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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. v 
    ISSN: 1573-269X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-269X
    Keywords: numerical simulation ; chaos ; cable ; resonances ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities. The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 63-74 
    ISSN: 1573-269X
    Keywords: backlash ; impacts ; discrete mappings ; Fokker-Planck equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Rattling in gears is a consequence of backlashes. It can be generated in those gear-wheels of a car transmission system, which are not under load. Models of rattling are established in three stages. from a straightforward patching method we proceed to a discrete mapping description and to a stochastic model applying the well-known Fokker-Planck equation. All three model stages are consistent and, moreover, they represent a good picture of the real world, which has been proven by tests.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 91-116 
    ISSN: 1573-269X
    Keywords: internal resonance ; random vibrations ; non-Gaussian closure experiments
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents the experimental results of random excitation of a nonlinear two-degree-of-freedom system in the neighborhood of internal resonance. The random signals of the excitation and response coordinates are processed to estimate the mean squares, autocorrelation functions, power spectral densities, and probability density functions. The results are qualitatively compared with those predicted by the Fokker-Planck equation together with a non-Gaussian closure scheme. The effects of system damping ratios, nonlinear coupling parameter, internal detuning ratio, and excitation spectral density level are considered in both studies except the effect of damping ratios is not considered in the experimental investigation. Both studies reveal similar dynamic features such as autoparametric absorber effect and stochastic instability of the coupled system. The experimental results show that the autocorrelation function of the coupled system has the feature of ultra narrow band process and degenerates to a periodic one as the internal detuning departs from the exact internal resonance condition. The measured probability density functions of the response of the main system suggests that the Gaussian representation is sufticient as long as the excitation level is relatively low in the neighborhood of the system internal resonance condition.
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  • 5
    ISSN: 1573-269X
    Keywords: Quenching ; parametric resonances ; subharmonic resonances ; adaptive control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An indirect adaptive quenching algorithm for a nonlinear single-degree-of-freedom system with unknown constant system parameters is presented. The system is subject to external or parametric sinusoidal disturbances and the resulting control signal is also sinusoidal. The quenching algorithm provides a reduction in the control effort required compared to direct disturbance cancellation. The disturbance sinusoid and the unknown parameters are incorporated into the system model and an extended Kalman filter (EKF) with modified update equations is used to estimate the system state and parameters. The estimates are then used to form the quenching signal. The adaptive quenching algorithm is found to work well inside a quenching region defined by the separatrices and suggests the use of a hybrid control law. The algorithm was verified by implementing it on an analog computer.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 243-264 
    ISSN: 1573-269X
    Keywords: Aeroelastic flutter ; influence of maneuvering ; chaotic vibrations ; routes to chaos ; dissipative dynamical systems ; computational methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The influence of maneuvering on the chaotic response of a fluttering buckled plate on an aircraft has been studied. The governing equations, derived using Lagrangian mechanics, include geometric non-linearities associated with the occurrence of tensile stresses, as well as coupling between the angular velocity of the maneuver and the elastic degrees of freedom. Numerical simulation for periodic and chaotic responses are conducted in order to analyze the influence of the pull-up maneuver on the dynamic behavior of the panel. Long-time histories phase-plane plots, and power spectra of the responses are presented. As the maneuver (load factor) increases, the system exhibits complicated dynamic behavior including a direct and inverse cascade of subharmonic bifurcations, intermittency, and chaos. Beside these classical routes of transition from a periodic state to chaos, our calculations suggest amplitude modulation as a possible new mode of transition to chaos. Consequently this research contributes to the understanding of the mechanisms through which the transition between periodic and strange attractors occurs in, dissipative mechanical systems. In the case of a prescribed time dependent maneuver, a remarkable transition between the different types of limit cycles is presented.
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  • 7
    ISSN: 1573-269X
    Keywords: Power systems ; chaos ; bifurcations ; loss of synchronism
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate some of the instabilities in a single-machine quasi-infinite busbar system. The system's behavior is described by the so-called swing equation, which is a nonlinear second-order ordinary-differential equation with additive and multiplicative harmonic terms having the frequency Ω. When Ω≈ω0, where ω0 is the linear natural frequency of the machine, we use digital-computer simulations to exhibit some of the complicated responses of the machine, including period-doubling bifurcations, chaotic motions, and unbounded motions (loss of synchronism). To predict the onset of these complicated behaviors, we use the method of multiple scales to develop an approximate first-order closed-form expression for the period-one responses of the machine. Then, we use various techniques to determine the stability of the analytical solutions. The analytically predicted period-one solutions and conditions for its instability are in good agreement with the digital-computer results.
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  • 8
    ISSN: 1573-269X
    Keywords: Space-platform ; system-modes ; dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper studies libration/vibration interaction dynamics of the space station Freedom during its evolutionary phases. To that end, a relatively general nonlinear formulation, applicable to a system of interconnected plate and beam-type structural members forming a tree topology is developed. System modes, obtained through a finite element analysis, are employed in the discretization process and the response study is purposely confined to the orbital plane to emphasize interactions between librational dynamics and flexibility. Finally, an approximate closed-form nonlinear solution of the problem is attempted using the variation of parameters method. Its validity is assessed through comparison with the numerically obtained results over a range of system parameters and initial conditions. Results provide information pertaining to the levels of librational and vibrational response and the associated acceleration tield, which may prove helpful in appropriately locating experiments and monitoring instruments. It may also aid in the planning of the control system. The analytical approach provides surprisingly close correlation with the more elaborate numerical procedure, thus promising better physical appreciation of the complex interactions as well as a considerable saving in the computational cost.
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 421-432 
    ISSN: 1573-269X
    Keywords: Rotating cable structures ; weight-excited oscillations ; snap-through phenomena ; experiment and theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The combined effect of gravity and a centrifugal field on the dynamics of a heavy cable with negligible bending stiffness is treated. The inextensible length of the cable is preassigned in such a way that the cross-sectional shape of the cable structure is circular if the contiguration rotates about its central axis with a constant speed and no external forces act. Under the additional influence of gravity, complicated nonlinear weight-excited vibrations occur. To understand the variety of vibrational phenomena. The dynamic system is studied by some experiments first. In order to explain the experimental results theoretically, the governing nonlinear boundary value problem is derived next. Subsequently, an appropriate variational formulation for application of a Rayleigh Ritz procedure is suggested. The resulting nonlinear ordinary differential equations approximately deseribe a part of the observed vibrational behaviour. Both the experiments and the calculations demonstrate that, for high speeds, nonlinear weight-excited vibrations about the circular reference configuration occur. On the other hand, for low velocitites, periodie and even chaos-like snap-through phenomena appear.
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 433-447 
    ISSN: 1573-269X
    Keywords: Multibody systems ; clastic bodies ; geometric nonlinearities ; centrifugal stiffening
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Systems of rigid and flexible bodies undergoing large rigid body motions but small elastic deformations are investigated. In order to get the correct equations of motion one has to consider geometric nonlinearities even in the elastic coordinates. Different possibilities of independently choosing these coordinates are presented. The flexible bodies are discretized using a Ritz-Galerkin approximation. This discretization leads to ordinary differential equations for the description of the clastic vibrations of the flexible bodies as well as for the description of the rigid body motions. The modelling of deformable bodies is one aspect of our investigations. Another aim of our research is the consideration of clastic bodies in existing multibody programs, in particular in the program AUTOLEV. This is a symbol manipulating program for the formulation of the equations of motion running on PCs. Two examples are considered showing that it is possible to treat clastic bodies in this and in similar programs with the concept of modelling presented above.
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