# Equivalence in Experimental Source Characterisation

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## Abstract

Transfer Path Analysis methods allow to describe the propagation of vibrations from an active source to a passive receiving structure. These methods are based in the experimental modelling domain, where the dynamics of the problem are described based on empirical observations from an experiment, as opposed to a numerical model based on natural and engineering laws. These methods describe the boundary conditions for the experiment, as well as how the obtained data can be used to deduce an equivalent force that describes the vibrations. Within the class of Transfer Path Analysis methods there is a specific class called the Component-based methods that are able to describe the vibrating source in such a way, that the assembled dynamical behaviour of the source attached to any other receiving structure can be predicted. This is by defining the characterisation in terms of an equivalent force on the interface. These methods are said to be a function of the source only. This is called source characterisation. In this thesis some of the assumptions of these Component-based Transfer Path Analysis are challenged. This will be done based on theoretical research as well as experiments on a numerical model. The reason this research is of interest, is that while these Component-based methods show to be valid in theory, their results in practice are not satisfactory. This is especially apparent when a characterised source is virtually assembled on a different receiving structure. Its predicted behaviour is not in line with what the real world counterpart. These Component-based Transfer Path Analysis methods involve solving an inverse problem to determine the equivalent force. Due to the experimental nature in which the data are obtained, the data has error-contamination. This happens because of interference effects causing noise on the analog sensors. This noise can have a profound effect on the solution of the inverse problem, since the problem is poorly conditioned. The main assumption of Component-based Transfer Path Analysis methods is that if an interface force can be defined that excites the interface in the same way as the source does, the interface force can be regarded as equivalent. Due to the linearity of the problem, this equivalent force can be added in negative on top of the source excitation. This would theoretically then cancel out all motion, since both the source excitation and equivalent force have the same effect on the receiving structure. This thesis proofs that it is not always possible to find an equivalent force that represents the source excitation. This is due to the position of the equivalent force, specifically that it is placed on a single point. Some dynamic modes will show a node of their motion on the interface, meaning the interface force can by definition never excite those modes. This limitation is based from the modelling choices of the interface. It will be uncovered that the modes with such nodal behaviour are the eigenfrequencies of the source for a perfectly fixed interface. The interface limitation can be expressed as a controllability measure, where the controllability defines to what extend the interface force can excite the possible modes crossing the interface from the source. This controllability is a function of the source, the receiving structure and the sensor placement, where the source determines at which frequencies the controllability problems occur and the receiving structure and sensor placement determine the size of the controllability measure. Both an upper and lower bound of controllability can be defined. It can be quantified what the effect of the incomplete interface description is on the source characterisation. This is based on comparing the motion from source excitation and the motion from the predicted dynamics from the equivalent force. It is shown that the interface motion cannot be reproduced by the equivalent force at the frequencies where there are controllability problems. This mismatch in interface motion has an effect on the motion of the receiving structure, where it will be clear that these mode occur at range of frequency bins. While a portion of the source cannot be characterised, a large part of the behaviour can be. To solve for this equivalent force, an inverse problem has to be solved. This problem is an over-determined problem with a full rank column space. Using a priori information about the levels of noise that are expected on each measurement, the problem can be altered. This makes it possible to reduce the propagation of noise from the measurement data to the solution. A set of guidelines are proposed that can help to dampen or emphasis individual modes of the inverse system. Alternatively, the entire spectrum of modes can be dampened or single modes can be truncated. The regularisation methods will all be judged on how they change the physical relevancy of the problem.