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.
Commercial off-the-shelf rubber isolators often come with no additional information other than the static stiffness in three translational directions. Hydraulic testing machines can be used to obtain ...frequency dependent dynamic stiffnesses of rubber isolators in translational degrees of freedom (DoF). Alternatively, dynamic substructuring based methods can be used, which can additionally identify the dynamic stiffness in rotational DoF while requiring only standard vibration testing equipment. Results of two substructuring methods will be compared to those from a hydraulic machine. Both of the presented methods use locally rigid fixtures, mounted to the bottom and top of the isolators. Frequency based substructuring (FBS) requires knowing the fixtures dynamics to decouple them. Inverse substructuring, also called in-situ decoupling, does not require knowing the fixtures dynamics, but is assuming negligible mass and a special stiffness matrix topology of the rubber isolator. Both methods produce accurate results for translational DoF up to the kilo Hertz range, which is confirmed by comparison to measurements on the hydraulic machine. However, FBS does not rely on specific assumptions about the isolator, like inverse substructuring. The limits of inverse substructuring's underlying assumptions are shown theoretically and in the measurements presented here. We propose two extensions to compensate for the assumptions and present their results. Nevertheless, the rubber model obtained with the FBS decoupling can provide better results when used in an assembly. This is illustrated by testing the experimental rubber element models, obtained with either method, in a substructuring prediction of coupled frequency response functions (FRFs) and comparing that to reference measurements.
A reliable experimental application of frequency based substructuring requires very accurate acquisition of frequency response functions (FRFs). Even a relatively small error introduced during the ...measurement can result in erroneous substructuring results. The measurement errors can be either random or systematic, with the latter often referred to as bias. Impact excitation is popular in dynamic substructuring due to the rapid FRF calculation for each separate location. However, deviations in the location of the excitation affect the FRFs across the whole frequency range. This paper proposes a novel methodology to characterize the bias errors in frequency based substructuring using the small deviations in impact excitation from typical experimental measurements. The small deviations are utilized to reconstruct a range of FRFs, which are directly used in the global sensitivity analysis. The sensitivity analysis is utilized to characterize how each impact location affects an arbitrary quality indicator, such as reciprocity or passivity. Therefore, the effect of the bias can be evaluated directly from a single series of measurements, without the need for a numerical model. The proposed approach is first shown on a synthetic numerical example, where the advantages and limitations are outlined. Finally, an application involving experimental frequency based substructuring on a beam-like structure is depicted.
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•Method to characterize bias error in frequency based substructuring directly from measurement.•Approach is suited for characterization of small deviations in impact excitation location.•Global sensitivity analysis is used to quantify bias error influence at each impact.•Sample set is provided by deducting functional dependency of FRF on relative location offset.•High sensitivity impacts are omitted from substructuring applications, improving results consistency.
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•Virtual point transformation is expanded to incorporate directly measured rotational response.•A novel formulation of a weighting matrix is proposed for including directly measured ...rotations.•Global sensitivity analysis is used to evaluate the expanded virtual point transformation.•Expanded transformation is less sensitive to deviations in sensor or impact location.•Practical applicability of the proposed expansion is shown on a simple experiment.
Dynamic substructuring methods serve as a powerful tool in the analysis of modern complex systems. The coupling of substructures has been successful with analytically obtained results. However, substructuring with experimentally obtained data remains challenging. One of the main problems associated with experimental substructuring is the coupling of the rotational degrees of freedom (RDoFs). A promising method where RDoFs are included implicitly is the virtual point transformation. Even though the transformation has been successfully used in the substructuring process, it is still highly susceptible to inaccuracies in the sensor sensitivity and positioning. In this paper an expansion to the virtual point transformation is proposed, which enables the projection of a directly measured rotation response on the interface deformation modes. A novel formulation of the weighting matrix is introduced to consistently include the measured rotations in the transformation. The proposed expansion is demonstrated on a numerical model of a simple beam-like structure and compared with the standard transformation. The effects of inaccuracies in the sensor sensitivity and placement on the overall quality of both transformation are analysed with a global sensitivity analysis. Finally, an experimental validation of the proposed expansion is carried out on a steel beam.
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•A piezoelectric rotational accelerometer is used for measuring rotations.•An expansion of the response model is based on the modal parameters.•A complete response model is used in ...the substructuring process.•The applicability of using a complete response model is shown in a case study.
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.
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.
► Presents a framework for decoupling methods based on dual assembly concept. ► Allows to derive new methods and classify existing ones. ► Different decoupling methods are applied to both an analytical and experimental decoupling problem.
•First description and application combining substructuring and blocked forces for acoustic design optimization.•First substructuring application of an experimental method for obtaining 12 DoF rubber ...isolator models.•Guidelines on how to properly formulate the cost function for finding robust design optima.•Auralization of predicted and experimentally validated sound.
The combination of frequency based substructuring (FBS) and blocked force Transfer Path Analysis (TPA) allows to perform parametric NVH design optimizations. Blocked forces are not dependent on one specific receiver structure, in contrast to interface forces of classical TPA. Blocked forces can therefore be used as a source description in design optimization. For optimizing the assembly, different substructures are virtually coupled to each other, where each substructure is described by the most appropriate modeling approach. Frequency based substructuring (FBS) allows coupling analytical, numerical or experimental models to each other. The transfer functions of the final assembly can thus be simulated by FBS. Numerical models are used for substructures which can be simulated with high accuracy. These are parametrized for optimization. Experimental substructure models are used for substructures that are hard to simulate accurately.
The application example is an electric climate compressor. Its excitation is characterized by means of blocked forces. The assembly consists of: a) a FEM model of the receiver, b) experimental models of different rubber isolators, c) a parametrized FEM model for the compressor support, and d) an analytical rigid body model for the compressor itself. The rubber isolator choice and the FEM model of the support, are iteratively optimized for minimal structure borne noise. Virtually coupling the substructures, and applying the compressors blocked forces to the assembly, makes it possible to simulate the resulting loudness for different design parameters. We discuss the formulation of an objective function and the applicability of different optimization algorithms on a minimal example first. Then we apply a genetic optimization algorithm to the objective function for the compressor design. The simulated predictions for the optimal parameters are validated with measurements on the physically built up design, including auralization of the results.
Frequency-based substructuring is a very popular approach to predict the vibroacoustic behaviour of built-up mechanical systems. Even if extensively used since a long time for sundry purposes, yet ...this method may encounter difficulties when attempting to include test-based substructure models. The reason is that the characteristics of the experimentally obtained subsystems are corrupted by unavoidable measurement inaccuracies. Such inaccuracies often drastically affect the prediction of the combined system behaviour, especially when the problem is badly conditioned. Here we will focus on two types of measurement induced inaccuracies: reciprocity and passivity violations. A new approach that enforces these two physical properties upon a test-based frequency response function (FRF) model is proposed. A well-behaving system is created while affecting the original corrupted FRF matrix as little as possible. Besides guaranteeing a physically consistent behaviour, the approach considerably reduces the impact of noise for coupling applications and hence decreases the quality requirements for experimental data.
In this work the method is presented and applied to a poorly-conditioned hybrid substructuring approach: a test-based plate model coupled through its rotational degrees of freedom with an analytical model of a damping layer. This is a particularly critical test case as the noise and physical anomalies already present in the experimentally obtained plate data turn out to be further amplified by the finite difference procedure needed to estimate its rotational degrees of freedom. It is shown that the removal of the physical inconsistencies from the plate data using the novel procedure, indeed yields more stable and reliable results after coupling.
•Small violations of physical properties within measured FRFs can generate large errors in the assembled system behaviour.•A new methodology to enforce passivity and reciprocity upon a test-based FRF model is presented.•A convex optimisation problem is adopted.•The methodology has been successfully applied to a hybrid substructuring problem containing experimental and numerical subsystems.•Noise appearing during substructuring is reduced and data quality requirements can be relaxed.
The dynamic substructuring design describes the dynamic responses in the modal, frequency and physical domains using coupling methods at joint nodes. This paper presents dynamic structuring methods ...to modify existing substructuring methods by merging the concept of interfacial forces in the satisfaction of compatibility conditions at joint nodes. The algorithms correspond to a type of model reduction approaches that use a few modes of substructures with the assumption of pseudo-masses at the joint nodes of the substructures. It was verified through numerical examples that the pseudo-masses rarely affected the dynamic responses of the synthesized structure. A frequency response function-based substructuring method was derived by applying interface forces at joint nodes to frequency response function (FRF). The proposed dynamic substructuring methods are expressed in explicit forms for synthesizing substructures without any numerical schemes. The validity of the proposed method was illustrated using two numerical examples. The limitations of this study are evaluated using numerical examples.
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