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  • nonlinear oscillations  (9)
  • nonlinear systems  (9)
  • 1
    ISSN: 1573-269X
    Keywords: Cables ; active control ; nonlinear oscillations ; bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 16 (1998), S. 127-151 
    ISSN: 1573-269X
    Keywords: Car-following models ; nonlinear systems ; delay-differential equations ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In order to describe the dynamics of a group of road vehicles travelling in a single lane, car-following models attempt to mimic the interactions between individual vehicles where the behaviour of each vehicle is dependent upon the motion of the vehicle immediately ahead. In this paper we investigate a modified car-following model which features a new nonlinear term which attempts to adjust the inter-vehicle spacing to a certain desired value. In contrast to our earlier work, a desired time separation between vehicles is used rather than simply being a constant desired distance. In addition, we extend our previous work to include a non-zero driver vehicle reaction time, thus producing a more realistic mathematical model of congested road traffic. Numerical solution of the resulting coupled system of nonlinear delay differential equations is used to analyse the stability of the equilibrium solution to a periodic perturbation. For certain parameter values the post-transient response is a chaotic (non-periodic) oscillations consisting of a broad spectrum of frequency components. Such chaotic motion leads to highly complex dynamical behaviour which is inherently unpredictable. The model is analysed over a range of parameter values and, in each case, the nature of the response is indicated. In the case of a chaotic solution, the degree of chaos is estimated.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 16 (1998), S. 321-336 
    ISSN: 1573-269X
    Keywords: Fractionally damped ; nonlinear systems ; Duffing's equation ; perturbation procedure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the harmonic oscillations of periodically excited nonlinear systems where hysteresis is simulated via fractional operator representations. Employing a diophantine version of the fractional operational powers, the energy constrained Lindstedt–Poincaré perturbation procedure is utilized to establish the harmonic solution. The constrained perturbation procedure was employed since it allows for the handling of strong damping and exciting forces over the full span of the driving frequency range. Based on the approach taken, the long time behavior of the fractionally damped Duffing's equation is studied in detail. Of special interest is the determination of the influence of fractional order on the frequency amplitude response behavior.
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  • 4
    ISSN: 1573-269X
    Keywords: Magnetic levitation ; fuzzy logic control ; nonlinear systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper deals with the control in both horizontal and vertical directions of a nonlinear magnetic spherical particle suspension. A pair of horizontal electromagnets were added to the typical magnetic levitation system in order to handle the horizontal unexpected displacements of the particle. A fuzzy logic controller was designed to bring the particle to the operating location situated on the symmetry axis. Simulations with different initial position conditions of the particle are provided. It was shown that the controller has a good behavior for almost any initial conditions.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 20 (1999), S. 309-317 
    ISSN: 1573-269X
    Keywords: nonlinear oscillations ; chaos ; control ; input-output linearization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The control of chaotic oscillations is investigated in this paper. A control methodology, termed input-output linearization, is modified by locally linearizing the nonlinear control law in the small neighborhood of the control goal. Its suitability for controlling chaotic oscillators is analyzed. The forced Duffing oscillator is treated as a numerical example of controlling chaotic motion to a given fixed point and a given period-2 motion. The control signals and time needed to achieve the desired goals of the modified method are compared with those of the original method. The robustness of the control law is demonstrated.
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  • 6
    ISSN: 1573-269X
    Keywords: Random vibration ; nonlinear systems ; harmonic and random excitations ; resonance ; stochastic averaging ; method of weighted residuals
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The probability distribution of the response of a nonlinearly damped system subjected to both broad-band and harmonic excitations is investigated. The broad-band excitation is additive, and the harmonic excitations can be either additive or multiplicative. The frequency of a harmonic excitation can be either near or far from a resonance frequency of the system. The stochastic averaging method is applied to obtain the Itô type stochastic differential equations for an averaged system described by a set of slowly varying variables, which are approximated as components of a Markov vector. Then, a procedure based on the concept of stationary potential is used to obtain the exact stationary probability density for a class of such averaged systems. For those systems not belonging to this class, approximate solutions are obtained using the method of weighted residuals. Application of the exact and approximate solution procedures are illustrated in two specific cases, and the results are compared with those obtained from Monte Carlo simulations.
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  • 7
    ISSN: 1573-269X
    Keywords: Ship dynamics ; nonlinear oscillations ; parametric excitation ; effects of bending and torsional elasticity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An enhanced mechanical model for simulating ship body oscillations and both the induced fluxural and twisting vibrations of the hull in the case of longitudinal seas is presented. The onset of parametric rolling, which may result from nonlinearly coupled heave-pitch-roll motions, and the effects of bending and torsional elasticity of the hull are considered in detail. It is shown that in the above sea conditions the flexural and/or twisting vibrations are likely to occur under a mechanism similar to that of parametric rolling.
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  • 8
    ISSN: 1573-269X
    Keywords: Strings ; nonlinear oscillations ; multiple modal interactions ; motion stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Nonlinear free transversal oscillations of an orbiting string satellite system are analyzed. They are governed by two partial integro-differential equations with quadratic nonlinearities. The system is weakly nonlinear but in practice works in conditions of nearly simultaneous internal resonance. The ability of truncated models to capture specific phenomena is discussed. By limiting the investigation to the planar motion with a one prevailing component perturbed out-of-plane, two different models with three modes in primary and secondary resonance are adopted. For increasing levels of the system energy, fundamental and bifurcated paths of fixed points are obtained and their stability is investigated. Moreover, periodically amplitude modulated planar motions and their stability for out-of-plane disturbances are studied.
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  • 9
    ISSN: 1573-269X
    Keywords: Identification ; nonlinear systems ; Gabor and Hilbert transforms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new technique for the identification of nonlinearity in multi-degree of freedom systems is presented. The technique is based on the joint application of the Gabor and the Hilbert transforms to the transient response of a system. The Gabor transform is used first to identify a time-variant matrix representing the spatial behaviour of the system. This matrix is then used to decouple the transient response into a set of uncoupled quasi-harmonic components. Finally the Hilbert transform is applied to identify the dissipative and restoring forces associated with each component which is equivalent to a single degree of freedom system. Numerical examples are supplied to help clarify the main advantages and the possible limitations of the method in the presence of strong nonlinearities and closely spaced frequencies.
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  • 10
    ISSN: 1573-269X
    Keywords: Parametric excitation ; nonlinear systems ; bifurcation ; bifurcation control ; stabilization ; nonlinear control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a parametrically excited Duffing system we propose a bifurcation control method in order to stabilize the trivial steady state in the frequency response and in order to eliminate jump in the force response, by employing a combined linear-plus-nonlinear feedback control. Because the bifurcation of the system is characterized by its modulation equations, we first determine the order of the feedback gain so that the feedback modifies the modulation equations. By theoretically analyzing the modified modulation equations, we show that the unstable region of the trivial steady state can be shifted and the nonlinear character can be changed, by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the suppression of the discontinuous bifurcation due to the change of the nonlinear character allows the elimination of the jump in the quasistationary force response. Furthermore, by performing numerical simulations, and by comparing the responses of the uncontrolled system and the controlled one, we clarify that the proposed bifurcation control is available for the stabilization of the trivial steady state in the frequency response and for the reduction of the jump in the nonstationary force response.
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