Application of the differential transformation method to the free vibrations of strongly non-linear oscillators

This paper adopts the differential transformation method to obtain the free vibration behavior of an oscillator with fifth-order non-linearities. The principle of differential transformation is briefly introduced, and is then applied in the derivation of a set of difference equations for the free vibration oscillator problem. The solutions are subsequently solved by a process of inverse transformation. The time responses of the oscillator are presented under different parameter conditions, and the current results are then compared with those derived from the established Runge–Kutta method in order to verify the accuracy of the proposed method. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the proposed differential transformation method is a powerful and efficient tool for solving non-linear problems.     

Effect of loading history on the free vibrations of an elastic-hereditary oscillator

The effect of the dynamic loading history on the free vibrations of an elastic-hereditary oscillator is examined. The response of the material is described by an integral relation of the heriditary type with an exponential-fractional relaxation kernel. An integrooperator representation of the starting equation makes it possible to obtain a closed solution to the problem of free vibrations of an elastic-heriditary oscillator for certain specific loading histories. The results are illustrated with reference to polyamide (kapron).Mekhanika Polimerov, Vol. 4, No. 2, pp. 222–226, 1968     

On the motion of an oscillator with a periodically time-varying mass

The stability of the motion of an oscillator with a periodically time-varying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [W.T. van Horssen, A.K. Abramian, Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. Sound Vibration 298 (2006) 1166–1172] to the forced case; second, for a weakly forced Duffing’s oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero.     

Using the Evolution Operator Method to Describe a Particle in a Homogeneous Alternating Field

Using the evolution operator method, we construct coherent states of a nonrelativistic free particle with a variable mass M(t) and a nonrelativistic particle with a variable mass M(t) in a homogeneous alternating field. Under certain physical conditions, they can be regarded as semiclassical states of particles. We discuss the properties (in particular, the completeness relation, the minimization of the uncertainty relations, and the time evolution of the corresponding probability density) of the found coherent states in detail. We also construct exact wave functions of the oscillator type and of the plane-wave type for the considered systems and compute the quantum Wigner distribution functions for the wave functions of coherent and oscillator states. We establish the unitary equivalence of the problems of a free particle and a particle in a homogeneous alternating field.     

Controlling bifurcation and chaos of a plastic impact oscillator

A two-degree-of-freedom plastic impact oscillator is considered. Based on the analysis of sticking and non-sticking impact motions of the system, we introduce a three-dimensional impact Poincaré map with dynamical variables defined at the impact instants. The plastic impacts complicate the dynamic responses of the impact oscillator considerably. Consequently, the piecewise property and singularity are found to exist in the three-dimensional map. The piecewise property is caused by the transitions of free flight and sticking motions of two masses immediately after impact, and the singularity of the map is generated via the grazing contact of two masses and the instability of their corresponding periodic motions. The nonlinear dynamics of the plastic impact oscillator is analyzed by using the Poincaré map. The simulated results show that the dynamic behavior of this system is very complex under parameter variation, varying from different types of sticking or non-sticking periodic motions to chaos. Suppressing bifurcation and chaotic-impact motions is studied by using an external driving force, delay feedback and damping control law. The effectiveness of these methods is demonstrated by the presentation of examples of suppressing bifurcations and chaos for the plastic impact oscillator.     

Stochastic Schrodinger Equation for a Quantum Oscillator with Dissipation

In this paper, we construct an exact solution of the stochastic Schrodinger equation for a quantum oscillator with possible dissipation of energy taken into account. Using the explicit form of the solution, we calculate estimates for the characteristic damping time of free damped oscillations. In the case of forced oscillations, we obtain formulas for the Q-factor of the system and for the variance of the coordinate and momentum of a quantum oscillator with dissipation. We obtain the quantum analog of the classical diffusion equation and explicitly show that the equations of motion for the mean value of the momentum operator following from the solution of the stochastic Schrodinger equation play the role of the quantum Langevin equation describing Brownian motion under the action of a stochastic force.     

Wave front set for solutions to Schrödinger equations with long-range perturbed harmonic oscillators

In this paper we consider a Schrödinger operator with variable coefficients and harmonic potential. The perturbation is assumed to be long-range in a sense similar to the work of Nakamura (2009) [13]. We construct a modified propagator, and then by using this propagator and also the propagator of the unperturbed free harmonic oscillator we characterize the propagation of singularities for solutions to the equations.     

Symmetry group classification of three-dimensional Hamiltonian systems

We present some results on the symmetry group classification for an autonomous Hamiltonian system with three degrees of freedom. The potentials considered are natural, i.e., depend on the position variables only and the symmetries considered are Lie point symmetries. With the exception of the harmonic oscillator or a free particle where the dimension is 24, we obtain all dimensions between 1 and 12, except 8.     

Multirate models for simulating a colpitts oscillator

A model based on multirate partial differential algebraic equations yields an efficient numerical simulation of electric circuits in radio frequency applications. Considering frequency modulation, free parameters of the model are determined appropriately by a minimisation strategy. We apply the multirate approach to simulate a modified version of a Colpitts oscillator, which exhibits frequency modulation at widely separated time scales. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classical propagators of quadratic quantum systems

A new quantum mechanical formalism based on the probability representation of states is applied to ions in traps and stimulated Raman scattering. Explicit expressions are found for the classical propagators of a free particle, a harmonic oscillator, an ion in the Paul trap, an ion in the Penning trap, and stimulated Raman scattering. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 285–296, November, 1999.     

Universal and nonperturbative behavior in the one-plaquette model of two-dimensional string theory

The one-plaquette Hamiltonian of large N lattice gauge theory offers a constructive model of a 1+1-dimensional string theory with a stable ground state. The free energy is found to be equivalent to the partition function of a string where the world sheet is discretized by even polygons with signature and the link factor is given by a non-Gaussian propagator. At large, but finite, N we derive the nonperturbative density of states from the WKB wave function and the dispersion relations. This is expressible as an infinite, but convergent, series with the inverse of the hypergeometric function replacing the harmonic oscillator spectrum of the 1+1-dimensional string. In the scaling limit, the series is shown to be finite, containing both the perturbative (asymptotic) expansion of the inverted harmonic oscillator model, and a nonperturbative piece that survives the scaling limit.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98. No. 3, pp. 414–429, March, 1994.     

Small amplitude, free longitudinal vibrations of a load on a finitely deformed stress-softening spring with limiting extensibility

A constitutive theory for a general class of incompressible, isotropic stress-softening, limited elastic rubberlike materials is introduced. The model is applied to study the small amplitude, free longitudinal vibrational frequency of a load about a suspended static equilibrium stretch of a finitely deformed, stress-softening spring with limiting extensibility. A number of physical results, including bounds on the frequency, are reported. It is proved, for example, that the normalized vibrational frequency for the ideally elastic neo-Hookean oscillator is a lower bound for the normalized frequency of every incompressible, isotropic stress-softening, limited elastic oscillator within the general class. All results are illustrated for the special limited elastic Gent and the purely elastic Demiray biomaterial models, both with stress-softening characterized by a Zú?iga–Beatty front factor damage function. The results for both stress-softening models are compared with experimental data for several gum rubbers and thoracic aortic tissue provided by others; and, overall, it is found that the stress-softening, limited elastic Gent model best characterizes the data.     

Small amplitude, free longitudinal vibrations of a load on a finitely deformed stress-softening spring with limiting extensibility

A constitutive theory for a general class of incompressible, isotropic stress-softening, limited elastic rubberlike materials is introduced. The model is applied to study the small amplitude, free longitudinal vibrational frequency of a load about a suspended static equilibrium stretch of a finitely deformed, stress-softening spring with limiting extensibility. A number of physical results, including bounds on the frequency, are reported. It is proved, for example, that the normalized vibrational frequency for the ideally elastic neo-Hookean oscillator is a lower bound for the normalized frequency of every incompressible, isotropic stress-softening, limited elastic oscillator within the general class. All results are illustrated for the special limited elastic Gent and the purely elastic Demiray biomaterial models, both with stress-softening characterized by a Zú?iga–Beatty front factor damage function. The results for both stress-softening models are compared with experimental data for several gum rubbers and thoracic aortic tissue provided by others; and, overall, it is found that the stress-softening, limited elastic Gent model best characterizes the data.     

The Ma�CTrudinger�CWang curvature for natural mechanical actions

The Ma?CTrudinger?CWang curvature??or cross-curvature??is an object arising in the regularity theory of optimal transportation. If the transportation cost is derived from a Hamiltonian action, we show its cross-curvature can be expressed in terms of the associated Jacobi fields. Using this expression, we show the least action corresponding to a harmonic oscillator has zero cross-curvature, and in particular satisfies the necessary and sufficient condition (A3w) for the continuity of optimal maps. We go on to study gentle perturbations of the free action by a potential, and deduce conditions on the potential which guarantee either that the corresponding cost satisfies the more restrictive condition (A3s) of Ma, Trudinger and Wang, or in some cases has positive cross-curvature. In particular, the quartic potential of the anharmonic oscillator satisfies (A3s) in the perturbative regime.     

Forced vibrations of an oscillator in the presence of dry friction

We study the vibrations of an oscillator with two degrees of freedom in the presence of dry friction. We compare the nature of the damping of the free oscillations in a straight line with the general case. For the forced vibrations we determine the way in which the critical values of friction at which there exist periodic motions depend on the parameters of external action in resonance mode.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 49–54.     

Bistable Resonance In Gravity-driven Film Flows

We study the flow of a viscous liquid down an inclined plane with a sinusoidal bottom profile. Applying the integral boundarylayer method leads to a nonlinear ordinary differential equation for the film thickness. Linear resonance between the free surface and the bottom contour is observed via a regular perturbation approach. Considering nonlinear contributions, the resonance curve shows a foldover effect which is typical for nonlinear oscillators. In order to understand the origin of this bistable behaviour we derive a simple model equation and show an analogy to the Duffing oscillator. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dissipation effects in infinite-dimensional Hamiltonian systems

We show that the potential coupling of classical mechanical systems (an oscillator and a heat bath), one of which (the heat bath) is linear and infinite-dimensional, can provoke energy dissipation in a finitedimensional subsystem (the oscillator). Under natural assumptions, the final dynamics of an oscillator thus reduces to a tendency toward equilibrium. D. V. Treschev previously obtained results concerning the dynamics of an oscillator with one degree of freedom and a quadratic or (under some additional assumptions) polynomial potential. Later, A. V. Dymov considered the case of a linear oscillator with an arbitrary (finite) number of degrees of freedom. We generalize these results to the case of a heat bath (consisting of several components) and a multidimensional oscillator (either linear or nonlinear).     

A new exact solution of a damped quadratic non-linear oscillator

In this paper, we derive a new exact solution of the damped quadratic nonlinear oscillator (Helmholtz oscillator) based on the developed solution for the undamped case by the Jacobi elliptic functions. It is interesting to see that both of the damped Duffing oscillator and Helmholtz oscillator possess solutions that follow closely to the undamped case, and even the solution procedures are almost the same.     

On the dynamics of a periodic Colpitts oscillator forced by periodic and chaotic signals

In this paper, we have examined effects of forcing a periodic Colpitts oscillator with periodic and chaotic signals for different values of coupling factors. The forcing signal is generated in a master bias-tuned Colpitts oscillator having identical structure as that of the slave periodic oscillator. Numerically solving the system equations, it is observed that the slave oscillator goes to chaotic state through a period-doubling route for increasing strengths of the forcing periodic signal. For forcing with chaotic signal, the transition to chaos is observed but the route to chaos is not clearly detectable due to random variations of the forcing signal strength. The chaos produced in the slave Colpitts oscillator for a chaotic forcing is found to be in a phase-synchronized state with the forced chaos for some values of the coupling factor. We also perform a hardware experiment in the radio frequency range with prototype Colpitts oscillator circuits and the experimental observations are in agreement with the numerical simulation results.     

Energy growth for a nonlinear oscillator coupled to a monochromatic wave

A system consisting of a chaotic (billiard-like) oscillator coupled to a linear wave equation in the three-dimensional space is considered. It is shown that the chaotic behavior of the oscillator can cause the transfer of energy from a monochromatic wave to the oscillator, whose energy can grow without bound.