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  • Chaos  (8)
  • method of multiple scales  (4)
  • 1990-1994  (12)
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 401-420 
    ISSN: 1573-269X
    Keywords: Chaos ; perturbation methods ; elliptic functions ; differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We investigate the system $$\ddot x - x\cos \varepsilon 1 + x^3 = 0$$ in which ε≪1 by using averaging and elliptie functions. It is shown that this system is applicable to the dynamics of the familiar rotating-plane pendulum. The slow foreing permits us to envision an ‘instantancous phase portrait’ in the $$x - \dot x$$ phase plane which exhibits a center at the origin when cos ε1≤0 and a saddle and associated double homoclinic loop separatrix when cos ɛ 1 〉 0. The chaos in this problem is related to the question of on which side (left (=L) or right (=R)) of the reappearing double homoclinic loop separatrix a motion finds itself. We show that the sequence of L's and R's exhibits sensitive dependence on initial conditions by using a simplified model which assumes that motions cross the instantancous separatrix instantancously. We also present an improved model which ‘patches’ a separatrix boundary layer onto the averaging model. The predictions of both models are compared with the results of numerical integration.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 2 (1991), S. 291-304 
    ISSN: 1573-269X
    Keywords: Chaos ; Fokker-Planck-Equation ; probability density ; global description of motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An observation of single trajectories exhibiting chaotic motion turns out to be disadvantageous because even smallest variations of the initial conditions grow exponentially in time and result in an unpredictable long-time behaviour. The paper gives a different approach based on a probability distribution of the state space variables which is invariant on the area of attraction and results in a global description of chaotic motion.
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  • 3
    ISSN: 1573-269X
    Keywords: Chaos ; unstable cycles ; embedding ; parameter variations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present study applies the recently developed ideas in experimental system modeling to both characterize the behavior of simple mechanical systems and detect variations in their parameters. First, an experimental chaotic time series was simulated from the solution of the differential equation of motion of a mechanical system with clearance. From the scalar time series, a strange attractor was reconstructed optimally by the method of delays. Optimal reconstructions of the attractors can be achieved by simultaneously determining the minimal necessary embedding dimension and the proper delay time. Periodic saddle orbits were extracted from the chaotic orbit and their eigenvalues were calculated. The eigenvalues associated with the saddle orbits are used to estimate the Lyapunov exponents for the steady state motion. An analysis of the associated one dimensional delay map, obtained from the chaotic time series, is made to determine the allowable periodic orbits and to yield an estimate of the topological entropy for the positive Lyapunov exponent. Sensitivity of the positions of the low order unstable periodic orbits (orbits of short period) of a chaotic attractor is used as a basis for detection of parameter variations in another unsymmetric bilinear system. For the experimental scalar time series generated by the dynamical system as a parameter varies, the chaotic attractors were again optimally reconstructed using the method of delays. The parameter variations were detected by the changes in location of the unstable periodic orbits extracted from the reconstructed attractors of the experimental scalar time series.
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  • 4
    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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  • 5
    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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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 4 (1993), S. 139-152 
    ISSN: 1573-269X
    Keywords: Chaos ; complex function ; Melnikov criterion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper presents an extension of Melnikov's theory for the differential equation with complex function. The sufficient condition for the existence of a homoclinic orbit in the solutions of a perturbed equation is given. The method shown in the paper is used to derive a precursor criterion for chaos. Suitable conditions are defined for the parameters of equations for which the equation possesses a strange attractor set. The analytical results are compared with numerical ones, and a good agreement is found between them.
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  • 7
    ISSN: 1573-269X
    Keywords: Chaos ; buckled beam ; parametric resonance ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A nonlinear analysis of the response of a simply-supported buckled beam to a harmonic axial load is presented. The method of multiple scales is used to determine to second order the amplitude- and phase-modulation equations. Floquet theory is used to analyze the stability of periodic responses. The perturbation results are verified by integrating the governing equation using both digital and analog computers. For small excitation amplitudes, the analytical results are in good agreement with the numerical solutions. The large-amplitude responses are investigated by using a digital computer and are compared with those obtained via an analog-computer simulation. The complicated dynamic behaviors that were found include period-multiplying and period-demultiplying bifurcations, period-three and period-six motions, jump phenomena, and chaos. In some cases, multiple periodic attractors coexist, and a chaotic attractor coexists with a periodic attractor. Phase portraits, spectra of the responses, and a bifurcation set of the many solutions are presented.
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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 6 (1994), S. 125-142 
    ISSN: 1573-269X
    Keywords: Chaos ; pendulum ; feedback control ; Melnikov method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study chaotic dynamics of a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small inductance, so that the feedback control system reduces to a periodic perturbation of a planar Hamiltonian system. This Hamiltonian system can possess multiple saddle points with non-transverse homoclinic and/or heteroclinic orbits. Using Melnikov's method, we obtain criteria for the existence of chaos in the pendulum motion. The computation of the Melnikov functions is performed by a numerical method. Several numerical examples are given and the theoretical predictions are compared with numerical simulation results for the behavior of invariant manifolds.
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  • 10
    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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