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  • Liapunov–Floquet transformation  (1)
  • Picard iteration  (1)
  • control  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Stability and Control of a Parametrically Excited Rotating System. Part II: Controls (1998)
    Springer
    Dynamics and control 8 (1998), S. 19-35 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008274728933
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  • 2
    Electronic Resource
    Electronic Resource
    Normal Forms and the Structure of Resonance Sets in Nonlinear Time-Periodic Systems (2000)
    Springer
    Nonlinear dynamics 23 (2000), S. 35-55 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008312424551
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  • 3
    Electronic Resource
    Electronic Resource
    A new numerical technique for the analysis of parametrically excited nonlinear systems (1993)
    Springer
    Nonlinear dynamics 4 (1993), S. 483-498 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF00053692
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