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  • bifurcation  (5)
  • method of multiple scales  (4)
  • 1990-1994  (9)
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 3 (1992), S. 261-271 
    ISSN: 1573-269X
    Keywords: Scaling behavior ; coupled nonlinear oscillator ; method of multiple scales ; Duffing equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The scaling of the solution of coupled conservative weakly nonlinear oscillators is demonstrated and analyzed through evaluating the normal modes and their bifurcation with an equivalent linearization technique and calculating the general solutions with a method of multiple seales. The scaling law for coupled Duffing oscillators is that the coupling intensity should be proportional to the total energy of the system.
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  • 2
    ISSN: 1573-269X
    Keywords: Nonlinear vibration of a beam ; three mode interaction ; mid-plane stretching ; method of multiple scales
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An analysis is presented for the primary resonance of a clamped-hinged beam, which occurs when the frequency of excitation is near one of the natural frequencies,ωn . Three mode interaction (ω2 ≈ 3ω1 and ω3 ≈ ω1 + 2ω2) is considered and its influence on the response is studied. The case of two mode interaction (ω2 ≈ 3ω1) is also considered to compare it with the case of three mode interaction. The straight beam experiencing mid-plane stretching is governed by a nonlinear partial differential equation. By using Galerkin's method the governing equation is reduced to a system of nonautonomous ordinary differential equations. The method of multiple scales is applied to solve the system. Steady-state responses and their stability are examined. Results of numerical investigations show that there exists no significant difference between both modal interactions' influences on the responses.
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  • 3
    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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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 251-268 
    ISSN: 1573-269X
    Keywords: Singular perturbation problems ; nonlinear effects ; turbulence via chaos techniques ; bifurcation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Flow-induced oscillations of rigid cylinders exposed to fully developed turbulent flow can be described by a fourth order autonomous system. Among the pertinent constants, the mass ratio is the control parameter governing the transition from limit cycle oscillations to chaotic vibrations. Particular attention is paid to the stability of the limit cycles: it has been found that they lose their stability at the point of appearance of quasi-periodic motion. The documentation of this transition is performed in terms of Lyapunov exponents, phase plots, Fourier spectra, bifurcation diagrams, and Poincaré maps. As opposed to the calculation of the Lyapunov exponents where remarkable numerical difficulties were encountered, the investigation of the remaining quantities shows clearly the passage of cylinder motions from limit cycle oscillations to more and more irregular vibrations, leading finally to chaos.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 389-408 
    ISSN: 1573-269X
    Keywords: Plate ; bifurcation ; chaos ; homoclinic orbit
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovačič and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.
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  • 6
    ISSN: 1573-269X
    Keywords: Slider-crank mechanism ; nonlinear resonance ; dynamic stability ; method of multiple scales
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The transverse vibrations of a flexible connecting rod in an otherwise rigid slider-crank mechanism are considered. An analytical approach using the method of multiple scales is adopted and particular emphasis is placed on nonlinear effects which arise from finite deformations. Several nonlinear resonances and instabilities are investigated, and the influences of important system parameters on these resonances are examined in detail.
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  • 7
    ISSN: 1573-269X
    Keywords: Shaft ; stability ; bifurcation ; analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dynamic stability and self-excited posteritical whirling of rotating transversally loaded shaft made of a standard material with elastic and viscous nonlinearities are analyzed in this paper using the theory of bifurcations as a mathematical tool. Partial differential equations of motion are derived under assumption that von Karman's nonlinearity is absent but geometric curvature nonlinearity is included. Galerkin's first-mode discretization procedure is then applied and the equations of motion are transformed to two third-order nonlinear equations that are analyzed using the theory of bifurcation. Condition for nontrivial equilibrium stability is determined and a bifurcating periodic solution of the second-order approximation is derived. The effects of dimensionless stress relaxation time and cubic elastic and viscous nonlinearities as well as the role of the transverse load are studied in the exemplary numerical calculations. A strongly stabilizing influence of the relaxation time is found that may eliminate self-excited vibration at all. Transition from super- to subcritical bifurcation is observed as a result of interaction between system nonlinearities and the transverse load.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 293-311 
    ISSN: 1573-269X
    Keywords: Oil whirl ; bifurcation ; unbalance ; instability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear behavior of an unbalanced rotor supported in a fluid film bearing is analyzed. A simplified two dimensional model is adopted which uses the long-bearing approximation with a π-film to account for cavitation. This model has been thoroughly studied by Myers [1] in the balanced case, where it is shown that the whirl instability is the result of a Hopf bifurcation. The implications of imbalance are studied in this paper. This leads to the study of a periodically perturbed Hopf bifurcation. It is shown that the dynamics in this situation can, especially under certain nonlinear resonance conditions, have an extremely complicated dependence on the system parameters and the rotor speed. Complete bifurcation diagrams are presented for a particular rotor model which demonstrate this dependence.
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  • 9
    ISSN: 1573-269X
    Keywords: Collocation ; stability ; bifurcation ; modal interaction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A methodology is first presented for analyzing long time response of periodically exited nonlinear oscillators. Namely, a systematic procedure is employed for determining periodic steady state response, including harmonic and superharmonic components. The stability analysis of the located periodic motions is also performed, utilizing results of Froquet theory. This methodology is then applied to a special class of two degree of freedom nonlinear oscillators, subjected to harmonic excitation. The numberical results presented in the second part of this study illustrate effects caused by the interaction of the modes as well as effects of the nonlinearities on the steady state response of these oscillators. In addition, sequences of bifurcations are analyzed for softening systems, leading to unbounded response of the model examined. Finally, the importance of higher harmonics on the response of systems with strongly nonlinear characteristics is investigated.
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