Structural response reconstruction for non-proportionally damped systems in the presence of closely spaced modes

2.1. Sifting process of the EMD method

A sifting process is used by the EMD method to filter out IMFs from a given signal. $z\left(t\right)$ is denoted as a time domain signal, the sifting process is summarized as follows: [13, 14, 22].

1. Construct the upper and lower envelopes of the signal by cubic-spline fitting. Compute the mean of both envelopes, which is denoted by $l_1\left(t\right)$.

2. Compute the first component $h_1\left(t\right)=z\left(t\right)-l_1\left(t\right)$ and check to see whether $h_1\left(t\right)$ is a monocomponent, referred to as an IMF. If it is not an IMF, continue the sifting process using $h_1\left(t\right)$ as the new signal data.

3. Construct the envelops of $h_1\left(t\right)$ and denote $l_{11}\left(t\right)$ as the mean of the envelops, and compute the component $h_{11}\left(t\right)=h_1\left(t\right)-l_{11}\left(t\right)$.

4. Repeat step 3 for $k$ times until the resulting $h_{1k}\left(t\right)$ ($h_{1k}\left(t\right)=h_{1(k-1)}\left(t\right)-l_{1k}\left(t\right)$) is an IMF.

5. Denote $w_1\left(t\right)=h_{1k}\left(t\right)$, which is the first IMF extracted from the signal data.

6. Repeat the above steps to get the second IMF $w_2\left(t\right)$ from the remaining signal of $z\left(t\right)-w_1\left(t\right)$.

7. Continue the above procedure to obtain another IMF from the remaining signal until the rest of the signal $r\left(t\right)$ ($r\left(t\right)=z\left(t\right)-\sum _{i=1}^n{w}_i\left(t\right)$) is the mean trend (or a constant) of the signal. $w_i\left(t\right)$ is the $i$th IMF and $r\left(t\right)$ is the residue.

2.2. The EMD method with intermittency criteria

IMFs obtained from the sifting process described above may contain more than one frequency, which is not an exact modal response. To obtain the structural modal responses, the intermittency criteria are used to ensure that each of the IMFs contains only one frequency component. The idea of the intermittency criteria in sifting process is to remove all frequency components lower and higher than ${\omega }_{int}$ (an intermittency frequency) from an IMF by using a band-pass filter [13, 14, 25]. Here, ${\omega }_{int}$ is just determined by the $i$th natural frequency ${\omega }_i$ of the structure. The procedure to obtain the modal responses is summarized as follows:

1. Estimate an approximate frequency range for ${\omega }_i$ (i.e. ${\omega }_{iL}<{\omega }_i<{\omega }_{iH}$) either by Fourier spectra of the measured signal $z\left(t\right)$ or FEM computations as the band-pass filter.

2. Process the signal $z\left(t\right)$ through the band-pass filter.

3. Use the sifting process described in Section 2.1 to process the data obtained from step 2. The first IMF is considered to be the approximation of modal response.

By repeating the above procedure with different frequency ranges for different natural frequencies, all the required modal responses can be obtained. It is important to note that the phase shift of the band-pass filter should be as small as possible. Using the sifting process with intermittency criteria, the original signal expression can be written as:

where $d_i\left(t\right)$ is the modal response (that is also an IMF) for the $i$th mode. $w_i\left(t\right)$ ($i=$1,…, $n-m$) are other IMFs but not modal responses.

Applying the EMD method with intermittency criteria to the measured signal, the modal responses can be extracted except for the closely spaced modal responses which the band-pass filter cannot separate.

3.1. Response reconstruction without closely spaced modes

Assume that the response of the structure doesn’t contain any closely spaced modes. According to the theory of modal analysis, the response can be expressed as the summation of $n$ modal responses. Define that $x_{a_1}\left(t\right)$, $x_{a_2}\left(t\right)$ (regard as the responses of DOF-$a_1$ and DOF-$a_2$) are the two responses of the known responses which can be obtained by sensor measurement, $x_u\left(t\right)$ is the vector of the unknown responses one wishes to know. The following equations can be obtained as:

where the subscript $i$ represents the $i$th mode, and ${\varphi }_{ai}$ represents the mode information for the $a$th DOF under $i$th mode. ${\mathbf{\varphi }}_{ui}$ represents the mode information for the DOFs whose responses are unknown. $q_i\left(t\right)$ is the response of the $i\text{th}$ mode in modal coordinate, and $d_{ai}\left(t\right)={\varphi }_{ai}{q}_i\left(t\right)+{\varphi }_{ai}^{*}{q}_i^{*}\left(t\right)$ is the $i$th modal response for DOF-$a$ in generalized coordinate. From Eqs. (7) and (8), we have:

Substituting Eq. (10) into Eq. (9), the following equation can be obtained as:

with the matrix ${\mathbf{S}}_{ua}$ and the vector ${\mathbf{d}}_a\left(t\right)$, which are respectively defined as:

where ${\mathbf{d}}_a\left(t\right)$ can be computed by using the EMD method discussed in Section 2, and ${\mathbf{S}}_{ua}$ can be obtained using the FEM. According to Eq. (11), it can be concluded that the unknown responses, e.g. at sensor inaccessible locations, can be computed from one response at a measurable location. Additionally, the exact mode shape is obtained from an accurate FE model for all the cases studied. Under realistic conditions, the estimated mode information from measurements or an inaccurate finite element model may be induced uncertainties in the reconstruction results [15].

3.2. Response reconstruction in the presence of closely spaced modes

When a structure exists with closely spaced modes, the method proposed in Section 3.1 will be not suitable for the response reconstruction of this structure. In this section, an improvement to the reconstruction method is proposed to handle this situation with closely spaced modes.

Assuming that $e$th and $f$th modes are closely spaced, the subscripts $m$ and $r$ represent the information of the closely spaced modes and the rest of the modes, respectively, the subscript $k$ and $u$ represent the information of the DOFs whose responses are known and unknown, respectively. The known-set and unknown-set responses can be written as:

where ${\mathbf{\varphi }}_{km}$ is the closely spaced mode shape matrix of the DOFs-$k$ at which responses are known, ${\mathbf{\varphi }}_{kr}$ is the rest of the mode shape matrix of the DOFs-$k$. ${\mathbf{\varphi }}_{km}$ and ${\mathbf{\varphi }}_{um}$ are the closely spaced mode shape matrices of the DOFs-$k$ and DOFs-$u$. ${\mathbf{q}}_m\left(t\right)$ and ${\mathbf{q}}_r\left(t\right)$ are the response vectors of the closely spaced modes and the rest of the modes in modal coordinate, respectively. They can be written as:

in which $q_e\left(t\right)$ and $q_f\left(t\right)$ are the responses of the two closely spaced modes in modal coordinate just as assumed before. $q_{r_1}\left(t\right)$, $q_{r_2}\left(t\right)$,…, $q_{r_r}\left(t\right)$ are the responses of the rest of the modes in modal coordinate, respectively.

From Eq. (14), the responses of the closely spaced modes in modal coordinate can be obtained as:

where ${\left[\begin{array}{ll}{\mathbf{\varphi }}_{km}& {\mathbf{\varphi }}_{km}^{*}\end{array}\right]}^{+}$ denotes the pseudo-inverse of matrix $\left[\begin{array}{ll}{\mathbf{\varphi }}_{km}& {\mathbf{\varphi }}_{km}^{*}\end{array}\right]$. It is should be noted that the number of the known responses should be at least 2 times equal or greater than the number of the closely spaced modes.

Substituting Eq. (18) into Eq. (15), the vector of unknown responses can be written as:

For Eq. (19), there are two methods to obtain the unknown responses vector ${\mathbf{x}}_u\left(t\right)$. One is to compute $\left({\mathbf{x}}_k\left(t\right)-\left[\begin{array}{ll}{\mathbf{\varphi }}_{kr}& {\mathbf{\varphi }}_{kr}^{*}\end{array}\right]\left(\begin{array}{l}{\mathbf{q}}_r\left(t\right)\\ {\mathbf{q}}_r^{*}\left(t\right)\end{array}\right)\right)$ directly using the EMD method, but it is more complicated than the following method, because it requires executing the EMD method for all of the known responses according to Eq. (19). The algorithm is illustrated in the following text.

Eq. (19) can be transformed into another form as following:

and several matrices are defined as:

in which the subscript $a$ represents the information of DOF-$a$ at which response is known, the subscript $r_1\text{,}$$r_2$,…, $r_r$ represent the information of the rest of the modes, and ${\mathbf{d}}_{ar}\left(t\right)$ is denoted as the vector of the rest of the modal responses of DOF-$a$. Therefore, Eq. (20) can be expressed as:

${\mathbf{T}}_{umk}$, ${\mathbf{S}}_{uar}$ and ${\mathbf{S}}_{kar}$ can be derived by FEM, and ${\mathbf{d}}_{ar}\left(t\right)$ can be computed by the EMD method with intermittency criteria. Eq. (25) can be called the reconstructed equation, and it is the same formulation as Eq. (21) in Ref. [15]. Note that a restriction of this proposed method is the number of the known responses $k$ must be greater or equal than the number of closely spaced modes $2m$.

When a structure contains multiple sets of closely spaced modes, Eq. (14) can be written as:

where $m_j$ represents the $j\text{th}$ set of closely spaced modes, and ${\mathbf{d}}_{k{m}_j}\left(t\right)=\left[\begin{array}{ll}{\mathbf{\varphi }}_{k{m}_j}& {\mathbf{\varphi }}_{k{m}_j}^{*}\end{array}\right]\left(\begin{array}{l}{\mathbf{q}}_{m_j}\left(t\right)\\ {\mathbf{q}}_{m_j}^{*}\left(t\right)\end{array}\right)$ is the corresponding modal responses of the known responses, which just can be obtained by the EMD method according to Fourier spectra. Then one can obtain that:

Eq. (15) can be easily transformed as following:

where:

Eq. (28) is also the reconstruction equation for the case of multiple sets of closely spaced modes. Note that the number of $k$ DOFs must be greater or equal than the number of 2×$\mathrm{m}\mathrm{a}\mathrm{x}\left(m_j\right)$ modes due to the pseudo-inverse of matrix ${\left[\begin{array}{ll}{\mathbf{\varphi }}_{k{m}_j}& {\mathbf{\varphi }}_{k{m}_j}^{*}\end{array}\right]}^{+}$. Taking each set of closely spaced modes as an integral part to obtain the modal responses is the key idea for dealing with this situation.

For stochastic or transient excitation, such as vibrations induced by ambient wind and earthquakes, the proposed method can reconstruct dynamic responses accurately and rapidly. Fig. 1 presents the overall reconstruction procedure for the methods presented in Section 3.1 and 3.2.

It should be noted that the method based on the modal superposition method presented in Section 3.1 and 3.2 are suitable for stochastic and transient excitation, and also for the process of both free vibration and steady forced vibration. Moreover, this proposed method is suitable for various types of dynamic responses, not only displacement responses but also velocity and acceleration responses.

Fig. 1Overall procedure of response reconstruction for non-proportionally damped system

4.1. Example 1: Response reconstruction without closely spaced modes

A 3-DOF theoretical model is used in this example. The mass, stiffness and damping matrices are presented as following:

The units are tone, N/mm and N/mm/s, respectively. No excitation force is applied, i.e. free vibration. The method proposed in section 3.1 is used to reconstruct the unknown responses. The displacement responses of DOF-1 and DOF-2 are used to be decomposed into modal responses (as ${\mathbf{d}}_a\left(t\right)$) by the EMD method.

The sampling frequency is 100 Hz, and the sampling time is 20 s. Fig. 2 shows the known displacement response of DOF-1. The Fourier spectra of the response is plotted in Fig. 3 where the three order frequencies are shown. The performance of the REMD method depends on the ability of the band-pass filter to separate different modes. Table 1 gives frequency ranges for each band-pass. Fig. 4 presents the filtered results of the rest of the modal responses. The reconstruction result and the theoretical prediction of DOF-1 are illustrated in Fig. 5. Fig. 6 is the partial time-history chart of Fig. 5 from 2 s to 10 s. From the two figures, it can be observed that the reconstruction result obtained from the proposed method is very close to theoretical prediction except for the beginning region. The discrepancy at the beginning of Fig. 5 is possible a result of the end boundary effect, which has been wildly discussed in literature as the intrinsic weakness of EMD [20]. This example proves that the proposed method is suitable for response reconstruction for free vibration.

Fig. 2Displacement response of DOF-1 with no noise terms

Fig. 3Fourier spectra of displacement response of DOF-1

Fig. 4The modal responses of DOF-1 obtained using the EMD method with intermittency criteria

Fig. 5Theoretical response and reconstructed response (using the REMD method) of DOF-3

Fig. 6Zoom-in view of a segment (2-10 s) of Fig. 5

4.2. Example 2: Response reconstruction in the presence of closely spaced modes

A six DOF mass-spring system, as shown in Fig. 7, is presented here to demonstrate the overall reconstruction procedure for the problems which involving closely spaced modes. And its characteristics are described in Table 2. Liang damping [26] is used in all the simulations of example 2, which is defined as:

where $\mathbf{R}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left(\right[0.32\mathrm{}\mathrm{}0.52\mathrm{}\mathrm{}0.46\mathrm{}\mathrm{}0.82\mathrm{}\mathrm{}0.78\mathrm{}\mathrm{}0.56\left]\right)$. It is easily verified that non-proportional damping ensures with the simple equation:

Table 3 shows the damping coefficients and the information of external excitations for all the cases.

Fig. 7The 6DOF model used in the simulations. Liang damping C=αM+βK+R is used

4.2.1. Case 1: Basic example

Sinusoidal excitation force is applied at DOF-2 for a duration of 0.05 s, which is:

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$f_1\left(t\right)=\left\{\begin{array}{ll}20\mathrm{s}\mathrm{i}\mathrm{n}\left(2\pi \times 45t\right),& \mathrm{t}\le 0.05\mathrm{}\text{s},\\ 0,& \mathrm{t}>0.05\mathrm{}\text{s}\text{.}\end{array}\right.$

The sampling frequency is 2000 Hz. Fig. 8 shows the known acceleration response of DOF-4 with no noise terms. The Fourier spectra of the response is plotted in Fig. 9 where the six theoretical natural frequencies are shown. It is observed that the 3rd and 4th modes are closely spaced. So the number of the known responses should be at least equal to 4 to satisfy the relationship of $k\ge 2m$.

Fig. 8Acceleration response of DOF-4 with no noise terms

Fig. 9Fourier spectra of acceleration response of DOF-4

The method proposed in Section 3.2 is used to reconstruct the unknown responses. It is assumed that the responses of DOF-1, 3, 4 and 6 are known. So the vectors of known and unknown responses are represented as:

${\mathbf{x}}_k\left(t\right)={\left[\begin{array}{llll}{x}_1\left(t\right)& x_3\left(t\right)& x_4\left(t\right)& x_6\left(t\right)\end{array}\right]}^T,{\mathbf{x}}_u\left(t\right)={\left[\begin{array}{ll}{x}_2\left(t\right)& x_5\left(t\right)\end{array}\right]}^{\mathrm{T}}.$

The vectors of the responses in modal coordinate of the system can be written as:

${\mathbf{q}}_m\left(t\right)={\left[\begin{array}{ll}{q}_3\left(t\right)& q_4\left(t\right)\end{array}\right]}^T,{\mathbf{q}}_r\left(t\right)={\left[\begin{array}{llll}{q}_1\left(t\right)& q_2\left(t\right)& q_5\left(t\right)& q_6\left(t\right)\end{array}\right]}^{\mathrm{T}}.$

The response of DOF-3 and 4 are used to be decomposed into modal responses to transfer the rest of the modal responses for the overall reconstruction procedure, i.e. $a_1=\text{3}$, $a_2=\text{4}$, according to Eqs. (21)-(24), the four critical matrices can be obtained as:

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${\mathbf{T}}_{umk}=\left(\begin{array}{llll}{\varphi }_{23}& {\varphi }_{23}^{\mathrm{*}}& {\varphi }_{24}& {\varphi }_{24}^{\mathrm{*}}\\ {\varphi }_{53}& {\varphi }_{53}^{\mathrm{*}}& {\varphi }_{54}& {\varphi }_{54}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{llll}{\varphi }_{13}& {\varphi }_{13}^{\mathrm{*}}& {\varphi }_{14}& {\varphi }_{14}^{\mathrm{*}}\\ {\varphi }_{33}& {\varphi }_{33}^{\mathrm{*}}& {\varphi }_{34}& {\varphi }_{34}^{\mathrm{*}}\\ {\varphi }_{43}& {\varphi }_{43}^{\mathrm{*}}& {\varphi }_{44}& {\varphi }_{44}^{\mathrm{*}}\\ {\varphi }_{63}& {\varphi }_{63}^{\mathrm{*}}& {\varphi }_{64}& {\varphi }_{64}^{\mathrm{*}}\end{array}\right)}^{-1},$

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${\mathbf{S}}_{uar}=\left[\begin{array}{llll}\left(\begin{array}{ll}{\varphi }_{21}& {\varphi }_{21}^{\mathrm{*}}\\ {\varphi }_{51}& {\varphi }_{51}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{31}& {\varphi }_{31}^{\mathrm{*}}\\ {\varphi }_{41}& {\varphi }_{41}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{22}& {\varphi }_{22}^{\mathrm{*}}\\ {\varphi }_{52}& {\varphi }_{52}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{32}& {\varphi }_{32}^{\mathrm{*}}\\ {\varphi }_{42}& {\varphi }_{42}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{25}& {\varphi }_{25}^{\mathrm{*}}\\ {\varphi }_{55}& {\varphi }_{55}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{35}& {\varphi }_{35}^{\mathrm{*}}\\ {\varphi }_{45}& {\varphi }_{45}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{26}& {\varphi }_{26}^{\mathrm{*}}\\ {\varphi }_{56}& {\varphi }_{56}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{36}& {\varphi }_{36}^{\mathrm{*}}\\ {\varphi }_{46}& {\varphi }_{46}^{\mathrm{*}}\end{array}\right)}^{-1}\end{array}\right],$

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${\mathbf{S}}_{kar}=\left[\begin{array}{llll}\left(\begin{array}{ll}{\varphi }_{11}& {\varphi }_{11}^{\mathrm{*}}\\ {\varphi }_{31}& {\varphi }_{31}^{\mathrm{*}}\\ {\varphi }_{41}& {\varphi }_{41}^{\mathrm{*}}\\ {\varphi }_{61}& {\varphi }_{61}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{31}& {\varphi }_{31}^{\mathrm{*}}\\ {\varphi }_{41}& {\varphi }_{41}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{12}& {\varphi }_{12}^{\mathrm{*}}\\ {\varphi }_{32}& {\varphi }_{32}^{\mathrm{*}}\\ {\varphi }_{42}& {\varphi }_{42}^{\mathrm{*}}\\ {\varphi }_{62}& {\varphi }_{62}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{32}& {\varphi }_{32}^{\mathrm{*}}\\ {\varphi }_{42}& {\varphi }_{42}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{15}& {\varphi }_{15}^{\mathrm{*}}\\ {\varphi }_{35}& {\varphi }_{35}^{\mathrm{*}}\\ {\varphi }_{45}& {\varphi }_{45}^{\mathrm{*}}\\ {\varphi }_{65}& {\varphi }_{65}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{35}& {\varphi }_{35}^{\mathrm{*}}\\ {\varphi }_{45}& {\varphi }_{45}^{\mathrm{*}}\end{array}\right)}^{-1}& \left(\begin{array}{ll}{\varphi }_{16}& {\varphi }_{16}^{\mathrm{*}}\\ {\varphi }_{36}& {\varphi }_{36}^{\mathrm{*}}\\ {\varphi }_{46}& {\varphi }_{46}^{\mathrm{*}}\\ {\varphi }_{66}& {\varphi }_{66}^{\mathrm{*}}\end{array}\right){\left(\begin{array}{ll}{\varphi }_{36}& {\varphi }_{36}^{\mathrm{*}}\\ {\varphi }_{46}& {\varphi }_{46}^{\mathrm{*}}\end{array}\right)}^{-1}\end{array}\right],$

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${\mathbf{d}}_{ar}\left(t\right)={\left[\begin{array}{llll}{\left(\begin{array}{l}{d}_{31}\left(t\right)\\ d_{41}\left(t\right)\end{array}\right)}^{\mathrm{T}}& {\left(\begin{array}{l}{d}_{32}\left(t\right)\\ d_{42}\left(t\right)\end{array}\right)}^{\mathrm{T}}& {\left(\begin{array}{l}{d}_{35}\left(t\right)\\ d_{45}\left(t\right)\end{array}\right)}^{\mathrm{T}}& {\left(\begin{array}{l}{d}_{36}\left(t\right)\\ d_{46}\left(t\right)\end{array}\right)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}.$

Thus, the unknown responses ${\mathbf{x}}_u\left(t\right)$ are easily computed associated to Eq. (25).

The modal responses of the rest of the modes which actually consisted of 1st, 2nd, 5th and 6th modes at DOF-3 and 4 are obtained by the EMD method with intermittency criteria. These four modal frequencies are used to design the band-pass filters. Frequency ranges for each band-pass filter are shown in Table 4. It is noted that the band-pass filter should have as small phase shift as possible [16]. Fig. 10 presents the filtered results of the rest of modal responses at DOF-4.

Table 4Frequency ranges for each band-pass filter

Mode	I	II	V	VI
Natural frequency	9.77	54.12	118.46	150.31
Passband corner frequency (Hz)	[6-9]	[45-52]	[102-114]	[132-140]
Stopband corner frequency (Hz)	[10.5-13]	[56-62]	[119-130]	[160-170]

Fig. 10The rest of the modal responses of DOF-4 obtained using the EMD method with intermittency criteria

Fig. 11a) Theoretical response and reconstructed response of DOF-5, b) Absolute error between the theoretical response and reconstructed response of DOF-5

a)

b)

All of the unknown responses can be obtained based on the reconstructed equation, i.e. Eq. (25). For example, the reconstructed acceleration response and the theoretical response of DOF-5 are shown in Fig. 11(a), and Fig. 11(b) shows the absolute error of these two signals. Fig. 12 is the partial time-history chart of Fig. 11(a) from 1 s to 1.3 s. The two curves of responses are almost overlapping, indicating very high reconstruction accuracy of the proposed method. Also from Fig. 11 and Fig. 12, it is observed that the reconstruction result is very close to theoretical prediction except for the beginning region. At the beginning of Fig. 11(b), the periodic sinusoidal force introduces a frequency component (45 Hz) in the Fourier spectra, which is not the natural modes of the structure itself, and the contribution of 45 Hz is significant in frequency domain with the force applied. On the other hand, the discrepancy at the beginning of Fig. 11(b) is possible a result of the end boundary effect, which just like the discrepancies found in example 1. This case proves that the proposed method is suitable for response reconstruction with transient excitation. Following this basic example, effects of the noise level, high damping ratio, multiple stochastic forces and two sets of closely spaced modes are studied as four cases in detail.

Fig. 12Zoom-in view of a segment (1-1.3 s) of Fig. 11(a)

Fig. 13Theoretical response and reconstructed response of DOF-2 (no noise)

4.2.2. Case 2: Effect of noise level

To investigate the effect of the noise level on the performance of the reconstruction method, numerical experiments with different noise levels are performed. The noise is defined as a normally distributed random noise with zero mean and unit standard deviation, which can be written as:

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$x_{noise}\left(t\right)=n{N}_r\mathrm{s}\mathrm{t}\mathrm{d}\left(x_{original}\left(t\right)\right),$

where $x_{noise}\left(t\right)$ is the added noise to the original measured signal; $n$ is the noise level; $N_r$ is a standard normal distribution vector with zero mean and unit standard deviation and $\mathrm{s}\mathrm{t}\mathrm{d}\left(x_{original}\right(t\left)\right)$ is the standard deviation of the original measured response.

In this case, the acceleration response of DOF-2 is reconstructed using the known acceleration responses. Three noise levels are calculated ($n=$0, 2 %, 5 %). The reconstruction results are shown in Fig. 13-15. It can be seen that the noise level increasing leads to a bigger discrepancies. However, all of the reconstruction curves are still very close to the theoretical predictions. Two kinds of error are used to measure the error between the theoretical value and the reconstruction value. The mean absolute error (MAE) and relative error (RE) of a reconstructed signal are defined as:

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$MAE\left(x\right)=\frac{1}{k}\sum _{t=1}^k\left|x\left(t\right)-\widehat{x}\left(t\right)\right|,$

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$RE\left(x\right)=\frac{\sum _{t=1}^k{\left(x\left(t\right)-\widehat{x}\left(t\right)\right)}^2}{\sum _{t=1}^k{\left(\widehat{x}\left(t\right)\right)}^2}\times 100\%.$

Respectively, where $k$ is the length of the reconstructed series, and $x\left(t\right)$ is the reconstructed value at time index $t$ and $\widehat{x}\left(t\right)$ is the theoretical value of the signal. MAE and RE values of the reconstructed responses under different noise levels are listed in Table 5. These two types of errors are very small that proving the high stability of the proposed method in the presence of measurement noise.

Fig. 14Theoretical response and reconstructed response of DOF-2 (2 % noise)

Fig. 15Theoretical response and reconstructed response of DOF-2 (5 % noise)

Table 5MAE and RE (%) values of each DOF in the response reconstruction under different noise levels

Noise level (%)	Error	DOF number
		2	5
0	MAE	1.61	0.76
	RE	2.75	2.53
2	MAE	2.11	1.28
	RE	6.26	2.64
5	MAE	3.17	2.21
	RE	7.48	3.42

4.2.3. Case 3: Effect of high damping ratio

Fig. 16Acceleration response of DOF-4 with high damping ratio

To investigate the effect of high damping ratio to the performance of the proposed response reconstruction method, a higher damping coefficient is used in this numerical case. The stiffness damping coefficient $\beta$ is taken as 1.0e-4 (10 times than before). Fig. 16 shows the acceleration response of DOF-4. The amplitude of the signal reduces rapidly due to the high damping ratio. Fig. 17 shows the theoretical prediction and reconstructed result of DOF-2. Fig. 18 is the zoom-in view of a segment (0.2 s-0.5 s) of Fig. 17. MAE and RE values of the reconstructed responses under different damping levels are listed in Table 6. It can be seen that RE of high damping system are larger than low damping system, however, all errors are very small, and the maximum RE is only 5.13 %. MAE of high damping system are less than low damping system due to the less absolute value of high damping system. From these figures and Table 6, good agreement can be observed between the theoretical acceleration and the reconstructed acceleration. This indicates that the proposed method has the capability for handling high damping ratio case.

Fig. 17Theoretical response and reconstructed response of DOF-5

Fig. 18Zoom-in view of a segment (0.2-0.5 s) of Fig. 17

Table 6MAE and RE (%) values in the response reconstruction under different damping levels

Damping level	Error	DOF number
		2	2
Low damping ratio ($\alpha =\text{1}$, $\beta =$1e-5)	MAE	1.61	1.61
	RE	2.75	2.75
High damping ratio ($\alpha =\text{1}$, $\beta =$1e-4)	MAE	1.31	1.31
	RE	5.13	5.13

4.2.4. Case 4: Effect of multiple forces

Fig. 19Theoretical response and reconstructed response of DOF-5 from 1-1.5 s (5 % noise)

The feasibility of the proposed method in the presence of multiple forces acting on different degrees of freedom is also investigated. In real life, persistent stochastic excitation is more common than transient excitation. In this case, two random forces $f_2\left(t\right)$, $f_3\left(t\right)$ are applied at DOF-1 and DOF-4 persistently, respectively. The reconstructed acceleration response and the theoretical response of DOF-5 from 1 s-1.5 s are shown in Fig. 19. The two time-curves are very close even with 5 % noise, indicating very high reconstruction accuracy of the reconstruction method. In addition, different levels of noise have been also added to the measured signals. MAE and RE under different noise levels are listed in Table 7. Results in Table 7 suggest an overall satisfactory agreement between the reconstructed responses and actual responses and they illustrate that the proposed method is still applicable in the presence of multiple forces acting on different degrees of freedom.

Table 7MAE and RE (%) values in the response reconstruction under different noise levels in Case 4

Noise level (%)	Error	DOF number	Noise level (%)
		2	5
0	MAE	1.88	0.86
	RE	2.85	2.51
2	MAE	2.35	1.32
	RE	2.98	2.71
5	MAE	3.41	2.22
	RE	3.80	3.56