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  • nonlinear oscillations  (9)
  • Hopf bifurcation  (7)
  • 1
    ISSN: 1573-269X
    Keywords: Cables ; active control ; nonlinear oscillations ; bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear oscillations of a controlled suspended elastic cable under in-plane excitation are considered. Active control realized by longitudinal displacement of one support is introduced in order to reduce the transverse in-plane and out-of-plane vibrations. Linear and quadratic enhanced velocity feedback control laws are chosen and their effects on the cable motion are investigated using a two degree-of-freedom model. Perturbation analysis is performed to determine the in-plane steady-state solutions and their stability under an out-of-plane disturbance. The analysis is extended to the bifurcated two-mode steady-state oscillations in the region of parametric excitation. The dependence of the control effectiveness on the system parameters is investigated in the case of the first symmetric mode and the range of oscillation amplitudes in which the proposed control ensures a dissipation of energy is determined. Although control based only on in-plane response quantities is effective in reducing oscillations with a prevailing in-plane component, addition of out-of-plane measures has to be considered when the motion is characterized by two comparable components.
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  • 2
    ISSN: 1573-269X
    Keywords: Vehicle dynamics ; Hopf bifurcation ; chaotic motion ; fractal basin boundaries
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear dynamics of a differential system describing the motion of a vehicle driven by a pilot is examined. In a first step, the stability of the system near the critical speed is analyzed by the bifurcation method in order to characterize its behavior after a loss of stability. It is shown that a Hopf bifurcation takes place, the stability of limit cycles depending mainly on the vehicle and pilot model parameters. In a second step, the front wheels of the vehicle are assumed to be subjected to a periodic disturbance. Chaotic and hyperchaotic motions are found to occur for some range of the speed parameter. Numerical simulations, such as bifurcation diagrams, Poincaré maps, Fourier spectrums, projection of trajectories, and Lyapunov exponents are used to establish the existence of chaotic attractors. Multiple attractors may coexist for some values of the speed, and basins of attraction for such attractors are shown to have fractal geometries.
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  • 3
    ISSN: 1573-269X
    Keywords: Ship dynamics ; nonlinear oscillations ; parametric excitation ; effects of bending and torsional elasticity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An enhanced mechanical model for simulating ship body oscillations and both the induced fluxural and twisting vibrations of the hull in the case of longitudinal seas is presented. The onset of parametric rolling, which may result from nonlinearly coupled heave-pitch-roll motions, and the effects of bending and torsional elasticity of the hull are considered in detail. It is shown that in the above sea conditions the flexural and/or twisting vibrations are likely to occur under a mechanism similar to that of parametric rolling.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 6 (1994), S. 301-316 
    ISSN: 1573-269X
    Keywords: Valve mechanism of internal combustion engine ; parametric and forcing excitation ; nonlinear oscillations ; local and global bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study in detail problems of nonlinear oscillations of valve mechanism at internal combustion engine. The practical measurement indicates that stiffness of valve mechanism is not constant but is a function of the rotational angle of the cam. For simplicity of analysis we replace valve mechanism of internal combustion engine with a nonlinear oscillator of single degree of freedom under combined parametric and forcing excitation. We use the method of multiple scales and normal form theory to study local and global bifurcations of valve mechanism at internal combustion engine.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 20 (1999), S. 309-317 
    ISSN: 1573-269X
    Keywords: nonlinear oscillations ; chaos ; control ; input-output linearization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The control of chaotic oscillations is investigated in this paper. A control methodology, termed input-output linearization, is modified by locally linearizing the nonlinear control law in the small neighborhood of the control goal. Its suitability for controlling chaotic oscillators is analyzed. The forced Duffing oscillator is treated as a numerical example of controlling chaotic motion to a given fixed point and a given period-2 motion. The control signals and time needed to achieve the desired goals of the modified method are compared with those of the original method. The robustness of the control law is demonstrated.
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  • 6
    ISSN: 1573-269X
    Keywords: 4WS vehicle ; time delay ; stability ; Hopf bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A mathematical model is presented for four-wheel-steeringvehicles, with the time delay in driver's response and the nonlinearityin lateral tyre forces taken into account. It is proved that thevehicle-driver system has a trivial steady state motion, as well aseight non-trivial steady state motions due to the nonlinearity of tyreforces. The asymptotic stability and Hopf bifurcation of the trivialsteady state are analyzed for two control strategies ofrear-wheel-steering. It is shown through the numerical simulations thatthe four-wheel-steering technique based on the bilinear control strategyworks better when the driver's response involves time delay.
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  • 7
    ISSN: 1573-269X
    Keywords: Beam ; gravity effect ; method of multiple scales ; nonlinear oscillations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A critical problem in designing large structures for space applications, such as space stations and parabolic antennas, is the limitation of testing these structures and their substructures on earth. These structures will exhibit very high flexibilities due to the small loads expected to be encountered in orbit. It has been reported in the literature that the gravitational sag effect under dead weight is of extreme importance during ground tests of space-station structural components [1–4]. An investigation of a horizontal, pinned-pinned beam with complete axial restraint and undergoing large-amplitude oscillations about the statically deflected position is presented here. This paper presents a solution for the frequency-amplitude relationship of the nonlinear free oscillations of a horizontal, immovable-end beam under the influence of gravity. The governing equation of motion used for the analysis is the Bernoulli-Euler type modified to include the effects of mid-plane stretching and gravity. Boundary conditions are simply supported such that at both ends there is no bending moment and no transverse and axial displacements. These boundary conditions give rise to an initial tension in the statically deflected shape. The displacement function consists of an assumed space mode using a simple sine function and unknown amplitude which is a function of time. This assumption provides for satisfaction of the boundary conditions and leads to an ordinary differential equation which is nonlinear, containing both quadratic and cubic functions of the amplitude. The perturbation method of multiple scales is used to provide an approximate solution for the fundamental frequency-amplitude relationship. Since the beam is initially deflected the small-amplitude fundamental natural frequency always increases relative to the free vibration situation provided in zero gravity. The nonlinear equation provides for interactions between frequency and amplitude in that both hardening and softening effects arise. The coefficient of the quadratic term in the nonlinear equation arises from the static (dead load) portion of the deflection. This quadratic term, depending upon its magnitude, introduces a softening effect that overcomes the hardening term (due to initial axial tension developed by deflection) for large slenderness ratios. For very large slender, immovable-end beams, the fundamental natural frequency is greater than that of beams without axial constraints undergoing small amplitude oscillations. This phenomenon is attributed to the stiffening effect of the statically-induced axial tension. However, the stiffening effect of axial tension in beams with slenderness ratios greater than approximately 392 undergoing large-amplitude symmetric-mode oscillations is overpowered by the presence of gravitational loading.
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  • 8
    ISSN: 1573-269X
    Keywords: Beams ; nonlinear oscillations ; flexural-torsional dynamics ; torsional response ; nonlinear resonance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Nonlinear coupling between torsional and both in-plane and out-of-plane flexural motion is examined for inextensional beams (or beam-like structures) whose torsional and flexural eigenfrequencies are of the same order. The analysis presented here is based on a consistent set of nonlinear differential equations which contain both curvature and inertia nonlinearities, and account for torsional dynamics. Response characteristics, including stability, are determined for cantilever beams subjected to a lateral periodic excitation. The beam's response in the presence of a one-to-one internal resonance involving a torsional frequency and an in-plane bending frequency is investigated in detail.
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  • 9
    ISSN: 1573-269X
    Keywords: grazing bifurcation ; impact oscillations ; pipe conveying fluid ; Hopf bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The transition from stable periodic nonimpacting motion to impacting motion, due to variations of parameters, is observable in a wide range of vibro-impact systems. Recent theoretical studies suggest a possible scenario for this type of transition. A key element in the proposed scenario is fulfilled if the oscillatory motion involved in the transition is born in a supercritical Hopf bifurcation. If the onset of impacting motion is close to the Hopf bifurcation, the impacting motion is likely to be chaotic. A numerical simulation of a system of articulated pipes conveying fluid is used to illuminate the theory. An experimental setup is presented, where a cantilevered pipe conveying fluid is unilaterally constrained. Results from experiments are found to be in good qualitative agreement with the theory.
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  • 10
    ISSN: 1573-269X
    Keywords: Rail ; vehicle ; hunting ; nonlinear ; Hopf bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An analytical investigation of Hopf bifurcation and hunting behavior of a rail wheelset with nonlinear primary yaw dampers and wheel-rail contact forces is presented. This study is intended to complement earlier studies by True et al., where they investigated the nonlinearities stemming from creep-creep force saturation and nonlinear contacts between a realistic wheel and rail profile. The results indicate that the nonlinearities in the primary suspension and flange contact contribute significantly to the hunting behavior. Both the critical speed and the nature of bifurcation are affected by the nonlinear elements. Further, the results show that in some cases, the critical hunting speed from the nonlinear analysis is less than the critical speed from a linear analysis. This indicates that a linear analysis could predict operational speeds that in actuality include hunting.
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