Strong Modal Interactions Research Articles

Mode Coupling in Dynamic Atomic Force Microscopy

Increasing the signal-to-noise ratio in dynamic atomic force microscopy plays a key role in nanomechanical mapping of materials with atomic resolution. In this work, we develop an experimental procedure for increasing the sensitivity of higher harmonics of an atomic-force-microscope cantilever without modifying the cantilever geometry but instead by utilizing dynamical mode coupling between its flexural modes of vibration. We perform experiments on different cantilevers and samples and observe that via nonlinear resonance frequency tuning we can obtain a frequency range where strong modal interactions lead to 7-fold and 16-fold increases in the sensitivity of the 6th and 17th harmonics while reducing sample indentation. We derive a numerical model that captures the observed physics and confirms that nonlinear mode coupling is the reason for the increase of the amplitude of higher harmonics during tip-sample interactions.

Nonlinear vibrations of a polar-orthotropic thin circular plate subjected to circularly moving point load

Here, we formulate and study nonlinear vibrations of a thin, polar-orthotropic circular plate subjected to a circularly moving point load at a fixed radius and constant angular velocity. The governing equations and boundary conditions are obtained following Kirchhoff’s plate theory, incorporating von Kármán nonlinearity, and employing extended Hamilton’s principle. The external damping is introduced via Rayleigh dissipation function. The governing equations are solved using mode summation procedure. The mode shapes and the natural frequencies of the polar-orthotropic circular plate are found using Frobenius series method. Mode summation procedure results in coupled nonlinear ordinary differential equations. These equations are solved using the Runge–Kutta method for time response and method of harmonic balance with the arc continuation method for frequency response. The spectrum of the undamped linear vibration response of isotropic and polar-orthotropic plates exhibits natural frequencies of plates and angular velocity of the rotating load. The damped response contains the frequency of the angular velocity of the rotating load only. The nonlinear transverse vibrations of the undamped and damped plate due to rotating point load reveal frequency rich spectrums. The frequency response function shows strong modal interactions.

Tunable-with-energy intense modal interactions induced by geometric nonlinearity

Modal interactions are distinct features of nonlinear systems that can be exploited in applications such as vibration and shock mitigation, targeted (irreversible) energy transfers (TET), and acoustic/stress wave tailoring. For such applications, different types of nonlinearities, e.g. hardening, softening, smooth, non-smooth, material or geometric, have been considered. In this work, we examine the geometric nonlinearity resulting from an initially inclined element consisting of a linear spring and a viscous damper connected in parallel, having an initial angle of inclination, [Formula: see text]. Because of its inclined configuration, this element possesses strong (and doubly tunable with respect to [Formula: see text] and energy) geometrically nonlinear stiffness and damping effects, despite the linear constitutive laws governing its constituent components. First, we consider a single-degree-of-freedom linearly grounded oscillator attached to the nonlinear inclined element. Omitting dissipative effects, we investigate the frequency–energy relation of this system by employing the canonical action-angle transformation and show that, depending on the initial angle of inclination and the energy-level, the resulting nonlinearity can be tuned to be softening, hardening or a combination of both. Next, we explore the efficacy of the geometric nonlinearity to induce strong modal interactions by considering a three-degree-of-freedom lightly damped primary system that is weakly coupled to a single-degree-of-freedom lightly damped attachment with the inclined nonlinear element, subjected to impulsive excitation. Varying [Formula: see text] and the input energy, we demonstrate strong modal energy-exchanges between the modes of the primary system and the nonlinear attachment over broad energy-dependent spans of [Formula: see text]. We show that the passive self-adaptiveness of the nonlinear damping and the hardening–softening geometric nonlinearity can induce narrowband or broadband frequency TET, including high-to-low frequency energy transfers. Interestingly, over a definitive range of [Formula: see text], these modal interactions may be limited only between the nonlinear mode of the attachment and the highest-frequency linear mode of the primary system, inducing strong high-frequency targeted energy transfer to the primary system.

Interaction-induced mode switching in steady-state microlasers.

We demonstrate that due to strong modal interactions through cross-gain saturation, the onset of a new lasing mode can switch off an existing mode via a negative power slope. In this process of interaction-induced mode switching (IMS) the two involved modes maintain their identities, i.e. they do not change their spatial field patterns or lasing frequencies. For a fixed pump profile, a simple analytic criterion for the occurrence of IMS is given in terms of their self- and cross-interaction coefficients and non-interacting thresholds, which is verified for the example of a two-dimensional microdisk laser. When the spatial pump profile is varied as the pump power is increased, IMS can be induced even when it would not occur with a fixed pump profile, as we show for two coupled laser cavities. Our findings apply to steady-state lasing and are hence different from dynamical mode switching or hopping. IMS may have potential applications in robust and flexible all-optical switching.

Internal resonance of axially moving laminated circular cylindrical shells

The nonlinear vibrations of a thin, elastic, laminated composite circular cylindrical shell, moving in axial direction and having an internal resonance, are investigated in this study. Nonlinearities due to large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structure damping. Differently from conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing Airy stress function is developed to study the nonlinear dynamics of thin shells. The system is discretized by Galerkin’s method while a model involving four degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the multi-degrees-of-freedom system. When the structure is excited close to a resonant frequency, very intricate frequency–response curves are obtained, which show strong modal interactions and one-to-one-to-one-to-one internal resonance phenomenon. The effects of different parameters on the complex dynamic response are investigated in this study. The stability of steady-state solutions is also analyzed in detail.

Suppression of interactions in multimode random lasers in the Anderson localized regime

Understanding random lasing is a formidable theoretical challenge. Unlike conventional lasers, random lasers have no resonator to trap light, they are highly multimode with potentially strong modal interactions, and they are based on disordered gain media, where photons undergo random multiple scattering. Interference effects notoriously modify the propagation of waves in such random media, but their fate in the presence of nonlinearity and interactions is poorly understood. Here, we present a semiclassical theory for multimode random lasing in the strongly scattering regime. We show that Anderson localization, a wave interference effect, is not affected by the presence of nonlinearities. To the contrary, its presence suppresses interactions between simultaneously lasing modes. Consequently, each lasing mode in a strongly scattering random laser is given by a single long-lived, Anderson localized mode of the passive cavity, the frequency and wave profile of which do not vary with pumping, even in the multimode regime when modes spatially overlap. Random lasing in the presence of nonlinearities and disordered gain media is still poorly understood. Researchers now present a semiclassical theory for multimode random lasing in the strongly scattering regime. They show that Anderson localization — a wave-interference effect — is not affected by the presence of nonlinearities, but instead suppresses interactions between simultaneously lasing modes.

Steady-stateab initiolaser theory: Generalizations and analytic results

We improve the steady-state ab initio laser theory (SALT) of T\"ureci et al. by expressing its fundamental self-consistent equation in a basis set of threshold constant flux states that contains the exact threshold lasing mode. For cavities with nonuniform index and/or nonuniform gain, the new basis set allows the steady-state lasing properties to be computed with much greater efficiency. This formulation of the SALT can be solved in the single-pole approximation, which gives the intensities and thresholds, including the effects of nonlinear hole-burning interactions to all orders, with negligible computational effort. The approximation yields a number of analytic predictions, including a ``gain-clamping'' transition at which strong modal interactions suppress all higher modes. We show that the single-pole approximation agrees well with exact SALT calculations, particularly for high-$Q$ cavities. Within this range of validity, it provides an extraordinarily efficient technique for modeling realistic and complex lasers.

Modified shooting approach to the non-linear periodic forced response of isotropic/composite curved beams

Large amplitude periodic forced vibration of curved beams under periodic excitation is investigated using a three-noded beam element. The element is based on the higher-order shear deformation theory satisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditions on the latter. The periodic responses are obtained using shooting technique coupled with Newmark time marching and arc length continuation algorithm developed. The second order governing differential equations of motion are solved without transforming to the first order differential equations thereby resulting in a computationally more efficient algorithm. The effects of excitation amplitude, support conditions and beam curvature on the frequency versus response amplitude relation are highlighted. The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented. Phenomenon of strong modal interactions is observed.

Passive targeted energy transfers and strong modal interactions in the dynamics of a thin plate with strongly nonlinear attachments

We study targeted energy transfers (TETs) and nonlinear modal interactions attachments occurring in the dynamics of a thin cantilever plate on an elastic foundation with strongly nonlinear lightweight attachments of different configurations in a more complicated system towards industrial applications. We examine two types of shock excitations that excite a subset of plate modes, and systematically study, nonlinear modal interactions and passive broadband targeted energy transfer phenomena occurring between the plate and the attachments. The following attachment configurations are considered: (i) a single ungrounded, strongly (essentially) nonlinear single-degree-of-freedom (SDOF) attachment—termed nonlinear energy sink (NES); (ii) a set of two SDOF NESs attached at different points of the plate; and (iii) a single multi-degree-of-freedom (MDOF) NES with multiple essential stiffness nonlinearities. We perform parametric studies by varying the parameters and locations of the NESs, in order to optimize passive TETs from the plate modes to the attachments, and we showed that the optimal position for the NES attachments are at the antinodes of the linear modes of the plate. The parametric study of the damping coefficient of the SDOF NES showed that TETs decreasing with lower values of the coefficient and moreover we showed that the threshold of maximum energy level of the system with strong TETs occured in discrete models is by far beyond the limits of the engineering design of the continua. We examine in detail the underlying dynamical mechanisms influencing TETs by means of empirical mode decomposition (EMD) in combination with wavelet transforms. This integrated approach enables us to systematically study the strong modal interactions occurring between the essentially nonlinear NESs and different plate modes, and to detect the dominant resonance captures between the plate modes and the NESs that cause the observed TETs. Moreover, we perform comparative studies of the performance of different types of NESs and of the linear tuned mass dampers (TMDs) attached to the plate instead of the NESs. Finally, the efficacy of using this type of essentially nonlinear attachments as passive absorbers of broadband vibration energy is discussed.

Forced large amplitude periodic vibrations of cylindrical shallow shells

Forced, large amplitude, periodic vibrations of open, linear elastic and isotropic, cylindrical shells are studied. A p-version finite element, which is derived following first-order shear deformation theory, is employed to define the model and an algorithm based on the shooting and Newton methods is used to solve the equations of motion. The influences that the amplitude and spatial distribution of the external excitation, the curvature to width ratio and the shell's thickness have on the dynamic response are investigated. To characterize the geometrically non-linear oscillations time plots, phase planes and Fourier spectra are employed. To characterize the stability of the solutions the Floquet multipliers are computed. Responses with even and odd harmonics to purely harmonic excitations and strong modal interactions due to the non-linearity of the system are found. The methods proposed constitute a good alternative to analyse geometrically non-linear periodic vibrations of shells because the number of approximations is reduced and the computational cost is, although significant, reasonable.

Influence of Stabilizing Cables on Seismic Response of a Multispan Cable‐Stayed Bridge

Abstract: A recent trend in the design of long‐span bridges is multispan cable‐stayed bridges with three or more towers. A critical problem of multispan cable‐stayed bridges is the stabilization of the central tower(s), which has resulted in increasing application of stabilizing cables. The Ting Kau Bridge in Hong Kong is one of a few multispan cable‐stayed bridges adopting stabilizing cables ever built. In this article, the dynamic properties of multispan cable‐stayed bridges with stabilizing cables and the effect of stabilizing cables on bridge seismic response are studied by referring to the Ting Kau Bridge. Based on a validated 3D finite‐element model, modal analysis is conducted which shows that the longitudinal stabilizing cables bring about a number of global modes with strong modal interaction among the deck, towers, and cables. The seismic response and internal force in the central tower are found to be much larger than those in the side towers. The longitudinal stabilizing cables are very effective in reducing the internal force in the central tower generated by longitudinal earthquake excitation, but insignificantly affect the seismic response in the bridge deck and side towers. As a whole, the stabilizing cables act favorably in the reduction of seismic response of the bridge.

Nonlinear Vibrations and Multiple Resonances of Fluid-Filled, Circular Shells, Part 2: Perturbation Analysis

The nonlinear ordinary differential equations describing the dynamics of a fluid filled circular cylindrical shell, obtained in Part 1 of the present study, is studied by using a second order perturbation approach and direct simulations. Strong modal interactions are found when the structure is excited with small resonant loads. Modal interactions arise in the whole range of vibration amplitude, showing that the internal resonance condition makes the system non-linearizable even for extremely small amplitudes of oscillation. Stationary and nonstationary oscillations are observed and the complex nature of modal interactions is accurately analyzed. No chaotic motion is observed in the case of 1:1:1:2 internal resonance studied. [S0739-3717(00)01304-0]

Experimental Investigation of Isolated and Simultaneous Internal Resonances in Suspended Cables

The near resonant response of suspended elastic cables driven by harmonic, planar excitation is investigated experimentally. Measurements of large amplitude cable motions confirm previous theoretical predictions of fundamental classes of internally-resonant responses. For particular magnitudes of equilibrium curvature, strong modal interactions arise through isolated (two-mode) or simultaneous (three-mode) internal resonances. Four qualitatively different periodic responses are observed: (1) pure planar response, (2) 2:1 internally resonant nonplanar response, (3) 1:1 internally resonant nonplanar response, and (4) simultaneous, 2:2:1 internally resonant nonplanar response. Quasiperiodic responses are also observed.

Mathematical Structure of Modal Interactions in a Spinning Disk-Stationary Load System

In a previous paper (Chen and Bogy, 1992) we studied the effects of various load parameters, such as friction force, transverse mass, damping, stiffness and the analogous pitching parameters, of a stationary load system in contact with the spinning disk on the natural frequencies and stability of the system when the original eigenvalues of interest are well separated. This paper is a follow-up investigation to deal with the situations in which two eigenvalues of the freely spinning disk are almost equal (degenerate) and strong modal interactions occur when the load parameters are introduced. After comparing an eigenfunction expansion with the finite element numerical results, we find that for each of the transverse and pitching load parameters, a properly chosen two-mode approximation can exhibit all the important features of the eigenvalue changes. Based on this two-mode approximation we study the mathematical structure of the eigenvalues in the neighborhood of degenerate points in the natural frequency-rotation speed plane. In the case of friction force, however, it is found that at least a four-mode approximation is required to reproduce the eigenvalue structure. The observations and analyses presented provide physical insight into the modal interactions induced by various load parameters in a spinning disk-stationary load system.