Nonlinear Damped Systems Research Articles

Periodic and quasiperiodic motion for complex non-linear systems

A damped complex non-linear system corresponding to two coupled non-linear oscillators with a periodic damping force is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Phase-locked solutions with period equal to the damping force period are possible only if the oscillators amplitudes are equal. On the contrary, if the oscillators amplitudes are different, periodic solutions exist only with a period different from the damping force period. These solutions are stable only for perturbations that conserve the phase difference and the square amplitude sum of the oscillators. Energy considerations are used in order to study existence and characteristics of quasiperiodic motion. We demonstrate that modulated motion can be also obtained for appropriate values of the detuning parameter and in this case an approximate analytic solution is easily constructed. If the detuning parameter decreases the modulation period increases and then diverges, an infinite-period bifurcation occurs and the resulting motion becomes unbounded. Analytic approximate solutions are checked by numerical integration.

Performance Evaluation of Passive Damping Systems for the Seismic Retrofit of Steel Moment-Resisting Frames Subjected to Near-Field Ground Motions

Passive friction and viscous damping systems for retrofitting steel moment-resisting frames located along the west coast of the United States are considered. First, preliminary design procedures are presented for friction as well as linear and nonlinear viscous damping systems. Thereafter, nonlinear dynamic analyses are performed on a six-story moment-resisting frame designed according to seismic provisions for California prior to the 1994 Northridge earthquake. A flexural strength degradation model is considered to account for the brittle behavior of pre-Northridge welded beam-to-column connections. The structure was subjected to three different earthquake ensembles including near-field records developed for major crustal earthquakes in California. The results of a parametric study indicate that, although both friction and viscous damping systems reduce significantly the response of the structure, they are unable by themselves to prevent fracture of welded beam-to-column joints. Connection retrofit measures of the types elaborated after the Northridge earthquake would still be required.

ANALYTICAL ESTIMATES OF THE EFFECT OF NONLINEAR DAMPING IN SOME NONLINEAR OSCILLATORS

This paper reports on the effect of nonlinear damping on certain nonlinear oscillators, where analytical estimates provided by the Melnikov theory are obtained. We assume general nonlinear damping terms proportional to the power of velocity. General and useful expressions for the nonlinearly damped Duffing oscillator and for the nonlinearly damped simple pendulum are computed. They provide the critical parameters in terms of the damping coefficient and damping exponent, that is, the power of the velocity, for which complicated behavior is expected. We also consider generalized nonlinear damped systems, which may contain several nonlinear damping terms. Using the idea of Melnikov equivalence, we show that the effect of nonlinear dissipation can be equivalent to a linearly damped nonlinear oscillator with a modified damping coefficient.

Damped stochastic system driven by colored noise: analytical solution by a path integral approach

We consider the nonlinear non-Markovian stochastic process associated with the damped nonlinear dynamical system driven by Ornstein-Uhlenbeck noise. An approximate Fokker-Planck-type equation governing the above stochastic process is derived using the path-integral approach. The stationary probability density function (SPDF) of the above process is then computed using the matrix continued fraction method. The SPDF compares favorably with the corresponding digital simulation results obtained by us.

Regular and chaotic dynamics of a simplified fly-ball governor

The dynamics of a simplified model of a fly-ball speed governor undergoing a harmonic variation about its rotational speed is studied in this paper. This system is a non-linear damped system subjected to parametric excitation. The harmonic balance method is applied to analyse the stability of period attractors and the behaviour of bifurcations. The time evolutions of the response of the non-linear dynamic system are described by time history, phase portraits and Poincaré maps. The regular and chaotic behaviour is observed by various numerical techniques such as power spectra, Lyapunov exponents and Lyapunov dimension. Finally, the domains of attraction of periodic and stranger attractors of the system are located by applying the interpolated cell mapping (ICM) method.

Quadratic optimal control problems for nonlinear damped second order systems in Hilbert spaces

Quadratic optimal control problems for nonlinear damped second order systems in Hilbert spaces

Analysis of the response of a strongly nonlinear damped system using a differential transformation technique

The differential transformation technique is applied to solve a nonlinear differential equation that describes the underdamped and the overdamped motion of a system subjected to external excitations. Numerical results are compared to those obtained by the Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method.

Finite element solution of the stability problem for nonlinear undamped and damped systems under nonconservative loading

A nonlinear finite element analysis of elastic structures which can be studied by a 3D beam theory, subjected to conservative as well as to nonconservative forces, is presented. The stability behaviour of the system is studied by means of an eigenvalue analysis. The stiffness matrix of the eigenvalue problem is asymmetric (i.e., non-self-adjoint system). The flutter and divergence modes of instability, as well as the values of the critical load, are identified for a number of numerical examples belonging to the benchmark tests proposed by NAFEMS (1990). The results demonstrate the reliability of this finite element formulation. In particular the effect of damping on the stability behaviour of such structures is investigated and the destabilizing effect of small damping is underlined. Finally, the need to define a number of benchmark tests for nonlinear-nonconservative analyses in presence of damping is included.

A New Approximate Analytical Analysis of the Response of Non-Linear Damped Systems Subjected to Step Function Excitation

A new approximate technique developed by Mansour and Hussein [1] has been used to solve a non-linear differential equation model that describes the underdamped and the overdamped motion of systems subjected to step function excitation. The analytical results obtained in this work have been compared with the numerical ones obtained by performing the Butcher Runge-Kutta method, and have been shown to be accurate in comparison.

A modified form of the central difference predictor scheme for damped nonlinear systems

The well-known central difference predictor method used for the analysis of nonlinear systems, subjected to transient dynamic loads, is modified by incorporating Rayleigh's damping into the governing incremental equation of motion. The new time-stepping scheme requires a starting algorithm for both the first and second time-steps, and can effectively be used to analyse the nonlinear damped systems in general.

Classical and Quantum Dynamics for a Class of Damped Nonlinear Systems

Classical and Quantum Dynamics for a Class of Damped Nonlinear Systems

Approximate Nonstationary Random Vibration Analysis for MDOF Systems

An approximate method for nonstationary random vibration analysis is presented. This method utilizes properties of the stationary solution for simplifying the analysis. This approach has previously been applied by the author to linear and nonlinearly damped SDOF systems. In the present paper the concept is extended to linear MDOF systems and applied to nonstationary earthquake-type loading. Comparisons with available exact solutions show very good agreement in numerical results with the additional benefit of reducing computer time by more than one order of magnitude.

Software of power series solution for non-linear free oscillations

A simple and accurate numerical method using a power series technique has been applied to obtain the free response of twenty different damped and undamped non-linear oscillatory systems. The derived results are compared with those previously derived by different methods.

Nonlinear vibrations of offshore structures under random wave excitation

An account is given of some mathematical methods which have been used to analyse the response of offshore structures to random wave excitation. The analysis of nonlinear phenomena and the assessment of related nongaussian probability distributions are emphasized. The following problems are analysed in some depth: probability distributions for Morison-type wave loading; response near resonance of nonlinearly damped systems to random excitation; parametric resonance and instability of nonlinearly damped systems with randomly fluctuating restoring force coefficient. Solutions to each of these problem areas are illustrated by practical examples.

On damped non-linear dynamic systems with many degrees of freedom

Non-linear mass-spring-damper systems with many degrees of freedom are studied; all springs and/or all dampers may be strongly non-linear. It is shown that the ultimate state of completely damped systems is always rest, that of incompletely damped systems may be either rest or a periodic normal mode motion. Necessary and sufficient conditions are given for the existence of classical normal mode motion in completely or incompletely damped systems. When the system is linear, these reduce to known results found by Rayleigh and generalized by Caughey and O'Kelly.

Response near resonance of non-linearly damped systems subject to random excitation with application to marine risers

Abstract In connection with questions concerning the resonant behaviour of certain types of offshore structures, the response of non-linearly damped systems to Gaussian excitation has been studied. Both a single and a multi-degree of freedom system excited predominantly at one natural frequency have been considered. Explicit descriptions of the response, partly in the time-domain and partly in terms of non-Gaussian probability distributions, have been derived for the limit in case the damping is small. The solutions have been used to indicate the effect of non-linear damping on the distribution of response maxima, on the expected fatigue damage and on the expected extreme response. Comparisons have been made with results obtained from equivalent linearisation methods and with results obtained from numerical time-domain simulations of the non-linearly damped response of a marine riser to random waves.

Seismic Design of Reinforced Earth Walls

Seismic design for reinforced earth retaining walls was developed largely on the results obtained from small laboratory scale walls on a shaking table, and is therefore tentative and must await verification from further analytical laboratory and field studies. The laboratory tests showed that the walls responded like a nonlinear damped elastic system to the input vibrations. From measurements of the peak tie forces, an empirical design force envelope was developed which is a function of input acceleration only. It is suggested that the design earth pressures for an actual wall subjected to earthquake loading be based on this design force envelope using a base acceleration determined by response spectra and modal participation factor techniques. Data are also presented of soil-tie friction under static and vibratory loading. The recommendations include data from which the required size and spacing of ties can be determined. Suggested factors of safety are given for tie pull out and tie breaking modes of failure.

An asymptotic analysis of the response of non-linear damped systems subjected to step function excitation

Two forms of a unified type of Krylov-Bogoliubov method are considered for the purpose of deriving approximate solutions of non-linear ordinary differential equations that describe the underdamped and overdamped motion of systems subjected to step function excitation. The method of analysis, based upon power series expansions of the phase and amplitude in terms of the non-linearity parameter e and, alternatively, the parameter μ=e/(e+1), explicitly accounts for the effect of linear damping, which is proportional to velocity, in such a way that the approximate solutions of the non-linear equations reduce in the limit to the exact solutions of the corresponding linear problems as the non-linearity parameter tends to zero. To assess the relative accuracies of the two variations of the fundamental method, numerical calculations were performed using (a) approximate formulas for the amplitude and phase and (b) the fourth order Runge-Kutta method of numerical integration. These numerical results are compared graphically, and the merits of the two approaches are discussed.

Differential tones in a damped mechanical system with quadratic and cubic non-linearities

The combination tones of differential type are studied in a non-linear damped mechanical system of two degrees of freedom with quadratic and cubic non-linearities and excited by two external harmonic forces with different frequencies. Approximate steady state solutions and the corresponding Galerkin approximations of high order are obtained and error bounds are given. For a certain frequency the existence of three exact periodic solutions is proved by Urabe's method.

Combination tones of summed type in a non-linear damped vibratory system with two degrees of freedom

The combination tones of summed type are studied in a non-linear damped vibratory system with two degrees of freedom, excited by two external harmonic forces of different frequencies. Equations are given for the response curves, the phase angles and the back bones of the summation tones. The stability of the summation tones is investigated by the method of varying amplitudes and phases. Numerical results and effects of the parameters are explained.