The effect of the number of nodal diameters on non-linear interactions in two asymmetric vibration modes of a circular plate

In order to investigate the effect of the number of nodal diameters on non-linear interactions in asymmetric vibrations of a circular plate, a primary resonance of the plate is considered. The plate is assumed to have an internal resonance in which the ratio of the natural frequencies of two asymmetric modes is three to one. The response of the plate is expressed as an expansion in terms of the linear, free oscillation modes, and its amplitude is considered to be small but finite, and the method of multiple scales is used. In view of the corrected solvability conditions for the responses, it has been found that in order for the modes to interact, the ratio of the numbers of nodal diameters of two modes must be either three to one or one to one. In this study the one-to-one case, in which the modes have the same number of nodal diameters, is examined. The non-linear governing equations are reduced to a system of autonomous ordinary differential equations for amplitude and phase variables by means of the corrected solvability conditions. The steady state responses and their stability are determined by using this system. The result shows very complicated interactions between two modes by telling existence of non-vanishing amplitudes of the mode not directly excited.     

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω₁and ω₂, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω?ω₁/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω?(m/n)ω₁, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.     

Effect of the Bloch-Siegert shift on the frequency responses of Rabi oscillations in the case of nutation resonance

The dynamics of a two-level spin system dressed by bichromatic radiation is studied under the conditions of double resonance when the frequency of one (microwave) field is equal to the Larmor frequency of the spin system and the frequency of the other (radio-frequency) field ω_rf is close to the Rabi frequency ω₁ in a microwave field. It is shown theoretically that Rabi oscillations between dressed-spin states with the frequency ? are accompanied by higher-frequency oscillations at frequencies nω_rf and nω_rf ± ?, where n = 1, 2,.... The most intense among these are the signals corresponding to n = 1. The counter-rotating (antiresonance) components of the RF field give rise to a shift of the dressed-state energy, i.e., to a frequency shift similar to the Bloch-Siegert shift. In particular, this shift is manifested as the dependence of the Rabi-oscillation frequency ? on the sign of the detuning ω₁ ? ω_rf from resonance. In the case of double resonance, the oscillation amplitude is asymmetric; i.e., the amplitude at the sum frequency ω_rf + ? increases, while the amplitude at the difference frequency ω_rf ? ? decreases. The predicted effects are confirmed by observations of the nutation signals of the electron paramagnetic resonance (EPR) of E′₁ centers in quartz and should be taken into account to realize qubits with a low Rabi frequency in solids.     

Radiative plasmon polaritons in multilayer structures with a two-dimensional electron gas

Two-dimensional plasmon polaritons are analyzed for a typical experimental configuration in which a layer of two-dimensional electrons with a finite mobility lies on the top of a dielectric waveguide formed by the substrate (a wafer of finite thickness). Two-dimensional plasmons couple strongly to the radiative modes of this dielectric waveguide. It is shown that, as a result of the competition between collisional and radiative processes, a family of eight quasi-stationary normal modes arises. Six of them decay carrying energy to infinity. The two remaining plasmon-polariton modes are nonradiative. One of these modes, the TM-type plasmon polariton, in the limiting case where retardation is disregarded corresponds to the conventional longitudinal two-dimensional plasmon. The other mode, the TE-type plasmon polariton, exists only for a finite thickness of the substrate. All of them are characterized by different dispersion relations of the complex frequency ω(q) = Reω + iImω and differ in both polarization (longitudinal and transverse) and symmetry with respect to the direction of decay (symmetric and asymmetric). The latter modes decay slowly, propagating into free space to plus or minus infinity. The conditions under which the Q factors of certain modes are arbitrarily high are found. In this case, Imω(q ₀) = 0, and dissipative losses in the two-dimensional electron gas are compensated by external sources. As a result, the reflection coefficient for a plane wave whose angle of incidence is determined by the vector q ₀ vanishes.     

Nonlinear susceptibilities of vapors and solutions of organic dyes

Results of theoretical studies of laser and Kerr nonlinear susceptibilities of vapors and solutions of organic dyes using a series of polycyclic arenes as an example are presented. Nonlinear susceptibilities of the third χ⁽³⁾ (?3ω; ω, ω, ω) and the fifth χ⁽⁵⁾ (?3ω; ω, ω, ω, ω, ?ω) orders of a series of organic dyes responsible for third harmonic generation of Nd:YAG laser radiation are calculated within the context of the free electron model. Results of calculations of their Kerr third-order nonlinear susceptibilities χ⁽³⁾ (?ω; ω, ?ω, ω) and non-linear refractive indices n ₂ are presented. The calculation results are compared with experimental data on third harmonic generation in naphthalene vapors and with χ⁽³⁾ (?ω; ω, ?ω, ω) as well as n ₂ of paraterphenyl and naphthalene solutions.     

Non-linear resonances in the forced responses of plates,part II: Asymmetric responses of circular plates

The dynamic analogue of the von Karman equations is used to study the forced response, including asymmetric vibrations and traveling waves, of a clamped circular plate subjected to harmonic excitations when the frequency of excitation is near one of the natural frequencies. The method of multiple scales, a perturbation technique, is used to solve the non-linear governing equations. The approach presented provides a great deal of insight into the nature of the non-linear forced resonant response. It is shown that in the absence of internal resonance (i.e., a combination of commensurable natural frequencies) or when the frequency of excitation is near one of the lower frequencies involved in the internal resonance, the steady state response can only have the form of a standing wave. However, when the frequency of excitation is near the highest frequency involved in the internal resonance it is possible for a traveling wave component of the highest mode to appear in the steady state response.     

Dispersion of the linear electrooptic coefficient in ZnS

The measured values and the analysis of the dispersion of the unclamped linear electrooptic coefficient r^T₄₁ in cubic ZnS single crystals operating as optical modulator are presented. The spectral dispersion of the non-linear optical coefficient d₄₁(ω, ω, 0) is also reported and the weak dependence on the light frequency, observed for d₄₁, is discussed by taking into account the opposite sign of ionic and electronic contributions.     

Nonlinear and multiquantum effects in dual-frequency EPR experiment: the intensities of sideband resonances

Two microwave frequencies ω₁ and ω₂ simultaneously exciting a paramagnetic spin probe near the resonance condition ω₁ = γB ₀ effect a pattern of resonances occurring at multiples of the difference frequency δ = ω₂ ? ω₁. Their intensities, measured in the absorption or dispersion mode in the magnetic field-sweep experiment, decrease rapidly as the distance from the centerband increases. Numerically solved Bloch equations are used to discuss the intensities of the transverse components of magnetization up to seven, harmonics of the frequency difference δ. In conclusion, it is suggested that the experimental investigation of these patterns can be used for the purpose of a continuous-wave monitoring the relaxation rates of spin probes.     

Exact solution of the asymmetric Mindlin's plate equations applied to a disk

An exact solution is presented for the static and dynamic asymmetric response of a disk governed by Mindlin's plate equations forced by a pressure that varies radially as r^m. The static solution agrees with a modal solution adopting the dynamic Mindlin's plate equations in the limit when excitation frequency vanishes. This solution is useful in sizing magnitude and shape of surface asymmetries on a disk from pressure loading with slight eccentricity and circumferential non-uniformity.     

Vibration and parametric excitation in asymmetric circular plates under moving loads

The natural frequencies and modes of transverse vibration of circular plates containing small imperfections are determined through a perturbation method. Incision of equally spaced, equal-size radial slots at the rim of the plate creates asymmetry in some, but not all, of the vibration modes, and it causes the repeated natural frequencies of these modes in the symmetric plate to split into two distinct values. These vibration modes are called the split modes, and those associated with the repeated natural frequencies are called the repeated modes. A relationship identifying the split and repeated modes for any configuration of slots is presented. The vibration of a plate containing any number of thin slots cut into it at the rim and with any number of rotating linear springs is analyzed. Parametric instability can be excited in the split modes of the plate by the springs rotating below critical speed, but it cannot be excited in the repeated modes. The response of the plate in forms such as traveling or standing waves at parametric resonance is discussed. The theoretical predictions of split and repeated vibration modes and of the excitation of parametric instability are confirmed by experiments.     

CHAOTIC AND NON-LINEAR DYNAMIC ANALYSIS OF A TWO-AXIS RATE GYRO WITH FEEDBACK CONTROL MOUNTED ON A SPACE VEHICLE

An analysis is presented in this paper for a two-axis rate gyro subjected to linear feedback control mounted on a space vehicle, which is spinning with uncertain angular velocity ω_z(t) about its spin of the gyro. For the autonomous case in which ω_z(t) is steady, the stability analysis of the system is studied by Routh-Hurwitz theory. For the non-autonomous case in which ω_z(t) is sinusoidal function, this system is a strongly non-linear damped system subjected to parametric excitation. By varying the amplitude of sinusoidal motion, periodic and chaotic responses of this parametrically excited non-linear system are investigated using the numerical simulation. Some observations on symmetry-breaking bifurcations, period-doubling bifurcations, and chaotic behavior of the system are investigated by various numerical techniques such as phase portraits, Poincaré maps, average power spectra, and Lyapunov exponents. In addition, some discussions about chaotic motions of this system can be suppressed and changed into regular motions by a suitable constant motor torque are included.     

Generalized thermoelastic Lamb waves in a plate bordered with layers of inviscid liquid

The propagation of thermoelastic waves in a homogeneous isotropic, thermally conducting plate bordered with layers of inviscid liquid or half-space of inviscid liquid on both sides is investigated in the context of generalized theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases. The different regions of secular equations are obtained and special cases, such as Lame modes, thin plate waves and short wavelength waves of the secular equations are also discussed. The secular equations for thermoelastic leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for an aluminum-epoxy composite and aluminum materials plate bordered with water. The dispersion curves for symmetric and antisymmetric thermoelastic wave modes and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement. The results have been deduced and compared with the relevant publications available in the literature at the relevant stages of the work.     

Double resonance in the system of coupled Josephson junctions

The effect of LC shunting on the phase dynamics of coupled Josephson junctions has been examined. It has been shown that additional (rc) branches appear in the current-voltage characteristics of the junctions when the Josephson frequency ω_J is equal to the natural frequency of the formed resonance circuit ω_rc. The effect of the parameters of the system on its characteristics has been studied. Double resonance has been revealed in the system at ω_J = ω_rc = 2ω_LPW, where ω_LPW is the frequency of a longitudinal plasma wave appearing under the parametric-resonance conditions. In this case, electric charge appears in superconducting layers in the interval of the bias current corresponding to the rc branch. The charge magnitude is determined by the accuracy with which the double resonance condition is satisfied. The possibility of the experimental implementation of the effects under study has been estimated.     

Nonlinear mixing of oppositely travelling surface plasmons

A theoretical treatment of the nonlinear mixing of two contra-directional surface plasmons of frequency ω₁ and ω₂ propagating on a semi-infinite metal surface is presented. The nonlinear interaction is analysed in terms of (a) nonlinear surface currents due to the breaking of inversion symmetry and the rapid variation of the normal electric field component at the surface, and (b) the nonlinear response of the electron gas in the bulk of the metal. These currents are treated as source terms for Maxwell's equations and electromagnetic fields are found which satisfy the driven wave equation and boundary conditions at the frequencies ω₁ ± ω₂. At the difference frequency ω₁ ? ω₂ the solution fields decay exponentially into both the air and the metal. However, at the sum frequency ω₁ + ω₂, coupling to transverse electromagnetic waves in both the air and the metal is predicted under appropriate wavevector and frequency conditions. The free-space radiation field is treated in detail and the feasibility of its experimental detection is discussed. No coupling to longitudinal bulk plasmons is predicted in this model of the nonlinear interaction.     

Effect of plasma on modal dispersion characteristics of elliptical Bragg waveguide

In this paper, the dispersion characteristics of a plasma filled elliptical Bragg waveguide is investigated. The modal characteristic equations of the proposed Bragg waveguide for both ω > ω_p and ω < ω_p are derived. The effects of plasma frequency, numbers of cladding layers and the eccentricity of elliptical Bragg waveguide on the dispersion characteristics are studied. The analysis shows that the introduction of plasma in the proposed waveguide allows to control the propagation of modes.     

Diffusion of vibrations in disordered systems

We consider diffusion of vibrations in random lattices with translational invariance. Above the frequency ω_IR corresponding to the Ioffe-Regel crossover (and depending on the strength of disorder), phonons cannot propagate through the lattice and transfer energy. At the same time, most of the vibrations in this range are not localized. We show that these delocalized excitations are similar to diffusons introduced by P. B. Allen, J. L. Feldman, J. Fabian, and F. Wooten (see, e.g., Phil. Mag. B 79, 1715 (1999)) to describe heat transport in glasses. In this range the energy in the lattice is transferred by means of diffusion of vibrational excitations. We have calculated the diffusivity of the modes D(ω) using both the direct numerical solution of Newton equations and the Edwards-Thouless formula. It is nearly constant above ω_IR and goes to zero at the localization threshold.     

The domain structure of a nematic liquid crystal layer in an oscillating poiseuille flow

A spatially periodic structure arising in a nematic liquid crystal layer with planar orientation under the effect of an oscillating Poiseuille flow is described theoretically. The effect is analyzed on the basis of hydrodynamic equations of nematic liquid crystals, from which a self-consistent set of equations for perturbations of hydrodynamic variables is separated. It is demonstrated that the structure type and the threshold parameters of the effect depend on the frequency and the layer thickness through the scaling combination ωh ². The dependence of the configuration of arising distortions on the value of viscosity α₃ is analyzed.     

Picosecond measurements of the third order susceptibility tensor in liquids

Single frequency Jamin interferometry is used for observation of non-linear susceptibility tensor measurements in the picosecond range (25 ps). At low density, when avoiding the polarization state instabilities, the displacement of the fringes leads to the measurement of two components of the nonlinear susceptibility tensor. It is shown that X_xyyx (ω = ω + ω - ω) decreases when the pulse duration is reduced, whereas X_{xxxx (ω = ω + ω - ω)} remains constant. This is interpreted by the difference between the orientational and vibrational molecular contributions.     

Natural vibration of unsymmetric cross-ply laminates

This study considers the linear vibration characteristics of square [0_n/90_n]_T laminates relative to their room-temperature static equilibrium configurations. A Rayleigh-Ritz approach combined with Hamilton's principle is used to provide approximate solutions to this vibration problem. The vibration mode shapes are assumed to have the same spatial dependence as used in past investigations to study the room-temperature configurations of these laminates, and are thus assumed to be perturbations on the static equilibrium configurations. Hamilton's principle then results in the so-called zero- and first-order equations. The zero-order equations lead to the classic static equilibrium results of past investigations, presented here in nondimensional form with analytical solutions at the bifurcation point. The first-order equations, combined with zero-order results, lead to the vibration characteristics for each zero-order static configuration. Interest centers on the lowest natural frequency and the associated mode shape for laminates clamped at their midpoints, with special attention as to how these vibration characteristics depend on the laminate side-length-to-thickness ratio. With an imaginary-valued frequency, the static saddle configuration for side-length-to-thickness ratios larger than the critical value is correctly assessed as unstable. A finite element model is also used to study the vibration characteristics and to compare with the findings for the developed analysis. The qualitative comparisons between the developed analysis and the finite element model are generally good, and the quantitative comparisons are also satisfactory.