Nonlinear Normal Modes Research Articles

Highly efficient nonlinear energy sink

The performance of the nonlinear energy sink (NES) that composed of a small mass and essentially nonlinear coupling stiffness with a linear structure is considerably enhanced here by including the negative linear and nonlinear coupling stiffness components. These negative linear and nonlinear stiffness components in the NES are realized here through the geometric nonlinearity of the transverse linear springs. By considering these components in the NES, very intersecting results for passive targeted energy transfer (TET) are obtained. The performance of this modified NES is found here to be much improved than that of all existing NESs studied up to date in the literature. Moreover, nearly 99 % of the input shock energy induced by impulse into the linear structures considered here has been found to be rapidly transferred and locally dissipated by the modified NES. In addition, this modified NES maintains its high performance of shock mitigation in a broadband fashion of the input initial energies where it keeps its high performance even for sever input energies. This is found to be achieved by an immediate cascade of several resonance captures at low- and high- nonlinear normal modes frequencies. The findings obtained here by including the negative linear and nonlinear stiffness components are expected to significantly enrich the application of these stiffness components in the TET field of such nonlinear oscillators.

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

This paper addresses the numerical computation of nonlinear normal modes defined as two-dimensional invariant manifolds in phase space. A novel finite-element-based algorithm, combining the streamline upwind Petrov–Galerkin method with mesh moving and domain prediction–correction techniques, is proposed to solve the manifold-governing partial differential equations. It is first validated using conservative examples through the comparison with a reference solution given by numerical continuation. The algorithm is then demonstrated on nonconservative examples.

A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models

The nonlinear normal modes of a dynamical system provide a modal framework in which the dynamics of a structure can be readily understood. Current numerical approaches use continuation to find a nonlinear normal mode branch that initiates at a low energy, linearized mode. The predictor-corrector based approach follows the periodic solutions as the response amplitude increases, forming the nonlinear normal mode. This method uses the Jacobian of the shooting function in a Newton–Raphson algorithm to find the initial conditions and integration period that result in a periodic response of the conservative equations of motion. Large scale finite element models require that the Jacobian be computed using finite differences since the closed form equations are not explicitly available. The Jacobian must be computed with respect to all of the states, making the algorithm prohibitively expensive for models with many degrees-of-freedom. In this paper, the initial conditions of each periodic solution are determined based on a subset of the linear modes of a geometrically nonlinear finite element model. The first approach, termed enforced modal displacement, sets the initial conditions as a linear combination of linear mode shapes. The second approach, here called the applied modal force method, applies a static load to the structure in a combination of applied forces that would excite a single linear mode, computes the static response to that load, and uses that to set the initial conditions. Both of these algorithms greatly reduce the number of variables that are iterated on during continuation. As a result, the cost of computing each solution along the nonlinear normal mode is only on the order of ten times the cost required to integrate the finite element model over one period of the response. The algorithm is initiated with only one linear mode and additional modes are added in a systematic way as they become important to the periodic solutions along the nonlinear mode branch. The approach is demonstrated on two geometrically nonlinear finite element models, showing a dramatic reduction in the computational cost required to obtain the nonlinear normal mode.

Classification of periodic orbits of two-dimensional homogeneous granular crystals with no pre-compression

In the present study we classify the periodic orbits of a squarely packed, uncompressed and undamped, homogeneous granular crystal, assuming that all elastic granules oscillate with the same frequency (i.e., under condition of 1:1 resonance); this type of Hamiltonian periodic orbits have been labeled as nonlinear normal modes. To this end we formulate an auxiliary system which consists of a two-dimensional, vibro-impact lattice composed of non-uniform “effective particles” oscillating in an anti-phase fashion. The analysis is based on the idea of balancing linear momentum in both horizontal and vertical directions for separate, groups of particles, whereby each such a group is represented by the single effective particle of the auxiliary system. It is important to emphasize that the auxiliary model can be defined for general finite, squarely packed granular crystals composed of n rows and m columns. The auxiliary model is successful in predicting the total number of such periodic orbits, as well as the amplitude ratios for different periodic regimes including strongly localized ones. In fact this methodology enables one to systematically study the generation of mode localization in these strongly nonlinear, highly degenerate dynamical systems. Good correspondence between the results of the theoretical model and direct numerical simulations is observed. The results presented herein can be further extended to study the intrinsic dynamics of the more complex granular materials, such as heterogeneous two-dimensional and three-dimensional granular crystals and multi-layered structures.

Nonlinear Oscillatory Acoustic Vacuum

A finite chain of particles with next-neighbor interactions, undergoing in-plane nonlinear oscillations with fixed-fixed boundary conditions is considered. In the most significant limiting case of low-energy predominantly transverse particle oscillations, the geometric nonlinearity in the system generates a nonlinear acoustic vacuum, whereby the governing equations of motion possess strongly nonlocal nonlinear terms, generated by the near-uniform axial tension, which, in turn, is caused linear acoustics and zero speed of sound (as defined in the classical sense); in that limit the strongly nonlocal terms become a time-dependent “effective speed of sound” for the medium, that is completely tunable with energy. Another unexpected finding is that, under the admitted assumptions, the nonlinear normal modes (NNMs) of the system are exactly identical to those of the corresponding linear chain, so their spatial shapes obey the same orthogonality conditions, which, in turn, can be used to get an uncoupled strongly nonlinear set of modal oscillators with frequencies completely tunable with energy. A rich structure of resonance manifolds of varying dimensions can then be defined, corresponding to resonance interactions between subsets of NNMs. The 1:1 resonance interactions are studied asymptotically and strong energy exchanges between modes are detected, with the most intense corresponding to a limiting phase trajectory. Moreover an additional interesting resonance motion such as time-periodic oscillations with characteristics of both standing (close to the boundaries) and traveling waves (away from them) are detected. In the view of the authors the nonlinear chain studied represents a new class of strongly nonlinear multidimensional mechanical systems whose degenerate dynamics and potential applications, such as shock and vibration absorption and mitigation, are worthy of exploring further.

Nonlinear modes of piecewise linear systems under the action of periodic excitation

The approach for calculation of nonlinear normal modes (NNM) of essential nonlinear piecewise linear systems’ forced vibrations is suggested. The combination of the Shaw–Pierre NNMs and the Rauscher method is the basis of this approach. Using this approach, the nonautonomous piecewise linear system is transformed into autonomous one. The Shaw–Pierre NNMs are calculated for this autonomous system. Torsional vibrations of internal combustion engine power plant are analyzed using these NNMs.

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

Abstract We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.

Reduced order modeling for the skin panels of hypersonic vehicles and nonlinear normal modes

The skin panels on concept hypersonic (Mach > 5) vehicles are subjected to intense acoustic and thermal loading. As a result, nonlinear structural dynamic models are needed to predict their response with the required level of accuracy. Vehicles such as this carry a large amount of fuel and so the structure must be very light if the overall vehicle is to meet its performance requirements. This talk will discuss the reduced order modeling frameworks that are being used to create models for these types of structures and discuss how nonlinear normal modes are being used to understand how the structure’s response changes with the loading amplitude. Nonlinear modes are also being used to evaluate the reduced order models in order to predict the frequency bandwidth and the range of forcing amplitude over which they will be valid. This approach allows the analyst to develop a considerable level of confidence in the reduced order model without having to compute time responses of the full nonlinear finite element model; time response simulations are far too expensive to be used in practice on the structures of interest.

A fully nonlinear dynamic formulation for rotating composite beams: Nonlinear normal modes in flapping

The geometrically exact equations of motion of pretwisted rotating composite beams parametrized by one space coordinate are derived from three-dimensional theory. The mechanical formulation is based on the special Cosserat theory of rods which does not place any restriction on the geometry of deformation besides the rigidity of the cross-sections. The constitutive relations for the composite beams are obtained within the context of three-dimensional theory. The Taylor expansion of the equations of motion – further specialized to the case of unshearability – is carried out about the equilibrium caused by centrifugal forces to obtain the linearized and third-order perturbed forms of the governing equations. The Galerkin approach is employed to project the linearized equations of motion and solve for the linear free vibration characteristics. Subsequently, the method of multiple scales is applied directly to the perturbed equations of motion to investigate the nonlinear flapping modes and their frequency variations with the amplitude and the angular speed. These modes are investigated away from internal resonance conditions found as the critical angular speeds for which nonlinear interactions between different modes may occur. Results obtained by the Galerkin method are compared with those obtained via a finite element discretization.

Vibration analog of a superradiant quantum transition

A vibrational analog of the superradiant quantum transition (SQT) in a classical system of weakly bound oscillators of van der Pole-Duffing (self-generators), in which the coupling element is a linear oscillator, is described. Such an analog is a strongly modulated oscillatory process of almost complete periodic energy exchange between the generators. This type of mode is alternative to nonlinear normal modes (NNM) and close in its character to limiting phase trajectories (LPTs), which have been introduced recently as applied to conservative systems, but in contrast to them, as the attractor. It is shown that the necessary condition of the transition to intense energy exchange in the classical system is the instability of one of the NNMs similarly to that when the condition of the superradiant transition is the instability of the ground state in a quantum model.

Complexification-averaging method via the averaged Lagrangian

In this paper, the complexification-averaging (CX-A) method for multi-DOF nonlinear vibratory systems is rederived in a new way based upon the averaged Lagrangian. The complex variables are introduced to represent the original displacements and velocities, and then the fast–slow decomposition of the complex variables is made. The time averaging of the Lagrangian over the fast variables is performed. Two different expressions for the kinetic energy are presented, and this results in two schemes for deriving the governing equations of the slow variables. For the scheme I, through the order analysis of the derivatives of the slow variables, it is shown that the second-order terms appeared in the averaged Lagrangian can be omitted, and thus a reduced averaged Lagrangian is obtained. Via the reduced averaged Lagrangian, the corresponding Lagrangian equations are derived. For the scheme II, through time averaging, the averaged Lagrangian is obtained, and then the corresponding equations for the slow variables can be obtained. Finally, two nonlinear vibratory systems with two-DOF and four-DOF, respectively, are given as examples to illustrate the new procedure for the CX-A method. The loci of nonlinear normal modes on the potential surface are studied in the first example, and the frequency-energy plot is investigated in the second example.

Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Analytical Study

We present an analytical study of the conservative and dissipative dynamics of a two-degree-of-freedom (DOF) system consisting of a linear oscillator coupled to a bistable light attachment. The main objective of the paper is to study the beneficial effect of the bistability on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to an appropriately designed attachment. As a numerical study of the problem has shown in a companion paper (Romeo, F., Sigalov, G., Bergman, L. A., and Vakakis, A. F., 2013, “Dynamics of a Linear Oscillator Coupled to a Bistable Light Attachment: Numerical Study,” J. Comput. Nonlinear Dyn. (submitted)) there is an essential difference in the system's behavior when compared to the conventional case of a monostable attachment. On the other hand, some similarity to the behavior of an oscillator with rotator attachment has been revealed. It relates, in particular, to the generation of nonconventional nonlinear normal modes and to the existence of two qualitatively different types of dynamics. We find that all numerical results can be explained in the framework of fundamental (1:1) and superharmonic (1:3) resonances (for large energies), as well as a subharmonic resonance (for small energies). This allows us to use the concept of limiting phase trajectories (LPTs) introduced earlier by one of the authors, and to derive accurate analytical approximations to the dynamics of the problem in terms of nonsmooth generating functions.

Superradiant transition and its classical analogue

The two-level quantum model proposed by Preparata, according to which the interaction of a material system with a electromagnetic field can cause the loss of stability of the lower level, is examined. As a result of this instability, the role of the ground state is played by a periodic process generating superradiance (in contrast to laser radiation induced by overpopulation of the upper level). A classical analogue of a super-radiant quantum transition, a system of van der Pol-Duffing oscillators (generators) weakly coupled via a linear oscillator, is for the first time described. Such an analogue is a strongly modulated oscillatory process of almost complete periodic energy exchange between the generators. It is shown that a necessary condition for the transition to intense energy exchange in the classical system is the instability of one of the nonlinear normal modes, similar to how the condition for a superradiant transition to occur is the instability of the ground state of a two-level quantum system coupled with a resonant electromagnetic field.

Nonlinear Modal Analysis of a Full-Scale Aircraft

Nonlinear normal modes, which are defined as a nonlinear extension of the concept of linear normal modes, are a rigorous tool for nonlinear modal analysis. The objective of this paper is to demonstrate that the computation of nonlinear normal modes and of their oscillation frequencies can now be achieved for complex, real-world aerospace structures. The application considered in this study is the airframe of the Morane–Saulnier Paris aircraft. Ground vibration tests of this aircraft exhibited softening nonlinearities in the connection between the external fuel tanks and the wing tips. The nonlinear normal modes of this aircraft are computed from a reduced-order nonlinear finite element model using a numerical algorithm combining shooting and pseudo-arclength continuation. Several nonlinear normal modes, involving, e.g., wing bending, wing torsion, and tail bending, are presented, which highlights that the aircraft can exhibit very interesting nonlinear phenomena. Specifically, it is shown that modes with distinct linear frequencies can interact and generate additional nonlinear modes with no linear counterpart.

Nonlinear dynamics of rectangular plates: investigation of modal interaction in free and forced vibrations

Nonlinear vibrations of thin rectangular plates are considered, using the von Karman equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure.

New type of synchronization of oscillators with hard excitation

It is shown that stable limiting cycles corresponding to nonlinear beats with complete energy exchange between oscillators can exist in a system of two weakly coupled active oscillators (generators). The oscillatory regime of this type, which implements a new type of synchronization in an active system, is an alternative to the well-studied synchronization in a regime close to a nonlinear normal mode. In this case, the ranges of dissipative parameters corresponding to different types of synchronization do not intersect. The analytic description of attractors revealed in analysis is based on the concept of limiting phase trajectories, which was developed earlier by one of the authors for conservative systems. A transition (in the parametric space) from the complete energy exchange between oscillators to predominant localization of energy in one of the oscillators can be naturally described using this concept. The localized normal mode is an attractor in the range of parameters in which neither the limiting phase trajectory nor any of the collective normal modes is an attractor.

Non-linear normal forced vibration modes in systems with internal resonance

The new method of the forced resonance vibrations construction in mechanical systems with internal resonance is represented. According to this approach, the generalized theory of non-linear normal vibration modes by Shaw and Pierre, the modified Rauscher method and the harmonic balance method are combined with a new iterative computation procedure.The proposed approach is used in analysis of the single-disk rotor system with the isotropic-elastic shaft and the non-linear supports of Duffing type. Gyroscopic effects, asymmetrical disposition of the disk on the shaft and internal resonance are also taken into account. The NNM approach allows reducing the 8-DOF problem of the rotor dynamics to the 2-DOF non-linear system for each non-linear normal mode. Both the model of massless supports and the model of supports with inertial effects are considered. It is shown that in last case all resonance regimes are separated into two different kinds. First kind corresponds to cyclic symmetric trajectories in a system's configuration space; the second kind corresponds to centrally symmetric ones. Regimes of the first kind can be evaluated by the use of the simplified mathematical model proposed in this work. Simplified model consists only of four generalized coordinates instead of the eight initial ones.

Nonlinear Normal Modes of a Continuous Cantilever Beam with Nonlinear Energy Sink Absorber

Using nonlinear energy sink absorber (NESA) is a good countermeasure for vibration suppression in wide board frequency region. The nonlinear normal modes (NNMs) are helpful in dynamics analysis for a NESA-attached system. Being a primary structure, a cantilever beam whose modal functions contain hyperbolic functions is surveyed, in case of being attached with NESA and subjected to a harmonic excitation. With the help of Galerkins method and Raushers method, the NNMs are obtained analytically. The comparison of analytical and numerical results indicates a good agreement, which confirms the existence of the nonlinear normal modes.

Determination of nonlinear normal modes of a planar nonlinear system with a constraint condition

To demonstrate in what a way nonlinear normal modes may influence or be used to predict system responses, this paper derives analytically the nonlinear normal modes with a constraint condition from the free vibration equation of the non-conservative nonlinear subsystem of a piecewise smooth system. Since the nonlinear subsystem describes the planar motion of an isotropic rotor with cross-coupling stiffness in contact with its stator, the nonlinear normal modes reflecting the two potential whirling directions within a plane possess both positive and negative modal frequencies. The results of this paper show that the stable and the unstable nonlinear normal modes affect the responses of the whole piecewise smooth system in different ways. Moreover, the nonlinear normal modes can also be used to predict some important characteristics of the self-excited backward whirl motion of the piecewise smooth system.

Nonlinear Free Vibrations of Multi-Disk Rotors on Ball Bearings

A model and method for investigation of the vibrations of a multi-disk rotor on two nonlinear supports with ball bearings are developed. This model considers the gyroscopic moments acting on disks and elastic properties of the shaft. The vibrations are studied using the Shaw–Pierre nonlinear normal modes. Skeleton curves describing the rotor vibrations are analyzed.