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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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... We propose an alternative, hybrid-based approach to modeling the dynamic properties of systems with a complex geometry (e.g., 3D-printed structures). The approach is based on the system equivalent model mixing (SEMM) method that integrates the experimental response model with a simplified numerical model [13]. The SEMM method is already applicable for joint identification [23], correction of noisy data from high-speed cameras and acoustic measurements [24,25] and data-consistency identification [26,27]. ...

... The SEMM method was introduced by Klaassen et al. [13] and is based on the dynamic substructuring approach using Lagrange multiplier frequency-based substructuring (LM FBS) [34]. While the LM FBS method tries to couple the response models of multiple different structures, the SEMM method assembles equivalent models on a single substructure. ...

... The basic SEMM method also has some extensions, increasing its robustness [13]. One of them is the ability to remove spurious peaks, which are a consequence of the conflicting dynamics between the models [36]. ...

... The toolbox was introduced to the community in [1]. Methods in FBS were implemented within the pyFBS, such as virtual point transformation [2] and system equivalent model mixing [3]. As with many open-source toolboxes, the pyFBS toolbox is subject to a ongoing development process. ...

... System equivalent model mixing is a method with which multiple equivalent response models can be mixed using the LM FBS [3]. A schematic representation of SEMM is schematically depicted in Fig. 5 properties and the parent model a DoF-set. ...

... Schematic representation of System Equivalent Model Mixing (SEMM)[3]. ...

pyFBS is an open-source Python package for frequency-based substructuring. The package implements an object-oriented approach for dynamic substructuring. This tutorial is intended to introduce structural dynamics and NVH engineers to the research toolbox in order to overcome vibration challenges in the future. The focus will be on experimental modeling and post-processing of datasets in the context of dynamic substructuring applications. The state-of-the-art methods of frequency-based substructuring, such as the virtual point transformation, the singular vector transformation, and system-equivalent model mixing, are available in pyFBS and will be presented. Furthermore, basic and application examples, as well as numerical and experimental datasets that are provided, are intended to familiarize users with the workflow of the package. pyFBS is demonstrated with two example structures. First, a simple beam-like structure is used to demonstrate how to start with the experimental modeling, FRF synthesis, virtual point transformation, and mixing of system equivalence models. Second, an automotive test structure is used to demonstrate the use of the pyFBS on a complex structure where in-situ transfer path analysis is used to characterize the blocked forces. This tutorial is intended to provide an informal overview of how research can be powered by open-source tools.

... On the other hand, expansion methods enable the expansion of the measurements at a limited number of points to a denser set of DoFs defined by the numerical model. Since the measured data are usually in the form of frequency-response functions (FRFs), Klassen et al. [1] introduced an expansion technique called System Equivalent Model Mixing (SEMM), fully defined in the frequency domain. With SEMM, equivalent numerical and experimental models of the same structure are dynamically coupled using the Dynamic Substructuring approach [2]. ...

... System Equivalent Model Mixing (SEMM) was introduced by Klaassen et al. [1] and forms a hybrid model by mixing the numerical and experimental FRFs. The method is based on the parent model (Fig. 1a), which provides the extensive DoF set and is usually of numerical nature. ...

... Conventionally, the SEMM method is intended to couple equivalent models of the same structure, whereby the numerical model is considered as a digital twin of the experimental model [1]. As we are often interested in the dynamic response of the certain sub-competent of the assembly only, we propose a different methodology to the SEMM formulation. ...

Accurate determination of the high-resolution dynamic response is a crucial step for the dynamic properties characterization in the development phase of a modern product. The conventional experimental procedure for the identification of the structure’s dynamic properties is an experimental modal analysis, where a high spatial resolution can be obtained by imposing a large number of response and excitation points at a structure. To avoid time-consuming experimental testing, measurements involving only a limited number of points at the structure can be expanded to the unmeasured points through model-based expansion techniques. They rely on the introduction of a numerical model, presenting the basics of the expansion. Recently, an expansion method called System Equivalent Model Mixing (SEMM) was proposed where a numerical DoF set is used to extend an experimental model consisting only of a limited number of measurement points. Using the dynamic substructuring approach, the equivalent experimental and numerical models are coupled so that the hybrid model inherits the dynamic properties of both. Although the method has been well adopted, there is still no comprehensive phenomenological analysis to determine the influence of the method’s parameters on the consistency of the hybrid model and thus on the accuracy of the expansion process. This paper addresses the issue by evaluating the accuracy of the SEMM expansion process, focusing on the influence of the regularity of the so-called equivalent numerical model. The introduction of quasi-equivalent numerical models into SEMM is analysed here, which can differ not only with respect to the mass and stiffness properties but also in terms of the geometry and boundary conditions. The parametric study was carried out on a real component of a household appliance, and the most influential parameters in terms of accuracy of the SEMM expansion process were identified.

... The latter was further augmented in [10] by applying truncated singular-value decomposition (TSVD), thus increasing the method's resilience to experimental errors. SEMM has been successfully employed in a variety of applications, such as dynamic coupling within frequency-based substructuring [8,11] and substructuring-based joint identification [12,13]. Although the framework is typically considered for experimental-numerical hybrid modeling, the same procedure can be applied for purely experimental applications, such as improving full-field high-speed [14] or acoustic [15] camera measurements. ...

... Spuriousities often occur in hybrid models, obtained with the basic formulation of frequency-based SEMM. This problem tends to be solved by applying the extended SEMM formulation, which typically smooths out the spurious peaks [8]. Modal implementation, however, makes it possible to completely disregard the spurious dynamics in the hybrid model, while maintaining low computational costs, which could prove beneficial when applied to real-life complex systems. ...

... and by pre-multiplication with L T m , the system of Eqs. (6)- (8) reduces to the primal formulation of the coupled equation of motion:M mξ +C mξ +K m ξ =f m , ...

A substructuring-based method System Equivalent Model Mixing (SEMM) has recently been introduced as a novel expansion method. Originally it was implemented in the frequency domain and has been proven to have a great potential. The objective of this paper is to introduce M-SEMM, the modal domain formulation of system equivalent model mixing. Its basic formulation is presented, considering either physical or modal constraints between substructures. When modal constraints are applied, the derivation reveals that the resulting M-SEMM formulation is equivalent to System Equivalent Reduction/Expansion Process (SEREP), one of the most established reduction/expansion methods in the modal domain. Further, when considering physical constraints, the resulting formulation can be seen as a potentially useful novel modal expansion method.KeywordsSEMMSEREPModel expansionHybrid modellingDynamic substructuring

... A general framework on dynamic substructuring is provided by De Klerk et al. in [2], where the approaches in physical, modal and frequency domain are presented supposing that the systems can be described by linear models. Substructuring techniques are broadly used because, among other advantages, they allow to combine experimental and numerical models [3,4]. Dealing with experimental data, the substructuring approach in frequency domain, called Frequency Based Substructuring (FBS) [5] is the most advantageous. ...

... Substructure decoupling can be used to perform the identification of the characteristics of a joint considered as a subsystem connected to the other physical subsystems of the assembly [12,13,14] In order to obtain the dynamic behaviour of the joint substructure at coupling DoFs, substructure decoupling requires the information of the assembled structure at least, on the coupling DoFs. This could be a problem if the joint interface is not accessible for measurements, but it can be overcome by using for example System Equivalent Model Mixing (SEMM) [4]. It performs coupling and decoupling operations between the numerical and the experimental model of the same component to obtain a hybrid model in which the experimental dynamics measured at accessible DoFs is expanded on inaccessible DoFs. ...

... • removed model [Y ] rem : it is a numerical condensed form of the parent model used to remove the parent dynamics from the component. In the so called "Extended SEMM" presented in [4], the removed model is defined on the global set of DoFs and coincides with the parent model: ...

Substructure decoupling techniques, defined in the frame of Frequency Based Substructuring, allow to identify the dynamic behaviour of a structural subsystem starting from the known dynamics of the coupled system and from information about the remaining components. The problem of joint identification can be approached in the substructuring framework by decoupling jointed substructures from the assembled system. In this case, information about the coupling DoFs of the assembled structure is necessary and this could be a problem if the interface is inaccessible for measurements. Expansion techniques can be used to obtain the dynamics on inaccessible (interface) DoFs starting from accessible (internal) DoFs. A promising technique is the System Equivalent Model Mixing (SEMM) that combines numerical and experimental models of the same component to obtain a hybrid model. This technique has been already used in an iterative coupling–decoupling procedure to identify the linear dynamic behaviour of a joint, with a Virtual Point description of the interface. In this work, a similar identification procedure is applied to the Brake Reus Beam benchmark to identify the linear dynamic behaviour of a three bolted connection at low levels of excitation. The joint is considered as a third independent substructure that accounts for the mass and stiffness properties of the three bolts, thus avoiding singularity in the decoupling process. Instead of using the Virtual Point Transformation, the interface is modelled by performing a modal condensation on remote points allowing deformation of the connecting surfaces between subcomponents. The purpose of the study is to highlight numerical and ill-conditioning problems that may arise in this kind of identification.KeywordsExperimental substructuringDecouplingJoint identificationModel mixingDeformable connecting surfaces

... The latter was further augmented in [10] by applying truncated singular-value decomposition (TSVD), thus increasing the method's resilience to experimental errors. SEMM has been successfully employed in a variety of applications, such as dynamic coupling within frequency-based substructuring [8,11] and substructuring-based joint identification [12,13]. Although the framework is typically considered for experimental-numerical hybrid modeling, the same procedure can be applied for purely experimental applications, such as improving full-field high-speed [14] or acoustic [15] camera measurements. ...

... Spuriousities often occur in hybrid models, obtained with the basic formulation of frequency-based SEMM. This problem tends to be solved by applying the extended SEMM formulation, which typically smooths out the spurious peaks [8]. Modal implementation, however, makes it possible to completely disregard the spurious dynamics in the hybrid model, while maintaining low computational costs, which could prove beneficial when applied to real-life complex systems. ...

... and by pre-multiplication with L T m , the system of Eqs. (6)- (8) reduces to the primal formulation of the coupled equation of motion:M mξ +C mξ +K m ξ =f m , ...

Hybrid modeling is an increasingly common procedure for predicting the structural dynamics of complex products. System equivalent model mixing (SEMM), a dynamic substructuring-based method, is a recent addition to the field. Its implementation within frequency-based substructuring has great potential. Therefore, it is reasonable to explore the options for implementing an equivalent substructuring framework in any of the other substructuring domains. The objective of this paper is to present M-SEMM, system equivalent model mixing in the modal domain. A theoretical derivation reveals that under certain constrained conditions, the proposed methodology is equivalent to the system equivalent reduction expansion process (SEREP), a well-established reduction/expansion technique. When considering the expansion of spatially sparse models with a high modal density, M-SEMM represents a novel expansion method, which can be seen as a potentially useful extension to SEREP. The proposed implementation offers certain advantages, the most notable being a superior ability to disregard spurious modes in the hybrid model. A study of the proposed methodology on a numerical and an experimental case demonstrates its applicability and provides a comparison with SEREP and the original implementation of SEMM in the frequency domain.

... The SEMM method was introduced by Klassen et al. [20] and is based on the dynamic substructuring approach using Lagrange multiplier frequency-based substructuring (LM FBS) [21]. While the LM FBS method tries to couple the response models of multiple different structures, the SEMM method assembles equivalent models on a single substructure. ...

... The basic SEMM method also has some extensions increasing its robustness [20]. One of them is the ability to remove spurious peaks, which are a consequence of the conflicting dynamics between the models [34]. ...

... Here, the procedure for expanding the measurements from a limited number of points to a full-field dynamic response using the SEMM method is presented. Conventionally, the SEMM method is intended for coupling equivalent models of the same structure, whereby the numerical model is considered as a digital twin of the experimental model [20]. ...

A high-resolution dynamic response is important for characterizing a system’s dynamic properties. Measurements involving a limited number of points on the structure can be expanded to unmeasured points through approximation or model-based expansion techniques that rely on the introduction of a numerical model. Recently, an expansion method called System Equivalent Model Mixing (SEMM) was proposed where a numerical DoF set is used to extend an experimental model with limited measurement points. The concept of SEMM is similar to the well-known SEREP and VIKING expansion methods, but it is defined in the frequency domain. Using the dynamic substructuring approach, the equivalent experimental and numerical models are coupled so that the hybrid model inherits the dynamic properties of both models. Although the method has been well adopted, there is still no comprehensive phenomenological analysis to determine the influence of the method’s parameters on the consistency of the hybrid model and thus on the accuracy of the expansion process. This paper addresses the issue by evaluating the accuracy of the SEMM expansion process, focusing on the influence of the regularity of the so-called equivalent numerical model. The introduction of quasi-equivalent numerical models into SEMM is analysed here, which can differ not only with respect to the mass and stiffness properties but also in terms of the geometry and boundary conditions. The parametric study was carried out on a real component of a household appliance, and the most influential parameters in terms of accuracy of the SEMM expansion process were identified. The analysis showed that accurate expansion results, with a small number of experimental points, is achieved if only those points are well scattered across the analysed system.

... Since the joint interfaces in the blade-roots are not accessible for measurements, the dynamics have to be expanded there. For this purpose, the technique System Equivalent Model Mixing (SEMM) [5] can be used to produce a frequency-domain hybrid model with the expanded dynamics. Essentially, the method mixes different model descriptions (numerical and experimental) of a component. ...

... A numerical FRF model obtained from a Finite Element (FE) analysis can provide the FRFs for all the DoF including the ones inaccessible (interface) and inconvenient (drive-point) for measurement. It can then provide all the elements in the FRF matrices of Eq. (4) and (5). This model is denoted by Y N,A . ...

... Using the above two models, one can construct a hybrid model by System Equivalent Model Mixing (SEMM) [5]. The method uses the DoF structure of the numerical model Y N,A and overlays the measured dynamics of the experimental model Y ov,A . ...

Joint identification of blade-root joints in typical bladed-disk assemblies is not possible with the classic decoupling methods due to inaccessibility of interface degrees-of-freedom. In a recent study, an attempt was made to identify such a joint by an expansion based decoupling strategy called System Equivalent Model Mixing (SEMM). The expanded sub-models of the connected substructures and their assembly can be influenced by the measurement errors and the discrepancies between the numerical and experimental sub-models. Therefore, the accuracy of the identified joint is compromised. In this work, we investigate some key factors to improve the expanded sub-models through a new measurement campaign on the unconstrained substructures and the assembly. These factors are i) expansion error, ii) interface type, and iii) singular value filtering. The resulting identified joint properties are validated by recoupling the joint with the respective substructures. It is shown that, by controlling these factors, the joint identification can be highly improved.

... The method can be referred to as hybrid and therefore represents a very powerful modeling methodology that can integrate high-resolution 70 spatial measurements using NAH with the accuracy and consistency provided by precise translation discrete measurements. The hybrid model is established using the recently developed System Equivalent Model Mixing (SEMM) method [28,29]. With SEMM, different dynamic models of the same system can be mixed into a single hybrid model using the Lagrange-multiplier frequency-based 75 substructuring (LM FBS) method [27,30]. ...

... The SEMM method was first introduced by Klaassen et al. [28] and is ...

... To remove dynamics of the parent from the newly formed model, removed model (a condensed version of parent model defined in the boundary DoF) is decoupled, resulting in hybrid model following dynamic properties provided by the overlay model. Therefore, SEMM can be treated as an expansion method [28]. The compatibility and the equilibrium conditions (Eq. ...

A reconstructed displacement field using near-field acoustic holography (NAH) serves as an alternative to conventional measurement methods when it comes to obtaining the high-resolution vibration response of a structure. The method is highly applicable as it enables direct, non-contact measurement of the 3D structural response based on a single acoustic measurement. Although useful, the method’s ill-posed nature limits its use in the field of structural dynamics. This problem can be effectively addressed by using regularization and/or field-separation techniques that can attenuate the noise and the presence of external acoustic sources. All these methods rely on the measurement of acoustic quantities; therefore, the reconstruction of structural admittances is based solely on the evaluation of the hologram(s). This article proposes an alternative approach to improving the accuracy of NAH-based structural admittances by integrating them with a few discrete response measurement on the structure itself. The formulation relies on the mixing of the high-resolution NAH measurement with accurate discrete measurements (e.g., accelerometer or laser vibrometer) using dynamic substructuring techniques. The proposed hybrid approach is a very powerful modeling methodology that can integrate high-resolution spatial measurements using NAH with the accuracy and consistency provided by precise translation discrete measurements. In order to mix two experimental response models System Equivalent Model Mixing (SEMM) method is proposed. An experimental case study on a T-shaped structure demonstrates the robustness and improved accuracy of the estimated structural admittances compared to the plain NAH formulation.

... Therefore, expansion methods are needed. Using the Frequency based Substructuring (FBS) framework [19], System Equivalent Model Mixing (SEMM) [20] allows expansion of the measured dynamics on the internal DoF to the interface DoF. The peculiarity of SEMM is that it uses different formulations of the same system, coupling the numerical model with experimental measurements performed on a limited number of locations. ...

... Using the numerical and experimental FRF models, the hybrid model [20] can be computed from the following single-line expression: ...

... where ( ) + represents the Moore-Penrose pseudo inverse. The equation is derived in [20] from the FBS framework [11]. The hybrid model has the following properties: ...

In mechanical systems coupled with joints, accurate prediction of the joint characteristics is extremely important. Despite years of research, a lot is yet to be learnt about the joints' interface dynamics. The problem becomes even more difficult when the interface Degrees-of-Freedom (DoF) are inaccessible for Frequency Response Function (FRF) measurements. This is, for example, the case of bladed-disk systems with dove-tail or fir-tree type joints. Therefore, an FRF based expansion method called System Equivalent Model Mixing (SEMM) is used to obtain expanded interface dynamics. The method uses numerical and experimental sub-models of each component and their assembly to produce the respective expanded or hybrid sub-models. By applying substructure decoupling to these sub-models, the joint can be identified. However, the joint can be noisy due to expansion and measurement errors which propagate to the hybrid sub-models. In this paper, a correlation based approach is proposed in the SEMM method wherein the quality of the expanded sub-models is improved. In this new approach, several expanded models are generated systematically using different combinations of the experimental FRFs and computing a parameter, Frequency Response Assurance Criteria (FRAC), to evaluate quality of the contribution of the different measurements. The lowest correlated channels or FRFs can be filtered out based on a certain threshold value of FRAC. Using the improved hybrid sub-models, the joint identification also shows a remarkable improvement. The test object for the method is an assembly of disk and one blade with a dove-tail joint.

... The method's application to complex, real-life engineering structures is often hindered by its notorious sensitivity to experimental errors [2]. It remains a challenge to extract a consistent dynamic model performed on a limited number of essentially non-collocated DoFs [5]. ...

... Here, however, the entire formulation is developed within the frequency domain. The expansion process is based on the System Equivalent Model Mixing (SEMM) method that was presented by Klaassen et al. [5]. Since the method is developed in the frequency domain, the transition to the modal domain, which can remove the physically relevant information about the real system, is not required. ...

... System Equivalent Model Mixing (SEMM) was first introduced by Klaassen et al. [5]. The method forms a hybrid structural dynamic model by mixing the numerical and experimental FRFs. ...

The dynamic properties of modern products are analysed using an experimental approach through the measurement of frequency-response functions (FRFs). For an individual measurement, the coherence offers an online check during the system acquisition. More general tools for determining the consistency of the complete measurement set are based on a comparison of the FRFs or the modal shapes with a numerical model. They are useful tools, but they rely on a comparison with a numerical model that might not reflect the behaviour of the actual system. This paper aims to develop a comprehensive experimental method to check the consistency of individual measurements based on comparisons with the complete experimental response model. The numerical model is introduced only to enable the experimental model to be expanded using the System Equivalent Model Mixing method. The entire formulation is developed in the frequency domain, so that the transition to the modal domain, which might remove the physically relevant information from the system, is not required. In the frequency domain, it is possible to assess the consistency of the FRF across the entire frequency range of interest and not only in the region of the natural frequencies. This is of great importance in the area of frequency-based substructuring, where even small inaccuracies in the substructure's FRFs (e.g., the position of the anti-resonance) can lead to erroneous coupling results due to the inversion process. The experimental case study demonstrates the efficiency of the proposed approach. By removing the identified inconsistent measurements, it was possible to significantly increase the accuracy of the final coupling process.

... Therefore, expansion methods are needed. Using the Frequency based Substructuring (FBS) framework [19] , System Equivalent Model Mixing (SEMM) [20] allows expansion of the measured dynamics on the internal DoF to the interface DoF. The peculiarity of SEMM is that it uses different formulations of the same system, coupling the numerical model with experimental measurements performed on a limited number of locations. ...

... In SEMM, an experimental FRF model of the structure is overlaid on its numerical model (to be discussed in the next subsection) to expand the measured dynamics on the unmeasured DoF [20] . This experimental model is called the overlay model Y ov and it can be obtained by setting it equal to Y exp or taking its subset. ...

... Using the numerical and experimental FRF models, the hybrid model [20] can be computed from the following single-line expression: ...

In mechanical systems coupled with joints, accurate prediction of the joint characteristics is extremely important. Despite years of research, a lot is yet to be learnt about the joints’ interface dynamics. The problem becomes even more difficult when the interface Degrees-of-Freedom (DoF) are inaccessible for Frequency Response Function (FRF) measurements. This is, for example, the case of bladed-disk systems with dove-tail or fir-tree type joints. Therefore, an FRF based expansion method called System Equivalent Model Mixing (SEMM) is used to obtain expanded interface dynamics. The method uses numerical and experimental sub-models of each component and their assembly to produce the respective expanded or hybrid sub-models. By applying substructure decoupling to these sub-models, the joint can be identified. However, the joint can be noisy due to expansion and measurement errors which propagate to the hybrid sub-models.
In this paper, a correlation based approach is proposed in the SEMM method wherein the quality of the expanded sub-models is improved. In this new approach, several expanded models are generated systematically using different combinations of the experimental FRFs and computing a parameter, Frequency Response Assurance Criteria (FRAC), to evaluate quality of the contribution of the different measurements. The lowest correlated channels or FRFs can be filtered out based on a certain threshold value of FRAC. Using the improved hybrid sub-models, the joint identification also shows a remarkable improvement. The test object for the method is an assembly of disk and one blade with a dove-tail joint.

... Therefore, the novel methodology presented in this paper proposes a hybrid approach to incorporate strong suits of full-field noisy measurements from a high-speed camera with accurate measurements from an accelerometer into a single model. This is referred to as the mixing of multiple equivalent experimental models of the same component into a hybrid model using dynamic substructuring techniques [25,26,27]. It represents a hybrid and therefore very powerful modeling approach that would not follow an updating scheme which could remove the physically relevant information of the system [28]. ...

... It represents a hybrid and therefore very powerful modeling approach that would not follow an updating scheme which could remove the physically relevant information of the system [28]. The hybrid model is established using the recently developed System Equivalent Model Mixing (SEMM) method [26]. With SEMM different dynamic models of the same system can be mixed into one hybrid model based on the Lagrange Multiplier Frequency-Based Substructuring (LM FBS) method [29]. ...

... System Equivalent Model Mixing (SEMM) [26] makes it possible to mix multiple equivalent-response models of the same system using Lagrange Multiplier Frequency-Based Substructuring (LM FBS) [29]. A schematic representation of SEMM is shown in Fig. 12. ...

The use of a high-speed camera for dynamic measurements is becoming a compelling alternative to accelerometers and laser vibrometers. However, the estimated displacements from a high-speed camera generally exhibit relatively high levels of noise. This noise has proven to be problematic in the high-frequency range, where the amplitudes of the displacements are typically very small. Nevertheless, the mode shapes of the structure can be identified even in the frequency range where the noise is dominant, by using eigenvalues from a Least-Squares Complex Frequency identification on accelerometer measurements. The identified mode shapes from the Least-Squares Frequency-Domain method can then be used to estimate the full-field FRFs. However, the reconstruction of the FRFs from the identified modeshapes is not consistent in the high-frequency range. In this paper a novel methodology is proposed for an improved experimental estimation of full-field FRFs using a dynamic substructuring approach. The recently introduced System Equivalent Model Mixing is used to form a hybrid model from two different experimental models of the same system. The first model is the reconstructed full-field FRFs that contribute the full-field DoF set and the second model is the accelerometer measurements that provide accurate dynamic characteristics. Therefore, no numerical or analytical model is required for the expansion. The experimental case study demonstrates the increased accuracy of the estimated FRFs of the hybrid model, especially in the high-frequency range, when compared to existing methods.

... On the other hand, frequency-based substructuring (FBS) methods provide a great advantage due to the fact that the directly measured FRFs are utilized without any modal parameters estimation. A recently developed expansion method, system equivalent model mixing (SEMM) [11], based on the FBS formulation provides a direct and convenient way to expand the measured FRFs over the numerical FRFs. It is a method of coupling (and decoupling) different equivalent models of the same (sub)structure, namely, parent, overlay, and removed models. ...

... Therefore, expansion methods can be used to extrapolate the dynamic information to those DoF. One such method is SEMM [11] based on the LM-FBS whose inputs are a parent model Y par , an overlay model Y ov , and a removed model Y rem , and the output is a hybrid model Y SEMM (or expanded numerical model). In this subsection, all the quantities belong to a substructure and not to a coupled structure. ...

... The dynamics of the overlay model are superposed linearly on the parent model's dynamics, and therefore, the latter's own dynamics need to be decoupled by choosing a removed model. Thus, the removed model is set as the parent model, as proposed in Ref. [11]. ...

A joint between two components can be seen as a means to transmit dynamic information from one side to the other. To identify the joint, a reverse process called decoupling can be applied. This is not as straightforward as the coupling, especially when the substructures have three-dimensional characteristics or sensor mounting effects are significant or the interface degrees-of-freedom (DOF) are inaccessible for response measurement and excitation. Acquiring frequency response functions (FRFs) at the interface DOF, therefore, becomes challenging. Consequently, one has to consider hybrid or expansion methods that can expand the observed dynamics on accessible DOF to inaccessible DOF. In this work, we attempt to identify the joint dynamics using the System Equivalent Model Mixing (SEMM) decoupling method with a virtual point description of the interface. Measurements are made only at the internal DOF of the uncoupled substructures and also of the coupled structure assuming that the joint dynamics are observable in the assembled state. Expanding them to the interface DOF and performing coupling and decoupling operations iteratively, the joint is identified. The substructures under consideration are a disk and blade - an academic test geometry which has a total of 18 blades but only one blade-to-disk joint is considered in this investigation. The joint is a typical dove-tail assembly. The method is shown to identify the joint without any direct interface DOF measurement.

... If, however, the DoF at the interface are not measured but calculated by means of an expansion method, these limitations no longer hold. System Equivalent Model Mixing (SEMM) is a method based on frequency based substructuring that can be used to expand a measurement's DoF-set by coupling the measurement-based model to an equivalent -yet not identical-model with the required boundary DoF [4]. Note that this model only needs to have the required DoF, and not the correct joint dynamics (since these are provided by the measurement). ...

... One can choose to formulate the removed model as the parent model itself, or as a reduced form of of the parent model which contain only the internal DoF i A , i B 2 . An explanation on the differences is omitted in this paper, but is given in [4]. In this application, it is chosen for the removed and parent model to be the same size and thus the same: ...

... It can be shown that from this solution the single-line equation of SEMM can be obtained. The derivation is omitted in this paper, but can also be found in [4]: ...

As the number of models created in a modular fashion increase, the need for accurate identification of real joint dynamics rises. Since joint dynamics are a consequence of component-to-component interaction, they are only present in the assembled state. Yet, it is in the assembled state that measuring the interface degrees of freedom is practically infeasible. Nevertheless, the effects of the joint are present in measurements throughout the component , i.e. the joint dynamics are observable. In this work, system equivalent model mixing is used to expand an experimental measurement with interface degrees of freedom-either rotational or translational-extracted from a numerical model. Subsequently, joint dynamics can be obtained by applying classic frequency based decoupling methods. The strength of this method lies in the ability to test different interface configurations from a single measurement campaign, limited only by the the actual number of sensor or impact locations. The paper shows that an updating scheme can be used to identify joint dynamics without directly measuring interfaces.

... With SEMM equivalent models (i.e. models of the same component yet founded on varying assumptions) are dynamically coupled following DS theory [6]. The result is a model that is comprised of dynamic properties from either one of the input models and which contains the DoF-space spanned by both input models. ...

... The paper does not intend to fully explain the complete working of SEMM, for this we refer to [6]. Instead, a main application of the method is tested and compared to a similar well-known techniques used throughout the community to offer a viable test-case. ...

... These can be solved using standard FBS, more precisely the Lagrange Multiplier Frequency Based Substructuring (LM-FBS) method [7]. It is shown in [6] that solving the LM-FBS problem and reformulating it to a primal notation results in a single line formulation for the SEMM model described by equation (9) Y ...

A method named System Equivalent Model Mixing (SEMM) is presented. SEMM allows for a mixing of two equivalent frequency based models which can be created by either numerical simulation or direct measurements. The resulting hybrid FRF model is full-rank, consists of the DoF of both input models, and contains a physically relevant weighted mix of input dynamics. It is demonstrated that with SEMM a numerical DoF-set can be used to extend an experimental model with limited measurement points; specifically, it is shown how complete interface dynamics can be obtained with just a handful of sensors. The purpose of SEMM is similar to the well-known SEREP and VIKING concepts, yet instead applies Frequency Based Substructuring (FBS) techniques to form a hybrid dynamic model.

... A hybrid model is the result of mixing the experimental model of an assembly part with its numerical model using the System Equivalent Model Mixing (SEMM) method [42]. ...

... The FRFs obtained with the experiment contained a substantial level of noise in a lower frequency range. For this reason they were blended with numerical results at lower frequencies, using the trust function as proposed in Klaassen et al. [42] to obtain the combined admittance tf . For a smoother transition between the numerical and experimental FRFs, the trust function was assigned the shape of the sigmoid function (Fig. 12). ...

Concerning the cost- and resource-saving maintenance of assembly products, it is vital to detect any potential malfunctions, defects or structural damage at the earliest-possible stage. For this reason, considerable efforts are being put into the development of Structural Health Monitoring, a field encompassing different approaches to damage identification and capable of preventing defects and even failure. Structural Health Monitoring is often supported by machine learning, a tool for rapid and effective damage identification that can recognize patterns or changes in the data received from the structure. Despite the advances machine learning has made in recent years, obtaining a suitable data set for the efficient training of machine learning algorithms within Structural Health Monitoring remains a challenge. Currently, the data are usually obtained experimentally, with numerical or analytical models. However, the experimental approach can often be time consuming, while the reliability of numerically obtained data relies heavily on the accuracy of the numerical models in capturing the true behavior of the structure. Analytical models may be constrained by the complexity of the observed object. In this paper an alternative approach based on an experimental–numerical (i.e., hybrid) modeling approach is proposed to build a training set for Structural Health Monitoring. Frequency Based Substructuring is utilized to determine the response model of the assembled system based on the properties of its components as well as to mix experimental and numerical models, while leveraging the advantages of each. This makes it possible to generate the samples of the training set in the form of hybrid models of the structure of interest, exhibiting the realistic properties of a physical structure, with a reasonable measurement effort. Here, the approach is demonstrated for the process of joint-damage identification.

... The nature of the spurious peaks obtained, as well as the influence of damping on their formation is also of interest. [1], which, by coupling and decoupling substructures with the Langrange multiplier frequency-based susbtructuring (LM-FBS) method [2], aims to obtain a hybrid dynamic model by mixing equivalent models -usually numerical and experimentalof the same structure. The goal of SEMM, therefore, is to expand the dynamics contained in a model with fewer DoFs, but higher fidelity, to a model with a denser space of DoFs. ...

... However, there was the occurrence of the spurious peaks. Therefore, applying the extended SEMM method, as demonstrated in [1] as a way of eliminating these peaks, Fig. 3 contains the Bode diagram for Y SEMM 3θ with the extended interface between the parent and removed models. Fig. 3, the effect of the extended interface was able to remove the spurious peaks and the response of the hybrid model becomes equal to that of the real model. ...

This paper reviews the system equivalent model mixing (SEMM) technique. This method is applied to obtain hybrid dynamic models, primarily by using a numerical model in order to expand the degree of freedom space of an experimental model. The formulation of this method is based on coupling and decoupling of substruc-tures, by means of the Lagrange multiplier frequency-based substructuring (LM-FBS) technique. Therefore, the SEMM technique will be applied in a half-vehicle model, where the dynamics of a parent model will be updated with the dynamics of an overlay model of an equivalent system, but with lower density of degrees of freedom (DoFs). This procedure aims to create a hybrid model, which expands the dynamics of the overlay model to the denser DoFs of the parent model. The influence of the interface size, damping and signal noise on the final hybrid model will be evaluated in this paper.

... This paper presents a new approach to the identification of inconsistent measurements. The basic idea is similar to the DCAF method, however, here the entire formulation is developed within the frequency domain using System Equivalent Model Mixing (SEMM) method [2] and coherence criteria. To present the capability of the introduced method, a coupling of two simple beam-like structures is presented. ...

... System Equivalent Model Mixing (SEMM) was first introduced by Klaassen et al. [2]. The method forms a hybrid structural dynamic model by mixing the numerical and experimental FRFs. ...

... In recent years, the Frequency Based Substructuring has gained an increased popularity, due to the proper formulation of the problem and better measurement hardware [6,7,8]. Also within the FBS, it is also possible to build hybrid models, in which experimentally characterized and numerically modelled parts can be combined [9]. This paper shows the capabilities of the pyFBS: an open-source Python package for frequency based substructuring and transfer path analysis. ...

... System equivalent model mixing is a method with which multiple equivalent response models can be mixed using the LM FBS [9]. A schematic representation of SEMM is schematically depicted in Fig. 7. ...

pyFBS is an open-source Python package for Frequency Based Substructuring. The package implements an object-oriented approach for dynamic substructuring. State-of-the-art methodologies in frequency based substructuring, such as virtual point transformation and system equivalent model mixing, are available within the pyFBS. Each method can be used as a standalone or interchangeably with others. Tools are provided to easily visualize components and configure the measurement setup. Also operational deflection shapes and mode shapes can be animated directly within the 3D display. Furthermore, basic and application examples are available, together with numerical and experimental datasets, to enable the user to get familiar with the work flow of the package. This paper showcases the use of the pyFBS on two example structures. Firstly, a simple beam-like structure is used to depict the use of the 3D display, FRF synthetization, virtual point transformation and system equivalent model mixing. Secondly, an automotive test-structure is used to show the use of the pyFBS on real-life complex structure, where the in-situ transfer path analysis is used to characterize blocked forces. The development of the pyFBS is an ongoing effort, as it is actively being used as a research tool. Additional features and new methods will be integrated within the pyFBS in the near future.

... To extend this concept, the round-trip equation can be applied recursively by nesting round-trip transfer FRFs within the identity in Eq. (16). Conceptually, the passive sub-component (B) of the source-receiver model may be sectioned by a virtual coupling interface ( ). ...

... Keeping the reduced amplifier settings unchanged, the matrix , ∈ C 12×2 is measured from shaker excitations on the accelerometer's faces at ( ). 2. The assembly matrices , ∈ C 6×24 and , ∈ C 12×24 are determined simultaneously by roving shaker excitation, still at reduced energy output. The shaker forces are directly applied to the 24 (yellow) accelerometer surfaces at ( ), depicted in the inset of Fig. 9. 3. The shorter transfer path segments are combined to determine the long distance transfer functions , ∈ C 6×2 using the generalised round-trip formulation in Eq. (16). * The ideal full length reference FRFs are determined at maximum shaker output ('high power') by excitation on the remote sensor faces at ( ), indicated by red arrows in Fig. 9. ...

In noise and vibration engineering, a structure’s passive dynamic properties are often quantified by frequency response functions (FRFs). This paper focuses on acquiring FRFs from experimental tests, considering both, translational (x, y, z) and rotational (e.g. moments around these axes) terms. In practical applications, test structures may not allow FRFs to be measured directly due to the impracticality of applying a controlled excitation in a particular direction (e.g. in-plane), the inability to measure rotational dynamics (e.g. moment excitation), insufficient signal-to-noise ratio (SNR) between excitation and response degrees of freedom, or simply due to restricted access. Methods exist to resolve some of the mentioned issues using indirect experimental techniques, such as the round-trip identity. However, these methods are limited to cases in which the driving-point FRFs are sought-after. The present paper extends previous work into a more generalised formulation of the round-trip identity feasible for reconstructing driving-point and transfer mobilities from in-situ measurements conducted in coupled assemblies. By using the round-trip identity, the excitation of moments and/or inaccessible points is avoided altogether and instead replaced by a number of applied forces remote to the points of interest. Manipulation of this round-trip identity yields a formulation for long distance transfer FRFs, expressed in terms of multiple shorter transfer path elements, which are less prone to insufficient SNR. These practical applications of the generalised round-trip concept are experimentally validated for multi-input multi-output assemblies.

... In the response based measurements, an expansion technique called System Equivalent Model Mixing (SEMM) exploits different equivalent models of the same component [11]. It is based on Frequency Based Substructuring [12]. ...

... It relies on three models, namely, an overlay, a parent and a removed model. In [11], its different interface formulations are presented. A recap of the models used in the SEMM extended interface formulations is given below. ...

Bladed-disks in turbo-machines experience high cycle fatigue failures due to high vibration amplitudes. Therefore, it is important to accurately predict their dynamic characteristics including the mechanical joints at blade-disk (root joint) or blade-blade (shroud) interfaces. These joints help in dampening the vibration amplitudes. Before the experimental identification of these joints, it is of paramount importance to accurately measure the interface degrees-of-freedom (DoF). However, they are largely inaccessible for the measurements. For this reason, expansion techniques are used in order to update the single components before their coupling. But the expansion can be affected adversely if the measurements are not properly correlated with the updated model or if they have significant errors.
Therefore, a frequency domain expansion method called System Equivalent Model Mixing (SEMM) is used to expand a limited set of measurements to a larger set of numerical DoF. Different measured models — termed the overlay models — are taken from an impact testing campaign of a blade and a disk and coupled to the numerical model according to the SEMM. The expanded models — termed the hybrid models — are then correlated with the validation channels in a round-robin way by means of Frequency Response Assurance Criteria (FRAC). The global correlations depict whether or not a measurement and the respective expansion is properly correlated. By this approach, the least correlated channels can be done away with from the measurements to have a better updated hybrid model.
The method is tested on both the structures (the blade and the disk) and it is successfully shown that removing the uncorrelated channels does improve the quality of the hybrid models.

... The classic experimental Lagrange Multiplier Frequency Based Substructuring (LM-FBS) has been shown to be quite challenging [1]. The main reasons remain i) measurement noise, ii) drive point FRFs, iii) rotational DoFs, iv) inaccessibility of the interface etc. Therefore, the techniques based on expansion System Equivalent Model Mixing (SEMM) [2] and filtering namely virtual point transformation [3] are employed for this purpose. The advantages are that i) inaccessible DoFs at the interfaces can be expanded based on the observed dynamics elsewhere, ii) unlike the modal domain expansion techniques the rank is full, iii) modal truncation error is avoided due to frequency domain formulation etc. among others. ...

... Note that here an extra step needs to be performed to receive the relevant DoF to truly obtain \bfY J . If a component has a finite element (FE) model as the numerical model providing a DoF structure and a model whose dynamics are important, for example from measurements, then a hybrid model can be created using System Equivalent Model Mixing (SEMM) [2] expressed in the following compact form: ...

Bladed disks are fundamental bricks of the rotating parts of a turbomachine, which can include many of them. Although each blade-disk sector can be considered identical, the presence of imperfections or misalignments or inhomogeneity (so-called mistuning) induces high amplification of vibration response. The case of blade-root joints is analysed as a source of mistuning of the contact. The test-case is an academic bladed-disk comprising of 18 blade-root joints. The root has a typical dovetail configuration with two-sided interface per sector. Its relatively long interface (compared to the blade length) makes the joint identification challenging yet interesting. Therefore, some non-classic identification techniques are employed. The joint identification is done by using System Equivalent Model Mixing (SEMM) and Virtual Point Transformation (VPT). The measured dynamics of the internal degrees of freedom (DoFs) are expanded to the interface DoFs by SEMM at the substructure level i.e. the disk and the blade. The substructure SEMM models are transformed to one or more virtual points by interface displacement modes (IDMs). The virtual point IDMs are decoupled from the assembled structure (one blade coupled to the disk at a time) to identify joint parameters. Since each disk and blade interface is considered unique, therefore, the process can be extended to the second blade coupled to the disk (having the first one removed). Such joint characterization is aimed for all the remaining blade to disk interfaces.

... Generally the removed model must contain, but is not limited to the boundary DoF. In the case of a full system decoupling, as applied in this work, the removed model is equal to the numerical parent model; more information can be found in [4]. Likewise, the overlay model and experimental parent model are identical; assuming the entirety of the experimental measurements DoF is deemed a trusted set. ...

... It is shown that from this solution the single line of SEMM can be obtained. The derivation is omitted in this paper, the interested reader is recommended to read [4]: ...

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.

... The reconstructed coupled FRFs based on decoupling are in reasonable agreement with the FRFs of the real assembly, except for spurious peaks, present across the observed frequency range, especially below 500 Hz. Performing additional simulations revealed that the spurious peaks are present at the eigenfrequencies of the assembly with fixed displacements at the joint DoFs, as described in [44]. The ANN-based reconstructed coupled FRFs match the true assembly's FRFs better; therefore, the joint model can describe the true joint dynamics in the observed frequency range. ...

The dynamic properties of assembled structures are governed by the substructure dynamics as well as the dynamics of the joints that are part of the assembly. It can be challenging to describe the physical interactions within the joints analytically, as slight modifications, such as static preload, temperature, etc. can lead to significant changes in the assembly's dynamic properties. Therefore, characterizing the dynamic properties of joints typically involves experimental testing and subsequent model updating. In this paper, a machine-learning-based approach to joint identification is proposed that utilizes a physics-based computational model of the joint. The idea is to combine the computational model of the joint with dynamic substructuring techniques to train the machine-learning model. The flexibility of dynamic substructuring permits the enforcement of compatibility and equilibrium conditions between the component models from the experimental and numerical domains, facilitating the development of machine-learning models that can predict the dynamic properties of joints. The proposed approach provides an accurate data-driven method for joint identification in real structures, while reducing the number of measurements needed for the identification. The approach permits the identification of a full 12-DoF joint, enabling the coupling of 3D dynamic models of substructures. Compared to the standard decoupling approach, no spurious peaks are present in the reconstructed assembly response. The proposed approach is validated numerically and experimentally by reconstructing the assembly response and comparing the results with known assembly dynamics.

... A combination of measured assembly data (for joint observation), updated numerical models, and optimization algorithm is used in a method proposed by Klaassen et. al [35], called System Equivalent Model Mixing (SEMM), to determine joint properties without the need of measuring DoFs at the interface. The performance and difficulties of SEMM were shown in [36]. ...

The tangential contact stiffness is an important parameter used in non-linear dynamic analyses of jointed structures since it can strongly affect the prediction of resonance frequencies. Many experimental techniques are available for contact stiffness estimations, but the reliability of such estimations remains unknown due to a lack of comparative studies. This paper proposes a comparative study of contact stiffness measurements obtained with two experimental techniques: hysteresis loop measurements and Frequency Based Substructuring (FBS). Hysteresis loops are traditionally measured with dedicated friction test rigs to provide, amongst others, contact stiffness estimations through local interface measurements. The assumption with hysteresis measurements is that the measured parameters are independent of the dynamics of the test rig and can therefore be used as input for analyses of other structures, as long as loading conditions and contact interfaces are comparable. An alternative approach to identify the contact stiffness is FBS, which uses information from the overall system dynamics. FBS has the advantage that it can be applied to any structure, without the need of building ad-hoc test rigs, consequently giving a structure-specific information. Despite this advantage over hysteresis measurements, it is as of yet not well understood how accurately FBS can extract contact stiffness values. This paper presents FBS measurements and hysteresis loop measurements performed simultaneously on the same contact interface of a traditional high-frequency friction rig during vibration, thus enabling a cross-validation of the results of both techniques. This novel comparison validates FBS approaches against local hysteresis measurements and shows the strengths and limitations of both experimental methods, making it possible to improve the current understanding of the contact stiffness of jointed structures.

... However, a precondition to applying these methods is having a reasonably accurate FEM model of the structure. If a FEM model is available, also the 'system equivalent model mixing' (SEMM) method [82] could be used for inferring rotational DoF from only a few measurements on the structure. To account implicitly for the RDoF on the coupling interface, some methods have used multiple connection points on the interface that are simultaneously coupled [36,103], which has been called 'equivalent multiple point connection' (EMPC) [86]. ...

[ Link to PhD defense video:
https://www.youtube.com/watch?v=IEVuF2rJOYs&t=6s ]
This thesis is the result of a 4-year collaboration between the Technical University of Munich and the BMW Group. The goal was to apply substructuring methods to the Noise Vibration Harshness (NVH) engineering needed for integrating electric climate compressors in upcoming vehicles. The compressor is one of the major contributors to the cabin noise in battery electric vehicles (BEVs). An accurate yet practical development process for its vehicle integration is crucial for industry. Specifically, the aim was to simulate the compressor noise in the cabin for different, virtual design variants of the isolation concept. Therefore, the methods from two broader fields were applied: First, the excitation of the compressor was modeled with component transfer path analysis (TPA) methods. Second, the full transfer path from the compressor to the driver’s ear is assembled from multiple subcomponent models, via dynamic substructuring (DS).
For accomplishing the above mentioned goals, different gaps in the current technology have been identified, which will be addressed in this thesis. With frequency based substructuring (FBS), a subclass of DS, it is possible to couple experimental and numerical substructure models in a virtual assembly. For the compressor, it was found that including rigid body models in the transfer path is a valuable addition. The proper formulation and integration of rigid body models in the framework of FBS will be presented. Another bottleneck at the onset of this project, was the proper modeling of rubber bushings in the transfer path. A novel method for experimentally identifying accurate substructure models of rubber isolators was developed. The rotating components in the compressor introduce gyroscopic effects that influence its dynamics. A novel substructuring method for virtually coupling gyroscopic terms to a component could prove that these effects are not relevant for the compressor case. The compressors excitation is described by blocked forces. Applying the blocked forces to the substructured transfer path of the assembly allows to simulate the sound in a virtual prototype. One goal was to make the simulated results audible to non-acoustic experts, which required the creation of sound files. This allowed for a subjective comparison
of different designs at an early development stage. Since the noise predictions with TPA
are typically in the frequency domain, some signal processing is required to create sound
files in the time domain. Different methods for auralization will be compared, which could
not be found in the existing TPA literature. Due to the inverse process for identifying the
blocked forces, measurement noise can be amplified to unacceptably high levels, which are audible in the sound predictions. Regularization methods have the potential to significantly suppress the noise amplification, which is explained and exemplified for blocked force TPA. Additionally, it was found that only the structure-borne sound transmission was not sufficient to describe the compressor noise in the cabin. The compressor is also directly radiating air-borne sound from its housing, which will be included in the NVH model by means of equivalent monopoles. The application examples at the thesis’ end are extending the current state-of-the-art, by showing how the modular vehicle models can be used for early phase, parametric design optimizations on a complex NVH problem.

... It relies on three models, namely, an overlay, a parent and a removed model. In [11], its different interface formulations are presented. A recap of the models used in the SEMM extended interface formulations is given below. ...

Bladed-disks in turbo-machines experience high cycle fatigue failures due to high vibration amplitudes. Therefore, it is important to accurately predict their dynamic characteristics including the mechanical joints at blade-disk interfaces. Before the experimental identification of these joints, it is of paramount importance to accurately measure the interface degrees-of-freedom (DoF). However, they are largely inaccessible for the measurements. For this reason, expansion techniques can be used in order to update the single components. But the expansion can be affected adversely if the measurements are not properly correlated with the updated model. Therefore, a frequency domain expansion method called System Equivalent Model Mixing (SEMM) is used to expand a limited set of measurements to a larger set of numerical DoF. Different measured models - termed the overlay models - are taken from an impact testing campaign of a blade and a disk and coupled to the numerical model according to the SEMM. The expanded models - termed the hybrid models - are then correlated with the validation channels in a round-robin way by means of Frequency Response Assurance Criteria (FRAC). The global correlations depict whether or not a measurement and the respective expansion is properly correlated. By this approach, the least correlated channels can be done away with from the measurements to have a better updated hybrid model. The method is tested on both the structures (the blade and the disk) and it is successfully shown that removing the uncorrelated channels does improve the quality of the hybrid models.

... For detailed information about SEMM, the authors refer to [1]. The results for the FRF magnitude at the reference ymeasurement point with y-excitation at the upper rotor shaft end are shown in Fig. 4. Here, the numerical approximation is opposed with the hybrid SEMM model prediction and validated by the experimental solution. ...

Creating holistic, efficient models for vibration-based monitoring applications containing rotor systems is still challenging. Reasons for these difficulties are application-dependent housing peripheries and inaccessible measurement points. Due to the complexity of these systems, numerical modeling is cumbersome and the application of experimental techniques only is restricted. With this contribution, we propose a solution approach for combining the experimental determined housing dynamics with a numerical rotor model. The method performs in the frequency domain, based on Lagrange Multiplier Frequency Based Substructuring (LM FBS) and System Equivalent Model Mixing (SEMM) as a closely related method. Our technique rests upon three parts: Firstly, a finite elements rotor model with reduced degrees of freedom (DoF) is created and the complete Frequency Response Function (FRF) matrix for all interface and input DoF is calculated. Secondly, the entire FRF-matrix is coupled with the simulatively determined transfer functions of the housing. Thirdly, FRFs of a collocated subset DoF of the rotor-assembly are experimentally measured. These are expanded to the FRF-matrix of the coupled model using the SEMM method. As a result, we get the complete FRF-matrix being full rank and containing dynamics of the entire, coupled system. Finally, the proposed methodology is experimentally validated based on an exemplary transfer function.

This paper shows a modular NVH engineering process based on Dynamic Substructuring and component TPA techniques, using experimental data obtained on a fully electric BMW i4 vehicle. Following the component TPA approach, the electric drive unit (EDU) of the BMW i4 is considered as the vibration source and is described by equivalent forces on the EDU. To describe the presence of a second vibration source, originating from the wheels running on the drums, a set of equivalent forces at the rear wheel hubs is included. The quality of the equivalent forces is evaluated using criteria as defined in a recent ISO standard on the topic [1]. Transfer paths from the EDU up to the targets in the cabin, i.e. sound pressure at the driver's ear and vibrations at the seat rail, are obtained through Dynamic Substructuring of the individual subsystem models using the Lagrange Multiplier Frequency Based Substructuring (LM-FBS) method. The subsystem models include multiple sets of rubber bushings, a rear axle carrier and the vehicle trimmed body. Transfer paths from the rear wheel hubs up to the targets in the vehicle are obtained from FRF measurements. The individual subsystem models are obtained through measurements using the Virtual Point Transformation in DIRAC, a software application specifically designed to generate subsystem models from FRF measurements using 3-or 6-DoF Virtual Points. The rubber bushings are modeled using the inverse-substructuring approach, which is also available in DIRAC. A second application, COUPLE, is then used to generate NVH predictions based on the modular subsystem models and the equivalent force descriptions.

Dynamic substructuring allows to describe an assembled structural system in terms of component subsystems. In experimental dynamic substructuring, the model of at least one (sub)system derives from experimental tests: this allows to consider systems that may be too difficult to model. The degrees of freedom (DoFs) of the assembled system can be partitioned into internal DoFs (not belonging to the couplings) and coupling DoFs. A possible application of experimental dynamic substructuring is substructure decoupling, i.e. the identification of the dynamic model of a structural subsystem embedded in a structural system known from experiments (assembled system) and connected to the rest of the system (residual subsystem) through a set of coupling DoFs. Coupling DoFs are often difficult to observe, either because they cannot be easily accessed or because they include rotational DoFs. However, whilst coupling DoFs and in particular rotational DoFs are needed when coupling together different subsystems, they are not essential in substructure decoupling, because the actions exchanged through the coupling DoFs are already included in the dynamic response of the assembled system. The most promising fields in substructure coupling are: coupling with configuration dependent interface and nonlinear coupling with localized nonlinearities. With reference to substructure decoupling, the most remarkable topics are: interface optimization, configuration dependent coupling conditions, and joint identification.KeywordsExperimental dynamic substructuringSubstructure decouplingInterface DoFs

This study proposes an improved dynamic substructuring model using the estimated frequency response function information at coupling points between substructures. An assembled system generally consists of two or more substructures connected by a bolt. Individual substructure evaluation excluding the effects of other components is important in the development stage of a general mechanical system because the vibro-acoustic performance of the system depends on the specific combination of substructures. Therefore, this study predicted the final coupling system performance using information from the initial evaluation of the individual substructures. Accurate measurements of the joint properties are required to accurately estimate the dynamic assembled system characteristics; however, physical constraints typically limit such measurements at actual coupling points. Accordingly, a method that utilizes generalized coupling properties to estimate the dynamic characteristics of a new coupling system based on the characteristics of an original substructure is proposed. Virtual point transformation is then used to estimate accurate frequency response functions at the coupling points of the assembled system based on convenient measurements. The proposed method was validated using a vehicle suspension that was hard mounted in a test jig and onto an actual vehicle body to estimate the vibration characteristics of the assembled system. The findings of this study contribute to the accurate estimation of the dynamic properties of many real-world bolt-assembled systems.

Vibration measurements with a high-speed camera are becoming a compelling alternative to accelerometers and laser vibrometers. However, estimated FRFs from the high-speed camera are usually displaying relatively high levels of noise. The noise has proven to be problematic especially in the higher frequency range, where the amplitude of the displacements are generally very small. Using the hybrid method the mode shapes can be identified even at higher frequencies, where the amplitude of the response is close to overall noise level. The identification is obtained by using eigenvalues from accelerometer in the Least Square Complex Frequency (LSCF) identification. The identified mode shapes were proven to be consistent; however, the synthesised full field FRFs after least-square frequency-domain LSFD method are erroneous, especially in the higher frequency range. In this paper the possibility of improving the estimation of full-field FRFs from noisy high-speed camera data using the System Equivalent Model Mixing (SEMM) is explored. The SEMM is normally used with a numerical model (as a parent model) with a large number of DoF which is then mixed with an experimental model which is a subset of the parent model DoF. With the proposed method the FRFs obtained from the high-speed camera data are used as a parent model and a few experimental FRFs from the accelerometers are used as an overlay model. Experimental research on the hybrid model shows an increased accuracy in the estimation of the FRFs.

Creating holistic, efficient models for vibration-based monitoring applications containing rotor systems is still challenging. Reasons for these difficulties are application-dependent housing peripheries and inaccessible measurement points. Due to the complexity of these systems, numerical modeling is cumbersome and the application of experimental techniques only is restricted. With this contribution, we propose a solution approach for combining the experimental determined housing dynamics with a numerical rotor model. The method performs in the frequency domain, based on Lagrange Multiplier Frequency Based Substructuring (LM FBS) and System Equivalent Model Mixing (SEMM) as a closely related method. Our technique rests upon three parts: Firstly, a finite elements rotor model with reduced degrees of freedom (DoF) is created and the complete Frequency Response Function (FRF) matrix for all interface and input DoF is calculated. Secondly, the entire FRF-matrix is coupled with the simulatively determined transfer functions of the housing. Thirdly, FRFs of a collocated subset DoF of the rotor-assembly are experimentally measured. These are expanded to the FRF-matrix of the coupled model using the SEMM method. As a result, we get the complete FRF-matrix being full rank and containing dynamics of the entire, coupled system. Finally, the proposed methodology is experimentally validated based on an exemplary transfer function.

The inverse sub-structuring method has been proposed and further developed for multi-component systems to predict the component-level frequency response functions (FRFs) only by the system-level FRFs. The previous methods are all based on the assumption that the system is linear. However, nonlinear systems are more common in engineering practices. This study aims to extend the linear inverse sub-structuring method for nonlinear systems, which can be applied to predict the FRFs of nonlinear substructure only from the system-level FRFs of the coupled nonlinear system. The formulations for nonlinear systems whose nonlinearity is previously known and only displacement-dependent is derived and verified by both lumped parameter models and FEA study. It should be noted that the nonlinear systems with known nonlinearity type and its location, and the nonlinearity can be modeled as a single nonlinear element between two coordinates. The results of the predicted nonlinear FRFs for the component are compared with those directly by measured, showing good agreement. The suggested new nonlinear inverse sub-structuring method is helpful for nonlinear component dynamics identification and/or nonlinear system vibration control.

For components that are difficult to model with conventional analytical or numerical tools, experimentally derived state-space models can instead be used in system synthesis. For successful state-space synthesis, a physically realistic model must be identified. For this purpose, a hybrid first- and second-order system description is used here as the basis for identification. In the identification procedure, a physically motivated rigid body rank constraint is imposed together with a reciprocity constraint. The two constraints are enforced during a re-estimation phase of the state-space matrices following after a traditional state-space subspace identification phase. In this paper, two complex and modally dense industrial components are combined into a dynamical system. An experimental model of a car body-in-white structure is identified. The identified subsystem model is coupled with a finite element model of a rear subframe in a system synthesis. The two subsystems are attached through four rubber bushings modelled by finite element procedures. It is shown that the experimental-analytical assembly successfully predicts the reference measured system, with higher accuracy than what could be achieved with a model based solely on finite elements. It is also shown that synthesis with individually calibrated rear subframe models can capture the variability in the coupled system.

This paper proposes a method to estimate the rotational stiffness at the coupled points of an assembled system. Conventional test-based rotational stiffness evaluation methods are sensitive to measurement errors and require a separate jig for testing. In contrast, as the proposed method uses the natural frequency shift phenomenon resulting from the addition of mass, the measurement error is relatively small, and accuracy is improved by excluding the interference of other modes. The proposed method also solves the complexity of the conventional method by changing the fixed condition of the system using frequency response function-based substructuring modeling; consequently, it does not require a dedicated jig for fixing. In this process, the concepts of trial mass, virtual mass, and virtual spring are introduced to systematically explain the proposed method and applied to the method using frequency shift. The results of experiments conducted on a vehicle shock absorber verify the utility of the proposed method.

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.

Dynamic Substructuring methods play a significant role in the analysis of today’s complex systems.
Crucial in Dynamic Substructuring is the correct definition of the interfaces of the subsystems and the connectivity between them. Although this is straightforward practice for numerical finite element models, the experimental equivalent remains challenging. One of the issues is the coupling of the rotations at the interface points that cannot be measured directly.
This work presents a further extension of the virtual point transformation that is based on the Equivalent Multi-Point Connection (EMPC) method and Interface Deformation Mode (IDM) filtering. The Dynamics Substructuring equations are derived for the weakened interface problem. Different ways to minimise the residuals caused by the IDM filtering will be introduced, resulting in a controllable weighting of measured Frequency Response Functions (FRFs). Also some practical issues are discussed related to the measurement preparation and post-processing. Special attention is given to sensor and impact positioning. New coherence-like indicators are introduced to quantify the consistency of the transformation procedures: sensor consistency, impact consistency and reciprocity.

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.

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.

The dynamic response of coupled structures is influenced by the joints connecting the individual substructures. The friction induced by the interfaces causes non-linear and damping-like effects, which need to be taken into account when applying Experimental Dynamic Substructuring techniques. This paper proposes a compliant interface model in the framework of substructuring, in order to account for the influence of jointed connections. Rather than modelling damping as a separate phenomenon, the proposed compliant interface model characterises (non-linear) damping as a function of the interface force and motion directly. As such the model fits into the Lagrange-Multiplier FBS method. In addition, the concept of complex power is adopted to characterise the effect of damping and isolate the contribution of the interface from the overall dissipation. The theory is illustrated with a test-case on a dedicated test structure. A successful attempt was made to identify damping parameters based on power dissipation of the structure.

This paper presents a validation study of several experimental Frequency Based Substructuring (FBS) techniques that were developed recently. Advances in the techniques up to 2009 were already applied and validated using the rear axle differential - vehicle interaction as a test case. Since then several advances were achieved by various researchers, such as the Modal Constraints for Fixtures and Subsystems (MCFS) and the method of Equivalent Multi-Point Connections (EMPC). So far these advances have been tested on relatively simple academic problems. This validation study is conducted at BMW and focuses on the complex interaction between the steering gear and the vehicle. First, the response of the vehicle is obtained experimentally while a simple substitute beam is installed in place of the steering gear. This beam is then decoupled and replaced numerically by the actual steering gear using Frequency Response Functions (FRF) obtained from finite element models. The whole procedure of decoupling and coupling is evaluated by comparison with a validation measurement. Different to the MCFS method, the decoupling and coupling is performed using Lagrange Multiplier FBS instead of component mode synthesis. This paper also presents a different derivation for the virtual point transformation that extends and generalises the EMPC method.

Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them. In this paper, a certain vision on substructuring methods is proposed, by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept. Thereafter, based on the dual and primal assembly of substructures, a general framework for the classification of the methods is presented. This framework allows us to indicate how the various classes of methods, proposed along the years, can be derived from a clear mathematical description of substructured problems. Current bottlenecks in experimental dynamic substructuring, as well as solution; found in literature, will also be briefly discussed.

This paper investigates methods for coupling analytical dynamic models of subcomponents with experimentally derived models in order to predict the response of the combined system, focusing on modal substructuring or Component Mode Synthesis (CMS), the experimental analog to the ubiquitous Craig-Bampton method. While the basic methods for combining experimental and analytical models have been around for many years, it appears that these are not often applied successfully. The CMS theory is presented along with a new strategy, dubbed the Maximum Rank Coordinate Choice (MRCC), that ensures that the constrained degrees of freedom can be found from the unconstrained without encountering numerical ill conditioning. The experimental modal substructuring approach is also compared with frequency response function coupling, sometimes called admittance or impedance coupling. These methods are used both to analytically remove models of a test fixture (required to include rotational degrees of freedom) and to predict the response of the coupled beams. Both rigid and elastic models for the fixture are considered. Similar results are obtained using either method although the modal substructuring method yields a more compact database and allows one to more easily interrogate the resulting system model to assure that physically meaningful results have been obtained. A method for coupling the fixture model to experimental measurements, dubbed the Modal Constraint for Fixture and Subsystem (MCFS) is presented that greatly improves the result and robustness when an elastic fixture model is used.

In this paper, suitable criteria are sought for an optimal replacement of unobservable coupling DoFs when performing substructure decoupling, that is the identification of a dynamic model of a substructure embedded in a known structure. The need arises since coupling DoFs are often difficult to observe, either because they cannot be easily accessed or because they include rotational DoFs. The substitution must be carried out both to preserve the information that would be lost when some coupling DoFs are not taken into account, and to avoid or minimize ill-conditioning problems.

Dynamic substructuring offers the possibility to simulate assembled systems efficiently. The coupling of the substructures can be established either by Component Mode Synthesis (CMS), or Frequency Response Functions (FRF) can be used to couple the substructures by Frequency Based Substructuring (FBS). In real systems, coupling is done by joints which can influence the dynamics of the assembled system significantly due to local damping and nonlinearities caused by friction. In this contribution the coupling of two beam-like substructures, which are assembled by a bolted joint, is considered using both coupling methods. While the substructures are linear, the implementation of the nonlinear friction forces requires special attendance in the equations of motion. The Harmonic Balance Method is therefore used to efficiently compute FRFs. Using FBS, the coupling is established directly in the frequency domain. The method provides the possibility to replace the dynamics of individual substructures by measured FRFs of the uncoupled system and combining numerical and experimental models. Alternatively, Component-Mode-Synthesis is used.

Expansion processes have been used for modal correction studies for some time now. In general, the expansion process was believed to be most accurate when there was fairly good correlation between the analytical and experimental mode for which expansion was to be performed to smooth and complete the experimental data using the analytical model. In essence this is similar to any least squares error minimization approach that is used to expand and complete data. If the correlation was not reasonably good then the expansion process would be tainted by the lack of adequate correlation. However, some recent work suggests that using many shape expansion functions simultaneously may have some merit as an expansion process. Using many shapes simultaneously is a very good alternate approach and overcomes the requirement of having well correlated modes for the expansion process. As such, a new approach for expansion called the Variability Improvement of Key Inaccurate Node Groups (VIKING) has been developed and used for the expansion and smoothing of measured data for system modeling studies as well as an assortment of different structural dynamic studies. The intent of this paper is to illustrate the VIKING technique's dependence on the set of analytical modes used in the expansion process. The U 12 matrix from the Structural Dynamic Modification methodology is utilized to identify the analytical vectors needed for proper formulation of the VIKING matrices. Several analytical cases are presented to clearly identify the number of modes and which modes can be predicted to cause the success of the VIKING technique.

Model reduction and model expansion play an important part in many aspects of structural dynamic modeling techniques. These include reduced models for structural response studies, correlation of analytical and experimental models, expansion of measured mode shapes for structural dynamic modification and system models, and expansion for analytical model improvement studies using modal vectors. A mapping is needed between the analytical and experimental models typically used in these studies that is provided by the model reduction and model expansion processes. Contributions to this important area made by John O'Callahan are presented in this paper. This paper is broken down into Part 1 covering the theoretical aspects and Part 2 which addresses some of the applications of this material.

Taking into account Rotational Degrees of Freedom (RDoF) in experimental Dynamic Substructuring is crucial for a successful coupling. The experimental determination of these RDoF is however very difficult. In this paper a solution is therefore presented where the RDoF information is only implicitly incorporated in the subsystem interface description. The method consists of measuring the subsystem's interface at multiple nodes in multiple directions. The coupling of the subsystems' interfaces is now performed on multiple nodes. As the number of FRFs used in this kind of coupling corresponds to the number of DoF describing the interface, rotational information can be implicitly accounted for. Indeed, a minimum of 6 coupling DoF at three nodes suffices to describe all motions of a rigid interface. Taking into account more interface DoF allows the description of more complex interface deformations. Notice that in many cases interfaces are rather rigid, meaning that the interfaces' local flexibility is commonly described quite well already by 6 rigid body motions only. A projection of the measured FRFs on such local rigid modes can therefore be shown to reduce measurement errors and enhance coupling results. Of course, this method can be extended to include other interface flexibility modes in case these contribute to the coupled interface deformation. The method is illustrated on the coupling of vehicle rear axle component models to the experimentally determined mech+anic-mechanic and mechanic-acoustic bodywork FRFs.

The paper deals with the identification of the dynamic behaviour of a structural subsystem, starting from the known dynamic behaviour of both the coupled system and the remaining part of the structural system (residual subsystem). This topic is also known as decoupling problem, subsystem subtraction or inverse dynamic substructuring. Whenever it is necessary to combine numerical models (e.g. FEM) and test models (e.g. FRFs), one speaks of experimental dynamic substructuring. Substructure decoupling techniques can be classified as inverse coupling or direct decoupling techniques. In inverse coupling, the equations describing the coupling problem are rearranged to isolate the unknown substructure instead of the coupled structure. On the contrary, direct decoupling consists in adding to the coupled system a fictitious subsystem that is the negative of the residual subsystem. Starting from a reduced version of the 3-field formulation (dynamic equilibrium using FRFs, compatibility and equilibrium of interface forces), a direct hybrid assembly is developed by requiring that both compatibility and equilibrium conditions are satisfied exactly, either at coupling DoFs only, or at additional internal DoFs of the residual subsystem. Equilibrium and compatibility DoFs might not be the same: this generates the so-called non-collocated approach. The technique is applied using experimental data from an assembled system made by a plate and a rigid mass.

In recent years, the structural dynamic community showed a renewed interest in dynamic substructuring (i.e. component coupling) techniques, especially in an experimental context. In this paper the reverse problem is addressed: the decoupling (or identification) of a substructure from an assembled system. This problem arises when substructures cannot be measured separately but only when coupled to neighboring substructures, a situation regularly encountered in practice.In this work we present a so-called dual approach to substructure (dis)assembly. In a transparent and straightforward manner, this dual framework allows imposing different equilibrium and compatibility conditions during decoupling in both the physical and modal space. Five substructure decoupling techniques will be derived and/or classified in a unified way by varying these conditions. Thereafter, the decoupling techniques will be applied to two case studies: the first is an academic example, the second a practical decoupling problem using measured data.

A scheme for synthesis of subsystem state-space models to be used for analysis of dynamic behaviour of built-up structures is presented. Using measurements on each component, subsystem models are identified adopting contemporary system identification methods. The subsystem state-space models are transformed into a coupling form, at which kinematic constraints and equilibrium conditions for the interfaces are introduced. The procedure is applied to a plane frame structure, which is built up of two components. It is found that the non-trivial model order determination constitutes a crucial step in the process. If the model order is incorrect at subsystem level, the synthesized model may radically fail to describe the properties of the built-up structure. It is also found that the identified subsystem models need to satisfy certain physically motivated constraints, e.g. reciprocity and passivity. Different approaches and methods to aid the model order determination and the estimation of physically consistent state-space models at subsystem level are discussed.

Modal substructuring or component mode synthesis (CMS) has been standard practice for many decades in the analytical realm, yet a number of significant difficulties have been encountered when attempting to combine experimentally derived modal models with analytical ones or when predicting the effect of structural modifications using experimental measurements. This work presents a new method that removes the effects of a flexible fixture from an experimentally obtained modal model. It can be viewed as an extension to the approach where rigid masses are removed from a structure. The approach presented here improves the modal basis of the substructure, so that it can be used to more accurately estimate the modal parameters of the built-up system. New types of constraints are also presented, which constrain the modal degrees of freedom of the substructures, avoiding the need to estimate the connection point displacements and rotations. These constraints together with the use of a flexible fixture enable a new approach for joining structures, especially those with statically indeterminate multi-point connections, such as two circular flanges that are joined by many more bolts than required to enforce compatibility if the substructures were rigid. Fixture design is discussed, one objective of which is to achieve a mass-loaded boundary condition that exercises the substructure at the connection point as it is in the built up system. The proposed approach is demonstrated with two examples using experimental measurements from laboratory systems. The first is a simple problem of joining two beams of differing lengths, while the second consists of a three-dimensional structure comprising a circular plate that is bolted at eight locations to a flange on a cylindrical structure. In both cases frequency response functions predicted by the substructuring methods agree well with those of the actual coupled structures over a significant range of frequencies.

A generalized frequency domain substructure synthesis methodology with special application to experimentally derived data is presented. This methodology affords efficient syntheses of total structure dynamics from those of its substructures. The implementing algorithms are inherently well-suited for combining finite element characteristics of structures with those experimentally derived. The connection pattern between substructures can be quite general and of relatively complicated nature; an application of graph theory is used to perform the synthesis procedure. The methodology is based on an implicit statement of the force and displacement continuity between substructures. The use of “connecting elements (springs, dampers etc.) between the substructures is provided for, and either simultaneous or sequential synthesis can be used. The methodology permits substructures (representing components such as gearboxes and transmissions) to be coupled by consideration of interface characteristics only. The methodology is thus ideally suited to the analysis of helicopter airframe vibration. Examples are given demonstrating the key features of the theory.

Integration of existing methods and user knowledge in a mimo identification algorithm for structures with high modal densities

- Etienne Balmès

Etienne Balmès, Integration of existing methods and user knowledge in a mimo identification algorithm for structures with high modal densities, in:
11th International Modal Analysis Conference -Modal Testing and Analysis -for Solutions that Fit, Proceedings of the 11th IMAC, A Conference and
Exposition on Structural Dynamics, 1993, pp. 613-619.

Frequency domain identification of structural dynamics using the pole/residue parametrization, in: IMAC XIV-14th International Modal Analysis Conference-Noise and Vibration Harshness (NVH)

- Etienne Balmès

Etienne Balmès, Frequency domain identification of structural dynamics using the pole/residue parametrization, in: IMAC XIV-14th International
Modal Analysis Conference-Noise and Vibration Harshness (NVH), Proceedings of the 14th IMAC, A Conference and Exposition on Structural
Dynamics, 1996.

Dynamic Substructuring Methodologies for Integrated Dynamic Analysis of Wind Turbines

- Sven Niels Voormeeren

Sven Niels Voormeeren, Dynamic Substructuring Methodologies for Integrated Dynamic Analysis of Wind Turbines, PhD thesis, Delft University of
Technology, The Netherlands, 2012.

Dennis de Klerk, A complex power approach to characterise joints in experimental dynamic substructuring, Dynamics of Coupled Structures

- Emiel Barten
- V Maarten
- Van Der Seijs

Emiel Barten, Maarten V. van der Seijs, Dennis de Klerk, A complex power approach to characterise joints in experimental dynamic substructuring,
Dynamics of Coupled Structures, Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, vol. 1, Springer, New York, 2014,
pp. 281-296, Chapter 27.

A benchmark structure for validation of experimental substructuring, transfer path analysis and source characterisation techniques, Dynamics of Coupled Structures

- M V Van Der Seijs
- E A Pasma
- D D Van Den
- M W F Bosch
- Wernsen

M.V. van der Seijs, E.A. Pasma, D.D. van den Bosch, M.W.F. Wernsen, A benchmark structure for validation of experimental substructuring, transfer path
analysis and source characterisation techniques, Dynamics of Coupled Structures, Conference Proceedings of the Society for Experimental Mechanics
Series, vol. 4, Springer, New York, 2017, pp. 295-305, Chapter 26.