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.     

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

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

Numerical and experimental investigations of a rotating nonlinear energy sink

In this work, passive nonlinear targeted energy transfer (TET) is addressed by numerically and experimentally investigating a lightweight rotating nonlinear energy sink (NES) which is coupled to a primary two-degree-of-freedom linear oscillator through an essentially nonlinear (i.e., non-linearizable) inertial nonlinearity. It is found that the rotating NES passively absorbs and rapidly dissipates a considerable portion of impulse energy initially induced in the primary oscillator. The parameters of the rotating NES are optimized numerically for optimal performance under intermediate and strong loads. The fundamental mechanism for effective TET to the NES is the excitation of its rotational nonlinear mode, since its oscillatory mode dissipates far less energy. This involves a highly energetic and intense resonance capture of the transient nonlinear dynamics at the lowest modal frequency of the primary system; this is studied in detail by constructing an appropriate frequency–energy plot. A series of experimental tests is then performed to validate the theoretical predictions. Based on the obtained numerical and experimental results, the performance of the rotating NES is found to be comparable to other current translational NES designs; however, the proposed rotating device is less complicated and more compact than current types of NESs.     

Vibration absorption of parallel-coupled nonlinear energy sink under shock and harmonic excitations

Nonlinear energy sink(NES) can passively absorb broadband energy from primary oscillators. Proper multiple NESs connected in parallel exhibit superior performance to single-degree-of-freedom(SDOF) NESs. In this work, a linear coupling spring is installed between two parallel NESs so as to expand the application scope of such vibration absorbers. The vibration absorption of the parallel and parallel-coupled NESs and the system response induced by the coupling spring are studied. The results show that the responses of the system exhibit a significant difference when the heavier cubic oscillators in the NESs have lower stiffness and the lighter cubic oscillators have higher stiffness. Moreover, the efficiency of the parallel-coupled NES is higher for medium shocks but lower for small and large shocks than that of the parallel NESs. The parallel-coupled NES also shows superior performance for medium harmonic excitations until higher response branches are induced. The performance of the parallel-coupled NES and the SDOF NES is compared. It is found that, regardless of the chosen SDOF NES parameters, the performance of the parallel-coupled NES is similar or superior to that of the SDOF NES in the entire force range.     

Phase locking control in the Circle Map

The phase-locking between two oscillators occurs when the ratio of their frequencies becomes locked in a ratio p/q of integer numbers over some finite domain of parameters values. Due to it, oscillators with some kind of nonlinear coupling may synchronize for certain set of parameters. This phenomenon can be better understood and studied with the use of a well-known paradigm, the Circle Map, and the definition of the winding number. Two diagrams related to this map are especially useful: the ‘Arnold tongues’ and the ‘devil’s staircase’. The synchronization that occurs in this map is described by the ‘Farey Series’. This property is the starting point for the development of control algorithms capable of locking the system under the action of an external excitation into a desired winding number. In this work, we discuss the main characteristics of the phase-locking phenomenon and consider three control algorithms designed to drive and keep the Circle Map into a desired winding number.     

Steady State Passive Nonlinear Energy Pumping in Coupled Oscillators: Theoretical and Experimental Results

We study theoretically and experimentally the effect that anonlinear energy sink (NES) has on the steady state dynamics of a weaklycoupled system. The NES possesses essentially nonlinear(nonlinearizable) stiffness nonlinearity of the third degree. We findthat, in contrast to the classical linear vibration absorber, the NES iscapable of absorbing steady state vibration energy from the linearoscillator over a relatively broad frequency range. This results inlocalization of the steady state vibration in the NES, away from thedirectly forced subsystem. For a forward frequency sweep the localizedbranch of steady state motions is suddenly eliminated by a jump to alinearized low-amplitude motion, whereas, for a backward frequency sweepa reverse jump occurs. The difference in the frequencies of the twojumps introduces a nonlinear hysteresis loop. This work extends to thesteady state case of earlier transient passive energy pumping results.The notion of passively transferring vibration energy to an a prioridetermined NES, weakly attached to a main structure, is novel. The useof nonlinear energy sinks for passively absorbing energy from a linearmain structure can form the basis of relatively simple and modularvibration and shock isolation designs of mechanical systems.     

Seismic base isolation by nonlinear mode localization

Summary In this paper, the performance of a nonlinear base-isolation system, comprised of a nonlinearly sprung subfoundation tuned in a 1∶1 internal resonance to a flexible mode of the linear primary structure to be isolated, is examined. The application of nonlinear localization to seismic isolation distinguishes this study from other base-isolation studies in the literature. Under the condition of third-order smooth stiffness nonlinearity, it is shown that a localized nonlinear normal mode (NNM) is induced in the system, which confines energy to the subfoundation and away from the primary or main structure. This is followed by a numerical analysis wherein the smooth nonlinearity is replaced by clearance nonlinearity, and the system is excited by ground motions representing near-field seismic events. The performance of the nonlinear system is compared with that of the corresponding linear system through simulation, and the sensitivity of the isolation system to several design parameters is analyzed. These simulations confirm the existence of the localized NNM, and show that the introduction of simple clearance nonlinearity significantly reduces the seismic energy transmitted to the main structure, resulting in significant attenuation in the response. This work was supported in part by the National Science Foundation Grant CMS 00-00060. The authors are grateful for this support.     

An analysis of subgrid-resolved scale interactions with use of results from direct numerical simulations

Subgrid nonlinear interaction and energy transfer are analyzed using direct numerical simulations of isotropic turbulence. Influences of cutoff wave number at different ranges of scale on the energetics and dynamics have been investigated. It is observed that subgrid-subgrid interaction dominates the turbulent dynamics when cut-off wave number locates in the energy-containing range while resolved-subgrid interaction dominates if it is in the dissipation range. By decomposing the subgrid energy transfer and nonlinear interaction into ‘forward’ and ‘backward’ groups according to the sign of triadic interaction, we find that individually each group has very large contribution, but the net of them is much smaller, implying that tremendous cancellation happens between these two groups.     

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     

Embedded-strain-sensor development for composite smart structures

The winding or layup procedure for fiber-reinforced composites lends itself to convenient installation of embedded sensors during fabrication. These permanently installed and protected sensors could be used during the service lifetime of the structure to monitor real-time conditions and determine when loading or vibration is excessive, and when damage has occurred. Such ‘smart or intelligent’ structures could be used to provide continuous ‘health monitoring’ of the structure as well as provide input for active vibration control. In the present study, two sizes of constantan wire (0.15-mm and 0.025-mm diameter) with a very thin but tough polyimide insulation were embedded in graphite-epoxy bars and tubes. The 25-mm by 2.5-mm by approximately 300-mm long bars were fabricated from hand-laid-up panels and subjected to static four-point bending and cantilever bending. The tubes (42-mm diameter by 1.25-m long) were subjected to static cantilever bending. Output from the constantan wire was monitored with conventional strain-gage indicators. Results indicate accurate tensile and compressive measurements of the integrated strain along the length of the constantan wire when compared with beam formulas and surface mounted strain gages. The constantan strain wire shows promise as an embedded sensor for ‘smart structures’.     

Takagi–Sugeno fuzzy generalized predictive control for a class of nonlinear systems

The new generations of compact high output power-to-weight ratio internal combustion engines generate broadband torsional oscillations, transmitted to lightly damped drivetrain systems. A novel approach to mitigate these untoward vibrations can be the use of nonlinear absorbers. These act as Nonlinear Energy Sinks (NESs). The NES is coupled to the primary (drivetrain) structure, inducing passive irreversible targeted energy transfer (TET) from the drivetrain system to the NES. During this process, the vibration energy is directed from the lower-frequency modes of the structure to the higher ones. Thereafter, vibrations can be either dissipated through structural damping or consumed by the NES. This paper uses a lumped parameter model of an automotive driveline to simulate the effect of TET and the assumed modal energy redistribution. Significant redistribution of vibratory energy is observed through TET. Furthermore, the integrated optimization process highlights the most effective configuration and parametric evaluation for use of NES.     

Targeted energy transfer between 2-D wing and nonlinear energy sinks and their dynamic behaviors

Nonlinear Dynamics - The flow-induced vibration of two-dimensional wing coupled with two nonlinear energy sinks (NESs) under freestream is studied by numerical methods, and the relationship between...     

An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells

The two-dimensional equations of a nonlinearly elastic ‘flexural’ shell have been recently identified and justified by V. Lods and B. Miara, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. These equations can be recast as a minimization problem for a ‘two-dimensional energy’ over a manifold of ‘admissible deformations’. The stored energy function is a quadratic expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one; the admissible deformations are those that preserve the metric of the undeformed middle surface and satisfy boundary conditions of clamping or of simple support. We establish here that this minimization problem has at least one solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Asymptotic expansions by Γ-convergence

Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.      

On a Generalized Nonlinear K–ε Model and the Use of Extended Thermodynamics in Turbulence

A resent extension of the nonlinear K–ε model is critically discussed from a basic theoretical standpoint. While it was said in the paper that this model was formulated to incorporate relaxation effects, it will be shown that the model is incapable of describing one of the most basic such turbulent flows as is obvious but is described for clarity. It will be shown in detail that this generalized nonlinear K–ε model yields erroneous results for the Reynolds stress tensor when the mean strains are set to zero in a turbulent flow – the return-to-isotropy problem which is one of the most elementary relaxational turbulent flows. It is clear that K–ε type models cannot describe relaxation effects. While their general formalism can describe relaxation effects, the nonlinear K–ε model – which the paper is centered on – cannot. The deviatoric part of the Reynolds stress tensor is predicted to be zero when it actually only gradually relaxes to zero. Since this model was formulated by using the extended thermodynamics, it too will be critically assessed. It will be argued that there is an unsubstantial physical basis for the use of extended thermodynamics in turbulence. The role of Material Frame-Indifference and the implications for future research in turbulence modeling are also discussed. Received 19 February 1998 and accepted 23 October 1998     

Invertibility and Non-Invertibility in Thin Elastic Structures

The nonlinear elastic energy of a thin film of thickness h is given by a functional E ^h . Friesecke, James and Müller derived the Γ-limits, as h → 0, of the functionals h ^−α E ^h for α ≧ 3. In this article we study the invertibility properties of almost minimizers of these functionals, and more generally of sequences with equiintegrable energy density. We show that they are invertible almost everywhere away from a thin boundary layer near the film surface. Moreover, we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages. We construct examples showing that these bounds are sharp. In particular, for all α ≧ 3 there exist Lipschitz continuous low energy deformations which are not locally invertible.     

Approximate MPA-based method for performing incremental dynamic analysis

In performance based earthquake engineering, it is important to accurately predict the seismic demand and capacities of structures. One recent estimation method is incremental dynamic analysis (IDA), which requires a series of nonlinear response history analyses (RHA) of the structure under various ground motions, each scaled to multiple levels of intensity, selected to cover entire range of structural response from elasticity, to yield and finally global dynamic instability. The implementation of IDA requires intensive computation and detailed knowledge of the nonlinear RHA of structures. In response to the complexity of IDA, an approximate method based on modal pushover analysis (MPA-based IDA) was developed. In MPA-based IDA, seismic demands are computed using the nonlinear RHA of the equivalent SDF systems instead of using nonlinear RHA of MDF systems. The objective of this study is to develop a simpler MPA-based IDA procedure that can avoid nonlinear RHA of equivalent SDF systems. For this purpose, MPA-based IDA employs the empirical equation of the inelastic displacement ratio (C _R ), defined as the ratio of peak displacement of the inelastic SDF system to that of the corresponding elastic SDF system given the strength ratio R, and that of the collapse strength ratio (R _c), which is the ratio of collapse intensity to yield strength. The proposed procedure is verified by comparing the seismic demands and capacities of 6-, 9-, and 20-story steel moment frames as determined by the proposed method and exact IDA.     

Vibration suppression and higher branch responses of beam with parallel nonlinear energy sinks

The effects of nonlinear energy sink (NES) on vibration suppression of a simply supported beam are investigated in this work. The slow flow equations of the system are derived by using complexification–averaging method, and the validity of the derivation is verified. By comparing the vibration absorption of single and parallel NESs of equal mass, it is found that the latter exhibits superior vibration absorption performance. In addition, the parallel NES can eliminate higher branch responses of the system under the harmonic load. Furthermore, it is found that parallel NES can eliminate the higher branches of the system more effectively by tuning nonlinear stiffness and damping. Moreover, the thermal effect on natural frequencies of the simply supported beam is considered, and the influences of the parallel NES’s parameters on the energy dissipation rate under shock load are investigated. The nonlinear responses of the simply supported beam with parallel NES under harmonic load and with the increase of temperature are described.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

Creep—Slip Capture as a Possible Source of Squeal During Decelerated Sliding

Friction-induced vibration of a two-degree-of-freedom mass-damper-spring system interacting with a decelerating rigid strip is investigated. The friction law is approximated by an analytical function to facilitate the analyses and numerical integrations. It is shown that, after a quasi-harmonic transient period, accompanied by viscous energy dissipation, a short period of intensive ‘creep-slip’ vibration occurs, which generates a series of ‘micro-impacts’ on the strip. Because of the impulsive character of such kind of loading, its Fourier spectrum is rich and quite broadband. Using an averaging technique, the ‘normal form’ equations of motion show that the out-of-phase vibration mode absorbs more energy from the decelerating strip when its natural frequency satisfies certain resonance conditions. The study is then applied to an automotive disc brake model to gain useful insight into the generation of squeal. It is shown that the out-of-phase creep-slip vibration (in the longitudinal direction) of the brake pads generates an impulsive bending moment on the decelerating strip (disc rotor). This impulsive load may be considered as a possible source for brake squeal. The technique developed in this paper may be extended to other ‘squealing systems’ including models for geophysical faults (earthquakes).