Nonlinear Internal Resonance Research Articles

Nonlinear resonance interaction of waves on a jet in a radial electric field

Nonlinear analytic asymptotic second-order calculations show that the motion of a charged jet relative to a material medium results in the appearance of rise-time periodic wave motions of the interface (Kelvin-Helmholtz instabilities), as well as an internal nonlinear resonance interaction of waves, whose parameters (the intensity and the characteristic time of interaction) depend on the physical parameters of the system: the electric charge density at the jet, its velocity relative the medium, the mass density, the values of the wavenumbers of the interacting waves, and the value of the surface tension coefficient of the boundary.

Nonlinear stability analysis for a jet moving in a material medium collinearly to an electrostatic field

Analytically, in the second order approximation, the nonlinear internal resonance energy exchange between waves on the surface of a cylindrical incompressible dielectric fluid jet that moves relative to a material medium in the longitudinal electrostatic field is investigated.

Internal nonlinear resonance on a charged jet

Nonlinear analytical asymptotic calculations of the second order of smallness show that the motion of a charged jet in a material medium generates periodic wave motions of the jet-medium interface (Kelvin-Helmholtz instabilities), which grow in time. In addition, the motion of the jet gives rise to nonlinear internal resonance interaction of waves. The parameters of this interaction (intensity and characteristic time) depend on the physical parameters of the system: electric charge density of the jet, its velocity in the medium, mass density, wavenumbers of interacting waves, and the interfacial tension coefficient.

Nonlinear asymptotic calculation of the Kelvin-Helmholtz instability

The problem of periodic capillary-gravitational wave motion on the uniformly charged interface between two ideal immiscible incompressible liquids is solved in the third order of smallness. The lower liquid is assumed to be ideally conducting, while the upper one is a dielectric executing translational motion parallel to the interface with a constant velocity. A nonlinear frequency correction in the resonance form is found. It is shown that the positions of internal nonlinear resonances depend on the sum of the field and Weber parameters, the density ratio of the liquids, and the wave number. When the upper liquid is denser than the lower one, resonances are absent.

Nonlinear multimode dynamics and internal resonances of the scan process in noncontacting atomic force microscopy

The focus of this paper is on the nonlinear multimode dynamics of a moving microbeam for noncontacting atomic force microscopy (AFM). An initial-boundary-value problem is consistently formulated, which includes both nonlinear dynamics of a microcantilever with a localized atomic interaction force, and a horizontal boundary condition for a constant scan speed and its control. The model considered is obtained using the extended Hamilton's principle, which yields two partial differential equations for the combined horizontal and vertical motions. The model incorporates, for the first time to our knowledge, two independent time-varying terms that depict the vertical base excitation of the AFM and the horizontal forcing term depicts the periodic scanning motion of the cantilever. Manipulation of these equations via a Lagrange multiplier enables construction of a modified equation of motion, which is reduced, via Galerkin's method, to a three-mode dynamical system, corresponding to finite amplitude AFM dynamics. The analysis includes a numerical study of the strongly nonlinear system culminating with a stability map describing an escape bifurcation threshold where the tip, at the free end of the microbeam, “jumps to contact” with the sample. Results include periodic, quasiperiodic, and non-stationary chaotic-like solutions corresponding to primary and secondary internal combination resonances, where the latter corresponds to energy balance between the cantilever modes.

Nonlinear analysis of the internal resonant interaction of nonaxisymmetric waves on the surface of a jet moving in reference to a medium in a collinear electrostatic field

The possibility was studied to realize the internal nonlinear resonance interaction of nonaxisymmetric waves on the surface of a jet that is moving in a medium collinear to a longitudinal electric field. For axisymmetric waves with the azimuth number equal to one and two, several resonance situations were realized.

Nonlinear analysis of a wave motion on the surface of a jet moving in a dielectric medium placed in a longitudinal electric field

The possibility of degenerate internal nonlinear resonance interaction between capillary waves with arbitrary symmetry (arbitrary azimuthal numbers) on the surface of a charged cylindrical jet of an ideal incompressible conducting liquid is demonstrated. The jet moves in an ideal incompressible dielectric medium collinearly with an external uniform electrostatic field. It is shown, in particular, that six different resonance situations take place for axisymmetric waves in which primary waves and waves due to the nonlinearity of the equations of hydrodynamics exchange energy.

Stability of the wave motion on the charged interface between two immiscible fluids in the presence of a tangential velocity field discontinuity

In the second-order approximation in the dimensionless wave amplitude, the problem of nonlinear periodic capillary-gravity wave motion of the uniformly charged interface between two immiscible ideal incompressible fluids, the lower of which is perfectly electroconductive and the upper, dielectric, moves translationally at a constant velocity parallel to the interface, is solved analytically. It is shown that on the uniformly charged surface of an electroconductive ideal incompressible fluid the positions of internal nonlinear degenerate resonances depend of the medium density ratio but are independent of the upper medium velocity and the surface charge density on the interface. All resonances are realized at densities of the upper medium smaller than the density of the lower medium. In the region of Rayleigh-Taylor instability with respect to density there is no resonant wave interaction.

Nonlinear resonance interaction between the oscillation modes of a spherical liquid layer on the surface of a melting hailstone

Evolutionary equations are derived and solved that describe the time dependence of the oscillation mode amplitudes on the surface of a charged conducting liquid layer resting on a solid core. It is assumed that the layer experiences a multimode initial deformation. The equations are solved asymptotically in the second order of smallness in the small dimensionless amplitude of capillary oscillations on the surface of the layer. Mechanisms behind internal nonlinear resonance interaction between the modes of the liquid layer oscillations and behind energy transfer between the modes both in degenerate and in secondary combination resonances are investigated. It is found that in the degenerate resonance interaction between oscillation modes, the energy may be transferred not only from lower to higher modes but also vice versa if the higher mode is excited at the zero time. This conclusion is valid not only for a liquid layer on the surface of a solid core but also for a drop.

Nonlinear analysis of governing laws of realization of the wave motion on the surface of a charged jet moving with respect to a material medium

On the base of analytic asymptotic calculations which are quadratic with respect to the ratio of the wave amplitude and the jet radius it is shown that the presence of a tangential jump in the velocity field on the jet surface leads to generation of a periodic wave motion on the interface between the media and has the destabilizing effect for both axisymmetric and bending and bending-deformation waves. It is found that there is a degenerate internal nonlinear resonance interaction between waves on the jet surface. This interaction may be of six different types in which the energy can be transferred between the interacting waves including waves of different symmetry. In the last case the energy is transferred from waves determining the initial deformation to axisymmetric waves.

Nonlinear analysis of interaction between finite-amplitude capillary waves in a layered inhomogeneous liquid

The nonlinear capillary wave motion in a two-layer liquid with a free surface is analytically investigated accurate to the second order of smallness in ratio of the wave amplitude to the layer thickness. The layers differ in physicochemical properties. A capillary analogue to the “dead water” effect is observed in the system in both linear and quadratic approximations. In the absence of an electric charge at the interfaces, internal nonlinear resonance interaction between capillary waves is also absent regardless of the place of their origination. When there is a charge at the interlayer boundary, capillary waves resonantly interact with each other.

Degenerated internal nonlinear resonance interaction of the waves on the surface of an uncharged dielectric jet in a longitudinal electrostatic field

Using the asymptotic analytical procedure of the second order of smallness with respect to the ratio between the amplitude of a wave and the radius of a jet for the calculation of the wave motion of a finite amplitude on the surface of a cylindrical jet of an ideal incompressible dielectric liquid, it was found that the nonlinear corrections for the jet’s profile and the potential of the fields of the rates and electrostatic potentials inside and outside of the jet have a resonance character. In a degenerate resonance interaction between the wave, which determines the initial deformation and the waves that become excited owning to the nonlinearity of the hydrodynamic equations, the waves with different symmetry (with different azimuthal numbers) can participate.

Nonlinear Dynamics and Internal Resonances of a Ship With a Rectangular Cross-Section in Head Seas

Finite amplitude dynamics of ships in head seas due to parametric instabilities is a subject of renewed interest with an increasing demand of operation in severe and variable environmental conditions. In this current study we investigate the nonlinear dynamics and internal resonances of a ship with a rectangular cross-section in head seas. We employ an asymptotic averaging method to obtain the slowly varying system evolution dynamics for the weakly nonlinear response, complemented by numerical integration in the strongly nonlinear regime. A weakly nonlinear frequency response is obtained analytically for a principal parametric resonance and a 1:1 roll—pitch internal resonance. Comparison of results with three degrees of freedom numerical simulations reveals a good fit. A strongly nonlinear numerical analysis reveals that beyond the stability thresholds, the system’s responses included quasiperiodic dynamics. This combined approach resolves both parametric instabilities and internal resonances induced for both weak and finite nonlinear interactions, and culminates with criteria for orbital stability thresholds describing the onset of quasiperiodic response and magnification of energy transfer between coupled pitch-heave and ship roll.

Nonlinear corrections to the oscillation mode frequencies of an ideal liquid charged jet

An analytical asymptotic expression is derived for the time evolution of an ideal incompressible conducting liquid jet with a uniformly charged surface that moves with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. The expression is obtained in the third order of smallness in jet oscillation amplitude under the conditions when the initial deformation of the equilibrium jet surface is specified by one (axisymmetric or nonaxisymmetric) mode. Analytical expressions are also found for the positions of internal nonlinear degenerate three-and four-mode resonances, which are typical of nonlinear corrections to the equilibrium shape of the jet, liquid velocity field potential in the jet, and electrostatic field near the jet, and also for a nonlinear frequency correction. It is established that the nonlinear oscillations of the jet take place about a surface depending on the type of initial (generally asymmetric) deformation rather than about the equilibrium cylindrical shape.

Nonlinear oscillations of a charged jet of a conducting liquid under multimode initial deformation of its surface

The subject of consideration is a uniformly charged jet of an ideal incompressible conducting liquid moving with a constant velocity along the symmetry axis of an undisturbed cylindrical surface. An evolutionary expression for the jet shape is derived accurate to the second order of smallness in oscillation amplitude for the case when the initial deformation of the equilibrium surface is a superposition of a finite number of both axisymmetric and nonaxisymmetric modes. The flow velocity field in the jet and the electric field distribution near it are determined. The positions of internal nonlinear secondary combined three-mode resonances are found, which are typical of nonlinear corrections to the analytical expressions for the jet shape, flow velocity field potentials, and electrostatic field in the vicinity of the jet.

On the internal nonlinear resonant three-mode interaction of charged drop oscillations

The difference between internal nonlinear three-mode degenerate and Raman resonances is found for the first time: in the former case, the energy spent on the initial deformation of a drop is only transferred from lower to higher modes; in the latter case, it is transferred in both directions. It turns out that degenerate resonances are slightly sensitive to the physical quantities that are responsible for the exact positions of the resonances (i.e., to the amount of electric charge). A deviation from the resonant value only changes the fraction of the energy the modes exchange and the time of resonant energy exchange: the interaction itself remains resonant.

Internal nonlinear resonance of charged drop capillary oscillations in a dielectric medium at multimode initial deformation of the interface

An analytical expression for the generatrix of the shape of a nonlinearly vibrating charged drop of a perfect incompressible conducting fluid immersed in an ideal incompressible dielectric medium is found in the second order of smallness in terms of perturbation theory. The drop experiences multimode initial deformation. The expression contains resonant (small-denominator) terms. With the effect of the environment taken into account, the number of resonant situations becomes dependent on the drop-to-environment density ratio and the resonant self-charge of the drop changes. It is shown that nonlinear vibrations may be of resonant character even if the charge of the drop is far away from exact resonant values. This is because Rayleigh subcritical values of the self-charge affect the frequencies of higher vibration modes insignificantly.

On the internal nonlinear resonance of capillary-gravitational waves on the charged surface of a deep viscous liquid

An analytical expression for the profile of a periodic wave of finite amplitude on the surface of a deep viscous conducting liquid is obtained for the first time. The formula admits the transition to a limiting case of the ideal liquid. It is shown that the position of an internal nonlinear resonance of these capillary-gravitational waves depends neither on the medium viscosity nor on the surface charging. It is established that, during the resonance interaction, the energy is pumped from longwave capillary-gravitational oscillations with the wavenumber \(k_* \equiv \sqrt {\rho g/2\gamma } \) to shortwave oscillations with \(k_0 \equiv \sqrt {\rho g/2\gamma } \).

The fundamental mode amplitude buildup in an oscillating drop under internal nonlinear resonance conditions

In a liquid drop charged below the critical level for realization of the instability with respect to the intrinsic charge, the fundamental oscillation mode amplitude can grow due to a nonlinear resonance interaction with higher modes.

Interaction of nonlinear resonances in cavity QED

Strong coupling of the internal and external degrees of freedom of a cold atom to each other and to the spatially periodic field of the standing light wave in a high-finesse cavity is responsible for the dynamic instability of the atomic center-of-mass motion. Due to a weak interaction of the internal nonlinear resonances in the standard model of cavity QED, a stochastic layer appears, whose width in the semiclassical approximation is estimated in terms of the main parameters of the system: atomic recoil frequency, mean number of excitations, and detuning from the resonance. As a result, the atomic motion in the absolutely regular potential has the fractal character, with long Levy flights alternating with small chaotic oscillations in potential wells.