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  • 1
    ISSN: 1573-8450
    Keywords: control ; systems with periodic coefficients ; Lyapunov-Floquet transformation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract In Part-I of this paper, the stability of a parametrically excited rotating system was analyzed. In this part the design of a feedback controller and an observer for the same mechanical system is considered. First, the time periodic system equations are transformed to a time invariant form which is suitable for an application of the standard techniques of linear control theory. A full-state feedback controller is designed in the transformed domain using the pole placement technique. Next, a Luenberger observer is constructed for estimating the unmeasurable states. Robustness of the observer is tested under the assumption that white noise is present in the measured states. Simulations for several combination of excitation and rotation parameters are provided.
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  • 2
    ISSN: 1619-6937
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
    Notes: Summary The stability of a uniform cantilever column supported by a Maxwell type viscoelastic foundation and subjected to a constant tangential force is investigated. Stability conditions are obtained for the entire range of system parameters through an application of Routh-Hurwitz criteria. Unlike the case of conservative loading, the Maxwell foundation is shown to produce a stabilizing effect on this nonconservative problem. Furthermore, an optimum combination of foundation parameters exist to yield the maximum flutter load. An approximate analysis is also presented through an application of the Galerkin's method. It is shown that a two-term approximation may not be adequate to yield meaning-ful results in a certain range of system parameters.
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  • 3
    ISSN: 1573-8450
    Keywords: time-varying periodic systems ; Floquet theory ; State Transition Matrix ; helicopter blades
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract This paper analyses the stability of a parametrically excited double pendulum rotating in the horizontal plane. The equations of motion for such a system contain time varying periodic coefficients. Floquet theory and the method of Hill's determinant are used to evaluate the stability of the linearized system. Stability charts are obtained for various sets of damping, parametric excitation, and rotation parameters. Several resonance conditions are found, and it is shown that the system stability can be significantly altered due to the rotation. Such systems can be used as preliminary models for studying the lag dynamics and control of helicopter blades and other gyroscopic systems.
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  • 4
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Description / Table of Contents: Zusammenfassung Die asymptotischen Stabilitätsresultate von Prichard für die Benard-und Taylor-Probleme, die mit Hilfe der Liapunov-Movchan-Theorie erhalten worden sind, werden durch Ungleichungen und Methoden der Variationsrechnung optimiert. Die Aequivalenz zwischen diesem Resultat und dem Ergebnis der Energiemethode wird nachgewiesen. Mögliche Anwendungen werden diskutiert, die sich auf die Symmetrie der Operatoren beziehen.
    Notes: Abstract The asymptotic stability result obtained by Pritchard for the Benard and Taylor problems employing the Liapunov-Movchan theory is optimized by using inequalities and variational techniques. The equivalence between this result and the one obtained by the energy theory is demonstrated. Future applications as related to the symmetry of the operators are discussed.
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  • 5
    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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  • 6
    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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  • 7
    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.
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  • 8
    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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  • 9
    ISSN: 1573-269X
    Keywords: dynamic instability ; autoparametric system ; experiment ; chaotic motion ; nonlinear motion ; symbolic computational technique ; Chebyshev polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Experimental and numerical investigations are carried out on anautoparametric system consisting of a composite pendulum attached to aharmonically base excited mass-spring subsystem. The dynamic behavior ofsuch a mechanical system is governed by a set of coupled nonlinearequations with periodic parameters. Particular attention is paid to thedynamic behavior of the pendulum. The periodic doubling bifurcation ofthe pendulum is determined from the semi-trivial solution of thelinearized equations using two methods: a trigonometric approximation ofthe solution and a symbolic computation of the Floquet transition matrixbased on Chebyshev polynominal expansions. The set of nonlineardifferential equations is also integrated with respect to time using afinite difference scheme and the motion of the pendulum is analyzed viaphase-plane portraits and Poincaré maps. The predicted resultsare experimentally validated through an experimental set-up equippedwith an opto-electronic set sensor that is used to measure the angulardisplacement of the pendulum. Period doubling and chaotic motions areobserved.
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  • 10
    ISSN: 1573-269X
    Keywords: normal forms ; time-periodic systems ; Liapunov–Floquet transformation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.
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