Perturbation Method Of Multiple Scales Research Articles

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

AbstractPaper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium‐sized inclusions in the framework of the Padé approximants.

Eliminate the Nonlinear Oscillations of the Modified duffing Equation by using the Nonlinear Integrated Positive Position Feedback

In this article, the nonlinear integrated positive position feedback (NIPPF) control adds to a nonlinear dynamical system modeled as the well known Duffng oscillators. This control is proposed to mitigate system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow- ow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. Before and after (NIPPF) control the nonlinear systems' steady-state amplitude are examined, the comparison is made to validate the closeness between the numerical solution and the analytical perturbative one at time-history and frequency response curves.

Quantum and Relativistic Effects on the KdV and Envelope Solitons in Ion-Plasma Waves

In this article, we investigate the linear and nonlinear behavior of ion-plasma waves in a two-component plasma containing ions and dust particles. We have studied the linear dispersion characteristics, nonlinear KdV solitons, and amplitude modulated electrostatic waves. Such ion-plasma wave modes in dust-ion plasma have been theoretically investigated by incorporating quantum diffraction and relativistic effects. Numerical analysis of the linear dispersion relation with variations of different parameters has been carried out. By employing the reductive perturbation technique the KdV equation describing the small amplitude solitary profile has been studied. Using the standard multiple scale perturbation method, a nonlinear Schrödinger (NLS) equation has been derived by including quantum and relativistic effects, which describes the nonlinear evolution of the ion-plasma waves in dust-ion plasma. The wave is found to become modulationally unstable beyond certain critical wavenumber. The critical wavenumber and the growth rate of modulational instability are shown to depend significantly on the relativistic motion of plasma particles and the quantum diffraction effect. The results presented in the article are expected to be helpful for understanding the propagation of finite amplitude ion-plasma waves in some laboratory and astrophysical plasma systems. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PACS</i> —52.59.Hq, 67.10.Jn, 52.20 −j, 52.25 −b, 52.30 −d, 52.35 Mw, 52.65 Vv

Characteristic behavior and modulation instability of Peregrine solitons of the ion acoustic waves: a perspective of Titan’s ionosphere

The propagational properties of solitons along with peregrine solitonic modulational instability of ion acoustic (IA) waves are investigated taking into account the plasma properties of the Titan’s ionosphere. The Titan’s ionosphere typically consists of positive (, , and ) and negative (, , and ) ionic species along with electrons. In this study, it is assumed that the positive ions and electrons follow the Lorentzian-like kappa distribution. The fluid model equations are considered for negative ions. The Korteweg-de Vries (KdV) equation is derived using the reductive perturbation method to study the propagational properties and the characteristic behavior of solitons. In addition, the multiple-scale perturbation method is also used to derive the non-linear Schrödinger (NLS) equation for the analyses of modulational instability along with the production of rogue waves considering Peregrine solitonic solution. The dark solitons appear for the smaller wave numbers, whereas the bright solitons appear for the larger wave numbers. On the other hand, instability due to modulation is excited when the wave number is greater than its critical value, whereas it becomes stable when the wave number is smaller. The Peregrine soliton carries a single pick that decays to the plane wave asymptotic background either in space or time domain.

All-round responses and boundaries of a shaft and dry friction damper assembly

A helicopter tailrotor driveline is equipped with dampers to suppress the resonance of transmission shafts. A dry friction damper and a slender shaft form an assembly The comprehensive understanding of different responses of the assembly provides the basis for a more reasonable parameter configuration of the damper. To this end, the assembly is modeled as a dynamic system with rub impact between a rotor with continuous parameter distribution and a suspended elastic stator. Boundaries of rub impact and condition of Hopf bifurcation are derived. Pinned natural frequencies and backward whirl frequencies are solved by analytical deduction. The critical speed for triggering a backward whirl with consideration of eccentric excitation is obtained by the multiple scale perturbation method. Then, all-round responses and boundaries are solved by both analytical and numerical approaches based on the parameters of a helicopter. The effects of friction coefficient, internal damping, suspension stiffness, variable impact stiffness, etc. on the boundaries are presented in detail with the discussion of relationships between physical reasons and analytical conditions. Results show that smaller impact stiffness difference and friction coefficient, a greater mass damping ring, moderate suspension stiffness, greater internal and external damping are the key factors for the damper design. This study has reference significance not only for the assembly in a helicopter but also for similar mechanical systems.

DYNAMICS ANALYSIS OF THREE-SPECIES REACTION-DIFFUSION SYSTEM VIA THE MULTIPLE SCALE PERTURBATION METHOD

In this paper, the general analysis of spatiotemporal dynamics of three-species reaction-diffusion system induced by Turing bifurcation is given. Firstly, by employing the Routh-Hurwitz criterion the conditions for Turing bifurcation in three-species reaction-diffusion equations are obtained. Secondly, through the tool of the multiple scale perturbation method the amplitude equations of Turing patterns are also given. Finally, we take a three-species predator-prey model as an example to illustrate the application of these general theoretical results, and meanwhile carry out many numerical simulations to depict spots pattern, stripes pattern and labyrinthine pattern and demonstrate the validity of these theories.

Analytical Approximations for Dry Friction-Induced Stick–Slip and Pure-Slip Vibration Amplitudes of a Self-Excited Smooth and Discontinuous Oscillator

Abstract An analytical and numerical investigation into pure-slip and stick–slip oscillations induced by dry friction between a rigid mass linked by an inclined spring, modeled by the archetypal self-excited smooth and discontinuous (SD) oscillator, and the classical moving rigid belt, is presented. The friction force between surface contacts is modeled in the sense of Stribeck effect to formulate the friction model that the friction force first decreases and then increases with increasing relative sliding speed. Some perturbation methods are considered into this system for establishing the approximate analytical expressions of the occurring conditions, vibration amplitudes, and base frequencies of dry friction-induced stick–slip and pure-slip oscillations. For pure-slip oscillations, two different approaches are applied to analyze this self-excited SD oscillator. One of them is the homotopy perturbation method by constructing the nonlinear amplitude and frequency. Based on the multiple-scales homotopy perturbation method, a nonlinear equation for amplitude of the analytical approximate solution is constructed, which containing all parameters of problem. For stick–slip oscillations, the analytical approximations for amplitude and frequency are obtained by perturbation methods for finite time intervals of the stick phase, which is linked to the subsequent slip phase under the conditions of continuity and periodicity. The accuracy of analytical approximations is verified by the comparison between analytical approximations and numerical simulations. These analytical expressions are needed for gaining a deeper understanding of dry friction-induced pure-slip and stick–slip oscillations for the friction system with geometric nonlinearity.

Effect of delay on pattern formation of a Rosenzweig–MacArthur type reaction–diffusion model with spatiotemporal delay

In this paper, a predator–prey reaction–diffusion model with Rosenzweig–MacArthur type functional response and spatiotemporal delay is investigated through using the tool of Turing bifurcation theories. First, by taking the average time delay as a bifurcation parameter, conditions of occurrence of Turing bifurcation are obtained through employing the Routh–Hurwitz criteria. Second, as the average time delay varies the amplitude equations of Turing bifurcation patterns including spots pattern and stripes pattern are also obtained through the multiple scale perturbation method. Finally, the two kinds of spatiotemporal evolution distributions of species such as spots pattern and stripes pattern are shown to illustrate theoretical results.

Flexural modes coupling in cantilever-type piezoelectric energy harvesters

The ability to harness the waste mechanical energy and convert it into useful electrical power has made kinetic energy harvesters a promising candidate to provide an everlasting energy source for wireless autonomous devices. Nonlinearities, whether introduced deliberately for the sake of bandwidth broadening or present intrinsically, can highly influence the dynamic response and output power behavior of these type of energy scavengers. This paper aims to investigate the effect of nonlinearity on multi-mode vibrational response of a harvester composed of a cantilevered piezoelectric composite beam with an attached mass of finite dimensions. To that end, first of all a 3-DoF lumped parameter coupled electromechanical model of the device is developed through a comprehensive mathematical approach and its mode shapes and natural frequencies are calculated. The perturbation method of multiple scales is then applied to obtain the steady state solutions to the extracted order-reduced governing equations of the system. Results indicate that a harvester with a cubic attached mass exhibits a simple Duffing-type resonance as the excitation frequency falls in the vicinity of each natural frequency. That occurs while for a U-shaped mass the vibration modes would be coupled through occurrence of an internal resonance. In this latter case, both flexural modes of the piezoelectric beam are stimulated by a single frequency excitation and contribute to the power generation leading to an enhancement of the total output power which is the major advantage of the proposed design in this paper compared to the other existing energy harvesters. The frequency response curves of the output power are found to be composed of four branches and include Hopf bifurcations and instability regions. To verify the results obtained from the analytical approach, they are compared to a numerical solution where a good agreement is observed between them.

Topology-Dependent Excitation Response of Networks of Linear and Nonlinear Oscillators

Abstract This paper investigates the near-resonance response to exogenous excitation of a class of networks of coupled linear and nonlinear oscillators with emphasis on the dependence on network topology, distribution of nonlinearities, and damping ratios. The analysis shows a qualitative transition between the behaviors associated with the extreme cases of all linear and all nonlinear oscillators, respectively, even allowing for such a transition under continuous variations in the damping ratios but for fixed topology. Theoretical predictions for arbitrary members of the network class using the multiple-scales perturbation method are validated against numerical results obtained using parameter continuation techniques. The latter include the tracking of families of quasi-periodic invariant tori emanating from saddle-node and Hopf bifurcations of periodic orbits. In networks in the class of interest with special topology, 1:1 and 1:3 internal resonances couple modes of oscillation, and the conditions to suppress the influence of these resonances are explored.

Adaptive versus Conventional Positive Position Feedback Controller to Suppress a Nonlinear System Vibrations

The nonlinear vibration control of a nonlinear dynamical system modeled as the well-known Duffing oscillators is investigated within this article. The conventional Positive Position Feedback (PPF) controller is proposed to mitigate the considered system nonlinear vibrations. The whole system mathematical model is analyzed by applying the multiple time scales perturbation method. The slow-flow modulation equations that govern the oscillation amplitudes of both the main system and controller are derived. The stability analysis is investigated based on Lyapunov’s first method. The effects of the different control parameters on both the main system and controller are explored. The obtained analytical and numerical results illustrated that the PPF controller can eliminate the main system nonlinear vibrations once the controller natural frequency is tuned to be the same value as the external excitation frequency, otherwise, the controller adds excessive vibrational energy to the main system rather than suppressing it. In addition, the PPF controller can destabilize the main system motion when excited by strong excitation force. Therefore, a modified version of the PPF controller named the Adaptive Positive Position Feedback (APPF) controller is proposed to overcome the main drawbacks of the conventional PPF controller. The idea is to track the external excitation frequency using an adaptive frequency measurement technique to update continuously the PPF controller natural frequency to become the same value of the excitation frequency. Based on this strategy, the system mathematical model is analyzed again by making the controller’s natural frequency equal to the external excitation frequency. The obtained analytical and numerical simulations showed that the adaptive positive position feedback controller can suppress the main system nonlinear vibration close to zero regardless of the excitation force amplitude and excitation frequency.

Response Analysis and Controlling the Nonlinear Vibration of Van Der-Pol Duffing Oscillator Connected to the NIPPF Controller

In this paper, we introduced response analysis and controlling the nonlinear vibration of van der pol duffing oscillator subject to parametric and external excitations by connecting the nonlinear integral positive position feedback (NIPPF). The controller combines the characteristics of the positive position feedback controller (PPF) and integral resonant control (IRC) and these controllers are viewed in a blocked scheme for the closed-loop system (system under control). The analytical solution and all resonance cases are obtained by applying the multiple time scales perturbation method (MSPM). The stability analysis is studied using the frequency response equations at the supposed worst resonance case. For numerical simulation, the system is examined before and after connecting the NIPPF controller using Matlab software program (package ode45). The influences of different parameters are examined numerically for the vibrating system and controller. The approximate analysis is confirmed with numerical simulation outcomes. Finally, making a comparison with previously related published work.

Optimal delayed control of primary and second resonances of an electrostatic driving double-sided micro-actuator

Abstract In this paper, an electrostatic driving double-sided micro-actuator was studied based on the fully clamped micro-beam with a uniform thickness. An analysis of primary, superharmonic and subharmonic resonances with two distinct time delays was investigated. The multiple scales perturbation method was used to solve the nonlinear equation without large deformation. Steady-state solutions and their stabilities under each resonance conditions were studied. The optimal time delay was obtained for controller design. The effect of the feedback gains and time delays on the nonlinear response of the system was discussed and the analysis shows that a suitable choice of the feedback gains and time delayed quantity can reduce the nonlinearity, resonance region and unstable multi-solutions, also improve the efficiency of the vibration control. It is believed that the results of this paper will be valuable in the design of optimal controllers.

ANALYSIS OF THE NONLINEAR AXIAL VIBRATIONS OF A CANTILEVERED PIPE CONVEYING PULSATING TWO PHASE FLOW

The parametric resonance of the axial vibrations of a cantilever pipe conveying harmonically perturbed two-phase flow is investigated using the method of multiple scale perturbation. The nonlinear coupled and uncoupled planar dynamics of the pipe are examined for a scenario when the axial vibration is parametrically excited by the pulsating frequencies of the two phases conveyed by the pipe. Away from the internal resonance condition, the stability regions are determined analytically. The stability boundaries are found to reduce as the void fraction is increasing. With the amplitude of the harmonic velocity fluctuations of the phases taken as the control parameters, the presence of internal resonance condition results in the occurrence of both axial and transverse resonance peaks due to the transfer of energy between the planar directions. However, an increase in the void fraction is observed to reduce the amplitude of oscillations due to the increase in mass content in the pipe and which further dampens the motions of the pipe.

Periodic water waves through suspended canopy

Small-amplitude water waves passing through a suspended canopy is studied. The area of suspended canopy is modeled by an array of vertical rigid cylinders with periodic spacing. Assuming that the diameter of cylinders and their spacing are much smaller than the typical incident wavelength, the homogenization theory (method of multiple-scale perturbation) is applied to create coupled micro-scale (cylinder spacing) and macro-scale (wavelength) problems. The micro-scale problem describes turbulent flows within a unit cell of the cylinder array, being driven by the macro-scale pressure gradients. Employing the concept of averaged energy balance over a wave period, the micro-scale flows determine the eddy viscosity, which damps the waves in the macro-scale flows. Eigenfunction expansions method is used to solve the macro-scale problem, in which a complex frequency dispersion relation is solved numerically by a multiple successive approximation technique. The potential decomposition method, which can avoid solving a complex frequency dispersion relation, is also employed to validate the accuracy of the eigenfunction expansions method. Both methods yield accurate solutions. However, the eigenfunction expansions method is relatively straightforward and can unify the solutions for suspended canopy and emergent vegetation. A new set of flume experiments of waves through suspended canopy is conducted and experimental data are used to check present solutions. Very good agreement has been observed. Finally, the effectiveness of suspended canopy, submerged and emergent vegetation on wave attenuation is discussed.

Effect of radially functionally graded materials on the primary resonances of large amplitude flexural vibration of in-extensional rotating shafts

This paper analytically studies the effect of functionally graded materials (FGMs) on the primary resonances of large amplitude flexural vibration of in-extensional rotating shafts with nonlinear curvature as well as nonlinear inertia. The constituent material is assumed to vary along the radial direction according to a power-law gradation. The governing differential equations and the corresponding boundary conditions are derived employing the variational approach. Then, the Galerkin method and the multiple scales perturbation method are utilized to obtain the frequency–response equation. In a numerical case study, the effects of the power-law index on the steady-state responses and locus of two saddle-node bifurcations for a heterogeneous functionally graded rotating shaft fabricated from the mixture of Aluminum oxide (Al2O3) and Stainless Steel (SUS304) are illustrated. The results of the present study are compared with the reported results for a homogeneous shaft as some kind of validation of the developed model. The results of this paper provide useful and practical information that can be helpful in the design of these new well-developed structures.

Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance

In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω1), detuning parameter (σ1), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.

Nonlinear forced vibration of sandwich plate with considering FG core and CNTs reinforced nano-composite face sheets

Nonlinear vibration of sandwich plate with functionally graded material (FGM) core and carbon nano tubes reinforced (CNTs) nano-composite layers by considering temperature-dependent material properties are studied in this paper. Base on Classical plate theory (CPT), the governing partial differential equations of motion for sandwich plate are derived using Hamilton principle. The Galerkin procedure and multiple scales perturbation method are used to find relation between nonlinear frequency and amplitude of vibration response. The dynamic responses of the sandwich plate are also investigated in both time and frequency domains. Then, the effects of nonlinearity, excitation, power law index of FG core, volume fraction of carbon nanotube, the function of material variations of FG core, temperature changes, scale transformation parameter and damping factor on the frequency responses are investigated.

Nonlinear Primary Frequency Response Analysis of Self-Sustaining Nanobeam Considering Surface Elasticity

This paper is focused to investigate the effects of nonlinear sources, including viscoelastic foundation and geometrical nonlinearity along with the surface elasticity and residual surface stress effects on the primary frequency response of a harmonically excited nanoscale Bernoulli-Euler beam. Due to large surface-area-to-volume ratio, the theory of surface elasticity as well as residual surface stress effects are taken into account within the beam models. The Galerkin approach accompanied by trigonometric shape functions is utilized to reduce the governing PDEs of the system to ODEs. The multiple scales perturbation method theory is applied to compute the nonlinear frequency response of nanobeam. The effects of linear and nonlinear viscoelastic damping coefficient of the medium, crystallographic directions of [100] and [111] of anodic alumina, geometrical nonlinear term and geometrical property on the nonlinear primary frequency response of nanoscale beam are investigated. The results show that theses parameters have a significant effect on the nonlinear frequency response of nanobeams in the case of primary resonance.

Nonlinear vibration of piezoelectric laminated nanobeams at higher modes based on nonlocal piezoelectric theory

The presented paper investigates the nonlinear vibration of a nanobeam with a piezoelectric layer bounded to its top surface considering the nonlocal piezoelectricity theory. To do this, Hamilton’s principle is implemented to derive the governing nonlinear vibration equations of the nanobeam by assumption of nonlocal piezoelectricity and a nonlinear strain–displacement relation. Then, the Galerkin separation method is applied to transform and simplify the partial differential equation of the nonlinear oscillation to an ordinary one with quadratic and cubic nonlinearities in the time domain. By implementing the multiple-scale perturbation method, an analytical relation for the nonlinear natural frequencies is obtained as a function of the oscillation amplitude and the nonlocal size scale parameter. Then, the nonlinear vibration characteristics of the nanobeam are investigated at higher modes of vibration and the size scale effects are reviewed comprehensively. It is observed that the nonlocal parameter decreases the nonlinear natural frequencies and becomes noticeable at higher modes of vibration. Moreover, by increasing the amplitude ratio, the nonlocal effects are decreased and the nonlocal nonlinear frequency approaches the local one. Also, the amplitude ratio has increasing effects on the nonlinear frequencies.