External Sinusoidal Force Research Articles

Exact solution of stable periodic one contact per N cycles motion of a damped linear oscillator contacting a unilateral elastic stop

An exact closed-form solution is developed for a damped single-degree-of-freedom oscillator contacting a unilateral stop once per N cycles of external sinusoidal force. It also includes the effect of a bias force. This solution was obtained by using two different displacement and velocity expressions for a two-region piecewise linear oscillator, continuity of displacement and velocity at the boundary of the two regions, and periodicity conditions. Contact duration is assumed as known and four simultaneous equations in four unknowns are solved exactly. The four unknowns are the amplitude of the sinusoidal force, phase angle of the assumed first contact, entry velocity, and exit velocity of mass. The stability of motion is checked using the closed-form expressions of elements of a 2×2 stability governing matrix. These elements are obtained by considering only first-order perturbations in periodic motion. Theoretical predictions based on exact solution agree with previous results and with results obtained using a numerical simulation approach. The relationships are studied in detail between input parameters (such as amplitude of external force, frequency of external force, bias force, gap, spring stiffness and damping constants) and output parameters (such as contact duration, phase angle at contact, entry velocity, exit velocity, maximum displacement and minimum displacement).

Dust Diffusion in Protoplanetary Disks by Magnetorotational Turbulence

We measure the turbulent diffusion coefficient of dust grains embedded in magnetorotational turbulence in a protoplanetary disk directly from numerical simulations and compare it to the turbulent viscosity of the flow. The simulations are done in a local coordinate frame comoving with the gas in Keplerian rotation. Periodic boundary conditions are used in all directions, and vertical gravity is not applied to the gas. Using a two-fluid approach, small dust grains of various sizes (with friction times up to Ω0τf = 0.02) are allowed to move under the influence of friction with the turbulent gas. We measure the turbulent diffusion coefficient of the dust grains by applying an external sinusoidal force field acting in the vertical direction on the dust component only. This concentrates the dust around the midplane of the disk, and an equilibrium distribution of the dust density is achieved when the vertical settling is counteracted by the turbulent diffusion away from the midplane. Comparing with analytical expressions for the equilibrium concentration, we deduce the vertical turbulent diffusion coefficient. The vertical diffusion coefficient is found to be lower than the turbulent viscosity and to have an associated vertical Schmidt number (vertical diffusion Prandtl number) of about 1.5. A similar radial force field also allows us to measure the radial turbulent diffusion coefficient. We find a radial Schmidt number of about 0.85 and also find that the radial turbulent diffusion coefficient is around 70% higher than the vertical. As most angular momentum transport happens through magnetic Maxwell stresses, both the vertical and the radial diffusion coefficients are found to be significantly higher than suggested by the angular momentum transport by Reynolds stresses alone. We also find evidence for trapping of dust grains of intermediate friction time in turbulent eddies.

Time Correlations and Power Spectra of the Duffing Equation

The time-correlation functions and power spectra of chaotic orbits of the Duffing equation are investigated numerically and theoretically. Because the system is driven by a sinusoidal external force with period T =2 π/ω0, the time-correlation function C(t) ≡� p(t)p(0)� of the momentum p(t), defined through the long-time average over a chaotic orbit, becomes a sinusoidal oscillation with period T as t →∞ . The phase φ(t )= ω0t + φ0 of the external force takes an initial value φ0 at times t = jT ˙

Analysis of valveless micropumps with inertial effects

Through flow field simplification, a set of differential equations governing the fluid flow and fluid–membrane coupling are obtained for a valveless micropump. The dimensional analysis on the equations reveals that the ratio of the inertial force of the fluid to the viscous loss is dependent on the size ratios among internal elements of the pump. For a micropump working at high frequencies, these two forces possess the same order of magnitude, and this phenomenon is independent of the excitation frequency and fluid type. For the liquid medium, the inertial force of the fluid is around O(102)–O(103) times as that of the plate membrane, and is also larger than the elastically deformed force of the plate when the excitation frequency is close to the plate fundamental frequency. For the case where there is no pressure difference between the inlet and the outlet, an approximate analytical solution is derived for the micropump under the action of an external sinusoidal excitation force. It shows that a phase shift lagging the excitation force exists in the vibration response. For certain combinations of micropump size and fluid–solid density ratios, the phase shift can come to 90° at a specific excitation frequency ω* due to the action of fluid inertia. Away from ω*, the phase shift becomes smaller. The amplitude response of coupling vibration changes nonlinearly with the excitation frequency and reaches maximum at another frequency ω** ≠ ω*. Due to the nonlinearity of viscous loss, resonance does not seem to occur at any frequency. To obtain a larger average flux, the two loss coefficients of the nozzle should be minimized while their difference should be maximized. Under the action of the fluid inertia, there exists an optimal working frequency (equal to ω*) at which the average flux is maximum. This optimal frequency is dependent on the size of the micropump, the material properties of the plate, the fluid properties and has no relation with the excitation force. For the case where a pressure difference between the inlet and the outlet exists, a constraint condition between the excitation force and the pressure difference is obtained.

The piecewise-linear model of the Van der Pol oscillator under external periodic force: complex dynamics, peculiarities of behaviour at the parameter plane

The electronic model of Van der Pol oscillator with piecewise-linear currentvoltage characteristics of nonlinear element is proposed. Experimental investigations of behaviour of the Van der Pol oscillator under external sinusoidal force were carried out. The parameter plane (external force amplitude - frequency) for the dynamical regimes of the oscillator was experimentally plotted. Numerical simulations of peculiarities of the oscillator model were carried out. In natural and numerical experiments it was shown that transition to chaos in the oscillator takes place through the period doubling cascade. It was revealed that the internal structure of the synchronization tongues looks typically as for the crossroad area.

Characteristic times and current inversion in inertial ratchets with colored thermal noise

We perform a systematical numerical study of the statistical behaviour of a particle immersed in a colored thermal bath moving with friction in a one-dimensional asymmetric and periodic potential and subjected to an external sinusoidal force. To this end, we characterize the problem in terms of a set of three characteristic times and two strength variables that determine the evolution of the system. We solve the corresponding equations of motion by using an iterative approach. We analyze different bath temperature regimes and study the appearance of a current. We also determine whether the evolution is diffusive or not. We find that current inversion is a very robust phenomenon appearing in an oscillatory fashion as a function of the frequency of the time-dependent external force.

Rocking ratchets with stochastic potentials

An analysis has been made for the motion of an overdamped Brownian particle in a periodic potential subjected to a position-dependent stochastic perturbation and a sinusoidal external force. In the presence of the stochastic potentials, it is noticed that with the increasing intensity of the stochastic potentials, the maximum of the current in general decreases, while it is shifted to the higher temperature, and moreover, the correlation length also strongly influences the magnitude of the current.

Dynamical origin of deterministic stochastic resonance.

We numerically demonstrate stochastic-resonance-like behavior in a deterministic chaotic oscillator system, using a modified Rössler system driven by a sinusoidal external force: intermittent 2 pi phase slips between the system and the external force synchronize with a periodic input signal that weakly modulates the external force in an appropriate parameter range. We show that the dynamical mechanism of this stochastic-resonance-like behavior is explained by a boundary crisis that depends on two bifurcation parameters.

Self-excited and Forced Oscillations with Solid Friction. 1st Report. Solutions and Stabilities by Averaging Method in Case of Combined Friction-Velocity Characteristics of Linear and Hyperbolic Curves.

In the present paper, the frictional oscillations in a mechanical driving system are studied theoretically on the self-excited and forced oscillations, when the friction-velocity characteristic curves are given as the combined curves of linear and hyperbolic functions and maximum static friction force is not an isolated point. Bodies acted upon by sinusoidal external force while sliding along a surface at a fixed velocity exhibit regular oscillations. We have analyzed these motions using an averaging method, without distinguishing between slipping and sticking. We have also examined the amplitude characteristic of motions found in the obtainable first approximations of the form of harmonic vibrations. Finally, we made a detailed investigation of the stability of these first approximations. Also, we have assembled the results of a comparison of the precise solutions for motions over a wide range of velocities where the kinetic velocity of the body exceeding the surface motion velocity, and we propose a usable theoretical analysis method, aimed at understanding the important aspects of regular harmonic solutions.

Transmission of Binary Information with a Chaos Coded Communication System using QDPSK-Modulation

Abstract Synchronisation of chaotic oscillators offers a way of practical application of the theory of Chaos in obtaining secure communication. In this work we introduce a nonautonomous chaotic system with sinusoidal external force for communication of binary signals. The information is applied to the phase position of the sinusoidal forcing signal of the chaotic oscillator using a quadrature difference phase shift keying (QDPSK) modulation. An inverse synchronisation system approach with direct modulation is applied. We describe the system in detail and discuss the requirements of a secure communication system. Issues related to bit error rate, transfer rate, signal to noise ratio, channel bandwidth, bandwidth efficiency and channel capacity are discussed, and the properties of the realized communication system are placed in relation to the requirements of a secure communication system.

High-dimensional chaotic neural network under external sinusoidal force

Controlling chaos by an external sinusoidal force is considered in an asymmetric analog neural network with time delay. Quantitative characteristics of the chaotic outputs (spectrum, correlation dimension and largest Lyapunov exponent) are studied. It is pointed out that the external sinusoidal force allows one to control the degree of chaos and to produce “chaos-order”, “order-chaos”, and “chaos-chaos” transitions. Possible mechanisms responsible for the different transitions are discussed. The results of a numerical simulation are compared with the experimental data on controlling chaos in the brain.

Self-Organization Processes in Chaotic Neural Networks Under External Periodic Force

Self-organization processes in an analog asymmetric neural network with time delay and under an external sinusoidal force are considered. Quantitative characteristics of the neuron outputs (spectrum, correlation dimension, largest Lyapunov exponent, Shannon entropy, normalized and renormalized Shannon entropies) are studied in dependence on the frequency and amplitude of the external force. It is shown that the external sinusoidal force allows the control of the degree of chaos and produces transitions "order–chaos", "chaos–order" and "chaos–chaos" with different quantitative characteristics. Information processing both by the individual neurons and by the neural network as a system is discussed. Chaotic neural network under an external force is considered as a qualitative model of the infra-frequencies action on the brain.

Long-time behavior of Navier-Stokes flow on a two-dimensional torus excited by an external sinusoidal force

In this paper we study the Navier Stokes flow on the two-dimensional torus S{sup 1} x S{sup 1} excited by the external force (k{sup 2} sin ky, 0) and find the long-time behavior for the flow starting from some states, where S{sup 1} = [ 0, 2{pi}](mod 2{pi}). Especially for the case k = 2, it follows from an analysis and computation that the Navier-Stokes flow with the initial state cos(mx + ny) or sin(mx + ny) will likely evolve through at most one step bifurcation to either a steady-state solution or a time-dependent periodic solution for any Reynolds number and integers m and n.

Viscoelastic Properties of Hyaluronic Acid Aqueous Solutions

Viscoelastic properties of aqueous solutions of hyaluronic acid extracted from cockscomb, were studied by make use of magnetic rheometer which was constructed by our laboratory under the conditions of: T (period)/s=180~11,520, F0(amplitude of external sinusoidal force)/N=10-7~10-8(10-5N≅1mg·wt), A0(amplitude of oscillation)/m≅10-4, Cp(polymer concentration)/wt%=0.5, and <Mη>=1.99×106. The results were analysed by the mechanical models, and were found to be well explained by applying the 3-element model (Voigt model with Gg(glass modulus) in series) of Gg/Pa>>Ge/Pa=8.78×10-3, η(Pa·s)=1.205×101 from the plot of J1, J2 vs. log10 ωλ; here J* (complex compliance)=J1-i·J2, ω/(rad·s)=2π/T and λ(retardation time)/s≡η/Ge. This result may be attributed to the experimental condition under very low frequency oscillations.

Statistical mechanics of soliton creation and annihilation in a driven nonlinear Schrödinger equation

We report numerical and theoretical results for the statistical mechanics of annihilation and creation processes of soliton- like pulses in a dissipative nonlinear Schrödinger equation driven strongly by an external sinusoidal force. It is shown by analyzing statistical mechanical quantities and a phenomenological stochastic theory that the soliton annihilation/creation process has sub-Poissonian character and is dominated by two-“solitons” and large amplitude radiation.

Ac-Driven quantum decay

We study the enhancement of the quantum decay rate out of a metastable state, via tunneling, in presence of an external sinusoidal force. It is shown that the Floquet picture of quantum mechanics, together with the complex scaling method, provides an adequate methodology to describe the periodically driven decay process in a nonperturbative way. In the limiting cases of extremely slow and fast external forces the numerical results are compared with simple semiclassical estimates. The decay near the fundamental resonance assumes a Lorentzian line shape in agreement with recent experiments on Josephson junctions in the deep quantum regime. For small forces the enhancement grows proportional to the square of the forcing strength and saturates above a threshold value. Additionally our results also exhibit secondary resonances: at higher frequency corresponding roughly to a second harmonic induced by the nonlinear potential shape, and at lower frequency, exactly at the half of the first resonance, revealing a two-photon transition.

Microslip of Wafer Pressed by Spring.

When an object is pressed by a spring and fixed with friction force, microslippage often occurs because of an external force lower than the predictable value. This is because the object is pressed not only by the normal force but also by the tangential force. This paper presents a microslip model in which the obiect supported by spring forces in the normal and tangential directions is subjected to sinusoidal external force. The study shows both analytically and experimentally that larger tangential force produces larger microslip.

Quantum mechanics of a molecular system adsorbed on a dielectric surface.

A molecular system adsorbed on a dielectric surface is modeled as a damped harmonic oscillator driven by a sinusoidal external force. The exact propagator and wave function through the Feynman path-integral method are obtained. The second quantization of this system is carried out. Expectation values of several physical quantities are evaluated. The amplitude for transitions between harmonic-oscillator states and damped driven oscillator states are obtained explicitly, and the result is applied to a two-level system.

Chaos from phase-locked loops. High-dissipation case

For pt.I see ibid., vol.35, no.8, p.987-1003, 1988. The numerical calculations of theorem 2 of pt.I which gives the homoclinicity condition for the non-Hamiltonian, i.e. dissipative, unperturbed case, are performed. In particular, many boundary curves which identify the homoclinic tangency and show that there exists a homoclinic orbit above the curves but no homoclinic orbit below them are obtained. Moreover, the associated Poincare maps (obtained by Runge-Kutta-Gill simulation) confirm that the homoclinicity condition predicted from these diagrams is correct. Finally, computer simulation is used to obtain the actual chaotic attractors observed from a very small external sinusoidal force. This corresponds exactly to the experimental results reported in pt.I that the chaotic phenomena observed from actual experiments in pt.I is indeed a horseshoe chaos based on a homoclinic orbit. >

Forced subharmonic oscillations of the simple pendulum

Symmetric subharmonic solutions of second order differential equation defining oscillations of the simple pendulum subjected to an external sinusoidal force are considered. In the case of small amplitude of the exciting force the subharmonic solutions are analytically determined and then continued in the domain of large amplitudes of that force, using numerical computations. Branching of derived solutions is investigated.