Coupling of structures using frequency response functions

. In the ﬁeld of structural dynamics is common to predict the behaviour of a structure regarding structural modiﬁcations. In this context, the frequency based substructuring method is well-known to perform structural modiﬁcations based on the coupling of structures. This process gives the possibility to perform the study of a structure at the level of its components and then assess the response of the coupled system. In practice, it is impossible to attain an experimental complete response model, although one can simulate all the responses of a structure using numerical models. Hence, the substructuring process can be enhanced by the combined use of experimental and numerical responses, as it was demonstrated using numerically obtained frequency response functions. This work presents the enhancement of the frequency based substructuring method using a method to expand experimental frequency response functions over the entire set of degrees of freedom in a ﬁnite element model. This expansion process, known as modiﬁed Kidder’s method, considers that if one can only measure translations due to exciting force, it is possible to obtain the complete response model, including the rotational frequency response functions due to exciting moments. The combined use of the frequency based substructuring and the modiﬁed Kidder’s methods has several advantages, as it avoids modal identiﬁcation or residual compensation. To evaluate the performance of the proposed procedure a numerical example of a beam structure is presented, and its results are discussed.


Introduction
Using structural modifications to predict the behaviour of a structure is a known practice in the field of structural dynamics. In this context, the frequency based substructuring (FBS) method [1] is a well-known method to perform structural modifications based on the coupling of substructures. This process gives the possibility to perform the study of a structure at the level of its components and then assess the response of the coupled system. To obtain a quality result in the coupling process, one should use a complete response model, which is experimentally unpractical or even impossible. However, one can simulate all the responses of a structure using numerical models. Usually, one can only measure few translation degrees of freedom (DoF) due to exciting forces. Hence, the rotational DoF are commonly disregarded in testing. On the other hand, if one considers the use of their numerical counter-parts, the https://doi.org/10.1051/matecconf/201821106005 VETOMAC XIV substructuring process can be enhanced, as it was demonstrated using numerically obtained frequency response functions (FRF). An approach to tackle this problem was proposed by Drozg et al. [2]. This paper presents the enhancement of the FBS method using the modified Kidder's method [3] to expand a set of experimental FRFs over the entire set of DoFs in a finite element model. This expansion process may consider that one can only acquire responses at translational DoFs due to exciting force, and it enables to attain the complete response model, including the rotational FRFs due to exciting moments. The combined use of the FBS method and the modified Kidder's one has several advantages, as it avoids modal identification or residual compensation. To evaluate the performance of the proposed procedure a numerical example is presented, and its results are discussed. A beam structure is divided in two substructures and then used to simulate the pseudo-experimental FRFs of the complete structure and of its substructures. Note that only translational FRFs due to forces are considered to be available.

Frequency based substructuring
Considering a structure C, composed by two substructures A and B and considering the sets of DoFs given in Figure 1, it is possible write the complete receptance matrix of the structure C as, The frequency dependence will be omitted for sake of simplicity. Note that DoFs i are the ones in the domain of A and k in B. The DoFs j denote DoFs shared by A and B in their interface. Hence, one has the receptance matrices of the substructures A and B given by, regarding that, in the interface DoFs j, both the force equilibrium and the compatibility of displacements must be fulfilled, one has, Therefore, eq. (1) can be recast, accordingly to the developed in [1], as, From eq. (4), it can be seen that the receptance matrix of the complete structure C can be obtained from the knowledge of the receptance matrices of both substructures A and B. However, in practice, it is almost impossible to acquire data that allows to built those matrices completely, mainly due to difficulties with the excitation with pure moments or the ability to measure some DoFs, namely the rotational ones. Hence, one should consider an expansion technique suitable to expand a set of experimental FRFs over the receptance matrix elements that were not acquired. The use of the modified Kidder's method [3] is here discussed.

Modified Kidder's method
A modified version of the Kidder's method, applied in the frequency domain, is here considered to provide an expansion technique suitable to use experimental FRFs as they are collected.
The modified Kidder's method is derived from the partitioned dynamic equilibrium equation written as, where the primary p and secondary s coordinates correspond to the measured and unmeasured DoFs, respectively, and I q is a boolean vector mapping a given force location q. From eq. (5) it can be shown that a FRF vector expanded for the unmeasured DoFs is given by, The expansion process is here achieved spectral line by spectral line for a given experimental frequency range, for each acquired FRF vector H p j (ω). In order to attain a complete FRF matrix from a set of measured FRFs, a two steps expansion strategy must be addressed. As given by Avitabile and O'Callahan [4] and adapted with the modified Kidder's method by Silva and Maia [3]. The expansion strategy relies on a careful definition of all the system DoFs, considering the specificities related to the response and excitation coordinates in the response model. Note that in a spatial model one has only primary and secondary coordinates, while in a response model the primary and secondary response coordinates have their dual in the excitation coordinate set. Therefore, consider a complete receptance matrix H partitioned wrt translational t and rotational θ response DoFs and DoFs where force f and moment τ excitations can be applied as, where t ≡ f and θ ≡ τ. Usually in practice, one can only measure a subset of t and excite part of f , and often these subsets are not equal. Note that the possibility of measuring rotational DoFs or applying pure moments is here completely rejected, although they can be included. Hence, one must recast eq. (7) as, where the subsets of t and f type DoFs present in the FRF matrix can be organized by their nature, using the subscripts: A for measured/forced DoFs, B for measured/unforced DoFs, tests. Hence, the results in Figure 4 were obtained by the FBS, considering that only few translational due to exciting force FRFs are available from both substructures (A and B). So, the data used as acquired data is limited to the following sets of DoFs for the expansion process: Note that all the FRFs in Figure 4 correspond to non-acquired FRFs, they were computed by the FBS method using the modified Kidder's method to expand the available data set. From Figure 4, one can observe a quite good match between FRFs, especially in the frequency range bellow 600 Hz. The deviation at higher frequencies is acceptable if compared to reference works, although it can be attenuated if more FRFs are known.

Conclusion
This paper presents the result of the combined implementation of the FBS method for coupling and the modified Kidder's method for the expansion of measured translational FRFs due to exciting forces. The expansion process is needed as the FBS method requires the complete receptance matrices of the substructures to be coupled and due to the fact that usually only translational due to exciting forces FRFs are experimentally acquired. A numerical example is used to demonstrate the capability of the proposed implementation and the obtained results show a high resemblance between the FRFs obtained with the FE method for the complete structure and the ones obtained with the coupling method, with incomplete receptance matrices. The experimental application of this approach has given also very good results, regarding the FRFs of the coupled structure. However, this case is out of the scope of this paper.