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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 20 (1999), S. 33-53 
    ISSN: 1573-269X
    Keywords: nonlinear vibrations ; cylindrical shells ; time domain ; random excitation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An analytical study of nonlinear flexural vibrations of cylindrical shells to random excitation is presented. Donnell's thin-shell theory is used to develop the governing equations of motion. Thermal effects for a uniform temperature rise through the shell thickness are included in the formulation. A Monte Carlo simulation technique of stationary random processes, multi-mode Galerkin-like approach and numerical integration procedures are used to develop nonlinear response solutions of simply-supported cylindrical shells. Numerical results include time domain response histories, root-mean-square values and histograms of probability density. Comparison of Monte Carlo results is made to those obtained by statistical linearization and the Fokker–Planck equation.
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  • 2
    ISSN: 1573-269X
    Keywords: Unbalanced shaft ; nonlinear vibrations ; limited power supply ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Rotation of an elastically mounted unbalanced shaft may be in general affected by its lateral vibration, due to a “vibrational torque”. This interaction is nonlinear, and can be neglected only in case of an unlimited power supply. Whenever the available power of the drive is comparable with power consumption due to vibration, various nonlinear phenomena may be observed, the most well-known of these being the so called Sommerfeld effect – slowing down or complete capture of the shaft at resonance. The corresponding steady-state motions and their stability can be studied by asymptotic methods, as applied to the governing nonlinear set of two second-order equations. However, study of transient motions in general requires numerical solution. This numerical solution is obtained here, and extensive parametric studies are performed of the Sommerfeld effect. In particular, the influence is evaluated of damping ratio and slope of the torque-speed curve of the drive on a passage/capture threshold. The results of numerical simulation, as well as experiments with a physical model, also demonstrate the effect of smooth passage through resonance with a limited power supply, based on using a “switch” of suspension stiffness from a certain artificially increased value to the design one. A brief description is presented also of time-variant components of the resonant amplitude and rotational frequency responses in the case of capture, as observed both in numerical simulation studies and in experiments. Whilst these components are small compared with the corresponding constant ones, i.e. steady-state vibration amplitude and rotational frequency, the above observations indicate the possibility for periodic or chaotic nonstationarity in the system's response.
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  • 3
    ISSN: 1573-269X
    Keywords: Experimental analysis ; cables ; nonlinear vibrations ; modal interaction ; bifurcations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A ‘local’ analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 19 (1999), S. 193-235 
    ISSN: 1573-269X
    Keywords: cylinder gyroscopes ; nonlinear vibrations ; energy transfer
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we model and analyze nonlinear oscillations which are known to exist in some Coriolis-based gyroscopes due to large amplitude excitation in the drive loop. A detailed derivation of a dynamic model for a cylinder gyroscope including geometric nonlinearities is given, and the energy transfer between the system's modes is analyzed using a perturbation theory and by proposing a simplified model. The model is also simulated, and the results are shown to give an accurate description of the experimental results. This work is carried out in order to gain a better understanding of the gyroscope's dynamics, and is intended to be a starting point for designing nonlinear observers and vibration controllers for the gyroscope in order to increase the performance.
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  • 5
    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.
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  • 6
    ISSN: 1573-269X
    Keywords: Symmetry ; structural dynamics ; nonlinear vibrations ; group theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show in this work that use of group-theoretic ideas can greatly simplify the procedure for finding both linear vibration modes and frequencies and locating solution branches for nonlinear vibration problems. These methods can be employed in both analytic and numerical settings, and they lead to dramatic increases in solving efficiency via block diagonalization. For the class of problems studied here (in which gyroscopic effects are present), we demonstrate that the original structural symmetry group can be employed to analyze the dynamics of the system despite the loss of reflection symmetry. This allows for a simpler and more intuitive implementation of the group representation techniques. In addition, we explain how ideas from bifurcation theory can be included with symmetry analysis to prove the existence of solution branches for nonlinear vibration problems. These examples provide great insight to the many advantages of using group-theoretic methods in structural vibrations problems.
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 14 (1997), S. 335-355 
    ISSN: 1573-269X
    Keywords: Normal forms ; beam on nonlinear supports ; nonlinear vibrations ; external and internal resonances
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.
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  • 8
    ISSN: 1573-269X
    Keywords: nonlinear vibrations ; parametric instability ; whirling motions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the ‘transmitted’ Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.
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