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  • Artikel  (43)
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  • chaos  (43)
  • Springer  (43)
  • 2020-2022
  • 1995-1999  (43)
  • Mathematik  (43)
  • 1
    Digitale Medien
    Digitale Medien
    Potentialised Partial Differential Equations in Economic Geography and Spatial Economics: Multiple Dimensions and Control (1998)
    Springer
    Acta applicandae mathematicae 51 (1998), S. 1-23 
    Details
    ISSN: 1572-9036
    Schlagwort(e): partial differential equations ; potentials ; chaos ; control
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The present study is a follow-up to previous recent publications in the field of theoretical economic geography and spatial economics. Earlier results are generalised and simulated in higher dimensions (in terms of variables and topological dimensions), and given possible undesirable outcomes of the process (which can behave chaotically), application of control methods to it is being studied.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1005848024409
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  • 2
    Digitale Medien
    Digitale Medien
    Local instability and localization of attractors. From stochastic generator to Chua's systems (1995)
    Springer
    Acta applicandae mathematicae 40 (1995), S. 179-243 
    Details
    ISSN: 1572-9036
    Schlagwort(e): 58F10 ; 58F13 ; Lyapunov stability ; orbital stability ; Zhukovskij stability ; instability ; system in variations ; Lyapunov characteristic exponent ; Lyapunov function ; positive invariant set ; frequency-domain theorem ; strange attractor ; chaos ; nonlocal reduction method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00992721
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  • 3
    Digitale Medien
    Digitale Medien
    The analysis of correlation integrals in terms of extremal value theory (1998)
    Springer
    Bulletin of the Brazilian Mathematical Society 29 (1998), S. 197-228 
    Details
    ISSN: 1678-7714
    Schlagwort(e): time series ; (deterministic) ; chaos ; correlation integral ; predictability ; extremal index
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In this paper we first give an overview of the methods of analysis of time series in terms of correlation integrals, which were developed for time series generated by deterministic systems. From the extremal value theory one obtains asymptotic information on the behaviour of the correlation integrals of time series generated by non-deterministic (mixing) systems. This leads to an analysis in terms of correlation integrals which is complementary to the estimation of dimension and entropy.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01237649
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  • 4
    Digitale Medien
    Digitale Medien
    Numerical approximation of homoclinic chaos (1997)
    Springer
    Numerical algorithms 14 (1997), S. 25-53 
    Details
    ISSN: 1572-9265
    Schlagwort(e): dynamical systems ; numerical methods ; homoclinic points for maps ; multihumped homoclinic orbits ; chaos ; 58F13 ; 58F15 ; 58F08
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: Abstract Transversal homoclinic orbits of maps are known to generate a Cantor set on which a power of the map conjugates to the Bernoulli shift on two symbols. This conjugacy may be regarded as a coding map, which for example assigns to a homoclinic symbol sequence a point in the Cantor set that lies on a homoclinic orbit of the map with a prescribed number of humps. In this paper we develop a numerical method for evaluating the conjugacy at periodic and homoclinic symbol sequences in a systematic way. The approach combines our previous method for computing the primary homoclinic orbit with the constructive proof of Smale's theorem given by Palmer. It is shown that the resulting nonlinear systems are well conditioned uniformly with respect to the characteristic length of the symbol sequence and that Newton's method converges uniformly too when started at a proper pseudo orbit. For the homoclinic symbol sequences an error analysis is given. The method works in arbitrary dimensions and it is illustrated by examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1019196426363
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  • 5
    Digitale Medien
    Digitale Medien
    Steady state motions of shallow arch under periodic force with 1∶2 internal resonance on the plane of physical parameters (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 625-635 
    Details
    ISSN: 1573-2754
    Schlagwort(e): shallow arch ; internal resonance ; steady state motion ; bifurcation ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1∶2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to the types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02452370
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  • 6
    Digitale Medien
    Digitale Medien
    The study on the chaotic motion of a nonlinear dynamic system (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 830-836 
    Details
    ISSN: 1573-2754
    Schlagwort(e): chaos ; Melnikov method ; Poincaré map ; phase portrait ; time-displacement diagram ; O343.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract In this paper, the system of the forced vibration $$\ddot T - \lambda _1 T + \lambda _2 T^2 + \lambda _3 T^3 = \varepsilon \left( {g\cos \omega t - \varepsilon '\dot T} \right)$$ is discussed, which contains square and cubic items. The critical condition that the system enters chaotic states is given by the Melnikov method. By Poincaré map, phase portrait and time-displacement history diagram, whether the chaos occurs is determined.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02452482
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  • 7
    Digitale Medien
    Digitale Medien
    Control of the Lorenz chaos by the exact linearization (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 67-73 
    Details
    ISSN: 1573-2754
    Schlagwort(e): Reyleigh number ; Lorenz system ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract Controlling chaos in the Lorenz system with a controllable Rayleigh number is investigated by the state space exact linearization method. Based on proving the exact linearizability, the nonlinear feedback is utilized to design the transformation changing the original chaotic system into a linear controllable one so that the control is realized. Numerical examples of control are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02458982
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  • 8
    Digitale Medien
    Digitale Medien
    The double mode model of the chaotic motion for a large deflection plate (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 360-364 
    Details
    ISSN: 1573-2754
    Schlagwort(e): buckled plate ; chaos ; Poincaré section
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract The primary aim of this paper is to study the chaotic motion of a large deflection plate. Considered here is a buckled plate, which is simply supported and subjected to a lateral harmonic excitation. At first, the partial differential equation governing the transverse vibration of the plate is derived. Then, by means of the Galerkin approach, the partial differential equation is simplified into a set of two ordinary differential equations. It is proved that the double mode model is identical with the single mode model. The Melnikov method is used to give the approximate excitation thresholds for the occurrence of the chaotic vibration. Finally numerical computation is carried out.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02458561
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  • 9
    Digitale Medien
    Digitale Medien
    Bifurcation in a nonlinear duffing system with multi-frequency external periodic forces (1998)
    Springer
    Applied mathematics and mechanics 19 (1998), S. 121-128 
    Details
    ISSN: 1573-2754
    Schlagwort(e): nonlinear frequency ; Floquet theory ; bifurcation ; chaos ; Duffing system with multi-frequency external periodic forces
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract By introducing nonlinear freqyency, using Floquet theory and referring to the characteristics of the solution when it passes through the transition boundaries, all kinds of bifurcation modes and their transition boundaries of Duffing equation with two periodic excitations as well as the possible ways to chaos are studied in this paper.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02457679
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  • 10
    Digitale Medien
    Digitale Medien
    Chaotic motion of a nonlinear thermoelastic elliptic plate (1999)
    Springer
    Applied mathematics and mechanics 20 (1999), S. 960-966 
    Details
    ISSN: 1573-2754
    Schlagwort(e): thermoelasticity ; chaos ; Melnikov function ; Poincaré mapping ; O343.5
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Maschinenbau , Mathematik , Physik
    Notizen: Abstract In the paper the nonlinear dynamic equation of a harmonically forced elliptic plate is derived, with the effects of large deflection of plate and thermoelasticity taken into account. The Melnikov function method is used to give the critical condition for chaotic motion. A demonstrative example is discussed through the Poincaré mapping, phase portrait and time history. Finally the path to chaotic motion is also discussed. Through the theoretical analysis and numerical computation some beneficial conclusions are obtained.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02459058
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  • 11
    Digitale Medien
    Digitale Medien
    Chaotic oscillations in pipes conveying pulsating fluid (1996)
    Springer
    Nonlinear dynamics 10 (1996), S. 333-357 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Pipes ; parametric excitation ; nonlinearity ; chaos ; multiple time scales ; harmonic balancing ; stability
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Chaotic motions of a simply supported nonlinear pipe conveying fluid with harmonie velocity fluetuations are investigated. The motions are investigated in two flow velocity regimes, one below and above the critical velocity for divergence. Analyses are carried out taking into account single mode and two mode approximations in the neighbourhood of fundamental resonance. The amplitude of the harmonic velocity perturbation is considered as the control parameter. Both period doubling sequence and a sudden transition to chaos of an asymmetric period 2 motion are observed. Above the critical velocity chaos is explained in terms of periodic motion about the equilibrium point shifting to another equilibrium point through a saddle point. Phase plane trajectories, Poincaré maps and time histories are plotted giving the nature of motion. Both single and two mode approximations essentially give the same qualitative behaviour. The stability limits of trivial and nontrivial solutions are obtained by the multiple time scale method and harmonic balance method which are in very good agreement with the numerical results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00045481
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  • 12
    Digitale Medien
    Digitale Medien
    Chaotic Oscillation of Saddle Form Cable-Suspended Roofs under Vertical Excitation Action (1997)
    Springer
    Nonlinear dynamics 12 (1997), S. 57-68 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Saddle form cable-suspended roofs ; truncated spectral model ; bifurcation ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The purpose of this paper is to investigate the dynamic behaviour of saddle form cable-suspended roofs under vertical excitation action. The governing equations of this problem are system of nonlinear partial differential and integral equations. We first establish a spectral equation, and then consider a model with one coefficient, i.e., a perturbed Duffing equation. The analytical solution is derived for the Duffing equation. Successive approximation solutions can be obtained in likely way for each time to only one new unknown function of time. Numerical results are given for our analytical solution. By using the Melnikov method, it is shown that the spectral system has chaotic solutions and subharmonic solutions under determined parametric conditions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008241710683
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  • 13
    Digitale Medien
    Digitale Medien
    Events Maps in a Stick-Slip System (1997)
    Springer
    Nonlinear dynamics 13 (1997), S. 99-115 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Stick-slip vibrations ; one-dimensional maps ; reconstruction ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This paper describes a one-dimensional map generated by a two degree-of-freedom mechanical system that undergoes self-sustained oscillations induced by dry friction. The iterated map allows a much simpler representation and a better understanding of some dynamic features of the system. Some applications of the map are illustrated and its behaviour is simulated by means of an analytically defined one-dimensional map. A method of reconstructing one-dimensional maps from experimental data from the system is introduced. The method uses cubic splines to approximate the iterated mappings. From a sequence of such time series the parameter dependent bifurcation behaviour is analysed by interpolating between the defined mappings. Similarities and differences between the bifurcation behaviour of the exact iterated mapping and the reconstructed mapping are discussed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008228120608
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  • 14
    Digitale Medien
    Digitale Medien
    Nonlinear Dynamics of Shells: Theory, Finite Element Formulation, and Integration Schemes (1997)
    Springer
    Nonlinear dynamics 13 (1997), S. 279-305 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Nonlinear shell dynamics ; integration schemes ; chaos ; finite elements
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008251113479
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  • 15
    Digitale Medien
    Digitale Medien
    Bifurcation and Chaos in the Duffing Oscillator with a PID Controller (1997)
    Springer
    Nonlinear dynamics 12 (1997), S. 251-262 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Bifurcation ; chaos ; Duffingoscillator ; fractal basin boundary ; PIDcontroller
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We discuss in this paper the bifurcation, stability and chaos of the non-linear Duffing oscillator with a PID controller. Hopf bifurcation can occur and we show that there is a global stable fixed point. The PID controller works well in some fields of the parameter space, but in other fields of the parameter space, or if the reference input is not equal to zero, chaos is common for hard spring type system and so is fractal basin boundary for soft spring system. The Melnikov method is used to obtain the criterion of fractal basin boundary.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008204332684
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  • 16
    Digitale Medien
    Digitale Medien
    Chaotic Dynamics of a Harmonically Excited Spring-Pendulum System with Internal Resonance (1997)
    Springer
    Nonlinear dynamics 14 (1997), S. 211-229 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Spring-pendulum system ; chaos ; Lyapunovexponent
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008256920441
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  • 17
    Digitale Medien
    Digitale Medien
    A study of the chaotic behaviour of the elastic-plastic Euler arch (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 217-229 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Nonlinear dynamics ; chaos ; elastic-plastic material ; Euler arch
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This paper deals with the dynamics of a truss structure, the Euler arch. The bars are made of elastic-plastic material, and the structure can exhibit large displacements. The aim of this paper is to give evidence of the possible chaotic behavior of this structure, even in the presence of a hardening plastic branch. The tools used are the diagrams of bifurcation, the measure of the dimension of the attractor, the Kolmogorov entropy, and the maximum Lyapunov exponent. This study emphasizes the sensitivity to the initial conditions by means of generalized basins of attraction.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00053709
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  • 18
    Digitale Medien
    Digitale Medien
    Parameter variation methods for cell mapping (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 273-284 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Cell mapping ; continuation ; basins of attraction ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In the study of nonlinear dynamic systems, the influence of system parameters on the long term behaviour plays an important role. In this paper, parameter variation methods are presented which can be used when investigating a nonlinear dynamic system by means of simple or interpolated cell mapping. In the case of coexisting attractors, the proposed methods determine the evolution of the basin boundaries when a system parameter is varied. Application of the methods to a modified Duffing equation is performed. It is concluded that the proposed methods are very efficient and accurate.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00046303
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  • 19
    Digitale Medien
    Digitale Medien
    Lie transformation method for dynamical systems having chaotic behavior (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 403-428 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Lie transformation (perturbation method) ; dynamical systems ; small divisors ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The paper persents recent developments in a singular perturbation method, known as the “Lie transformation method” for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of “small divisors” in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00121106
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  • 20
    Digitale Medien
    Digitale Medien
    Chaos in a mapping describing elastoplastic oscillations (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 111-139 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Elastoplastic oscillator ; piecewise linear map ; chaos ; Smale horseshoe ; symbolic dynamics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We study the local and global dynamical behavior of a two dimensional piecewise linear map which describes the asymptotic motions of a single degree of freedom, parametrically excited, elastoplastic oscillator after it has settled down to purely elastic oscillations. We give existence and stability conditions for periodic orbits and prove that chaos, in the form of a Smale horseshoe, exists at specific, but representative, parameter values. We interpret simulations of the elastoplastic oscillator itself in the light of these results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00045009
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  • 21
    Digitale Medien
    Digitale Medien
    A Modified Exact Linearization Control for Chaotic Oscillators (1999)
    Springer
    Nonlinear dynamics 20 (1999), S. 309-317 
    Details
    ISSN: 1573-269X
    Schlagwort(e): nonlinear oscillations ; chaos ; control ; input-output linearization
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008312908664
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  • 22
    Digitale Medien
    Digitale Medien
    Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part II: Experimental Analysis (1999)
    Springer
    Nonlinear dynamics 19 (1999), S. 387-397 
    Details
    ISSN: 1573-269X
    Schlagwort(e): rotor dynamics ; chaos ; conical whirl ; restoring forces
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In Part I, theoretical analysis of the dynamic behaviour of a rigid rotor with nonlinear elastic restoring forces was carried out. In this part (Part II), an experimental confirmation of the theoretical data from that analysis was sought. With this aim, an experimental model was set up consisting mainly of a practically rigid rotor clamped onto a small diameter piano wire symmetrical to the wire supports. These supports were rigid and equipped with roller bearings and a device that made it possible to adjust the initial tension in the wire so as to make the elastic restoring forces less or more linear. The rotor was dynamically unbalanced and was driven by an asynchronous motor regulated by means of an inverter in order to adjust the rotor speed. A series of tests was performed on this rig with different values of the initial tension in the wire, and the trajectories of two points on the rotor axis were recorded in the course of the tests. These trajectories were obtained, under the hypothesis of similarity, from the orbits covered by two given sections of the wire and detected with two pairs of capacitive transducers. The collected data was compared with the theoretical results from Part I of the present investigation. Comparison of the collected data with the corresponding theoretical results made it possible to infer that system nonlinearity in the presence of small damping can give rise to motions that are periodic, whether synchronous or not, or quasi-periodic, but never chaotic.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008358319488
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  • 23
    Digitale Medien
    Digitale Medien
    Three routes to chaos for a three-degree-of-freedom articulated cylinder system subjected to annular flow and impacting on the outer pipe (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 429-450 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Articulated cylinder ; chaos ; period-doubling ; intermittency ; quasi-periodicity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In this paper, the dynamics of a cantilevered articulated system of rigid cylinders interconnected by rotational springs, within a pipe containing fluid flow is studied. Although the formulation is generalized to any number of degrees-of-freedom (articulations), the present work is restricted to three-degree-of-freedom systems. The motions are considered to be planar, and the equations of motion, apart from impacting terms, are linearized. Impacting of the articulated cylinder system on the outer pipe is modelled by either a cubic spring (for analytical convenience) or, more realistically, by a trilinear spring model. The critical flow velocities, for which the system loses stability, by flutter (Hopf bifurcation) or divergence (pitchfork bifurcation) are determined by an eigenvalue analysis. Beyond these first bifurcations, it is shown that, for different values of the system parameters, chaos is obtained through three different routes as the flow is incremented: a period-doubling cascade, the quasiperiodic route, and type III intermittency. The dynamical behaviour of the system and differing routes to chaos are illustrated by phase-plane portraits, bifurcation diagrams, power spectra, Poincaré sections, and Lyapunov exponent calculations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00121107
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  • 24
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    Digitale Medien
    A simple feedback control system: Bifurcations of periodic orbits and chaos (1996)
    Springer
    Nonlinear dynamics 9 (1996), S. 391-417 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Bifurcation ; chaos ; pendulum ; feedback control ; second-order averaging ; Melnikov method ; experiment
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01833363
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  • 25
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    Digitale Medien
    An experimental study of bifurcation, chaos, and dimensionality in a system forced through a bifurcation parameter (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 467-489 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Experimental ; bifurcation ; chaos ; fractal dimension ; parametric excitation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00045709
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  • 26
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    Digitale Medien
    The analytical predictive criteria for chaos and escape in nonlinear oscillators: A survey (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 129-147 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Nonlinear oscillations ; chaos ; escape ; predictive criteria
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The purpose of this paper is to provide a brief summary of the various analytical predictive criteria in order for “strange phenomena” to occur in a class of softening nonlinear oscillators, oscillators which may exhibit escape from a potential well. Implications of Melnikov's criteria are discussed first and transient chaos in the twin-well potential oscillator is illustrated. Three different heuristic criteria for steady state chaos or escape solution, proposes by F. Moon, G. Schmidt and W. Szemplińskia-Stupnicka, are then presented and compared to computer simulation results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00053705
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  • 27
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    Digitale Medien
    Nonlinear interactions in a parametrically excited system with widely spaced frequencies (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 195-216 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Energy transfer ; bifurcations ; chaos ; crises
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00053708
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  • 28
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    Digitale Medien
    Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium (1998)
    Springer
    Nonlinear dynamics 15 (1998), S. 83-102 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Bifurcation ; chaos ; Lyapunov exponents ; stable and unstable manifolds ; Latté's method ; convective flow
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008220923483
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  • 29
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    Digitale Medien
    Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams (1998)
    Springer
    Nonlinear dynamics 15 (1998), S. 31-61 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Beams ; internal resonance ; parametric resonance ; bifurcations ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008218009139
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  • 30
    Digitale Medien
    Digitale Medien
    Deteriorated Geometrical Stiffness for Higher Order Finite Elements with Application to Panel Flutter (1999)
    Springer
    Nonlinear dynamics 18 (1999), S. 89-103 
    Details
    ISSN: 1573-269X
    Schlagwort(e): higher order finite elements ; panel flutter ; nonlinear ; postbuckling ; chaos ; dynamic response
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Higher order elements were first design for linear problems where, in certain situations, they present advantages over the lower order elements. A method to efficiently extend their use to geometrical nonlinear problems as panel flutter and postbuckling behavior is presented. The chaotic and limit-cycle oscillations of an isotropic plate are obtained based on direct integration of the discretized equation of motion. The plate is modeled using the von Karman theory and the geometrical nonlinearities are separated in a nonlinear term of the first kind which manifests especially in the prebuckling and buckling regimes, and a nonlinear term of the second kind which is responsible for the postbuckling behavior. A fifth order, fully compatible element has been used to model thin plates while the inplane loads where introduced through a membrane element. The aerodynamics was modeled using the first order 'piston theory’. The method introduces the concept of a deteriorated form of the second geometric matrix which is equivalent to neglecting higher order terms in the strain energy of the plate. This allows for a drastic reduction in the computational effort with no observable loss of accuracy. Well established results in the literature are used to validate the method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008348332176
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  • 31
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    Digitale Medien
    Chaos in the Imbalance Response of a Flexible Rotor Supported by Oil Film Bearings with Non-Linear Suspension (1998)
    Springer
    Nonlinear dynamics 16 (1998), S. 71-90 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Oil film journal bearing-rotor system ; power spectrum ; chaos ; fractaldimension
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The dynamics of flexible rotors associated with fluid film bearings have been studied since the 1950s. Most of the literature has assumed rigid, undamped bearing support with linear elastic restoring force. For a more precise description of fluid film bearing-rotor systems, a non-linearly supported model is proposed in this paper, where a linear damping force and a non-linear elastic restoring force are assumed. Numerical results show that due to non-linear factors, though the dynamic equations of the bearing center and the rotor center are coupled, the trajectory of the rotor center demonstrates steady-state symmetric motion even when the trajectory of the bearing center is in a state of disorder. Poincaré maps, bifurcation diagrams, and power spectra are used to analyze the behavior of the bearing center in the horizontal and vertical directions under different operating conditions. The fractal dimension concept is used to determine whether the system is in a state of chaotic motion. Numerical results find that the dimension of the bearing center trajectory is fractal and greater than two in some operating conditions, indicating that the system is in a state of chaotic motion. Chaotic behavior was found in an intermediate speed range, disappearing at higher speeds. It is suggested that this is a characteristic of all fluid film systems. It is suggested that a number of existing life-critical fluid film bearing systems are possibly being operated in this chaotic region and that such systems should be reevaluated in terms of this new observation.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008239206153
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  • 32
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    Digitale Medien
    A Nonlinear Temporal Headway Model of Traffic Dynamics (1998)
    Springer
    Nonlinear dynamics 16 (1998), S. 127-151 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Car-following models ; nonlinear systems ; delay-differential equations ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008279031113
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  • 33
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    Digitale Medien
    Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System (1999)
    Springer
    Nonlinear dynamics 19 (1999), S. 313-332 
    Details
    ISSN: 1573-269X
    Schlagwort(e): double pendulum system ; double Hopf bifurcation ; stability ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two-dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008347523779
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  • 34
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    Digitale Medien
    Periodic and Homoclinic Motions in Forced, Coupled Oscillators (1999)
    Springer
    Nonlinear dynamics 20 (1999), S. 319-359 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Melnikov method ; bifurcation ; chaos ; weak coupling
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We study periodic and homoclinic motions in periodically forced, weakly coupled oscillators with a form of perturbations of two independent planar Hamiltonian systems. First, we extend the subharmonic Melnikov method, and give existence, stability and bifurcation theorems for periodic orbits. Second, we directly apply or modify a version of the homoclinic Melnikov method for orbits homoclinic to two types of periodic orbits. The first type of periodic orbit results from persistence of the unperturbed hyperbolic periodic orbit, and the second type is born out of resonances in the unperturbed invariant manifolds. So we see that some different types of homoclinic motions occur. The relationship between the subharmonic and homoclinic Melnikov theories is also discussed. We apply these theories to the weakly coupled Duffing oscillators.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008336402517
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  • 35
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    Digitale Medien
    Global bifurcations in externally excited two-degree-of-freedom nonlinear systems (1995)
    Springer
    Nonlinear dynamics 8 (1995), S. 85-109 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Resonance ; global bifurcations ; homoclinic orbits ; Melnikov ; Silnikov ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In this study we examine the global dynamics associated with a generic two-degree-of-freedom (2-DOF), coupled nonlinear system that is externally excited. The method of averaging is used to obtain the second order approximation of the response of the system in the presence of one-one internal resonance and subharmonic external resonance. This system can describe a variety of physical phenomena such as the motion of an initially deflected shallow arch, pitching vibrations in a nonlinear vibration absorber, nonlinear response of suspended cables etc. Using a perturbation method developed by Kovačič and Wiggins (1992), we show the existence of Silnikov type homoclinic orbits which may lead to chaotic behavior in this system. Here two different cases are examined and conditions are obtained for the existence of Silnikov type chaos.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00045008
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  • 36
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    Digitale Medien
    Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis (1997)
    Springer
    Nonlinear dynamics 14 (1997), S. 57-87 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Rotor dynamics ; rigid rotors ; chaos ; conical whirl ; journal bearings
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the π-film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008282014350
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  • 37
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    Digitale Medien
    Dynamics of an Unbalanced Shaft Interacting with a Limited Power Supply (1997)
    Springer
    Nonlinear dynamics 13 (1997), S. 171-187 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Unbalanced shaft ; nonlinear vibrations ; limited power supply ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Rotation of an elastically mounted unbalanced shaft may be in general affected by its lateral vibration, due to a “vibrational torque”. This interaction is nonlinear, and can be neglected only in case of an unlimited power supply. Whenever the available power of the drive is comparable with power consumption due to vibration, various nonlinear phenomena may be observed, the most well-known of these being the so called Sommerfeld effect – slowing down or complete capture of the shaft at resonance. The corresponding steady-state motions and their stability can be studied by asymptotic methods, as applied to the governing nonlinear set of two second-order equations. However, study of transient motions in general requires numerical solution. This numerical solution is obtained here, and extensive parametric studies are performed of the Sommerfeld effect. In particular, the influence is evaluated of damping ratio and slope of the torque-speed curve of the drive on a passage/capture threshold. The results of numerical simulation, as well as experiments with a physical model, also demonstrate the effect of smooth passage through resonance with a limited power supply, based on using a “switch” of suspension stiffness from a certain artificially increased value to the design one. A brief description is presented also of time-variant components of the resonant amplitude and rotational frequency responses in the case of capture, as observed both in numerical simulation studies and in experiments. Whilst these components are small compared with the corresponding constant ones, i.e. steady-state vibration amplitude and rotational frequency, the above observations indicate the possibility for periodic or chaotic nonstationarity in the system's response.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008205012232
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  • 38
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    Digitale Medien
    On a New Route to Chaos in Railway Dynamics (1997)
    Springer
    Nonlinear dynamics 13 (1997), S. 117-129 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Period-doubling bifurcations ; chaos ; quasiperiodicity ; railway dynamics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Cooperrider's mathematical model of a railway bogie running on a straight track has been thoroughly investigated due to its interesting nonlinear dynamics (see True [1] for a survey). In this article a detailed numerical investigation is made of the dynamics in a speed range, where many solutions exist, but only a couple of which are stable. One of them is a chaotic attractor. Cooperrider's bogie model is described in Section 2, and in Section 3 we explain the method of numerical investigation. In Section 4 the results are shown. The main result is that the chaotic attractor is created through a period-doubling cascade of the secondary period in an asymptotically stable quasiperiodic oscillation at decreasing speed. Several quasiperiodic windows were found in the chaotic motion. This route to chaos was first described by Franceschini [9], who discovered it in a seven-mode truncation of the plane incompressible Navier–Stokes equations. The problem investigated by Franceschini is a smooth dynamical system in contrast to the dynamics of the Cooperrider truck model. The forcing in the Cooperrider model includes a component, which has the form of a very stiff linear spring with a dead band simulating an elastic impact. The dynamics of the Cooperrider truck is therefore “non-smooth”. The quasiperiodic oscillation is created in a supercritical Neimark bifurcation at higher speeds from an asymmetric unstable periodic oscillation, which gains stability in the bifurcation. The bifurcating quasiperiodic solution is initially unstable, but it gains stability in a saddle-node bifurcation when the branch turns back toward lower speeds. The chaotic attractor disappears abruptly in what is conjectured to be a blue sky catastrophe, when the speed decreases further.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008224625406
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  • 39
    Digitale Medien
    Digitale Medien
    Phase-Locking and Bifurcations of the Sinusoidally-Driven Double Scroll Circuit (1998)
    Springer
    Nonlinear dynamics 17 (1998), S. 119-139 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Phase-locking ; bifurcations ; chaos ; electronic circuits
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The dynamic alterations of an electronic circuit in a chaotic regime, described by the Double Scroll attractor, subjected to sinusoidal perturbation are numerically investigated. Parameter diagrams of the circuit phase-locking oscillations in terms of the driving amplitude and frequency are computed. These diagrams have highly interleaved and complex structures, part of them Cantor-like fractals. However, a Cantor-like fractal structure is also observed. In addition, the power spectrum analysis is used to find and characterize three ways of phase-locking the Double Scroll circuit, and to determine how this process depends on the driving parameters. Furthermore, the dynamics of bifurcation phenomena, as chaotic attractor entrainment, Arnold's tongues, coexistence of attractors, and hysteresis are identified in the parameter space.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008284804398
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  • 40
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    Digitale Medien
    Influence of Initial Conditions on the Postcritical Behavior of a Nonlinear Aeroelastic System (1998)
    Springer
    Nonlinear dynamics 15 (1998), S. 63-81 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Aeroelasticity ; nonlinear vibration ; panel flutter ; bifurcation ; attractor ; chaos ; attraction basin
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract The behavior of a nonlinear, non-Hamiltonian system in the postcritical (flutter) domain is studied with special attention to the influence of initial conditions on the properties of attractors situated at a certain point of the control parameter space. As a prototype system, an elastic panel is considered that is subjected to a combination of supersonic gas flow and quasistatic loading in the middle surface. A two natural modes approximation, resulting in a four-dimensional phase space and several control parameters is considered in detail. For two fixed points in the control parameter space, several plane sections of the four-dimensional space of initial conditions are presented and the asymptotic behavior of the final stationary responses are identified. Amongst the latter there are stable periodic orbits, both symmetric and asymmetric with respect to the origin, as well as chaotic attractors. The mosaic structure of the attraction basins is observed. In particular, it is shown that even for neighboring initial conditions can result in distinctly different nonstationary responses asymptotically approach quite different types of attractors. A number of closely neighboring periodic attractors are observed, separated by Hopf bifurcations. Periodic attractors also are observed under special initial conditions in the domains where chaotic behavior is usually expected.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008204409853
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  • 41
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    Digitale Medien
    Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part II: Experimental Analysis (1997)
    Springer
    Nonlinear dynamics 14 (1997), S. 157-189 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Rotor dynamics ; rigid rotors ; chaos ; conical whirl ; journal bearings
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In the first part of the present investigation [9], the dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. In the present paper an experimental confirmation of the theoretical results is sought. The steel rotor of the experimental rig was given a constant circular cross section in order to fix in an easy way the two distances between supports corresponding, respectively, to the values of the λ parameter assigned in [9]. Two steel rings, each one with a series of holes and a clamping screw, were mounted onto the rotor with a small clearance. This arrangement made it possible to fix the positions of the rings and their holes respect to the rotor, so as to realize a pre-estabilished unbalance. The two bronze journal bearings were characterised by a relatively low length/diameter ratio, and a relatively high value of the radial clearance and were lubricated with oil delivered from a thermostatic tank. In this way, despite the relative lightness of the rotor, the dimensionless static eccentricity εs was given the high values that were apt to realize the operating conditions assumed in the theoretical analysis. The rotor was driven by means of a d.c. motor connected to a toothed belt-drive. Varying the rotor speed in the range 1000 ÷ 10000 r.p.m., made it possible to assign the values of the modified Sommerfeld number assumed in the theoretical analysis. Three pairs of eddy-current probes were mounted in order to detect the trajectories of three points (C1, C and C2) suitably fixed along the rotor axis. These orbits were finally put in comparison with the corresponding ones previously obtained through numerical analysis. The comparison pointed out that the experimental data were in good agreement with the theoretical predictions, despite the approximations that characterise the theoretical model and the unavoidable errors affecting measures in the course of the experimental test.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008275231189
    Standort Signatur Erwartet Verfügbarkeit
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  • 42
    Digitale Medien
    Digitale Medien
    Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis (1999)
    Springer
    Nonlinear dynamics 19 (1999), S. 359-385 
    Details
    ISSN: 1573-269X
    Schlagwort(e): rotor dynamics ; chaos ; conical whirl ; restoring forces
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1008336006400
    Standort Signatur Erwartet Verfügbarkeit
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  • 43
    Digitale Medien
    Digitale Medien
    Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator (1995)
    Springer
    Nonlinear dynamics 7 (1995), S. 249-272 
    Details
    ISSN: 1573-269X
    Schlagwort(e): Numerical and geometrical techniques ; local and global bifurcations ; chaos
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics. Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF00053711
    Standort Signatur Erwartet Verfügbarkeit
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