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  • parametric excitation  (6)
  • method of multiple scales  (4)
  • 1990-1994  (10)
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  • 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.
    Type of Medium: Electronic Resource
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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: 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.
    Type of Medium: Electronic Resource
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  • 4
    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.
    Type of Medium: Electronic Resource
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  • 5
    ISSN: 1573-269X
    Keywords: Nonlinear dynamic systems ; parametric excitation ; numerical integration ; Picard iteration ; Chebyshev polynomials ; periodic and aperiodic solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.
    Type of Medium: Electronic Resource
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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.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 131-141 
    ISSN: 1573-269X
    Keywords: Ship dynamics ; nonlinear vibrations ; parametric excitation ; heave-pitch-roll motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two different models for simulating the ship motion in longitudinal or oblique seas are presented and studied in detail. Particular attention is devoted to the parametrically induced rolling which may be established by means of the nonlinear coupling between both heave-roll and/or pitch-roll motions. It is proved that the phenomenon is likely to occur with this mechanism when the roll frequency is subharmonic of the encounter wave frequency and when the vertical motions become resonant.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 1 (1990), S. 359-377 
    ISSN: 1573-269X
    Keywords: Vibrations ; parametric excitation ; Kane's equations ; Floquet theory ; axial motions ; cantilever beam
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Studied in this work are the formulation of equations of motion and the response to parametric excitation of a uniform cantilever beam moving longitudinally over a single bilateral support. The equations of motion are generated by using assumed modes to discretize the beam, by regarding the support as a kinematic constraint, and by employing an alternate form of Kane's method that is particularly well suited to systems subject to constraints. Instability information is developed using the results of perturbation analysis for harmonic longitudinal motions of small amplitude and with Floquet theory for general periodic motions of any amplitude. Results demonstrate that definitive instability information can be obtained for a beam moving longitudinally over supports based on the frequencies of free transverse vibration of a beam that is longitudinally fixed.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 3 (1992), S. 41-56 
    ISSN: 1573-269X
    Keywords: Ship dynamics ; nonlinear oscillations ; parametric excitation ; effect of torsional elasticity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new mechanical model for simulating both the ship oscillations and the induced twisting of the hull in the case of longitudinal seas is presented. Particular attention is given to the onset of parametric rolling, which may result from non-linearly coupled heave-pitch-roll motions. It is shown that in these sea conditions the phenomenon of twisting is likely to occur under a mechanism similar to that of parametric rolling.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 6 (1994), S. 317-329 
    ISSN: 1573-269X
    Keywords: Global transient stability ; integrity measures ; parametric excitation ; basins of attraction ; chaotic transients ; fractal control boundaries
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system. We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.
    Type of Medium: Electronic Resource
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