions. For any time-local quantum master equation, whether markovian or non-markovian, in Lindblad form or not, we give a general procedure to construct analytical approximations to the corresponding dynamical map and, consequently, to the temporal evolution of the density matrix. As a simple illustrative example of the implementation of the method, we study an atom-cavity system described by a dissipative Jaynes-Cummings model. Performing a multiple-scale analysis we obtain approximate analytical expressions for the strong and weak coupling regimes that allow us to identify characteristic time scales in the state of the physical system.

The effectiveness of nonlinear integral positive position feedback control on a duffing oscillator system based on primary and super harmonic resonances

In this paper, we applied three different control methods; Positive Position Feedback (PPF), Integral Resonance Control (IRC) and Nonlinear Integrated Positive Position Feedback (NIPPF) added to a Duffing oscillator system subjected to harmonic force. An analytic solution is introduced using the multiple scales perturbation technique (MSPT) to solve the nonlinear differential equations, which simulate the system with NIPPF controller. Before and after control at the primary and superharmonic resonances, the nonlinear systems’ steady-state amplitude and stability are studied and examined. The influences of various parameters of the system after being connected to NIPPF are illustrated. Optimum working conditions for the NIPPF controller are obtained at internal resonance ratio 1:1. A Comparison is also made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves (FRC). Finally, a comparison with the available results in the literature is presented. From this comparison, we find that the best control to the system is via the NIPPF controller.

Local nonlinear dynamics of MEMS arches actuated by fringing-field electrostatic actuation

The nonlinear dynamic behavior of a resonant MEMS arch microbeam actuated by fringing electric actuation is investigated in this paper. The arch microbeam is loaded with DC and AC harmonic electric load and the ground electrodes placed at either side of the beam. The curvature caused imbalance distribution of field lines and results in a resultant force. Euler–Bernoulli beam equation was used in accordance with the shallow beam theory to get the equation of motion. Method of multiple scales (MMS) perturbation technique was used to perturb the beam near the fundamental frequency of the microbeam to study its resonant behavior in the local vicinity of the considered frequency. Different midpoint elevations were investigated to study the effect of curvature on the resonance frequency of the beam. The MMS reveals that a beam with low initial midpoint elevation shows initially a softening behavior for small DC voltage excitation. Then, the beam undergoes a hardening when the DC voltage is increased. Finally, the beam returns back to softening behavior when the DC load is further increased to higher values with a curling in the frequency response graph. When the midpoint elevation is increased, the beam is known to undergo symmetry breaking when the DC load is increased before regaining symmetry at even further higher DC load. The MMS shows that there is softening behavior in the initial region prior to symmetry breaking. The effective nonlinearity even drops further to more negative values making the beam more softened in the region of symmetry breaking. After the beam regains symmetry, the effective nonlinearity experiences a more significant drop now causing the beam to be highly softened in this region of high DC load.

Amplitude Modulation of Weakly Nonlinear Electrostatic Solitary Waves in Ultrarelativistic Degenerated Semiconductor Quantum Plasma

The amplitude modulation of weakly nonlinear electrostatic solitary waves is investigated using standard quantum hydrodynamic model with multiple scale perturbation technique by deriving a nonlinear Schrodinger equation for ultrarelativistic degenerate semiconductor quantum plasma. Due to nonlinear development of amplitude of electrostatic solitary waves, stable and unstable wave packets are generating and it is numerically analyzed that critical wave number separating stable and unstable wave packets is significantly dependent upon ultrarelativistic parameter, quantum Bohm potential, degenerate pressure, hole-electron mass ratio, and hole-electron temperature ratio. It is seen that critical wave number is varying in a different semiconductor material. Further instability growth characteristic is also examined for different semiconductors. In linear analysis, a dispersion relation has been evaluated analytically and the nature of two wave modes (slow frequency and fast frequency) is studied by varying different plasma parameters.

Outcome of special vibration controller techniques linked to a cracked beam

• Analytical solution and numerical simulation of non-linear differential equations. • Multiple scale perturbation technique is used to obtained the first order approximation. • PPF, IRC and NIPPF controllers to reduce the vibration. • Stability study subjected to external excitation. • Effect of different parameters in the system. This paper presents a comparison between three different controller methods added to a cracked beam under the action of a harmonic excitation. Those three controllers are Positive Position Feedback (PPF), Integral Resonant Control (IRC) and Nonlinear Integral Positive Position Feedback (NIPPF) which be added to the measured system. The multiple scales method (MSM) is applied for getting the approximate solution on behalf of measured design. This method is effective to solve the major equations of measured system. Stability and effect of different coefficients of the system are demonstrated. The approximate solution response is established via numerical simulation outcome. NIPPF controller is the best one gives better results compared to the other two controllers in decreasing the high amplitude of the system. Comparison between mathematical solution and numerical simulation are considered. Relationship of formerly available papers is considered.

Size-dependent nonlinear vibration analysis of shear deformable microarches using strain gradient theory

Non-classical continuum theories are able to capture the size-dependent behavior of microscale structures. Among them, strain gradient modeling is one of the most accurate ones. The objective of this paper is to develop for the first time a nonlinear strain gradient formulation for the vibrational analysis of electrostatically actuated shear deformable microarches. The normalized equations of motion and the corresponding boundary conditions are derived based on Hamilton’s principle. Novel closed form expressions are derived for the mode shapes of the system using the perturbation technique. These mode shapes are then utilized in a Galerkin projection method to reduce the partial differential equations of motion to time domain nonlinear ordinary differential equations. Multiple time scales perturbation technique is then employed to derive analytical solutions for the nonlinear vibration of the system resulting from a DC step voltage excitation. The analytical results are compared with numerical simulations, and excellent agreement is observed. The formulations and the results presented in this paper can be utilized for design, analysis and synthesis of microscale bistable structures with pre-deformed thick microbeams as their main operating element.

An analytical approach for modeling nonlinear vibration of doubly clamped functionally graded Timoshenko microbeams using strain gradient theory

In the current study, the nonlinear free and forced vibration of functionally graded strain gradient Timoshenko microbeam are analytically investigated. The nonlinear equations governing the motion of the microbeam and the corresponded boundary conditions are derived using variational approach. A novel analytical approach is proposed to derive closed-form expressions for the mode shapes of the system. The analytical mode shapes are then closely verified via numerical simulations and the effect of different parameters on the fundamental natural frequency of the beam is studied. Multiple time scales (MTS) perturbation technique is employed to model nonlinear free vibration of the system based on a single mode approximation. The accuracy of the time history results are confirmed by numerical calculations. The nonlinear forced vibration of the Timoshenko beam is also simulated using MTS. More specifically, the primary resonance and the frequency responses of the system are derived analytically and the effect of different design parameters on their behavior is studied in detail. The results and approaches presented in this paper can be further employed for suggesting analytical predictions in more complex micro structures.

The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading

This paper studied the nonlinear vibrations of top-tensioned cantilevered pipes conveying pressurized steady two-phase flow under thermal loading. The coupled axial and transverse governing partial differential equations of motion of the system were derived based on Hamilton’s mechanics, with the centerline assumed to be extensible. Using the multiple-scale perturbation technique, natural frequencies, mode shapes, and first order approximate solutions of the steady-state response of the pipes were obtained. The multiple-scale assessment reveals that at some frequencies the system is uncoupled, while at some frequencies a 1:2 coupling exists between the axial and the transverse frequencies of the pipe. Nonlinear frequencies versus the amplitude displacement of the cantilever pipe, conveying two-phase flow at super-critical mixture velocity for the uncoupled scenario, exhibit a nonlinear hardening behavior; an increment in the void fractions of the two-phase flow results in a reduction in the pipe’s transverse vibration frequencies and the coupled amplitude of the system. However, increases in the temperature difference, pressure, and the presence of top tension were observed to increase the pipe’s transverse vibration frequencies without a significant change in the coupled amplitude of the system.

Nonlinear analysis of functionally graded piezoelectric energy harvesters

This study deals with nonlinear vibrational analysis of functionally graded piezoelectric energy harvesters (FGPEH) carrying concentrated mass at free end, while taking into account the geometrical nonlinear terms as well as electromechanical couplings. Hamilton’s principle and Rayleigh’s beam theory are utilized to obtain the coupled electromechanical governing equations. The first mode shape of the system is derived analytically which is also validated by comparing it with numerical findings of a three-dimensional finite element simulation performed in COMSOL Multiphysics commercial software. The Galerkin projection method is employed along with a single mode approximation to derive the nonlinear coupled electromechanical governing equations which are then solved analytically using the method of multiple time scales perturbation technique for both cases of free and forced vibrations. Time domain responses of free vibration are compared with those of numerical simulations carried out in MATLAB and excellent agreement is observed. Moreover, the nonlinear frequency responses of the forced vibration are obtained for the case of near resonance excitation. A parametric study is also accomplished to reveal the effects of power low index of the FG material as well as the scaled amplitude of the input displacements of the base on the shape of the frequency response.

Breather structures in degenerate relativistic non-extensive plasma

We examine the excitation of breather structures in a degenerate relativistic plasma consisting of non-extensive electrons and cold ions. For this purpose, the multiple time scale perturbation technique is used to obtain a nonlinear Schrödinger equation (NLSE). We then consider different localized solutions regarding analytical breather solutions of the NLSE, and examine their properties in the frame of the present plasma system, i.e. a degenerate relativistic non-extensive plasma. The results of the present investigation may be useful for the understanding of the basic features of the nonlinear excitations that may occur in dense astrophysical plasmas.

The Vibration reduction of a cantilever beam subjected to parametric excitation using time delay feedback

In this paper, dynamical behavior of a cantilever beam subject to parametric excitation under state feedback control with time delay is analyzed. The method of multiple scale perturbation technique is applied to obtain the solution up to the first order approximation. We obtain equations for the amplitude and phase. We studied all resonance cases numerically. Stability of the steady state solution for the selected resonance case is studied applying Rung-Kutta fourth method and frequency response equation via Matlab 7.0 and maple 16. From the results, it can be seen that the frequency and amplitude responses for the selected resonance case can be affected by the time delayed control. Effects of different parameters of the system are studied.

Nonlinear vibration control of a horizontally supported Jeffcott-rotor system

A positive position feedback (PPF) controller is proposed to control the nonlinear vibrations of a horizontally supported Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The controller is coupled to the system with 1:1 internal resonance. A second order approximate solution to the system governing equations is constructed by applying the multiple scales perturbation technique (MSPT). The bifurcation analyses of the Jeffcott-rotor system before and after control are conducted. The effects of the different controller parameters on the system frequency–response curves are investigated. Optimum working conditions of the controlled system are extracted to be used in the design of such systems. Numerical simulations demonstrated a good agreement with the approximate results that obtained by MSPT. A comparison is provided with already published work.

Hydro-acoustic frequencies of the weakly compressible mild-slope equation

We present a novel analytical solution for hydro-acoustic waves in a weakly compressible fluid flow over a slowly varying bottom. Application of a multiple-scale perturbation technique and matched asymptotic analysis leads to a uniform analytical solution of the depth-averaged governing equations in three dimensions. We show that the slow depth variation has a leading-order effect on the evolution of the normal mode amplitude and direction. This dynamics is much richer than the two-dimensional limit analysed in previous studies. For tsunamigenic disturbances, we show that the hydro-acoustic wave field is made up by longshore trapped and offshore propagating components, which were not explicated in previous work. For a plane beach, we find an exact analytical solution of the model equation in terms of integrals of Bessel functions. Our model offers a physical insight into the evolution of hydro-acoustic waves of interest for the design of tsunami early warning systems.

On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force

This paper deals with the vibration analysis of a horizontally supported Jeffcott rotor system. Both nonlinear restoring force and the rotor weight are considered in the system modeling. The model shows a small difference between the natural frequencies of the vertical and horizontal mode. The multiple scales perturbation technique is utilized to obtain a second-order approximate solution at the simultaneous resonance case. The bifurcation analyses are conducted. The stability of the obtained solution is investigated by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. The Effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. At the large rotor eccentricity, the analysis revealed the following: (1) the existence of three different stable solutions in an interval of the rotational speed. (2) The disk exposed to two consecutive jumps if its speed crossed the resonant speed. (3) For a soft spring, localized and nonlocalized oscillation in both the horizontal and vertical mode occurs. (4) For a hard spring, nonlocalized oscillation occurs in the two directions in addition to the localized motion in the vertical direction only (5) The system is very sensitive to initial conditions. Then, numerical simulations are performed to confirm the accuracy of the approximate results. It is found that the predictions from the analytical solutions are in a good agreement with the numerical simulations. Finally, a comparison with previously published work is included.

Vibration Control in MEMS Resonator Using Positive Position Feedback (PPF) Controller

In this paper, the vibration of a micro-electromechanical resonator with positive position feedback controller is studied. The analytical results are obtained to the first order approximation by using the multiple scale perturbation technique. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. The effects of the controller and some system parameters on the vibrating system are studied numerically. The main result of this paper indicates that it is possible to reduce the vibration for the resonator system.

Effect of doping on amplitude modulation of space-charge wave in semiconductor quantum plasma

To describe the modulational instability of space-charge waves in an n-type compensated semiconductor plasma, a nonlinear Schrodinger equation has been derived by using quantum hydrodynamical model and standard multiple scale perturbation technique. It has been shown that compensation factor (i.e. relative proportion of donor, acceptor and intrinsic carrier concentrations) and quantum diffraction parameter play important role in generating bright and dark envelope solitons within the semiconductor. Instability growth rate is also found to depend sensitively on the compensation factor and quantum diffraction parameter. From the linear dispersion relation it has been found that inclusion of quantum parameter gives rise to two new wave modes of purely quantum origin. Further the effect of compensation factor and quantum diffraction parameter on the linear dispersion characteristics has been analyzed. It has also been found that due to parabolicity of conduction band the group velocity of space-charge wave becomes dependent on compensation factor and quantum diffraction parameter.