Nonlinear Vibration Modes Research Articles

Delocalized nonlinear vibrational modes in fcc metals

Nonlinear lattices support delocalized nonlinear vibrational modes (DNVMs) that are exact solutions to the dynamical equations of motion dictated by the lattice symmetry. Since only lattice symmetry is taken into consideration for derivation of DNVMs, they exist regardless the type of interaction between lattice points, and for arbitrary large amplitude. Here, considering space symmetry group of the fcc lattice, we derive all one-component DNVMs, whose dynamics can be described by single equation of motion. Twelve such modes are found and their dynamics are analyzed for Cu, Ni, and Al based on ab initio and molecular dynamics simulations with the use of two different interatomic potentials. Time evolution of atomic displacements, kinetic and potential energy of atoms, and stress components are reported. Frequency–amplitude dependencies of DNVMs obtained in ab initio simulations are used to assess the accuracy of the interatomic potentials. Considered interatomic potentials (by Mendelev et al. and Zhou et al.) for Al are not as accurate as for Cu and Ni. Potentials by Mendelev can be used for relatively small vibration amplitudes, not exceeding 0.1 Å, while potentials by Zhou are valid for larger amplitudes. Overall, the presented family of exact solutions of the equations of atomic motion can be used to estimate the accuracy of the interatomic potentials of fcc metals at large displacements of atoms.

Discrete breathers in a triangular β-Fermi-Pasta-Ulam-Tsingou lattice.

A practical approach to the search for (quasi-) discrete breathers (DBs) in a triangular β-FPUT lattice (after Fermi, Pasta, Ulam, and Tsingou) is proposed. DBs are obtained by superimposing localizing functions on delocalized nonlinear vibrational modes (DNVMs) having frequencies above the phonon spectrum of the lattice. Zero-dimensional and one-dimensional DBs are obtained. The former ones are localized in both spatial dimensions, and the latter ones are only in one dimension. Among the one-dimensional DBs, two families are considered: the first is based on the DNVMs of a triangular lattice, and the second is based on the DNVMs of a chain. We speculate that our systematic approach on the triangular β-FPUT lattice reveals all possible types of spatially localized oscillations with frequencies bifurcating from the upper edge of the phonon band as all DNVMs with frequencies above the phonon band are analyzed.

Two-dimensional Intrinsic Localized Mode Solutions in Hexagonal Fermi–Pasta–Ulam Lattice

By an iterative method, the Generalized Minimal Residual method, numerically well-convergent solutions of spatially localized and nonlinear vibrational modes are obtained to the two-dimensional hex...

Effect of Discrete Breathers on the Specific Heat of a Nonlinear Chain

A nonlinear chain with six-order polynomial on-site potential is used to analyze the evolution of the total to kinetic energy ratio during development of modulational instability of extended nonlinear vibrational modes. For the on-site potential of hard-type (soft-type) anharmonicity, the instability of $q=\pi$ mode ($q=0$ mode) results in the appearance of long-living discrete breathers (DBs) that gradually radiate their energy and eventually the system approaches thermal equilibrium with spatially uniform and temporally constant temperature. In the hard-type (soft-type) anharmonicity case, the total to kinetic energy ratio is minimal (maximal) in the regime of maximal energy localization by DBs. It is concluded that DBs affect specific heat of the nonlinear chain and for the case of hard-type (soft-type) anharmonicity they reduce (increase) the specific heat.

Properties of one-dimensional nonlinear vibrational modes in triangular lattice with Lennard-Jones interactions

Crystal lattices support delocalized nonlinear vibrational modes (DNVMs), which are determined solely by the lattice point symmetry, and are exact solutions of the equations of atomic motion for any interatomic potential. DNVMs can be used for setting initial conditions to excite spatially localized vibrational modes called discrete breathers (DBs). In this study, DNVMs derived for instability can result in the formation of localized vibrational modes concentrating a significant part of the lattice energy. In some cases, localized vibrational modes can be obtained by imposing localizing functions upon DNVM.

Delocalized nonlinear vibrational modes of triangular lattices

All possible one- and two-component delocalized nonlinear vibrational modes (DNVMs) in triangular lattice are analyzed. DNVMs are obtained considering solely upon the symmetry of the triangular lattice, and thus, they exist regardless the type of interactions between particles. In this work, the nearest-neighbor inter-particle interactions are described by the $$\beta $$ -FPU potential. The one-component DNVMs are periodic in time, while the two-component ones are superposition of the two vibrational modes with, generally speaking, incommensurate frequencies. For many (but not for all) two-component DNVMs, time-periodic solutions can be obtained by a proper choice of the amplitudes of the two constitutive modes. For each DNVM, frequency and energy per particle are reported as the functions of vibration amplitude together with other characteristics.

Transient Vibro-Thermography and Nonlinear Resonant Modes

Abstract Transient vibro-thermography for nondestructive evaluation and super-resolution imaging of material defects invariably employs nonlinear contact dynamics involving the ultrasonic actuator (horn) and the surface of the target structure. It produces nonlinear resonant modes of vibration in the target structural component. Vibration-induced heat generation is one phenomenon involved here. However, the contribution of nonlinear vibration on the thermal signature is poorly understood. In this study, we consider a metallic component with a thin-walled cavity as a representative sharp feature tuned to the main excitation frequency of the ultrasonic actuator. We have developed a mathematical model to simulate transient thermal signature of structural discontinuity/cavity/defect. The model incorporates a coupled thermo-viscoelastic heat generation process in the bulk material based on the Helmholtz free energy formulation. To capture the source of nonlinear resonant modes, we incorporate the stick-separation contact dynamics due to the ultrasonic horn and the target structural component. Commercial finite element simulation (comsol multiphysics) is used to quantitatively understand the nonlinear vibration response and the thermal transport behavior of the target structure with the cavity. The proposed model accounts for the effects of both the normal and the shear components of deformation contributing on heat generation and captures the nonlinear modal contribution on the heat flux map. The study shows how the geometric feature and material parameters produce an evolution of the nonlinear subsuper harmonics along with the primary harmonics tuned to the excitation frequency. Results obtained from numerical simulations are compared with the experimental results.

Two-dimensional discrete breathers in fcc metals

The interest in discrete breathers (DB), i.e. a time-periodic and spatially localized vibrational mode in a defect free nonlinear lattice, is related to their ability to localize vibrational energy of the order of several eV. In the present work, for the first time, a systematic study of eight nonlinear vibrational modes localized in one spatial dimension and delocalized in the two other dimensions is performed by means of molecular dynamics simulations in defect-free single crystals of fcc metals (Al, Cu and Ni). For this goal, the standard embedded atom method potentials are employed. Calculations are performed at a zero temperature in three-dimensional computational cells. Excitation of DBs takes place via displacement of the specified atoms from their equilibrium lattice sites according to the patterns corresponding to the delocalized nonlinear vibrational modes (DNVMs) found earlier for a two-dimensional triangular lattice. It is revealed that only four out of the eight studied DNVMs can support stable two-dimensional DBs with the lifetimes in the range of 24–47 ps and accumulate vibrational energy of the order of 2 eV per atom. All excited DBs are characterized by a hard type of nonlinearity, i.e. the frequency increases with the increasing oscillation amplitude. Obtained results indicate the presence of a wide variety of nonlinear, spatially localized vibrational modes in three-dimensional lattices of fcc metals.

Spherically localized discrete breathers in bcc metals V and Nb

Discrete breathers (DBs), also called intrinsic localized modes (ILMs), are spatially localized, large-amplitude vibrational modes in nonlinear lattices. Rod-like discrete breathers (DBs) have been reported in fcc Ni and bcc Nb by Haas et al. in 2011 based on molecular dynamics simulations. Here we use a general approach to find new type DBs in bcc V and Nb. In this approach, we firstly find the lattice symmetry dictated exact solutions to the equations of atomic motion in the form of delocalized nonlinear vibrational modes (DNVMs). Secondly, a localizing function with spherical symmetry is imposed over the DNVMs. Parameter of the localizing function is chosen such that the obtained DB has a long lifetime. The results presented in this work demonstrate that pure metals can support a variety of DBs. Interatomic potentials have a strong effect on the DB lifetime. Maximal DB lifetime in V is two orders of magnitude larger than that in Nb. The results of this study are interesting for the theory of DBs, and also they will help to better understand the impact of DBs on the physical properties of metals.

Stability of delocalized nonlinear vibrational modes in graphene lattice

Crystal lattices support delocalized nonlinear vibrational modes (DNVMs), which are determined solely by the lattice point symmetry, and are exact solutions of the equations of atomic motion for any interatomic potential. DNVMs are interesting for a number of reasons. In particular, DNVM instability can result in the formation of localized vibrational modes concentrating a significant part of the lattice energy. In some cases, localized vibrational modes can be obtained by imposing localizing functions upon DNVM. In this regard, stability of DNVMs is an important issue. In this paper, molecular dynamics is employed to address stability of all four delocalized modes in a graphene lattice in the presence of small perturbations both in the plane and normal to the plane of the lattice. When DNVM amplitude is above the stability threshold, atom trajectories deviate from the mode pattern exponentially in time. Critical exponents are calculated for the in- and out-of-plane displacements. Stability threshold amplitudes are established. Interestingly, in three of the studied DNVMs the in-plane displacements diverge faster, but in one of them the instability develops through the out-of-plane displacements. This result can be explained by the difference in atomic vibration patterns of DNVMs. Reported results refine our understanding of the nonlinear dynamics of graphene lattice and can be useful in the design of electro-mechanical resonators and sensors.

Adaptive point-to-point control system design using friction-based optimal state feedback

Abstract This paper presents an adaptive control system design for high-precision positioning of a point-to-point (PTP) servo actuator. These mechanisms are characterized by nonlinear friction behaviour, vibrational modes, and dead times which are limitations to precise motion. In the proposed strategy, adaptation of a primary controller (PID) efforts is obtained depending on an optimal state feedback. As a new methodology, estimating the states of a PTP servo actuator was carried out based on the observed friction forces besides the control signal. Performances of the proposed control systems were verified using numerical simulations and compared with that of the primary controller. The results demonstrate that, the proposed scheme achieved high-precision positioning of the PTP servo actuator regardless of the presence of its dynamics problems. Moreover, the simplicity of the proposed control technique and its effectiveness for getting high-precision motion support its applicability to variety of applications.

Delocalized Nonlinear Vibrational Modes in Graphene: Second Harmonic Generation and Negative Pressure

With the help of molecular dynamics simulations, delocalized nonlinear vibrational modes (DNVM) in graphene are analyzed. Such modes are dictated by the lattice symmetry, they are exact solutions to the atomic equations of motion, regardless the employed interatomic potential and for any mode amplitude (though for large amplitudes they are typically unstable). In this study, only one‐ and two‐component DNVM are analyzed, they are reducible to the dynamical systems with one and two degrees of freedom, respectively. There exist 4 one‐component and 12 two‐component DNVM with in‐plane atomic displacements. Any two‐component mode includes one of the one‐component modes. If the amplitudes of the modes constituting a two‐component mode are properly chosen, periodic in time vibrations are observed for the two degrees of freedom at frequencies ω and 2ω, that is, second harmonic generation takes place. For particular DNVM, the higher harmonic can have frequency nearly two times larger than the maximal frequency of the phonon spectrum of graphene. Excitation of some of DNVM results in the appearance of negative in‐plane pressure in graphene. This counterintuitive result is explained by the rotational motion of carbon hexagons. Our results contribute to the understanding of nonlinear dynamics of the graphene lattice.

Non-linear modes of vibration of single-layer non-local graphene sheets

Single-layer graphene sheets (SLGSs) with dimensions of the order of a few nanometres are relatively new, but expected to have several applications. When SLGSs experience displacements that are large in comparison with their extremely small thickness, the membrane forces that develop lead to non-linear behaviour. Knowing the modes of vibration of SLSGs is important, because these modes provide a picture of what one may expect not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of flat single-layer graphene sheets are investigated. For that purpose, a Galerkin type formulation, based on classic plate theory with Von Kármán non-linear terms and resorting to Airy’ s stress function, is implemented. The formulation takes into account non-local effects, which are thought to be important in very small structural elements. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM), with several harmonics, and are subsequently solved by an arc-length continuation method. The combined importance of non-local effects and of the geometrical non-linearity on the non-linear modes of vibration is analysed. They result in alterations of the natural frequencies, variations in the degrees of hardening, changes in the frequency content of the free vibrations, and alterations in shapes assumed along a period of vibration. The main outcome of this work is the finding that the small scale has a major effect on interactions between the first and higher order modes, interactions which are induced by the geometrical non-linearities. It turns out to be possible, e.g., for non-local effects to considerably change the frequencies at which internal resonances occur, or even to eliminate those internal resonances.

Stability of in-plane delocalized vibrational modes in triangular Morse lattice

Delocalized nonlinear vibrational modes (DNVMs) play a very important role in the study of the dynamics of a nonlinear lattice in solid state physics. Such modes are exact solutions to the equations of motion of atoms dictated by the lattice space symmetry. If the amplitude of DNVM is above the threshold value, it is modulationally unstable. In the present study, we consider the stability of DNVMs in a two-dimensional (2D) triangular lattice with atoms interacting via the Morse potential. Critical exponents are calculated numerically as functions of the DNVM amplitude. Extrapolation to the zero value of the critical exponent gives an estimation of the DNVM amplitude, below which it is stable.

Two-dimensional discrete breathers in hcp titanium

Recently delocalized nonlinear vibrational modes (DNVMs) have been derived for a two-dimensional (2D) triangular lattice and their frequencies as the functions of amplitude have been calculated for a 2D Morse crystal. A 3D hcp lattice is formed by certain stacking of 2D triangular lattices. In the present work, we attempt to excite eight DNVMs with in-plane atomic displacements in a single plane of hcp titanium, modelled using many-body interatomic potentials. It is found that three of eight DNVMs produce long-lived 2D discrete breathers (DBs) in a titanium crystal. A 2D DB is delocalized in two spatial dimensions and localized in the third one, so that atoms of only one atomic plane have large vibrational amplitudes. The fFrequencies of the newly found DBs lie above the small-amplitude phonon spectrum of titanium and grow with increasing amplitude. Thus, we report on three novel 2D DBs in hcp metal. Our results open the route to finding new types of DBs in pure metals.

The Influence of Nonlinear Mode on Generalized Aerodynamic Spectrum in the Downwind

Combined with the coherence function of fluctuating wind load, the average wind speed profile index and the mode index, the method of numerical integration was used to calculate the generalized aerodynamic spectrum of nonlinear modes for high-rise buildings in the downwind. The fitting formula for that value was derived by multivariate linear regression analysis, whose reliability was confirmed by the wind tunnel test. At the same time, the effect of nonlinear vibration modes on the generalized aerodynamic spectrum of high-rise buildings was discussed with the vibration index as the parameters:When the vibration mode index is less than 1, the correction coefficient is more than 1, that is, the calculated generalized aerodynamic force spectrum by the traditional method is too small, which is not safe for the structural design, and on the contrary, it is less than 1. Base on the generalized aerodynamic spectrum of the linear mode, it is recommended to use a correction factor of 1.1 in the basic mode of the load code for the design of building structures.

Symmetries of nonlinear vibrations in tetrahedral molecular configurations

We study nonlinear vibrational modes of oscillations for tetrahedral configurations of particles. In the case of tetraphosphorus, the interaction of atoms is given by bond stretching and van der Waals forces. Using the equivariant gradient degree, we present a topological classification of the spatio-temporal symmetries of the periodic solutions with finite Weyl's group. This procedure describes all the symmetries of the nonlinear vibrations for general force fields.

Interaction of longitudinal phonons with discrete breather in strained graphene

We numerically analyze the interaction of small-amplitude phonon waves with standing gap discrete breather (DB) in strained graphene. To make the system support gap DB, strain is applied to create a gap in the phonon spectrum. We only focus on the in-plane phonons and DB, so the issue is investigated under a quasi-one-dimensional setup. It is found that, for the longitudinal sound waves having frequencies below 6 THz, DB is transparent and thus no radiation of energy from DB takes place; whereas for those sound waves with higher frequencies within the acoustic (optical) phonon band, phonon is mainly transmitted (reflected) by DB, and concomitantly, DB radiates its energy when interacting with phonons. The latter case is supported by the fact that, the sum of the transmitted and reflected phonon energy densities is noticeably higher than that of the incident wave. Our results here may provide insight into energy transport in graphene when the spatially localized nonlinear vibration modes are presented.

Periodic solutions of a two-degree-of-freedom autonomous vibro-impact oscillator with sticking phases

This contribution explores the free dynamics of a simple two-degree-of-freedom vibro-impact oscillator. One degree-of-freedom is unilaterally constrained by the presence of a rigid obstacle and periodic solutions involving one sticking phase per period (1-SPP) are targeted. A solution method to obtain such orbits is proposed: it provides conditions on the existence of 1-SPP as well as closed-form solutions. It is shown that 1-SPP might not exist for a given combination of masses and stiffnesses. The set of 1-SPP is at most a countable set of isolated periodic orbits. The construction of 1-SPP requires numerical developments that are illustrated on a few relevant examples. Comparison with nonlinear modes of vibration involving one impact per period (1-IPP) is also considered. Interestingly, an equivalence between 1-SPP and a special set of isolated 1-IPP is established. It is also demonstrated that the prestressed system features sticking phases of infinite duration.

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.