Multiharmonic Multiple-Point Collocation: A Method for Finding Periodic Orbits of Strongly Nonlinear Oscillators

Ardeh, Hamid A.; Allen, Matthew S.

doi:10.1115/1.4031286

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…Comparing the rates shown in Table 6 with those shown in Table 4, it can be seen that the computational time per solution for MMC increased by about an order of magnitude. This is larger than expected, since early results [7,14] suggested that the increase was sub-linear with the number of degrees of freedom.…”

Section: Application To 10-dof Systemmentioning

confidence: 61%

“…These results are not fully understood yet. Computation time for numerical integration is known to scale with N DOF 2 or more, and since MMC had been previously found to scale sub-linearly with N DOF [14] and so one would expect to see the performance gap between MMC and NNMcont increase rather than decreasing. The Matlab implementation of MMC will be improved and these issues will be explored further in future works.…”

Section: Discussionmentioning

confidence: 99%

“…One can solve for the unknown period and the coefficients A jk and B jk that define the periodic response using a Newton-Raphson approach. Note that Ardeh & Allen also presented a steepest descent algorithm in [14] but that algorithm is not employed in this work. The Newton Raphson scheme is implemented by defining a set of M collocation points that are distributed between 0 and 2 in phase as follows,…”

Section: Review Of MMCmentioning

confidence: 99%

“…In fact, the MMC algorithm does not integrate the response over any time interval, so it has the potential to dramatically reduce the cost required to compute the periodic responses of a system. In their initial work, Ardeh and Allen showed cases where the MMC algorithm was orders of magnitude faster than shooting [14]. Also, because MMC uses a Fourier series description, one may be able to truncate the description to avoid computing higher order internal resonances.…”

Section: Introductionmentioning

confidence: 99%

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Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

Kelly

Allen

Ardeh

2015

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control

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Fast numerical approaches have recently been proposed to compute the undamped nonlinear normal modes (NNMs) of discrete and structural systems, and these have enabled remarkable insights into the nonlinear behavior of more complicated systems than are tractable analytically. Ardeh et al. recently proposed the multi-point, multi-harmonic collocation (MMC) method, which finds a truncated harmonic description of a structure’s NNMs. The MMC algorithm resembles harmonic balance but doesn’t require any analytical pre-processing nor the computational expense of the alternating time-frequency method. In a previous work this method showed significantly faster performance than the shooting method and it also allows the possibility of truncating the description of the NNM to avoid having to compute every internal resonance. This work presents a pseudo-arc length continuation version of the MMC algorithm which can compute a branch of NNMs and compares its performance with the shooting/pseudo-arc length continuation algorithm presented by Peeters and Kerschen in 2009. The algorithms are compared for two systems, a two degree-of-freedom (DOF) spring-mass system and a 10-DOF model of a simply-supported beam.

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Section: Application To 10-dof Systemmentioning

confidence: 61%

Section: Discussionmentioning

confidence: 99%

Section: Review Of MMCmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

Kelly

Allen

Ardeh

2015

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control

Self Cite

View full text Add to dashboard Cite

show abstract

“…The methods for calculating periodic solutions can be mainly classified into two different categories, namely, the frequency domain (Liao, 2016;Liao and Sun, 2013;Liao andWu, 2018a, 2018b;Zhao et al, 2016) and time domain approaches (Ardeh and Allen, 2016;Liao andWu, 2018a, 2018b;Mousavi at al., 2020). Many researchers have applied the shooting method to solve nonlinear dynamics problems.…”

Section: Introductionmentioning

confidence: 99%

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

Liao

Gao

2020

Journal of Vibration and Control

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A continuation method for bifurcation tracking is presented based on the proposed optimization problem formulation which is designed to locate the bifurcation periodic solution. The bifurcation detection problem is formulated as a constrained optimization problem. The nonlinear constraints of the optimization problem are imposed on the shooting function and bifurcation conditions derived from the Floquet theory whereas the objective function associated with the pseudo-arclength correlation equation is devised to solution continuation. The proposed optimization formulation is integrated with the prediction–correction strategy to achieve bifurcation tracking. Two numerical examples about the Jeffcott rotor and the nonlinear tuned vibration absorber are illustrated to validate the effectiveness of the proposed methodology. Numerical results have demonstrated that the proposed method offers a convenient scheme to follow bifurcation periodic solution.

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Continuation of nonlinear normal modes using reduced-order models based on generalized characteristic value decomposition

Stein,

Chelidze

2024

Nonlinear Dyn

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Over the past two decades, data-driven reduced-order modeling (ROM) strategies have gained significant traction in the nonlinear dynamics community. Currently, several challenges in physical interpretation and data availability remain overlooked in current methodologies. This work proposes a novel ROM methodology based on a newly proposed generalized characteristic value decomposition (GCVD) to address these obstacles. The GCVD-ROM approach proposes a new perspective toward data-driven ROMs via characterization of the dynamics before any ROM considerations are made. In doing so, a significant degree of versatility is inherited in the GCVD-ROM strategy, allowing our models to reproduce the full-scale dynamics in different regions of the parameter space at the cost of a single training data set. Our approach utilizes computationally efficient free-decay data sets alongside a windowed-decomposition scheme, allowing us to extract energy-dependent modal structures for use in model-order reduction. This is accomplished using the physically insightful characteristic values provided by the GCVD, which are shown to be directly related to the system poles at a particular response amplitude. This natural metric, paired with a resonance tracking scheme, allows us to address the difficulties associated with physical interpretation and data availability without sacrificing the convenient aspects of linear projection-based model order reduction. A computational framework for the continuation and bifurcation analysis using linear projection-based ROMs is also presented, permitting us to deploy rigorous analysis and bifurcation studies to verify that our ROMs reproduce the intrinsic complexity of full-scale systems. A detailed walk-through of the GCVD-ROM approach is demonstrated on a simple system where important practical considerations and implementation details are discussed using a concrete example. The discretized von Kármán beam and shallow arch partial differential equations are also used to explore complicated scenarios involving modal coupling across disparate time scales and internal resonances.

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Multiharmonic Multiple-Point Collocation: A Method for Finding Periodic Orbits of Strongly Nonlinear Oscillators

Cited by 5 publications

References 19 publications

Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

Multi-Point Multi-Harmonic Collocation With Continuation to Compute Branches of Nonlinear Modes of Structural Systems

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

Continuation of nonlinear normal modes using reduced-order models based on generalized characteristic value decomposition

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