Nonlinear Damped Systems Research Articles

Parameter Identification of a New Metal Rubber Damper System

The parameter identification of a new hysteretic nonlinear metal rubber damper system is studied in this paper. Firstly, the advantage of the new metal rubber damper which composed by two components is proved by the static and dynamic test comparing to the damper included only one metal rubber. Secondly, the constitutive of bilinear hysteresis model and parameter identification equation are obtained. Then the parameter of the new damper system is identified by generalized least squares estimation. In the last, the simulation result is compared with the real values, it shows that the method is simple, effective, high precision, and it is also reliable and applicable to the corresponding engineering.

Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator

In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviors are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively.

Optimum Vibration Control of a Nonlinear Damper System under Multi-Axis Excitations

This paper presents an investigation of the vibration control performance of a nonlinear damper system under multi-axis excitations, focusing on its optimum strategy. The nonlinear damper system is composed of a particle damper and a primary structure. Cross-correlation functions and auto-correlation functions are shown to be suitable means to interpret the physics involved in the behavior of such a nonlinear system. Using different types of random excitations, the optimum operating regions are all determined, within which particles move in a plug flow pattern and correlation functions decay much faster than that under the inefficient operating conditions.

Parametric Studies of the Vibration Control Effects of a Nonlinear Damper System under Multi-Axis Excitations

This paper studies the influence of system parameters to the vibration control effects of a nonlinear damper system under multi-axis excitations. The nonlinear damper system is composed of a particle damper and a primary structure. Based on numerical simulations, it is shown that: increasing the mass ratio can improve the dampers effectiveness, but only up to a certain level; applying particles with a high value of the coefficient of restitution can result in a broader range of acceptable response levels; a lightly-damped primary system can achieve a considerable reduction in its response with a small weight penalty; and that a cylindrically-shaped container provides a higher level of effectiveness than a rectangularly-shaped one.

Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations

AbstractIn this article, we study a streamline diffusion‐based discontinuous Galerkin approximation for the numerical solution of a coupled nonlinear system of Schrödinger equations and extend the resulting method to a multiscale variational scheme. We prove stability estimates and derive optimal convergence rates due to the maximal available regularity of the exact solution. In the weak formulation, to make the underlying bilinear form coercive, it was necessary to supply the equation system with an artificial viscosity term with a small coefficient of order proportional to a power of mesh size. We justify the theory by implementing an example of an application of the time‐dependent Schrödinger equation in the coupled ultrafast laser. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Stability of Implicit Midpoint Algorithm Applied to Structures with Nonlinear Damping

Stability of implicit midpoint algorithm applied to solving the equation of motion with nonlinear damping. The analytical and numerical study exhibit the unconditional stability of implicit midpoint scheme for nonlinear damping systems. The work of the present paper provides theoretic base for further application of this implicit midpoint approach.

A statistically accurate modified quasilinear Gaussian closure for uncertainty quantification in turbulent dynamical systems

We develop a novel second-order closure methodology for uncertainty quantification in damped forced nonlinear systems with high dimensional phase-space that possess a high-dimensional chaotic attractor. We focus on turbulent systems with quadratic nonlinearities where the finite size of the attractor is caused exclusively by the synergistic activity of persistent, linearly unstable directions and a nonlinear energy transfer mechanism. We first illustrate how existing UQ schemes that rely on the Gaussian assumption will fail to perform reliable UQ in the presence of unstable dynamics. To overcome these difficulties, a modified quasilinear Gaussian (MQG) closure is developed in two stages. First we exploit exact statistical relations between second order correlations and third order moments in statistical equilibrium in order to decompose the energy flux at equilibrium into precise additional damping and enhanced noise on suitable modes, while preserving statistical symmetries; in the second stage, we develop a nonlinear MQG dynamical closure which has this statistical equilibrium behavior as a stable fixed point of the dynamics. Our analysis, UQ schemes, and conclusions are illustrated through a specific toy-model, the forty-modes Lorenz 96 system, which despite its simple formulation, presents strongly turbulent behavior with a large number of unstable dynamical components in a variety of chaotic regimes. A suitable version of MQG successfully captures the mean and variance, in transient dynamics with initial data far from equilibrium and with large random fluctuations in forcing, very cheaply at the cost of roughly two ensemble members in a Monte-Carlo simulation.

Non Oscillatory Nonlinear Differential Systems with Slowly Varying Coefficients in Presence on Certain Damping Forces

Krylov-Bogoliubov-Mitropolskii method is modified and applied to certain damped nonlinear systems with slowly varying coefficients. The results obtained by this method show excellent coincidence with those obtained by numerical method. The method is illustrated by an example.

Application of Three Bioinspired Optimization Methods for the Design of a Nonlinear Mechanical System

The present work focuses on the optimal design of nonlinear mechanical systems by using heuristic optimization methods. In this context, the nonlinear optimization problem is devoted to a two-degree-of-freedom nonlinear damped system, constituted of a primary mass attached to the ground by a linear spring and a secondary mass attached to the primary system by a nonlinear spring. This arrangement forms a nonlinear dynamic vibration absorber (nDVA), which is used in this contribution as a representative example of a nonlinear mechanical system. The sensitivity analysis of the suppression bandwidth, namely, the frequency range over which the ratio of the main mass displacement amplitude to the amplitude of the forcing function is less than unity, with respect to the design variables that characterize the nonlinear system based on the first order finite differences is presented. For illustration purposes the optimization problem is written as to maximize the suppression bandwidth by using three recent bioinspired optimization methods: Bees Colony Algorithm, Firefly Colony Algorithm, and Fish Swarm Algorithm. The results are compared with other evolutionary strategies.

A Mathematical Modeling of Resonances of the Nonlinear Tilted Support Spring System under Harmonic Excitation

A novel nonlinear tilted support spring damping system is proposed in this paper. The nonlinear dynamic equation is developed under the foundation displacement excitation and the resonance frequency of the system is also analyzed. Furthermore, different support angles affecting the vibration amplitude and the natural frequency of the system are researched. The model of the tilted support spring damping system is analyzed by ANSYS, and modal analysis and harmonic response analysis are made under harmonic excitation. The study shows that the supporting angle makes a great impact on the amplitude of the system and the response amplitude of the system is also reduced with the decrease of support angle. Through the analysis and comparison of the example, it provides a theoretical basis of the analysis for the tilted support spring damping system support angle’s optimal design.

A Study on Vibration Characteristics and Stability of the Ambulance Nonlinear Damping System

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.

Approximate Solution of Fourth Order Near Critically Damped Nonlinear Systems with Special Conditions

A perturbation technique has been developed based on the Krylov-Bogoliubov-Mitropolskii (KBM) method to investigate the solution of fourth order near critically damped nonlinear systems in the case of ?1??2, ?4 ? ?3 + 2?1 but ?4<2?3 among the eigenvalues ?1, ?2, ?3, ?4. The solutions obtained by this technique were compared with those obtained by numerical method. The method has been explained by an example. DOI: http://dx.doi.org/10.3329/jbas.v36i2.12962 Journal of Bangladesh Academy of Sciences, Vol. 36, No. 2, 187-197, 2012

Nonexistence of global solutions for a nonlocal nonlinear hyperbolic system with linear damping

This article concerns the Cauchy problem for the damped nonlinear hyperbolic system where ϵ > 0 is a small parameter, 0 < α ≤ 1,0 < β ≤ 1,p,q ≥ 1 satisfying pq > 1 , and N ≥ 1 is an integer.It is proved that if N ∕ 2α < max((p + 1) ∕ (pq − 1),(q + 1) ∕ (pq − 1)), then every weak solution does not exist globally whenever the initial data satisfy or . Copyright © 2012 John Wiley & Sons, Ltd.

Application of non-linear damping to vibration isolation: an experimental study

In a previous study, the authors have proved that in theory the introduction of a cubic non-linear damping can produce ideal vibration isolation such that the system force transmissibility over the resonant frequency region is modified, but the transmissibility over the non-resonant regions remain unaffected. The present study is concerned with both an experimental verification of this theoretical finding and the selection of the cubic damping characteristic parameter required to achieve a desired performance for a single degree of freedom vibration isolation system. These results provide an important basis for the design and practical application of non-linearly damped vibration isolation systems in engineering practice.

Power law damping parameter identification

The parameter estimation of a nonlinear power damping system is studied. The parameter identification method used here assumes a priori the equation of motion describing the system dynamics. The method, which is based on the measured data (acceleration), was applied to the free and forced vibrations. The identification procedure was found to be robust on the guessed value of parameters at the numerical experimentation. The parameter values were estimated with a good accuracy for both modes of system operation (free and forced) if only the measured time history was sampled at a high enough rate for the noise level contained within. It was shown that the steady state of the harmonically excited system is not the best region for the parameter identification with this method. During the experimentation the method was applied to the free vibrations in different media (air and water). The results obtained by the parameter identification method were compared to the ones obtained by separate tests and good agreement was found. The identification procedure was found to work fine for all models under consideration and the models' responses correspond well to the measured acceleration time histories.

An analytical technique to find approximate solutions of nonlinear damped oscillatory systems

Combining Krylov–Bogoliubov–Mitropolskii (KBM) and harmonic balance methods, an analytical technique is presented to determine approximate solutions of nonlinear oscillatory systems with damping. The first approximate perturbation solutions in which the unperturbed solutions contain two harmonic terms agree with numerical solutions nicely even if the damping force is significant. With suitable examples it has been shown that the combination of classical KBM and harmonic balance methods sometimes fails to measure satisfactory results; but the combination of extended KBM method (by Popov) and harmonic balance method always give the desired results. The method is illustrated by several examples and the solutions are compared to some existing solutions.

A new method for measuring nonharmonic periodic excitation forces in nonlinear damped systems

The purpose of this study is to identify an external loading of long time duration, which is nonharmonic but periodic, acting on a nonlinear dynamic system with nonlinear restoring as well as nonlinear damping. A new procedure is proposed for the force identification through an inverse formalism. However, this involves a Volterra-type nonlinear integral equation of the first kind, which lacks solution stability. Therefore, the nonlinear dynamic system under investigation is transformed into a linear relation between forces and pseudo-displacements. The lack of solution stability is resolved by applying available regularization methods. The feasibility of the force identification is demonstrated through a numerical example.

Free vibration of the nonlinear pendulum using hybrid Laplace Adomian decomposition method

The pendulum system is definitely nonlinear in the real world of physics and has been considered a fundamental subject to the nonlinear oscillators. In this paper, a hybrid method of the Laplace Adomian decomposition method combined with Pade approximant, named the LADM-Pade approximant technique is proposed to solve the nonlinear undamped and damped pendulum systems to demonstrate efficient and reliable results without small angular displacement assumption or linearization. Three examples here in are given to show the accuracy and convergence in comparison with the fourth-order Runge–Kutta solutions. Copyright © 2009 John Wiley & Sons, Ltd.

Sharp energy estimates for nonlinearly locally damped PDEs via observability for the associated undamped system

We consider the problem of sharp energy decay rates for nonlinearly damped abstract infinite-dimensional systems. Direct methods for nonlinear stabilization generally rely on multiplier techniques, and thus are valid under restrictive geometric conditions compared to the optimal geometric optics condition of Bardos et al. (1992) [10]. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely an observability estimate for the corresponding undamped system. One of the main advantage of these results is that they allow to combine optimal geometric conditions, as for instance that of Bardos et al. (1992) [10] and the optimal-weight convexity method of the first author (Alabau-Boussouira, 2010 [6], Alabau-Boussouira, 2005 [2]) to deduce very simple and quasi-optimal energy decay rates for nonlinearly locally damped systems. We also show that using arguments based on Russell's principle (Russell, 1978 [24]), one can deduce sharp energy decay rates from the exponential stabilization of the linearly damped system. Our results extend to nonlinearly damped systems, those of Haraux (1989) [14] and Ammari and Tucsnak (2001) [9] which concern linearly damped systems.

Global solution and blow-up solution for a nonlinear damped beam with source term

A nonlinear damped system with boundary input and output, which also has source term, is studied in this paper. It is proved that under some conditions the system has global solution and blow-up solution.