Revisited Harmonic Balance Trim Solution Method for Periodically-Forced Aerospace Vehicles

Saetti, Umberto; Rogers, Jonathan

doi:10.2514/1.g005553

Cited by 14 publications

(4 citation statements)

References 21 publications

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“…The perturbation method requires weak nonlinearity, which leads to limited applications. Thus, the HB-like methods play an important role in the periodic solution of nonlinear systems [19][20][21][22][23] due to their high accuracy, multidisciplinary applications and ability to handle strong nonlinearities.…”

Section: Hdhbmentioning

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: Hdhbmentioning

confidence: 99%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

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“…Numerical integration methods can be used to find periodic nonlinear solutions [14][15][16][17], but limited by i) undesired simulation time for transient motions; ii) small step-size to constrain accumulated computational error; and iii) inability to compute unstable periodic solutions. Credited to the seminal work of Blondel [18], the HB method has been the most popular method for nonlinear periodic solutions in various fields during the last 100 years [19][20][21][22][23]. It is free from the above restrictions, and works by presuming a Fourier expansion for the desired periodic solution and then obtaining resultant nonlinear algebraic equations (NAEs) of the Fourier coefficients through balancing harmonics up to the truncation order.…”

Section: Introductionmentioning

confidence: 99%

Collocation-based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

Dai¹,

Zipu²,

Wang³

et al. 2022

Preprint

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Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely-used approaches for such systems, but are either confined to loworder approximations or impaired by aliasing and improper-sampling problems. Here we propose a collocation-based harmonic balance framework to successfully unify and reconstruct the HB-like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequencydomain and time-domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high-order (>100) non-aliasing solutions, previously deemed outof-reach, for a range of complex nonlinear systems including the cavitation bubble equation and the three-body problem. We show that the present method is 2-3 orders of magnitude more accurate and simultaneously much faster than the state-of-the-art method. Hence, it has immediate applications in multi-disciplinary problems where highly accurate periodic solutions are sought.

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“…Numerical integration methods can be used to find periodic nonlinear solutions, 16‐19 but limited by (i) undesired simulation time for transient motions; (ii) small step‐size to constrain accumulated computational error; and (iii) inability to compute unstable periodic solutions. Credited to the seminal work of Blondel, 20 the HB method has been the most popular method for nonlinear periodic solutions in various fields during the last 100 years 21‐25 . It is free from the above restrictions, and works by presuming a Fourier expansion for the desired periodic solution and then obtaining resultant nonlinear algebraic equations (NAEs) of the Fourier coefficients through balancing harmonics up to the truncation order.…”

Section: Introductionmentioning

confidence: 99%

“…Credited to the seminal work of Blondel, 20 the HB method has been the most popular method for nonlinear periodic solutions in various fields during the last 100 years. [21][22][23][24][25] It is free from the above restrictions, and works by presuming a Fourier expansion for the desired periodic solution and then obtaining resultant nonlinear algebraic equations (NAEs) of the Fourier coefficients through balancing harmonics up to the truncation order. Essentially the HB method is a Galerkin method with the same trial and test functions that are both Fourier series (trigonometric functions).…”

Section: Introductionmentioning

confidence: 99%

Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

Dai

Zipu

Wang

et al. 2022

Numerical Meth Engineering

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Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely‐used approaches for such systems, but are either confined to low‐order approximations or impaired by aliasing and improper‐sampling problems. Here we propose a collocation‐based harmonic balance framework to successfully unify and reconstruct the HB‐like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency‐domain and time‐domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high‐order (>100) nonaliasing solutions, previously deemed out‐of‐reach, for a range of complex nonlinear systems including the cavitation bubble dynamics, the three‐body problem and the two dimensional airfoil dynamics. We show that the present method is 2–3 orders of magnitude more accurate and simultaneously much faster than the state‐of‐the‐art method. Hence, it has immediate applications in multidisciplinary problems where highly accurate periodic solutions are sought.

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Revisited Harmonic Balance Trim Solution Method for Periodically-Forced Aerospace Vehicles

Cited by 14 publications

References 21 publications

Harmonic Balance Methods: A Review and Recent Developments

Harmonic Balance Methods: A Review and Recent Developments

Collocation-based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

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