Abstract:
The Duffing equation describes the oscillations of a damped nonlinear oscillator [1]. Its non-linearity is confined to a one coordinate-dependent cubic term. Its applications describing a mechanical system is limited e.g. oscillations of a theoretical weightless-spring. We propose generalizing the mathematical features of the Duffing equation by including in addition to the cubic term unlimited number of odd powers of coordinate-dependent terms. The proposed generalization describes a true mass-less magneto static-spring capable of performing highly non-linear oscillations. The equation describing the motion is a super non-linear ODE. Utilizing Mathematica [2] we solve the equation numerically displaying its time series. We investigate the impact of the proposed generalization on a handful of kinematic quantities. For a comprehensive understanding utilizing Mathematica animation we bring to life the non-linear oscillations.

Abstract:
The original idea of quantum optical spring arises from the requirement of quantization of the frequency of oscillations in Hamiltonian of harmonic oscillator. This purpose is achieved by considering a spring whose constant (and so its frequency) depends on the quantum states of another system. Recently, it is realized that by the assumption of frequency modulation of $\omega$ to $\omega\sqrt{1+\mu a^\dagger a}$ the mentioned idea can be established. In the present paper we generalize the approach of quantum optical spring (has been called by us as nonlinear quantum optical spring) with attention to the {\it dependence of frequency to the intensity of radiation field} that {\it naturally} observed in nonlinear coherent states. Then, after the introduction of the generalized Hamiltonian of nonlinear quantum optical spring and it's solution, we will investigate the nonclassical properties of the obtained states. Specially, typical collapse and revival in the distribution functions and squeezing parameters as particular quantum features will be revealed.

Abstract:
The method of mathematical modeling of valve spring with the help of hull girder is presented. The technique of solution of equation of engine valve spring coil oscillations using numerical integration is generally considered. The analysis of theoretical and experimental research of VAZ valve spring coil oscillations by the example of rigid valve actuating gear is also given.

Abstract:
Leaf springs are special kind of springs used in automobile suspension systems. The advantage of leaf spring over helical spring is that the ends of the spring may be guided along a definite path as it deflects to act as a structural member in addition to energy absorbing device. The main function of leaf spring is not only tosupport vertical load but also to isolate road induced vibrations. It is subjected to millions of load cycles leading to fatigue failure. Static analysis determines the safe stress and corresponding pay load of the leaf spring and also to study the behavior of structures under practical conditions. The present work attempts to analyze the safeload of the leaf spring, which will indicate the speed at which a comfortable speed and safe drive is possible. A typical leaf spring configuration of TATA-407 light commercial vehicle is chosen for study. Finite element analysis has been carried out to determine the safe stresses and pay loads.

Abstract:
Quantum oscillations of nonlinear resistance are investigated in response to electric current and magnetic field applied perpendicular to single GaAs quantum wells with two populated subbands. At small magnetic fields current-induced oscillations appear as Landau-Zener transitions between Landau levels inside the lowest subband. Period of these oscillations is proportional to the magnetic field. At high magnetic fields different kind of quantum oscillations emerges with a period,which is independent of the magnetic field. At a fixed current the oscillations are periodic in inverse magnetic field with a period that is independent of the dc bias. The proposed model considers these oscillations as a result of spatial variations of the energy separation between two subbands induced by the electric current.

Abstract:
The nonlinear oscillations of a scalar field are shown to have cosmological equations of state with $w = p / \rho$ ranging from $-1 < w < 1$. We investigate the possibility that the dark energy is due to such oscillations.

Abstract:
The spring-slider is a simple dynamical system consisting in a massive block sliding with friction and pulled through a spring at a given velocity. Understanding the block motion is fundamental for studying more complex phenomena of frictional sliding, such as the seismogenic fault motion. We analyze the dynamical properties of the system, subject to rate- and state-dependent friction laws and forced at a constant load velocity. In particular we study the limits within which the quasi-static model can be used. The latter model approximates the complete model of the system without taking into account the inertia effects. The system parameters are here found to be grouped into three characteristic times of the three dynamics present in the complete model. A necessary condition for the quasi-static approximation to hold is that the characteristic time of the inertial equation is much smaller than the other two characteristic times. We have studied a modification of one of the classical forms of the rate- and state-dependent friction laws. Subsequently we have developed a linear analysis in the neighbourhood of the equilibrium point of the system. For the quasi-static model we rigorously found, by means of a nonlinear analysis, a supercritical Hopf bifurcation, a dynamical property of the complete model. The classical form of the friction laws can be obtained as a particular case of the one we considered, but fails to preserve the Hopf bifurcation in the quasi-static approximation. We conclude that to have a good quasi-static approximation of the system, even in nonlinear conditions, the form of the friction laws considered is a critical factor.

Abstract:
The dynamics in a nonlinear Schrodinger chain in an homogeneous electric field is studied. We show that discrete translational invariant integrability-breaking terms can freeze the Bloch nonlinear oscillations and introduce new faster frequencies in their dynamics. These phenomena are studied by direct numerical integration and through an adiabatic approximation. The adiabatic approximation allows a description in terms of an effective potential that greatly clarifies the phenomenon.

Abstract:
A new simple method to measure the spatial distribution of the electric field in the plasma sheath is proposed. The method is based on the experimental investigation of vertical oscillations of a single particle in the sheath of a low-pressure radio-frequency discharge. It is shown that the oscillations become strongly nonlinear and secondary harmonics are generated as the amplitude increases. The theory of anharmonic oscillations provides a good qualitative description of the data and gives estimates for the first two anharmonic terms in an expansion of the sheath potential around the particle equilibrium.

Abstract:
Effects of the static electric field on the splitting and annihilation widths of the levels of antiprotonic hydrogen with a large principal quantum number (n=30) are studied. Non-trivial aspects of the consideration is related with instability of (p\bar{p})^*-atom in ns and np-states due to coupling of these states with the annihilation channels. Properties of the mixed nl-levels are investigated depending on the value of external static electric field. Specific resonance-like dependence of effective annihilation widths on the strength of the field is revealed.