# Experimental Verification of Modal Identification of a High-rise Building Using Independent Component Analysis

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## Abstract

Independent component analysis is one of the linear transformation methods based the techniques for separating blind sources from the output signals of the system. Recently, the method has been analytically applied to the identification of mode shapes and modal responses from the output signal of structures. This study aims to experimentally validate the blind source separation using ICA method and propose a novel method for identification of the modal parameters from the decomposed modal responses. The result of the experimental testing on the three-story steel scale model shows that the mode shapes obtained by ICA method are in good agreement with those by the analytical and peak-picking method in the frequency domain. Based on the robust mathematical model, ICA can calculate the natural frequency and damping ratio effectively using the probability distribution function of the instantaneous natural frequency determined by Hilbert transform of the decomposed modal responses and the change in the output covariance. Finally, the validity of the proposed method paves the way for more effective output-only modal identification for assessment of existing steel-concrete buildings.

## Keywords

independent component analysis mode decomposition Hilbert transform instantaneous natural frequency derivative of covariance output-only modal analysis## Abbreviations

*α*basis

*λ*eigenvalue

- ξ
damping ratio

*ϕ*spectral mode shape

- \(\bar{\varPhi }\)
analytical mode shape

- ω
natural frequency

- A
matrix consisting for column vectors

*α*(*i*)*C*transform matrix between output y and state variable

*x**D*diagonal matrix comprised of eigen values

*λ**E*orthogonal matrix comprised of eigen vectors

*kurt*kurtosis

- m
the number of signals

*MAC*mode assurance criteria

- \(\widehat{N}\)
noise

*P*expectation of

*xx*^{T}*P*_{y}covariance of the output

- \(P_{{y_{0} }}\)
unknown initial value of the covariance

*P*_{y}- \(\dot{P}_{y}\)
derivative of covariance of the output

*Re*real part of complex number

*s*original signal

*Sy*standard deviation

*t*time step

*W*linear transform matrix

*x*mixed system response signal

*x*_{H}Hilbert transform of state variable

*x**X*modal response

- \(\bar{x}\)
mean value of

*x**x*^{′}zero mean value of

*x*- \(\dot{x}\)
State variable

*xo*unknown initial value

*y*output signal

*Y*structure’s response

*y*_{H}Hilbert transform of output

*y*

## 1 Introduction

Modal parameters are vital to understand the structure’s behavior. By decomposing the global response into the equivalent SDOF system’s responses corresponding to each mode, the mode shape, natural frequency and damping can be estimated. The modal parameters of existing buildings are employed to update and refine the numerical model for structural analysis and, subsequently, enable localization and assessment of the damage by comparing pre-and post-damage state (Alvandi and Cremona 2006). As for calculating the effective mass of the damper and the input for control algorithm of the active mass damper, the dynamic identification technique is of great importance. For these reasons, a number of modal identification methods have been studied and applied to many fields of engineering. The recent advance of structural health monitoring also encouraged development of Operational Modal Analysis for maintenance of existing buildings using ambient vibration (Zhang and Brincker 2005; Reynders 2012). OMA, also referred to as Output-only Modal Analysis based on mathematical robustness is extensively being exploited to determine the dynamic parameters of structures using only the ambient vibration as input. The classical frequency-domain techniques that employ the contribution ratio of the specific mode to the overall output at the sensor locations are widely used (Brincker et al. 2002). However, the frequency-domain technique has intrinsic uncertainty in the identification of modal parameters since determining modal participation ratio relies on engineer’s decision. One of most well-known methods, Stochastic subspace identification is based on the assumption that the external excitation is represented as a white noise and the variable in its procedure such as determining the order of the system relying on engineering judgement also leads to uncertainties (Peeters and De Roeck 1999). Other recent development in system identification schemes involving structural health monitoring may be found in (Guo and Kareem 2015, 2016a, b; Guo et al. 2016; Hwang et al. 2018).

Principal Component Analysis is another approach to decompose the response signal into modal responses by the linear transformation of the output. PCA employs the normal orthogonal basis determined by the covariance of structure’s response to extract the mode shapes of the structure (Feeny 2002; Han and Feeny 2003). Since PCA also uses the gaussian excitation for input, robustness of linear transform of the method is degraded with respect to non-gaussian excitation and underdetermined case.

Over the last two decades, Independent Component Analysis which is one of the most popular Blind Source Separation techniques has recently become a focused topic of research work due to its high potential in mode decomposition of non-gaussian structural response. ICA was introduced in Lee (1998), Hyvaerinen and Oja (2000), and Hyvaerinen et al. (2001). The method that makes use of cumulative density other than covariance allows one to obtain independent components comprising the structure’s response and determine modal parameters at the location of the limited number of sensors. It has recently become the focus of intensive research work due to its high potential in many applications. The extensive application of ICA can be found in image processing (Fortuna and Capson 2002), biomedical data analysis (Cichocki 2004), and telecommunication (Madhow 1998). Several applications in structural dynamics are presented in the literature. Zang et al. (2004) demonstrated the result of simulated damage detection of the truss and frame structure using ICA. Poncelet et al. (2007) presented the robustness of the proposed BSS methods for the simple and moderately damped systems. Zhou and Chelidze (2007) proposed BSS-based mode shape extraction and illustrated its performance by comparing its result to that of the time-domain analysis. Hazra et al. (2010) pointed out the limited performance of ICA under the certain level of damping presence and proposed a new method based on modified cross-correlation. As a statistical measure of independence of the components, kurtosis is usually employed to separate independent components (McNeill and Zimmerman 2010; Wu 2011). However, ICA faces difficulties with closely spaced modes and the highly damped system cases subject to non-stationary ambient excitation. Yang and Nagarajaiah (2013) and Nagarajaiah and Yang (2013) proposed the improved ICA techniques employing complexity pursuit algorithm, short time Fourier transform in time–frequency domain, respectively. The further modification of ICA for the particular case of the non-proportionally damped structures is verified in (Nagarajaiah and Yang 2015).

In most OMA techniques, extracting modal properties other than the mode shape requires a post-process. To address this challenge, revised fixed-point complex ICA is presented in (Yang et al. 2013). Another application of ICA is found in Structural Health monitoring and damage identification. The long-term monitoring response is processed in wavelet-domain before ICA to capture the time varying modal parameters (Yang and Nagarajaiah 2014) and the SHM data is compressed and transferred using Fast ICA algorithm (Yang et al. 2015). The latest development and application of ICA based modal identification methods are summarized in Sadhu et al. (2017).

In this paper, the algorithm of ICA for modal parameters extraction and the experimental modal analysis of the high-rise building subjected to strong wind load is discussed within the ICA framework. Since the analytical approach is limited to be applied to the response containing nonlinearity and low signal-to-noise ratio, the experimental evaluation is crucial to examine the applicability of ICA to the real structure’s response.

First, a three-story scale model made of steel elements are used as a preliminary validation of the ICA algorithm. Assuming the input excitation is unknown, the mode decomposition of the response measured at each floor by the accelerometers is carried out. At the same time, the experimental testing shows that the linear combination matrix used in mode decomposition is equivalent to the mode shape of the structure. Subsequently, the natural frequency and damping ratio is obtained from the decomposed modal responses. The same procedure is repeated for the accelerations measured at the top floor of the high-rise building to examine the close modes separation performance of the ICA technique. Lastly, the modal parameters identified by ICA are compared with those by a conventional method.

This paper is organized as follows. The principle of ICA is explained in Sect. 2. Section 3 presents the modal analysis of the scale model and tall building followed by introducing and applying the effective modal parameter extraction method (Lee et al. 2017; Du et al. 2017). In Session 4, finally, the results of the modal analysis on the examples are discussed comparing with other classical dynamic identification methods and the concluding remarks are presented.

## 2 Basic Principles of ICA

### 2.1 Fundamentals of ICA

*α*(

*i*) is basis and component of \(A = \, [ \alpha \left( 1 \right), \alpha \left( 2 \right), \ldots \ldots , \alpha \left( m \right)]\). In short, ICA is a solution for the inverse matrix

*A*

^{−1}using the observed data that is the measured mixed signal

*x*. The procedure of ICA is displayed in Fig. 2. As shown, the observed output of the mixer is available, while the mixing characteristic

*A*and the original input

*s*are unknown. The result of the analysis shall satisfy the fact that the input coincides with the output

*y*,

*W*and

*A*satisfying the prior condition is given by

*y*must be statistically independent to each other and the orthogonality condition does not hold. Consequently, original signal

*s*is approximated by determining

*W*which can be calculated by optimization of its associated objective function. To facilitate the ICA, the pre-processed mixed signal considering the first and second-order statistical correlation is employed.

### 2.2 Pre-processing of ICA

*x*identity, namely

*E*(

*xx*

^{T}) =

*I*.

*V*is defined as

*D*is the diagonal matrix comprised of eigen values

*λ*and

*E*is the orthogonal matrix comprised of eigen vectors.The advantage of PCA whitening is that this can be realized by the well-known commercial software and, moreover, it performs well in estimating the number of independent components that are the original individual signals. In this study, those processes are used in ICA to extract the displacement of the structure and mode shape.

### 2.3 Algorithm of ICA

*X*is the vector composed of the non-Gaussian random variables representing the time history of the independent components and

*E(O)*is the expectation operator. In this study, ICA method that employs Kurtosis as the objective function and its application to mode decomposition from the structure’s response are discussed. The linear relation between the structure’s response and modal response is as bellow

*Y*is the structure’s response,

*X*is the modal response, and

*W*is the linear transform matrix (mode shape). The key of ICA method is that the unknown variables

*X*and

*Y*are determined by the measurement

*Y*. If

*W*is a unitary square matrix and the inverse matrix

*W*

^{−1}exists, modal response

*X*can be expressed as bellow

*W*and

*X*can be determined by an iterative method minimizing the objective function in (6) until each mode is mutually independent. Equation (9) presents the updated matrix

*W*.

_{i+1}is the updated matrix W from the past W

_{i}and the cubed term (3) in (9) stands for the cubed each element in the row vector \(Y^{T} W_{i}\). Based on the assumption that the each of decomposed modes are equivalent to the response of the corresponding single degree of freedom systems, the natural frequencies and damping ratio are calculated. The viability of the ICA algorithm is evaluated in the following session with two examples.

## 3 Experimental Testing

### 3.1 Three-story Steel Frame

Dynamic parameters of the structure.

Floor mass | m = 18.62 kg | ||

Floor stiffness |
= 7.7945 kN/m (L = 400 mm) = 11.635 kN/m (L = 350 mm) | ||

Mass matrix | \(\left[ {\begin{array}{*{20}c} m & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & m \\ \end{array} } \right]\) | Stiffness matrix | \(\left[ {\begin{array}{*{20}c} {2k} & { - k} & 0 \\ { - k} & {2k} & { - k} \\ 0 & { - k} & k \\ \end{array} } \right]\) |

Modal parameters | Mode 1 | Mode 2 | Mode 3 |

Natural frequency | |||

L = 400 | 1.4492 Hz | 4.0605 Hz | 5.8676 Hz |

L = 350 | 1.7706 Hz | 4.9610 Hz | 7.1689 Hz |

Mode shape normalized to the top floor | 0.4450 | − 1.2467 | 1.8026 |

0.8021 | − 0.5547 | − 2.2474 | |

1.0000 | 1.0000 | 1.0000 |

The analytical modal analysis of the structure represents different natural frequencies, but uniform mode shapes. The errors between the results with two different column lengths for the first three modes are uniform equal to approximately 20%. The result is consistent with the stiffness increased by about 50% when the short columns are taken into account without change in mass. The uniform stiffness of all floors accounts for the identical mode shapes for two column lengths.

Accelerometer and impact hammer.

Sensitivity | 0.23 mV/N |

Frequency range | 0–5 kHz |

Amplitude range | 22 kN |

Hammer mass | 0.32 kg |

Head diameter | 25 mm |

Hammer view |

Modal parameters by spectral analysis.

Mode 1 | Mode 2 | Mode 3 | |
---|---|---|---|

Natural frequency | 1.7322 Hz | 4.9824 Hz | 7.2504 Hz |

Normalized mode shape | 0.4649 | − 1.2203 | 1.4396 |

0.8278 | − 0.4459 | − 2.0374 | |

1.0000 | 1.0000 | 1.0000 |

*Φ*is spectral mode shape and \(\bar{\varPhi }\) is analytical mode shape. The MAC values of the 1st, 2nd and 3rd mode shape are 1.6% and 3.26% and 12.11%, respectively. The peak picking method, however, in frequency domain holds significant uncertainties in practice. In this study, thus, Hilbert transform (Feldman 2011) together with ICA is exploited to determine the modal frequency and subsequently the robustness of this method is evaluated comparing with the classical methods such and pick peaking.

*Y*is the relative acceleration generated by the impact on the top floor of the steel structure with respect to the support at the bottom. The values of elements in the linear transform matrix

*W*in Eq. (9) are those that are converged within the tolerance in the iterative process of updating the linear transform matrix

*W*. In Table 4, the mode shapes of the structure are presented.

*W*that consists of the mode shapes is unitary matrix and the column vectors are normalized for elements corresponding to the top floor to have unity. The difference in MAC values between ICA and analytical method is greater than that between ICA and spectral analysis and the discrepancy increases in the higher modes. Figure 6 displays the modal responses and spectral contents around each mode obtained by ICA. The decomposed time history by the three methods mentioned in this example are in good agreement with each other despite a small discrepancy in the mode shape amongst the methods. Meanwhile, the analytical spectrum shows slight discrepancy from those of the spectral analysis and ICA. However, those three methods identify three different modes effectively since the log-scale amplitude difference in three peaks lies between two and three orders in each mode’s power spectrum plot.

The mode shapes by ICA method.

Method | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

ICA (1st–2nd–3rd floor) | 0.4702 | − 1.2009 | 1.4306 |

0.8216 | − 0.4192 | − 1.8688 | |

1.0000 | 1.0000 | 1.0000 | |

Analytical | 0.4450 | − 1.2467 | 1.8026 |

0.8021 | − 0.5547 | − 2.2474 | |

1.0000 | 1.0000 | 1.0000 | |

MAC ICA Vs. Analytical | 1.46% | − 4.62% | − 16.36% |

Peak picking(PP) | 0.4649 | − 1.2203 | 1.4396 |

0.8278 | − 0.4459 | − 2.0374 | |

1.0000 | 1.0000 | 1.0000 | |

MAC ICA Vs. PP | − 0.14% | − 1.33% | − 4.93% |

Next, the natural frequency of identified modes is calculated. In order to calculate modal parameters, it is very useful to understand that the response of decomposed mode is equivalent to the SDOF system. In this sense, the natural frequency is obtained from the well-distributed peaks in the spectra in Fig. 6. However, it is not always simple to pick the peaks in practice due to the intrinsic uncertainties in the spectrum. Thus, Hilbert transform that employs the probability distribution of time-varying frequencies is used to determine the natural frequency.

The identified natural frequencies.

Method | Mode 1 [Hz] | Mode 2 [Hz] | Mode 3 [Hz] |
---|---|---|---|

ICA (Cumulative distribution) | 1.7357 | 4.8043 | 7.1150 |

Peak-picking | 1.7322 | 4.9824 | 7.2504 |

Error | 0.2% | − 3.57% | − 1.87% |

*x*

_{o}is the unknown initial value,

*A*is the system matrix expressed as

*ω*and

*ξ*are the natural frequency and damping ratio of each mode, respectively. If the covariance of the system response

*x*is \(P = E [xx^{T} ]\), the relationship between

*P*and

*A*can be written as

*y*can be expressed as

*y*and the state variable

*x*can be also expressed as

*λ*of the system matrix that satisfies the relationship \(CAx = \lambda Cx\) is present, the covariance including the Hilbert transform can be expressed as

*Re*(

*λ*) is the real part of

*λ*and is equal to −

*ωλ*. Consequently, the covariance of the output

*P*

_{y}and its square root which is standard deviation

*Sy*can be obtained.

*P*

_{y}.

Damping estimation using the derivative of the covariance.

Method | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|

Covariance | 0.25% | 0.35% | 0.3% |

Logarithmic decrement | 0.24% | 0.36% | 0.29% |

Error | + 4% | − 3% | 3% |

### 3.2 High-rise Building Subjected to Strong Wind Load

In recent tall buildings, coupled torsional-translational behavior is observed. As the structure is subjected to unidirectional translational wind load, for instance, usually two translational responses in parallel and perpendicular direction to the load and torsional response are observed. To be specific, a few lower modes are blended in torsional-translational modes and those individual modes other have different contribution to the global behavior of the structure. The natural frequencies of those modes comprising the coupled behavior are often so close that the modal identification requires more delicate approach to separate them clearly (Kareem 1985).

In this example, the ICA technique discussed above is applied to decompose the close modes in the coupled motion of the existing 39-story building and validation of the technique is performed. For dynamic identification, the acceleration signals recorded at three different locations on the top floor while a typhoon took place are used.

^{2}and it is observed that waveforms in X and Y direction are slightly different. Next, X- and Y-direction vibrations at two locations recorded in the afternoon are displayed in Fig. 12. Instead of the vertical Z-direction response, two Y-direction time history were recorded to observe the torsional behavior of the plane. As shown, the amplitude of the Y-direction responses is greater than that in X direction and the Y-direction acceleration of A near the edge is greater than that of B. The peak acceleration in this period is 3.94 cm/sec

^{2}.

*A*is required for the outcome of ICA to be converted into the acceleration with unit. The components of transform matrix A and corresponding mode shapes are presented in Table 7 and Fig. 16, respectively. Very little effect of torsion is found in mode 1 and 2. Regarding translational behavior, mode 1 represents almost equal effect on both X-and Y-direction response, while the effect in Y direction is pronounced in the second mode shape. As predicted, a remarkable torsional mode shape is shown in mode 3.

Transform matrix and mode shape.

Mode 1 | Mode 2 | Mode 3 | ||
---|---|---|---|---|

Transform matrix A | X | − 0.2036 | − 0.1065 | 0.0649 |

Y1 at B | − 0.2011 | 0.2060 | − 0.1505 | |

Y2 at A | − 0.2058 | 0.2187 | − 0.3624 | |

Transform relationship | Z = AS where S: dimensionless response by ICA |

## 4 Conclusions

- 1)
Based on the robust mathematical model, ICA is an effective method to evaluate mode shapes from the output-only signal of the structure. The mode shapes obtained by ICA method agree with those by the analytical and peak picking methods.

- 2)
Two modal identification examples show that the ICA technique allows one to decompose structural response into individual modes even if the modes are very close. The natural frequency and damping ratio can be also calculated from each identified mode by a statistical approach. Closely spaced modes in a coupled torsional translational behaviour induced by non-gaussian excitation such as strong wind loads on a high-rise building are successfully identified. The ICA and Hilbert transform-based scheme can identify explicitly modal parameters of existing buildings.

- 3)
Another advantage of using ICA is verified by reversing the separate modes obtained by ICA. The restoring of the outcome of ICA results in the original output signal without any deterioration in the data quality.

- 4)
Complementary study to reduce the statistical error in the estimation of natural frequency due to unknown dynamic properties the excitation.

## Notes

### Authors’ contribution

JH improved the ICA method and programmed its algorithm for the analysis used in the examples. JH, JN and SL conducted experimental studies, outlined the structure of the paper and drafted the manuscript. AK participated in the review and verification of the proposed methods in this paper and drafting the manuscript. All authors read and approved the final manuscript.

### Acknowledgements

This research was financially supported by the Basic Science Research Program of the National Research Foundation of Korea (NRF), Grant No. NRF-2017R1D1A1B03031265 and No. NRF-2015R1A2A1A10054506. The third author was in part supported by NSF Grant No. GMMI1612843.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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