Abstract
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
A frequency response function (FRF) describes the relationship between the excitation input and response output of a dynamic system as a function of frequency. The FRF provides information on how a structure responds to harmonic excitations across different frequencies. For translational and rotational degrees of freedom (DOFs), the FRFs explain how translational and rotational displacements respond to a harmonic force input at various frequencies. FRF data expansion techniques consider various aspects, such as applications in structural health monitoring and model updating.
Data on moment excitations and the corresponding rotation angles cannot be collected simply by measuring them through experiments. The rotational FRFs (RFRFs), which represent the response of the system to rotational inputs at different frequencies, are predicted using FRF data expansion techniques.
Several techniques for FRF data expansion including rotational DOFs have been proposed [1]-[4]. Avitabile and O’ Callahan [5] introduced data expansion processes for impedance-based system models, including rotational DOFs. Mirza et al. [6] proposed a modified form of a frequency-based substructuring (FBS) technique by estimating the unmeasured FRFs, including rotational DOFs. Gibbons et al. [7] described the rotational dynamic behavior of a structure using a finite element algorithm, and introduced a generalized analytical error analysis to balance the numerical errors. Silva and Pereira [8] presented a technique for expanding translational FRFs into rotational FRFs using Kidder’s method. Ozguven [9] estimated the FRFs of modified structures using measured FRFs.
FRFs can be applied to data expansion or FBS techniques by using some measured FRF data as constraints. Substructures can be synthesized using numerical and experimental FRF data [10]-[13]. Dynamic substructuring was performed based on the FRF data collected through numerical experiments. Asma and Bouazzouni [14] presented an updating method based on the measured FRFs by minimizing the difference between the measured and analytical frequency responses. Kidder [15] presented an appropriate back-transformation relationship to complement the Guyan reduction method. Maia and Silva [16] proposed a process for expanding a set of translational FRFs to estimate the entire receptance matrix by using Kidder’s method and the principle of reciprocity. Klerk et al. [17] utilized Lagrange multipliers to define the interface forces between substructures, and developed the Lagrange multiplier FBS method to assemble substructures expressed by dynamic admittance. Carne and Dohrmann [18] presented an admittance modeling process to minimize the measurement noise contained in the FRFs of a combined system.
Existing FRF data expansion techniques are limited to constraints based on displacement responses, and depend mostly on numerical schemes, such as the Lagrange multiplier method. This study develops an explicit FRF expansion method by minimizing the difference between analytical and estimated FRF matrices with FRF constraints. The feasibility of the proposed method was verified using numerical examples. The proposed FRF expansion method accurately predicted the unmeasured FRFs, including the RFRFs. An FBS algorithm was developed by incorporating the FRFs of each substructure, and the compatibility conditions were transformed into the FRFs. This algorithm introduced a pseudomass at the joint nodes to disassemble an entire structure and synthesize the substructures. The limitations and solutions of the proposed FBS technique are discussed, along with the results of numerical experiments.
Formulation
The FRF is a type of black box used to predict the physical information and extract the modal characteristics of dynamic systems. In contrast to updating physical parameter matrices, FRF expansion techniques utilize the measured FRF data and the correlation of FRF responses at some locations as constraints.
It is impractical to measure the FRFs for all the DOFs. It is difficult to acquire measurement data at locations where excitation or response measurements are impossible in limited environments. Thus, the FRFs measured for a small number of DOFs should be expanded by incorporating an expansion technique. The cost function for the FRF expansion is established as
where \({\mathbf{H}}_{a}\) and \({\mathbf{H}}_{c}\) are the n × n analytical and predicted FRF matrices, respectively. The FRF is a mathematical form that shows how a system responds to sinusoidal inputs at different frequencies and all nodes in the system model.
This expansion technique estimates the FRFs for a full set of DOFs using the constraints of the FRF data. Two types of constraints expressed by the measured FRFs were considered. They are expressed by the FRFs measured at several nodes owing to excitations at all DOFs and the FRFs measured at several nodes owing to excitations at several nodes.
FRF Constraints Measured at Several Nodes Owing to Excitations at all DOFs
The FRFs measured at several nodes owing to excitations at all DOFs can be expressed as
where\(\mathbf{A}\) is the m × n coefficient matrix used to define the measurement DOFs, and \(\mathbf{B}\) indicates the measured \(m\times n\) FRF matrix owing to excitations at all nodes. Equation (2) represents the effects of excitations at all the DOFs.
where \({\mathbf{U}}_{\varvec{c}}\) and \(\mathbf{F}\) represent the \(n\times 1\) displacement and unit force vectors, respectively, in the frequency domain.
The solution of Eq. (2) with respect to the predicted FRF matrix cannot be explicitly obtained. By utilizing the generalized inverse matrix and minimizing the cost function in Eq. (1) to satisfy the constraints in Eq. (2), the predicted and expanded FRF matrices are derived. Equation (2) was modified to apply to Eq. (1) as
Substituting \(\mathbf{R}=\mathbf{A}{\mathbf{H}}_{a}^{1/2}\) in Eq. (4) and solving for \({{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\), the following equation is obtained:
where \(\mathbf{Z}\) denotes the n × n arbitrary matrix, and + denotes the generalized inverse matrix. Minimizing the right-hand side of Eq. (5) in accordance with the cost function yields
Solving Eq. (6) for Z using the generalized inverse matrix with \({\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]}^{+}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\), \({\mathbf{R}}^{+}\mathbf{R}{\mathbf{R}}^{+}={\mathbf{R}}^{+}\), and \({\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]}^{+}\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\), the following equation is obtained:
where \(\mathbf{Y}\) represents another n × n arbitrary matrix. Substituting Eq. (7) into Eq. (5) yields an expanded FRF matrix for all the DOFs.
Equation (8) represents the expanded FRF matrix when the constraints of the same form as Eq. (2) are provided. In the following section, an expanded FRF technique is derived using the measured FRFs at small DOFs, rather than displacements, as constraints.
FRF Constraints Measured at Several Nodes
In cases where measurement cannot be easily performed, such as for RFRFs, estimation is performed using data expansion. This section considers the FRF constraints measured at some DOFs applicable to this case. The FRFs are expanded using the same cost function as in Eq. (1) and the FRFs measured at specific nodes as constraints. The measured FRFs can be expressed as
where \(\mathbf{L}\) is the m × n Boolean matrix used to define the FRF measurement DOFs, and D denotes the m × m FRF matrix measured at the corresponding DOFs. The constraint equation in Eq. (9) is modified as follows:
Using \(\mathbf{R}=\mathbf{L}{\mathbf{H}}_{a}^{1/2}\) in Eq. (10) and solving with respect to \({{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{L}}^{\varvec{T}}\) yields
where Z is an arbitrary matrix. By utilizing the condition to minimize Eq. (1) into Eq. (11) and solving with respect to an arbitrary matrix, the following equation is obtained:
where \(\mathbf{Y}\) denotes an arbitrary matrix. Subsequently, substituting Eq. (12) into Eq. (11) yields
Again, solving Eq. (13) with respect to \({{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}\) with \(\mathbf{Q}={{\mathbf{H}}_{a}^{1/2}\mathbf{L}}^{\varvec{T}}\), the following equation is obtained:
where \(\text{X}\) denotes an arbitrary matrix. By applying the condition to minimize Eq. (1) to Eq. (14), the arbitrary matrix \(\text{X}\) can be expressed as
Substituting Eq. (15) into Eq. (14), the expanded FRF matrix can be expressed as
Equation (16) expresses the FRF matrix expanded to a full set of DOFs using the FRFs measured at some nodes.
FBS Using FRF Compatibility Conditions at Joint Nodes
The substructures were synthesized according to the deformation compatibility conditions at the joint nodes. The displacements at the joint nodes were transformed into FRFs, which were expanded to a full set of DOFs. In the FBS process, pseudomasses that are linearly divided and synthesized at joint nodes are utilized for disassembly and synthesis, respectively.
The substructures were synthesized using compatibility conditions in the frequency domain. The displacements U at the joint nodes of adjacent independent substructures must be identical, as follows:
where subscripts 1 and 2 denote the 1st and 2nd substructures, respectively, and b represents the boundary DOFs. By utilizing the FRF matrix and the force vector in Eq. (17), the following equation is obtained:
where \({\mathbf{H}}_{ib}\)\(\left(i=1, 2\right)\) represent the FRFs at the boundary nodes of the i-th substructure, and \({\mathbf{F}}_{i}\) denotes the excitations at all nodes of the i-th substructure. Substituting \({\mathbf{H}}_{ib}={\mathbf{A}}_{i}{\mathbf{H}}_{i}\) into Eq. (18) yields
where \({\mathbf{A}}_{i}, \left(i=1, 2\right)\) represents the m × n coefficient matrix used to define the boundary nodes of substructures 1 and 2, \(\mathbf{F}\) denotes an \(n\times 1\) unit vector \(\mathbf{F}=\left\{1\right\}={\left[1 \dots 1\right]}^{T}\), and \({\mathbf{H}}_{\varvec{c}}\) represents the predicted and expanded FRF matrices.
Equation (19) represents the compatibility conditions expressed by the FRFs at the joint nodes, and is modified as
Considering \(\mathbf{R}=\mathbf{A}{\mathbf{H}}_{a}^{1/2}\) in Eq. (20), the solution with respect to \({{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{F}\)
is given by
where Z denotes an arbitrary matrix obtained by applying the condition to minimize the cost function of Eq. (1). By utilizing the condition to minimize Eq. (1) into Eq. (21) and solving it, the arbitrary matrix can be derived as
where \(\mathbf{Y}\) denotes an arbitrary matrix. Substituting Eq. (22) into Eq. (21) yields
Solving Eq. (23) with respect to \({{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}\) with \(\mathbf{Q}={\mathbf{H}}_{a}^{1/2}\mathbf{F}\), the following equation is obtained:
where \(\mathbf{X}\) denotes an arbitrary matrix. By applying the condition to minimize Eq. (1) to Eq. (24), the arbitrary matrix \(\mathbf{X}\) can be obtained as
Substituting Eq. (25) into Eq. (24) and pre- and post-multiplying the result by \({\mathbf{H}}_{a}^{\frac{1}{2}}\), the expanded FRF matrix can be expressed as
Equation (26) represents the full set of the FRF matrices expanded using the compatibility conditions expressed by the FRFs at the joint nodes.
The adequacy of the expansion of the FRFs and the effect of the pseudomass according to the constraints of the compatibility conditions must be evaluated. Additionally, the limitations and improvements of the proposed FBS technique are discussed through numerical experiments.
Numerical Experiments
Application of FRF Data Expansion Technique in a Both-Ends-Fixed Beam
The FRF dataset of a beam member was collected through impact hammer testing or numerical simulations, and the dynamic characteristics of the member were extracted. A structural beam is described by the vertical displacement and rotation responses owing to vertical forces and moment excitations. Although it is possible to measure the displacement FRFs (DFRFs) to describe the vertical displacement through impact hammer testing, it is difficult to apply moment excitations and measure the rotation angle. The DFRFs measured at small DOFs should be expanded to estimate the DFRFs at the remaining unmeasured nodes and the RFRFs at all nodes. This example considers the FRF expansion to satisfy some measured DFRFs.
This example considers the FRF data expansion in a both-ends-fixed beam, as shown in Fig. 1. The beam elements in the element analysis had two DOFs at each node. A beam with a length of \(2.4 \text{m}\), a cross section of \(b\times h=100\times 40 \text{m}\text{m}\), and a weight per unit volume of \(\text{7,860} \text{k}\text{g}/{\text{m}}^{3}\) was modeled with 40 elements. It was assumed that the second moment of inertia of the eight beam elements (8, 11, 15, 18, 23, 27, 31, and 37) deteriorated by 10%. Rayleigh damping was assumed with the stiffness matrix multiplied by 0.00001.
Experimental data are often contaminated by external noise and measurement errors. Assuming the containment of external noise during the measurement, the measured FRF data utilized in this example are given by
where \({\mathbf{H}}_{o}\) represents the FRFs obtained via a computer analysis of the damaged beam, and \(\mathbf{H}\) denotes the measurement FRFs, including the external noise. \(\alpha\) denotes the relative magnitude of the error, and \(\sigma\) is a random number varying in the range \(\left[-1, 1\right]\). The magnitude of \(\alpha\) was taken as 0.1 in this example. The FRF receptance curves were plotted from 0.01 Hz to 20 Hz in increments of 0.02 Hz.
The \(5\times 5\) DFRF matrix corresponding to the inputs and outputs at nodes 4, 12, 20, 28, and 36 is expanded to an entire FRF matrix, including unmeasured DFRFs and RFRFs, using the proposed technique. The constraints are expressed as Eq. (9). Figure 2 shows a comparison of the predicted FRF receptance curves on a logarithmic scale using Eq. (16) and the analytical FRF curves before and after imposing the constraints under vertical excitation. The dynamic responses of the beam are described by 78 DOFs from nodes 1 to 39. \({\text{H}}_{x,y}\) represents the response at DOF x owing to the excitation at DOF y. The plots show that the resonance frequency and FRF receptance magnitude of the damaged beam changed slightly owing to the presence of the damaged elements. The RFRFs predicted using the proposed method are very close to the analytical RFRFs of the damaged beam, except for minor discrepancies owing to the presence of external noise. The proposed technique accurately estimated the FRFs at unmeasured DOFs.
Application to a Dynamic System Constrained by Measured FRFs
Assuming damage to an eight-DOF system, as shown in Fig. 3, this example considers the expansion of the measured FRFs to unmeasured FRFs. The material properties of the initial system for the numerical experiments were as follows:
\({m}_{1}=3\), \({m}_{2}=4\), \({m}_{3}=3\), \({m}_{4}=4\), \({m}_{5}=4\), \({m}_{6}=5\), \({m}_{7}=5\), \({m}_{8}=5\),
\({k}_{1}=880\), \({k}_{2}=830\), \({k}_{3}=690\), \({k}_{4}=1020\), \({k}_{5}=360\), \({k}_{6}=730\), \({k}_{7}=920\), \({k}_{8}=550\), \({k}_{9}=940\), \({k}_{10}=570\), \({k}_{11}=720\,.\)
Rayleigh damping was assumed when the stiffness matrix was multiplied by 0.002. These values were used to determine the FRF matrix (\({\mathbf{H}}_{a})\)at the intact state.
Assume that the system is partially deteriorated by external factors, and the stiffness deteriorates as
Damage to this system results in variations in the dynamic characteristics and responses. Unmeasured FRFs were predicted using the proposed method because the responses at all nodes could not be measured.
The surgically simulated FRFs for the damaged system were used as constraints for data expansion. The FRF difference between nodes 3 and 7 was considered as a constraint condition of the same form as in Eq. (2).
The constraint corresponds to the relationship between the sums of displacements at nodes 3 and 7 owing to excitations at all nodes. This is consistent with the displacement constraint between nodes 3 and 7.
where \(\varDelta U\) indicates the displacement difference at measurement.
The FRFs measured at the two nodes were expanded to the FRFs at the remaining nodes using the proposed method. The magnitude of α in Eq. (27) was taken as 0.1, and the FRF receptance magnitude was calculated from 0.01 Hz to 10 Hz in increments of 0.02 Hz. Figure 4 compares the predicted and analytical FRFs with and without constraints. The plots include the FRF receptance curves at node 3 owing to the excitation from nodes 1 to 4 and at node 7 owing to the excitation at node 4. As in the previous example, small variations in the FRF curves were observed owing to damage to the system and the existence of the constraint. The predicted and analyzed FRF curves of the damaged system were almost identical. This example illustrates that the FRFs for a full set of nodes can be explicitly expanded using the proposed expansion technique without any numerical scheme, despite the existence of external noise.
Application to FBS Technique
After the disassembly of the entire system in Fig. 3 into two subsystems at the mass positions of the joint nodes, as shown in Fig. 5, this example attempts to synthesize them using the proposed technique. The two substructures can be synthesized into an entire system by applying compatibility conditions at the joint nodes. Substructures 1 and 2 comprise six and five nodes, respectively. There are infinite ways to divide the masses into two subsystems. It is expected that the division rate of the masses during synthesis will considerably affect the subsequent analysis results. This example predicts the FRFs at all nodes to satisfy the compatibility conditions at the joint nodes. The predicted FRF curves were compared based on the mass division rates.
The physical properties used in this example have the same values as those in Example 3.2. The deformation compatibility conditions were transformed into FRFs, which are expressed as follows:
where \(H\) and \(\widehat{H}\) represent the FRFs of substructures 1 and 2, respectively. The displacement responses of the subsystems 1 and 2 can be expressed, respectively, by \({\mathbf{u}}_{1}={\left[\begin{array}{ccc}\begin{array}{cc}{u}_{1}& {u}_{2}\end{array}& {u}_{3}& \begin{array}{ccc}{u}_{4}& {u}_{5}& {u}_{6}\end{array}\end{array}\right]}^{T}\) and \({\mathbf{u}}_{1}={\left[\begin{array}{ccc}{u}_{4{\prime }}& {u}_{5{\prime }}& \begin{array}{ccc}{u}_{6{\prime }}& {u}_{7}& {u}_{8}\end{array}\end{array}\right]}^{T}\). The coefficient matrices in Eq. (19a) can be expressed as
By substituting the FRFs calculated from each substructure and the constraints in Eq. (26), the FRFs of the entire synthesized system were estimated. The FRFs of the independent substructures can be expressed as
where \({\mathbf{H}}_{1}\) and \({\mathbf{H}}_{2}\) denote the FRF matrices of independent substructures 1 and 2, respectively. This example considers three different cases of mass division: 0.2/0.8 (\({m}_{i}=0.2{m}_{i}\) and \({m}_{{i}^{{\prime }}}=0.8{m}_{{i}^{{\prime }}}\)), 0.8/0.2 (\({m}_{i}=0.8{m}_{i}\) and \({m}_{{i}^{{\prime }}}=0.2{m}_{{i}^{{\prime }}}\)), and 0.4/0.6 (\({m}_{i}=0.4{m}_{i}\) and \({m}_{{i}^{{\prime }}}=0.6{m}_{{i}^{{\prime }}}\)), \(i=4, 5, 6\).
Substructuring of an entire structure in Fig. 3
The dynamic responses of the entire structure to the synthesis of the disassembled substructures must match the dynamic responses of the intact original structure. The FRFs of the entire synthesized structures were estimated and compared based on the division rates of the masses.
Consider the disassembled substructures at nodes 4, 5, and 6, as shown in Fig. 3. The torn masses at the joint nodes are called pseudomasses. Figures 6, 7 and 8 display the predicted FRF curves of the synthesized structure according to the mass division rates and the FRFs of the original entire structure within the frequency range from 0.01 Hz to 10 Hz in increments of 0.02 Hz. These plots include FRF curves corresponding to the excitation and response measurements at the same node. The mass division rates of substructures 1 and 2 were 0.2/0.8, 0.4/0.6, and 0.8/0.2. The division rates were observed to affect the synthesis of FRF curves. Discrepancies between the predicted and analyzed FRF curves were observed. The discrepancies gradually increased and remained constant above 6 Hz. The 0.4/0.6 FRF curves were closest to the analyzed FRF curve and had similar shapes up to approximately 2 Hz. Comparing the 0.2/0.8 and 0.8/0.2 FRF curves, the latter exhibited larger discrepancies. The pseudomasses must be divided by considering the proportion of mass distributed to the divided substructure to obtain closer FRFs when disassembling an asymmetric intact structure. In these plots, \({H}_{\text{6,8}}\) represents the FRF curves at a DOF not included in the joint node. This curve had a different shape and larger discrepancies than the FRF curves at joint nodes. The effect of the constraints was so minimal that the FRF at this node could rarely be controlled.
The discrepancy among these plots was due to the linear and unclear division rates of masses at splitting, insufficient information owing to the limited number of FRF relationships included in the constraints, and some constraints in the synthesis process. The proposed FBS technique can be enhanced by combining it with other dynamic assembly techniques and supplementing it with additional information, such as the measured FRFs.
Comparison of FRF curves applying FBS approach \(({m}_{i}=0.4{m}_{i}\,\,\text{a}\text{n}\text{d}\,\,{m}_{{i}^{{\prime }}}=0.6\,{m}_{{i}^{{\prime }}},\,i=4, 5, 6):\) (a) \({H}_{\text{4,4}}\) and \({H}_{4{\prime },4{\prime }}\), (b) \({H}_{\text{5,5}}\) and \({H}_{5{\prime },5{\prime }}\), (c) \({H}_{\text{6,6}}\) and \({H}_{6{\prime },6{\prime }}\), (d) \({H}_{\text{6,8}}\)
Comparison of FRF curves applying FBS approach \(({m}_{i}=0.2{m}_{i}\,\,\text{a}\text{n}\text{d}\,\,{m}_{{i}^{{\prime }}}=0.8{m}_{{i}^{{\prime }}},\,i=4, 5, 6):\) (a) \({H}_{\text{4,4}}\) and \({H}_{4{\prime },4{\prime }}\), (b) \({H}_{\text{5,5}}\) and \({H}_{5{\prime },5{\prime }}\), (c) \({H}_{\text{6,6}}\) and \({H}_{6{\prime },6{\prime }}\), (d) \({H}_{\text{6,8}}\)
Comparison of FRF curves applying FBS approach \(({m}_{i}=0.8{m}_{i}\,\,\text{a}\text{n}\text{d}\,\,{m}_{{i}^{{\prime }}}=0.2{m}_{{i}^{{\prime }}},\,i=4, 5, 6):\)\(\left(\mathbf{a}\right) {H}_{\text{4,4}}\) and \({H}_{4{\prime },4{\prime }}\), (b) \({H}_{\text{5,5}}\) and \({H}_{5{\prime },5{\prime }}\), (c) \({H}_{\text{6,6}}\) and \({H}_{6{\prime },6{\prime }}\), (d) \({H}_{\text{6,8}}\)
Conclusions
This paper proposed expansion techniques for estimating FRFs at unmeasured nodes and an FBS technique for synthesizing substructures using FRF compatibility conditions. The algorithms proposed herein were expressed in explicit mathematical forms to develop the FBS algorithm without depending on numerical schemes, such as the Lagrange multiplier method. The results of this study are summarized as follows.
[1] In this study, FRF expansion techniques were derived without any numerical schemes by minimizing the difference between the analytical and estimated FRF matrices to satisfy constraints, such as the measured FRFs.
[2] Numerical experiments demonstrated that RFRFs, for which it is difficult to measure moment forces or rotation angle responses, could be properly estimated using the proposed expansion method.
[3] In the FBS technique, the division rate of the mass at the joint nodes of each substructure affected the FRFs of the synthesized structure. The pseudomasses must be divided by considering the proportion of the mass distributed to the divided substructure to obtain closer FRFs when disassembling an asymmetric intact structure.
[4] A discrepancy between the analytical and predicted curves was observed when the FBS algorithm was applied. This discrepancy was due to the division rates of masses at splitting, insufficient information owing to the limited number of FRF relationships included in the constraints, and some constraints in the synthesis process.
[5] The FBS technique can be improved by combining it with other dynamic assembly techniques and supplementing it with additional information at more nodes, such as the measured FRFs.
Data Availability
Data will be made available on request.
References
Montalvao D, Ribeiro AMR, Maia NMM, Silva JMM (2004) Estimation of the rotational terms of the dynamic response matrix. Shock Vib 11:333–350
Kim J, Lee J, Kim K, Kang J, Yang M, Kim D, Lee S, Jang J (2021) Estimation of the frequency response function of the rotational degree of freedom. Appl Sci 11(18). https://doi.org/10.3390/app11188527
Hosoya N, Ozawa S, Kajiwara I (2019) Frequency response function measurements of rotational degrees of freedom using a non-contact moment excitation based on nanosecond laser ablation. J Sound Vib 456:239–253
Silva TAN, Maia NMM, Urgueira APV, Riscado P (2016) Rotational frequency response functions: model based estimation and experimental assessment. Proceedings of ISMA 2016
Avitabile P, O’callahan J (2003) Frequency response function expansion for unmeasured translation and rotation dofs for impedance modelling applications. Mech Syst Signal Process 17(4):723–745
Mirza WIIWI, Kyprianou A, Silva TAN, Rani MNA (2023) Frequency based substructuring and coupling enhancement using estimated rotational frequency response functions. Exp Tech. https://doi.org/10.1007/s40799-023-00670-0
Gibbons TJ, Ozturk E, Sims ND (2018) Rotational degree-of-freedom synthesis: an optimized finite difference method for non-exact data. J Sound Vib 412:207–221
Silva T, Pereira J (2018) Coupling of structures using frequency response functions. MATEC Web Conf 211. https://www.matec-conferences.org/articles/matecconf/pdf/2018/70/matecconf_vetomacxiv2018_06005.pdf
Ozguven HN (1990) Structural modifications using frequency response functions. Mech Syst Signal Process 4(1):53–63
Batista FC, Maia NMM (2012) Estimation of unmeasured frequency response functions. 19th International Congress on Sound and Vibration (ICVS 19)
Lee D (2023) Updating of the complete joint characteristics of finite element model via FRF-based substructuring of complex structures. J Mech Sci Technol 37:3437–3444
Elliott A, Moorhouse A, Meggitt J (2018) Identification of coupled degrees of freedom at the interface between sub-structures. The Proceedings of INTER-NOISE-CON Congress and Conference: 672–680
Cepon G, Drozg A, Boltezar (2019) Introduction of line contact in frequency-based substructuring process using measured rotational degrees of freedom. Proc J Physics: Conf Ser 1264:012025
Asma F, Bouazzouni A (2005) Finite element model updating using FRF measurements. Shock Vib 12(5), 2005: 377–388
Kidder RL (1973) Reduction of structural frequency equations. AIAA J 11(6):892
Maia NMM, Silva TAN (2021) An expansion technique for the estimation of unmeasured rotational frequency response functions. Mech Syst Signal Process 156. https://doi.org/10.1016/j.ymssp.2021.107634
De Klerk D, Rixen D, Jong JD (2006) The frequency based substructuring (FBS) method reformulated according to the dual domain decomposition method. 24th International Modal Analysis Conference. St.Louis, MO
Carne TG, Dohrmann CR (2006) Improving experimental frequency response function matrices for admittance modeling. Shock Vib Dig 38(6):546
Funding
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (RS-2023-00242973).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The authors have reported that there is no conflict of interest.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Kim, S., An, JH. & Eun, HC. Expansion Technique of Frequency Response Function Data and its Applications. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01428-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s42417-024-01428-7