Damage Detection of a large space structure using a model obtained by Modal Coupling

In the case we have to analyze a complex structure, as the ISS might be, many factors must be thoroughly kept in mind, in order for the whole work to be efficient. One of the criterions should be the consideration of the expenses that a full scale entire structure model, along with the excitation and measuring devices may bring about. To ease this problem, it is a common procedure in the analysis of somewhat cumbersome or big structures, to resort to substructuring. The idea behind substructuring is to be able to reach conclusions concerning the properties of the assembled structures, only working with the models of its components. Clearly, this still leaves us with many degrees of freedom in order to shape this process. For the sake of clarity, it is better to describe at least a few alternatives. Coupled loads analysis can stem from either modal models or response models, clearly yielding different results. It is obvious that the convenience of using one or the other will mainly depend on the case at hand, and that there is no such a thing as a rule in choosing which strategy to adopt, because, in practical cases, the hardware that we can dispose of is of crucial importance.

Response methods aim at computing the assembled structure's frequency response function matrix starting from the components' matrices, by means of enforcing displacement compatibility and force equilibrium at the coupling degrees of freedom. This means that the first step of the process must be to find the matrix of the frequency response functions relating all the active degrees of freedom of each component alone, with the boundaries at the coupling degrees of freedom free. Active degrees of freedom are here defined as those degrees of freedom that we decide to include in our analysis in order to represent the behaviour of the single component – as a matter of facts, dealing with coupled structure analysis, using all the degrees of freedom for each component would demand very high costs for the hardware to be used. So, since not all the degrees of freedom need to be used either as excitation or as measurement stations – as they are of no or little interest – in order to reduce the size of the models, it is preferred to make a choice as to which ones to use, rather than reducing or condensing the model, as this would imply an approximation. Then, displacement compatibility and force equilibrium can be enforced at those degrees of freedom, and the resulting matrix can be computed. It's important to highlight pros and cons (bright sides and drawbacks) of this method. On one hand, simplicity is allowed by the fact that modal tests' results are used directly in this calculation, without undergoing further treatment, as we use the frequency response function matrices. On the other hand, measurements are repeated as many times as are the frequencies of interest to the final assembled structure, so this implies processing a large amount of data – but, once said that, what this process gives us is the global structure's matrix, at each of these frequencies. Practically, problems arise from the size of the “world” we work in. First, if the result has to be the matrix for the whole structure, this will mean that calculations will be quite onerous, as all matrices involved include the (active) degrees of freedom of the system. Then again, usually, in modal testing, not all the frequency response function matrix is measured; in fact, down to one row of measurements can be enough (even though often more than just one is measured). This brings about the fact that either we measure each one of the matrix elements, and this means really a lot of measurements, or we have a way to derive all the matrix from the measurements we have done. This is a crucial point, as for many structures the matrix being symmetric justifies going no further with measurements or calculations, but, conversely, if we cannot say the matrix is symmetric, then this can mean a lot more work.

Modal methods aim at producing a modal model of the assembled structure, starting from the modal models of its components. First of all, the evident difference with respect to the previous method is the fact that modal analysis is carried out on each component. While this is done in order to have a “lighter” (smaller) model of the substructures to work with afterwards, the modal analysis of each one of them is in fact a burden brought about by this method. Then, strictly speaking, the power of modal models is that they reduce the order of the problem, but this inevitably requires a truncation of the number of modes, and this poses a problem: it is difficult to understand how many modes we must include in the substructures we have, in order to cover up to a certain range of the coupled structure. In this work, we will be focusing on this second kind of methods. Modal coupling can be carried out based on a number of techniques. Mainly, the choice of the path to follow is dependent on which set or sets of modes we decide to use. The alternatives include the normal modes, which can be further divided into fixed-interface, free-interface, or loaded-interface normal modes, or static modes, such as constraint modes, rigid modes and attachment modes. Once it is chosen to undertake the task of modal coupling, the most well-known methods are the Craig and Bampton's method and the fictitious (boundary) masses method. Craig and Bampton's method is based on the use of a particular set of modes. This set includes the normal modes of the component with clamped coupling boundary (fixed-interface normal modes mentioned above), and the static modes obtained by imposing a unit displacement to each one of the coupling degrees of freedom, one by one, while clamping all the others of the coupling interface and leaving all the remaining degrees of freedom free (constraint modes from the above list). While this method is commonly used for numerical models, it is extremely hard to develop in practice, due to the difficulties in imposing the displacement fields required by the constraint modes at the coupling interface. Modal coupling through fictitious masses, instead, is based on the use of loaded-interface normal modes, and can (will) be shown to be able to predict both the behaviour of an assemble structure, given its substructures, and the behaviour of the single component with the coupling interface clamped.