54636408
Transition scenario to turbulence in thin vibrating plates
A thin plate, excited by a harmonic external forcing of increasing amplitude, shows transitions from a periodic response to a chaotic state of wave turbulence. By analogy with the transition to turbulence observed in fluid mechanics as the Reynolds number is increased, a generic transition scenario for thin vibrating plates, first experimentally observed, is here numerically studied. The von Karman equations for thin plates, which include geometric non-linear effects, are used to model large amplitude vibrations, and an energy-conserving finite difference scheme is employed for discretisation. The transition scenario involves two bifurcations separating three distinct regimes. The first regime is the periodic, weakly non-linear response. The second is a quasiperiodic state where energy is exchanged between internally resonant modes. It is observed only when specific internal resonance relationships are fulfilled between the eigenfrequencies of the structure and the forcing frequency; otherwise a direct transition to the last turbulent state is observed. This third, or turbulent, regime is characterized by a broadband Fourier spectrum and a cascade of energy from large to small wavelengths. For perfect plates including cubic non-linearity, only third-order internal resonances are likely to exist. For imperfect plates displaying quadratic nonlinearity, the energy exchanges and the quasiperiodic states are favored and thus are more easily obtained. Finally, the turbulent regime is characterized in the light of available theoretical results from wave turbulence theory.
120882372
Dissipative Structures and Weak Turbulence
We present a brief overview of the current understanding of temporal and spatio-temporal chaos, both termed weak turbulence according to the context [1]. The process which allows one to reduce the primitive problem to a low-dimensional dynamical system is discussed. It turns out to be appropriate as long as confinement effects are sufficiently strong to freeze the space dependence of unstable modes, hence temporal chaos only. Otherwise modulated patterns arise, yielding genuine space-time chaos. The corresponding theory rests on envelope equations providing a useful framework for weak turbulence in a globally super-critical setting. spatio-temporal intermittency analyzed next is the relevant scenario in the sub-critical case. Finally, the connection with hydrodynamic turbulence and the more general relevance of some of the ideas developed here are examined.
52036028
Wave turbulence and intermittency
In the early 1960s, it was established that the stochastic initial value problem for weakly coupled wave systems has a natural asymptotic closure induced by the dispersive properties of the waves and the large separation of linear and nonlinear time scales. One is thereby led to kinetic equations for the redistribution of spectral densities via three- and four-wave resonances together with a nonlinear renormalization of the frequency. The kinetic equations have equilibrium solutions which are much richer than the familiar thermodynamic, Fermi–Dirac or Bose–Einstein spectra and admit in addition finite flux (Kolmogorov–Zakharov) solutions which describe the transfer of conserved densities (e.g. energy) between sources and sinks. There is much one can learn from the kinetic equations about the behavior of particular systems of interest including insights in connection with the phenomenon of intermittency. What we would like to convince you is that what we call weak or wave turbulence is every bit as rich as the macho turbulence of 3D hydrodynamics at high Reynolds numbers and, moreover, is analytically more tractable. It is an excellent paradigm for the study of many-body Hamiltonian systems which are driven far from equilibrium by the presence of external forcing and damping. In almost all cases, it contains within its solutions behavior which invalidates the premises on which the theory is based in some spectral range. We give some new results concerning the dynamic breakdown of the weak turbulence description and discuss the fully nonlinear and intermittent behavior which follows. These results may also be important for proving or disproving the global existence of solutions for the underlying partial differential equations. Wave turbulence is a subject to which many have made important contributions. But no contributions have been more fundamental than those of Volodja Zakharov whose 60th birthday we celebrate at this meeting. He was the first to appreciate that the kinetic equations admit a far richer class of solutions than the fluxless thermodynamic solutions of equilibrium systems and to realize the central roles that finite flux solutions play in non-equilibrium systems. It is appropriate, therefore, that we call these Kolmogorov–Zakharov (KZ) spectra. © 2001 Elsevier Science B.V. All rights reserved.
23823161
Turbulence of capillary waves.
A numerical model for direct simulation of the surface of ideal ﬂuid based on the expansion of theHamiltonian of the surface up to terms of fourth order is developed. For the case of capillary wavewe observe the formation of powerlike spectrum of spatial elevations close to one predicted by weak-turbulent theory
12419937
Weak turbulent Kolmogorov spectrum for surface gravity waves.
We study the long-time evolution of surface gravity waves on deep water excited by a stochastic external force concentrated in moderately small wave numbers. We numerically implemented the primitive Euler equations for the potential flow of an ideal fluid with free surface written in Hamiltonian canonical variables, using the expansion of the Hamiltonian in powers of nonlinearity of terms up to fourth order. We show that because of nonlinear interaction processes a stationary Fourier spectrum of a surface elevation close to <|eta(k)|(2)> approximately k(-7/2) is formed. The observed spectrum can be interpreted as a weak-turbulent Kolmogorov spectrum for a direct cascade of energy.
12849511
Observation of gravity-capillary wave turbulence.
We report the observation of the crossover between gravity and capillary wave turbulence on the surface of mercury. The probability density functions of the turbulent wave height are found to be asymmetric and thus non-Gaussian. The surface wave height displays power-law spectra in both regimes. In the capillary region, the exponent is in fair agreement with weak turbulence theory. In the gravity region, it depends on the forcing parameters. This can be related to the finite size of the container. In addition, the scaling of those spectra with the mean energy flux is found in disagreement with weak turbulence theory for both regimes.
49266088
Weak Langmuir turbulence
Abstract Weak turbulence theory presents a regular method for a statistical description of nonlinear wave interactions. The present review deals with an application of weak turbulence theory to Langmuir wave turbulence. Our main attention is devoted to a plasma with comparable ion and electron temperatures, both magnetized and unmagnetized. In this practically important situation ion-sound motions are heavily damped, which simplifies the physics of nonlinear phenomena. We will demonstrate that the turbulence spectra are highly anisotropic and take the form of “jets” in k-space, and that the onset of a steady state is nontrivial and sometimes does not occur at all. On the base of the jet-like spectra approach it is possible to find the turbulence spectra, to evaluate the anomalous absorption rate and to determine the comparable role of the different absorption mechanisms for a number of practical problems: the excitation of waves by powerful electromagnetic radiation or by electron and ion beams. We demonstrate also that the range in which pure weak turbulence is valid is pretty narrow. The jet-like spectra structure stimulates a modulation instability and after that wave self-focusing and collapse. Then, weak and strong turbulence coexist. The final part of the review deals with the turbulence of nonisothermal plasmas when additional degrees of freedom are excited. We demonstrate that the ideas, models and methods, presented in this review, give us a chance to advance greatly in the understanding of turbulence patterns.
122504946
Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schro¨dinger equation
The nonlinear Schrodinger (NLS) equation iΨt + ∇2Ψ + α⋎Ψ⋎sΨ = 0 is a canonical and universal equation which is of major importance in continuum mechanics, plasma physics and optics. This paper argues that much of the observed solution behavior in the critical case sd = 4, where d is dimension and s is the order of nonlinearity, can be understood in terms of a combination of weak turbulence theory and condensate and collapse formation. The results are derived in the broad context of a class of Hamiltonian systems of which NLS is a member, so that the reader can gain a perspective on the ingredients important for the realization of the various equilibrium spectra, thermodynamic, pure Kolmogorov and combinations thereof. We also present time-dependent, self-similar solutions which describe the relaxation of the system towards these equilibrium states. We show that the number of particles lost in an individual collapse event is virtually independent of damping. Our numerical simulation of the full governing equations is the first to show the validity of the weak turbulence approximation. We also present a mechanism for intermittency which should have widespread application. It is caused by strongly nonlinear collapse events which are nucleated by a flow of particles towards the origin in wavenumber space. These highly organized events result in a cascade of particle number towards high wavenumbers and give rise to an intermittency and a behavior which violates many of the usual Kolmogorov assumptions about the loss of statistical information and the statistical independence of large and small scales. We discuss the relevance of these ideas to hydrodynamic turbulence in the conclusion.
55425956
Non-local MHD turbulence
We consider an example of strongly non-local interaction in incompressible magnetohydrodynamic (MHD) turbulence which corresponds to the case where the Alfven waves travelling in the opposite directions have essentially different characteristic wavelengths. We use two approaches to the dynamics of turbulent Alfvenic wavepackets: the first is a geometrical WKB theory [Phys. Lett. A 165 (1992) 330] and the second one is a three-wave kinetic equation derived for weakly turbulent waves [J. Plasma Phys., in press]. We show that these theories have a common limit of weak turbulence with scale separation in which they both predict the same Fokker–Planck equation for the wave power spectrum. In both cases the packet wavenumbers (and therefore the Lagrangian field-line separations) are allowed to experience order 1 changes. The WKB theory developed here formalises an intuitive geometrical argument of Goldreich and Sridhar [ApJ 485 (1997) 680] and allows one to see where such an intuition leads to a wrong conclusion about the inapplicability of the three-wave kinetic equation for order 1 wavepacket distortions. We show that the exponent of the constant flux non-local spectrum matches the value previously found for local turbulence at the boundary of the locality interval. The relationship between the WKB theory and the weak turbulence theory found in this paper for an ensemble of Alfven waves seems to be general for three-wave systems. © 2001 Elsevier Science B.V. All rights reserved.
73618530
Lyapunov exponents from experimental time series. Application to cymbal vibrations
Lyapunov exponents are among the most relevant and most informative invariants for detecting and quantifying chaos in a dynamical system. This method is applied here to the analysis of cymbal vibrations. The advantage of using a quadratic fit for determining the Jacobian of the dynamics is presented. In addition, the interest of using a time step for the evolution of the neighbourhood not equal to the timelag used for the reconstruction of the phase space is underlined. The robustness of the algorithm used yields a high degree of confidence in the characterization and in the quantification of the chaotic state. 
To illustrate these features in the case of cymbal vibrations, transitions from quasiperiodicity to chaos are exhibited. The quasiperiodic state of the system is characterized together by the power spectrum of the experimental signal and by calculation of the Lyapunov spectrum.
121292656
Nonlinear vibrations and chaos in gongs and cymbals
This paper summarizes some results obtained in the last few years for the modeling of nonlinear vibrating instruments such as gongs and cymbals. Linear, weakly nonlinear and chaotic regimes are successively examined. A theoretical mechanical model is presented, based on the nonlinear von Karman equations for thin shallow spherical shells. Modal projection and Nonlinear Normal Mode (NNM) formulation leads to a subset of coupled nonlinear oscillators. Current developments are aimed at using this subset for sound synthesis purpose.
59882402
Numerical Sound Synthesis: Finite Difference Schemes and Simulation in Musical Acoustics
Digital sound synthesis has long been approached using standard digital filtering techniques. Newer synthesis strategies, however, make use of physical descriptions of musical instruments, and allow for much more realistic and complex sound production and thereby synthesis becomes a problem of simulation. This book has a special focus on time domain finite difference methods presented within an audio framework. It covers time series and difference operators, and basic tools for the construction and analysis of finite difference schemes, including frequency-domain and energy-based methods, with special attention paid to problems inherent to sound synthesis. Various basic lumped systems and excitation mechanisms are covered, followed by a look at the 1D wave equation, linear bar and string vibration, acoustic tube modelling, and linear membrane and plate vibration. Various advanced topics, such as the nonlinear vibration of strings and plates, are given an elaborate treatment. Key features: Includes a historical overview of digital sound synthesis techniques, highlighting the links between the various physical modelling methodologies. A pedagogical presentation containing over 150 problems and programming exercises, and numerous figures and diagrams, and code fragments in the MATLAB programming language helps the reader with limited experience of numerical methods reach an understanding of this subject. Offers a complete treatment of all of the major families of musical instruments, including certain audio effects. Numerical Sound Synthesis is suitable for audio and software engineers, and researchers in digital audio, sound synthesis and more general musical acoustics. Graduate students in electrical engineering, mechanical engineering or computer science, working on the more technical side of digital audio and sound synthesis, will also find this book of interest.
45802670
Nonlinearity, chaos, and the sound of shallow gongs
Experimental studies on several orchestral gongs of the tamtam and cymbal families suggest that two separate nonlinear mechanisms contribute to the evolution of the sound. The first mechanism is an upward cascade of energy from the low‐frequency modes initially excited into high‐frequency modes, caused by coupling between tension and shear stresses at regions of sharp change in shape of the gong. The second is a transition from simple periodic nonlinear modal motion to multiple fractional subharmonics, or even chaotic motion, which fills out the radiated spectrum at frequencies between those of the normal linear modes. Each of these mechanisms has considerable hysteresis, so that the spectrum of the radiated sound evolves over a period of several seconds. Measurements using high‐level sinusoidal excitation have elucidated some of the features of this behavior.
2364674
Spatio-Temporal Chaos and solitons Exhibited by von kÁrmÁn Model
Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so-called von Karman form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h4) and 0(h2) (in respect to the spatial coordinates). The ODEs are solved using the Runge–Kutta fourth order method, whereas the AEs are solved using either the Gauss or relaxation methods. The analysis and identification of spatio-temporal oscillations are carried out by investigation of the series wij(t), wt,ij(t), phase portraits wt,ij (wij) and wtt,ij(wt,ij, wij) and the mode portraits in the planes wx,ij(wij), wy,ij (wij) and in the space wxx(wx,ij,wij), FFT as well as the Poincare sections and pseudo-sections.
120786918
Chaotic Vibrations of Sector-Type Spherical Shells
In this work, chaotic vibrations of shallow sector-type spherical shells are studied. A sector-type shallow shell is understood as a shell defined by a sector with associated boundary conditions and obtained by cutting a spherical shell for a given angle θ k , or it is a sector of a shallow spherical cap associated with the mentioned angle. Both static stability and complex nonlinear dynamics of the mentioned mechanical objects subjected to transversal uniformly distributed sign-changeable load are analyzed, and the so-called vibration charts and scales regarding the chosen control parameters are reported. In particular, scenarios of transition from regular to chaotic dynamics of the mentioned shells are investigated. A novel method to control chaotic dynamics of the studied flexible spherical shells driven by transversal sign-changeable load via synchronized action of the sign-changeable antitorque is proposed and applied. All investigations are carried out within the fields of qualitative theory of differential equations and nonlinear dynamics.
122080519
Chaotic oscillations of a shallow cylindrical shell with a concentrated mass under periodic excitation
This paper presents effects of a concentrated mass on chaotic oscillations of a shallow cylindrical shell under gravity and periodic acceleration. The rectangular shell is simply supported and is elastically in-plane constrained. Assuming mode functions, the Donnell equation with inertia force is reduced to non-linear coupled differential equations by the Galerkin method. The chaotic response is calculated numerically and is examined by the maximum Lyapunov exponent. Dominant chaotic responses are generated within restricted frequency regions of sub-harmonic resonance of 1/2 order. As the concentrated mass increases, the chaotic response is shifted to the lower frequency region. The increment of the concentrated mass decreases the maximum Lyapunov exponent.
122168436
Experiments and analysis on chaotic vibrations of a shallow cylindrical shell-panel
Abstract Detailed experimental results and analytical results are presented on chaotic vibrations of a shallow cylindrical shell-panel subjected to gravity and periodic excitation. The shallow shell-panel with square boundary is simply supported for deflection. In-plane displacement at the boundary is elastically constrained by in-plain springs. In the experiment, the cylindrical shallow shell-panel with thickness 0.24 mm, square form of length 140 mm and mean radius 5150 mm is used for the test specimen. All edges around the shell boundary are simply supported by adhesive flexible films. First, to find fundamental properties of the shell-panel, linear natural frequencies and characteristics of restoring force of the shell-panel are measured. These results are compared with the relevant analytical results. Then, geometrical parameters of the shell-panel are identified. Exciting the shell-panel with lateral periodic acceleration, nonlinear frequency responses of the shell-panel are obtained by sweeping the frequency of periodic acceleration. In typical ranges of the exciting frequency, predominant chaotic responses are generated. Time histories of the responses are recorded for inspection of the chaos. In the analysis, the Donnell equation with lateral inertia force is introduced. Assuming mode functions, the governing equation is reduced to a set of nonlinear ordinary differential equations by the Galerkin procedure. Periodic responses are calculated by the harmonic balance method. Chaotic responses are integrated numerically by the Runge–Kutta–Gill method. The chaotic responses, which are obtained by the experiment and the analysis, are inspected with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the Lyapunov dimension. It is found that the dominant chaotic responses of the shell-panel are generated from the responses of the sub-harmonic resonance of 1 2 order and of the ultra-sub-harmonic resonance of 2 3 order. By the convergence of the maximum Lyapunov exponent to the embedding dimension, the number of predominant vibration modes which contribute to the chaos is found to be three or four. Fairly good agreements are obtained between the experimental results and the analytical results.
123329958
Modal interaction in chaotic vibrations of a shallow double-curved shell-panel
Abstract Experimental results and analytical results are presented on chaotic vibrations of a shallow double-curved shell-panel subjected to gravity and periodic excitation. Modal interactions in the chaotic responses are discussed. The shell-panel with square boundary is simply supported for deflection. In-plane displacement at the boundary is elastically constrained. In the experiment, time histories of the chaotic responses at the spatial multiple positions of the shell-panel are measured for the inspection of modal interaction. In the analysis, the shallow shell-panel is assumed to have constant curvatures along to orthogonal directions and geometric initial imperfection. The Donnell–Mushtari–Vlasov type equation is used as governing equation with lateral inertia force. Assuming deflection with multiple modes of vibration, the governing equation is reduced to a set of nonlinear ordinary differential equations by the Bubnov–Galerkin procedure. Chaotic responses are integrated numerically. The chaotic responses, which are obtained by the experiment and the analysis, are inspected with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the Lyapunov dimension. Contribution of modes of vibration to the chaotic responses is analyzed by the principal component analysis, i.e., Karhunen–Loeve transformation.
3092630
Characterizing the Dynamic Response of a Thermally Loaded, Acoustically Excited Plate
Abstract In this work the dynamic response is considered of a homogeneous, fully clamped rectangular plate subject to spatially uniform thermal loads and narrow-band acoustic excitation. In both the pre-and post-buckled regimes, the small amplitude, linear response is confirmed. However, the primary focus is on the large amplitude, non-linear, snap-through response, because of the obvious implications for fatigue in aircraft components. A theoretical model is developed which uses nine spatial modes and incorporates initial imperfections and non-ideal boundary conditions. Because of the higher order nature of this model, it is inherently more complicated than a one-mode buckled beam equation (Duffing's equation). An experimental system was developed to complement the theoretical results, and also to measure certain system parameters for the model which are not available theoretically. Several analysis techniques are used to characterize the response. These include time series, power spectra and autocorrelation functions. In addition, the fractal dimension and Lyapunov exponents for the response are computed to address the issue of spatial dimension and temporal complexity (chaos), respectively. Comparisons between theory and experiment are made and show considerable agreement. However, these comparisons also serve to point out difficulties in computing the fractal dimension and Lyapunov exponents from experimental data.
120763886
Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates
The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Karman equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincare maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.
9924414
Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum?
We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at long times with a slow transfer of energy from one mode to another. We derive a kinetic equation for the spectral transfer in terms of the second order moment. We show that such a theory describes the approach to an equilibrium wave spectrum and represents also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations that confirm this scenario.
375112
Observation of wave turbulence in vibrating plates.
The nonlinear interaction of waves in a driven medium may lead to wave turbulence, a state such that energy is transferred from large to small length scales. Here, wave turbulence is observed in experiments on a vibrating plate. The frequency power spectra of the normal velocity of the plate may be rescaled on a single curve, with power-law behaviors that are incompatible with the weak turbulence theory of Düring et al. [Phys. Rev. Lett. 97, 025503 (2006)10.1103/PhysRevLett.97.025503]. Alternative scenarios are suggested to account for this discrepancy -- in particular the occurrence of wave breaking at high frequencies. Finally, the statistics of velocity increments do not display an intermittent behavior.
16618456
Statistics of power injection in a plate set into chaotic vibration
A vibrating plate is set into a chaotic state of wave turbulence by either a periodic or a random local forcing. Correlations between the forcing and the local velocity response of the plate at the forcing point are studied. Statistical models with fairly good agreement with the experiments are proposed for each forcing. Both distributions of injected power have a logarithmic cusp for zero power, while the tails are Gaussian for the periodic driving and exponential for the random one. The distributions of injected work over long time intervals are investigated in the framework of the fluctuation theorem, also known as the Gallavotti-Cohen theorem. It appears that the conclusions of the theorem are verified only for the periodic, deterministic forcing. Using independent estimates of the phase space contraction, this result is discussed in the light of available theoretical framework.
18590088
Are there waves in elastic wave turbulence?
An thin elastic steel plate is excited with a vibrator and its local velocity displays a turbulentlike Fourier spectrum. This system is believed to develop elastic wave turbulence. We analyze here the motion of the plate with a two-point measurement in order to check, in our real system, a few hypotheses required for the Zakharov theory of weak turbulence to apply. We show that the motion of the plate is indeed a superposition of bending waves following the theoretical dispersion relation of the linear wave equation. The nonlinearities seem to efficiently break the coherence of the waves so that no modal structure is observed. Several hypotheses of the weak turbulence theory seem to be verified, but nevertheless the theoretical predictions for the wave spectrum are not verified experimentally.
123088324
Fourier analysis of wave turbulence in a thin elastic plate
The spatio-temporal dynamics of the deformation of a vibrated plate is measured by a high speed Fourier transform profilometry technique. The space-time Fourier spectrum is analyzed. It displays a behavior consistent with the premises of the Weak Turbulence theory. A isotropic continuous spectrum of waves is excited with a non linear dispersion relation slightly shifted from the linear dispersion relation. The spectral width of the dispersion relation is also measured. The non linearity of this system is weak as expected from the theory. Finite size effects are discussed. Despite a qualitative agreement with the theory, a quantitative mismatch is observed which origin may be due to the dissipation that ultimately absorbs the energy flux of the Kolmogorov-Zakharov casade.
122825779
A family of conservative finite difference schemes for the dynamical von Karman plate equations
Numerical methods for nonlinear plate dynamics play an important role across many disciplines. In this article, the focus is on numerical stability for numerical methods for the von Karman system, through the use of energy-conserving methods. It is shown that one may take advantage of structure particular to the system to construct numerical methods, which are provably numerically stable, and for which computer implementation is simplified. Numerical results are presented. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007
54712898
Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: Experiments
This article is devoted to an experimental validation of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced vibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the vibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driving frequency are compared to the theoretical ones, leading to a quantitative validation of the predictions given by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relevance of the approach. © 2003 Elsevier Science Ltd. All rights reserved.
122717335
Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance
This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.
5068330
Non-linear vibrations of imperfect free-edge circular plates and shells
Large-amplitude, geometrically non-linear vibrations of free-edge circular plates with geometric imperfections are addressed in this work. The dynamic analog of the von Karman equations for thin plates, with a stress-free initial deflection, is used to derive the imperfect plate equations of motion. An expansion onto the eigenmode basis of the perfect plate allows discretization of the equations of motion. The associated non-linear coupling coefficients for the imperfect plate with an arbitrary shape are analytically expressed as functions of the cubic coefficients of a perfect plate. The convergence of the numerical solutions are systematically addressed by comparisons with other models obtained for specific imperfections, showing that the method is accurate to handle shallow shells, which can be viewed as imperfect plate. Finally, comparisons with a real shell are shown, showing good agreement on eigenfrequencies and mode shapes. Frequency-response curves in the non-linear range are compared in a very peculiar regime displayed by the shell with a 1:1:2 internal resonance. An important improvement is obtained compared to a perfect spherical shell model, however some discrepancies subsist and are discussed.
59332203
Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance
This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Karman's theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied. © 2004 Elsevier Ltd. All rights reserved.
28388197
ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY
In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von Karman equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency $\omega_n$ of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to $\omega_n$, which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper.
120513655
Nonlinear Vibrations and Stability of Shells and Plates
Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.
118129164
Vibration of Plates
Abstract : Contents: Fundamental Equations of Classical Plate Theory; Circular Plates; Elliptical Plates; Rectangular Plates; Parallelogram Plates; Other Quadrilateral Plates; Triangular Plates; Plates of Other Shapes; Anisotropic Plates; Plates With Inplane Forces; Plates With Variable Thickness; and Other Considerations.
122702757
Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties
Abstract This study is devoted to the derivation of some properties of the von Karman equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions—with vanishing prescribed forces and displacements—are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called “triple self-adjointness”, is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.
121562322
Computational formulation for periodic vibration of geometrically nonlinear structures—part 1: Theoretical background
Abstract A general computational formulation for geometrically nonlinear structures excited by harmonic forces and executing periodic motion in a steady-state is presented. The equations of both continuous and discretized models are reformulated to obtain the motion equation in a more suitable form to a further analysis. The multi-harmonic solution of motion equation is written in a form of truncated Fourier series. Next, the Galerkin, Ritz and the harmonic balance method are discussed in a context of their equivalency in derivation of the matrix amplitude equation. The matrix amplitude equation as well as the associated tangent matrix are given in an explicit form. The stability of steady-state solution is discussed by using the Floquet theory. The numerical algorithm and an example application are described in a companion paper by Lewandowski [Lewandowski, R. Computational formulation for periodic vibration of geometrically nonlinear structures—Part 2: Numerical strategy and numerical examples. International Journal of Solids and Structures . (in preparation)].
121641221
Computational formulation for periodic vibration of geometrically nonlinear structures—part 2: Numerical strategy and examples
Abstract In this paper the numerical strategy for solving the matrix amplitude equation with parameter is discussed in detail. The matrix amplitude equation is the result of application of the Galerkin method for analysis of periodic solutions of the geometrically nonlinear structures. A theoretical background of the proposed method of analysis is given in a companion paper by Lewandowski [Lewandowski, R. (1996). Computational formulation for periodic vibration of geometrically nonlinear structures—Part 1: theoretical background. International Journal of Solids and Structures34(15), 1925–1947]. An example application of general theory is also presented. Numerical results are given and discussed to show some interesting behaviors of nonlinear structures and to illustrate the numerical efficiency and validity of the suggested method.
34493719
Nonlinear normal modes, Part I: A useful framework for the structural dynamicist
The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part II. Because there is virtually no application of the NNMs to large-scale engineering structures, these papers are an attempt to highlight several aspects that might drive their development in the future. Specifically, we support that (i) numerical methods for the continuation of periodic solutions pave the way for an effective and practical computation of NNMs, and (ii) time–frequency analysis is particularly suitable for the analysis of the resulting dynamics. Another objective of the present paper is to describe, in simple terms, and to illustrate the fundamental properties of NNMs. This is achieved to convince the structural dynamicist not necessarily acquainted with them that they are a useful framework for the analysis of nonlinear vibrating structures.
53979325
Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques
The concept of nonlinear normal modes (NNMs) is discussed in the present paper and its companion, Part I. One reason of the still limited use of NNMs in structural dynamics is that their computation is often regarded as impractical. However, when resorting to numerical algorithms, we show that the NNM computation is possible with limited implementation effort, which paves the way to a practical method for determining the NNMs of nonlinear mechanical systems. The proposed algorithm relies on two main techniques, namely a shooting procedure and a method for the continuation of NNM motions. The algorithm is demonstrated using four different mechanical systems, a weakly and a strongly nonlinear two-degree-of-freedom system, a simplified discrete model of a nonlinear bladed disk and a nonlinear cantilever beam discretized by the finite element method.
118174323
Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems
Volume I: Preface Abbreviations 1 Introduction 2 Preliminary Concepts, Methodologies and Techniques 2.1 Nonlinear Normal Modes (NNMs) 2.2 Energy Localization in Nonlinear Systems 2.3 Internal Resonances, Transient and Sustained Resonance Captures 2.4 Averaging, Multiple Scales and Complexification 2.5 Methods of Advanced Signal Processing 2.5.1 NumericalWavelet Transforms 2.5.2 Empirical Mode Decompositions and Hilbert Transforms 2.6 Perspectives on Hardware Development and Experiments 3 Nonlinear Targeted Energy Transfer in Discrete Linear Oscillators with Single-DOF Nonlinear Energy Sinks 3.1 Configurations of Single-DOF NESs 3.2 Numerical Evidence of TET in a SDOF Linear Oscillator with a SDOF NES 3.3 SDOF Linear Oscillators with SDOF NESs: Dynamics of the Underlying Hamiltonian Systems 3.3.1 Numerical Study of Periodic Orbits (NNMs) 3.3.2 Analytic Study of Periodic Orbits (NNMs) 3.3.3 Numerical Study of Periodic Impulsive Orbits (IOs) 3.3.4 Analytic Study of Periodic and Quasi-Periodic IOs 3.3.5 Topological Features of the Hamiltonian Dynamics 3.4 SDOF Linear Oscillators with SDOF NESs: Transient Dynamics of the Damped Systems 3.4.1 Nonlinear Damped Transitions Represented in the FEP 3.4.2 Dynamics of TET in the Damped System 3.5 Multi-DOF (MDOF) Linear Oscillators with SDOF NESs: Resonance Capture Cascades and Multi-frequency TET 3.5.1 Two-DOF Linear Oscillator with a SDOF NES 3.5.2 Semi-Infinite Chain of Linear Oscillators with an End SDOF NES 4 Targeted Energy Transfer in Discrete Linear Oscillators with Multi-DOF NESs 4.1 Multi-Degree-of-Freedom(MDOF) NESs 4.1.1 An AlternativeWay for Passive Multi-frequency Nonlinear Energy Transfers 4.1.2 Numerical Evidence of TET in MDOF NESs 4.2 The Dynamics of the Underlying Hamiltonian System 4.2.1 System I: NES with O(1) Mass 4.2.2 System II: NES with O(e) Mass 4.2.3 Asymptotic Analysis of Nonlinear Resonant Orbits 4.2.4 Analysis of Resonant Periodic Orbits 4.3 TRCs and TET in the Damped and Forced System 4.3.1 Numerical Wavelet Transforms 4.3.2 Damped Transitions on the Hamiltonian FEP 4.4 Concluding Remarksl Index. Volume 2: 5 Targeted Energy Transfer in Linear Continuous Systems with Singlean Multi-DOF NESs 5.1 Beam of Finite Length with SDOF NES 5.1.1 Formulation of the Problem and Computational Procedure 5.1.2 Parametric Study of TET 5.2 Rod of Finite Length with SDOF NES 5.2.1 Formulation of the Problem, Computational Procedure and Post-Processing Algorithms 5.2.2 Computational Study of TET 5.2.3 Damped Transitions on the Hamiltonian FEP 5.3 Rod of Semi-Infinite Length with SDOF NES 5.3.1 Reduction to Integro-differential Form 5.3.2 Numerical Study of Damped Transitions 5.3.3 Analytical Study 5.4 Rod of Finite Length with MDOF NES 5.4.1 Formulation of the Problem and FEPs 5.4.2 Computational Study of TET 5.4.3 Multi-Modal Damped Transitions and Multi-Scale Analysis 5.5 Plate with SDOF and MDOF NESs 5.5.1 Case of a SDOF NES 5.5.2 Case of Multiple SDOF NESs 5.5.3 Case of a MDOF NES 5.5.4 Comparative Study with Linear Tuned Mass Damper 6 Targeted Energy Transfer in Systems with Periodic Excitations 6.1 Steady State Responses and Generic Bifurcations 6.1.1 Analysis of Steady State Motions 6.1.2 Numerical Verification of the Analytical Results 6.2 Strongly Modulated Responses (SMRs) 6.2.1 General Formulation and Invariant Manifold Approach 6.2.2 Reduction to One-DimensionalMaps and Existence Conditions for SMRs 6.2.3 Numerical Simulations 6.3 NESs as Strongly Nonlinear Absorbers for Vibration Isolation 6.3.1 Co-existent Response Regimes 6.3.2 Efficiency and Broadband Features of the Vibration Isolation 6.3.3 Passive Self-tuning Capacity of the NES 7 NESs with Non-Smooth Stiffness Characteristics 7.1 Systems with Multiple NESs Possessing Clear
32019158
Modulation instability: The beginning
Abstract We discuss the early history of an important field of “sturm and drang” in modern theory of nonlinear waves. It is demonstrated how scientific demand resulted in independent and almost simultaneous publications by many different authors on modulation instability, a phenomenon resulting in a variety of nonlinear processes such as envelope solitons, envelope shocks, freak waves, etc. Examples from water wave hydrodynamics, electrodynamics, nonlinear optics, and convection theory are given.
7227838
Modulational instability: first step towards energy localization in nonlinear lattices
We study the modulational instability in discrete lattices and we show how the discreteness drastically modifies the stability condition. Analytical and numerical results are in very good agreement. We predict also the evolution of a linear wave in the presence of noise and we show that modulational instability is the first step towards energy localization.
32103952
Modulation instability and capillary wave turbulence
Formation of turbulence of capillary waves is studied in laboratory experiments. The spectra show multiple exponentially decreasing harmonics of the parametrically excited wave which nonlinearly broaden with the increase in forcing. Spectral broadening leads to the development of the spectral continuum which scales as ∝f− 2.8, in agreement with the weak turbulence theory (WTT). Modulation instability of capillary waves is shown to be responsible for the transition from discrete to broadband spectrum. The instability leads to spectral broadening of the harmonics, randomization of their phases, it isolates the wave field from the wall, eventually allows the transition from 4- to 3-wave interactions as the dominant nonlinear process, thus creating the prerequisites assumed in WTT.
40718746
Space-time resolved wave turbulence in a vibrating plate.
Wave turbulence in a thin elastic plate is experimentally investigated. By using a Fourier transform profilometry technique, the deformation field of the plate surface is measured simultaneously in time and space. This enables us to compute the wave-vector-frequency (k, omega) Fourier spectrum of the full space-time deformation velocity. In the 3D (k, omega) space, we show that the energy of the motion is concentrated on a 2D surface that represents a nonlinear dispersion relation. This nonlinear dispersion relation is close to the linear dispersion relation. This validates the usual wave-number-frequency change of variables used in many experimental studies of wave turbulence. The deviation from the linear dispersion, which increases with the input power of the forcing, is attributed to weak nonlinear effects. Our technique opens the way for many new extensive quantitative comparisons between theory and experiments of wave turbulence.
10159687
Linear versus nonlinear response of a forced wave turbulence system.
A vibrating plate is set into a chaotic state of wave turbulence by a forcing having periodic and random components. Both components are weighted in order to explore continuously intermediate forcing from the periodic to the random one, but keeping constant its rms value. The transverse velocity of the plate is measured at the application point of the force. It is found that whatever the detail of the forcing is, the velocity spectra exhibit a universal cascade for frequencies larger than the forcing frequency range. In contrast, the velocity spectra strongly depend on the nature of the forcing within the range of forcing frequencies. The coherence function is used to extract the contribution of the velocity fluctuations that display a linear relationship with the forcing. The nonlinear contribution to the velocity fluctuations is found to be almost constant, about 55% of the total velocity fluctuations whatever the nature of the forcing from random to periodic. On the other hand, the nonlinear contribution to the fluctuations of the injected power depends on the nature of the forcing; it is significantly larger for the periodic forcing (60%) and decreases continuously as the randomness is increased, reaching a value of 40% for the pure random forcing. For all the cases of intermediate forcing from random to periodic, a simple model of the velocity response recovers in a fairly good agreement the probability density function of the injected power. The consequence of the existence of a linear-response component is discussed in the context of the fluctuation-dissipation theorem validation in experiments of out-of-equilibrium systems.
2719333
Effective dissipation and turbulence in spectrally truncated euler flows.
A new transient regime in the relaxation towards absolute equilibrium of the conservative and time-reversible 3D Euler equation with a high-wave-number spectral truncation is characterized. Large-scale dissipative effects, caused by the thermalized modes that spontaneously appear between a transition wave number and the maximum wave number, are calculated using fluctuation dissipation relations. The large-scale dynamics is found to be similar to that of high-Reynolds number Navier-Stokes equations and thus obeys (at least approximately) Kolmogorov scaling.