Standing shear waves in rubberlike layered media

Standing shear waves arising in layered media the shear modulus of which varies in a stepwise manner at the plain boundaries between the layers are considered. A general solution is obtained for the shear wave amplitudes in a resonator with an N-layer structure the lower boundary of which performs harmonic vibrations while a finite-mass plate is attached to the upper boundary. Results of calculations and measurements are presented for a resonator with a structure in which nondeformable metal layers alternate with elastic rubberlike polymer layers. It is shown that the resonance frequencies of such a resonator can be controlled by changing the number of layers and their thicknesses. It is demonstrated, both experimentally and theoretically, that, from the resonance curve of a resonator with a two-layer structure, it is possible to determine the shear modulus of one of the layers under the condition that the elasticity of the other layer is known. The method of separation into a finite number of layers is used to analyze the resonance characteristics of a one-dimensional resonator filled with a rubberlike medium the properties of which continuously vary in the direction perpendicular to the shear displacements. The choice of the number of layers depending on the type of inhomogeneity is analyzed.     

Measurement of shear elasticity and viscosity of rubberlike materials

The method and results of measuring the shear elastic modulus of a rubberlike polymer by the deformation of a plane elastic layer are described. For shear deformations not exceeding 0.5 of the layer thickness, the shear modulus is constant and its value is in agreement with the value determined by pressing a rigid ball against the polymer layer. For deformations exceeding 0.5 of the layer thickness, the stress-strain dependence becomes nonlinear. The coefficient of shear viscosity is determined from the shear wave form generated by focused ultrasound in a homogeneous polymer sample.     

Finite-amplitude standing acoustic waves in a cubically nonlinear medium

The acoustic field in a resonator filled with a cubically nonlinear medium is investigated. The field is represented as a linear superposition of two strongly distorted counterpropagating waves. Unlike the case of a quadratically nonlinear medium, the counterpropagating waves in a cubically nonlinear medium are coupled through their mean (over a period) intensities. Free and forced standing waves are considered. Profiles of discontinuous oscillations containing compression and expansion shock fronts are constructed. Resonance curves, which represent the dependences of the mean field intensity on the difference between the boundary oscillation frequency and the frequency of one of the resonator modes, are calculated. The structure of the profiles of strongly distorted “forced” waves is analyzed. It is shown that discontinuities are formed only when the difference between the mean intensity and the detuning takes certain negative values. The discontinuities correspond to the jumps between different solutions to a nonlinear integro-differential equation, which, in the case of small dissipation, degenerates into a third-degree algebraic equation with an undetermined coefficient. The dependence of the intensity of discontinuous standing waves on the frequency of oscillations of the resonator boundary is determined. A nonlinear saturation is revealed: at a very large amplitude of the resonator wall oscillations, the field intensity in the resonator ceases depending on the amplitude and cannot exceed a certain limiting value, which is determined by the nonlinear attenuation at the shock fronts. This intensity maximum is reached when the frequency smoothly increases above the linear resonance. A hysteresis arises, and a bistability takes place, as in the case of a concentrated system at a nonlinear resonance.     

Calculation of the Viscosity of Polymer Melts Based on Measurements of the Recovered Rubber-Like Deformation

On the basis of a model of polymer flow, considering the forces of entropic elasticity of extended macromolecules within the Eyring's concept, the relationships between the shear rate, shear stress, viscosity, and recovered rubber-like deformation were derived. The reduction of activation energy of the flow, by an amount proportional to the recovered rubber-like deformation, leads to an exponential decrease of viscosity with increasing shear rates; this nonlinear dependence of viscosity on shear rate (and shear stress) is defined as the viscosity anomaly of polymers. The measurement of deformation recovery after the cessation of polymer flow in the mode of constant shear rate or shear stress on a rotational viscometer confirmed the validity of the theoretical dependences.     

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.     

Self-Oscillating Mode of a Nanoresonator

The paper proposes a new graphene resonator circuit which operates on the principle of a self-oscillator and has no drawbacks typical of nanoresonators as mass detectors and associated with their law quality factor, eigenfrequency errors (measurements from resonance curves), and dependence of quench frequency on oscillation frequency (curves with quenching for nonlinear systems). The proposed circuit represents a self-oscillator comprising an amplifier, a graphene resonator, and a positive feedback loop with a graphene oscillation transducer, and its major advantage is in self-tuning to resonance frequency at slowly varying resonator parameters, compared to oscillation periods. The graphene layer with a conducting substrate beneath it forms a capacitor which is recharged by a dc voltage source as its capacitance varies due to graphene deformation, and the recharge current is an oscillation- dependent signal transmitted from the transducer to the amplifier input. The graphene layer is placed in a magnetic field and is deformed when a current from the amplifier output is passed through. By properly choosing the magnetic field direction and the amplifier gain, it is possible to provide swinging oscillation whose amplitude is limited by the amplifier nonlinearity. For the proposed system we present an electromechanical model, dimensionless equations of motion, and numerical data demonstrating the generation of steady-state oscillations with eigenfrequency. Also presented is an analysis showing that the system can have only one limit cycle and that this cycle is always stable. The proposed resonator circuit can be used as a mass detector which determines the added mass from a change in self-oscillation frequency.     

Influence of resonator shape on nonlinear acoustic field in a thermoacoustic engine

The influence of the resonator shape on nonlinear acoustic field in a thermoacoustic engine is studied.The resonator of themoacoustic engine is boundary driving by a piston at one end,and the other end of it is rigid closed.A one-dimensional wave equation that accounts for gas dynamic nonlinearities and viscous dissipation in the resonator is established based on the governing equations of viscous hydromechanics.The nonlinear wave equation is solved using approximate Galerkin method.The nonlinear acoustic field in four different types of shaped resonators including hyperbolical,exponential,conical and sinusoidal are obtained and compared with that of a cylindrical resonator.It is found that the amplitude and waveform of the pressure are strongly affected by the resonator shape,the driving amplitude and the oscillation frequency of the piston.Waveform distortion,resonance frequency shift and hysteresis are observed,when the piston oscillation amplitude is large enough.The advantages of shaped resonator for thermoacoustic engine lie in inhibition of higher order harmonics and improvement of pressure ratio,etc.     

Symmetry-breaking instabilities on a fluid surface

A sequence of symmetry-breaking instabilities leading to a chaotic state has been discovered in the surface deformations of a fluid layer subjected to a vertical oscillation. For driving amplitudes above a critical value, a primary instability leads to circularly symmetric standing waves at half the driving frequency. A second instability at a higher threshold breaks the circular symmetry and leads to a slow precession of the pattern, so that the overall motion is quasiperiodic. Beyond a third threshold, azimuthal modulations produce chaotic time dependence A fourth instability leads discontinuously to a spatially disordered flow. The spatial structure associated with each instability has been determined qualitatively, and the frequency spectrum of the local surface deformation has been measured using a sensitive laser deflection technique.     

谐振管形状对热声发动机内非线性声场的影响

研究了谐振管一端受活塞声源激励,另一端刚性封闭条件下,管道形状对热声发动机谐振管内部非线性声场的影响。基于流体力学基本方程建立了渐变截面谐振管内一维非线性声场的模型,考虑了黏性耗散及非线性效应的影响。利用伽辽金法数值求解了该模型的速度势方程,分析了谐振管形状、活塞振动速度及激励频率对管内声场的影响。将双曲形、指数形、锥形、正弦形等四种变截面谐振管内的非线性声场与圆柱形直管的情况进行了比较。结果反映了谐振管内声场的压力波动受活塞振动速度及谐振管形状的影响;显示了当活塞振动幅度较大时,谐振管内出现的波形畸变、频率曲线偏移、共振频率滞后等非线性现象;揭示了变截面谐振管在抑制管内的高阶谐波及提高压比等方面的优越性。      

Standing waves in an elastic layer loaded with a finite mass

Standing shear waves in a plane-parallel rubberlike layer fixed without slippage between two rigid plates with finite masses are investigated. The lower plate, which underlies the layer, oscillates in the direction parallel to its surface under an external harmonic force, whereas the upper plate freely overlies the layer. It is shown both theoretically and experimentally that such a system exhibits resonances at frequencies the values of which depend on the mass of the free plate and the shear modulus of the layer. The shapes of the resonance curves are calculated and measured for different values of parameters of the layer and different masses of the upper plate. From the measured resonance curves, it is possible to determine the dynamic shear modulus and the shear viscosity of the rubberlike material.     

Transmission of oscillations through a layer of a nonlinear elastic medium

The transmission of shear one-dimensional periodic perturbations through a layer of a nonlinearly elastic medium under the conditions close to resonance is considered. The layer separates two half-spaces consisting of a medium that is much more rigid, as compared to the medium in the layer. A system of differential equations is obtained for describing the slow variations in the amplitude and waveform of nonlinear strain and stress oscillations at the fixed boundary that occur because of the nonlinear properties of the medium while the other boundary performs arbitrary periodic motions in its plane. The period of these oscillations is close to the period of natural oscillations of the layer. It is shown that, in addition to continuous strain variations at the fixed boundary, strain variations containing strong discontinuities are possible. Relations at the discontinuities are obtained. The analogy between the equations derived for the case under study and the equations describing the propagation of strain waves in a homogeneous anisotropic elastic medium is pointed out.     

Shear standing waves in a resonator with a rubberlike inhomogeneous medium

A modification of the finite-element method is proposed for calculating shear standing waves in a resonator filled with an incompressible elastic medium with allowance for the finite dimensions of the resonator and inhomogeneities of the shear modulus. Resonance curves are calculated for resonators with inhomogeneities in the form of cavities and elastic inclusions. Numerical calculations are compared with experimental data.     

Low-frequency shear parameters of liquid viscoelastic materials

An experimental study of the shear parameters of viscoelastic liquids is carried out by the acoustic resonance method based on the changes in the natural frequency and Q factor of a piezoelectric quartz resonator. The liquid to be studied is placed between a stationary quartz strap and the piezoelectric quartz crystal vibrating at the resonance frequency. For a set of drilling muds, the values of the real and imaginary shear moduli are obtained at a frequency of 74 kHz. The measurements are performed with a liquid layer thickness much smaller than the shear wavelength. It is shown that the shear modulus decreases with increasing strain amplitude. A cluster model based on the Isakovich-Chaban nonlocal diffusion theory is proposed for explaining the low-frequency viscoelastic relaxation process.     

Nonlinear degenerate resonance interaction of waves on a charged liquid surface

The physics of nonlinear degenerate resonance energy exchange between waves on the flat free charged surface of a conducting liquid is analytically (asymptotically) studied up to the second order of smallness. A set of differential equations for the evolution of the amplitudes of nonlinearly resonantly interacting waves is derived. It turns out that nonlinear computations (taking into account the dependence of the wave frequency on the finite amplitude) yield an infinite number of degenerate resonances, although computations based on frequencies found in the linear theory give a finite number of resonances. In nonlinear computations, the positions of the degenerate resonances depend on the surface charge density (or on the external electric field normal to the free surface of the liquid) in contrast to the results of linear computations (based on frequencies found in the linear theory). It is found that as the wavenumber of an exact degenerate resonance is approached (that is, in the vicinity of this number), the direction of energy transfer changes sign: now the energy is transferred from a shorter wave to a longer one and not the reverse.     

On the interaction of a shear wave with a moving domain wall in a nonlinear response of the spin subsystem

A solution of the problem of parametric interaction between a plane monochromatic shear wave and a uniformly moving 180°-domain wall of a garnet-ferrite crystal is obtained in the exchangeless magnetostatic approximation by using the perturbation method under the conditions of a nonlinear response of the spin subsystem. It is shown that in a ferromagnetic resonance with magnetostatic oscillation of stray fields, the nonlinearity of the spin subsystem leads to the excitation of shear waves of triple frequency, which may have amplitudes comparable with that of the incident wave for oscillations doubly localized by a domain wall.     

Numerical simulation of nonlinear ionization nonuniformities innonequilibrium disk MHD generator

The formation and evolution of the ionization nonuniformities from initial disturbances of finite amplitude in the nonequilibrium Ar-Cs plasma in a disk magnetohydrodynamic (MHD) generator is studied by the numerical simulation, The simulations are carried out in the wide interval of electron temperatures corresponding to the region at which the seed partially ionizes, the region of the linear plasma stability at the fully ionized seed, and the region of the instability corresponding to the partial ionization of Ar at high electron temperatures. Initial disturbances of finite amplitude in electron temperature and density are introduced at the time t=0 into the homogeneous plasma distribution, and the critical amplitudes determining the development of the instability are calculated. The initial disturbances are constructed using random functions with different spatial scales, The results are compared with the calculation of the critical amplitudes from the nonlinear theory of the plane ionization waves, It is found that at electron temperatures lower than 5500 K, the temperature dependence of the critical amplitudes and the structure of the nonlinear waves agree well with the nonlinear theory, In the electron temperature region corresponding to the partial ionization of the noble gas (T_e>5500 K), the finite ionization rate of argon atoms is essential for analysis of the instability, In this region the margin of the plasma stability is wider than it is predicted by the nonlinear theory, The nonuniformity in the argon ion number density plays the dominating role in the instability development at high electron temperatures (T_e>5500 K) in comparison with the nonuniformity in T_e in the initial disturbances,     

Electromechanical models of nanoresonators

The goal of this study is to construct simple electromechanical models of nanoresonators as mass detectors. A major obstacle in the achievement of sufficient measurement accuracy for the resonant frequency associated with the adsorption of additional mass onto the graphene layer is a low quality factor of the oscillatory system containing the graphene layer. A graphene resonator can be considered as an elastic system with distributed parameters. The application of the Galerkin method to study nearly resonant vibrational modes reduces the problem to considering an oscillatory system with a few degrees of freedom with pronounced nonlinear properties. These properties are, first of all, due to the nonlinear dependence of the forces produced by the electric field on the graphene deflection and, second, due to the nonlinear dependence of the graphene layer tension on its deflection. Taking into account the nonlinear properties leads to the appearance of characteristic drops in the resonance curve which allow for a more accurate resonant frequency measurement. Resonance curves with such characteristic drops can be obtained using a demonstration experimental macromodel of the resonator. Two absolutely new layouts are proposed, such as a differential resonator and resonator with parametric excitation. The oscillations excited in the differential resonator that contains two graphene layers resemble beats. In this case, small changes in the mass of the main layer correspond to significant changes in the frequency of the envelope. This effect is illustrated by oscillograms obtained for an experimental macromodel of the differential resonator. The parametric resonator has one graphene layer between two conducting surfaces. Parametric excitation of steady-state high amplitude oscillations is possible in this resonator only in a narrow frequency band close to the eigenfrequency. The band width reduces with a decrease in the quality factor of the oscillatory system. The latter fact can be useful for the improvement of eigenfrequency measurement accuracy at a low quality factor of the oscillatory system.     

Slow nonlinearity kinetics of the viscoelastic properties of oil during shear vibrations

Earlier we discovered the slow evolution of viscoelastic moduli of heavy crude oil. The shear modulus was measured at frequencies of 0.5, 5, and 50 Hz at different temperatures over 72 h. New studies of the dependence of the complex shear modulus on the strain amplitude revealed a logarithmic increase in the nonlinearity parameter as a function of time for this oil sample. It was experimentally established that the complex shear modulus is a linear function of the amplitude of shear perturbations. This is possible in the case of a linear dependence of values of the viscoelastic characteristics on the medium modulus of deformation.     

Shear elasticity of fluids at low-frequent shear influence

The visco-elastic properties of liquids have been investigated using acoustical resonance method. Piezoquatrz performed tangential oscillations on the main resonance frequency of 74 kHz contacts by the one end of horizontal surface with the studied liquid layer covered by quartz cover-plate. So the stagnant shear waves are installed in layer. The solution of interaction of piezoquartz-liquid layer-cover-plate gives three methods of determination of the real shear modulus (G) and the tangent of mechanical loss angle (tan theta) of liquid. The first method is realized at smaller thickness of liquid layer then the length of shear wave. Liquids of different classes have been studied using this method: polymer liquids, oils, glycols and alcohols. The second method is connected with the propagation of shear wave in liquid layer, parameters of which are determined the G and tan theta. And the third method is based on the determination of limit shift of resonance frequencies at completes damping of shear wave in thick layer of liquid. All these three methods give satisfactory agreement of results.