Proceedings of the IMAC-XXVIII February 1–4, 2010, Jacksonville, Florida USA ©2010 Society for Experimental Mechanics Inc. Practical Aspects of Dynamic Substructuring in Wind Turbine Engineering S.N. Voormeeren,¤ P.L.C. van der Valk and D.J. Rixen Delft University of Technology, Faculty of Mechanical, Maritime and Materials Engineering Department of Precision and Microsystem Engineering, section Engineering Dynamics Mekelweg 2, 2628CD, Delft, The Netherlands [email protected] ABSTRACT In modern day society concern is growing about the use of fossil fuels to meet our constantly rising energy demands. Although wind energy certainly has the potential to play a signi¯cant role in a sustainable future world energy supply, a number of challenges are still to be met in wind turbine technology. One of those challenges concerns the correct determination of dynamic loads caused by structural vibrations of the individual turbine components (such as rotor blades, gearbox and tower). Thorough understanding of these loads is a prerequisite to further increase the overall reliability of a wind turbine. Hence, improved insight in the component structural dynamics could eventually lead to more cost-e®ective wind turbines. This paper addresses the use of dynamic substructuring (DS) as an analysis tool in wind turbine engineering. The bene¯ts of a component-wise approach to structural dynamic analysis are illustrated, as well as a number of practical issues that need to be tackled for successful application of substructuring techniques in an engineering setting. Special attention is paid to the modeling of interfaces between components. The potential of the proposed approach is illustrated by a DS analysis on a Siemens SWT-2.3-93 turbine yaw system. NOMENCLATURE M { Mass matrix C { Damping matrix K { Sti®ness matrix u { Vector of degrees of freedom f { External force vector g { Connection force vector B { Compatibility matrix (Boolean) L { Localization matrix (Boolean) ¸ { Vector of Lagrange multipliers R { Reduction matrix ?+ { Generalized (pseudo) inverse 1 INTRODUCTION At present there are few topics as heavily debated as \sustainability". On a daily basis the media are full of items on climate change, oil prices, CO2 reductions, rising energy consumptions and so on. Regardless of one's opinion on the subject, a fact of the matter is that more sustainable ways of power generation need to be found simply because the currently used resources will some day be exhausted. One of the more promising ways of generating \green" electricity on a large scale is provided by wind energy. As a result, the wind turbine industry has undergone a huge transition: from a small group of (mainly Danish) enthusiasts in the early 1980's, the modern wind power industry now has grown to a globalized multi billion dollar industry.1 However, to enable ¤This research is supported by Siemens Wind Power A/S 1From 2002 onwards, the wind power industry has seen an annual growth of no less than 25%. wind power to truly ful¯ll a signi¯cant role in a sustainable future energy supply, a number of technological challenges are still to be met. One of those challenges concerns the correct modeling and analysis of the structural dynamic behavior of the wind turbine. 1.1 Structural Dynamics in Wind Turbine Engineering Naturally a wind turbine, with its large and relatively slender structure and the complex excitations, exhibits all kinds of structural dynamic behavior. The dynamic loading and structural vibrations sometimes can cause problems, from cracking blades, breaking gearboxes to \singing" towers. These problems have not been limited to a single manufacturer, but simply seem inherent to the structure of a modern wind turbine. To cope with these dynamic e®ects, wind turbine manufacturers, research institutes and universities have developed many di®erent aero-elastic codes [18]. These advanced codes are perfectly suited to analyze the global dynamics of a wind turbine, taking into account aerodynamic loads and coupling, possibly wave loads (for o®shore turbines), and hence are commonly used for certi¯cation purposes. Driven by today's highly competitive wind turbine market, manufacturers are searching for ways to optimize their turbine designs and hence save costs. An important way of achieving this is by reducing the total weight of turbine, by optimizing the design of each individual component. This causes a chain reaction of bene¯ts as less material is used, transport and installation is made easier, a smaller foundation can be used and so on. On the downside, these optimized turbine designs generally introduce more flexibility to the structure. As a result, components start to exhibit local dynamic behavior, which can lead to increased component loading and decreased reliability. In some cases the local dynamic e®ects can interact with the global dynamics of the turbine, or vice versa. Thorough understanding of these dynamics is a prerequisite to further increase the overall reliability of a wind turbine. However, the aero-elastic models commonly used in wind turbine engineering are often incapable of predicting these local dynamic e®ects and their interaction with the global dynamics, due to their relatively few degrees of freedom and geometric simpli¯cations. Therefore, a need exists for more detailed structural dynamic analysis tools, without losing generality and versatility. In this paper we propose to use the paradigm of dynamic substructuring (DS) to ¯ll this need. 1.2 Paper Outline The remainder of the paper is organized as follows. The next section will introduce the concept of dynamic substructuring and discuss the details of substructure assembly. Although the basic principles of the DS methodology were established already some decades ago, implementing the dynamic substructuring approach in an industrial setting requires solving a number of practical issues. These issues and a number of solutions will be discussed in section 3. Section 4 thereafter presents a case study of the methodology on the yaw system of modern 2.3MW Siemens wind turbine. The paper is ended with some conclusions and recommendations in section 5. 2 INTRODUCTION TO DYNAMIC SUBSTRUCTURING The theory of dynamic substructuring (DS) is about performing a dynamic analysis of a complex structure by dividing it into a number of smaller, less complex ones. These parts of the system are called substructures, subsystems or components, and their dynamic behavior is in general easier to determine than that of the complete system. When the dynamic properties of all the subsystems are known, DS techniques allow to construct the dynamic behavior of the complete system by coupling the subsystems together. Performing the analysis of a structural system component-wise has some important advantages over global methods where the entire problem is handled at once: ² It allows the evaluation of the dynamical behavior of structures that are too large or complex to be analyzed as a whole. For experimental analysis this is true for large and complex systems such as aircrafts. For numerical models this holds when the number of degrees of freedom is such that solution techniques cannot ¯nd results in a reasonable time. ² By analyzing the subsystems, local dynamic behavior can be recognized more easily than when the entire system is analyzed. Thereby, DS allows identifying local problems and performing e±cient local optimization. ² Dynamic substructuring gives the possibility to combine modeled parts (discretized or analytical) and experimentally identi¯ed components. ² It allows sharing and combining substructures from di®erent project groups. Dynamic substructuring methods have been long established; the ¯rst contributions in the literature date from over six decades ago [10; 23]. At the end of the 1960's the DS methodology saw rapid development with the rise of component mode synthesis methods [6; 15; 22], driven by the desire to reduce the complexity and size of computational structural dynamic models. Since then the methodology has seen many new developments, especially in the ¯eld of assembly of experimental component models, but the basic theory has remained the same. This basic theory of dynamic substructuring will be presented in this section, based on the discussion in [8]. 2.1 Component Models and Interfacing The starting point for the treatment of DS theory in this paper are the equations of motion in the physical domain. In this domain, the system is described by its mass, damping, and sti®ness matrices as obtained from its mechanical and geometrical properties. Note however that the following discussion is also valid for substructure models in the frequency domain (where the component is seen through its frequency response functions) and the modal domain (where the dynamic behavior of a structure is interpreted as a combination of modal responses). The equations of motion of a discrete/discretized and linear(ized) dynamic subsystem s in the physical domain may be written as: M (s)uÄ(s) (t) + Cu_ (s) (t) + K(s)u(s) (t) = f (s) (t) + g(s) (t) (1) Here M (s), C(s) and K(s) are the mass, damping and sti®ness matrices of substructure s, u(s) denotes its vector of degrees of freedom (DoF), f (s) is the external force vector and g(s) is the vector of connecting forces with the other substructures. Suppose now that n substructure models of the form shown above are to be coupled. In order to simplify the notation, the equations of motion of these n substructures can be rewritten in a block diagonal format as: MuÄ + Cu_ + Ku = f + g (2) With: 2 3 M (1) ¢ ¢ ¡ ¢ (1) (n) 6 . 7 M , diag M ;:::; M = 4 ¢ .. ¢ 5 ¢ ¢ M (n) ¡ ¢ C , diag C(1) ;:::; C(n) ¡ ¢ K , diag K(1) ;:::; K(n) 2 3 2 3 2 3 u(1) f (1) g(1) 6 . 7 6 . 7 6 . 7 u , 4 . 5 ; f , 4 . 5 ; g , 4 . 5 u(n) f (n) g(n) For the sake of simplicity, the explicit time dependence has been omitted here.