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

$${\mathbf{P}=\mathbf{H}}_{a}^{-1/2}\left({\mathbf{H}}_{a}-{\mathbf{H}}_{c}\right){\mathbf{H}}_{a}^{-1/2}\,,$$
(1)

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

$$\mathbf{A}{\mathbf{H}}_{\varvec{c}}=\mathbf{B}\,,$$
(2)

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.

$$\mathbf{A}{\mathbf{U}}_{\varvec{c}}=\mathbf{B}\mathbf{F}\,,$$
(3)

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

$$\mathbf{A}{\mathbf{H}}_{a}^{1/2}{{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}=\mathbf{B}\,.$$
(4)

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:

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}={\mathbf{R}}^{+}\mathbf{B}+\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\mathbf{Z}\,,$$
(5)

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

$$\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\mathbf{Z}=-{\mathbf{R}}^{+}\mathbf{B}+{\mathbf{H}}_{a}^{1/2}\,.$$
(6)

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:

$$\mathbf{Z}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\left(-{\mathbf{R}}^{+}\mathbf{B}+{\mathbf{H}}_{a}^{1/2}\right)+{\mathbf{R}}^{+}\mathbf{R}\mathbf{Y}\,,$$
(7)

where \(\mathbf{Y}\) represents another n × n arbitrary matrix. Substituting Eq. (7) into Eq. (5) yields an expanded FRF matrix for all the DOFs.

$${\mathbf{H}}_{\varvec{c}}={\mathbf{H}}_{a}+{\mathbf{H}}_{a}^{1/2}{\left(\mathbf{A}{\mathbf{H}}_{a}^{1/2}\right)}^{+}\left(\mathbf{B}-\mathbf{A}{\mathbf{H}}_{a}\right)$$
(8)

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

$$\mathbf{L}{\mathbf{H}}_{\varvec{c}}{\mathbf{L}}^{T}=\mathbf{D}\,,$$
(9)

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:

$$\mathbf{L}{{\mathbf{H}}_{a}^{1/2}{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{L}}^{\varvec{T}}=\mathbf{D}\,.$$
(10)

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

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{L}}^{\varvec{T}}={\mathbf{R}}^{+}\mathbf{D}+\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\mathbf{Z}\,,$$
(11)

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:

$$\mathbf{Z}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]{\mathbf{H}}_{a}^{1/2}{\mathbf{L}}^{T}+{\mathbf{R}}^{+}\mathbf{R}\mathbf{Y}\,,$$
(12)

where \(\mathbf{Y}\) denotes an arbitrary matrix. Subsequently, substituting Eq. (12) into Eq. (11) yields

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{L}}^{\varvec{T}}={\mathbf{H}}_{a}^{1/2}{\mathbf{L}}^{T}+{\mathbf{R}}^{+}\left(\mathbf{D}-\mathbf{R}{\mathbf{H}}_{a}^{1/2}{\mathbf{L}}^{T}\right)\,.$$
(13)

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:

$$\eqalign{ H_a^{( - 1/2)}{H_c}H_a^{( - 1/2)} = \cr & [H_a^{(1/2)}{L^T} + {R^ + }(D - RH_a^{(1/2)}{L^T})]{Q^ + } \cr & + X(I - Q{Q^ + }) \cr \cr}$$
(14)

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

$$\mathbf{X}=\mathbf{I}-\mathbf{Q}{\mathbf{Q}}^{+}+\mathbf{Y}\mathbf{Q}{\mathbf{Q}}^{+}\,.$$
(15)

Substituting Eq. (15) into Eq. (14), the expanded FRF matrix can be expressed as

$${\mathbf{H}}_{\varvec{c}}={\mathbf{H}}_{a}+\left[{\mathbf{H}}_{a}{\mathbf{L}}^{T}+{{\mathbf{H}}_{a}^{\frac{1}{2}}\mathbf{R}}^{+}\left(\mathbf{D}-\mathbf{R}{\mathbf{H}}_{a}^{\frac{1}{2}}{\mathbf{L}}^{T}\right)-{\mathbf{H}}_{a}^{\frac{1}{2}}\mathbf{Q}\right]{\mathbf{Q}}^{+}{\mathbf{H}}_{a}^{\frac{1}{2}}\,.$$
(16)

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:

$${\mathbf{U}}_{1b}={\mathbf{U}}_{2b}\,,$$
(17)

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:

$${\mathbf{H}}_{1b}{\mathbf{F}}_{1}={\mathbf{H}}_{2b}{\mathbf{F}}_{2}\,,$$
(18)

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

$$\left[\begin{array}{cc}{\mathbf{A}}_{1}& {-\mathbf{A}}_{2}\end{array}\right]\left[\begin{array}{cc}{\mathbf{H}}_{1}& 0\\ 0& {\mathbf{H}}_{2}\end{array}\right]\left[\begin{array}{c}{\mathbf{F}}_{1}\\ {\mathbf{F}}_{2}\end{array}\right]=0\,,$$
(19a)
$$\mathbf{A}{\mathbf{H}}_{\varvec{c}}\mathbf{F}=0\,,$$
(19b)

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

$$\mathbf{A}{{\mathbf{H}}_{a}^{1/2}{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{F}=0\,.$$
(20)

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

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{F}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]\mathbf{Z}\,,$$
(21)

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

$$\mathbf{Z}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]{\mathbf{H}}_{a}^{1/2}\mathbf{F}+{\mathbf{R}}^{+}\mathbf{R}\mathbf{Y}\,,$$
(22)

where \(\mathbf{Y}\) denotes an arbitrary matrix. Substituting Eq. (22) into Eq. (21) yields

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}{\mathbf{H}}_{a}^{1/2}\mathbf{F}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]{\mathbf{H}}_{a}^{1/2}\mathbf{F}\,.$$
(23)

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:

$${{\mathbf{H}}_{a}^{-1/2}\mathbf{H}}_{\varvec{c}}{\mathbf{H}}_{a}^{-1/2}=\left[\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right]{\mathbf{H}}_{a}^{1/2}\mathbf{F} {\mathbf{Q}}^{+}+\mathbf{X}\left(\mathbf{I}-\mathbf{Q}{\mathbf{Q}}^{+}\right)\,,$$
(24)

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

$$\mathbf{X}=\mathbf{I}-\mathbf{Q}{\mathbf{Q}}^{+}+\mathbf{Y}\mathbf{Q}{\mathbf{Q}}^{+}\,.$$
(25)

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

$${\mathbf{H}}_{\varvec{c}}={\mathbf{H}}_{a}+\left[{\mathbf{H}}_{a}^{\frac{1}{2}}\left(\mathbf{I}-{\mathbf{R}}^{+}\mathbf{R}\right)-{\mathbf{H}}_{a}^{\frac{1}{2}}\right]\mathbf{Q}{\mathbf{Q}}^{+}{\mathbf{H}}_{a}^{\frac{1}{2}}\,.$$
(26)

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.

Fig. 1
figure 1

Both-ends-fixed beam model

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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

$$\mathbf{H}={\mathbf{H}}_{o}\left(\mathbf{I}+\alpha \sigma \right)\,,$$
(27)

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.

Fig. 2
figure 2

FRF receptance magnitude curves: (a) \({H}_{\text{2,3}}\), (b) \({H}_{\text{3,3}}\), (c) \({H}_{\text{16,15}}\), (d) \({H}_{\text{16,19}}\), (e) \({H}_{\text{8,23}}\), (f) \({H}_{\text{16,23}}\)

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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

$${k}_{1}=0.9{k}_{1}, {k}_{7}=0.95{k}_{7}, {k}_{10}=0.82{k}_{10}.$$

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).

$${H}_{3,i}-{H}_{7,i}={B}_{i}, i=1, 2,\dots , 8$$

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.

$${U}_{3}-{U}_{7}=\varDelta U\,,$$

where \(\varDelta U\) indicates the displacement difference at measurement.

Fig. 3
figure 3

An eight DOF system

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Fig. 4
figure 4

FRF receptance magnitude curves: (a) \({H}_{\text{3,1}}\), (b) \({H}_{\text{3,2}}\), (c) \({H}_{\text{3,3}}\), (d) \({H}_{\text{3,4}}\), (e) \({H}_{\text{7,4}}\)

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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:

$$\sum _{i=1}^{6}{H}_{4,i}=\sum _{j=1}^{5}{\widehat{H}}_{1,j}\,,$$
$$\sum _{i=1}^{6}{H}_{5,i}=\sum _{j=1}^{5}{\widehat{H}}_{2,j}\,,$$
$$\sum _{i=1}^{6}{H}_{6,i}=\sum _{j=1}^{5}{\widehat{H}}_{3,j}\,,$$

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

$$\mathbf{A}=\left[\begin{array}{ccc}\begin{array}{ccc}0& 0& 0\end{array}& \begin{array}{ccc}1& 0& 0\end{array}& \begin{array}{ccc}-1& 0& \begin{array}{ccc}0& 0& 0\end{array}\end{array}\\ \begin{array}{ccc}0& 0& 0\end{array}& \begin{array}{ccc}0& 1& 0\end{array}& \begin{array}{ccc}0& -1& \begin{array}{ccc}0& 0& 0\end{array}\end{array}\\ \begin{array}{ccc}0& 0& 0\end{array}& \begin{array}{ccc}0& 0& 1\end{array}& \begin{array}{ccc}0& 0& \begin{array}{ccc}-1& 0& 0\end{array}\end{array}\end{array}\right]\,,$$
$$\mathbf{F}={\left[\begin{array}{ccc}\begin{array}{ccc}1& 1& 1\end{array}& \begin{array}{ccc}1& 1& 1\end{array}& \begin{array}{ccc}1& 1& \begin{array}{ccc}1& 1& 1\end{array}\end{array}\end{array}\right]}^{T}\,.$$

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

$${\mathbf{H}}_{a}=\left[\begin{array}{cc}{\mathbf{H}}_{1}& 0\\ 0& {\mathbf{H}}_{2}\end{array}\right]\,,$$

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 \({m}_{{i}^{{\prime }}}=0.6{m}_{{i}^{{\prime }}}\)), \(i=4, 5, 6\).

Fig. 5
figure 5

Substructuring of an entire structure in Fig. 3

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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.

Fig. 6
figure 6

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) \({H}_{\text{6,8}}\)

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Fig. 7
figure 7

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) \({H}_{\text{6,8}}\)

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Fig. 8
figure 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) \({H}_{\text{6,8}}\)

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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.