Dynamic substructuring

Dynamic Substructuring (DS) is an engineering tool used to model and analyse the dynamics of mechanical systems by means of its components or substructures. Using the dynamic substructuring approach one is able to analyse the dynamic behaviour of substructures separately and to later on calculate the assembled dynamics using coupling procedures. Dynamic substructuring has several advantages over the analysis of the fully assembled system: Dynamic Substructuring (DS) is an engineering tool used to model and analyse the dynamics of mechanical systems by means of its components or substructures. Using the dynamic substructuring approach one is able to analyse the dynamic behaviour of substructures separately and to later on calculate the assembled dynamics using coupling procedures. Dynamic substructuring has several advantages over the analysis of the fully assembled system: Dynamic substructuring is particularly tailored to simulation of mechanical vibrations, which has implications for many product aspects such as sound / acoustics, fatigue / durability, comfort and safety. Also, dynamic substructuring is applicable to any scale of size and frequency. It is therefore a widely used paradigm in industrial applications ranging from automotive and aerospace engineering to design of wind turbines and high-tech precision machinery. The roots of dynamic substructuring can be found in the field of domain decomposition. In 1890 the mathematician Hermann Schwarz came up with an iterative procedure for domain decomposition which allows to solve for continuous coupled subdomains. However, many of the analytical models of coupled continuous subdomains do not have closed-form solutions, which led to discretization and approximation techniques such as the Ritz method (which is sometimes called the Raleigh-Ritz method due the similarity between Ritz's formulation and the Raleigh ratio) the boundary element method (BEM) and the finite element method (FEM). These methods can be considered as 'first level' domain decomposition techniques. The finite element method proved to be the most efficient method and the invention of the microprocessor made it possible to easily solve a large variety of physical problems. In order to analyse even larger and more complex problems, methods were invented to optimize the efficiency of the discretized calculations. The first step was replacing the direct solvers by iterative solvers such as the conjugate gradient method. The lack of robustness and slow convergence of these solvers did not make them an interesting alternative in the beginning. The rise of parallel computing in the 1980s however sparked their popularity. Complex problems could now be solved by dividing the problem into subdomains, each processed by a separate processor, and solving for the interface coupling iteratively. This can be seen as a second level domain decomposition as is visualized in the figure. The efficiency of dynamic modelling could be increased even further by reducing the complexity of the individual subdomains. This reduction of the subdomains (or substructures in the context of structural dynamics) is realized by representing substructures by means of their general responses. Expressing the separate substructures by means of their general response instead of their detailed discretization led to the so-called dynamic substructuring method. This reduction step also allowed for replacing the mathematical description of the domains by experimentally obtained information. This reduction step is also visualized by the reduction arrow in the figure. The first dynamic substructuring methods were developed in the 1960s and were more commonly known under the name component mode synthesis (CMS). The benefits of dynamic substructuring were quickly discovered by the scientific and engineering communities and it became an important research topic in the field of structural dynamics and vibrations. Major developments followed, resulting in e.g. the classic Craig-Bampton method. Due to improvements in sensor and signal processing technology in the 1980s, substructuring techniques also became attractive for the experimental community. Methods dealing with structural dynamic modification were created in which coupling techniques were directly applied to measured frequency response functions (FRFs). Broad popularity of the method was gained when Jetmundsen et al. formulated the classical frequency-based substructuring (FBS) method, which laid the ground work for frequency-based dynamic substructuring. In 2006 a systematic notation was introduced by De Klerk et al. in order to simplify the difficult and elaborate notation that had been used prior. The simplification was done by means of two Boolean matrices that handle all the 'bookkeeping' involved in the assembly of substructures Dynamic substructuring can best be seen as a domain-independent toolset for assembly of component models, rather than a modelling method of its own. Generally, dynamic substructuring can be used for all domains that are well suited to simulate multiple input/multiple output behaviour. Five domains that are well suited for substructuring are: The physical domain concerns methods that are based on (linearised) mass, damping and stiffness matrices, typically obtained from numerical FEM modelling. Popular solutions to solve the associated system of second order differential equations are the time integration schemes of Newmark and the Hilbert-Hughes-Taylor scheme. The modal domain concerns component mode synthesis (CMS) techniques such as the Craig-Bampton, Rubin and McNeal method. These methods provide efficient modal reduction bases and assembly techniques for numerical models in the physical domain. The frequency domain is more popularly known as Frequency Based Substructuring (FBS). Based on the classic formulation of Jetmundsen et al. and the reformulation of De Klerk et al., it has become the most commonly used domain for substructuring, because of the ease of expressing the differential equations of a dynamical system (by means of Frequency Response Functions, FRFs) and the convenience of implementing experimentally obtained models. The time domain refers to the recently proposed concept of Impulse Based Substructuring (IBS), which expresses the behaviour of a dynamic system using a set of Impulse Response Functions (IRFs). The state-space domain, finally, refers to methods proposed by Sjövall et al. that employ system identification techniques common to control theory.