Dynamic analysis by direct superposition of Ritz vectors

Wilson, Edward L.; Yuan, Mingxuan; Dickens, John M.

doi:10.1002/eqe.4290100606

Cited by 332 publications

(116 citation statements)

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“…Previous approaches deriving the Ritz functions from the linearization of the equations of motion were probably initiated by Nickell [31], who extended the use of modal superposition methods for systems with nonlinear dynamics. Wilson et al [38] later introduced the so-called load-dependent vectors. A recent overview of these techniques was given by Leger and Dussault [19], and this method was also used in [15,16].…”

Section: Previous Workmentioning

confidence: 99%

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

138

112

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This paper focuses on methods of constructing of reduced-order models of mechanical systems which preserve the Lagrangian structure of the original system. These methods may be used in combination with standard spatial decomposition methods, such as the Karhunen-Loève expansion, balancing, and wavelet decompositions. The model reduction procedure is implemented for three-dimensional finite-element models of elasticity, and we show that using the standard Newmark implicit integrator, significant savings are obtained in the computational costs of simulation. In particular simulation of the reduced model scales linearly in the number of degrees of freedom, and parallelizes well.

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Section: Previous Workmentioning

confidence: 99%

Structure-preserving model reduction for mechanical systems

Lall¹,

Krysl²,

Marsden³

2003

Physica D: Nonlinear Phenomena

138

112

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“…The most common methods are mode superposition methods [1], in which a limited number of free vibration modes of the structure is used to represent the displacement pattern [25]. There are also improvements of the original mode superposition method by the addition of different vectors to the expansion procedure, such as the mode acceleration or modal truncation augmentation [1,26]. Mode superposition methods are generally considered for the complete structure.…”

Section: Mode Displacement Methodsmentioning

confidence: 99%

“…This means that the reduced-order transfer function corresponding to system (27), which results from applying matrices V and W to the original system matrices, has the property (26). To ensure the satisfaction of the moment matching property (26), one can choose V and W such that the columns of these matrices span so-called Krylov subspaces. The k-th Krylov subspace induced by a matrix P and a vector r is defined as…”

Section: Krylov Subspace Based Model Order Reductionmentioning

confidence: 99%

“…The modal truncation augmentation form presented in Section 2.1.3 includes only the zerothorder correction as indicated by the basis (61). Higher-order corrections as suggested in the Krylov sequence (60) can also be included in the reduction space as proposed in [54,26,55,56,57,33], which guarantees matching higher-orders moments and thus leads to an approach similar to the moment matching technique. Higher-order correction modes have also been used in the context of substructuring and mode component synthesis [32,58,59].…”

Section: Moment Matching and Model Truncation Augmentationmentioning

confidence: 99%

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A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

Besselink

Tabak

Lutowska

et al. 2013

Journal of Sound and Vibration

230

129

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Document VersionPublisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publicationCitation for published version (APA): Besselink, B., Lutowska, A., Tabak, U., Wouw, van de, N., Nijmeijer, H., Hochstenbach, M. E., ... Rixen, D. J. (2013). A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control. (CASA-report; Vol. 1327). Eindhoven: Technische Universiteit Eindhoven. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractIn this paper, popular model reduction techniques from the fields of structural dynamics, numerical mathematics and systems and control are reviewed and compared. The motivation for such a comparison stems from the fact the model reduction techniques in these fields have been developed fairly independently. In addition, the insight obtained by the comparison allows for making a motivated choice for a particular model reduction technique, on the basis of the desired objectives and properties of the model reduction problem. In particular, a detailed review is given on mode displacement techniques, moment matching methods and balanced truncation, whereas important extensions are outlined briefly. In addition, a qualitative comparison of these methods is presented, hereby focussing both on theoretical and computational aspects. Finally, the differences are illustrated on a quantitative level by means of application of the model reduction techniques to a common example.

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“…This is easily achieved by adding an additional attachment mode a corresponding to the external force f to the reduction basis T DCB . Such an additional attachment mode a is often called a load dependent Ritz vector [13,14,15] or modal truncation vector [16,17]. For the case of a structure decomposed into two substructures and an external force f with force application point on substructure 2, the augmented reduction matrix of the dual Craig-Bampton method is:…”

Section: Modal Truncation Vectors Related To External Force Excitationsmentioning

confidence: 99%

Time Integration of Dual Craig-Bampton Reduced Systems

Gruber¹,

Gille²,

Rixen³

2017

Proceedings of the 6th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COM

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Abstract. In this paper, time integration procedures are demonstrated and investigated for dual Craig-Bampton reduced systems. The dual Craig-Bampton method for the reduction and successive coupling of dynamic systems employs free interface vibration modes, attachment modes and rigid body modes to build the reduction bases of the substructures, but assembles the substructures using interface forces. Thereby, the interface kinematic conditions are transformed, allowing for incompatibilities associated with the equilibrium residual in the substructure. Hence, the eigenvalues of the reduced-order model are not guaranteed to be upper bounds for the unreduced system's eigenvalues. Furthermore, the dual Craig-Bampton reduced system will always have as many negative eigenvalues as interface coupling conditions. The reduced system is unstable, rendering a straightforward time integration of the dual Craig-Bampton reduced system impossible.The feasibility of a reliable time integration of dual Craig-Bampton reduced systems is demonstrated and investigated in detail. The unstable behavior when time-integrating such systems without further modifications is illustrated and two approaches to overcome this instability are suggested: on the one hand, a modal analysis of the reduced system is performed as a subsequent step to the dual Craig-Bampton reduction. Only modes corresponding to positive eigenvalues are thereby kept for transient analysis. This allows for a stable time integration. On the other hand, a modal interface reduction during the dual Craig-Bampton reduction process is performed and only interface modes corresponding to positive eigenvalues are kept. This makes the final reduced system also positive definite. The accuracy using these two approaches is demonstrated in examples with either different initial conditions or varying external periodic excitations.

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Dynamic analysis by direct superposition of Ritz vectors

Cited by 332 publications

References 2 publications

Structure-preserving model reduction for mechanical systems

Structure-preserving model reduction for mechanical systems

A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control

Time Integration of Dual Craig-Bampton Reduced Systems

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