Nonlinear Harmonic Components Research Articles

Interfacial magneto-plasticity in partially oxidized magnetite inferred from linear and nonlinear dielectric response

In partially oxidized polycrystalline magnetite, defects/dislocations are formed at the grain/interphase boundaries among the remaining magnetite and oxidized phases. Here, the studied sample is a pellet with the composition of Fe3O4 and α-Fe2O3(3:7), which is obtained by annealing partially air-oxidized polycrystalline Fe3O4. To probe the dislocation behavior in it under the magnetic field, linear and nonlinear dielectric spectroscopy has been utilized under three conditions: before applying the magnetic field, under a magnetic field after cooling at the same field, and after removing the magnetic field. An excessive qualitative and quantitative difference between the dielectric data—both linear and nonlinear harmonic components—taken in the first two protocols has been observed. The dielectric data feature remains like the field-cooled measurement after magnetic field withdrawal. In the field-cooled dielectric frequency spectrum, grain boundary relaxation shows a glass transition above 110 K. This phenomenon occurs mainly due to the plastic deformation through the dislocation movements at the boundaries by the magnetostrictive strain produced in the magnetite grains below the Verwey transition (TV). The dislocations freeze into a glassy phase below the TV of magnetite. The interaction among the dislocations and the boundaries are responsible for showing the nonlinear features.

An experimental decomposition of nonlinear forces on a surface-piercing column: Stokes-type expansions of the force harmonics

Wave loading on marine structures is the major external force to be considered in the design of such structures. The accurate prediction of the nonlinear high-order components of the wave loading has been an unresolved challenging problem. In this paper, the nonlinear harmonic components of hydrodynamic forces on a bottom-mounted vertical cylinder are investigated experimentally. A large number of experiments were conducted in the Danish Hydraulic Institute shallow water wave basin on the cylinder, both on a flat bed and a sloping bed, as part of a European collaborative research project. High-quality data sets for focused wave groups have been collected for a wide range of wave conditions. The high-order harmonic force components are separated by applying the ‘phase-inversion’ method to the measured force time histories for a crest focused wave group and the same wave group inverted. This separation method is found to work well even for locally violent nearly-breaking waves formed from bidirectional wave pairs. It is also found that the $n$th-harmonic force scales with the $n$th power of the envelope of both the linear undisturbed free-surface elevation and the linear force component in both time variation and amplitude. This allows estimation of the higher-order harmonic shapes and time histories from knowledge of the linear component alone. The experiments also show that the harmonic structure of the wave loading on the cylinder is virtually unaltered by the introduction of a sloping bed, depending only on the local wave properties at the cylinder. Furthermore, our new experimental results reveal that for certain wave cases the linear loading is actually less than 40 % of the total wave loading and the high-order harmonics contribute more than 60 % of the loading. The significance of this striking new result is that it reveals the importance of high-order nonlinear wave loading on offshore structures and means that such loading should be considered in their design.

Nonlinear post-flutter behavior and self-excited force model of a twin-side-girder bridge deck

The nonlinear aeroelastic behavior of a typical bluff bridge section, i.e., open twin-side-girder bridge deck, was investigated through a series of spring-suspended sectional model (SSSM) tests in the post-flutter state. When the reduced wind speed exceeded linear aeroelastic limits, the sectional model underwent significant limit cycle oscillation (LCO) characterized by a quasi-harmonic torsional vibration with slight heave-torsion coupling effect. The stable amplitudes of post-flutter LCOs increased in a roughly linear way with reduced wind speed. The nonlinear self-excited force (NSEF) was measured by a novel force measurement technique and validated by comparing the calculated post-flutter responses with experiments. The measured NSEF contained significant 2nd and 3rd order nonlinear harmonic components. Based on the measured NSEF signals and equivalent energy principle, a NSEF model suitable for large amplitude of vibration was proposed and then verified by comparing its predicted response with experimental results. The identified aerodynamic parameters further indicated that the mechanism of post-flutter LCOs was due to the negative aerodynamic damping provided by the linear term as an energy source to drive the growth of vibration amplitude and the nonlinear positive aerodynamic damping provided by the 3rd order angular velocity term as a stabilizing factor of self-limitation phenomenon.

Necessary Conditions for Nonlinear Ultrasonic Modulation Generation Given a Localized Fatigue Crack in a Plate-Like Structure.

It has been shown that nonlinear ultrasonics can be more sensitive to local incipient defects, such as a fatigue crack, than conventional linear ultrasonics. Therefore, there is an increasing interest in utilizing nonlinear ultrasonics for structural health monitoring and nondestructive testing applications. While the conditions, which are the necessary conditions that should be satisfied for the generation of nonlinear harmonic components, are extensively studied for distributed material nonlinearity, little work has been done to understand the necessary conditions at the presence of a localized nonlinear source such as a fatigue crack. In this paper, the necessary conditions of nonlinear ultrasonic modulation generation in a plate-like structure are formulated specifically for a localized nonlinear source. Then, the correctness of the formulated necessary conditions is experimentally verified using ultrasounds obtained from aluminum plates.

On the use of non-linear vibrations and the anti-resonances of Higher-Order Frequency Response Functions for crack detection in pipeline beam

The identification of new scientific challenges, as well as the increasing high-performance computing support, indicates that the benefits of applying novel nonlinear techniques for crack detection will continue to grow. So, significant effort has been invested in recent years to develop effective techniques to detect crack in mechanical structures. The objective of this paper is to discuss and propose a robust diagnostic of damage based on non-linear vibrational measurements with particular regard to the Higher-Order Frequency Response Functions. An important observation is that the appearances of the non-linear harmonic components and the emerging anti-resonances in Higher-Order Frequency Response Functions can provide useful information on the presence of cracks and may be used on an on-line crack monitoring system for small levels of damage. Efficiency of the proposed methodology is illustrated through numerical examples for a pipeline beam including a breathing crack.

Simultaneous Measurements of Harmonic Waves at Fatigue-Cracked Interfaces

Nonlinear harmonic waves generated at cracked interfaces are investigated theoretically and experimentally. A compact tension specimen is fabricated and the amplitude of the transmitted wave is analyzed as a function of position along the fatigued crack surface. In order to measure as many nonlinear harmonic components as possible, broadband lithium niobate (LiNbO3) transducers are employed together with a calibration technique for making absolute amplitude measurements with fluid-coupled receiving transducers. Cracked interfaces are shown to generate high acoustic nonlinearities, which are manifested as harmonics in the power spectrum of the received signal. The first subharmonic f/2 and the second harmonic 2f waves are found to be dominant nonlinear components for an incident toneburst signal of frequency f. To explain the observed nonlinear behavior, a partially closed crack is modeled by planar half interfaces that can account for crack parameters, such as crack opening displacement and crack surface conditions. The simulation results show reasonable agreement with the experimental results.

Nonlinear harmonic components in dielectric relaxation of polyelectrolytes

The problem of the nonlinear ac dielectric response in polyelectrolytes is carried out by considering these electrically charged systems acted on by a strong dc bias electric field superimposed on a weak ac field. An infinite set of coupled (translation-rotation) kinetic equations is derived, and by using a perturbation procedure, we establish a new expression for the complex dielectric increment of an assembly of nonpolar rodlike macroions surrounded by small counterions. By considering the stationary regime, we show that the electric polarization of the nonpolar macroion is given by calculating the expectation value of P(1)(cos theta)H(1)(x), where P(1) and H(1) stand for the first Legendre and Hermite polynomials, respectively. The results so obtained provide new relaxation features, which are presented in the form of analytic expressions and illustrated by Cole-Cole-like diagrams and three-dimensional dispersion and absorption plots in order to show the influence of the coupling parameter a measuring the importance of the rotational motion over the translational one.

Determination of the nonlinear dielectric increment in the Cole-Davidson model

The problem of the nonlinear dielectric relaxation of complex liquids is tackled in the context of the Cole-Davidson [J. Chem. Phys. 19, 1484 (1951)] model. By using an appropriate time derivative of noninteger order, an infinite hierarchy of differential-recurrence relations for the moments (expectation values of the Legendre polynomials) is obtained. The solution is established for the stationary regime of an ensemble of polar and symmetric-top molecules acted on by a strong dc bias electric field superimposed on a weak ac electric field. The results for the first three nonlinear harmonic components of the electric susceptibility are analytically established and illustrated with the help of Argand diagrams for the nonlinear dielectric increment and three-dimensional dispersion and absorption spectra for the second and the third harmonic components as a function of the anomalous exponent beta</=1, the value of which gives rise to skewed arcs (Argand plots) and asymmetric shapes (loss spectra) in the high-frequency domain.

Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids.

The problem of the nonlinear dielectric response due to the application of a strong electric field is reconsidered in the context of fractional kinetic equations. To accomplish that, we start from a fractional noninertial Fokker-Planck equation and restrict ourselves to the case of anomalous subdiffusive processes characterized by the critical exponent alpha ranging from 0 to 1, the limit of normal diffusion. In particular, we evaluate the first- and third-order nonlinear harmonic components of the electric polarization in the case of either a pure ac field or a strong dc bias field superimposed on a weak ac field. The stationary regime is therefore calculated from an infinite set of differential recurrence relations by using a perturbation method. The results so obtained are illustrated by three-dimensional dispersion and absorption plots in order to show the influence of alpha. Cole-Cole diagrams are also presented, allowing one to see that the arcs become more and more flattened as alpha-->0, and corresponding to a broadening of the absorption peaks as effectively observed in complex liquids. The theoretical model is supported by comparison with experimental data of the third-order nonlinear dielectric permittivity of a ferroelectric liquid crystal.

Nonlinear harmonic components of the electric polarization of symmetric-top molecules

The nonlinear dielectric relaxation of an assembly of polar molecules submitted to the action of a dc field superimposed on an ac electric field is studied theoretically in the framework of the noninertial rotational diffusion model. It is also assumed that the permanent dipole moment vector makes an arbitrary angle with respect to the long molecular axis of each rodlike molecule. By calculating the ensemble averages of two Wigner rotation matrix elements 〈D0,01(Ω)〈(t) and 〈D0,11(Ω)〉(t), it is possible to express the first three harmonic components of the electric polarization varying in ω, 2ω and 3ω in analytic form, together with the steady-state frequency-dependent component. Three-dimensional dispersion and absorption plots are presented in order to show the influence of the diffusion anisotropy parameter Δ. Cole–Cole diagrams of the second and third harmonic components are also presented allowing one to distinguish two relaxation processes along the short and the long molecular axes, respectively.