Multiple-scale Technique Research Articles

Rotor Active Magnetic Bearings System Control via a Tuned Nonlinear Saturation Oscillator

The proposed work focuses on improving the performance of the traditional nonlinear saturation controller (NSC) algorithm. In this work, a tuning mechanism can be implemented by providing the NSC control unit with the measured speed $\Omega $ of the rotor from a shaft encoder device in order to introduce a tuned NSC (TNSC). In other words, we are going to tune the NSC’s natural frequency ${\omega _{c}}$ at one half of $\Omega $ such that $\Omega =2{\omega _{c}}$ . This TNSC is adopted to reduce the vibratory amplitudes of a 16-pole constant-stiffness rotor-Active magnetic bearings AMBs) system for a wide range of speeds and eccentricities. The whole controlled system is studied mathematically to seek its approximate solutions via the multiple scales technique. Different relations between the rotor’s amplitudes and its parameters are plotted so as to verify the proposed tuning mechanism and its crucial role in improving the NSC’s performance.

In-Plane Nonlinear Vibration of Overhead Power Transmission Conductors With Coupling Effects of Wind and Rain

In this paper a small sag-to-span ratio conductor, belonging to overhead power transmission conductors, is studied. The single conductor is assumed to withstand a prescribed motion at a single conductor ends caused by suspended insulator swing and placed in the simultaneous wind and rain condition. It is modeled as a planar, nonlinear, one-dimensional continuum, and its ends are assumed to undergo the suspended insulator swing of given amplitude and frequency. Raindrops are blowing by uniformed wind flow, hitting the single conductor and may form rivulets on its surface. Besides rain-wind flow induces aerodynamic instability, the motions of the suspended insulator swing contribute to external and parametric excitation. Vertical motion equation of the single conductor is discretized with Galerkin method of which first order in-plane symmetric mode is took as trial function. Multiple-scale perturbation technique and computation methods are introduced to study nonlinear dynamic responses of the single conductor under parametric excitation or force excitation, or both excitations. Effects of key parameters, such as total damping ratio, upper rivulet angle, suspended insulator swing amplitude and frequency on dynamic characteristics of the single conductor are discussed. The results indicated that those key-parameters have obviously affected the dynamic characteristics of high-voltage conductor.

Breather Structures and Peregrine Solitons in a Polarized Space Dusty Plasma

In this theoretical investigation, we have examined the combined effects of nonthermally revamped polarization force on modulational instability MI of dust acoustic waves DAWs and evolution of different kinds of dust acoustic (DA) breathers in a dusty plasma consisting of negatively charged dust as fluid, Maxwellian electrons, and ions obeying Cairns’ nonthermal distribution. The nonthermality of ions has considerably altered the strength of polarization force. By employing the multiple-scale perturbation technique, the nonlinear Schrödinger equation NLSE is derived to study modulational MI instability of dust acoustic waves DAWs. It is noticed that influence of the polarization force makes the wave number domain narrow where MI sets in. The rational solutions of nonlinear Schrödinger equation illustrate the evolution of DA breathers, namely, Akhmediev breather, Kuznetsov–Ma breather, and Peregrine solitons (rogue waves). Further, the formation of super rogue waves due to nonlinear superposition of DA triplets rogue waves is also discussed. It is analyzed that combined effects of variation in the polarization force and nonthermality of ions have a comprehensive influence on the evolution of different kinds of DA breathers. It is remarked that outcome of present theoretical investigation may provide physical insight into understanding the role of nonlinear phenomena for the generation of various types of DA breathers in experiments and different regions of space (e.g., the planetary spoke and cometary tails).

Cross-Diffusion-Driven Instability in a Predator-Prey System with Fear and Group Defense

In this paper, a reaction-diffusion prey-predator system including the fear effect of predator on prey population and group defense has been considered. The conditions for the onset of cross-diffusion-driven instability are obtained by linear stability analysis. The technique of multiple time scales is employed to deduce the amplitude equation near Turing bifurcation threshold by choosing the cross-diffusion coefficient as a bifurcation parameter. The stability analysis of these amplitude equations leads to the identification of various Turing patterns driven by the cross-diffusion, which are also investigated through numerical simulations.

Hopf-Hopf bifurcation in the delayed nutrient-microorganism model

In view of time delay in the transport of nutrients, a delayed reaction-diffusion system with homogeneous Neumann boundary conditions is presented to understand the formation of the heterogeneous distribution of bacteria and nutrients in the sediment. With the effects of time delay and diffusion, the system will experience various dynamical behaviors, such as stability, the Turing instability, successive switches of stability of equilibria, the Hopf and the Hopf-Hopf bifurcations. To further understand the dynamics of the Hopf-Hopf bifurcation, the multiple time scale (MTS) technique is employed to derive the amplitude equations at this co-dimensional bifurcation point, and the dynamical classification near such bifurcation point is also identified by analyzing the obtained amplitude equations. Some numerical simulations are carried out to demonstrate the validity of the theoretical analysis.

On the tuning of static pull-in instability and nonlinear vibrations of functionally graded micro-resonators with three different configurations

In this paper, the vibrational behavior of a functionally graded (FG) micro-resonator is investigated for three different configurations based on the Euler–Bernoulli beam model and the Von Karman nonlinear strain assumption. The three configurations include: (1) an FG micro-resonator with a fixed foundation, (2) piezoelectric layers added to the first model, and (3) a second fixed foundation added to the first model. These investigations are performed under the effect of electrostatic forces, the forces caused by the deformations of piezoelectric layers due to the applied voltage, Casimir forces, and uniform temperature changes. The equations governing the vibrational behaviors are obtained using the Hamilton’s principle and the modified couple stress theory. Static deformations and the fundamental vibrational mode are calculated using the differential quadrature method in the spatial domain for all three configurations. Furthermore, the dynamic equations are obtained from the Galerkin discretization and solved with multiple time scales technique. The results show that the frequency behavior of a micro-resonator can be adjusted using the three models.

Frequency response curves for a Mooney-Rivlin hyperelastic microbubble oscillating as a contrast agent in an acoustic pressure field

In this work, we have developed numerical simulations and weakly nonlinear analysis based on the multiple-scales perturbation technique for a coated microbubble that performs radial pulsations subject to an acoustic pressure disturbance in the far-field and whose encapsulated hyperelastic material obeys the Mooney-Rivlin equation. Departing from an elastic coating as a hyperelastic shell of finite thickness, we assume eventually that the shell is of very small thickness in comparison with the microbubble radius. Under this condition, we then perform weakly nonlinear analysis, to identify resonance conditions for small pressure disturbances of the acoustic field. In parallel and also for the limit of small thickness, we have carried out numerical simulations of the radial motion of the microbubble, identifying the onset of limit cycles via the construction of Poincare maps. Under both schemes, we have recognized the importance of two dimensionless hyperelastic parameters that dictate the main behavior of the oscillations: α∗ and β∗. Decreasing the values of these parameters, the resonance conditions are drastically amplified, which is an expected result because of the weak rigidity of the hyperelastic solid, prevails. In this manner, we suggest that moderate values for these previous parameters can be widely advisable when, in medical diagnostic applications, we are applying microbubbles as contrast agents. Therefore, we recommend widely the use of shell softens, because in this case the amplitude of radial pulsation is always amplified.

Modified multiple scale technique for the stability of the fractional delayed nonlinear oscillator

In the present proposal, the familiar method of the parameter expansion is combined with the multiple scales to study the stability behaviour of the Riemann–Liouville fractional derivative applied to the cubic delayed Duffing oscillator. The analysis of the modified multiple scale perturbation leads to a system of nonlinear differential-algebraic equations governing the solvability conditions. The nonlinear differential equation was reduced to the linear differential equation with the help of the algebraic one. The stability attitude of the periodic motion is determined by the steady-state analysis. Such a periodic motion is needed to better understand the dynamics of the fractional cubic delayed Duffing oscillator.

Homoclinic bifurcations and chaotic dynamics of non-planar waves in axially moving beam subjected to thermal load

The homoclinic bifurcations and nonplanar chaotic waves in axially moving beam (AMB) under thermal excitation are investigated. By the multiple scale technique, the equivalent nonlinear system is derived to explore qualitatively the dynamical characteristics of AMB system for the case of primary resonance. Using Melnikov approach as well as geometric analysis, the criterion for homoclinic chaos and complex nonplanar motions for AMB system is discussed. The theoretical predictions are tested by the numerical approach. For the design and application of the AMB, some inspiration and guidance are provided by the results from theory and simulation.

Thermal analysis of heat transfer in pipe cooling concrete structure by a meshless RBF-FD method combined with an indirect model

Numerical simulation of heat conduction in the concrete structure with cooling pipes is meaningful and full of challenges. In this paper, the radial basis function finite difference method (RBF-FD) combined with a self-correcting prediction model (SCM) is proposed for thermal analysis of heat transfer in concrete structures with pipe cooling system. Using the Multiquadric radial basis function (MQ-RBF), the SCM is proposed to give full play to advantages of the localized meshless method. This technique allows use of more number of nodes in the whole domain to solve the ill-conditioning problem associated with multiple regions of rapid variation and increases the accuracy of models. A multiple-scale technique is applied to determine the shape parameter in the MQ-RBF and the fictitious nodes method is modified by using just one fictitious node outside the domain to deal with boundary conditions of the third kind in tiny holes. To show the flexibility and efficiency of the proposed scheme, we consider several examples with different number, size, and positions of pipes. Meanwhile, a guidance of determination of effected area near a tiny hole is provided and a simple optimization design of cooling pipe system is performed. The results demonstrate advantages of the proposed model which can effectively handle heat conduction problem with multiple holes subjected to the third kind boundary conditions.

Numerical and Experimental Stability Analysis Predicting Natural Laminar Flow Extension on a Realistic Swept Wing

It is well known that the stability theory, coupled with eN method, offers a rational approach to predict transition from laminar to turbulent flow condition in several aerodynamic applications. It is also known that stability analyses are affected by the accuracy of the mean flow that strongly depends from an adequate resolution in the boundary layer region. Unfortunately, Navier-Stokes analyses for stability investigations are often not affordable since the large number of points normal to the wall, necessary in the viscous layer to have desiderata boundary layer accuracy, makes RANS computations extremely time consuming. Furthermore, the classical definition of the boundary layer edge (the height where the boundary layer velocity reaches 99% of the no-viscous velocity) is not a straightforward task. For these reasons, in the framework of EU-funded Research Project RECEPT, a three-step approach has been applied to perform stability analyses. The first step consists of Euler computations to evaluate the mean flow. In the second step, a 3-D boundary-layer code, starting from the no-viscous flow field, allows to compute the boundary layer data necessary to perform the linear stability analysis and, finally, a 2.5D linear stability code is used to predict transition location. The reference geometry used to test this approach is the laminar wing developed by CIRA and Piaggio Aero Industries S.p.A. (PAI) in the framework of an Italian national program called VITAS (Vettore Innovativo per il Trasporto AeroSostenibile). This wing has been selected because an extended set of experimental data is available. Among the available experimental results, four different conditions have been selected and numerical analyses were performed. The selection has been driven preferring the data that presented both the best IR (Infra-Red) camera results and the pressure distributions. Euler computations have been performed using the commercial CFD (Computational Fluid Dynamic) code CFD++ and the boundary layer computations have been performed using the ONERA (Office National d'Études et de Recherches Aérospatiales) 3C3D laminar/turbulent boundary layer code. Stability computations have been performed using the compressible code NOLLI (NOn Local Linear Instability code) developed in house by CIRA and based on the application of the multiple scale technique and ray tracing theory. For stability analyses the fixed span- wise wavenumber strategy has been used.

Nonlinear stability of electro-visco-elastic Walters’ B type in porous media

This paper investigates the nonlinear instability of a non-Newtonian fluid of the Walters’ B type. The fluids fill the regions inside and outside a vertical circular cylinder. An axial electric field of uniform strength is pervaded along the axis of the jet. The fluids are saturated in porous media. Typically, the nonlinear analysis is based on solving the linear governing equations of motion, then applying the convenient nonlinear boundary conditions. This methodology yields a nonlinear characteristic equation which governs the behavior of the interface deflection. As the nonlinear terms are omitted, a linear dispersion relation arises. Therefore, the stability criteria are analytically analyzed and numerically confirmed. The nonlinear approach depends on the multiple time scale technique together with the support of the Taylor theory. This approach resulted in a Ginzburg–Landau equation. Consequently, the stability criteria are achieved in both analytical and numerical analysis. Furthermore, by means of the expanded frequency analysis, a bounded approximate solution of the amplitude of the surface waves is accomplished. The homotopy perturbation method (MPM) is utilized to obtain an approximate distribution of the conducted artificial frequency. Additionally, the generating function of the interface is graphically represented. Several special cases are reported upon convenient data choices. Regions of stability and instability are addressed. In the stability profile, the electric field intensity is plotted versus the wave number. The influences of the parameters on the stability are identified. The nonlinear stability approach divides the phase plane into several parts of stability/instability.

Systematically Improving Espresso: Insights from Mathematical Modeling and Experiment

Summary Espresso is a beverage brewed using hot, high-pressure water forced through a bed of roasted coffee. Despite being one of the most widely consumed coffee formats, it is also the most susceptible to variation. We report a novel model, complimented by experiment, that is able to isolate the contributions of several brewing variables, thereby disentangling some of the sources of variation in espresso extraction. Under the key assumption of homogeneous flow through the coffee bed, a monotonic decrease in extraction yield with increasingly coarse grind settings is predicted. However, experimental measurements show a peak in the extraction yield versus grind setting relationship, with lower extraction yields at both very coarse and fine settings. This result strongly suggests that inhomogeneous flow is operative at fine grind settings, resulting in poor reproducibility and wasted raw material. With instruction from our model, we outline a procedure to eliminate these shortcomings.

Динамика двух лазеров, связанных взаимной модуляцией добротности

В приближении скоростных уравнений построена математическая модель синхронизации двух одномодовых полупроводниковых лазеров. Модель представляет собой систему нелинейных обыкновенных дифференциальных уравнений для концентраций электронов и плотностей фотонов в обоих лазерах. Особенность модели состоит в характере связи между лазерами — она устроена так, что удельные потери каждого лазера пропорциональны интенсивности излучения другого. Помимо того, в уравнениях учитываются нелинейные (квадратичные) потери излучения. Система исследована методами теории бифуркаций и многих масштабов. Показано, что модель проявляет различное поведениепри сильной и слабой связи между лазерами. При сильной связи в системе нет колебаний, и в то же время возможны бистабильность и гистерезис. При слабой связи в системе возможно возникновение предельного цикла и пичковых релаксационных колебаний. Выявлено, что синхронные колебания плотности фотонов в лазерах происходят в антифазе. Сделан вывод о возможности применения предложенной схемы синхронизации для получения длиннопериодических осцилляций.

Asymptotic Description of Forced Response Vibration Saturation by Friction Forces

Abstract The estimation of the final vibration amplitude of a turbomachinery bladed disk is of extreme practical importance; it is an essential information for the prediction of the level of high cycle fatigue of the blades and for the subsequent estimation of its operative life span. The forced response vibration is saturated by the nonlinear damping introduced by the friction forces at the interfaces between blade and disk (and/or at the included dampers). The computation of the final amplitude of the limit cycle oscillation requires to solve a quite complicated nonlinear problem. In the case of a tuned bladed disk, this problem can be reduced to a single sector calculation with phase lag boundary conditions. The solution of this one-sector problem requires to consider many harmonics in order to capture the details of the nonlinear time periodic oscillation that sets in. If the small unavoidable differences among blades (mistuning) are also taken into account, then the situation becomes even more complicated because the solution of the mistuned vibration problem requires to consider not only a single sector but also the complete bladed disk. The possibility of applying multiple scales techniques to drastically simplify this problem is explored in this paper. The idea is to exploit the fact that all relevant effects present (forcing, nonlinear friction, and mistuning) are, in most practical situations, small effects that develop in a time scale that is much longer than that associated with the natural elastic vibration frequency of the tuned system. A mass-spring model with microslip nonlinear friction is used to represent the forced bladed disk. The multiple scales method is used to asymptotically derive simplified models for both tuned and mistuned configurations. The results of the asymptotic model are compared with those from the mass-spring system and used to analyze the particular characteristics of the nonlinear friction effects on the final vibration states.

The MAPS with polynomial basis functions for solving axisymmetric time-fractional equations

This study presents a parallel meshless approach for time-fractional equations in the axisymmetric geometry. In the present parallel meshless approach, the method of approximate particular solutions (MAPS), in conjunction with Laplace transform, extended precision arithmetic (EPA) and multiple scale technique (MST) is implemented. In this application, the Laplace transform is employed to eliminate the dependence on time. The MAPS is used to obtain the solution of time-independent equation in Laplace space. Furthermore, the solution of time dependent equation is restored by Stehfest’s algorithm. Here the multiple scale technique (MST) and extended precision arithmetic (EPA) are applied to alleviate the effect of an ill-conditioned matrix on a numerical inverse Laplace transform. Then multi-cores parallel computing can improve effectively computing speed of the proposed meshless approach. In the numerical implementation, both the 2D axisymmetric equations and the original 3D equations are considered and compared.

A Meshfree Computational Approach Based on Multiple-Scale Pascal Polynomials for Numerical Solution of a 2D Elliptic Problem with Nonlocal Boundary Conditions

A two-dimensional (2D) elliptic problem with nonlocal boundary conditions on both regular and irregular domains is solved numerically by Pascal polynomial basis unified with multiple-scale technique. Very accurate numerical solutions and quite reasonable condition numbers are obtained with the proposed method which is also a truly meshfree method since difficult meshing processes or numerical integrations over domains are not needed for considered problems. Four test problems are solved to show the accuracy and efficiency of the proposed method. Also stability of the method is studied against large noise effect.

Periodic solution of the nonlinear Sitnikov restricted three-body problem

The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time.

Nonlinear structural behavior of a size-dependent MEMS gyroscope assuming a non-trivial shaped proof mass

In this paper, a size-dependent based non-classical mechanics model for the structural behavior of a MEMS (micro-electro-mechanical systems) gyroscope is investigated. The micro-cantilever based gyroscope is possessing a proof mass at its free end, assumed to hold a non-negligible length as compared to the micro-cantilever’s length. The proof mass is triggered through an actuating electrode and at the same time is assuming a sensing electrode together assuming parallel-plates capacitive arrangements. The governing equations of the micro-gyroscope system are derived within the framework of a modified couple stress non-classical mechanics theory. Based on the resulting equations, the static and dynamic analyses of the system are performed to estimate the pull-in instability voltages, natural frequency, and dynamic responses of the sensing electrodes near primary resonance that are caused by the use of point mass assumption and classical theory. A mixed based method involving both the Galerkin modal expansion procedure along with the method of multiple scales (MMS) technique is utilized to carry out the micro-gyroscope primary resonance analysis through plotting its respective frequency responses near its fundamental mode. Simulated results show that the proof mass dimension and its respective size effects play a substantial role in drastically changing the initiation of the system pull-in instability along its actuation direction, its fundamental natural frequency, and accordingly its dynamic amplitude along its corresponding sensing direction.

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