Nonlinear Modal Interactions Research Articles

Systematic higher order nonlinear analysis of power systems operating under perturbation conditions

In this paper, the modal series method is revisited and an extension on the method to include higher order terms is proposed. This proposal is based on the introduction of the multidimensional Laplace transform and association of variables theorems to deduce the analytical closed-form solution when it is applied to the analysis of a nonlinear power system model. The method is systematic and can incorporate higher order terms to the modal analysis to determine nonlinear modal interaction. When the power system is operating under stressed conditions, such as an increase in load demand, it results very important to consider the oscillations due to its nonlinear nature. Thus, the method is carefully exemplified with the application to the synchronous machine-infinite busbar power system operating under stress conditions. The oscillations produced during changes in its operation are analyzed as well as the nonlinear interaction through nonlinear indices and nonlinear participation factors. The time domain responses are compared between linear approximation, modal series, normal forms method and the direct numerical full solution of the nonlinear power system model.

A Broadband Internally Resonant Vibratory Energy Harvester

The objective of this paper is twofold: first to illustrate that nonlinear modal interactions, namely, a two-to-one internal resonance energy pump, can be exploited to improve the steady-state bandwidth of vibratory energy harvesters; and, second, to investigate the influence of key system’s parameters on the steady-state bandwidth in the presence of the internal resonance. To achieve this objective, an L-shaped piezoelectric cantilevered harvester augmented with frequency tuning magnets is considered. The distance between the magnets is adjusted such that the second modal frequency of the structure is nearly twice its first modal frequency. This facilitates a nonlinear energy exchange between these two commensurate modes resulting in large-amplitude responses over a wider range of frequencies. The harvester is then subjected to a harmonic excitation with a frequency close to the first modal frequency, and the voltage–frequency response curves are generated. Results clearly illustrate an improved bandwidth and output voltage over a case which does not involve an internal resonance. A nonlinear model of the harvester is developed and validated against experimental findings. An approximate analytical solution of the model is obtained using perturbation methods and utilized to draw several conclusions regarding the influence of key design parameters on the harvester’s bandwidth.

Modeling and design of direct nonlinear velocity feedback for modal self-excitation in a class of multi degrees-of-freedom mechanical systems

The paper presents a new nonlinear control design methodology for inducing modal, self-excited oscillation in a class of multi degrees-of-freedom mechanical systems. The system is assumed to be fully actuated and the controller operates in a centralized fashion based on the direct nonlinear velocity feedback. The linear part of the controller is designed to assign negative modal damping in the mode to be excited and positive modal damping in other modes. The nonlinear modal interaction is investigated using averaging analysis and different excitation regimes are delineated in the control parameter space. The nonlinear part of the controller is optimized to minimize the control cost. Numerical simulations of the control system are performed to substantiate the analytical results and validate the accuracy of the design. The control method is also shown to have some degree of robustness against parametric variations over the nominal values.

Vector plasmonic lattice solitons in nonlinear graphene-pair arrays.

We investigate the vector plasmonic lattice solitons (PLSs) in nonlinear graphene-pair arrays (GPAs) consisting of periodically arranged double graphene sheets, which are spatially separated. There are two dispersion bands for the Bloch modes in the array due to the coupling of surface plasmon polaritons (SPPs) between the graphene pairs. The vector PLSs composed of two components originate from the nonlinear interaction of Bloch modes in different bands. Both components undergo mutual self-trapping through the balance between diffraction and self-focusing nonlinearity of graphene. Thanks to the strong confinement of SPPs, the vector PLSs can be squeezed into a lateral width of ∼λ/100. The study provides a promising approach to all-optical control on a deep-subwavelength scale.

Nonlinear modal interactions in composite thin-walled beam structures with simultaneous 1:2 internal and 1:1 external resonances

Nonlinear dynamic characteristics of a composite aircraft wing structure modeled by a geometrically nonlinear anisotropic thin-walled beam in the presence of simultaneous 1:2 internal and 1:1 external resonances are investigated. Some prominent non-classical effects such as of transverse shear strain, warping inhibition, and three-dimensional strain are considered in the beam model. Moreover, circumferentially asymmetric stiffness lay-up configuration is adapted to generate the transverse bending-twisting elastic coupling. The solution methodology is based on the Extended Galerkin’s Method, and the method of multiple scales is applied to the system in order to obtain the equations of amplitude and modulation. Steady-state solutions and their stability are investigated. The peculiarity of the internal resonances and the conditions for saturation and jump phenomenon during the modal interactions are discussed and the commercial code ABAQUS is used to validate the theoretical results we have obtained. Finally, the prominent features of modal interactions in composite thin-walled beam structures are summarized and pertinent suggestions concerning safe design of the wing structures are given.

On the effect of mechanical non-linearities on vortex-induced lock-in vibrations

Vortex-induced vibrations at lock-in conditions are modeled through generalized van der Pol-Duffing oscillators endowed with frequency-dependent coefficients, taking inspiration from fluid-elastic models. Accordingly, it is found that the limit-cycle amplitude and the non-linear frequency are mutually dependent (feedback effect), differently from the classic oscillator behavior. Consequently, the mechanical non-linearities, which are often believed to be unimportant, do affect the amplitude of motion. Examples concerning an ideal one degree-of-freedom van der Pol-Duffing oscillator and a two degree-of-freedom model, coarsely representative of a tower building, confirm the importance of this approach also from a technical point of view. Thus, non-linear geometric terms and modal interaction (even in non-resonant cases) can lead to non-negligible modifications of purely aeroelastic problems.

Nonlinear modal interactions in parity-time (PT) symmetric lasers.

Parity-time symmetric lasers have attracted considerable attention lately due to their promising applications and intriguing properties, such as free spectral range doubling and single-mode lasing. In this work we discuss nonlinear modal interactions in these laser systems under steady state conditions, and we demonstrate that several gain clamping scenarios can occur for lasing operation in the -symmetric and -broken phases. In particular, we show that, depending on the system’s design and the external pump profile, its operation in the nonlinear regime falls into two different categories: in one the system is frozen in the phase space as the applied gain increases, while in the other the system is pulled towards its exceptional point. These features are first illustrated by a coupled mode formalism and later verified by employing the Steady-state Ab-initio Laser Theory (SALT). Our findings shine light on the robustness of single-mode operation against saturation nonlinearity in -symmetric lasers.

Experimental validation of a nonlinear derating technique based upon Gaussian-modal representation of focused ultrasound beams.

A technique useful for performing derating at acoustic powers where significant harmonic generation occurs is illustrated and validated with experimental measurements. The technique was previously presented using data from simulations. The method is based upon a Gaussian representation of the propagation modes, resulting in simple expressions for the modal quantities, but a Gaussian source is not required. The nonlinear interaction of modes within tissue is estimated from the nonlinear interaction in water, using appropriate amounts of source reduction and focal-point reduction derived from numerical simulations. An important feature of this nonlinear derating method is that focal temperatures can be estimated with little additional effort beyond that required to determine the focal pressure waveforms. Hydrophone measurements made in water were used to inform the derating algorithm, and the resulting pressure waveforms and increases in temperature were compared with values directly measured in tissue phantoms. For a 1.05 MHz focused transducer operated at 80 W and 128 W, the derated pressures (peak positive, peak negative) agreed with the directly measured values to within 11%. Focal temperature rises determined by the derating method agreed with values measured using a remote thermocouple technique with a difference of 17%.

Coherence loss of partially mode-locked fibre laser.

Stochastically driven nonlinear processes limit the number of amplified modes in a natural system due to competitive mode interaction, which is accompanied by loss of coherence when increasing the complexity of the system. Specifically, we find that modulation instability, which exhibits great fluctuations when it spontaneously grows from noise in conservative systems, may possess a high degree of coherence in dissipative laser system with gain. Nonlinear mode interactions can be competitive or cooperative: adjusting the intracavity polarization state controls the process of loss of coherence. Single-shot spectra reveal that, first, the fibre laser redistributes its energy from the center wavelength mode into sidebands through parametric instabilities. Subsequently, longitudinal modes are populated via cascaded four-wave-mixing. Parametric frequency conversion populates longitudinal modes with a random distribution of position, intensity and polarization, resulting in partially (rather than highly) coherent pulses. These dynamics unveil a new route towards complex pattern formation in nonlinear laser systems, and they may be also beneficial for the understanding of supercontinuum, Kerr-combs phenomena, and optical rogue waves.

On the efficacy of friction damping in the presence of nonlinear modal interactions

This work addresses friction-induced modal interactions in jointed structures, and their effects on the passive mitigation of vibrations by means of friction damping. Under the condition of (nearly) commensurable natural frequencies, the nonlinear character of friction can cause so-called nonlinear modal interactions. If harmonic forcing near the natural frequency of a specific mode is applied, for instance, another mode may be excited due to nonlinear energy transfer and thus contribute considerably to the vibration response. We investigate how this phenomenon affects the performance of friction damping. To this end, we study the steady-state, periodic forced vibrations of a system of two beams connected via a local mechanical friction joint. The system can be tuned to continuously adjust the ratio between the first two natural frequencies in the range around the 1:3 internal resonance, in order to trigger or suppress the emergence of modal interactions. Due to the re-distribution of the vibration energy, the vibration level can in fact be reduced in certain situations. However, in other situations, the multi-harmonic character of the vibration has detrimental effects on the effective damping provided by the friction joint. The resulting response level can be significantly larger than in the absence of modal interactions. Moreover, it is shown that the vibration behavior is highly sensitive in the neighborhood of internal resonances. It is thus concluded that the condition of internal resonance should be avoided in the design of friction-damped systems.

Cascaded third-harmonic generation in hybrid graphene-semiconductor waveguides

We study cascaded harmonic generation of hybrid surface plasmons in integrated planar waveguides composed of a graphene layer and a doped-semiconductor slab. We derive a comprehensive model of cascaded third harmonic generation through phase-matched nonlinear interaction of fundamental, second harmonic and third harmonic plasmonic modes supported by the structure. We show that hybrid graphene-semiconductor waveguides can simultaneously phase-match these three interacting harmonics, increasing the total third-harmonic output by a factor of 5 compared to the non-cascaded regime.

Inertial convection in a rotating narrow annulus: Asymptotic theory and numerical simulation

An important way of breaking the rotational constraint in rotating convection is to invoke fast oscillation through strong inertial effects which, referring to as inertial convection, is physically realizable when the Prandtl number Pr of rotating fluids is sufficiently small. We investigate, via both analytical and numerical methods, inertial convection in a Boussinesq fluid contained in a narrow annulus rotating rapidly about a vertical symmetry axis and uniformly heated from below, which can be approximately realizable in laboratory experiments [R. P. Davies-Jones and P. A. Gilman, “Convection in a rotating annulus uniformly heated from below,” J. Fluid Mech. 46, 65-81 (1971)]. On the basis of an assumption that inertial convection at leading order is represented by a thermal inertial wave propagating in either prograde or retrograde direction and that buoyancy forces appear at the next order to maintain the wave against the effect of viscous damping, we derive an analytical solution that describes the onset of inertial convection with the non-slip velocity boundary condition. It is found that there always exist two oppositely traveling thermal inertial waves, sustained by convection, that have the same azimuthal wavenumber, the same size of the frequency, and the same critical Rayleigh number but different spatial structure. Linear numerical analysis using a Galerkin spectral method is also carried out, showing a quantitative agreement between the analytical and numerical solutions when the Ekman number is sufficiently small. Nonlinear properties of inertial convection are investigated through direct three-dimensional numerical simulation using a finite-difference method with the Chorin-type projection scheme, concentrating on the liquid metal gallium with the Prandtl number Pr = 0.023. It is found that the interaction of the two counter-traveling thermal inertial waves leads to a time-dependent, spatially complicated, oscillatory convection even in the vicinity of the onset of inertial convection. The nonlinear properties are analyzed via making use of the mathematical completeness of inertial wave modes in a rotating narrow annulus, suggesting that the laminar to weakly turbulent transition is mainly caused by the nonlinear interaction of several inertial wave modes that are excited and maintained by thermal convection at moderately supercritical Rayleigh numbers.

$$N-1$$ N - 1 modal interactions of a three-degree-of-freedom system with cubic elastic nonlinearities

In this paper the $$N-1$$ nonlinear modal interactions that occur in a nonlinear three-degree-of-freedom lumped mass system, where $$N=3$$ , are considered. The nonlinearity comes from springs with weakly nonlinear cubic terms. Here, the case where all the natural frequencies of the underlying linear system are close (i.e. $$\omega _{n1}: \omega _{n2}:\omega _{n3} \approx 1:1:1$$ ) is considered. However, due to the symmetries of the system under consideration, only $$N-1$$ modes interact. Depending on the sign and magnitude of the nonlinear stiffness parameters, the subsequent responses can be classified using backbone curves that represent the resonances of the underlying undamped, unforced system. These backbone curves, which we estimate analytically, are then related to the forced response of the system around resonance in the frequency domain. The forced responses are computed using the continuation software AUTO-07p. A comparison of the results gives insights into the multi-modal interactions and shows how the frequency response of the system is related to those branches of the backbone curves that represent such interactions.

Nonlinear vibrations and mode interactions for a continuous rod with microstructure

Natural longitudinal vibrations of a periodically heterogeneous rod embedded into a nonlinear elastic matrix are considered. The governing macroscopic dynamical equation was obtained by the higher-order asymptotic homogenization method. An asymptotic solution is developed by the method of multiple time scales. The effects of internal resonances and modes coupling are predicted. The specific objective of the paper is to analyse how the presence of the microstructure influences on the processes of mode interactions. It is shown that depending on a scaling relation between the amplitude of the vibrations and the size of the unit cell different scenarios of the modes coupling can be realized.

Non-linear mode interaction between spin torque driven and damped modes in spin torque nano-oscillators

The influence of dynamic coupling in between magnetic layers of a standard spin torque nano-oscillator composed of a synthetic antiferromagnet (SyF) as a polarizer and an in-plane magnetized free layer has been investigated. Experiments on spin valve nanopillars reveal non-continuous features such as kinks in the frequency field dependence that cannot be explained without such interactions. Comparison of experiments to numerical macrospin simulations shows that this is due to non-linear interaction between the spin torque (STT) driven mode and a damped mode that is mediated via the third harmonics of the STT mode. It only occurs at large applied currents and thus at large excitation amplitudes of the STT mode. Under these conditions, a hybridized mode characterized by a strong reduction of the linewidth appears. The reduced linewidth can be explained by a reduction of the non-linear contribution to the linewidth via an enhanced effective damping. Interestingly, the effect depends also on the exchange interaction within the SyF. An enhancement of the current range of reduced linewidth by a factor of two and a reduction of the minimum linewidth by a factor of two are predicted from simulation when the exchange interaction strength is reduced by 30%. These results open directions to optimize the design and microwave performances of spin torque nano-oscillators taking advantage of the coupling mechanisms.

Quantum state engineering with circuit electromechanical three-body interactions.

We propose a hybrid system with quantum mechanical three-body interactions between photons, phonons, and qubit excitations. These interactions take place in a circuit quantum electrodynamical architecture with a superconducting microwave resonator coupled to a transmon qubit whose shunt capacitance is free to mechanically oscillate. We show that this system design features a three-mode polariton-mechanical mode and a nonlinear transmon-mechanical mode interaction in the strong coupling regime. Together with the strong resonator-transmon interaction, these properties provide intriguing opportunities for manipulations of this hybrid quantum system. We show, in particular, the feasibility of cooling the mechanical motion down to its ground state and preparing various nonclassical states including mechanical Fock and cat states and hybrid tripartite entangled states.

The nonlinear interaction of convection modes in a box of a saturated porous medium

A plethora of convection modes may occur within a confined box of porous medium when the associated dimensionless Rayleigh number R is above some critical value dependent on the geometry. In many cases the crucial Rayleigh number Rc for onset is different for each mode, and in practice the mode with the lowest associated Rc is likely to be the dominant one. For particular sizes of box, however, it is possible for multiple modes (typically three) to share a common Rc. For box shapes close to these special geometries the modes interact and compete nonlinearly near the onset of convection. Here this mechanism is explored and it is shown that generically the dynamics of the competition takes on one of two possible structures. A specific example of each is described, while the general properties of the system enables us to compare our results with some previous calculations for particular box dimensions.

Nonlinear Mode Interactions in the Wake of a Medium Height Roughness Element

The present study is devoted to nonlinear (resonant) mode interactions in the wake of a medium height, cylindrical roughness element, which is placed in a laminar, airfoil boundary layer. The roughness element causes a considerable mean flow distortion in the near wake centerline region and two counter-rotating vortex pairs arise in the outer spanwise domain. In the streamwise evolution the mean flow stabilizes and a spanwise uniform flow is recovered in the far wake. Upstream of the roughness controlled, low-amplitude Tollmien-Schlichting (TS) wave modes are excited in the upper branch unstable region. The interference of the excited, 2D fundamental modes with the roughness results in the excitation of oblique modes in a broad spanwise wave number range. Nonlinear interactions in-between the (oblique) fundamental modes in the destabilized near wake lead to the generation of primary, low-frequency modes at the difference frequencies of the fundamental modes, before the fundamental modes can recover linear stability characteristics in the far wake. In contrast, the primary, subharmonic-type modes experience a nonlinear growth, which is a result of frequency and spanwise wave number detuned resonant mode interactions with the 2D fundamental modes. The resonant growth of the primary interaction modes initiates a symmetrization of the spectrum in the low-frequency, subharmonic range. That is, symmetric subharmonic-type (secondary) modes are resonantly amplified in the far wake. Therefore, although the fundamental modes recover linear stability characteristics with the stabilization of the mean flow, resonant mode interactions in the low-frequency, subharmonic range initiate the onset of boundary layer transition in the far wake of the medium height roughness element.

Nonlinear stable steady solutions to the Ostroumov problem

We investigate the linear and weakly nonlinear solutions to a convection problem that was first studied by Ostroumov in 1947. The problem pertains to the stability of the equations governing convective motion in an infinite vertical fluid layer that is heated from below. Ostroumov’s linear stability analysis yields instability threshold conditions that are characterized by zero wavenumber for the Fourier mode in the vertical direction and by eigenfunctions that are independent of the vertical coordinate. Thus, any undertaking at determining the supercritical nonlinear solutions and their stability through a small amplitude expansion fails. This failure is attributed to the fact that the nonlinear interaction of the linear modes vanish identically. In this paper, we put forth exact and stable similarity type solutions to the Ostroumov problem. These solutions are characterized by the same linear threshold conditions as Ostroumov’s solutions. Moreover, we are able to extend the analysis to the supercritical regime through a small amplitude analysis to obtain steady two-dimensional solutions for a small range of Prandtl numbers. These solutions are found to be stable to general two-dimensional, time-dependent disturbances. Furthermore, when the analysis is extended to the case where the fluid layer thickness is also allowed to be infinite, we find that the infinite two-dimensional fluid region becomes linearly unstable through a Batchelor–Nitsche (BN) instability mechanism. Thus, the nonlinear solutions are obtained through a long wavelength expansion, and consequently our analysis provides the nonlinear development of the BN instability for the case of an unstable and uniform density profile. Finally, numerical solutions of the nonlinear problem are also presented which shed light on the flow patterns and temperature distributions for both the vertical channel and the infinitely extended stratified region problems.

Modelling, Simulation and Experimental Validation of Nonlinear Dynamic Interactions in an Aramid Rope System

Vibration phenomena taking place in lifting and hoist installations may influence the dynamic performance of their components. For example, in an elevator system they may affect ride quality of a lift car. Lateral and longitudinal vibrations of suspension ropes and compensating cables may result in an adverse dynamic behaviour of the entire installation. Thus, there is a need to develop reliable mathematical and computer simulation models to predict the dynamic behaviour of suspension rope and compensating cable systems. The aim of this paper is to develop a model of an aramid suspension rope system in order to predict nonlinear modal interactions taking place in the installation. A laboratory model comprising an aramid suspension rope, a sheave/ pulley assembly and a rigid suspended mass has been studied. Experimental tests have been conducted to identify modal nonlinear couplings in the system. The dynamic behaviour of the model has been described by a set of nonlinear partial differential equations. The equations have been solved numerically. The numerical results have been validated by experimental tests. It has been shown that the nonlinear couplings may lead to adverse modal interactions in the system.