Modal truncation vectors for reduced dynamic substructure models

Voormeeren

Journal of Computational and Nonlinear Dynamics

et al. 2015

One of the main mechanisms for driving down the cost of offshore wind energy is to install ever larger wind turbines in larger wind farms. At the same time, these turbines are placed further offshore in deeper waters. As a result traditional monopile foundations are not always feasible and multi-membered foundations, such as jackets and tripods, are required.Typically thousands of load cases need to be simulated for the design and certification of offshore wind turbines. As models of such foundations are significantly larger than their monopile counterparts, model reduction is often applied to limit the computational costs. Additionally, the foundation design is generally done by a specialized company, which bases its design on the results of the load simulations. Hence, an accurate estimation of the stresses in load simulation is essential to predict the integrity and the lifetime of different designs. The effect on the load accuracy of both the model reduction as well as the post-processing method used by foundation designers are investigated in this paper. A case study is performed on a jacket-based wind turbine model to verify and quantify the findings.Firstly, it is observed that the effect of the reduced foundation model on the wind turbine loads is negligible. However, both the reduction method and the post-processing method applied by the foundation designer have a large influence on the fatigue loading in the jacket. It is shown that the popular Guyan reduction results in significant errors on the fatigue damage and that a static post-processing analysis leads to serious underestimations of the fatigue loads. Finally, an outlook is given into future developments in the field of load calculations for offshore wind turbine foundation design. Nomenclature B Signed Boolean operator C Damping matrix D el i Fatigue damage in element i f Array of external forces g Array of connection forces Journal of Computational and Nonlinear Dynamics. Downloaded From: http://computationalnonlinear.asmedigitalcollection.asme.org/ on 02/20/2015 Terms of Use: http://asme.org/terms A c c e p t e d M a n u s c r i p t N o t C o p y e d i t e dH s Significant wave height K Stiffness matrix L Boolean assembly operator M Mass matrix p Array of nonlinear internal damping and elastic forces q Array of reduced degrees of freedom R Reduction matrix r Array of residual forces T Orthogonal projector u Array of displacement degrees of freedom v a Average wind speeds α Array of modal force amplitudes η Array of modal amplitudes λ Array of Lagrange multipliers φ Normal mode ψ C Constraint mode ω Frequency [i] Associated to internal DoF [b] Associated to boundary DoF [W T ] Associated to the wind turbine substructure [F] Associated to the foundation substructurė First time derivativë Second time derivativē Primal assembled Reduced variable CB Craig-Bampton method CMS Component mode synthesis DC Displacement controlled DEL Damage equivalent load DLC Design load case DoF Degree of freedom FC Force controlled FD Foundation designer GR Guyan red...

Section: Augmented Craig-bampton Methodsmentioning

confidence: 99%

Section: Accepted Manuscript N O T C O P Y E D I T E Dmentioning

confidence: 99%

Dynamic Models for Load Calculation Procedures of Offshore Wind Turbine Support Structures: Overview, Assessment, and Outlook

Voormeeren

Journal of Computational and Nonlinear Dynamics

et al. 2015

Journal of Sound and Vibration

“…Its main principle depends on the use of the static correction as an additional direction for the truncation expansion [32,33,34]. Inclusion of the correction in a modal expansion results in the modal truncation augmentation method, such that q is approximated as…”

Section: Modal Truncation Augmentation Methodsmentioning

confidence: 99%

“…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]. Note that the high-order corrections for structural problems can be obtained as a by-product of the Lanczos algorithm used to compute the eigenmodes [60] and that one can also consider quasi-static corrections (i.e.…”

Section: Moment Matching and Model Truncation Augmentationmentioning

confidence: 99%

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

Besselink

Tabak

Lutowska

et al. 2013

245

129

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.

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

“…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

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.