The current article is concerned with a comprehensive investigation in achieving approximate solutions of coupled Nonlinear Oscillations with high Nonlinearity. These equations are highly Nonlinear second-order ordinary differential equations. Via a coupling of the Homotopy perturbation method and Laplace transforms, which is so-called the He-Laplace method, traditional approximate solutions involving the secular terms are accomplished. On the other hand, in order to cancel the secular terms, an expanded frequency technique is adapted to accomplish periodic approximate solutions. Therefore, a Nonlinear frequency, for each differential equation, is achieved. Furthermore, for more convenience, these solutions are pictured to indicate their behavior. The multiple time-scales with the aid of the Homotopy concept are utilized to judge the stability criteria. The analyses reveal the resonance as well as the non-resonant cases. Additionally, numerical calculations are carried out, graphically, to address the regions that guaranteed the bounded solutions. It is found that the latter method, is the most powerful mathematical tool in extracting the stability analysis of the considered system.

Purpose: This paper proposes the use of different analytical methods in obtaining approximate solutions for Nonlinear differential equations with Oscillations.Methods: Three methods are considered in this paper: Lindstedt-Poincare method, the Krylov-Bogoliubov first approximate method, and the differential transform method.Results: Figures that are given in this paper give a strong evidence that the proposed methods are effective in handling Nonlinear differential equations with Oscillations.Conclusions: This study reveals that the differential transform method provides a remarkable precision compared with other perturbation methods.

A viscoelastic microcantilever beam is analytically analyzed based on the modi ed strain gradient theory. Kelvin-Voigt scheme is used to model beam viscoelasticity. By applying Euler-Bernoulli inextensibility of the centerline condition based on Hamilton's principle, the Nonlinear equation of motion and the related boundary conditions are derived from shortening e ect theory and discretized by Galerkin method. Inner damping, Nonlinear curvature e ect, and Nonlinear inertia terms are also taken into account. In the present study, the generalized derived formulation allows modeling any Nonlinear combination such as Nonlinear terms that arise due to inertia, damping, and sti ness, as well as modeling the size e ect using modi ed coupled stress or modi ed strain gradient theories. First-mode Nonlinear frequency and time response of the viscoelastic microcantilever beam are analytically evaluated using multiple time scale method and then, validated through numerical  ndings. The obtained results indicate that Nonlinear terms have an appreciable e ect on natural frequency and time response of a viscoelastic microcantilever. Moreover, further investigations suggest that due to the size e ects, natural frequency would drastically increase, especially when the thickness of the beam and the length scale parameter are comparable. The  ndings elaborate the signi cance of size e ects in analyzing the mechanical behavior of small-scale structures.

During the ultrasound imaging process, the ultrasound contrast agents (UCAs) are beating near the blood vessel wall. Therefore, the purpose of the present simulation study is to investigate the effect of the presence of an elastic wall on the radial and frequency acoustic response of a UCA microbubble oscillating in a Nonlinear regime. For this reason, the numerical simulation of the dynamic behavior of a coated microbubble was performed using coding in MATLAB and a Rayleigh-Plesset equation modified by Doinikov. To study the Nonlinear bubble Oscillations, its compression-only behavior and the sub-harmonic Nonlinear component are taken from a Nonlinear shell model presented by Marmottant et al. Initially, coated bubble Oscillations in two linear and Nonlinear regimes were investigated for two types of shell models, and it was observed that presence of the elastic wall affects the bubble's compression-only behavior. Finally, due to the importance of the subharmonic component in the Nonlinear oscillation of the coated bubble, the threshold of the appearance of subharmonic components for a coated bubble near an elastic wall was investigated using the Fast Fourier Transform (FFT) and compared with the oscillation in the infinite fluid.

Stability and damping Oscillations during fault occurrence are two important challenges which have attracted the attention of the power systems designers. In this paper Improvement of the Damping of Multi Machine Power System Oscillations is considered by using UPFC. The applied control algorithm for the damping Oscillations is the Multi-input Nonlinear Back-stepping Controller. The injection model of UPFC is used for modeling of UPFC, Using of this model decreases 4 input control of UPFC to two inputs and the performance of power system is improved by this control method. Performance of this control algorithm in comparison with classical method is shown, more robustness, more independent to the work point and using of whole control variables of UPFC. The proposed control method, is simulated for a single machine connected to infinite bus and a multi-machine(9 bus IEEE) power system. The results of simulation, is illustrated that the performance of designed multi-input Back-stepping controller rather is better than conventional controller.

In this paper, Nonlinear Oscillations of a charged spherical oscillator under a magnetic field are examined. The oscillator is under the influence of gravity and magnetic fields, and air resistance is considered a damping effect. Using the Hamiltonian principle, the two-dimensional equations of motion are extracted, and by considering a constant velocity in one dimension, they are transformed into a one-dimensional Nonlinear equation. To obtain an analytical solution, the differential transform method (DTM) is employed. Due to the oscillatory nature of the problem, the solution obtained by the DTM is improved using the Padé approximation technique. The proposed method is then applied to the oscillator for various system parameters and initial conditions. The obtained results are compared with the numerical fourth-order Runge-Kutta method, and good agreement is observed. The proposed method is simple in implementation and provides a high level of accuracy. Hence, the method is suggested for the vibration analysis of Nonlinear oscillators including damping effect

Nonlinear acoustic Oscillations of large amplitude created in a gas-filled tube under the action of two pistons located at the ends of the pipe are numerically investigated. The pistons oscillate according to the harmonic law at one of the natural frequencies and with different values of phase shift. The movement of the gas is described by mathematical equations of conservation for the main determining relations for the flow, which are estimated by applying the finite volume method based on OpenFOAM package. The non-stationary forced oscillatory motion of a gas inside an axisymmetric tube from a state of rest to a periodic steady motion is investigated. The features of Nonlinear acoustic fluctuations of gas in cylindrical duct under the action of two pistons are found. The effect of the phase shift value has a strong effect on the oscillation amplitude of gas, when pistons oscillating at equal natural frequencies, in turn, when the pistons oscillate at different natural frequencies, the effect is very small. In particular, resonant Oscillations are detected when the pistons vibrate at the same frequency values equal to odd values of their own higher harmonics in the absence of a phase shift value. In the case when the frequency values are equal to even values of the natural harmonics, resonant Oscillations occur when the pistons move in anti-phase. The numerical method appears to work well and would be hoped for practical computations of different resonators.

This work is a study of the Earth's free Oscillations considering a merge of solid and liquid model. At the turn of 19th century Geophysicists presented the theory of the free Oscillations for a self-gravitating, isotropic and compressible sphere. Assuming a steel structure for an Earth size sphere, they predicted a period of oscillation of about 1 hour. About 50 years later, the free Oscillations of stars was studied by Cowling and others. They classified the oscillation modes of the stars into acoustic and gravity modes on the basis of their driving forces. These are pressure and buoyancy forces respectively. The earliest measurements for the period of the free Oscillations of the Earth were made by Benyove from a study of Kamchathca earthquake. Since then, the Geophysicists have been trying to provide a theoretical basis for these measurements.  Recently, the theory concerning Oscillations of celestial fluids is extended by Sobouti to include the possible Oscillations of the Earthlike bodies. Using the same technique, we study the free Oscillations of a spherically symmetric, non-rotating and elastic model for the Earth.We used the actual data of the Earth's interior structure in our numerical calculations. Numerical results show that there exist three distinct oscillation modes namely acoustic, gravity and toroidal modes. These modes are driven by pressure, buoyancy and shear forces respectively. The shear force is due to the elastic properties of the solid part of the Earth. Our numerical results are consistent with the seismic data recorded from earthquake measurements.

Solar magnetic tornadoes are known to be one of the mass and energy transport mech-anisms from the lower solar atmosphere into the upper layers of the solar corona. A bright spiral structure with two arms is observed using high-cadence EUV images of 171, 193 and 304  A channels of Atmospheric Imaging Assembly (AIA) aboard the So-lar Dynamics Observatory (SDO) on 10th of July 2011 for three hours. The structure studied here looks bright in front of dark background emissions. The rotational energy budget of this tornado is estimated using three di erent approaches. Results showed the rotational energy ranges between 1: 95 1018 erg to 3: 36 1019 erg. After correcting for the solar di erential rotation, the oscillatory behaviour of the structure is studied using FFT technique. Results show four di erent regimes of Oscillations: 3-min, 5-6 min, 8-11 min, and 15-17 min. The showed origin of each of these Oscillations are discussed.