DEFECT AND IMPROVEMENT OF TWO-TIME EXPANSION METHOD IN NONLINEAR OSCILLATIONS

In this paper, the defect of the Two-Time Expansion method is indicated and an improvement of this method is suggested. Certain examples.in which the present method is used, are given. Moreover, the paper shows the equivalence of the improved Two-Time Expansion Method and the method of KBM(Kryloy-Bogoliuboy-Mitropolski).

Nonlinear Oscillations of a Pair of Charged Rings

Two electrically charged rings of different sizes are assembled along their common vertical symmetry axis through their centers. The bottom ring is secured on a horizontal support while the top one is loose. For a set of practical values characterizing the charged rings we envision a scenario where the mutual electric repulsion between the rings and the weight of the top ring results in stable nonlinear oscillations. To quantify the characteristics of the oscillations, we utilize a Computer Algebra System specifically Mathematica [1]. We accompany the analysis with a simulation for a comprehensive visual understanding.

Impact of Eccentricity on Nonlinear Oscillations of a Point-Like Charge in the Electric Field of a Curvature-Dependent Elliptic Charged Ellipse

Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly Mathematica [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.

Nonlinear Oscillations of Semigeostrophic Eady Waves in the Presence of Diffusivity

Analyses are performed to examine the physical processes involved innonlinear oscillations of Eady baroclinic waves obtained from viscous semigeostrophic models withtwo types of boundary conditions (free-slip and non-slip). By comparing with previous studies forthe case of the free-slip boundary condition, it is shown that the nonlinear oscillations areproduced mainly by the interaction between the baroclinic wave and zonal-mean state (totalzonal-mean flow velocity and buoyancy stratification) but the timescale of the nonlinearoscillations is largely controlled by the diffusivity. When the boundary condition is non-slip, thenonlinear oscillations are further damped and slowed by the diffusive process. Since the free-slip(non-slip) boundary condition is the zero drag (infinite drag) limit of the more realistic dragboundary condition, the nonlinear oscillations obtained with the two types of boundary conditionsare two extremes for more realistic nonlinear oscillations.

Nonlinear Oscillations Analysis of the Elevator Cable in a Drum Drive Elevator System

This paper aims to investigate nonlinear oscillations of an elevator cable in a drum drive.The governing equation of motion of the objective system is developed by virtue of Lagrangian’s method.A complicated term is broached in the governing equation of the motion of the system owing to existence of multiplication of a quadratic function of velocity with a sinusoidal function of displacement in the kinetic energy of the system.The obtained equation is an example of a well-known category of nonlinear oscillators,namely,non-natural systems.Due to the complex terms in the governing equation,perturbation methods cannot directly extract any closed form expressions for the natural frequency.Unavoidably,different non-perturbative approaches are employed to solve the problem and to elicit a closed-form expression for the natural frequency.Energy balance method,modified energy balance method and variational approach are utilized for frequency analyzing of the system.Frequencyamplitude relationships are analytically obtained for nonlinear vibration of the elevator’s drum.In order to examine accuracy of the obtained results,exact solutions are numerically obtained and then compared with those obtained from approximate closed-form solutions for several cases.In a parametric study for different nonlinear parameters,variation of the natural frequencies against the initial amplitude is investigated.Accuracy of the three different approaches is then discussed for both small and large amplitudes of the oscillations.

Straightforward Method for Nonlinear Oscillators

It is challenging to predict the frequency property of a nonlinear vibration system conveniently and efficiently.Especially,an invalid or physically irrelevant result might be obtained by some advanced methods.Therefore,predicting the frequency lacks an expedient and efficient method.This challenge is addressed by developing a straightforward and effective frequency formulation that reliably predicts the frequency-amplitude relationship.This study provides a one-step approach which can fast determine the periodic properties of any conservative oscillators and also provides a reference for other similar studies.

C- METHOD AND ITS APPLICATION TO ENGINEERING NONLINEAR DYNAMICAL PROBLEMS

The C-L method was generalized from Liapunov-Schmidt reduction method, combined with theory of singularities, for study of non-autonomous dynamical systems to obtain the typical bifurcating response curves in the system parameter spaces. This method has been used, ar an example, to analyze the engineering nonlinear dynamical problems by obtaining the bifurcation programs and response curves which are useful in developing techniques of control to subharmonic instability of large rotating machinery.

ON INTERNAL RESONANCE OF NONLINEAR VIBRATING SYSTEMS WITH MANY DEGREES OF FREEDOM

The problem of periodic solutions of nonlinear autonomous systems with many degrees of freedom is considered. This is made possible by the development of a modified version of the KBM method[1]. The method can be used to generate limit cycle phase portrait, amplitude, period and to indicate stability of the limit cycle.

Coulomb, Yukawa, and Hooke’s Oscillations

Oscillations due to three different forces in three areas of physics: electrostatic, nuclear, and mechanics, are analyzed. The electrostatic long-range Coulomb force has a different character than the nucleonic short-range Yukawa force. Both are different from the linear Hooke’s force. The equation of motion of each case is solved applying a Computer Algebra System (CAS). It is shown that these oscillations have similarities and differences. Phase diagrams of all three cases are compared.

A hyperbolic Lindstedt-Poincare method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators

A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Lienard oscillator is studied in detail, and the present method＇s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.

NONLINEAR TIME TRANSFORMATION METHOD FOR STRONG NONLINEAR OSCILLATION SYSTEMS

In this paper,a nonlinear time transformation method is presented for the analysis of strong nonlinear oscillation systems.This method can be used to study the limit cycle behavior of the autonomous systems and to analyze the forced vibration of a strong nonlinear system.

Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

In the optimal control problem of nonlinear dynamical system,the Hamiltonian formulation is useful and powerful to solve an optimal control force.However,the resulting Euler-Lagrange equations are not easy to solve,when the performance index is complicated,because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations.To be a numerical method,it is hard to exactly preserve all the specified conditions,which might deteriorate the accuracy of numerical solution.With this in mind,we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator,which can exactly satisfy all the required conditions for the minimality of the performance index.A new idea of shape functions method(SFM)is introduced,from which we can transform the optimal control problems to the initial value problems for the new variables,whose initial values are given arbitrarily,and meanwhile the terminal values are determined iteratively.Numerical examples confirm the high-performance of the iterative algorithms based on the SFM,which are convergence fast,and also provide very accurate solutions.The new algorithm is robust,even large noise is imposed on the input data.

Coupled large earthquakes in the Baikal rift system: Response to bifurcations in nonlinear resonance hysteresis

The current lithospheric geodynamics and tectonophysics in the Baikal rift are discussed in terms of a nonlinear oscillator with dissipation.The nonlinear oscillator model is applicable to the area because stress change shows up as quasi-periodic inharmonic oscillations at rifting attractor structures （RAS）.The model is consistent with the space-time patterns of regional seismicity in which coupled large earthquakes,proximal in time but distant in space,may be a response to bifurcations in nonlinear resonance hysteresis in a system of three oscillators corresponding to the rifting attractors.The space-time distribution of coupled MLH ＞ 5.5 events has been stable for the period of instrumental seismicity,with the largest events occurring in pairs,one shortly after another,on two ends of the rift system and with couples of smaller events in the central part of the rift.The event couples appear as peaks of earthquake ‘migration＇ rate with an approximately decadal periodicity.Thus the energy accumulated at RAS is released in coupled large events by the mechanism of nonlinear oscillators with dissipation.The new knowledge,with special focus on space-time rifting attractors and bifurcations in a system of nonlinear resonance hysteresis,may be of theoretical and practical value for earthquake prediction issues.Extrapolation of the results into the nearest future indicates the probability of such a bifurcation in the region,i.e.,there is growing risk of a pending M ≈ 7 coupled event to happen within a few years.

A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.

Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the non- linearity of the airfoil section＇s freeplay. There are two crit- ical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincar6 map is constructed for the limit cycle os- cillations by using piecewise-linear solutions with and with- out contact in the system. Through analysis of the Poincar6 map, a series of equations which can determine the frequen- cies of period-1 limit cycle oscillations at any flight veloc- ity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agree- ment is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. More- over, there exist multi-periods limit cycle oscillations and even complicated ＂chaotic＂ oscillations may occur, which are usually found in smooth nonlinear dynamic systems.

Nonlinear oscillation characteristics of magnetic microbubbles under acoustic and magnetic fields

Microbubbles loaded with magnetic nanoparticles(MMBs) have attracted increasing interests in multimode imaging and drug/gene delivery and targeted therapy. However, the dynamic behaviors generated in diagnostic and therapeutic applications are not clear. In the present work, a novel theoretical model of a single MMB was developed, and the dynamic responses in an infinite viscous fluid were investigated under simultaneous exposure to magnetic and acoustic fields. The results showed that the amplitude reduces and the resonant frequency increases with the strength of the applied steady magnetic field and the susceptibility of the magnetic shell. However, the magnetic field has a limited influence on the oscillating. It is also noticed that the responses of MMB to a time-varying magnetic field is different from a steady magnetic field. The subharmonic components increase firstly and then decrease with the frequency of the magnetic field and the enhanced effect is related to the acoustic driving frequency. It is indicated that there may be a coupling interaction effect between the acoustic and magnetic fields.

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

The periodic motion and stability for a class of two-degree-of-freedom nonlinear oscillating systems are studied by using the method of Liapunov function. The sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions are obtained.

Homotopy Analysis Method for a Conservative Nonlinear Oscillator with Fractional Power

In this paper, the homotopy analysis method is applied to deduce the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to u1/3. By introducing the auxiliary linear operator and the initial guess of solution, the homotopy analysis solving is set up. By choosing the suitable convergence-control parameter, the accurate high-order approximations of solution and frequency for the whole range of initial amplitudes can easily be obtained. Comparison of the results obtained using this method with those obtained by different methods reveals that the former is more accurate, effective and convenient for these types of nonlinear oscillators.

High frequency forcing on nonlinear systems

High-frequency signals are pervasive in many science and engineering fields.In this work,the effect of high-frequency driving on general nonlinear systems is investigated,and an effective equation for slow motion is derived by extending the inertial approximation for the direct separation of fast and slow motions.Based on this theory,a high-frequency force can induce various phase transitions of a system by changing its amplitude and frequency.Numerical simulations on several nonlinear oscillator systems show a very good agreement with the theoretic results.These findings may shed light on our understanding of the dynamics of nonlinear systems subject to a periodic force.