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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 15 (1998), S. 225-244 
    ISSN: 1573-269X
    Keywords: Localization ; cyclic systems ; disorder ; nonlinear
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The dynamics of weakly coupled, nonlinear cyclic assemblies are investigated in the presence of weak structural mistuning. The method of multiple scales is used to obtain a set of nonlinear algebraic equations which govern the steady-state, synchronous (‘modal-like’) motions for the structures. Considering a degenerate assembly of uncoupled oscillators, spatially localized modes are computed corresponding to motions during which vibrational energy is spatially confined to one oscillator (strong localization) or a subset of oscillators (weak localization). In the limit of weak substructural coupling, asymptotic solutions are obtained which correspond to (i) spatially extended, (ii) strongly localized, and (iii) weakly localized modes for fully coupled systems. Throughout the analysis, the influence of structural mistunings on the resulting solutions are discussed. Additionally, numerical solutions (including linearized stability characteristics) are obtained for prototypical two- and three-degree-of-freedom (DoF) systems with various structural mistunings. The numerical results provide insight into the strong influence of structural irregularities on the dynamical behavior of nonlinear cyclic systems, and demonstrate that the combined influences of structural mistunings and nonlinearities do not lead to uniform improvement of motion confinement characteristics.
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  • 2
    ISSN: 1573-269X
    Keywords: nonlinear ; time-periodic systems ; local bifurcations ; versal deformation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 3 (1992), S. 375-384 
    ISSN: 1573-269X
    Keywords: Roller-coaster ; experimental ; nonlinear
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the dynamics of ‘roller-coaster’ type experimental models used as analog devices for nonlinear oscillators. It is shown how to chose the shape of the track in order to achieve a desired oscillator equation, in terms of the are length coordinate or its projection onto the horizontal. Explicit calculations are carried out for the linear oscillator, the so-called ‘escape equation’, the two-well Duffing oscillator, and the pendulum.
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  • 4
    ISSN: 1573-269X
    Keywords: Symbolic computation ; stability ; bifurcation ; nonlinear ; time-periodic
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincaré map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.
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  • 5
    ISSN: 1573-269X
    Keywords: higher order finite elements ; panel flutter ; nonlinear ; postbuckling ; chaos ; dynamic response
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Higher order elements were first design for linear problems where, in certain situations, they present advantages over the lower order elements. A method to efficiently extend their use to geometrical nonlinear problems as panel flutter and postbuckling behavior is presented. The chaotic and limit-cycle oscillations of an isotropic plate are obtained based on direct integration of the discretized equation of motion. The plate is modeled using the von Karman theory and the geometrical nonlinearities are separated in a nonlinear term of the first kind which manifests especially in the prebuckling and buckling regimes, and a nonlinear term of the second kind which is responsible for the postbuckling behavior. A fifth order, fully compatible element has been used to model thin plates while the inplane loads where introduced through a membrane element. The aerodynamics was modeled using the first order 'piston theory’. The method introduces the concept of a deteriorated form of the second geometric matrix which is equivalent to neglecting higher order terms in the strain energy of the plate. This allows for a drastic reduction in the computational effort with no observable loss of accuracy. Well established results in the literature are used to validate the method.
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Nonlinear dynamics 22 (2000), S. 393-413 
    ISSN: 1573-269X
    Keywords: piecewise linear ; nonlinear ; vibration ; absorber
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper explores the potential of the piecewise linearvibration absorber in a system subject to narrow band harmonic loading.Such a spring is chosen because the design of linear springs is commonknowledge among engineers. The two-degrees-of-freedom system is solvedby using the Incremental Harmonic Balance method, and response aspectssuch as stiffness crossing frequency and jump behaviour are discussed.The effects of mass, stiffness, natural frequency ratios, and stiffnesscrossing positions on the suppression zone are probed. It is shown thata hardening absorber can deliver a wider bandwidth than a linear oneover a range of frequencies. The absorber parameters needed to producegood designs have been determined and the quality of the realizedsuppression zone is discussed. Design guidelines are formulated to aidthe parameter selection process.
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  • 7
    ISSN: 1573-269X
    Keywords: Fluidelasticity ; stability ; nonlinear ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, the planar dynamics of a nonlinearly constrained pipe conveying fluid is examined numerically, by considering the full nonlinear equation of motions and a refined trilinear-spring model for the impact constraints—completing the circle of several studies on the subject. The effect of varying system parameters is investigated for the two-degree-of-freedom (N=2) model of the system, followed by less extensive similar investigations forN=3 and 4. Phase portraits, bifurcation diagrams, power spectra and Lyapunov exponents are presented for a selected set of system parameters, showing some rather interesting, and sometimes unexpected, results. The numerical results are compared with experimental ones obtained previously. It is found that in the parameter space that includesN, there exists a subspace wherein excellent qualitative, and reasonably good (N=2) to excellent (N=4) quantitative agreement with experiment. In the latter case, excellent agreement is not only obtained in the threshold flow velocities (u) for the key bifurcations, but the inclusion of the nonlinear terms improves agreement with experiment in terms of amplitudes of motion and by capturing features of behaviour not hitherto predicted by theory.
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  • 8
    ISSN: 1573-269X
    Keywords: Laminated spherical caps ; static and dynamic analysis ; transverse shear deformation ; nonlinear
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Based on Timoshenko-Mindlin kinematic hypothesis, the shallow shell theory is extended to include the transverse shear deformation for the nonlinear axisymmetric dynamic analysis of the symmetric cross-ply shallow spherical shell. Using the orthogonal point collocation method and the Newmark scheme, an iterative solution is formulated. The numerical results for the nonlinear static and dynamic responses and dynamic buckling of these shallow spherical shells with circular holes under uniformly distributed static or dynamic normal impact loads are presented and compared with available data.
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  • 9
    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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  • 10
    ISSN: 1573-269X
    Keywords: Time-periodic systems ; nonlinear ; time-invariant forms ; critical systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.
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