In experimental dynamic substructuring, coupling of substructures sharing a line- or surface-like interface proves to be a challenge due to the difficulties in interface modelling. Modelling a high ...number of degrees of freedom at the common interface can be too stringent when imposing compatibility and equilibrium conditions, thereby causing redundancy and ill-conditioning. To mitigate the effects of overdetermination and experimental errors, that can lead to a high error amplification, several techniques have been developed, proposing different reduction spaces to weaken the interface conditions. This work investigates reduction space definitions in dynamic substructuring for coupling continuous interfaces. In particular, a comparative investigation of three established techniques, namely the frequency-based modal constraints for fixture and subsystem, singular vector transformation, and virtual point transformation, is conducted within the frequency domain. The feasibility of all approaches is supported by an experimental case study, which can guide practitioners in selecting a suitable approach for their specific needs.
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•Dynamic coupling of line- or surface-like interfaces is investigated.•Various reduction bases are compared based on the experimental case study.•Guidelines are provided that guide practitioners in selecting a suitable reduction basis.
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.
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.
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•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.
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.
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•Improved experimental estimation of full-field FRFs from noisy high-speed camera data.•Hybrid model is formed by mixing two equivalent models using a substructuring ...approach.•Full-field data from a high-speed camera is mixed with accurate accelerometer data to create a single model.•Experiment demonstrates increased consistency of the hybrid model over the whole frequency range.
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.
•Polynomial and interpolation models allow for interface displacement reconstruction.•Complex polynomial models outperform the rigid model using the same indicators.•The proposed approaches can be ...combined for an extended capability.•The method is numerically validated in the context of substructuring.
Frequency response functions (FRF) are valuable structure characterizations that are used for the study of noise and vibration, load identification, modal analysis or model updating. They are usually estimated as mobilities by exciting the structure and measuring the normalized response for each frequency. However, rotational degrees of freedom (DOF) are typically difficult or impossible to measure directly. Previous studies have shown that rotational DOFs can be critical for the successful application of different techniques, such as dynamic substructuring. Consequently, different approaches have been proposed to take them into account, such as the finite differences method. Frequency Based Substructuring (FBS) is a dynamic substructuring technique that allows for the coupling of the frequency response functions of separate subsystems in order to synthesize the coupled system's FRFs without the physical assembly. This technique can be sensitive to inconsistencies between the substructures or incomplete information in the interface characterization. The inclusion of rotational degrees of freedom in the FRFs of the interfaces for the application of substructuring has been extensively discussed in literature, but there are still some limitations in frequency due to the assumptions that are required, especially the rigidity assumption of the interfaces. The goal of this paper is to present an experimental approach to relax the rigidity assumption and increase the complexity of the force and displacement field characterization of interfaces. This would extend the value of the estimated FRFs to higher frequency ranges, in which the local deformations start to have an important effect. To particularize the general approach, different possible models are proposed and discussed, highlighting their benefits and disadvantages. Some of these models are validated numerically and experimentally. Finally, the performance of the approach for substructuring applications is also analyzed on a numerical model.
•The paper introduces a novel approach to experimental substructure decoupling: Singular Vector Transformation.•The theory is thoroughly explained and the mathematical details are elaborated.•A ...physical interpretation and understanding is given with an open comparison with other techniques.•Numerical and experimental examples are provided.
Substructure decoupling is the process of identifying the dynamic behavior of one component by removing the dynamic influence of the second component from the assembled system. In experimental practice, several techniques have been developed to address the decoupling problem. In this context, measurements errors of random and systematic nature remain a major hindrance to a successful implementation of the methodology. For this reason, approaches such as extended interface, Virtual Point Transformation and truncated Singular Value Decomposition are commonly adopted on top of a standard interface decoupling procedure. This paper introduces the Singular Vector Transformation. The idea is to weaken the interface problem by using the Singular Value Decomposition to extract reduction spaces directly from the measured dynamics. A least square smoothing minimizes random errors and outliers, thereby improving the conditioning of the interface matrix inversion. No geometrical or analytical model is required. The reduction basis are frequency-dependent and can include flexible interface behavior, if properly controlled and observed. Further understanding and interpretation of the interface problem in frequency-based decoupling is provided. Numerical and experimental examples show the potential of the proposed technique in comparison with state-of-the-art approaches.
•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.
System equivalent model mixing Klaassen, Steven W.B.; van der Seijs, Maarten V.; de Klerk, Dennis
Mechanical systems and signal processing,
05/2018, Volume:
105
Journal Article
Peer reviewed
Open access
•The paper introduced a new method called System Equivalent Model Mixing.•The theory is thoroughly explained and several extensions are provided.•A physical interpretation and comparison to other ...techniques is provided.•A practical test-case validates the method.
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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