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Dávid, Alexandra; Sinha, Subhash C.

doi:10.1023/a:1008330023291

Cited by 20 publications

(5 citation statements)

References 11 publications

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“…In all cases, even though the HB-like methods can find a solution, they do not account for the stability of solution [174]. The stability of the solution can be determined using the Lyapunov-Floquet transform [66,[175][176][177][178]. When the transformation matrix has an eigenvalue whose modulus exceeds 1, instability occurs; when the eigenvalue is less than −1, there is a period-doubling bifurcation.…”

Section: Stability Analysismentioning

confidence: 99%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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The harmonic balance (HB) method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems. However, it is confined to low-order approximations due to complex symbolic operations. Many variants have been developed to improve the HB method, among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation. So far, there are two problems remaining to be addressed. i) A dozen of different versions of HB-like methods, in frequency domain or time domain or in hybrid, have been developed; unfortunately, misclassification pervades among them due to the unclear borderlines of different methods. ii) The time domain HB-like methods suffer from non-physical solutions, which have been shown to be caused by aliasing (mixture of the high-order into the low-order harmonics). Although a series of dealiasing techniques have been developed over the past two decades, the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community. This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences. In particular, the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed. KEYWORDSHarmonic balance; frequency domain HB-like method; time domain HB-like method; dealiasing technique; HB algebraic equation NomenclatureThe commonly used abbreviations of the computing methods are shown below. Abbreviations AFT-HBAlternating frequency-time harmonic balance ANM Asymptotic numerical method ETDC Extended time domain collocation GOIA Global optimal iterative algorithm HB Harmonic balance

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Section: Stability Analysismentioning

confidence: 99%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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“…(23): By defining small increments on the bifurcation parameter as η, we can write e k ¼ e c þ η k to represent the k disparate sets of bifurcation parameter in the neighborhood of the critical parameter e c ¼ 0:2. We employ the least squares, curve fitting technique proposed by [42] to obtain the relationship between μ 1 and η,as μ 1 ¼ 1:47476 þ i0:301628 ðÞ η À 1:82052 þ i0:414608 ðÞ η 2 . NB values of C 1 and C 2 in Eq.…”

Section: State-augmented Systemmentioning

confidence: 99%

“…36 Solutions of the versal deformation equations enable investigation of the postbifurcation steady-state behavior in the small neighborhood of the bifurcation point. However, as observed by [42], this method is only useful for local analysis. This is because minor errors introduced by back transformation close to the bifurcation points significantly grow as you move further away.…”

Section: L-f Transformed Systemmentioning

confidence: 99%

Analysis and Control of Nonlinear Attitude Motion of Gravity-Gradient Stabilized Spacecraft via Lyapunov-Floquet Transformation and Normal Forms

Waswa¹,

Redkar²

2020

Advances in Spacecraft Attitude Control

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This chapter demonstrates analysis and control of the attitude motion of a gravity-gradient stabilized spacecraft in eccentric orbit. The attitude motion is modeled by nonlinear planar pitch dynamics with periodic coefficients and additionally subjected to external periodic excitation. Consequently, using system state augmentation, Lyapunov-Floquet (L-F) transformation, and normal form simplification, we convert the unwieldy attitude dynamics into relatively more amenable schemes for motion analysis and control law development. We analyze the dynamical system's periodicity, stability, resonance, and chaos via numerous nonlinear dynamic theory techniques facilitated by intuitive system state augmentation and Lyapunov-Floquet transformation. Versal deformation of the normal forms is constructed to investigate the bifurcation behavior of the dynamical system. Outcome from the analysis indicates that the motion is quasi-periodic, chaotic, librational, and undergoing a Hopf bifurcation in the small neighborhood of the critical point-engendering locally stable limit cycles. Consequently, we demonstrate the implementation of linear and nonlinear control laws (i.e., bifurcation and sliding mode control laws) on the relatively acquiescent transformed attitude dynamics. By employing a two-pronged approach, the quasiperiodic planar motion is independently shown to be stabilizable via the nonlinear control approaches.

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“…We cannot compute the value of a owing to the unknown control gains, but this form provides the necessary information about the stability in order to design the controller. Now, to study the dynamics in the neighbourhood of the bifurcation point, versal deformation of the normal form is constructed as (Dávid & Sinha 2000) _ v Z mðaÞv C av 3 ; ð5:10Þ…”

Section: (A ) Flip (Period Doubling) Bifurcationmentioning

confidence: 99%

“…An excellent review can be found in Chen et al (2000). Recent results obtained by Dávid & Sinha (2000, 2003 show that the idea of nonlinear bifurcation control can be extended to systems with time-periodic coefficients. The method is based on the construction of dynamically equivalent timeinvariant normal forms.…”

Section: Introductionmentioning

confidence: 99%

Control of chaos in nonlinear systems with time-periodic coefficients

Sinha

Dávid

2006

Phil. Trans. R. Soc. A.

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In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

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Untitled

Cited by 20 publications

References 11 publications

Harmonic Balance Methods: A Review and Recent Developments

Harmonic Balance Methods: A Review and Recent Developments

Analysis and Control of Nonlinear Attitude Motion of Gravity-Gradient Stabilized Spacecraft via Lyapunov-Floquet Transformation and Normal Forms

Control of chaos in nonlinear systems with time-periodic coefficients

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