Frequency Based Substructuring Method Research Articles

Expansion Technique of Frequency Response Function Data and its Applications

This paper presents frequency response function (FRF) expansion techniques that minimize the difference between the analytical and predicted FRF matrices to satisfy the FRF constraints. The measured FRF relationships at a small number of locations were used as constraints. The expansion method is useful for estimating the rotational FRFs that are difficult to measure or apply using frequency-based substructuring (FBS) techniques. The validity of the proposed method, including the effects of external noise, was confirmed using numerical examples. An FBS algorithm was also derived by incorporating the FRFs of each substructure, and the compatibility conditions were transformed into FRFs with pseudomasses at the joint nodes. A discrepancy between the FRFs of the synthesized system estimated using the proposed method and the analysis results of the entire system was observed in the numerical example, and the reasons for the discrepancy were investigated and discussed. It is estimated that the proposed FBS method can be enhanced by combining it with other dynamic substructuring techniques and supplementing additional information, such as measured FRFs.

Reformulation of Frequency Based Substructuring Method Considering Elastic Joints According to Sherman-Morrison-Woodbury Formula

Reformulation of Frequency Based Substructuring Method Considering Elastic Joints According to Sherman-Morrison-Woodbury Formula

Estimating Rotational Frequency Response Function Using Mode Expansion and Frequency Response Function Synthesis Method

The rotational frequency response function (RFRF) plays a crucial role in increasing the accuracy of the calculated results of the frequency-based substructuring method. However, RFRFs are often omitted due to the difficulties in the measurement process and limitations of the equipment. This paper presents a scheme of estimating the rotational FRF of an irregular plate structure using the FE model reduction and expansion method. The reduced FE model was introduced using the improved reduction system (IRS) and expanded to the experimental modal model (EMA model) using the system reduction and the expansion (SEREP) method. The FRF expanded method was then employed to derive the translational and rotational FRFs from the expanded EMA model. The accuracy of the expanded FRFs was evaluated with the EMA model of the irregular plate. It was found that the translational and rotational FRFs estimated from the proposed scheme were in good agreement with the EMA counterparts. Furthermore, the patterns of the estimated RFRFs were well correlated with the EMA RFRFs. This work shows that the proposed scheme may offer an attractive alternative way of accurately determining the RFRs of complex structures or structural components.

Accuracy and Efficiency of Updated FRF Coupling for the Dynamic Behaviour Investigation of an Assembled Structure

The frequency based substructuring (FBS) method allows the combination between the experimental and analytical frequency response function (FRF). The combined FRF is then used for calculation of the dynamic behaviour of an assembled structure which usually consists of a large number structural components or substructures. However, the accuracy of the dynamic behaviour calculated using the FBS method relies heavily on the quality of experimental FRF data of the interfaces on which, in practice, it is very problematic to be attained. Furthermore, in most cases, some parts of rotational FRF data are not included in the FBS method due to the complex measurement process. The exclusion of the rotational FRF data will usually lead to numerous errors in the combined FRF data. Therefore, this paper proposes a new frequency response function (FRF) coupling of the FBS method that may uniquely address the difficulties and improve the quality of predicted results of the FBS method. The proposed coupling was formulated based on the finite element method, model updating method and FRF synthesize method. The finite element model of the physical test substructure was developed and reconciled based on the mode shapes obtained from experimental modal analysis. The FRF synthesize method was performed on the updated finite element model to obtain a complete matrix with a full degree of freedom FRF data containing all the translational and rotational FRF data required. It was found that the proposed coupling has allowed generating full translational and rotational FRF data. The generation has improved significantly the FRF coupling process and the quality of the predicted dynamic behaviour of the FBS method.

Alternative scheme for frequency response function measurement of experimental-analytical dynamic substructuring

The accuracy of the predicted dynamic behaviour of an assembled structure using the frequency based substructuring (FBS) method is often found to be diverged from the experimental counterparts. The divergence which has become the paramount concern and major issue for structural dynamicists is because of the unreliable experimental FRF data of the interfaces of substructures, arising from the limited resources of appropriate excitation points and accelerometer attachments in the vicinity of the interfaces. This paper presents an alternative scheme for FRF measurement of the experimental FRF data of substructures. In this study, an assembled structure consisting of two substructures were used, namely substructure A (Finite element model) and substructure B (Experimental model). The FE model of substructure A was constructed by using 3D elements and the FRFs were derived via the FRF synthesis method. Specially customised bolts were used to allow the attachment of accelerometers and excitation to be made at the interfaces of substructure B, and the FRFs were measured by using impact testing. Both substructures A and B were then coupled by using the FBS method and the coupled FRF was validated with the measured FRF counterparts. This work revealed that the proposed scheme with specially customized bolts has led to a significant enhancement and improvement in the FBS predicted results.

The Prediction of the Dynamic Behaviour of a Structure Using Model Updating and Frequency Based Substructuring

Combining analytical with experimental data to predict the dynamic behaviour of an assembled structure via the frequency based substructuring (FBS) method, is a common practice in the field of structural dynamics. However, the accuracy of the dynamic behaviour predicted from the FBS method relies heavily on the quality of experimental frequency response function (FRF) of the interfaces on which, in practice, it is very difficult to obtain. In addition, the accuracy of the FBS method is highly dependent on experimental rotational degrees of freedom (DOF) which are always found to be very difficult to measure accurately. Therefore, this paper proposes a new frequency response function (FRF) coupling scheme that may uniquely address the difficulties and improve the quality of predicted results of the FBS method. The scheme is formulated based on the finite element method, model updating technique and experimental modal analysis. A simplified finite element model of a physical test substructure is developed to generate rotational FRF data by reconciling the initial FE model using the model updating technique. It was found that the scheme adopted allows generating full translational and rotational degrees of freedom data and leads to a significant improvement in the FRF coupling process between the analytical and experimental model in the FBS method.

Frequency Based Substructuring for Structure with Double Bolted Joints: A Case Study

Adopting dynamic substructuring schemes is a common practice in the field of structural dynamics, where the dynamic behaviour of a structure is predicted by combining the multiple subsystems that are analysed individually. This paper investigated the dynamic behaviour of a structure with doubled bolted joints using the frequency based substructuring (FBS) method. In the attempt, the substructures were combined with the frequency domain that can be numerically derived or experimentally measured. However, the applicability of this method suffered from several issues where most of them were related to the frequency response function (FRF) of the rotational degree of freedom (DOF) at the subsystem’s interface. In some cases, the system’s interface vibrated in the rotary motion of certain modes, for instance, a car side rear view mirror. Therefore, excluding the rotational FRF during the coupling process could lead to a completely different result. This paper presents the use of the FBS method for a structure with double bolted joints by using the equivalent multipoint constraint (EMPC) method through which rotational DOFs can be completely neglected. The actual tested structure for this study was an assembled structure consisting of two substructures: a simple beam and an irregular plate steel structure. The FRFs of both substructures were derived from using the FRF synthesis from finite element models and combined together via the FBS method. This study reveals that the use of the FBS method with the EMPC method for a structure double bolted joints has led to very promising results.

Full-degrees-of-freedom frequency based substructuring

Abstract 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.

An Impulse Based Substructuring method for coupling impulse response functions and finite element models

In the field of Dynamic Substructuring (DS), large and complex structures are divided into several smaller and simpler components. The linear substructures are subsequently described in their dominant dynamics and reassembled, allowing one to compute the coupled dynamic behavior. DS methods are often classified into two distinct families, the Frequency Based Substructuring (FBS) methods and Component Mode Synthesis (CMS) techniques. In the former substructures are assembled whose dynamics are described in terms of frequency response functions (FRFs) and the latter are used to reduce and assemble the substructure finite element (FE) models. Lately a new substructuring method has been proposed, one that does not fit the framework of the FBS and CMS methods. The method, named Impulse Based Substructuring (IBS), was first used to obtain the coupled response of a system by assembling its component impulse response functions (IRFs). In this paper the IBS method is extended, thereby allowing one to determine the coupled behavior of structures that are composed of both substructure FE models and substructure IRFs. The method can be regarded as an extension to the normal time integration methods used for obtaining the time responses of FE models. As the linear substructures (described in their IRFs) are fully condensed on the interface of the FE model, one can significantly reduce the computational cost required for time integrating otherwise large FE models. However, as the linear (IRF) domains are exactly accounted for, the IBS method can be seen as a dynamic condensation on the interface, but not as a reduction method in the classical sense. Nonetheless, one can regard IRFs as a sort of “superelements in time” and the IBS method can therefore serve as an attractive alternative to CMS methods in case these are not available in the applied FE modeling programs, or if a high spectral bandwidth of the substructure is required. The method proposed in this work is based on the generalized-α time integration scheme and it is analytically proven that it can be applied in such a way that the simulation results are identical to the responses obtained from a monolithic integration of the full system, thereby guaranteeing its stability and accuracy. The method is demonstrated using a numerical test case, where a wind turbine FE model is coupled with a the IRFs of a marine foundation.