Energy Pumping with Various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small nonlinear attachment. The essential nonlinearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment. Different nonlinear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to civil engineering. As a specific application, attenuation of vibrations of a building is studied with two building models. In particular, the case of stochastic excitations is analyzed to examine if it is possible to process energy pumping when a seism occurs and an indicator of efficiency has been introduced.     

Bifurcations of Nonlinear Normal Modes of Linear Oscillator with Strongly Nonlinear Damped Attachment

Linear oscillator coupled to damped strongly nonlinear attachment with small mass is considered as a model design for nonlinear energy sink (NES). Damped nonlinear normal modes of the system are considered for the case of 1:1 resonance by combining the invariant manifold approach and multiple scales expansion. Special asymptotical structure of the model allows a clear distinction between three time scales. These time scales correspond to fast vibrations, evolution of the system toward the nonlinear normal mode and time evolution of the invariant manifold, respectively. Time evolution of the invariant manifold may be accompanied by bifurcations, depending on the exact potential of the nonlinear spring and value of the damping coefficient. Passage of the invariant manifold through bifurcations may bring about destruction of the resonance regime and essential gain in the energy dissipation rate.     

Theoretical and Experimental Study of Multimodal Targeted Energy Transfer in a System of Coupled Oscillators

The purpose of this study is the theoretical and experimental investigation of targeted energy transfers from a two-degree-of-freedom primary structure to a nonlinear energy sink (NES). It is demonstrated that an NES can resonate with and extract energy from both modes of the primary structure. By facilitating these energy transfers, notably through excitation of appropriate periodic and quasi-periodic orbits, one can promote dissipation of a major portion of externally induced energy in the nonlinear attachment.     

Contribution to efficiency of irreversible passive energy pumping with a strong nonlinear attachment

The present study deals with nonlinear energy pumping which consists in passive irreversible transfer of energy from a linear structure to a nonlinear one. Various results (theoretical, numerical, and experimental) about energy pumping based on recent works are given. Thus, the phenomenon is studied for different excitations: transient and periodical. Moreover, advantages of such a system are carried out in particular efficiency of this phenomenon. That is why the robustness and comparison with classical tuned mass damper are analyzed. An application is considered with physical experiment using a reduced scale building.     

非线性非自治系统拓扑结构分析的能量法

利用能量原理,提出一种以能量为状态参数的分析非线性非自治系统拓扑结构的定性方法。分析了能量参数确定系统状态的唯一性,揭示了能量方法的正确性和有效性以及反映系统运动状态信息的丰富性。以非自治Duffing方程为对象,采用能量状态参数,应用提出的能量方法在相空间和运动空间展开了一般的定性分析,给出了能量相图和能量时间历程、能量Poincaré截面图、能量参数分岔图,分析了其中呈现出的运动规律和捕捉到的有价值信息,并与现有的方法进行了比较。     

Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator

We present a theoretical study of the dynamics of the coupled system of Jiang, McFarland, Bergman, and Vakakis. It comprises a harmonically excited linear subsystem weakly coupled to an essentially nonlinear oscillator. We explored the rich dynamics exhibited by this coupled system by determining its periodic responses and their bifurcations. Not surprisingly, we found a lot of interesting dynamics over a broad frequency range: cyclic-fold, Hopf, symmetry-breaking, and period-doubling bifurcations; phase-locked motions; regions with multiple coexisting solutions; hysteresis; and chaos. We did not find any occurrence of energy transfer via modulation (also known as zero-to-one internal resonance); theoretically, the possibility of its occurrence was ruled out for systems with weak nonlinearity and damping. Finally, we investigated the ef fectiveness of the so-called nonlinear energy sink (NES) in vibration attenuation of forced linear structures. We found that the NES results in an increase in the vibration amplitude of the linear subsystem, especially when the damping is low, contrary to the claim made by Jiang et al. Also, we did not find any indication of nonlinear energy pumping or localization of energy in the NES, away from the directly forced linear subsystem, indicating that the NES is not ef fective for controlling the vibrations of forced linear structures.     

Attractor for a Degenerate Nonlinear Diffusion Problem with Nonlinear Boundary Condition

In this paper we prove the existence of a compact attractor in L () for a degenerate nonlinear diffusion problem with nonlinear flux on the boundary. In order to formulate the equation as a dynamical system, some existence and uniqueness results for weak solutions are proved.     

Collapse of Shock Waves upon Their Interaction with a Local Energy Source

Numerical analysis is performed of the interaction of a shock wave with a local energy source and the wake behind it. It is shown that for specified shock-wave intensity and flow parameters there is a threshold value of the energy release starting with which the shock wave collapses. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 3–7, March–April, 2006.     

Linear and Nonlinear Formulation of Phase Field Model with Generalized Polynomial Degradation Functions for Brittle Fractures

The classical phase field model has wide applications for brittle materials,but nonlinearity and inelasticity are found in its stress-strain curve.The degradation function in the classical phase field model makes it a linear formulation of phase field and computationally attractive,but stiffness reduction happens even at low strain.In this paper,generalized polynomial degradation functions are investigated to solve this problem.The first derivative of degradation function at zero phase is added as an extra constraint,which renders higher-order polynomial degradation function and nonlinear formulation of phase field.Compared with other degradation functions(like algebraic fraction function,exponential function,and trigonometric function),this polynomial degradation function enables phase in[0,1](should still avoid the first derivative of degradation function at zero phase to be 0),so there is no Γ convergence problem.The good and meaningful finding is that,under the same fracture strength,the proposed phase field model has a larger length scale,which means larger element size and better computational efficiency.This proposed phase field model is implemented in LS-DYNA user-defined element and user-defined material and solved by the Newton-Raphson method.A tensile test shows that the first derivative of degradation function at zero phase does impact stress-strain curve.Mode Ⅰ,mode Ⅱ,and mixed-mode examples show the feasibility of the proposed phase field model in simulating brittle fracture.     

A Method for Obtaining Approximate Analytic Periods for a Class of Nonlinear Oscillators

This paper deals with nonlinear oscillations of conservative single-degree-of-freedom systems with odd nonlinearity. By combining the linearization of the governing equation with the method of harmonic balance, we establish two approximate analytic formulas for the period. These two formulas are valid for small as well as large amplitudes of oscillation. Three examples are used to illustrate that the proposed formulas can give very satisfactory approximate results. Sommario. Questo lavoro tratta il problema delle oscillazioni nonlineari di sistemi conservativi ad un grado di libertà con nonlinearità simmetriche. Combinando opportunamente la tecnica di linearizzazione dellequazione del moto con il metodo del bilancio armonico si perviene a due formule analitiche approssimate per il periodo. Le formule ottenute sono valide sia per piccole che per grandi ampiezze di oscillazione. Si utilizzano tre esempi classici di oscillatori nonlineari per illustrate lefficacia del metodo nel produrre risultati approssimati soddisfacenti.     

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

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

Energy Method for Computing Periodic Solutions of Strongly Nonlinear Autonomous Systems with Multi-Degree-of-Freedom

In this paper with the use of conservation of average energy, a newmethod for computing the periodic solutions of strongly nonlinearautonomous systems with multi-degree-of-freedom is suggested. Thismethod cannot only decide the existence, but also give the approximateexpressions of the periodic solutions.     

An Explicit Formula for the Minimum Free Energy in Linear Viscoelasticity

A general explicit formula for the maximum recoverable work from a given state is derived in the frequency domain for full tensorial isothermal linear viscoelastic constitutive equations. A variational approach, developed for the scalar case, is here generalized by virtue of certain factorizability properties of positive-definite matrices. The resultant formula suggests how to characterize the state in the sense of Noll in the frequency domain. The property that the maximum recoverable work represents the minimum free energy according to both Graffi's and Coleman-Owen's definitions is used to obtain an explicit formula for the minimum free energy. Detailed expressions are presented for particular types of relaxation function. This revised version was published online in August 2006 with corrections to the Cover Date.     

Energy Method for Approximate Periodic Solution of Strongly Nonlinear Nonautonomous Systems

In this paper, based upon the concept of the conservation of average energy, a theorem for the periodic solution of strongly nonlinear nonautonomous systems is proved. By using this theorem not only the necessary and sufficient conditions for the existence and stability of the periodic solutions can be obtained but, at the same time, the approximate expressions for those periodic solutions can also be obtained.     

Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation

This paper deals with the most important characteristics of a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation. We use an asymptotic perturbation method based on Fourier expansion and time rescaling and demonstrate through a second order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion, when a second low frequency appears, in addition to the forcing frequency. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.     

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

The work analyzes energy relations for nonholonomic systems, whose motion is restricted by nonlinear nonholonomic constraints. For the mechanical systems with linear constraints, the analysis of energy relations was carried out in [1], [2], [3], [4], [5], [6] …. On the basis of corresponding Lagrange’s equations, a general law of the change in energy dε/dt is formulated for mentioned systems by the help of which it is shown that there are two types of the laws of conservation of energy, depending on the structure of elementary work of the forces of constraint reactions. Also, the condition for existing the second type of the law of conservation of energy is formulated in the form of the system of partial differential equations. The obtained results are illustrated by a model of nonholonomic mechanical system.     

Singular Nonlinear Eigenvalue Problem for a Second-Order Differential Equation with Energy Dissipation

We consider an eigenvalue problem for a singular nonlinear ordinary differential operator of the second order on the half line. We find sufficient conditions under which this problem has a solution that has a prescribed number of zeros and vanishes at infinity.     

An Energy Analysis of the Local Dynamics of a Delayed Oscillator Near a Hopf Bifurcation

Hopf bifurcation exists commonly in time-delay systems. The local dynamics of delayed systems near a Hopf bifurcation is usually investigated by using the center manifold reduction that involves a great deal of tedious symbolic and numerical computation. In this paper, the delayed oscillator of concern is considered as a system slightly perturbed from an undamped oscillator, then as a combination of the averaging technique and the method of Lyapunov's function, the energy analysis concludes that the local dynamics near the Hopf bifurcation can be justified by the averaged power function of the oscillator. The computation is very simple but gives considerable accurate prediction of the local dynamics. As an illustrative example, the local dynamics of a delayed Lienard oscillator is investigated via the present method.     

Dynamic Behavior and Vibration Suppression of a Generally Restrained Pre-pressure Beam Structure Attached with Multiple Nonlinear Energy Sinks

Beam structures are extensively used in many engineering branches.For marine engineering,the ship shafting system is generally simplified as a vibration model with single or multiple beam structures connected by the coupling stiffness.In engineering,multiple nonlinear energy sinks(NESs)can be arranged on the premise of sufficient installation space to ensure their vibration suppression effect.Considering engineering practice,this study investigates the dynamic behavior and vibration suppression of a generally restrained pre-pressure beam structure with multiple uniformly distributed NESs,where the pre-pressure is typically caused by thrust bearings,installation ways,and others.System governing equations are derived through the generalized Hamiltonian principle and the variational procedure.Dynamic responses of the pre-pressure beam structure are predicted by the Galerkin truncation method.The effect of NESs on dynamic responses and vibration suppression of the pre-pressure beam structure is studied and discussed.Suitable parameters of NESs have a beneficial effect on the vibration suppression at both ends of the pre-pressure beam structure.NESs can modify the vibration frequency and energy transmission characteristics of the vibration system.For different boundary conditions,the optimized parameters of NESs significantly suppress the vibration energy of the pre-pressure beam structure.     

Nonlinear analysis of a rub-impact rotor-bearing system with initial permanent rotor bow

A general model of a rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given. The nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory. The rubbing model is assumed to consist of the radial elastic impact and the tangential Coulomb type of friction. Through numerical calculation, rotating speeds, initial permanent bow lengths and phase angles between the mass eccentricity direction and the rotor permanent bow direction are used as control parameters to investigate their effect on the rub-impact rotor-bearing system with the help of bifurcation diagrams, Lyapunov exponents, Poincaré maps, frequency spectrums and orbit maps. Complicated motions, such as periodic, quasi-periodic even chaotic vibrations, are observed. Under the influence of the initial permanent bow, different routes to chaos are found and the speed when the rub happens is changed greatly. Corresponding results can be used to diagnose the rub-impact fault in this kind of rotor systems and this study may contribute to a further understanding of the nonlinear dynamics of such a rub-impact rotor-bearing system with initial permanent bow.