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The development process of mechanical systems involves the evaluation of its modes of vibrations in the frequency range of interest. In general, a linear modal analysis is sufficient to determine whether the system can operate in dynamic conditions. However, in some cases the assembly is composed of many subsystems connected through nonlinear connections which make the response depend on the amplitude and frequency of the excitation. In those cases, Linear Normal Modes (LNMs) are not sufficient to fully describe the dynamics of the system and Nonlinear Normal Modes (NNMs) must be used. Using substructuring techniques it is possible to treat nonlinear joints as independent subsystems. However, a reliable nonlinear model is needed to use this approach. The experimental characterization is the only way to correctly estimate the nonlinear behavior of the connection. Thus, the aim of this work is to develop and validate the experimental characterization to build an experimental nonlinear modal model of a strongly nonlinear element that can be used to connect different linear subsystems and can be regarded as a localized source of nonlinearity. The NNMs identification is performed using the single-point single-harmonic phase resonance method, and the experimental nonlinear modal model of the NLCE is obtained by retaining only the first harmonic term of each NNM in order to get nearly uncoupled modal equations. • Experimental identification of nonlinear normal modes of strongly nonlinear elements. • New concept design of a nonlinear connecting element. • Quantification of super-harmonics contributions of the response around resonance. • Validation of the identified model of the strongly nonlinear element. [ABSTRACT FROM AUTHOR]

The dynamic behavior of a rotating machine consists not only of the response of the shaft, but it also contains information of all supporting structures and equipment interacting with the rotor. These support structures, also named foundations, can add substantial influence on the machine operational behavior, hence it becomes necessary to dynamically characterize its components. This study presents the application of substructuring techniques in assembling a supporting structure model for a rotating machine and presents a novel formulation of the mixed coordinates method, which is used to connect rotor and foundation. This foundation model is validated against the actual physical foundation and provides an efficient way to analyze and optimize the behavior of such systems. The robustness of the substructuring technique is demonstrated by comparing the responses of the global system solved with both the dynamically assembled and experimentally measured supporting structures in free-free and clamped conditions. The simulations reveal important phenomena of the complete system, including the presence of structural modes in the Campbell diagrams and in displacement responses. The technique allows flexibility in the design of numerical foundations without requiring physical mounting and measurements. [ABSTRACT FROM AUTHOR]

The Hurty/Craig-Bampton method in structural dynamics represents the interior dynamics of each subcomponent in a substructured system with a truncated set of normal modes and retains all of the physical degrees of freedom at the substructure interfaces. This makes the assembly of substructures into a reduced-order system model relatively simple, but means that the reduced-order assembly will have as many interface degrees of freedom as the full model. When the full-model mesh is highly refined, and/or when the system is divided into many subcomponents, this can lead to an unacceptably large system of equations of motion. To overcome this, interface reduction methods aim to reduce the size of the Hurty/Craig-Bampton model by reducing the number of interface degrees of freedom. This research presents a survey of interface reduction methods for Hurty/Craig-Bampton models, and proposes improvements and generalizations to some of the methods. Some of these interface reductions operate on the assembled system-level matrices while others perform reduction locally by considering the uncoupled substructures. The advantages and disadvantages of these methods are highlighted and assessed through comparisons of results obtained from a variety of representative linear FE models. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Modeling joint dynamics is the bottleneck for precise predictive models of machines. Bolted joints are especially relevant due to their common use in mechanical engineering. Stiffness and damping properties of bolted joints are highly influenced by contact parameters (geometry, surface treatment, preload, ...). For high amplitude vibrations, when non-appropriate joint closure is present, even frictional effects can play a role. Substructuring techniques offer a lean solution for isolating dynamical or quasi-static components of joint dynamics. It may be used for an identification of linearized contact parameters for multiple degrees of freedom (dofs). The main limitation of the method is finding a robust workflow to guarantee a proper controllability and observability of the desired joint dynamics, as well as dealing with disturbance/noise in the measurements. In this contribution a robust procedure for the identification of a bolted joint using frequency-based substructuring is presented for a contact in an experimental scenario. The dynamics of the joint are isolated from the assembled system using different substructuring techniques treating the joint as a quasi-static or dynamic component. A simple physical model of the joint is parametrized from the experimental joint dynamics. A validation of the methodology is given by using the identified joint parameters on a modified assembled system. • Robust procedure for the multi-dof identification of bolted joints. • Theoretical and experimental comparison of quasi-static and dynamic decoupling in frequency domain. • Practical experimental application of the proposed joint identification. • Inverse substructuring seems to be a good fit for bolted connections. [ABSTRACT FROM AUTHOR]

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. [Display omitted] • A neural-network-based approach to joint identification is proposed. • A physics-based computational model is used to generate training dataset of joints. • Training dataset of assembly admittances are computed using dynamic substructuring. • Measuring the assembly's interface dynamics is not required for the identification. • The proposed joint identification approach is tested in an experimental case-study. [ABSTRACT FROM AUTHOR]

Successful online hybrid (numerical/physical) dynamic substructuring simulations have shown their potential in enabling realistic dynamic analysis of almost any type of non-linear structural system (e.g., an as-built/isolated viaduct, a petrochemical piping system subjected to non-stationary seismic loading, etc.). Moreover, owing to faster and more accurate testing equipment, a number of different offline experimental substructuring methods, operating both in time (e.g. the impulse-based substructuring) and frequency domains (i.e. the Lagrange multiplier frequency-based substructuring), have been employed in mechanical engineering to examine dynamic substructure coupling. Numerous studies have dealt with the above-mentioned methods and with consequent uncertainty propagation issues, either associated with experimental errors or modelling assumptions. Nonetheless, a limited number of publications have systematically cross-examined the performance of the various Experimental Dynamic Substructuring (EDS) methods and the possibility of their exploitation in a complementary way to expedite a hybrid experiment/numerical simulation. From this perspective, this paper performs a comparative uncertainty propagation analysis of three EDS algorithms for coupling physical and numerical subdomains with a dual assembly approach based on localized Lagrange multipliers. The main results and comparisons are based on a series of Monte Carlo simulations carried out on a five-DoF linear/non-linear chain-like systems that include typical aleatoric uncertainties emerging from measurement errors and excitation loads. In addition, we propose a new Composite-EDS (C-EDS) method to fuse both online and offline algorithms into a unique simulator. Capitalizing from the results of a more complex case study composed of a coupled isolated tank-piping system, we provide a feasible way to employ the C-EDS method when nonlinearities and multi-point constraints are present in the emulated system. [ABSTRACT FROM AUTHOR]

Dividing the whole system into multiple subsystems and a separate dynamic analysis is common practice in the field of structural dynamics. The substructuring process improves the computational efficiency and enables an effective realization of the local optimization, modal updating and sensitivity analyses. This paper focuses on frequency-based substructuring methods using experimentally obtained data. An efficient substructuring process has already been demonstrated using numerically obtained frequency-response functions (FRFs). However, the experimental process suffers from several difficulties, among which, many of them are related to the rotational degrees of freedom. Thus, several attempts have been made to measure, expand or combine numerical correction methods in order to obtain a complete response model. The proposed methods have numerous limitations and are not yet generally applicable. Therefore, in this paper an alternative approach based on experimentally obtained data only, is proposed. The force-excited part of the FRF matrix is measured with piezoelectric translational and rotational direct accelerometers. The incomplete moment-excited part of the FRF matrix is expanded, based on the modal model. The proposed procedure is integrated in a Lagrange Multiplier Frequency Based Substructuring method and demonstrated on a simple beam structure, where the connection coordinates are mainly associated with the rotational degrees of freedom. [ABSTRACT FROM AUTHOR]

Model updating using measured system dynamic response has a wide range of applications in system response evaluation and control, health monitoring, or reliability and risk assessment. In this paper, we are interested in model updating of a linear dynamic system with non-classical damping based on incomplete modal data including modal frequencies, damping ratios and partial complex mode shapes of some of the dominant modes. In the proposed algorithm, the identification model is based on a linear structural model where the mass and stiffness matrix are represented as a linear sum of contribution of the corresponding mass and stiffness matrices from the individual prescribed substructures, and the damping matrix is represented as a sum of individual substructures in the case of viscous damping, in terms of mass and stiffness matrices in the case of Rayleigh damping or a combination of the former. To quantify the uncertainties and plausibility of the model parameters, a Bayesian approach is developed. A new Gibbs-sampling based algorithm is proposed that allows for an efficient update of the probability distribution of the model parameters. In addition to the model parameters, the probability distribution of complete mode shapes is also updated. Convergence issues and numerical issues arising in the case of high-dimensionality of the problem are addressed and solutions to tackle these problems are proposed. The effectiveness and efficiency of the proposed method are illustrated by numerical examples with complex modes. [ABSTRACT FROM AUTHOR]

Virtual sensing techniques use information available from a limited set of physical sensors together with the finite element model to calculate an estimate of the quantity of interest. In structural dynamics applications, analytical mode shapes from the finite element model are typically used as a basis to estimate the response at unmeasured locations by an expansion algorithm. An alternative is to model only the interesting part of the structure using substructuring techniques, in which the natural modes are replaced by component modes consisting of a selected number of fixed interface modes plus the interface constraint modes. They are mutually independent and compose a valid subspace for estimating the unmeasured response. If the number of interface degrees of freedom is large, interface reduction is applied. The main advantage of the proposed approach is that the modelling effort can be substantially decreased, because only part of the structure is modelled and the modelling uncertainties, non-linearities, or changes in the omitted structure can be ignored. The method is validated by numerical simulations of three different structures under unknown excitation. Different types and locations of virtual sensors are studied. Also, the effects of noise and model errors are investigated. The most accurate estimation is obtained if the virtual sensor is located away from the interface and close to a physical sensor. [ABSTRACT FROM AUTHOR]

Noise legislations and the increasing customer demands determine the Noise Vibration and Harshness (NVH) development of modern commercial vehicles. In order to meet the stringent legislative requirements for the vehicle noise emission, exact knowledge of all vehicle noise sources and their acoustic behavior is required. Transfer path analysis (TPA) is a fairly well established technique for estimating and ranking individual low-frequency noise or vibration contributions via the different transmission paths. Transmission paths from different sources to target points of interest and their contributions can be analyzed by applying TPA. This technique is applied on test measurements, which can only be available on prototypes, at the end of the designing process. In order to overcome the limits of TPA, a numerical transfer path analysis methodology based on the substructuring of a multibody system is proposed in this paper. Being based on numerical simulation, this methodology can be performed starting from the first steps of the designing process. The main target of the proposed methodology is to get information of noise sources contributions of a dynamic system considering the possibility to have multiple forces contemporary acting on the system. The contributions of these forces are investigated with particular focus on distribute or moving forces. In this paper, the mathematical basics of the proposed methodology and its advantages in comparison with TPA will be discussed. Then, a dynamic system is investigated with a combination of two methods. Being based on the dynamic substructuring (DS) of the investigated model, the methodology proposed requires the evaluation of the contact forces at interfaces, which are computed with a flexible multi-body dynamic (FMBD) simulation. Then, the structure-borne noise paths are computed with the wave based method (WBM). As an example application a 4-cylinder engine is investigated and the proposed methodology is applied on the engine block. The aim is to get accurate and clear relationships between excitations and responses of the simulated dynamic system, analyzing the noise and vibrational sources inside a car engine, showing the main advantages of a numerical methodology. [ABSTRACT FROM AUTHOR]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: The integration of a model for longitudinal hydroacoustic fluid damping in thin hydraulic pipes in 3D finite element models is presented in this paper. In order to perform quantitative prediction of the vibroacoustic behavior and resulting noise levels of such fluid–structure coupled system due to hydraulic excitation, an accurate frequency-dependent fluid damping model including friction effects near the pipe wall is required. This step is achieved by matching complex wave numbers from analytical derivation into a parameterized damped wave equation and consecutive translation into finite element modeling. Since the friction effect close to the pipe wall changes locally with the inner pipe radius, the fluid damping model is applied segment-wise in order to model the influence of cross-sectional discontinuity, such as orifices, on the oscillating pressure pulsations. A component synthesis approach, which uses pipe segments as substructures, allows a simple model generation and fast computation times. The numerical harmonic results are compared to experimental frequency response functions, which are performed on a hydraulic test bench driven by a dynamic pressure source in the kHz-range. [Copyright &y& Elsevier]

Abstract: Frequency based substructuring approaches have been used for the generation of system models from component data. While numerical models show successful results, there have been many difficulties with actual measurements in many instances. Previous work has identified some of these typical problems using simulated data to incorporate specific measurement difficulties commonly observed along with approaches to overcome some of these difficulties. This paper presents the results using actual measured data for a laboratory structure subjected to both analytical and experimental studies. Various commonly used approaches are shown to illustrate some of the difficulties with measured data. A new approach to better condition the measured functions and purge commonly found data measurement contaminants is utilized to provide dramatically improved results. Several cases are explored to show the difficulties commonly observed as well as the improved conditioning of the measured data to obtain acceptable results. [Copyright &y& Elsevier]

Abstract: Numerical conditioning of the subsystems’ interface flexibility matrix is an important issue in experimental frequency based substructuring (FBS) methods. As this matrix needs to be inverted, it is believed that ill numerical conditioning could severely magnify even small random errors in experimentally obtained subsystems, yielding erroneous FRFs of the coupled system. In this paper a method is introduced with which the uncertainty of the coupled system''s FRFs can be quantified based on the uncertainties of the subsystem FRFs. This uncertainty propagation method is based on the statistical moment method and uses the measured (time) data of the subsystems as input. The method will be applied to a numerical example, where it is shown that the uncertainty on substructure FRFs can significantly influence the accuracy of the coupled system. Furthermore, the application shows that the method introduced is computationally efficient and highlights some interesting phenomena that can be encountered in experimental dynamic substructuring. [Copyright &y& Elsevier]

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. • Contact stiffness measurements and guidelines for use in nonlinear dynamic analysis. • Comparison of FBS and hysteresis loops for contact stiffness estimations. • FBS can provide structure-dependent contact stiffness estimations. • Joint resonances strongly influence contact stiffness over a wide frequency range. • The shape of stuck hysteresis loops can indicate (unknown) structural resonances. [ABSTRACT FROM AUTHOR]

• A covariance-based approach is presented for estimating the complex FRF random uncertainty in impact testing. • The measurement uncertainty is propagated through common experimental substructuring techniques. • A study is conducted on the assumptions underlying the proper application of the propagation formula. • The practical implementation of the methodology is illustrated with an experimental case study. A proper acquisition of FRFs is a prerequisite for a successful implementation of experimental substructuring techniques in the frequency domain. In this context, the study of uncertainty associated with measurements is of particular interest due to the precision standards required by industrial practice. This paper aims to provide a practical and reliable methodology for the quantification and propagation of the random measurement uncertainty in Frequency Based Substructuring applications. Extending previous studies, the framework presents a covariance-based approach for quantifying the complex-valued random uncertainty on measured FRFs and analytical methods for propagating it through interface modeling and substructures coupling approaches. The assumptions underlying the correct application of the method are investigated. An optimal number of impacts for an appropriate Gaussianity of the FRF distribution is computed based on empirical data. Experimental testing reveals encouraging results in the validation of the small error approximation. Considerable correlation effects between FRFs are found, although their impact on the coupled FRFs uncertainty seems to be limited. The methodology is applied to an experimental example. [ABSTRACT FROM AUTHOR]