An analytical formulation in 3D domain for the nonlinear response of piezoelectric slabs under weak electric fields

Piezoceramic materials exhibit different types of nonlinearities depending upon the magnitude of the mechanical and electric field strength within the body. Some of the nonlinear phenomena observed under weak electric fields near resonance frequency excitation are the presence of superharmonics in the response spectra and the jump phenomena etc. In this work, an analytical solution for the nonlinear response of rectangular piezoceramic slabs have been obtained by Rayleigh–Ritz method and perturbation technique in the 3-D domain using a generalized nonlinear electric enthalpy density function. Forced vibration experiments (excitation with electric field) have been conducted on a rectangular piezoceramic slab at varying electric field amplitudes and the analytical solutions have been shown to compare very well with the experimental results.     

Dynamic Electrical-Mechanical Energy Coupling in Electrolyte-Mineral Systems

Electrical-mechanical and mechanical-electrical coupling phenomena have been observed in geo-materials. Underlying internal phenomena are not always clear. In this paper, we identify potential microscale processes on the bases of previously published research; fundamental physical principles and analytical models are included. Second, we notice important inconsistencies between theoretically predicted values or high-frequency laboratory data with respect to low-frequency field observations. To explore these differences, we conduct a laboratory experimental study that simulates field conditions, placing emphasis on low frequency coupling, that is, less than 10 kHz. Both, mechanical-electrical and electrical-mechanical effects are observed. Finally, hypotheses are presented in an attempt to explain observed differences.     

Coupling between high-frequency modes and a low-frequency mode: Theory and experiment

An analytical and experimental investigation into the response of a nonlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotropic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the response consisted of the first, third, and fourth modes. In these cases, the modulation frequency of the amplitudes and phases of the third and fourth modes was equal to the response frequency of the first mode. Subsequently, we developed an analytical model to explain the interactions between the widely separated modes observed in the experiments. We used a three-mode Galerkin projection of the partial-differential equation governing a thin, isotropic, inextensional beam and obtained a sixth-order nonautonomous system of equations by using an unconventional coordinate transformation. In the analytical model, we used experimentally determined damping coefficients. From this nonautonomous system, we obtained a first approximation of the response by using the method of averaging. The analytically predicted responses and bifurcation diagrams show good qualitative agreement with the experimental observations. The current study brings to light a new type of nonlinear motion not reported before in the literature and should be of relevance to many structural and mechanical systems. In this motion, a static response of a low-frequency mode interacts with the dynamic response of two high-frequency modes. This motion loses stability, resulting in oscillations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.     

Nonlinear behaviour of piezoceramics under weak electric fields. Part-II: Numerical results and validation with experiments

When excited near resonance in the presence of weak electric fields, piezoceramic materials exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In an accompanying paper, a generalized nonlinear 3D finite element formulation has been developed incorporating quadratic and cubic terms in the electric enthalpy density function and the virtual work done by damping forces. In this paper, the formulation has been validated by conducting experiments on test pieces of various geometries and of three different materials (in all, four case studies). Both proportional damping and nonlinear damping formulations have been used to predict the frequency response of these systems. Newmark-β method has been used to obtain the dynamic response of the systems using FE analysis. It is demonstrated that the nonlinear finite element model is able to predict the responses of the various test cases studied and the results match very well with those of experimental observations.     

Nonlinear behaviour of piezoceramics under weak electric fields: Part-I: 3-D finite element formulation

Piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak electric fields, they exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In order to model such nonlinearities, a nonlinear electric enthalpy density function (using quadratic and cubic terms) valid for a general 3-D piezoelectric continuum has been proposed in this work. Linear (i.e. proportional) and nonlinear damping models have also been proposed. The coupled nonlinear finite element equations have been derived using variational formulation. The classical linearization technique has been used to derive the linearized stiffness and damping matrices which helps in assembling the nonlinear matrices and solution of resulting nonlinear equation. The general 3-D finite element formulation is discussed in this paper. In a companion paper by Samal et al., numerical results on various typical examples are shown to match very well with the experimental observations.     

Measurement of the acoustic velocity field of nonlinear standing waves using the synchronized PIV technique

The motion of gas within an air-filled rigid-walled square channel subjected to acoustic standing waves is experimentally investigated. The synchronized particle image velocimetry (PIV) technique has been used to measure the acoustic velocity fields at different phases over the excitation signal period. The acoustic velocity measurements have been conducted for two different acoustic intensities in the quasi-nonlinear range (in which the nonlinear effects can be neglected in comparison with the dissipation effects), and one acoustic intensity in the finite-amplitude nonlinear range (in which both the nonlinear term and the dissipative term play a role in the wave equation). The experimental velocity fields for the quasi-nonlinear cases are compared with the analytical results obtained from the time-harmonic solution of the wave equation. Good agreement between the experimental and analytical velocity fields proves the ability of the synchronized PIV technique to accurately measure both temporal and spatial variations of the acoustic velocity fields. The verified technique is then used to measure the acoustic velocity fields of the finite-amplitude nonlinear case at different phases.     

Acoustic resonance phenomena in air bleed channels in aviation engines

The existence of axial–radial acoustic resonance oscillations of the basic air flow in bleed channels of aviation engines is demonstrated theoretically and experimentally. Numerical and analytical methods are used to determine the frequency of acoustic resonance oscillations for the lowest modes of open and closed bleed channels of the PS-90A engine. Experimental investigations reveal new acoustic resonance phenomena arising in the air flow in bleed channel cavities in the core duct of this engine owing to instability of the basic air flow. The results of numerical, analytical, and experimental studies of the resonance frequencies reached in the flow in bleed channel cavities in the core duct of the PS-90A engine are found to be in reasonable agreement. As a result, various types of resonance oscillations in bleed channels can be accurately described.     

Non-lane-discipline-based car-following model incorporating the electronic throttle dynamics under connected environment

We investigate the nonlinear dynamics of a system of generalized Duffing-type MEMS resonator in the frame of simple analog electronic circuit. A mathematical model formed for the proposed generalized Duffing-type MEMS oscillator in which nonlinearities arising out of two different sources such as mid-plane stretching and electrostatic force can lead to variety of nonlinear phenomena such as period-doubling route, transient chaos and homo-/heteroclinic oscillations. These phenomena were confirmed through detailed numerical investigations such as phase portraits, bifurcation diagram, Poincaré map, Lyapunov exponent spectrum and finite-time Lyapunov exponent. The analog circuit realization for the Duffing-type MEMS resonator is constructed. The numerically simulated results are confirmed in the laboratory experimental observations which are closely matched with each other. The experimentally observed chaotic attractor confirmed through FFT spectrum, 0–1 test and Poincaré cross section. In addition, the robustness of the signal strength is confirmed through signal-to-noise ratio.     

Interaction between Low and High-Frequency Modes in a Nonlinear System: Gas-Filled Cylinder Covered by a Movable Piston

A simple mechanical system containing a low-frequency vibration mode andset of high-frequency acoustic modes is considered. The frequencyresponse is calculated. Nonlinear behaviour and interaction betweenmodes is described by system of functional equations. Two types ofnonlinearities are taken into account. The first one is caused by thefinite displacement of a movable boundary, and the second one is thevolume nonlinearity of gas. New mathematical models based on nonlinearequations are suggested. Some examples of nonlinear phenomena arediscussed on the base of derived solutions.     

Parametric interaction of acoustic waves in micro-inhomogeneous media with hysteretic nonlinearity and relaxation

A theoretical investigation of parametric processes that arise as a result of the interaction of powerful and weak longitudinal acoustic waves in micro-inhomogeneous media with hysteretic nonlinearity and relaxation was carried out. The case of degenerate interaction between a powerful high-frequency wave and a weak low-frequency one was considered. The nonlinear damping coefficient and the carrier frequency phase delay of the weak wave propagating under the action of the powerful wave were determined.     

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.     

Stability of Gaussian-type light bullets in the cubic-quintic-septimal nonlinear media with different diffractions under $${\mathcal {}}$$-symmetric potentials

From the governing equation \(-(3+1)\)-dimensional nonlinear Schrödinger equation with cubic-quintic-septimal nonlinearities, different diffractions and \({\mathcal {PT}}\)-symmetric potentials, we obtain two kinds of analytical Gaussian-type light bullet solutions. The septimal nonlinear term has a strong impact on the formation of light bullets. The eigenvalue method and direct numerical simulation to analytical solutions imply that stable and unstable evolution of light bullets against white noise attributes to the coaction of cubic-quintic-septimal nonlinearities, dispersion, different diffractions and \({\mathcal {PT}}\)-symmetric potential.     

Nonlinear LCO “amplitude–frequency” characteristics for plates fluttering at supersonic speeds

The nonlinear aeroelastic behavior of isotropic rectangular plates in supersonic gas flow is examined. Quadratic and cubic aerodynamic nonlinearities as well as cubic geometrical nonlinearities are considered in this study. While the aerodynamic nonlinearities are the results of the expansion of the nonlinear piston-theory aerodynamics loading up to the third-order, the geometrical nonlinearities are due to stiffening effects from the panel out-of-plane deformation consistent with the von Karman’s nonlinear plate theory. While in vacuum the typical nonlinear hardening frequency vs. oscillation amplitude, one characterized by monotonically increasing amplitudes at increasing frequencies, exists, in the presence of a high-speed flow, qualitative and quantitative changes of the nonlinear relationship are expected. This paper shows how the thin-plate behavior is influenced by the high-speed flows providing the “amplitude–frequency” dependency, which describes the nonlinear oscillations of the considered aeroelastic system.     

Three-wave interactions between disturbances in the hypersonic boundary layer on impermeable and porous surfaces

The interaction between disturbances in the hypersonic boundary layer on impermeable and porous surfaces is considered within the framework of weakly-nonlinear stability theory. It is established that on the impermeable surface nonlinear interactions between different waves (acoustic and vortex) occur in the parametric resonance regime. The role of pumping wave is played by a plane acoustic wave. The nonlinear interactions take place over a wide frequency range and can lead to the packet growth of Tollmien-Schlichting waves. On the porous surface the analogous interactions are fairly weak and result in a slight decay of the acoustic mode and a slight amplification of the vortex mode. This leads to the dragging out of the laminar flow regime and the regions of linear disturbance growth. In this situation the low-frequency spectrum of the vortex modes may be filled on account of the nonlinear processes occurring in the three-wave systems between the vortex components.     

压电超声电机的非线性动力学研究进展

压电超声电机 (ultrasonic motor, USM) 是一个具有闭环控制的机电耦合强非线性系统,在大功率情况下,会表现出明显的非线性现象．其非线性因素主要表现为: (1) 定子与动子之间的接触非线性; (2) 定子材料非线性; (3) 温度、磨损等因素引起的缓变因素; (4) 驱动电网络中的非线性因素．由于诸多非线性因素的存在,使得 USM 系统具有跳跃、滞后、共振频率漂移、死区和饱和等非线性现象,同时,给 USM 的非线性动力学建模带来了困难．而 USM 闭环控制要求有一个合理的动力学模型,因此,研究 USM 的非线性动力学建模是 USM 研究中的一个重要内容．针对超声电机系统中普遍存在的非线性因素和非线性动力学建模方法进行了归纳,对已有文献中有关超声电机非线性动力学的研究进行了总结,指出了有关该方面的若干研究方向．      

Nonlinear vibration of stringer shell by means of extended Hamiltonian approach

In this paper, the nonlinear free vibration of a stringer shell is studied. The mathematical model of the string shell, which is the most convenient for frequency analysis, is considered. Due to the geometrical properties of the vibrating shell, strong nonlinearities are evident. Approximate analytical expressions for the nonlinear vibration are provided by introducing the extended version of the Hamiltonian approach. The method suggested in the paper gives the approximate solution for the differential equation with dissipative term for which the Lagrangian exists. The aim of this study is to provide engineers and designers with an easy method for determining the shell nonlinear vibration frequency and nonlinear behavior. The effects of different parameters on the ratio of nonlinear to linear natural frequency of shells are studied. This analytical representation gives excellent approximations to the numerical solutions for the whole range of the oscillation amplitude, reducing the respective error of the angular frequency in comparison with the Hamiltonian approach. This study shows that a first-order approximation of the Hamiltonian approach leads to highly accurate solutions that are valid for a wide range of vibration amplitudes.     

Sound absorbing properties of spiral metasurfaces inspired by micro-perforated plates

As a kind of classical low-frequency sound-absorbing material, the microperforated plate (MPP) has been widely used. Here, we inspired by the sound absorption mechanism of the MPP, a spiral metasurface (SM) is designed and the analytical solution of acoustic impedance and sound absorption coefficient are obtained. The relationship between the sound absorption properties of the MPP and the SM with their own structures is systematically studied, and the analytical solutions are used to optimise the structure. It is concluded that the MPP and the SM of the same thickness achieve effective absorption in the frequency range between 390-900 Hz and 1920-4266 Hz, with a total thickness less than 1/6 of the wavelength. Meanwhile, the numerical calculation shows that the MPP and SM can match well with the background medium in the effective rang. Our study provides new insights into the design methods of sound-absorbing materials and is potentially suitable for many acoustic engineering applications.     

Bulk and surface acoustic wave phenomena in crystals: Observation and interpretation

Crystal acoustics is a field that has engaged the attention of theoreticians and experimentalists alike for decades and more. Many striking effects have been revealed, and elegant analytical techniques applied to their interpretation. This article is oriented towards the experimental aspects of the field and the interpretation of the phenomena that have been observed. Particular attention is given to reviewing the techniques that have probed the intricacy of acoustic wave propagation in crystals, including phonon imaging, laser- and capillary-fracture-generated ultrasound, transmission acoustic microscopy and surface Brillouin scattering, and a selection of results obtained with these techniques is presented. Some of these studies pertain to bulk waves and others to surface acoustic waves. The interpretation of far-field observations is carried out within the ray approximation, and elastodynamic Green’s functions are invoked in the interpretation of near-field results. Extensive use is made of the acoustic-slowness and wave surfaces, in particular features such as acoustic axes, with their attendant polarization singularities, and folds in the wave surface.