A popular strategy in structural dynamic modelling is breaking the structure down into separable, manageable substructures. One can choose the most efficient way of modelling the substructures, before synthesizing the full system model. System Equivalent Model Mixing (SEMM) is a new method that allows mixing of frequency-based models, either of numerical or experimental nature, to form a hybrid structural dynamic model. The method expands measured data onto a numerical mode manifold using Lagrange-Multiplier Frequency Based Substructuring (LM-FBS). Hence, SEMM combines the DoF-space of the numerical model with the dynamic properties of the measured substructure. In this paper, SEMM is applied to a complex vehicle component. Frequency Response Function (FRF) measurements on the component are used to enrich the uncalibrated Finite Element Model of the component. The resulting hybrid model comprises interfaces in six degrees of freedom, which is required for the connectivity to neighboring structures in the FBS framework.

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In this contribution, Frequency Based Substructuring is applied to a well-known benchmark structure with special attention to correct modelling of the interface problem. Various approaches with the Virtual Point Transformation are used to incorporate the effect of interface flexibility that is present in the clamped connections. An extension of the rigid displacement basis with flexible interface displacement modes is proposed and evaluated. Furthermore, a practical approach using a Transmission Simulator is used in order to improve the quality of the interface description.@en

This paper introduces SEMM: a method based on Frequency Based Substructuring (FBS) techniques that enables the construction of hybrid dynamic models. With System Equivalent Model Mixing (SEMM) frequency based models, either of numerical or experimental nature, can be mixed to form a hybrid model. This model follows the dynamic behaviour of a predefined weighted master model. A large variety of applications can be thought of, such as the DoF-space expansion of relatively small experimental models using numerical models, or the blending of different models in the frequency spectrum. SEMM is outlined, both mathematically and conceptually, based on a notation commonly used in FBS. A critical physical interpretation of the theory is provided next, along with a comparison to similar techniques; namely DoF expansion techniques. SEMM's concept is further illustrated by means of a numerical example. It will become apparent that the basic method of SEMM has some shortcomings which warrant a few extensions to the method. One of the main applications is tested in a practical case, performed on a validated benchmark structure; it will emphasize the practicality of the method.

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This paper presents a practical study on popular Experimental Dynamic Substructuring topics. A series of substructures is designed of such complexity to fit in right between “real life” structures as often found in industrial applications and “academic” structures which are typically the simplest models to identify a particular phenomenon. The designed benchmark structure comprises an active side with a vibration source, a passive side and a test rig for source characterisation. The connectivity is scalable in complexity, meaning that a single-point, two-point and continuous interface can be established. Substructuring-compatible component models are obtained from impact measurements using the Virtual Point Transformation. The vibration source on the active structure is characterised on the test rig using the in-situ TPA concept. Hereafter the component TPA method is applied to simulate the response on the passive side of the coupled structure, in turn obtained using dynamic substructuring.

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Component-based Transfer Path Analysis allows us to analyse and predict vibration propagation between an active source and passive receiver structures. The forces that characterise the active source are determined using sensors placed on the connected passive substructure. These source characterisation forces, often called blocked or equivalent forces, are an inherent and unique property of the source, allowing to predict vibration levels in assemblies with different connected passive structures. In order to obtain a unique and accurate characterisation, accurate measurements are of key importance. The success of the characterisation is not only dependent on the hammer skill of the experimentalist, but also relates to sensor placement, overdetermination and matrix conditioning. In this paper the effects of each of these influences are studied using theoretical approaches, numerical studies and measurements on a benchmark structure designed for in-situ source characterisation. An assembly of two substructures is tested, representing an active substructure with a source and a passive substructure. In order to determine a criterion for the placement of indicator sensors, the effect of the various influences on the in-situ characterisation is compared. Using the results, a structured approach for the use of indicator sensors for in-situ blocked force TPA is proposed.@en

Sound and vibration have a defining influence on our perception of product quality. They are especially well-known aspects in the automotive industry; a branch which sees, besides safety and driving comfort, ever-increasing expectations of the acoustic experience. After all, a smooth and silent driving experience appeals to a feeling of premiumness, a connotation no longer reserved to the top segment in the industry. While traditional combustion engines are gradually getting replaced by hybrid or full-electric drive-lines, other electromechanical (so-called mechatronic) systems make their entrance. As a consequence, the sound experience shifts from low-frequent engine roar to high-frequent humming and whining – a yet unfamiliar experience that calls for redefinition of the soundscape. To support such change, it is necessary that sound and vibration aspects can be considered in an early phase of development by means of simulations. This poses a true challenge: although state-of-art numerical modelling techniques can simulate the low-frequent dynamics fairly well, they often fail to provide reliable answers for the higher acoustic frequency range.

This thesis presents techniques that aim to implement measurements of structural dynamics and active vibration sources into development processes. By characterising the passive and active dynamics of yet available components by means of measurements and combining those with numerical models, a hybrid simulation emerges that may provide answers to high-frequent problems in an early phase of development. This hybrid simulation is facilitated by use of Experimental Dynamic Substructuring: a methodology that determines structural dynamic aspects of complete products based on individually measured components.Part one of this thesis presents a variety of methods for simulation and substructuring that form the basic toolbox for generation, analysis, coupling and decoupling of dynamic models. Pivotal is the experimental approach, which means that dynamic models are obtained from measurements rather than numerical modelling efforts. To transform such measurements into a model that is compatible for coupling with other (numerical) models, the virtual point transformation is proposed. This method considers measured responses and applied forces around (user-chosen) points as locally rigid displacements and forces. Doing so, every connection point of a component can be described by three translations and three rotations with respect to a global reference frame, perfectly suited for substructuring. At the same time, the quality of the measurement and transformed frequency response functions can be quantified objectively using the proposed consistency functions. Altogether, the virtual point method bridges the gap between experimental and numerical modelling activities and enables us to exploit substructuring effectively for complex high-frequency systems.

Part two presents a comprehensive study of Transfer Path Analysis (TPA); a collection of methods that contemplate a vibration problem as a source, transmission and receiver. A general framework for TPA is presented by re-interpreting eleven methods from the perspective of substructuring. It is shown that these methods can be categorised into three families, that in turn differ in the nature of characterisation of the source. The component-based TPA is regarded the most promising family, which allows to characterise a source independent of the environment in which it has been measured. The vibrations of the active source can be replaced by equivalent force spectra that, multiplied with the (simulated) FRFs of the assembled vehicle, predict what this source would sound like in the vehicle. Several practical methods are discussed to determine such equivalent forces: from forces measured against a blocked boundary, using free velocities, based on measurements on a compliant test bench or using the so-called in-situ and pseudo-forces methods. For further generalisation, a notation is presented that governs the abovementioned principles and facilitates the application and comparison of component-based TPA methods. In particular, it is shown that controllability and observability – concepts adopted from control theory – are strongly related to TPA; proper understanding of these principles yields interesting opportunities for analysis and simulation.

The developed methods have been applied to analyse the vibrations of the electric power-assisted steering (EPS) system, which is reported on in part three. It is demonstrated that the virtual point transformation is able to determine accurate FRFs in a frequency range up to 6000 Hertz. Substructuring is applied to simulate the FRFs of a vehicle by applying the principle of substitute coupling, which employs a substitute beam during measurement in the vehicle to represent the dynamic effects of the steering system to couple. For the purpose of characterisation of the steering system’s excitations, several testing environments are discussed: a stiff test bench, more compliant test benches and the vehicle itself. Each configuration is accompanied by a specific method for source characterisation, for which it is demonstrated that the equivalent forces are indeed an environment-independent description of the active excitations of the steering system. It is shown that these forces can be used for the prediction of sound and vibrations in the vehicle. The presented applications offer, with understanding of substructuring and TPA theory, insights in the practical aspects of the methodology. This opens interesting opportunities for early-phase development of sound and vibration.@en

Transfer Path Analysis (TPA) designates the family of test-based methodologies to study the transmission of mechanical vibrations. Since the first adaptation of electric network analogies in the field of mechanical engineering a century ago, a multitude of TPA methods have emerged and found their way into industrial development processes.
Nowadays the TPA paradigm is largely commercialised into out-of-the-box testing products, making it difficult to articulate the differences and underlying concepts that are paramount to understanding the vibration transmission problem. The aim of this paper is to derive and review a wide repertoire of TPA techniques from their conceptual basics, liberating them from their typical field of application. A selection of historical references is provided to align methodological developments with particular milestones in science. Eleven variants of TPA are derived from a unified framework and classified into
three categories, namely classical, component-based and transmissibility-based TPA.
Current challenges and practical aspects are discussed and reference is made to related fields of research.@en