Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities

We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic stiffness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes—NNMs), as well as, asynchronous periodic motions (elliptic orbits—EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets ‘captured’ in the neighborhood of a damped NNM before ‘escaping’ and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations confirm the analytical predictions.     

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.     

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.     

Investigation of a non-linear system under partially prescribed random excitation

The problem of non-linear systems excited by random forces with known power spectral density functions and unspecified probability structure is considered. Sufficient, but not necessary, conditions on the input under which the response can be a Gaussian process are investigated. The approach is illustrated by investigating the hardening spring cubic oscillator under wide and narrow band excitations. The non-Gaussian probability density of the input that leads to Gaussian response is determined.     

Attractors of harmonically forced linear oscillator with attached nonlinear energy sink. II: Optimization of a nonlinear vibration absorber

This paper is the second one in the series of two papers devoted to detailed investigation of the response regimes of a linear oscillator with attached nonlinear energy sink (NES) under harmonic external forcing and assessment of possible application of the NES for vibration absorption and mitigation. In this paper, we study the performance of a strongly nonlinear, damped vibration absorber with relatively small mass attached to a periodically excited linear oscillator. We present a nonlinear absorber tuning procedure in the vicinity of (1:1) resonance which provides the best total system energy suppression, using analytical and numerical tools. A linear absorber is also tuned according to the same criterion of total system energy suppression as the nonlinear one. Both optimally tuned absorbers are compared under common parameters of damping, external forcing but different absorber stiffness characteristics; certain cases for which nonlinear absorber is preferable over the linear one are revealed and confirmed numerically.     

Effects of excitation probability distribution on system responses

For a system subjected to a random excitation, the probability distribution of the excitation may affect behaviors of the system responses. Such effects are investigated for a variety of dynamical systems, including a linear oscillator, an oscillator of cubic non-linearity in both damping and stiffness, and a non-linear oscillator of the van der Pol type. The random excitations are assumed to be stationary stochastic processes, sharing the same spectral density, but with different probability distributions. Each excitation process is generated by passing a Brownian motion process through a non-linear filter, which is governed by an Ito stochastic differential equation. Monte Carlo simulations are carried out to obtain the transient and stationary properties of the system response in each case. It is shown that, under different excitations, the transient behaviors of the system response can be markedly different. The differences tend to reduce, however, as time of exposure to the excitations increases and the system reaches the stationary state.     

A Theoretical and Experimental Implementation of a Control Method Based on Saturation

A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

Resonant non-linear normal modes. Part II: activation/orthogonality conditions for shallow structural systems

The general conditions, obtained in Lacarbonara and Rega (Int. J. Non-linear Mech. (2002)), for orthogonality of the non-linear normal modes in the cases of two-to-one, three-to-one, and one-to-one internal resonances in undamped unforced one-dimensional systems with arbitrary linear, quadratic and cubic non-linearities are here investigated for a class of shallow symmetric structural systems. Non-linear orthogonality of the modes and activation of the associated interactions are clearly dual problems. It is known that an appropriate integer ratio between the frequencies of the modes of a spatially continuous system is a necessary but not sufficient condition for these modes to be non-linearly coupled. Actual activation/orthogonality of the modes requires the additional condition that the governing effective non-linear interaction coefficients in the normal forms be different/equal to zero. Herein, a detailed picture of activation/orthogonality of bimodal interactions in buckled beams, shallow arches, and suspended cables is presented.     

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.     

Trispectrum for the response of a non-linear oscillator

Simulation is used to obtain information about non-Gaussian aspects of the absolute response acceleration of a bi-linear hysteretic oscillator with an excitation which is Gaussian white noise. Attention is focused on the frequency content of the fourth cumulant of the response. This frequency content is studied by consideration of the trispectrum and also by the simplified technique of looking at the coefficient of excess for the response of a narrowband linear system mounted on the non-linear oscillator. Attempts are also made to model the non-Gaussian response of the non-linear oscillator by a filtered delta correlated (FDC) process, but it is shown that no process of this type can exhibit some of the significant characteristics of the non-linear response. In particular, the trispectrum of the non-linear response appears to be more narrowband than the power spectral density, and also it sometimes does not have the same sign at every point in the three-dimensional frequency space, and this behavior is distinctly different from that of any FDC process. Modifications of the FDC model are suggested in order to obtain improved approximations of the non-linear response.     

Interaction of elastic system with snap-through vibration absorber

Interaction of essentially non-linear vibration absorber (snap-through truss) with primary 1DOF linear oscillator is investigated. The study covers both free and forced (external excitation applied to the linear oscillator) responses of the system. The analysis is performed by complexification-averaging technique, describing a slow variation of amplitudes of principal harmonic components. Periodic and quasiperiodic oscillations are revealed and investigated. It turns out that for realistic parameters of the system vast majority of the responses, including those suitable for the vibration absorption, is quasiperiodic. The results are verified by means of a direct numeric simulation.     

Energy flux equalization: An open boundary condition for non-linear free surface flows

In the simulation of non-linear free surface flows in a finite domain a major concern is with the radiation condition to be applied at the ‘open’ boundary. No general theoretical radiation conditions are known to exist. In this paper a new open boundary condition is formulated based on energy flux equalization between the non-linear inner domain and a linear outer domain. The non-linear flow in the inner domain is solved numerically using a semi-Lagrangian procedure. The energy flux arriving at the open boundary is removed using a new open boundary condition which acts as a linear wave absorber. For the cases studied the performance of this boundary condition is found to be quite good.     

Energy Based Stochastic Estimation of Nonlinear Oscillators with Parametric Random Excitation

A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.     

Suppression of super-harmonic resonance response using a linear vibration absorber

Super-harmonic resonances may appear in the forced response of a weakly nonlinear oscillator having cubic nonlinearity, when the forcing frequency is approximately equal to one-third of the linearized natural frequency. Under super-harmonic resonance conditions, the frequency-response curve of the amplitude of the free-oscillation terms may exhibit saddle-node bifurcations, jump and hysteresis phenomena. A linear vibration absorber is used to suppress the super-harmonic resonance response of a cubically nonlinear oscillator with external excitation. The absorber can be considered as a small mass-spring-damper oscillator and thus does not adversely affect the dynamic performance of the nonlinear primary oscillator. It is shown that such a vibration absorber is effective in suppressing the super-harmonic resonance response and eliminating saddle-node bifurcations and jump phenomena of the nonlinear oscillator. Numerical examples are given to illustrate the effectiveness of the absorber in attenuating the super-harmonic resonance response.     

Dynamical behavior of a mechanical system including Saint-Venant component coupled to a non-linear energy sink

Multi-scale vibratory energy exchange between a main oscillator including Saint-Venant term and a cubic non-linear energy sink is studied. Analytically obtained invariant manifold of the system at a fast time scale and detected fixed points and/or fold singularities at a first slow time scale let us predict and explain different regimes that the system may face during the energy exchange process. The paper will be accompanied by some numerical results confirming our analytical predictions.     

Usefulness of passive non-linear energy sinks in controlling galloping vibrations

The suppression of vibration amplitudes of an elastically-mounted square prism subjected to galloping oscillations by using a non-linear energy sink is investigated. The non-linear energy sink consists of a secondary system with linear damping and non-linear stiffness. A representative model that couples the transverse displacement of the square prism and the non-linear energy sink is constructed. A linear analysis is performed to determine the impacts of the non-linear energy sink parameters (mass, damping, and stiffness) on the coupled frequency and onset speed of galloping. It is demonstrated that increasing the damping of the non-linear energy sink can result in a significant increase in the onset speed of galloping. Then, the normal form of the Hopf bifurcation is derived to identify the type of instability and to determine the effects of the non-linear energy sink stiffness on the performance of the aeroelastic system near the bifurcation. The results show that the non-linear energy sink can be efficiently implemented to significantly reduce the galloping amplitude of the square prism. It is also shown that the multiple stable responses of the coupled aeroelastic system are obtained as well as the periodic responses, which are dependent on the considered non-linear energy sink parameters.     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.