ALBERT

All Library Books, journals and Electronic Records Telegrafenberg

Your email was sent successfully. Check your inbox.

An error occurred while sending the email. Please try again.

Proceed reservation?

Export
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 12 (1997), S. 1-37 
    ISSN: 1573-269X
    Keywords: Chaos ; dynamic bifurcations ; matchedasymptotic expansions ; averaging ; nonlinearoscillator ; perturbation methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A study is made of the dynamics of oscillating systems with a slowly varying parameter. A slowly varying forcing periodically crosses a critical value corresponding to a pitchfork bifurcation. The instantaneous phase portrait exhibits a centre when the forcing does not exceed the critical value, and a saddle and two centres with an associated double homoclinic loop separatrix beyond this value. The aim of this study is to construct a Poincaré map in order to describe the dynamics of the system as it repeatedly crosses the bifurcation point. For that purpose averaging methods and asymptotic matching techniques connecting local solutions are applied. Given the initial state and the values of the parameters the properties of the Poincaré map can be studied. Both sensitive dependence on initial conditions and (quasi) periodicity are observed. Moreover, Lyapunov exponents are computed. The asymptotic expressions for the Poincaré map are compared with numerical solutions of the full system that includes small damping.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 2
    ISSN: 1573-269X
    Keywords: Chaos ; numerical simulation ; spacecraft ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dynamics of a simplified model of a spinning spacecraft with a circumferential nutational damper is investigated using numerical simulations for nonlinear phenomena. A realistic spacecraft parameter configuration is investigated and is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Such a torque, in practice, may arise in the platform of a dual-spin spacecraft under malfunction of the control system or from an unbalanced rotor or from vibrations in appendages. The equations of motion of the model are derived with Lagrange's equations using a generalisation of the kinetic energy equation and a linear stability analysis is given. Numerical simulations for satellite parameters are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. The motion is studied by means of time history, phase space, frequency spectrum, Poincaré map, Lyapunov characteristic exponents and Correlation Dimension. For sufficiently large values of torque amplitude, the behaviour of the system was found to have much in common with a two well potential problem such as a Duffing oscillator. Evidence is also presented, indicating that the onset of chaotic motion was characterised by period doubling as well as intermittency.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 3
    ISSN: 1573-269X
    Keywords: Chaos ; rubbing ; rotor dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents an application for chaotic motion identification in a measured signal obtained in an experiment. The method of state space reconstruction with delay co-ordinates with the dynamic evolution described by a map is used. Poincaré diagrams, correlation dimensions and Lyapunov exponents are obtained as tools for deciding about the existence of chaotic behaviour. The method is applied to measurements of the lateral displacement of a vertical rotor experiencing rubbing and in some signals chaos is observed. The work concludes that the possibility of chaotic motion is well determined with the observation of Poincaré diagrams and computation of Lyapunov exponents. Correlation dimensions computations, strongly influenced by noise, are not convenient tools for investigation of chaotic behaviour in signals generated by mechanical systems.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 5 (1994), S. 421-432 
    ISSN: 1573-269X
    Keywords: Chaos ; unbalance ; journal ; bearings
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The behaviour of non-linear systems often yield unexpected phenomena which are extremely sensitive to initial conditions. The hydrodynamic journal bearing is a common machine element which is strongly nonlinear for large excursions within the clearance space. A simple model of a rigid journal, supported hydrodynamically using a short bearing theory is shown to behave chaotically when the rotating unbalance force exceeds the gravitational load. At these values of the force ratio the time history of the response is very sensitive to initial conditions and a spectral analysis demonstrates a significant broadening from the expected peak at the rotational frequency. A once per revolution sampling of the time history (Poincaré plot) revealed an apparent aperiodic pattern. An estimate of the fractal dimension using the Grasberger-Procaccia algorithm resulted in a lower bound of 2.15, a typical result for low dimensional systems with significant dissipative action. The required levels of unbalance are only an order of magnitude greater than acceptable levels for rotating machinery and thus could be achieved with in-service erosion or minor damage. The subsequent non-synchronous response could result in fatigue and potential shaft failure.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 6 (1994), S. 125-142 
    ISSN: 1573-269X
    Keywords: Chaos ; pendulum ; feedback control ; Melnikov method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 401-420 
    ISSN: 1573-269X
    Keywords: Chaos ; perturbation methods ; elliptic functions ; differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate the system $$\ddot x - x\cos \varepsilon 1 + x^3 = 0$$ in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the $$x - \dot x$$ phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 〉 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 291-304 
    ISSN: 1573-269X
    Keywords: Chaos ; Fokker-Planck-Equation ; probability density ; global description of motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An observation of single trajectories exhibiting chaotic motion turns out to be disadvantageous because even smallest variations of the initial conditions grow exponentially in time and result in an unpredictable long-time behaviour. The paper gives a different approach based on a probability distribution of the state space variables which is invariant on the area of attraction and results in a global description of chaotic motion.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 139-152 
    ISSN: 1573-269X
    Keywords: Chaos ; complex function ; Melnikov criterion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper presents an extension of Melnikov's theory for the differential equation with complex function. The sufficient condition for the existence of a homoclinic orbit in the solutions of a perturbed equation is given. The method shown in the paper is used to derive a precursor criterion for chaos. Suitable conditions are defined for the parameters of equations for which the equation possesses a strange attractor set. The analytical results are compared with numerical ones, and a good agreement is found between them.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 9
    ISSN: 1573-269X
    Keywords: Chaos ; buckled beam ; parametric resonance ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
  • 10
    ISSN: 1573-269X
    Keywords: Chaos ; unstable cycles ; embedding ; parameter variations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present study applies the recently developed ideas in experimental system modeling to both characterize the behavior of simple mechanical systems and detect variations in their parameters. First, an experimental chaotic time series was simulated from the solution of the differential equation of motion of a mechanical system with clearance. From the scalar time series, a strange attractor was reconstructed optimally by the method of delays. Optimal reconstructions of the attractors can be achieved by simultaneously determining the minimal necessary embedding dimension and the proper delay time. Periodic saddle orbits were extracted from the chaotic orbit and their eigenvalues were calculated. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents for the steady state motion. An analysis of the associated one dimensional delay map, obtained from the chaotic time series, is made to determine the allowable periodic orbits and to yield an estimate of the topological entropy for the positive Lyapunov exponent. Sensitivity of the positions of the low order unstable periodic orbits (orbits of short period) of a chaotic attractor is used as a basis for detection of parameter variations in another unsymmetric bilinear system. For the experimental scalar time series generated by the dynamical system as a parameter varies, the chaotic attractors were again optimally reconstructed using the method of delays. The parameter variations were detected by the changes in location of the unstable periodic orbits extracted from the reconstructed attractors of the experimental scalar time series.
    Type of Medium: Electronic Resource
    Location Call Number Expected Availability
    BibTip Others were also interested in ...
Close ⊗
This website uses cookies and the analysis tool Matomo. More information can be found here...