Shear waves in a resonator with cubic nonlinearity

Shear waves with finite amplitude in a one-dimensional resonator in the form of a layer of a rubber-like medium with a rigid plate of finite mass at the upper surface of the layer are investigated. The lower boundary of the layer oscillates according to a harmonic law with a preset acceleration. The equation of motion for particles in a resonator is determined using a model of a medium with a single relaxation time and cubical dependence of the shear modulus on deformation. The amplitude and form of shear waves in a resonator are calculated numerically by the finite difference method at shifted grids. Resonance curves are obtained at different acceleration amplitudes at the lower boundary of a layer. It is demonstrated that, as the oscillation amplitude in the resonator grows, the value of the resonance frequency increases and the shape of the resonance curve becomes asymmetrical. At sufficiently large amplitudes, a bistability region is observed. Measurements were conducted with a resonator, where a layer with the thickness of 15 mm was manufactured of a rubber-like polymer called plastisol. The shear modulus of the polymer at small deformations and the nonlinearity coefficient were determined according to the experimental dependence of mechanical stress on shear deformation. Oscillation amplitudes in the resonator attained values when the maximum shear deformations in the layer were 0.4–0.6, which provided an opportunity to observe nonlinear effects. Measured dependences of the resonance frequency on the oscillation amplitude corresponded to the calculated ones that were obtained at a smaller value of the nonlinear coefficient.     

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.     

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.     

Analytical approach for free vibration analysis of two-layer Timoshenko beams with interlayer slip

In this paper, an analytical procedure for free vibrations of shear-deformable two-layer beams with interlayer slip is developed. The effect of transverse shear flexibility of two layers is taken into account in a general way by assuming that each layer behaves as a Timoshenko beam element. Therefore, the layers have independent shear strains that depend indeed on their own shear modulus. This is the main improvement of the proposed model compared to existing models where the transverse shear flexibility is ignored or taken into account in a simplified way in which the shear strains of both layers are assumed to be equal whatever the shear modulus of the layers. In the proposed model, the two layers are connected continuously and the partial interaction is considered by assuming a continuous relationship between the interface shear flow and the corresponding slip. Based on these key assumptions, the governing differential equation of the problem is derived using Hamilton's principle and is analytically solved. The solutions for the eigenfrequencies and eigenmodes of four single span two-layer beams with classical Euler boundary conditions, i.e. pinned-pinned, clamped-clamped, clamped-pinned and clamped-free, are presented. Next, some numerical applications dealing with these four beams are carried out in order to compare the eigenfrequencies obtained with the proposed model against two existing models which consider different kinematic assumptions. Finally, a parametric study is conducted with the aim to investigate the influence of varying material and geometric parameters on the eigenfrequencies, such as shear stiffness of the connectors, span-to-depth ratios, flexural-to-shear moduli ratios and layer shear moduli ratios.     

Elastic moduli inheritance and the weakest link in bulk metallic glasses

We show that a variety of bulk metallic glasses (BMGs) inherit their Young's modulus and shear modulus from the solvent components. This is attributed to preferential straining of locally solvent-rich configurations among tightly bonded atomic clusters, which constitute the weakest link in an amorphous structure. This aspect of inhomogeneous deformation, also revealed by our in situ neutron diffraction studies of an elastically deformed BMG, suggests a rubberlike viscoelastic behavior due to a hierarchy of atomic bonds in BMGs.     

The pure-shear mode solidly mounted resonator based on c-axis oriented ZnO film

In this paper, we fabricate a pure-shear mode film bulk acoustic resonator based on c-axis oriented ZnO film. The resonator is consisted of an in-plane electrode, a highly c-axis oriented ZnO film and a SiO₂/W Bragg reflector. The shear mode wave is excited by the lateral electric field. The resonator works in a pure-shear mode with the resonance frequency near 1.5 GHz and the Q-factor of 479 in air. There is no obvious longitudinal mode resonance in the frequency response, which can be explained that the electric field component normal to the surface is very weak and the Bragg reflector has the effective frequency selectivity for the shear mode. Importantly for sensors, the immersion into de-ionized water and glycerol liquid still allows for a Q-factor up to 335 and 220, respectively. This resonator shows the potential as mass loading sensors for biochemical application.     

Electric impedance and amplitude-frequency response of a one-dimensional ultrasonic liquid-filled resonator with flat piezoelectric plates

The impedance method is used to determine the electric impedance of a resonator. The amplitude-frequency response of a one-dimensional liquid-filled ultrasonic resonator is calculated by directly solving the wave equations and piezoelectric effect equations under the corresponding boundary conditions. An analysis of the amplitude-frequency response shows that the simple analytical expression obtained from the aforementioned solution is in good agreement with experimental data. An anomalous variation of the electric current in the radiating piezoelectric plate versus the excitation frequency is theoretically revealed near the high-Q resonance peaks. This effect is confirmed experimentally. It gives rise to errors in the measured absorption coefficient and multiply broadens the resonance peaks when the measurements are performed near the resonance frequencies of the piezoelectric plates.     

边界层效应对定程干涉法声速测量的影响

定程干涉法是精度最高的气相声速测量方法,本文再现了定程干涉法共振频率的导出过程,在此基础上复现了边界层修正式的导出过程。最后以声速数据丰富的Ar为例,分析了边界层对这两种定程干涉法测量的影响规律。结果表明:边界层对定程干涉法声速测量有着显著的影响;边界层的影响随温度升高而增大、随压力升高而降低;适当增加共鸣腔体的几何尺寸有利于减小边界层效应的影响。     

Nonlinear phenomena accompanying the development of oscillations excited in a layer of a linear dissipative medium by finite displacements of its boundary

A method of describing oscillations in resonators on the basis of evolution equations is proposed. The latter are obtained by simplifying the functional equations under the assumption that the distortions of travelling waves within the resonator length are small, that the Mach number for the moving boundary oscillations is small, and that the frequency is close to one of the natural frequencies of the resonator. The problems of nonstationary oscillations of a layer with a moving boundary are solved. The law that should govern the wall oscillations to provide the development of steady-state linear resonance oscillations is determined. The shape of the resonance curve formed in the presence of a boundary nonlinearity is calculated. The method of matching of asymptotics is applied to the singularly perturbed problem with small dissipation. It is shown that a boundary nonlinearity leads to a distortion of the temporal profile of the standing wave and to the generation of higher harmonics in the process of the development of steady-state oscillations. In contrast to the classical linear problems where the resonance occurs at the coincidence of the external force frequency with one of the natural frequencies, in the case under study the resonance behavior is observed in frequency bands, which are wider the higher the amplitude of the boundary oscillations is.     

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.     

Hopf saddle-node bifurcation for fixed points of 3D-diffeomorphisms: Analysis of a resonance ‘bubble’

The dynamics near a Hopf saddle-node bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a two-parameter model map G is constructed, such that at the central bifurcation the derivative has two complex conjugate eigenvalues of modulus one and one real eigenvalue equal to 1. To investigate the effect of resonances, the complex eigenvalues are selected to have a 1:5 resonance. It is shown that, near the origin of the parameter space, the family G has two secondary Hopf saddle-node bifurcations of period five points. A cone-like structure exists in the neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf bifurcations. Quasi-periodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantor-like boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasi-periodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasi-periodic bifurcations.     

On the Helmholtz resonator

This paper develops the theory of the excitation of a Helmholtz resonator by external disturbances located arbitrarily close to the mouth of the resonator. The classical approach of Rayleigh is thereby extended to situations in which the disturbance at the mouth is not necessarily equivalent to a uniform, time dependent pressure perturbation. The analysis involves the derivation of the Green function of the resonator in a manner similar to that described in an earlier paper. The use of the Green function is illustrated by two examples in which the resonator is excited by a low Mach number stream of air. In the first case the air stream has a periodic large scale structure such as may be caused by a Kelvin-Helmholtz type of instability. The second example models the case of excitation by a shear layer possessing a continuous spectrum of turbulent eddies. In both of these applications the orders of magnitude of the sound pressure levels involved are illustrated for a typical resonator.     

High friction from a stiff polymer using microfiber arrays

High dry friction requires intimate contact between two surfaces and is generally obtained using soft materials with an elastic modulus less than 10 MPa. We demonstrate that high-friction properties similar to rubberlike materials can also be obtained using microfiber arrays constructed from a stiff thermoplastic (polypropylene, 1 GPa). The fiber arrays have a smaller true area of contact than a rubberlike material, but polypropylene's higher interfacial shear strength provides an effective friction coefficient of greater than 5 at normal loads of 8 kPa. At the pressures tested, the fiber arrays showed more than an order of magnitude increase in shear resistance compared to the bulk material. Unlike softer materials, vertical fiber arrays of stiff polymer demonstrate no measurable adhesion on smooth surfaces due to high tensile stiffness.     

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.     

Nonlinear vibration of viscoelastic embedded-DWCNTs integrated with piezoelectric layers-conveying viscous fluid considering surface effects

This article studies the nonlinear vibration of viscoelastic embedded nano-sandwich structures containing of a double walled carbon nanotube (DWCNT) integrated with two piezoelectric Zinc oxide (ZnO) layers. DWCNT and ZnO layers are subjected to magnetic and electric fields, respectively. This system is conveying viscous fluid and the related force is calculated by modified Navier–Stokes relation considering slip boundary condition and Knudsen number. Visco–Pasternak model with three parameters of the Winkler modulus, shear modulus, and damp coefficient is used for simulation of viscoelastic medium. The nano-structure is simulated as an orthotropic Timoshenko beam (TB) and the effects of small scale, structural damping and surface stress are considered based on Eringen's, Kelvin-voigt and Gurtin–Murdoch theories. Energy method and Hamilton's principle are employed to derive motion equations which are then solved using differential quadrature method (DQM). The detailed parametric study is conducted, focusing on the combined effects of small scale effect, fluid velocity, thickness of piezoelectric layer, boundary condition, surface effects, van der Waals (vdW) force on the frequency and critical velocity of nano-structure. Results indicate that the frequency and critical velocity increases with assume of surface effects.     

Zur spektralen Verteilung der Turbulenzenergie und ihrer Komponenten in turbulenten Grenzschichten

On the Spectral Distribution of Turbulent Energy and Components in Turbulent Boundary Layers KOLMOGOROV 's theory of the inertial subrange of energy spectrum is presented for turbulent boundary layers. As a consequence KOVASZNAY 's formula for the transfer of kinetic energy is confirmed. It is shown that the structure of the turbulent transfer of momentum in boundary layers is analogous to the structure of the transfer of kinetic energy in the spectrum. For the spectral distribution of the turbulent shear stress in the subrange a k^?3 law (k wave number) – apparently in agreement with measurements – is derived from dimensional arguments. It follows that the exchange of energy among the components of turbulent energy in the subrange is some orders of magnitude smaller than the latter.     

Calculation of acoustic energy trapping in resonators based on isotropic and nanoceramic materials

The problem of acoustic energy trapping in a microwave resonator structure operating on the basis of acoustic waves and containing a relatively thick nanoceramic plate, which has a piezoelectric film with electrodes on its surface, is solved. For a composite resonator structure made on the basis of isotropic substrates and nanoceramics, formulas are derived that allow one to choose the thicknesses of its layers to obtain a high Q factor at a desired frequency.     

Tunable bulk acoustic wave resonators with the induced piezoelectric effect in a ferroelectric

An electromechanical model of the piezoelectric effect induced in an acoustic resonator based on a ferroelectric film under the action of a dc or weak ac voltage is developed. The basic equation is obtained by expansion of the free energy in a series with respect to the electric induction and the mechanical deformation. The system of electromechanical equations for variable components of the induction and the mechanical deformation involves all linear terms along with the component of the electrostriction nonlinear with respect to the mechanical deformation. These electromechanical equations made it possible to obtain a one-dimensional approximation for the effective parameters of the material: the piezoelectric modulus and the elastic modulus as a function of the strength of the electric field applied to the acoustic layer. Expressions for the controlled electromechanical coupling coefficient and resonance frequencies of the tunable acoustic resonator are found. It is shown that the most significant parameter responsible for the tuning is the nonlinear electros-triction coefficient M, whose magnitude and sign were evaluated from the available experimental data.     

A subwavelength MIM waveguide resonator with an outer portion smooth bend structure

We systematically investigate bends in metal–insulator–metal (MIM) subwavelength plasmonic waveguides resonator, realized in a two-dimensional (2D) plasmon polariton metal using a finite-difference time-domain (FDTD) method with perfectly matched layers (PMLs) boundary conditions. We apply the outer portion of the bend structure in resonator which can lead to remarkably good bending transmission characteristics. We discuss the existence conditions of different modes which affect the device performance, and analyze coupling efficiency of the outer portion of the bend structure resonator in detail. Meanwhile, we find that the first dip of the outer portion smooth bend resonator nearly linear shifts toward longer wavelengths with the increase of the effective waveguiding length. In addition, add/drop directional couplers are realizable using the present resonator structure.     

A tunable hybrid graphene-metal metamaterial absorber for sensing in the THz regime

In this paper, a graphene-based metamaterial absorber is proposed and investigated numerically, in which the interaction between a split ring resonator (SRR) and graphene results in a high-Q absorption. To make a better understanding of the resonance mechanism, the electric and the magnetic fields, and the surface currents at the resonance frequency are investigated. In order to ease the analysis of the structure, an equivalent circuit model is introduced using the transmission line theory, and the accuracy of the proposed model is verified by the full-wave simulation. Finally, different aspects of the designed metamaterial are discussed as a potential label-free sensor for chemical and biomedical sensing. It is shown that by using this structure, a sensor with a sensitivity of 597 GHz/RIU can be achieved.