Dual mode vibration isolation based on non-linear mode localization

We study a non-linear vibration isolation system capable of (a) isolating its upper part (the ‘machine’) from periodic disturbances generated at its base; and (b) simultaneously isolating its base from periodic disturbances generated at the level of the machine. By making use of essentially non-linear (e.g. non-linearizable) stiffness elements we completely eliminate resonances close to linearized modes, thus achieving vibration isolation over an extended frequency range. Instead, we prove the existence of branches of localized steady state motions in the frequency domain. Indeed, these localized forced motions are principally responsible for fulfilling the dual mode vibration isolation objective of this work. The method of analysis followed is based on complexification and separation of the dynamics into ‘slow’ varying and ‘fast’-varying parts. Direct numerical simulations confirm the analytical predictions. An analytical method is then developed for determining the placement of the localized branches in the frequency domain as the system parameters vary; this permits the design of the vibration isolation system for best performance in a specified frequency range. The vibration isolation performance achieved by the non-linear system considered has no counterpart in linear theory.     

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ??slow?? and ??fast?? motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.     

Experimental study of non-linear energy pumping occurring at a single fast frequency

Experimental verification of passive non-linear energy pumping in a two-degree-of-freedom system comprising a damped linear oscillator coupled to an essentially non-linear attachment is carried out. In the experiments presented the non-linear attachment interacts with a single linear mode and, hence, energy pumping occurs at a single ‘fast’ frequency in the neighborhood of the eigenfrequency of the linear mode. Good agreement between simulated and experimental results was observed, in spite of the strongly (essentially) non-linear and transient nature of the dynamics of the system considered. The experiments bear out earlier predictions that a significant fraction of the energy introduced directly to a linear structure by an external impulsive (broadband) load can be transferred (pumped) to an essentially non-linear attachment, and dissipated there locally without spreading back to the system. In addition, the reported experimental results confirm that (a) non-linear energy pumping in systems of coupled oscillators can occur only above a certain threshold of the input energy, and (b) there is an optimal value of the energy input at which a maximum portion of the energy is absorbed and dissipated at the NES.     

Nonlinear motions of beam-mass structure

We study the planar dynamic response of a flexible L-shaped beam-mass structure with a two-to-one internal resonance to a primary resonance. The structure is subjected to low excitation (mili g) levels and the resulting nonlinear motions are examined. The Lagrangian for weakly nonlinear motions of the undamped structure is formulated and time averaged over the period of the primary oscillation, leading to an autonomous system of equations governing the amplitudes and phases of the modes involved in the internal resonance. Later, modal damping is assumed and modal-damping coefficients, determined from experiments, are included in the analytical model. The locations of the saddle-node and Hopf bifurcations predicted by the analysis are in good agreement, respectively, with the jumps and transitions from periodic to quasi-periodic motions observed in the experiments. The current study is relevant to the dynamics and modeling of other structural systems as well.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment

We analyze complex, multi-frequency, non-linear modal interactions in the damped dynamics of a viscously damped dispersive finite rod coupled to a multi-degree-of-freedom essentially non-linear attachment. We perform a parametric study to show that the attachment can be an effective broadband energy absorber and dissipater of shock energy from the rod. It is shown that strong targeted energy transfer from the rod to the attachment occurs when there is strong stiffness asymmetry in the attachment. For weak viscous dissipation, a clear understanding of dynamical transitions in the integrated rod-non-linear attachment system can be gained by wavelet transforming the time series and superimposing the resulting wavelet spectra in the frequency-energy plot (FEP) of the periodic orbits of the underlying Hamiltonian system. Two distinct NES configurations are analyzed in detail, and their damped responses are analyzed by the Hilbert-Huang transform (HHT). We show that the HHT is capable of analyzing even complex non-linear damped transitions, by providing the dominant frequency components (or equivalently, time scales) at which the non-linear phenomena take place, and clarifying the series of non-linear resonance captures between the rod and attachment dynamics that are responsible for the broadband energy exchanges in this system.     

Effect of 1:3 resonance on the steady-state dynamics of a forced strongly nonlinear oscillator with a linear light attachment

We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency–energy domain by computing forced frequency–energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.     

Physical mechanism of the destabilizing effect of damping in continuous non-conservative dissipative systems

The paper attempts to give a physical explanation of the mechanism behind the so-called destabilizing effect of small internal damping in the dynamic stability of Beck's column. Both internal (material) and external (viscous fluid) damping are considered. An energy equation is derived for the balance between the work done by the non-conservative ‘follower force’ and the energy dissipated by the internal and external damping forces. Evaluated at the critical load, where a flutter instability is initiated, this equation explicitly shows the influence of damping upon flutter frequency, phase angle, and vibration amplitude. The gradient of the phase angle, evaluated at the free end of the column, is found to be the ‘valve’, which controls how much work the follower force can do on the column during each period of oscillation. And a large change in this gradient with increasing—but still small—internal damping is found to be responsible for the destabilizing effect.     

Energy Transfers in a System of Two Coupled Oscillators with Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits

The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy pumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest, a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understand the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the theoretical developments are discussed. This study was carried out while the author was a postdoctoral fellow at the National Technical University of Athens and at the University of Illinois at Urbana-Champaign.     

Non-linear longitudinal vibrations of piezoceramics excited by weak electric fields

Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relationship between excitation voltage and vibration amplitude as well as jump phenomena are observed in experiments with piezoceramics excited at resonance by weak electric fields. These non-linear effects can be observed for both the piezoelectric 31- and the 33-effect. In contrast to the well-known non-linear effects exhibited by piezoceramics in the presence of strong electric fields, these effects are not described in detail in the literature.In this paper, we attempt to model these phenomena using an electric enthalpy density to capture the cubic-like effects observed in the experiments. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. The ‘non-linear’ parameters are identified and the numerical results are compared to those obtained experimentally. The effects described herein may have a significant influence in structures excited close to resonance frequencies via piezoelectric elements.     

Complex dynamics and targeted energy transfer in linear oscillators coupled to multi-degree-of-freedom essentially nonlinear attachments

We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency–energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers. PACS numbers: 05.45.Xt, 02.30.Hq     

Global and bifurcation analysis of a structure with cyclic symmetry

This article combines the application of a global analysis approach and the more classical continuation, bifurcation and stability analysis approach of a cyclic symmetric system. A solid disc with four blades, linearly coupled, but with an intrinsic non-linear cubic stiffness is at stake. Dynamic equations are turned into a set of non-linear algebraic equations using the harmonic balance method. Then periodic solutions are sought using a recursive application of a global analysis method for various pulsation values. This exhibits disconnected branches in both the free undamped case (non-linear normal modes, NNMs) and in a forced case which shows the link between NNMs and forced response. For each case, a full bifurcation diagram is provided and commented using tools devoted to continuation, bifurcation and stability analysis.     

Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping

The dynamics of a system of coupled oscillators possessing strongly nonlinear stiffness and damping is examined. The system consists of a linear oscillator coupled to a strongly nonlinear, light attachment, where the nonlinear terms of the system are realized due to geometric effects. We show that the effects of nonlinear damping are far from being purely parasitic and introduce new dynamics when compared to the corresponding systems with linear damping. The dynamics is analyzed by performing a slow/fast decomposition leading to slow flows, which in turn are used to study transient instability caused by a bifurcation to 1:3 resonance capture. In addition, a new dynamical phenomenon of continuous resonance scattering is observed that is both persistent and prevalent for the case of the nonlinearly damped system: For certain moderate excitations, the transient dynamics “tracks” a manifold of impulsive orbits, in effect transitioning between multiple resonance captures over definitive frequency and energy ranges. Eventual bifurcation to 1:3 resonance capture generates the dynamic instability, which is manifested as a sudden burst of the response of the light attachment. Such instabilities that result in strong energy transfer indicate potential for various applications of nonlinear damping such as in vibration suppression and energy harvesting.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

KBM unified method for solving an nth order non-linear differential equation under some special conditions including the case of internal resonance

The Krylov-Bogoliubov-Mitropolskii (KBM) unified method is used for obtaining the approximate solution of an nth order (n?4) ordinary differential equation with small non-linearities when a pair of eigen-values of the unperturbed equation is multiple (approximately or perfectly) of the other pair or pairs. The general solution can be used arbitrarily for over-damped, damped and undamped cases. In a damped or undamped case, one of the natural frequencies of the unperturbed equation may be a multiple of the other. Thus, the solution also covers the case of internal resonance which is an interesting and important part of non-linear oscillation. The determination of the solution is very simple and easier than the existing procedures developed by several authors (both in methods of averaging and multiple time scales) especially to tackle the case of internal resonance. The method is illustrated by an example of a fourth-order differential equation. The solution shows a good agreement with numerical solution in all of the three cases, e.g. over-damped, damped and undamped.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator

In this paper, the analytical dynamics of asymmetric periodic motions in the periodically forced, hardening Duffing oscillator is investigated via the generalized harmonic balance method. For the hardening Duffing oscillator, the symmetric periodic motions were extensively investigated with the aim of a good understanding of solutions with jumping phenomena. However, the asymmetric periodic motions for the hardening Duffing oscillators have not been obtained yet, and such asymmetric periodic motions are very important to find routes of periodic motions to chaos in the hardening Duffing oscillator analytically. Thus, the bifurcation trees from asymmetric period-1 motions to chaos are presented. The corresponding unstable periodic motions in the hardening Duffing oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out as well. This investigation provides a comprehensive understanding of chaos mechanism in the hardening Duffing oscillator.     

Fundamental and subharmonic resonances in a system with a ‘1-1’ internal resonance

The fundamental and subharmonic resonances of a two degree-of-freedom oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-seales averaging analysis. The system is in a 1–1 internal resonance, i.e., it has two equal linearized eigenfrequencies, and it possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies. The analytical results are verified by a numerical integration of the equations of motion, and a discussion of the effects of the mode bifurcation on the dynamics of the system is given.     

The construction of non-linear normal modes for systems with internal resonance

A numerical method, based on the invariant manifold approach, is presented for constructing non-linear normal modes for systems with internal resonances. In order to parameterize the non-linear normal modes of interest, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are then constrained to these ‘seed’, or master, variables, resulting in a system of non-linear partial differential equations that govern the constraint relationships, and these are solved numerically. The computationally-intensive solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two non-linear normal modes is constructed, resulting in a reduced order model that accurately captures the system dynamics. The methodology is then applied to a larger order system, specifically, an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the non-linear two-mode reduced order model is verified by comparing time-domain simulations of the two DOF model and the full system equations of motion.     

Periodically forced delay limit cycle oscillator

This paper investigates the dynamics of a delay limit cycle oscillator under periodic external forcing. The system exhibits quasiperiodic motion outside of a resonance region where it has periodic motion at the frequency of the forcer for strong enough forcing. By perturbation methods and bifurcation theory, we show that this resonance region is asymmetric in the frequency detuning, and that there are regions where stable periodic and quasiperiodic motions coexist.