Nonlinear Normal Modes Research Articles

Exact solutions of nonlinear dynamical equations for large-amplitude atomic vibrations in arbitrary monoatomic chains with fixed ends

Intermode interactions in one-dimensional nonlinear periodic structures have been studied by many authors, starting with the classical work by Fermi, Pasta, Ulam, and Tsingou (FPUT) in the middle of the last century. However, symmetry selection rules for the energy transfer between nonlinear vibrational modes of different symmetry, which lead to the possibility of excitation of some bushes of such modes, were not revealed. Each bush determines an exact solution of nonlinear dynamical equations of the considered system. The collection of modes of a given bush does not change in time, while there is a continuous energy exchange between these modes. Bushes of nonlinear normal modes (NNMs) are constructed with the aid of group-theoretical methods and therefore they can exist for the case of large amplitude atomic vibrations and for any type of interatomic interactions. In most publications, bushes of NNMs or similar dynamical objects in one-dimensional systems are investigated under periodic boundary conditions. In this paper, we present a detailed study of the bushes of NNMs in monoatomic chains for the case of fixed boundary conditions, which sheds light on a series of new properties of the intermode interactions in such systems. We prove some theorems that justify a method for constructing bushes of NNMs by continuation of conventional normal modes to the case of large atomic oscillations. Our study was carried out for FPUT chains, for the chains with the Lennard-Jones interatomic potential, as well as for the carbon chains (carbynes) in the framework of the density functional theory. For one-dimensional bushes (Rosenberg nonlinear normal modes), the amplitude–frequency diagrams are presented and the possibility of their modulational instability is briefly discussed. We also argue in favor of the fact that our methods and main results are valid for any monoatomic chain.

Frequency-Energy Analysis of Coupled Linear Oscillator With Unsymmetrical Nonlinear Energy Sink

Abstract The underlying nonlinear dynamical behavior of a linear oscillator (LO) attached with nonlinear energy sink (NES) is usually revealed by studying the LO-NES system on the frequency-energy plot (FEP). Accordingly, different kinds of periodic motions on the backbone curves of the nonlinear normal modes (NNMs) and their associated subharmonic branches are analyzed. Here, the considered unsymmetrical nonlinear energy sink (UNES) incorporates a cubic stiffness element in one direction of the motion and a weak linear restoring coupling stiffness in both directions of the motion. This UNES is attached with the LO in the LO-UNES system which is studied here on the FEP. Therefore, the obtained FEP of the LO-UNES system is generated and analyzed. Accordingly, several unsymmetrical backbone curves of periodic oscillations between the UNES mass and the LO mass at 1:1 resonance have been obtained for broad range of nonlinear frequency levels.

Modal balancing for ball bearing-rotor system using nonlinear normal modes

Rotor dynamic balancing procedure relies on the linearisation hypothesis of the rotor system. However, some mechanical elements constituting a rotor system behave in various nonlinearities, such as the contact force and radial internal clearance between rolling balls and races for a rotor-bearing system. To balance a nonlinear rotor system, the traditional modal balancing method must be extended. This paper investigates the modal balancing method for a rotor-bearing system based on nonlinear normal modes (NNMs). The proposed algorithm includes three main steps. First, compute the NNMs and express nonlinear modal shapes using complex Fourier series. Second, describe the unbalance response as a complex amplitude containing higher harmonics. Third, estimate the correction coefficient of each harmonic, then compute the unbalance correction of each mode of interest until the total unbalance corrections are obtained. Hertz contact force and two different sizes of radial internal clearances of the ball bearings are considered during the balancing simulation. The responses after balancing with NNMs are present and compared with the linear modal method to demonstrate the accuracy and effectiveness of the proposed method.

Physics-constrained deep learning of nonlinear normal modes of spatiotemporal fluid flow dynamics

In this study, we present a physics-constrained deep learning method to discover and visualize from data the invariant nonlinear normal modes (NNMs) which contain the spatiotemporal dynamics of the fluid flow potentially containing strong nonlinearity. Specifically, we develop a NNM-physics-constrained convolutional autoencoder (NNM-CNN-AE) integrated with a multi-temporal-step dynamics prediction block to learn the nonlinear modal transformation, the NNMs containing the spatiotemporal dynamics of the flow, and reduced-order reconstruction and long-time future-state prediction of the flow fields, simultaneously. In test cases, we apply the developed method to analyze different flow regimes past a cylinder, including laminar flows with low Reynolds number in transient and steady states (RD = 100) and high Reynolds number flow (RD = 1000), respectively. The results indicate that the identified NNMs are able to reveal the nonlinear spatiotemporal dynamics of these flows, and the NNMs-based reduced-order modeling consistently achieves better accuracy with orders of magnitudes smaller errors in construction and prediction of the nonlinear velocity and vorticity fields, compared to the linear proper orthogonal decomposition (POD) method and the Koopman-constrained-CNN-AE using the same number or dimension of modes. We perform an analysis of the modal energy distribution of NNMs and find that compared to POD modes, the few fundamental NNMs capture a very high level of total energy of the flow, which is advantageous for reduced-order modeling and representation of the complex flows. Finally, we discuss the potentials and limitations of the presented method.

Blue-Sky Catastrophic Bifurcations Behind Emergence and Disappearance of Quasiperiodic Rubbing Oscillations in a Piecewise Smooth Rotor–Stator System

In this paper, an approach combining the methods of multigrid Point Mapping under Cell Reference and Stagger-and-step is proposed, which can efficiently determine both attractors and unstable saddle-type invariant sets in higher-dimensional dynamical systems. Then, the bifurcations around quasiperiodic partial rub oscillations of a piecewise smooth rotor–stator system are investigated, which are categorized into two classes depending on whether Nonlinear Normal Modes (NNMs) exist or not. It is found that the quasiperiodic rubbing oscillations, within the parameter region with an NNM, emerge through a Hopf bifurcation from a periodic rubbing oscillation and disappear through a collision with a saddle torus with an accompanying variation of system parameters. Within the parameter region without NNM, the bifurcation scenarios of two sequences of quasiperiodic rubbing oscillations, which emerge subsequently with the increasing of the rotating speed, are similar but somewhat complicated. That is, the quasiperiodic rubbing oscillation emerges through an inverse doubling of torus from a chaotic attractor, which is produced from a chaotic saddle after chaotic crisis. The quasiperiodic rubbing oscillation disappears through a collision between the stable and the unstable quasiperiodic tori. The insight into blue-sky catastrophic bifurcations in the quasiperiodic rubbing oscillations of the piecewise smooth rotor–stator rubbing system is thus obtained through this study.

Exact augmented perpetual manifolds as a tool for the determination of similar rigid body modes: Corollary for determining backbone curves of autonomous systems

Perpetual points in mathematics have been defined recently, and their significance in describing the dynamics of systems is ongoing research. In mechanical engineering, the perpetual points of linear unforced mechanical systems are associated with rigid body motions, and they form the perpetual manifolds. The systems that admit perpetual manifolds of rigid body motions are called perpetual mechanical systems. The concept of perpetual manifolds has been extended to the exact augmented perpetual manifolds, whereas all the accelerations are equal but not necessarily zero. A theorem defines the conditions of external forces that lead to rigid body motion of externally forced mechanical systems. Herein, based on this theorem, a corollary is proven that, through a very simple framework, defines specifications to mechanical systems resulting in the existence of rigid body similar modes and can serve as a tool in the nonlinear normal modes (NNMs) theory for the determination of their backbone curves of dissipative mechanical systems. Further, on these, backbone curves can be used for passive vibration control design of N-degree-of-freedom (dof) mechanical systems. The verification of the theory has been done with the design of a 5-dof nonlinear perpetual mechanical system with three types of boundaries, linear, nonlinear smooth, and nonsmooth boundaries. The numerical simulations results are in excellent agreement with the theory. Although 5-dof mechanical systems have been examined, the corollary is valid for N-degree-of freedom systems. The corollary provides a simplified framework for the design of mechanical systems to admit rigid body modes for systems with a general configuration that is not obvious using traditional techniques. This work is rather significant in mechanical engineering because it provides a new tool firstly for determining similar NNMs of dissipative mechanical systems and secondly for the design of passive vibration control devices.

Hamiltonian dynamics and targeted energy transfer of a grounded bistable nonlinear energy sink

A two-degree-of-freedom nonlinear mechanical system describing a damped linear oscillator weakly coupled to a lightweight grounded bistable nonlinear energy sink is proposed to investigate the targeted energy transfer (TET) under asymmetric local nonlinear potentials. The bifurcations of equilibria are discussed to reflect the bistable and asymmetric characteristics of the system. Through projections of isoenergetic manifolds, the topological features of the Hamiltonian dynamics under 1:1 internal resonance have been investigated, which induces conditions for energy localization and exchange. In the general case, the topological features of the Hamiltonian dynamics related to the nonlinear normal modes (NNMs) are classified by the Poincaré map, the relations between the NNMs and the Hamiltonian dynamics have been interpreted through the frequency energy plots. The results show that introducing the asymmetric characteristics may enhance the TET and realize the complete TET.

A receptance based method for the calculation of nonlinear normal modes of large ordered structures with distributed localized nonlinearities

In recent years, the concept of nonlinear normal modes (NNMs) has gained interest for interpreting a broad range of nonlinear dynamic phenomena. Present study introduces a simple and efficient computational framework to compute NNMs of large order nonlinear structures overcoming difficulties associated with commonly used time-integration and discretization methods. In this paper, a Receptance Based Nonlinear Normal Mode Calculation Method (RBNM) is developed to determine the NNMs of large ordered realistic finite element models in frequency domain. Describing Function Method (DFM), which makes it possible to write the nonlinear internal forcing vector as a nonlinear displacement dependent stiffness matrix multiplied by displacement vector, is used to model the internal nonlinear forcing vector in frequency domain. A unique matrix manipulation based on dynamic stiffness and receptance concepts is used, as a result of which the number of governing nonlinear equations becomes independent from the total number of degrees of freedom of the model and it only depends on the number of degrees of freedom associated with the nonlinear elements. This brings a significant computational advantage over the classical reduced order modeling techniques. A simple 2-degree of freedom (DOF) system, the results of which are available in the literature, is used to verify the proposed method in determining NNMs. A 20-DOF lumped parameter system model and later a complex finite element model of a large structure are used to demonstrate the applicability of RBNM in determining NNMs of larger systems with several different types of nonlinear elements distributed among the degrees of freedom.

Effects of centrifugal stiffening and spin softening on nonlinear modal characteristics of cyclic blades with impact–friction coupling

This paper aims to interpret the coupling modal properties of cyclic blades under impact–friction interactions and their evolution mechanism versus operating points. Therefore, a coupling analytical model of cyclic blades is developed based on a Lagrange method and the assumed mode method (AMM), after considering centrifugal stiffening, spin softening, stagger angle, and twist angle. Then a mixed modal analysis method (MMAM) for this analytical model is extended by combining the extended periodic motion concept (EPMC) with AMM. Wherein a classic alternating frequency/time method (AFT) and the continuation method are employed to overcome the numerical divergence problem. Then damped nonlinear normal modes (dNNMs), including eigenfrequencies, modal damping ratios, and mode shapes, of the coupling system with shroud joints are finally computed and discussed under different excitation levels and contact conditions through a modal synthesis algorithm. After that, the influence laws of centrifugal stiffening and spin softening on the dNNMs are explored to reveal its evolution mechanism versus operation speeds. Finally, the Campbell diagrams of dNNMs are successfully obtained to discuss the effects of the impact–friction coupling on critical speeds (CSs) of the shrouded blades system.

Modeling and simulations of a nonlinear granular metamaterial: application to geometric phase-based mass sensing

Dynamical simulations of an externally harmonically driven model granular metamaterial composed of four linearly and nonlinearly coupled granules show that the nonlinear normal mode can be expressed in a linear normal mode orthonormal basis with time dependent complex coefficients. These coefficients form the components of a state vector that spans a 22 dimensional Hilbert space parametrically with time. Local π jumps in the phase of these components occurring periodically are indicative of topological features in the manifold spanned by the geometric phase of the vibrational state of the metamaterial. We demonstrate that these topological features can be exploited to realize high sensitivity mass sensor. The effect of dissipation on sensitivity is also reported. Nonlinear granular metamaterials with very low dissipation could serve as mass sensors with considerable sensitivity to small mass changes via large changes in geometric phase.

Instability dynamics of nonlinear normal modes in the Fermi–Pasta–Ulam–Tsingou chains

Nonlinear normal modes are periodic orbits that survive in nonlinear chains, whose instability plays a crucial role in the dynamics of many-body Hamiltonian systems toward thermalization. Here we focus on how the stability of nonlinear modes depends on the perturbation strength and the system size to observe whether they have the same behavior in different models. To this end, as illustrating examples, the instability dynamics of the N/2 mode in both the Fermi–Pasta–Ulam–Tsingou-α and -β chains under fixed boundary conditions are studied systematically. Applying the Floquet theory, we show that for both models the stability time T as a function of the perturbation strength λ follows the same behavior; i.e., , where λ c is the instability threshold. The dependence of λ c on N is also obtained. The results of T and λ c agree well with those obtained by the direct molecular dynamics simulations. Finally, the effect of instability dynamics on the thermalization properties of a system is briefly discussed.

Low-to-high frequency targeted energy transfer using a nonlinear energy sink with softening-hardening nonlinearity

This paper investigates the performance of a nonlinear energy sink (NES) equipped with a softening–hardening (SH) element for mitigating vibrations that occur below its linear, low-energy frequency. First, an SH elastic struct is constructed, tested quasi-statically, and the resulting restoring force is modeled using a polynomial. Next, the SH polynomial is non-dimensionalized and a theoretical system composed of a linear oscillator (LO) and NES is studied where the NES has a linear, low-energy frequency above that of the LO. The underlying nonlinear normal modes (NNMs) of the system are studied by using harmonic excitation and tracking the changes in the frequency response functions. The transient performance of the NES is investigated and compared with that of a NES with the same linear stiffness but only a cubic nonlinearity. The theoretical performance of the NES is verified using a comparable experimental system that incorporates the SH elastic struct built in the beginning of the paper. The results of this work demonstrate that an SH NES is capable of mitigating vibrations that occur at frequencies below its own linear, low-energy frequency.

Modal interactions of a dual-joint cylindrical shell system via Nonlinear Normal Modes

Bolted flange joints are commonly encountered in engineering structures. One specific challenge that is regularly encountered is the modeling of connecting interfaces between flanges. Indeed, for large amplitudes of vibration, due to discontinuity of the structure and mechanical contact of the connecting interface, clearances may occur between flanges and cause local stiffness loss, exhibiting complicated nonlinear dynamic behavior. In this paper, a simplified dynamic model for bolted flange joint via structural static analysis is proposed, in which the stiffness nonlinearity due to the non-smoothness of jointed interface is characterized by piecewise-smooth springs. The proposed simplified model has demonstrated good accuracy and precision in capturing the nonlinear dynamic characteristics of bolted flange connections. Following the simplified modeling technique and focusing on the stiffness nonlinearities, an analytical dual-joint system with free–free boundary conditions is established capturing the interaction of a longitudinal mode with two bending modes. The Harmonic Balance Method combined with Asymptotic Numerical Method (HBM–ANM) is utilized to estimate Nonlinear Normal Modes (NNMs) of the system, which provide an interpretation of the underlying nonlinear dynamic behavior, including bifurcation analysis, internal resonances characterization, and stability properties. Especially, internal resonances between the three modes can be triggered at high energy (oscillation amplitude) level, without having commensurate linear natural frequencies, leading to qualitative changes in periodic motions and internal loads environment of the system. The influences of linear frequency ratios and stiffness nonlinearity of each joint on the dynamics of the system are discussed in terms of NNMs. It is interesting to reveal that reducing the stiffness nonlinearity of joint may grow the complexity of the nonlinear dynamics of the system, from the perspective of motion stability, and trigger internal resonance with a complex topology. The studies in this paper provide an in-depth understanding of the nonlinear dynamic behavior of dual-joint systems, and the NNMs certainly represent a useful framework that could lead to a better dynamic design of jointed structures.

Localized and non-localized nonlinear normal modes in a system of two coupled pendulums under a magnetic field

The dynamics of two coupled pendulums under a magnetic field are considered. Inertial components of the pendulums are essentially different, and a ratio of masses is chosen as a small parameter. Padé approximations are used to describe magnetic excitation with good accuracy. The small parameter method is used to construct two nonlinear normal modes (NNMs) one of them is the coupled vibration mode, and the second is the localized one. The influence of the masses ratio and other system parameters to the NNMs for small and not small initial angles of the pendulums is studied.

On the dynamic properties of statistically-independent nonlinear normal modes

Much attention has been given to the study of nonlinear normal modes (NNMs), a nonlinear extension to the eminently useful framework for the analysis of linear dynamics provided by linear modal analysis (LMA). In the literature, several approaches have gained traction, with each able to preserve a subset of the useful properties of LMA. A recently-proposed framework (Worden and Green, 2017) casts nonlinear modal analysis as a problem in machine learning, viewing the NNM as directions in a latent modal coordinate space within which the modal dynamics are statistically uncorrelated. Thus far, the performance of this framework has been measured in a largely qualitative way. This paper presents, for the first time, an exploration into the underlying dynamics of the statistically-independent NNMs using techniques from nonlinear system identification (NLSI) and higher-order frequency-response functions (HFRFs).In this work, the statistically-uncorrelated NNMs are found for two simulated nonlinear cubic-stiffness systems using a recently-proposed neural-network based approach. NLSI models are fitted to both physical and modal displacements and the HFRFs of these models are compared to theoretical values. In particular, it is found for both systems that the modal decompositions here permit an independent single-input single-output (SISO) representation that can be projected back onto the original displacements with low error. It is also shown via the HFRFs that the underlying linear natural frequencies of the modal dynamics lie very close to the underlying linear natural frequencies of the nonlinear systems, indicating that a true nonlinear decomposition has been identified.

Evaluation on spectral submanifold based reduced models of a rotor/stator rubbing system with cross-coupling stiffness

The methodology for reduced order models of nonlinear dynamical systems is always of great interest in dynamics community. Recently, spectral submanifolds (SSMs) related to hyperbolic fixed points have been constructed and successfully used to build reduced order models for nonlinear systems with a large number of degrees of freedom. There are, however, few works on the spectral submanifolds associated with periodic orbits in literature. In this paper, SSM associated to the periodic orbit of a nonlinear normal mode (NNM) of a rotor/stator rubbing system with cross-coupling stiffness is first constructed following the parameterization method. The SSM with a two-dimensional slow sub-stable invariant manifold attached to the stable NNM with forward whirl also serves as a reduced order model of the system, whose dynamics on the parameter space conjugates the dynamics of the full system. Unlike the previous works, where reduced order models are mainly used to obtain backbones of conservative autonomous systems or FRCs of light damped forcing systems as the non-autonomous perturbation, the reduced order model derived from autonomous system are used to predict the self-excited dominant forcing responses of the rotor/stator rubbing system. As can be seen, the reduced order model is valid with good accuracy in some range of forcing amplitudes. Moreover, the modification of the reduced order model by considering forcing perturbation is also built and works well even in a larger range of forcing amplitudes.

Nonlinear Normal Modes of a Three Degrees-of-Freedom Cyclically Symmetric Piecewise Linear System

AbstractThis article investigates the nonlinear normal modes (NNMs) of a three degrees-of-freedom cyclically symmetric system with bilinear contact springs. Further, two different cases of zero and nonzero clearance are assumed for the contact model. These systems are found to exhibit qualitatively different characteristics. Interestingly, the former system admits positive-scaling of solutions, which is reminiscent of linear systems. Four classes of NNMs are found for both zero and nonzero clearance. The NNM solutions are calculated using analytical or numerical techniques. The analytical solution involves solving the equations of motion using appropriate ansatzes, while the numerical solution uses a shooting-based continuation scheme. The NNMs' features and their variations with contact spring stiffness and system energy are also studied. This study highlights certain simplifications applicable to cyclically symmetric structures with contact. These simplifications will be of practical interest for designing real-life structures.

Efficient energy balancing across multiple harmonics of nonlinear normal modes

Predicting the forced responses of nonlinear systems is a topic that attracts extensive studies. The energy balancing method considers the net energy transfer in and out of the system over one period and establishes connections between forced responses and nonlinear normal modes (NNMs). In this paper, we consider the energy balancing across multiple harmonics of NNMs for predicting forced resonances. This technique is constructed by combining the energy balancing mechanism with restrictions (established via excitation scenarios) on external forcing and harmonic phase-shifts; a semi-analytical framework is derived to achieve both accurate/robust results and efficient computations. With known inputs from NNM solutions, the required forcing amplitudes to reach NNMs at resonances, along with their discrepancy, i.e. the harmonic phase-shifts, are computed via a one-step scheme. Several examples are presented for different excitation scenarios to demonstrate the applicability of this method and to show its capability in accurately predicting the existence of an isola where multiple harmonics play a significant part in the response.

Breaking reciprocity to realize extreme energy isolation in coupled oscillators

This research investigates the isolation achieved by breaking the reciprocity of between two coupled oscillators. The two oscillators have equal mass and the first one is linearly grounded and called the linear oscillator (LO). The second oscillator is nonlinearly coupled to the LO and is termed the nonlinear oscillator (NO). By breaking dynamical reciprocity using asymmetry and nonlinearity, the LO–NO system is shown to exhibit regimes of extreme energy isolation in only one of the oscillators as well as regimes where energy is exchanged between them. These regimes are shown to arise under both impulsive and harmonic excitation. The resulting system is governed by two nonlinear normal modes (NNMs), which can interact with each other under internal resonance of different ratios. Under different loading scenarios, different energy isolations are illustrated. This research starts with analytical study using numerical simulations that assess how energy distributes in the structure under varying loads. The analytical predictions are validated experimentally for both impulsive and harmonic excitations. The results of this research demonstrate that there remains much to learn about energy transfer in general and the breaking of dynamic reciprocity may lead to new types of acoustic and vibrational metamaterials.

Experimental Closed-Loop Excitation of Nonlinear Normal Modes on an Elastic Industrial Robot

Adding elastic elements to the mechanical structure should enable robots to perform efficient oscillatory tasks. Still, even characterizing natural oscillations in nonlinear systems is a challenge in itself, which nonlinear modal theory promises to solve. Therein eigenmanifolds generalize eigenspaces to mechanical systems with non-Euclidean metrics and thus characterize families of oscillations that are autonomous evolutions of the robot. Eigenmanifolds likewise provide a framework for deriving feedback controllers to excite and sustain these oscillations. Nevertheless, these results have been so far essentially theoretical. They have been applied on relatively low dimensional systems and almost exclusively in simulation. We aim to bridge the theory to the real-world gap with the present work and show that we can excite nonlinear modes in complex systems. To this end, we propose control strategies that can simultaneously stabilize numerically evaluated eigenmanifolds and sustain oscillations in the presence of dissipation. We then focus on the KUKA iiwa with simulated parallel springs as an example of the highly nonlinear and articulated system. We calculate all the nonlinear modes of the system, and we use the proposed strategies to excite the associated natural oscillations.