Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

Sinha, Subhash C.; Redkar, Sangram; Deshmukh, Venkatesh; Butcher, Eric A.

doi:10.1007/s11071-005-2822-z

Cited by 19 publications

(9 citation statements)

References 20 publications

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“…Due to this better practical performance, the postprocessing technique has been applied to the study of nonlinear shell vibrations [37], as well as to stochastic differential parabolic equations [38]. Also, it has been effectively applied to reduce the order of practical engineering problems modeled by nonlinear differential systems [42,43].…”

Section: In the Present Paper We Analyze The Errors U(t N ) −ũ (N) H mentioning

confidence: 99%

Postprocessing Finite-Element Methods for the Navier–Stokes Equations: The Fully Discrete Case

Frutos¹,

García-Archilla²,

Novo³

2009

SIAM J. Numer. Anal.

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Abstract. An accuracy-enhancing postprocessing technique for finite-element discretizations of the Navier-Stokes equations is analyzed. The technique had been previously analyzed only for semidiscretizations, and fully discrete methods are addressed in the present paper. We show that the increased spatial accuracy of the postprocessing procedure is not affected by the errors arising from any convergent time-stepping procedure. Further refined bounds are obtained when the timestepping procedure is either the backward Euler method or the two-step backward differentiation formula.

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Section: In the Present Paper We Analyze The Errors U(t N ) −ũ (N) H mentioning

confidence: 99%

Postprocessing Finite-Element Methods for the Navier–Stokes Equations: The Fully Discrete Case

Frutos¹,

García-Archilla²,

Novo³

2009

SIAM J. Numer. Anal.

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“…1, also known as the Ziegler column [32,40,42]. The rigid and inextensible bars of length 2l have a mass m. The two bars are allowed to rotate at points O and B thanks to elastic hinges characterized by a rotational stiffness k. At rest, the biarticulated structure is lying in the horizontal direction (O, x).…”

Section: Nonlinear Equation Of Motionmentioning

confidence: 99%

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column

Bentvelsen

Lazarus

2017

Nonlinear Dyn

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We present a spectral method to compute the transverse vibrational modes, or Floquet Forms (FFs), of a 2D bi-articulated bar in periodic elastic state due to an end harmonic compressive force. By changing the directional nature of the applied load, the trivial straight Ziegler column exhibits the classic instabilities of stationary states of dynamical system. We use this simple structure as a numerical benchmark to compare the various spectral methods that consist in computing the FFs from the spectrum of a truncated Hill matrix. We show the necessity of sorting this spectrum and the benefit of computing the fundamental FFs that converge faster. Those FFs are almost-periodic entities that generalize the concept of harmonic modal analysis of structures in equilibria to structures in periodic states. Like their particular harmonic relatives, FFs allow to get physical insights in the bifurcations of periodic stationary states. Notably, the local loss of stability is due to the frequency lock-in of the FFs for certain modulation parameters. The presented results could apply to many structural problems in mechanics, from the vibrations of rotating machineries with shape imperfections to the stability of periodic limit cycles or of any slender structures with tensile or compressive periodic elastic stresses.

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“…(6) and using a fixed-point iteration approach [18] (setting 2  equal to zero on the right side of eq. (21)), 2  can be approximated as …”

Section: The Analytically Reduced Dimension Systemsmentioning

confidence: 99%

A novel order reduction method for nonlinear dynamical system under external periodic excitations

Cao

Wang

Huang

2010

Sci. China Technol. Sci.

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The concept of approximate inertial manifold (AIM) is extended to develop a kind of nonlinear order reduction technique for non-autonomous nonlinear systems in second-order form in this paper. Using the modal transformation, a large nonlinear dynamical system is split into a 'master' subsystem, a 'slave' subsystem, and a 'negligible' subsystem. Accordingly, a novel order reduction method (Method I) is developed to construct a low order subsystem by neglecting the 'negligible' subsystem and slaving the 'slave' subsystem into the 'master' subsystem using the extended AIM. As a comparison, Method II accounting for the effects of both 'slave' subsystem and the 'negligible' subsystem is also applied to obtain the reduced order subsystem. Then, a typical 5-degree-of-freedom nonlinear dynamical system is given to compare the accuracy and efficiency of the traditional Galerkin truncation (ignoring the contributions of the slave and negligible subsystems), Method I and Method II. It is shown that Method I gives a considerable increase in accuracy for little computational cost in comparison with the standard Galerkin method, and produces almost the same accuracy as Method II. Finally, a 3-degree-of-freedom nonlinear dynamical system is analyzed by using the analytic method for showing predominance and convenience of Method I to obtain the analytically reduced order system. Galerkin method, nonlinear systems, dimension reduction, post-processed method, model truncation Citation:Cao D Q, Wang J L, Huang W H. A novel order reduction method for nonlinear dynamical system under external periodic excitations.

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Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

Cited by 19 publications

References 20 publications

Postprocessing Finite-Element Methods for the Navier–Stokes Equations: The Fully Discrete Case

Postprocessing Finite-Element Methods for the Navier–Stokes Equations: The Fully Discrete Case

Modal and stability analysis of structures in periodic elastic states: application to the Ziegler column

A novel order reduction method for nonlinear dynamical system under external periodic excitations

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