- Email: [email protected]
Experimental identification of closely spaced modes using NExT-ERA
Journal of Sound and Vibration 412 (2018) 116e129
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Experimental identification of closely spaced modes using NExT-ERA S.A. Hosseini Kordkheili a, *, S.H. Momeni Massouleh a, S. Hajirezayi a, H. Bahai b a
Center of Research and Development in Space Science and Technology, Aerospace Engineering Department, Sharif University of Technology, Azadi Avenue, P.O. Box: 11365-9567, Tehran, Iran b Department of Mechanical, Aerospace & Civil Engineering, Brunel University, Uxbridge, Middlesex, UB8 3pH, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 May 2016 Received in revised form 17 September 2017 Accepted 26 September 2017
This article presents a study on the capability of the time domain OMA method, NExT-ERA, to identify closely spaced structural dynamic modes. A survey in the literature reveals that few experimental studies have been conducted on the effectiveness of the NExT-ERA methodology in case of closely spaced modes specifically. In this paper we present the formulation for NExT-ERA. This formulation is then implemented in an algorithm and a code, developed in house to identify the modal parameters of different systems using their generated time history data. Some numerical models are firstly investigated to validate the code. Two different case studies involving a plate with closely spaced modes and a pulley ring with greater extent of closeness in repeated modes are presented. Both structures are excited by random impulses under the laboratory condition. The resulting time response acceleration data are then used as input in the developed code to extract modal parameters of the structures. The accuracy of the results is checked against those obtained from experimental tests. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Closely spaced modes Time domain OMA method NExT-ERA method Correlation function Experimental identification
1. Introduction Repeated modes or closely spaced modes which occur in structures such as circular plates, gears or bladed-discs that have some degrees of symmetry. From an experimental point of view, closely spaced modes are two close modes that are sometimes misinterpreted as one single mode. Failure in identification of closely spaced modes, which are normally associated with pseudo-repeated root problem [1], during dynamic analysis may lead to critical effects in dynamic systems such as pogo-effect in Titan 2 space rocket [2]. Therefore the possibility of occurrence of closely spaced modes during modal analysis process is inevitable. Moreover, in model updating process one needs an experimental mode corresponding to each of those identified analytically. This requirement dictates that all the existing modes, including those which are closely spaced, have to be extracted employing more adapted techniques. During the last four decades different algorithms have been proposed to extract modal parameters of structures by researchers. These algorithms, which are basically developed in time, frequency, or space domains, are categorized into single input-single output, single input-multi output or Multi Input-Multi Output (MIMO) methods. Among these, frequency and time domains are considered to be more practical. Time domain methods which directly use experimentally recorded time
* Corresponding author. E-mail address: [email protected] (S.A. Hosseini Kordkheili). https://doi.org/10.1016/j.jsv.2017.09.038 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
117
history data avoid some of the problems (such as leakage) which are normally associated with the Fourier transform in the frequency domain methods. On the other hand, in time domain methods a reasonable signal to noise ratio suffices to extract modal properties without the necessity of measuring excitation force functions [3]. Avoiding FRF calculations in time domain algorithms together with its superior damping estimation [4] makes these methods more suitable to analyze closely spaced modes of structures. However, some frequency domain methods have been successfully employed to extract the closely spaced modes of structures [5e7]. As reported in Ref. [8], taking into account the response at different DOFs, MIMO procedures are better candidates to analyze complex structures with closely spaced or even repeated modes. In 1982, Vold et al. [9] proposed Polyreference Complex Exponential (PRCE) method which is considered as the first developed MIMO algorithm. Two other well-known time domain MIMO modal identification techniques which were developed later are the Extended Ibrahim Time Domain (EITD) [10] and Eigensystem Realization Algorithm (ERA) [11]. It has been reported that Experimental Modal Analysis (EMA) methods, even the MIMO ones, face difficulty when dealing with close modes [5]. Because in these methods due to the point excitation, the identification of closely spaced modes highly depends on the excitation technique and the excited region. Moreover, the methods have shown inconsistency in calculating precise damping ratios when it comes to close modes. While, unlike methods in EMA, due to the broadband random excitation in real loading situation, which acts through different points of the structure, OMA methods can excite all vibration modes. Additionally, because of their MIMO nature, the OMA methods are more accurate in studying the closely spaced modes of structures. As mentioned earlier, in time domain methods there is no need to record excitation data. This notion, accompanied with the fact that Operational Modal Analysis (OMA) only requires output data due to natural or ambient loading, was the motivation in this study to employ OMA in conjunction with time domain schemes. For modal identification of structures, OMA methods may be employed for the whole structure at once without any need to elaborate excitation equipment. In 1992 the Natural Excitation Technique (NExT) was introduced [12] to process the random response data of the structure, so that the results may be used as an input to MIMO-type time domain EMA algorithms such as EITD, ERA and PRCE. Indeed, NExT in combination with any MIMO time domain algorithm can be considered as an OMA method in its own right. For example Caicedo [13,14] used NExT-ERA to extract the modal parameters of four-story steel structures. Among all the time domain methods in modal analysis, ERA has proved to have great accuracy and is considered as one of the most robust and successful methodologies for engineering practices [15e17]. Several studies have also been conducted to evaluate the accuracy of the procedure for various systems, as well. Siringoringo [17] exploited the method for system identification of a suspension bridge. Chiang and Lin [18] studied the accuracy of a modified version of ERA method for coupled simulated mass spring systems. Zhang et al. [19] used the methodology for extracting the modal properties of a bridge model. The method has also proved useful for structural health monitoring purposes [13,20]. Moreover, many researches have been conducted to study the effect of close modes on performance of different methods in modal analysis. The efficacy of FDD [5] was investigated in the case of close modes by Brinker et al., just when it was developed. and Agneni [21] extracted the close modes of a AB-204 helicopter blade using Hilbert transform method and frequency domain decomposition. Ranieri [22] also studied the effectiveness of OMA time domain method and second order blind identification in case of close modes. Brincker et al. [23] also introduced a modified version of EITD to extract repeated modes of numerical structures using output data only. Chen [24] studied the efficiency of the analytical mode decomposition method in case of closely-spaced modes, as well. The concept of closely-spaced modes has proved so critical today that every new system identification technique is evaluated in the case of close modes just when it is proposed [6,7,25,26]. NExT-ERA has also been employed for identification of closely-spaced modes of structures. Caicedo [14] managed to extract the lowfrequency close modes of a benchmark structure by NExT-ERA. In Siringoringo's study [17] the method was also employed for system identification of a suspension bridge. In his study, the structure had some low frequency closely-spaced natural frequencies which were successfully identified by NExT-ERA. This research aims to assess the accuracy of the time domain method ERA in corporation with NExT for identification of close modes in higher frequencies, where noise is a more critical issue and makes the identification process more difficult. Moreover, for symmetric plate-like structures there is some difficulty in identifying accurate mode-shapes [28], the NExT-ERA method is also evaluated in such a case, as well. For this purpose, first a simulated mass-spring system and a numerically modeled beam are studied to verify the procedure and formulation. The effectiveness of the method is then evaluated for a plate having closely spaced modes. Finally two separate experiments are carried out on a plate and a pulley ring with closely spaced modes and the effectiveness of the method is studied experimentally.
2. Eigensystem Realization Algorithm ERA was developed by Juang and Pappa [11] based on the concepts that predominantly originated form the control theory, so that it differs from the usual developments found in the EMA literature. In ERA technique, using dynamic equation of _ equilibrium for an N-DOF viscously damped system together with the state vector fuðtÞg ¼ fyðtÞ; yðtÞg, in which yðtÞ is displacement vector, the output of the system at kth time sample is written as fxðkÞg ¼ ½RfuðkÞg, where fuðkÞg ¼ ½Afuðk 1Þg þ ½Bfdðk 1Þg, dðkÞ is input vector at q locations and [A] and [B] are discrete-time state-space matrices which are determined in a process called realization. {x(k)} is a vector of dimension p which is the number of output
118
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
locations for the experiments. Now, consider that the structure is excited at q coordinates by separate impulses. Here the Markov parameter matrix [X(k)] is defined for the response of structure at kth time step as Xij ðkÞ ¼ xij ðkÞ [27], where xij ðkÞ is the displacement, velocity or acceleration response at ith coordinate due to the impulse at jth coordinate at kth time sample. Assuming stationary structure initial condition as well as impulse input excited at t ¼ 0 i.e. dðtÞ ¼ dð0Þ in state space we obtain
½XðkÞ ¼ ½R ½Ak1 ½B |fflfflffl{zfflfflffl} |{z} |fflfflffl{zfflfflffl} |{z} pq
(1)
p2N 2N2N 2Nq
where [R] is transformation matrix. According to this process these matrices are estimated from the response of the structure based on a minimum realization algorithm. As the next step, the Hankel matrix is constructed having the response data of the structure at different times as shown below
2
½XðkÞ 6 ½Xðk þ 1Þ 6 Hðk 1Þ ¼ 4 « |fflfflfflfflfflffl{zfflfflfflfflfflffl} ½Xðk þ r 1Þ prqs
½Xðk þ 1Þ ½Xðk þ 2Þ « ½Xðk þ rÞ
3 … ½Xðk þ s 1Þ 7 … ½Xðk þ sÞ 7 5 1 « … ½Xðk þ r þ s 2Þ
(2)
This matrix plays a major role in most time domain algorithms. In Eq. (2), r and s are large enough integers so that the rank of matrix cannot be increased by increasing their numbers further. Using an algorithm proposed in Ref. [27], it can be shown that the matrices [R], [A] and [B] can be defined as follows
½R ¼
h
i T 1 Ep ½U2N ½S2N 2
(3)
h i 1 1 ½A ¼ ½S2N 2 ½U2N T ½Hð1Þ½V2N ½S2N 2 h
1 ½B ¼ ½S2N 2 ½V2N T Eq
(4)
i
(5)
where,
T T Ep ¼ ½ ½I ½0 … ½0 ; Eq ¼ ½ ½I ½0 … ½0 |fflffl{zfflffl} |fflffl{zfflffl}
ppr
(6)
qqs
also ½U2N , ½V2N and ½S2N are matrices containing only the first 2N columns of matrices ½U, ½V and ½S which are calculated using singular value decomposition of matrix [H (0)], i. e.
½Hð0Þ ¼ ½U ½S ½VT |fflfflffl{zfflfflffl} |{z} |{z} |ffl{zffl} prqs
(7)
prpr prps psqs
Finally solving the eigenvalue problem ½AfJu g ¼ lfJu g leads to the modal parameters of the system. The mode shapes of the structure at output coordinates can be computed using fJx g ¼ ½R fJu g . |fflfflffl{zfflfflffl} |{z} |fflfflffl{zfflfflffl} p1
p2N 2N1
3. Natural Excitation Technique NExT is one of the OMA methods that can be used in conjugation with EMA time domain identification algorithms to extract modal properties of structures. According to NExT, the correlation (COR) function of the structure response under ambient loading or white noise can be written as the summation of decaying sinusoids. The modal parameters of each sinusoid i.e. natural frequency, modal damping and mode shape coefficient are equivalent to the modal parameters of each mode of the structure. Thus, the conventional MIMO time domain identification algorithms in EMA, such as ERA, EITD and PRCE use the NExT product to extract the modal parameters in a way that COR functions are used instead of impulse response function. In this method the response at coordinate i due to input force f(t) at kth coordinate can be written as
xik ðtÞ ¼
N X
Z 4ir 4kr
r¼1
t
∞
f ðtÞgr ðt tÞdt
(8)
where gr ðtÞ ¼ m1ud ezr ur t sinðudr tÞ and 4ir is the ith component of rth mode shape and udr is the rth damped natural frequency. n
r
r
N denotes the total number of modes. When f(t) is a random white noise function, the correlation function Rij between the
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
119
Fig. 1. Simulated mass spring system.
response of output coordinates i and j is defined as Rij ðTÞ ¼ E½xik ðt þ TÞ xjk ðtÞ, where E is the expectation operator. Using the algorithm proposed in Refs. [12,17] it can be deduced that
Rij ðTÞ ¼
N X 4ir Ajr r¼1
mr udr
n ezr ur T sin udr T þ qr
(9)
It can be seen that the correlation function can be written as summation of decaying sinusoids as the same form of Impulse Response Function. Thus, COR function Rij ðTÞ between the response coordinates i and j can be considered as the response at output coordinate i due to impulse applied at coordinate j for MIMO time domain algorithms. For discrete time responses of two output coordinates the correlation function can be calculated from
Rij ðmÞ ¼
1 M
Mmþ1 X
xi ðnÞxj ðx þ m 1Þ
(10)
n¼1
where M is the total number of time history data and increasing in M yields better accuracy in COR function calculation. The COR function for NExT can also be calculated using cross-spectral density function and taking its inverse Fast Fourier transform [14]. Based on the discussions made above the step-by-step algorithm for employing NExT-ERA is as follows: Recording the structure response due to the ambient excitation in time domain at different measurement points Calculating the correlation function (Rij ) using NExT Treating the correlation functions like the impulse response functions for utilization in ERA (or any other MIMO time domain method) Averaging and extracting the modal parameters
4. Numerical results and discussions The previously presented formulation is applied in a code developed to identify the modal parameters of different systems using their experimentally or numerically generated time history data. In order to examine the capability of the developed code some investigations on discrete and continuous numerical systems are conducted and the results are reported. 4.1. Simulated mass spring system For this part a simple mass spring system, consisting of 5 masses as shown in Fig. 1, is simulated. Each mass weighs 1 kg, the springs connecting the mass have a stiffness of 1000 N/m and a damping of 1 N s/m. The system is excited by a couple of impulses which acted through some random DOFs of the system and the time history response data of the masses are
Table 1 Modal parameters of the mass spring system. Method
Actual
NExT-ERA (Present Work)
Actual
NExT-ERA (Present Work)
Modal parameter
u (rad/s)
u (rad/s)
Err (%)
z (%)
z (%)
Err (%)
Mode Mode Mode Mode Mode
28.46 83.08 130.97 168.25 191.90
28.46 83.08 130.97 167.25 191.90
0.00 0.00 0.00 0.00 0.00
0.14 0.42 0.65 0.84 0.96
0.25 0.42 0.65 0.84 0.96
73.33 0.00 0.00 0.00 0.00
1 2 3 4 5
120
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Table 2 Damping ratios extracted by NExT-EITD and NExT-PRCE using different recording length. Method
NExT-EITD
Number of Data 500
NExT-PRTD 2000
8000
Parameter
z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%)
Mode Mode Mode Mode Mode
1.4230 0.6027 0.6740 0.8464 0.9626
900.40 45.08 2.92 0.60 0.32
160.99 0.29 0.29 0.43 0.29
8.22 0.07 0.29 0.43 0.29
0.49 0.07 0.29 0.37 0.29
780.67 33.40 1.91 0.52 0.32
168.80 0.29 0.29 0.44 0.31
1.69 0.07 0.29 0.44 0.31
0.00 0.07 0.29 0.44 0.31
1 2 3 4 5
2000
8000
0.3714 0.4166 0.6568 0.8450 0.9623
0.1540 0.4157 0.6568 0.8450 0.9623
15000
0.1430 0.4157 0.6568 0.8444 0.9623
500
1.2532 0.5541 0.6674 0.8457 0.9626
0.3825 0.4166 0.6568 0.8450 0.9625
0.1447 0.4157 0.6568 0.8450 0.9625
15000
0.1423 0.4157 0.6568 0.8450 0.9625
Fig. 2. MAC diagram for identified mode shapes of the mass spring system by NExT-ERA vs. exact mode shapes.
recorded by sampling rate of 500 Hz. A total number of 8000 time history displacement data is used in computing the correlation function. The resulting modal parameters are shown in Table 1. As seen from this table, the modal damping coefficient for the first mode is not calculated with an acceptable accuracy. Because this is the first natural frequency of the system which has a relatively low value, the recording time needs to be increased to get more accurate result. Subsequently, instead of 8000 data, 24000 displacement data with the same sampling rate is used for the identification procedure. Implementing this number of data into the developed code leads to a damping ratio z ¼ 0:1483 for the first mode which means that the deviation from actual value is decreased to about 4.22%. In accordance with the results presented here, ref [14] points that NExT-ERA can lead to inaccurate results in identifying the low frequency modes if the data length is not long enough. Moreover, other methods such as EITD [23] and PRCE [9] have also shown such a trend in calculating damping ratios. The associated results for these methods are represented in Table 2. The correlation between the actual mode shapes and those extracted by NExT-ERA is also studied by drawing a MAC diagram. Fig. 2 depicted this diagram for the extracted mode shapes and correlation between the resulted mode shapes is verified. 4.2. Numerical case of a randomly excited cantilever beam In order to evaluate the accuracy of modes extracted by the developed code for NExT-ERA, a cantilever steel beam is modeled using a finite element software i.e. Patran. The beam has a length of 1 m and a square cross-section of 20 mm 20 mm. A hundred beam elements are used in the FE simulation. The modal parameters of the beam are computed first by imposing a conventional normal mode solution. Then the beam, for which a structural damping coefficient of 1.2% is Table 3 Modal parameters of the beam. Method
Actual
NExT-ERA (Present Work)
No. of Response nodes
100
No. of acceleration history data
1000
Actual
1500
NExT-ERA (Present Work)
5
100
1500
1000
5 1500
1500
Modal parameter
u (Hz)
u (Hz)
Err (%)
u (Hz)
Err (%)
u (Hz)
Err (%)
z (%)
z (%) Err (%) z (%) Err (%) z (%) Err (%)
Mode Mode Mode Mode
16.76 104.86 292.91 571.99
16.72 104.86 292.91 572.01
0.24 0.00 0.00 0.00
16.77 104.86 292.91 572.01
0.06 0.00 0.00 0.00
16.84 104.86 292.90 572.16
0.48 0.00 0.00 0.03
0.60 0.60 0.60 0.60
1.52 0.62 0.60 0.60
1 2 3 4
153.33 3.33 0.00 0.00
1.35 0.60 0.60 0.60
125.00 0.00 0.00 0.00
1.91 0.60 0.60 0.61
218.33 0.00 0.00 1.67
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
121
Table 4 Modal parameters of plate (C denotes close mode). Method
Actual
NExT-ERA (Present Work)
Actual
Modal Parameter
u (Hz)
u (Hz)
Err (%)
z (%)
z (%)
Err (%)
Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode
20.29 28.99 36.84 50.95 52.12 87.69 90.83 93.31 98.21 112.77 147.17
20.29 28.99 36.84 50.95 52.12 87.69 90.83 93.31 98.21 112.77 147.17
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
0.72 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
2.86 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 2 3 4C 5C 6 7C 8C 9 10 11
NExT-ERA (Present Work)
Fig. 3. MAC diagram for identified mode shapes by NExT-ERA vs. FEM mode shapes (numerical plate).
considered, is excited by a couple of successive random impulses and the transient response of the beam is recorded by a sampling frequency of 2000 Hz. For this part of the analysis a transient response solution is used by applying an impulse excitation with equilateral triangle shape with 100 N maximum amplitude and 0.002 s duration. A total number of 1000 acceleration history data, at 100 nodes of the model is used for calculating COR function. Inserting these recorded data into the developed code results in the identified modal parameters of the beam as shown in Table 3. It can be seen again that the modal parameters of the basic modes of structure are not calculated accurately. As demonstrated in the first example increasing the number of response data will contribute to achieving more accurate damping ratios for the first two modes of the structure. Accordingly, the number of acceleration time history data is increased to 1500 with the same sampling rate. In this case, as can be observed in Table 3, the results for damping coefficients of the first
Fig. 4. eTwo repeated mode shapes of the tested plate identified by finite element analysis.
122
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Fig. 5. Schematics of (a) EMA and (b) OMA test setups.
two modes are converging to actual values. Practically measurement of responses at 100 different locations in a beam is impossible during real experiments. Consequently, we adopted another numerical simulation for the same beam with only 5 output coordinates. According to the results of this solution which are shown in Table 3, the modal parameters are still estimated in an acceptable range. 4.3. Numerical case of a randomly excited plate with close modes In this section a plate with closely spaced modes is modeled using the same FEA software. The structure analyzed is a square 400 mm 400 mm steel plate with a thickness of 1 mm and a 50 g mass at a corner. The plate is modeled for free-free boundary condition. Being symmetric, the plate will definitely possess a few pairs of repeated eigenvalues (and hence repeated natural frequencies and damping ratios) with coupled mode shapes. The added mass would disturb the symmetry of structure slightly so that the paired and formerly similar natural frequencies will diverge from each other by a small amount, creating closely spaced modes. The plate is modeled by 1.6 1.6 cm isometric QUAD shell elements in Patran, so the total number of 2D elements reaches 525. To ensure the existence of close modes, the modal properties of the model are identified
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
123
Table 5 Modal parameters of the tested plate. Method
EMA (Experiment)
NExT-ERA (Present Work)
EMA (Polymax) (Experiment)
Modal parameter
u (Hz)
u (Hz)
z (%)
z (%)
Mode Mode Mode Mode Mode
117.94 178.64 201.71 293.96 296.82
117.60 178.09 201.10 292.64 295.83
0.38 0.06 0.17 0.17 0.14
0.32 0.05 0.14 0.17 0.13
1 2 3 4 5
NExT-ERA (Present Work)
Fig. 6. eCoherence diagrams obtained by EMA test of the pulley ring.
using a simple normal modes analysis first. Next, the plate is subjected to several successive impulses at random coordinates and the response of structure is obtained by the use of transient solution. The acceleration response data are recorded at a rate of 2000 Hz. A structural damping coefficient of 1.4% is also used in the transient response analysis. The results shown in Table 4 and Fig. 3 are obtained by using 5000 time history data at 16 response points of the model. In order to assess the extent of closeness of two modes the Modal Overlap Factor (MOF) mn ¼ un zn =ðun un1 Þ is considered. It is noted that two modes with MOF value above 0.17 can be considered moderately close [22]. In the case of our numerical plate the MOF value is about 0.31 which is higher than the closeness criteria. So there are two pairs of close modes among the first eleven modes of the structure which are determined with acceptable accuracy. Again, increasing the number of recorded data will lead to more accurate results in lower frequency modes. 5. Experimental results and discussions 5.1. Test setup and results Tests are conducted on two structures, a plate and a pulley ring both of which are tested under a free-free boundary condition and are suspended by simple strings. A 34 channel NI PXIe 1082 data acquisition system along with PCB piezoelectric accelerometers (with 100 mV/g sensitivity and 10 kHz frequency range) and a PCB force sensor (with 1124 mV/kN sensitivity and 4.448 kN measurement range) are used to measure the response of the structure. The first test consisted of a 300 mm 305 mm rectangular steel plate which is studied in modal laboratory condition. The thickness of the plate is 3 mm and since the sides of the structure only slightly differ (5 mm), the plate will have closely spaced modes. In order to confirm this assumption, a simple finite element model of the plate is created and the approximate natural frequencies and mode shapes of the structure are identified. It is revealed that this structure does possess pairs of closely Table 6 Modal parameters of the tested pulley ring. Method
EMA (Experiment)
NExT-ERA (Present Work)
EMA (Polymax) (Experiment)
Modal parameter
u (Hz)
u (Hz)
z (%)
NExT-ERA (Present Work)
z (%)
Mode 1 Mode 2 Mode 3
1549.59 1635.52 1645.14
1549.31 1633.09 1643.39
0.05 0.19 0.15
0.03 0.17 0.13
124
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
spaced modes or repeated mode shapes (Fig. 4). Now, an FRF based roving hammer test is performed to measure the actual modal properties of the plate structure. Schematic of this test is depicted in Fig. 5a. Sixteen points on the plate are marked and hammer impacts are acted through these sixteen locations of the plate. For the EMA test a simple hammer, to which the force sensor is attached, is used to excite the structure with a steel tip. Modal View software is also utilized to analyze the collected data in this test. The damping ratios of the plate are extracted by Polymax method. Based on the results attained by EMA hammer test (Table 5) the MOF for the close modes of the structure is about 0.17, so it can be deduced that the modes are moderately close. In addition, to perform the OMA NExT-ERA test as the structure is quite light attaching sixteen concurrent accelerometers would not be appropriate since the added mass will definitely change the modal properties of the structure. Therefore, fifteen separate experiments were carried out by using only two accelerometers as shown in Fig. 5 (b). The first accelerometer is fixed at point one as a reference point, whilst the other is shifted through the other fifteen locations in each experiment. For each experiment, the structure is excited by a couple of random hammer impulses and the response is recorded. Table 5 is also listed the results from this data.
Fig. 7. (a) Pulley ring dimensions, (b) OMA test setup for pulley ring.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
125
Fig. 8. An example of recorded data for NExT-ERA procedure.
In the second test a steel pulley ring with more closely repeated modes is tested, so that the accuracy of the method can be studied in a more critical situation. A finite element model of the ring is analyzed initially to ensure that the repeated modes do indeed exist. Next, in order to assess the results of NExT/ERA method, an FRF based hammer test is executed to identify the close modes of the structure. Fig. 6 shows the coherence of measured data for the EMA test which somehow ensures the validity of the results in the vicinity of natural frequencies. According to the results obtained by hammer test (Table 6), the MOF value is calculated about 0.32 which demonstrates a greater extent of closeness in comparison with the plate model. The modal properties are extracted using the response data at eight locations of the ring. Next, the OMA test is performed at seven steps using two accelerometers in each experiment as before. Since the structure (Fig. 7a) is quite light, during the tests the movement of the location of accelerometer sensor causes a change in the modal properties of the structure in each experiment. In order to reduce the effect of sensor movement, six weights (Fig. 7b) with the same weight as accelerometer sensor, are added to the pulley ring to maintain the symmetry of the structure. Each experiment is repeated three times to obtain average data. This number for averaging is achieved by experience and is a commonlyused number during modal analysis tests. In each of the experiments, the ring is excited by use of several random hammer impulses while the structure response is being recorded (Fig. 8). 5.2. Discussions on test results for the plate Using fifteen pairs of time history acceleration data which are extracted by a sampling frequency of 2500 Hz and a recording length of about 10 s, the average natural frequencies and damping ratios for the plate are estimated by employing the NExT-ERA method. Table 5 represents these results as well as those obtained by simple hammer test. Based on these results, it can be concluded that the accuracy of NExT-ERA in calculating the closely spaced modes of the plate is reasonable. Also, the OMA time domain method predicts the modal damping ratios with an acceptable accuracy.
Fig. 9. MAC diagram for mode shapes identified by NExT-ERA vs. simple hammer test of the plate.
126
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Fig. 10. AutoMAC diagram for mode shapes of the plate identified by simple hammer test.
Fig. 11. AutoMAC diagram for mode shapes of the plate identified by NExT-ERA.
Fig. 12. AutoMAC diagram for mode shapes of the pulley ring identified by EMA hammer test.
The MAC diagram for the mode shapes identified by NExT-ERA vs. mode shapes identified by hammer test is also plotted in Fig. 9. Reference [28] notes that computing viable MAC values for axis-symmetric structures with repeated modes, is associated with major challenges. In these cases, usually one of the identified repeated mode shapes is a combination of two true mode shapes. Indeed each of the identified repeated mode shapes may be rotated with respect to the corresponding analytical mode shapes. In other words the repeated modes may exhibit a coupling with respect to each other. Accordingly,
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
127
Fig. 13. AutoMAC diagram for mode shapes of the pulley ring identified by NExT-ERA.
Table 7 Modal parameters of the tested pulley ring by various time domain methods. Method
NExT-ERA
NExT-PRCE
SSI
NExT-ERA
NExT-PRCE
SSI
Modal parameter
u (Hz)
u (Hz)
u (Hz)
z (%)
z (%)
z (%)
Mode 1 Mode 2 Mode 3
1549.31 1633.09 1643.39
1549.24 1633.00 1642.33
1548.88 1633.08 1642.75
0.03 0.17 0.13
0.03 0.17 0.23
0.03 0.18 0.17
Fig. 9 indicates that the mode shapes of the repeated modes are not reasonably consistent with those extracted by hammer test. Also plotting the AutoMAC diagrams for the mode shapes calculated by each method (as shown in Fig. 10 and Fig. 11) shows that each set of mode shapes is well correlated within itself. Therefore both methods showed great robustness to identify well-correlated mode shapes for the plate. 5.3. Discussions on test results for the pulley ring Average modal parameters extracted by NExT-ERA with 10 s recording time and 5000 Hz sampling frequency are shown in Table 6. It can be seen that NExT-ERA has reasonable accuracy in identifying the close modes of the structure and their related parameters. For both methods the AutoMAC diagrams are plotted in Fig. 12 and Fig. 13. It can be deduced that the repeated
Fig. 14. AutoMAC diagram for mode shapes of the pulley ring identified by (a) SSI (b) NExT-PRCE (c) NExT-ERA.
128
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
mode shapes identified by NExT-ERA are in much better correlation in comparison with those identified by the hammer test. In fact, the AutoMAC diagram for the EMA mode shapes has serious problems. Yet, as it was depicted in Fig. 6 the measured data for the EMA test entailed adequate quality, therefore the reliability of the performed test cannot be blamed for the detected deficiency in the AutoMAC diagram. It is possible that increasing the output coordinates of the ring may result in extracting mode shapes with better correlation. However, it seems that the EMA method shows some shortcomings when used in calculating the finely-correlated mode shapes when the natural frequencies are too close, whilst the NExT-ERA method avoids such problems. Moreover, in order to assess the accuracy of NExT-ERA with other leading time domain algorithms in modal identification, the results obtained by the method are compared with PRCE and EITD in combination with NExT, as well as Stochastic Subspace Identification (SSI) for the pulley ring (Table 7). Fig. 14 also depicts the AutoMAC diagrams for these methods. In order to use NExT-EITD method, the response from all coordinates should be recorded at the same time, or similar excitation should be performed for each step. Since only two sets of accelerometer sensors are available for data recording the experiment is conducted at several steps, therefore NExT-EITD cannot extract valuable modal parameters in this case and excludes during comparison. However, from Table 7 and Fig. 14 it can be deduced that other time domain methods are also capable to identify closely-spaced modes with reasonable accuracy. 6. Conclusions In this paper the accuracy of the time domain OMA method, NExT-ERA, is studied for several cases. Firstly, the method showed very good accuracy for a simulated 5-DOFs mass-spring system. It was however concluded that a sufficiently long recording time is also required to extract the damping ratios with acceptable accuracy. The effectiveness of the method when implemented in the finite element models of the beam and plate was extremely good. In these cases, the method shows a great capability in identifying the modal properties related to the closely-spaced modes as well. Under the test condition, the natural frequencies and the mode shapes for the repeated modes of the structure were extracted with extremely good accuracy by NExT-ERA. The damping coefficients obtained by the method were also close to those identified by the FRF based impact test. In the case of the pulley ring, which possessed a pair of natural frequencies with greater extent of closeness, the mode shapes identified by the hammer test lacked adequate correlation. Yet, the mode shapes identified by NExT-ERA were in much better correlation, which proves NExT-ERA as a more robust method compare to EMA method for identifying mode shapes of structures, especially in case of close modes. A comparison study is also conducted between NExT-ERA with other reliable time domain methods in OMA. The comparison shows that all other considered OMA methods identify close modes of the structure in a reasonable manner. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R.J. Allemang, A.W. Phillips, Spatial information in autonomous modal parameter estimation, J. Shock Vib. 404132 (2014). S. Tan, Q. Wang, Z. Wu, Effects of damping ratio and critical coupling strength on pogo instability, J. Spacecr. Rockets 53 (2016) 370e379. K. D'Souza, B.I. Epureanu, Noise rejection for two time-based multi-output modal analysis techniques, J. Sound Vib. 330 (2011) 1045e1051. R.M. Lin, Development of a new and effective modal identification method e mathematical formulations and numerical simulations, J. Vib. Control 17 (2010) 741e758. R. Brincker, L. Zhang, P. Andersen, Modal identification of output-only systems using frequency domain decomposition, Smart Mater. Struct. 10 (2001) 441e445. S.K. Au, F.L. Zhang, Y.C. Ni, Bayesian operational modal analysis: theory, computation, practice, Comput. Struct. 126 (2013) 3e14. W.J. Yan, L.S. Katafygiotis, A two-stage fast Bayesian spectral density approach for ambient modal analysis, Mech. Syst. Signal Process. 55 (2016) 139e155. L. Zhang, T. Wang, Y. Tamura, A frequency-spatial domain decomposition (FSDD) technique for operational modal analysis, Mech. Syst. Signal Process. 24 (2010) 1227e1239. H. Vold, J. Kundrat, G. Rocklin, R. Russell, A multi-input modal estimation algorithm for mini-computers, SAE Tech. Pap. 820194 (1982). S.R. Ibrahim, Large modal survey testing using the ibrahim time domain identification technique, AIAA J. Spacecr. Rockets 19 (1982) 459e465. J.N. Juang, R. Pappa, An Eigensystem realization algorithm for modal parameter identification and model reduction, Journal of Guidance, Control Dyn. 8 (1985) 620e627. G.H. James, T.G. Carne, J.P. Lauffer, A.R. Nard, The natural excitation technique (NExT) for modal parameter extraction from operating structures, Modal Anal.: Int. J. Anal. Exp. Modal Anal. 10 (1995) 260e277. J.M. Caicedo, S.J. Dyke, E.A. Johnson, Natural excitation technique and eigensystem realization algorithm for phase I of the IASC-ASCE benchmark problem: simulated data, J. Eng. Mech. 130 (2004) 49e60. J.M. Caicedo, Practical guidelines for the natural excitation technique (NExT) and the Eigensystem realization algorithm (ERA) for modal identification using ambient vibration, Exp. Tech. 35 (2011) 52e58. K.A. Petsounis, S.D. Fassois, Parametric time domain methods for the identification of vibrating structures e a critical comparison and assessment, Mech. Syst. Signal Process. 15 (2001) 1031e1060. S.J. Hu, W.L. Yang, F.S. Liu, H.J. Li, Fundamental comparison of time-domain experimental modal analysis methods based on high- and first-order matrix models, J. Sound Vib. 333 (2014) 6869e6884. D.M. Siringoringo, Y. Fujino, System identification of suspension bridge from ambient vibration response, Eng. Struct. 30 (2008) 462e477. D.Y. Chiang, C.S. Lin, Identification of modal parameters from ambient vibration data using eigensystem realization algorithm with correlation technique, J. Mech. Sci. Technol. 24 (12) (2010) 2377e2382. G. Zhang, J. Ma, Z. Chen, R. Wang, Automated eigensystem realization algorithm for operational modal analysis, J. Sound Vib. 333 (15) (2014) 3550e3563. D.M. Siringoringo, Y. Fujino, Experimental study of laser Doppler vibrometer and ambient vibration for vibration-based damage detection, Eng. Struct. 28 (2006) 1803e1815. A. Agneni, L.B. Crema, G. Coppotelli, Output-only analysis of structures with closely spaced poles, J. Mech. Syst. Signal Process. 24 (5) (2010) 1240e1249.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
129
[22] C. Rainieri, G. Fabbrocino, Performance assessment of selected OMA techniques for dynamic identification of geometrical systems and closely spaced structural modes, J. Theor. Appl. Mech. 49 (3) (2011) 825e839. [23] A. Malekjafarian, R. Brincker, M.R. Ashory, M.M. Khatibi, Identification of closely spaced modes using Ibrahim Time Domain method, in: Proceeding of 4th IOMAC, 2012. [24] G. Chen, Z. Wang, A signal decomposition theorem with Hilbert transform and its application to narrowband time series with closely spaced frequency components, Mech. Syst. Signal Process. 28 (2012) 258e279. [25] Y. Guo, A. Kareem, System identification through nonstationary data using timeefrequency blind source separation, J. Soundand Vib. 371 (2016) 110e131. [26] T. Wang, O. Celik, F.N. Catbas, L.M. Zhang, A frequency and spatial domain decomposition method for operational strain modal analysis and its application, Eng. Struct. 114 (2016) 104e112. [27] N.M.M. Maia, Extraction of Valid Modal Properties from Measured Data in Structural Vibrations, PhD thesis, Imperial College, Department of Mechanical Engineering, London SW7, UK, 1988. [28] R. Brincker, M.L. Aenlle, Mode shape sensitivity of two closely spaced eigenvalues, J. Sound Vib. 334 (2015) 377e387.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Experimental identification of closely spaced modes using NExT-ERA S.A. Hosseini Kordkheili a, *, S.H. Momeni Massouleh a, S. Hajirezayi a, H. Bahai b a
Center of Research and Development in Space Science and Technology, Aerospace Engineering Department, Sharif University of Technology, Azadi Avenue, P.O. Box: 11365-9567, Tehran, Iran b Department of Mechanical, Aerospace & Civil Engineering, Brunel University, Uxbridge, Middlesex, UB8 3pH, UK
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 May 2016 Received in revised form 17 September 2017 Accepted 26 September 2017
This article presents a study on the capability of the time domain OMA method, NExT-ERA, to identify closely spaced structural dynamic modes. A survey in the literature reveals that few experimental studies have been conducted on the effectiveness of the NExT-ERA methodology in case of closely spaced modes specifically. In this paper we present the formulation for NExT-ERA. This formulation is then implemented in an algorithm and a code, developed in house to identify the modal parameters of different systems using their generated time history data. Some numerical models are firstly investigated to validate the code. Two different case studies involving a plate with closely spaced modes and a pulley ring with greater extent of closeness in repeated modes are presented. Both structures are excited by random impulses under the laboratory condition. The resulting time response acceleration data are then used as input in the developed code to extract modal parameters of the structures. The accuracy of the results is checked against those obtained from experimental tests. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Closely spaced modes Time domain OMA method NExT-ERA method Correlation function Experimental identification
1. Introduction Repeated modes or closely spaced modes which occur in structures such as circular plates, gears or bladed-discs that have some degrees of symmetry. From an experimental point of view, closely spaced modes are two close modes that are sometimes misinterpreted as one single mode. Failure in identification of closely spaced modes, which are normally associated with pseudo-repeated root problem [1], during dynamic analysis may lead to critical effects in dynamic systems such as pogo-effect in Titan 2 space rocket [2]. Therefore the possibility of occurrence of closely spaced modes during modal analysis process is inevitable. Moreover, in model updating process one needs an experimental mode corresponding to each of those identified analytically. This requirement dictates that all the existing modes, including those which are closely spaced, have to be extracted employing more adapted techniques. During the last four decades different algorithms have been proposed to extract modal parameters of structures by researchers. These algorithms, which are basically developed in time, frequency, or space domains, are categorized into single input-single output, single input-multi output or Multi Input-Multi Output (MIMO) methods. Among these, frequency and time domains are considered to be more practical. Time domain methods which directly use experimentally recorded time
* Corresponding author. E-mail address: [email protected] (S.A. Hosseini Kordkheili). https://doi.org/10.1016/j.jsv.2017.09.038 0022-460X/© 2017 Elsevier Ltd. All rights reserved.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
117
history data avoid some of the problems (such as leakage) which are normally associated with the Fourier transform in the frequency domain methods. On the other hand, in time domain methods a reasonable signal to noise ratio suffices to extract modal properties without the necessity of measuring excitation force functions [3]. Avoiding FRF calculations in time domain algorithms together with its superior damping estimation [4] makes these methods more suitable to analyze closely spaced modes of structures. However, some frequency domain methods have been successfully employed to extract the closely spaced modes of structures [5e7]. As reported in Ref. [8], taking into account the response at different DOFs, MIMO procedures are better candidates to analyze complex structures with closely spaced or even repeated modes. In 1982, Vold et al. [9] proposed Polyreference Complex Exponential (PRCE) method which is considered as the first developed MIMO algorithm. Two other well-known time domain MIMO modal identification techniques which were developed later are the Extended Ibrahim Time Domain (EITD) [10] and Eigensystem Realization Algorithm (ERA) [11]. It has been reported that Experimental Modal Analysis (EMA) methods, even the MIMO ones, face difficulty when dealing with close modes [5]. Because in these methods due to the point excitation, the identification of closely spaced modes highly depends on the excitation technique and the excited region. Moreover, the methods have shown inconsistency in calculating precise damping ratios when it comes to close modes. While, unlike methods in EMA, due to the broadband random excitation in real loading situation, which acts through different points of the structure, OMA methods can excite all vibration modes. Additionally, because of their MIMO nature, the OMA methods are more accurate in studying the closely spaced modes of structures. As mentioned earlier, in time domain methods there is no need to record excitation data. This notion, accompanied with the fact that Operational Modal Analysis (OMA) only requires output data due to natural or ambient loading, was the motivation in this study to employ OMA in conjunction with time domain schemes. For modal identification of structures, OMA methods may be employed for the whole structure at once without any need to elaborate excitation equipment. In 1992 the Natural Excitation Technique (NExT) was introduced [12] to process the random response data of the structure, so that the results may be used as an input to MIMO-type time domain EMA algorithms such as EITD, ERA and PRCE. Indeed, NExT in combination with any MIMO time domain algorithm can be considered as an OMA method in its own right. For example Caicedo [13,14] used NExT-ERA to extract the modal parameters of four-story steel structures. Among all the time domain methods in modal analysis, ERA has proved to have great accuracy and is considered as one of the most robust and successful methodologies for engineering practices [15e17]. Several studies have also been conducted to evaluate the accuracy of the procedure for various systems, as well. Siringoringo [17] exploited the method for system identification of a suspension bridge. Chiang and Lin [18] studied the accuracy of a modified version of ERA method for coupled simulated mass spring systems. Zhang et al. [19] used the methodology for extracting the modal properties of a bridge model. The method has also proved useful for structural health monitoring purposes [13,20]. Moreover, many researches have been conducted to study the effect of close modes on performance of different methods in modal analysis. The efficacy of FDD [5] was investigated in the case of close modes by Brinker et al., just when it was developed. and Agneni [21] extracted the close modes of a AB-204 helicopter blade using Hilbert transform method and frequency domain decomposition. Ranieri [22] also studied the effectiveness of OMA time domain method and second order blind identification in case of close modes. Brincker et al. [23] also introduced a modified version of EITD to extract repeated modes of numerical structures using output data only. Chen [24] studied the efficiency of the analytical mode decomposition method in case of closely-spaced modes, as well. The concept of closely-spaced modes has proved so critical today that every new system identification technique is evaluated in the case of close modes just when it is proposed [6,7,25,26]. NExT-ERA has also been employed for identification of closely-spaced modes of structures. Caicedo [14] managed to extract the lowfrequency close modes of a benchmark structure by NExT-ERA. In Siringoringo's study [17] the method was also employed for system identification of a suspension bridge. In his study, the structure had some low frequency closely-spaced natural frequencies which were successfully identified by NExT-ERA. This research aims to assess the accuracy of the time domain method ERA in corporation with NExT for identification of close modes in higher frequencies, where noise is a more critical issue and makes the identification process more difficult. Moreover, for symmetric plate-like structures there is some difficulty in identifying accurate mode-shapes [28], the NExT-ERA method is also evaluated in such a case, as well. For this purpose, first a simulated mass-spring system and a numerically modeled beam are studied to verify the procedure and formulation. The effectiveness of the method is then evaluated for a plate having closely spaced modes. Finally two separate experiments are carried out on a plate and a pulley ring with closely spaced modes and the effectiveness of the method is studied experimentally.
2. Eigensystem Realization Algorithm ERA was developed by Juang and Pappa [11] based on the concepts that predominantly originated form the control theory, so that it differs from the usual developments found in the EMA literature. In ERA technique, using dynamic equation of _ equilibrium for an N-DOF viscously damped system together with the state vector fuðtÞg ¼ fyðtÞ; yðtÞg, in which yðtÞ is displacement vector, the output of the system at kth time sample is written as fxðkÞg ¼ ½RfuðkÞg, where fuðkÞg ¼ ½Afuðk 1Þg þ ½Bfdðk 1Þg, dðkÞ is input vector at q locations and [A] and [B] are discrete-time state-space matrices which are determined in a process called realization. {x(k)} is a vector of dimension p which is the number of output
118
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
locations for the experiments. Now, consider that the structure is excited at q coordinates by separate impulses. Here the Markov parameter matrix [X(k)] is defined for the response of structure at kth time step as Xij ðkÞ ¼ xij ðkÞ [27], where xij ðkÞ is the displacement, velocity or acceleration response at ith coordinate due to the impulse at jth coordinate at kth time sample. Assuming stationary structure initial condition as well as impulse input excited at t ¼ 0 i.e. dðtÞ ¼ dð0Þ in state space we obtain
½XðkÞ ¼ ½R ½Ak1 ½B |fflfflffl{zfflfflffl} |{z} |fflfflffl{zfflfflffl} |{z} pq
(1)
p2N 2N2N 2Nq
where [R] is transformation matrix. According to this process these matrices are estimated from the response of the structure based on a minimum realization algorithm. As the next step, the Hankel matrix is constructed having the response data of the structure at different times as shown below
2
½XðkÞ 6 ½Xðk þ 1Þ 6 Hðk 1Þ ¼ 4 « |fflfflfflfflfflffl{zfflfflfflfflfflffl} ½Xðk þ r 1Þ prqs
½Xðk þ 1Þ ½Xðk þ 2Þ « ½Xðk þ rÞ
3 … ½Xðk þ s 1Þ 7 … ½Xðk þ sÞ 7 5 1 « … ½Xðk þ r þ s 2Þ
(2)
This matrix plays a major role in most time domain algorithms. In Eq. (2), r and s are large enough integers so that the rank of matrix cannot be increased by increasing their numbers further. Using an algorithm proposed in Ref. [27], it can be shown that the matrices [R], [A] and [B] can be defined as follows
½R ¼
h
i T 1 Ep ½U2N ½S2N 2
(3)
h i 1 1 ½A ¼ ½S2N 2 ½U2N T ½Hð1Þ½V2N ½S2N 2 h
1 ½B ¼ ½S2N 2 ½V2N T Eq
(4)
i
(5)
where,
T T Ep ¼ ½ ½I ½0 … ½0 ; Eq ¼ ½ ½I ½0 … ½0 |fflffl{zfflffl} |fflffl{zfflffl}
ppr
(6)
qqs
also ½U2N , ½V2N and ½S2N are matrices containing only the first 2N columns of matrices ½U, ½V and ½S which are calculated using singular value decomposition of matrix [H (0)], i. e.
½Hð0Þ ¼ ½U ½S ½VT |fflfflffl{zfflfflffl} |{z} |{z} |ffl{zffl} prqs
(7)
prpr prps psqs
Finally solving the eigenvalue problem ½AfJu g ¼ lfJu g leads to the modal parameters of the system. The mode shapes of the structure at output coordinates can be computed using fJx g ¼ ½R fJu g . |fflfflffl{zfflfflffl} |{z} |fflfflffl{zfflfflffl} p1
p2N 2N1
3. Natural Excitation Technique NExT is one of the OMA methods that can be used in conjugation with EMA time domain identification algorithms to extract modal properties of structures. According to NExT, the correlation (COR) function of the structure response under ambient loading or white noise can be written as the summation of decaying sinusoids. The modal parameters of each sinusoid i.e. natural frequency, modal damping and mode shape coefficient are equivalent to the modal parameters of each mode of the structure. Thus, the conventional MIMO time domain identification algorithms in EMA, such as ERA, EITD and PRCE use the NExT product to extract the modal parameters in a way that COR functions are used instead of impulse response function. In this method the response at coordinate i due to input force f(t) at kth coordinate can be written as
xik ðtÞ ¼
N X
Z 4ir 4kr
r¼1
t
∞
f ðtÞgr ðt tÞdt
(8)
where gr ðtÞ ¼ m1ud ezr ur t sinðudr tÞ and 4ir is the ith component of rth mode shape and udr is the rth damped natural frequency. n
r
r
N denotes the total number of modes. When f(t) is a random white noise function, the correlation function Rij between the
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
119
Fig. 1. Simulated mass spring system.
response of output coordinates i and j is defined as Rij ðTÞ ¼ E½xik ðt þ TÞ xjk ðtÞ, where E is the expectation operator. Using the algorithm proposed in Refs. [12,17] it can be deduced that
Rij ðTÞ ¼
N X 4ir Ajr r¼1
mr udr
n ezr ur T sin udr T þ qr
(9)
It can be seen that the correlation function can be written as summation of decaying sinusoids as the same form of Impulse Response Function. Thus, COR function Rij ðTÞ between the response coordinates i and j can be considered as the response at output coordinate i due to impulse applied at coordinate j for MIMO time domain algorithms. For discrete time responses of two output coordinates the correlation function can be calculated from
Rij ðmÞ ¼
1 M
Mmþ1 X
xi ðnÞxj ðx þ m 1Þ
(10)
n¼1
where M is the total number of time history data and increasing in M yields better accuracy in COR function calculation. The COR function for NExT can also be calculated using cross-spectral density function and taking its inverse Fast Fourier transform [14]. Based on the discussions made above the step-by-step algorithm for employing NExT-ERA is as follows: Recording the structure response due to the ambient excitation in time domain at different measurement points Calculating the correlation function (Rij ) using NExT Treating the correlation functions like the impulse response functions for utilization in ERA (or any other MIMO time domain method) Averaging and extracting the modal parameters
4. Numerical results and discussions The previously presented formulation is applied in a code developed to identify the modal parameters of different systems using their experimentally or numerically generated time history data. In order to examine the capability of the developed code some investigations on discrete and continuous numerical systems are conducted and the results are reported. 4.1. Simulated mass spring system For this part a simple mass spring system, consisting of 5 masses as shown in Fig. 1, is simulated. Each mass weighs 1 kg, the springs connecting the mass have a stiffness of 1000 N/m and a damping of 1 N s/m. The system is excited by a couple of impulses which acted through some random DOFs of the system and the time history response data of the masses are
Table 1 Modal parameters of the mass spring system. Method
Actual
NExT-ERA (Present Work)
Actual
NExT-ERA (Present Work)
Modal parameter
u (rad/s)
u (rad/s)
Err (%)
z (%)
z (%)
Err (%)
Mode Mode Mode Mode Mode
28.46 83.08 130.97 168.25 191.90
28.46 83.08 130.97 167.25 191.90
0.00 0.00 0.00 0.00 0.00
0.14 0.42 0.65 0.84 0.96
0.25 0.42 0.65 0.84 0.96
73.33 0.00 0.00 0.00 0.00
1 2 3 4 5
120
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Table 2 Damping ratios extracted by NExT-EITD and NExT-PRCE using different recording length. Method
NExT-EITD
Number of Data 500
NExT-PRTD 2000
8000
Parameter
z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%) z (%)
Err (%)
Mode Mode Mode Mode Mode
1.4230 0.6027 0.6740 0.8464 0.9626
900.40 45.08 2.92 0.60 0.32
160.99 0.29 0.29 0.43 0.29
8.22 0.07 0.29 0.43 0.29
0.49 0.07 0.29 0.37 0.29
780.67 33.40 1.91 0.52 0.32
168.80 0.29 0.29 0.44 0.31
1.69 0.07 0.29 0.44 0.31
0.00 0.07 0.29 0.44 0.31
1 2 3 4 5
2000
8000
0.3714 0.4166 0.6568 0.8450 0.9623
0.1540 0.4157 0.6568 0.8450 0.9623
15000
0.1430 0.4157 0.6568 0.8444 0.9623
500
1.2532 0.5541 0.6674 0.8457 0.9626
0.3825 0.4166 0.6568 0.8450 0.9625
0.1447 0.4157 0.6568 0.8450 0.9625
15000
0.1423 0.4157 0.6568 0.8450 0.9625
Fig. 2. MAC diagram for identified mode shapes of the mass spring system by NExT-ERA vs. exact mode shapes.
recorded by sampling rate of 500 Hz. A total number of 8000 time history displacement data is used in computing the correlation function. The resulting modal parameters are shown in Table 1. As seen from this table, the modal damping coefficient for the first mode is not calculated with an acceptable accuracy. Because this is the first natural frequency of the system which has a relatively low value, the recording time needs to be increased to get more accurate result. Subsequently, instead of 8000 data, 24000 displacement data with the same sampling rate is used for the identification procedure. Implementing this number of data into the developed code leads to a damping ratio z ¼ 0:1483 for the first mode which means that the deviation from actual value is decreased to about 4.22%. In accordance with the results presented here, ref [14] points that NExT-ERA can lead to inaccurate results in identifying the low frequency modes if the data length is not long enough. Moreover, other methods such as EITD [23] and PRCE [9] have also shown such a trend in calculating damping ratios. The associated results for these methods are represented in Table 2. The correlation between the actual mode shapes and those extracted by NExT-ERA is also studied by drawing a MAC diagram. Fig. 2 depicted this diagram for the extracted mode shapes and correlation between the resulted mode shapes is verified. 4.2. Numerical case of a randomly excited cantilever beam In order to evaluate the accuracy of modes extracted by the developed code for NExT-ERA, a cantilever steel beam is modeled using a finite element software i.e. Patran. The beam has a length of 1 m and a square cross-section of 20 mm 20 mm. A hundred beam elements are used in the FE simulation. The modal parameters of the beam are computed first by imposing a conventional normal mode solution. Then the beam, for which a structural damping coefficient of 1.2% is Table 3 Modal parameters of the beam. Method
Actual
NExT-ERA (Present Work)
No. of Response nodes
100
No. of acceleration history data
1000
Actual
1500
NExT-ERA (Present Work)
5
100
1500
1000
5 1500
1500
Modal parameter
u (Hz)
u (Hz)
Err (%)
u (Hz)
Err (%)
u (Hz)
Err (%)
z (%)
z (%) Err (%) z (%) Err (%) z (%) Err (%)
Mode Mode Mode Mode
16.76 104.86 292.91 571.99
16.72 104.86 292.91 572.01
0.24 0.00 0.00 0.00
16.77 104.86 292.91 572.01
0.06 0.00 0.00 0.00
16.84 104.86 292.90 572.16
0.48 0.00 0.00 0.03
0.60 0.60 0.60 0.60
1.52 0.62 0.60 0.60
1 2 3 4
153.33 3.33 0.00 0.00
1.35 0.60 0.60 0.60
125.00 0.00 0.00 0.00
1.91 0.60 0.60 0.61
218.33 0.00 0.00 1.67
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
121
Table 4 Modal parameters of plate (C denotes close mode). Method
Actual
NExT-ERA (Present Work)
Actual
Modal Parameter
u (Hz)
u (Hz)
Err (%)
z (%)
z (%)
Err (%)
Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode Mode
20.29 28.99 36.84 50.95 52.12 87.69 90.83 93.31 98.21 112.77 147.17
20.29 28.99 36.84 50.95 52.12 87.69 90.83 93.31 98.21 112.77 147.17
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
0.72 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70 0.70
2.86 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
1 2 3 4C 5C 6 7C 8C 9 10 11
NExT-ERA (Present Work)
Fig. 3. MAC diagram for identified mode shapes by NExT-ERA vs. FEM mode shapes (numerical plate).
considered, is excited by a couple of successive random impulses and the transient response of the beam is recorded by a sampling frequency of 2000 Hz. For this part of the analysis a transient response solution is used by applying an impulse excitation with equilateral triangle shape with 100 N maximum amplitude and 0.002 s duration. A total number of 1000 acceleration history data, at 100 nodes of the model is used for calculating COR function. Inserting these recorded data into the developed code results in the identified modal parameters of the beam as shown in Table 3. It can be seen again that the modal parameters of the basic modes of structure are not calculated accurately. As demonstrated in the first example increasing the number of response data will contribute to achieving more accurate damping ratios for the first two modes of the structure. Accordingly, the number of acceleration time history data is increased to 1500 with the same sampling rate. In this case, as can be observed in Table 3, the results for damping coefficients of the first
Fig. 4. eTwo repeated mode shapes of the tested plate identified by finite element analysis.
122
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Fig. 5. Schematics of (a) EMA and (b) OMA test setups.
two modes are converging to actual values. Practically measurement of responses at 100 different locations in a beam is impossible during real experiments. Consequently, we adopted another numerical simulation for the same beam with only 5 output coordinates. According to the results of this solution which are shown in Table 3, the modal parameters are still estimated in an acceptable range. 4.3. Numerical case of a randomly excited plate with close modes In this section a plate with closely spaced modes is modeled using the same FEA software. The structure analyzed is a square 400 mm 400 mm steel plate with a thickness of 1 mm and a 50 g mass at a corner. The plate is modeled for free-free boundary condition. Being symmetric, the plate will definitely possess a few pairs of repeated eigenvalues (and hence repeated natural frequencies and damping ratios) with coupled mode shapes. The added mass would disturb the symmetry of structure slightly so that the paired and formerly similar natural frequencies will diverge from each other by a small amount, creating closely spaced modes. The plate is modeled by 1.6 1.6 cm isometric QUAD shell elements in Patran, so the total number of 2D elements reaches 525. To ensure the existence of close modes, the modal properties of the model are identified
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
123
Table 5 Modal parameters of the tested plate. Method
EMA (Experiment)
NExT-ERA (Present Work)
EMA (Polymax) (Experiment)
Modal parameter
u (Hz)
u (Hz)
z (%)
z (%)
Mode Mode Mode Mode Mode
117.94 178.64 201.71 293.96 296.82
117.60 178.09 201.10 292.64 295.83
0.38 0.06 0.17 0.17 0.14
0.32 0.05 0.14 0.17 0.13
1 2 3 4 5
NExT-ERA (Present Work)
Fig. 6. eCoherence diagrams obtained by EMA test of the pulley ring.
using a simple normal modes analysis first. Next, the plate is subjected to several successive impulses at random coordinates and the response of structure is obtained by the use of transient solution. The acceleration response data are recorded at a rate of 2000 Hz. A structural damping coefficient of 1.4% is also used in the transient response analysis. The results shown in Table 4 and Fig. 3 are obtained by using 5000 time history data at 16 response points of the model. In order to assess the extent of closeness of two modes the Modal Overlap Factor (MOF) mn ¼ un zn =ðun un1 Þ is considered. It is noted that two modes with MOF value above 0.17 can be considered moderately close [22]. In the case of our numerical plate the MOF value is about 0.31 which is higher than the closeness criteria. So there are two pairs of close modes among the first eleven modes of the structure which are determined with acceptable accuracy. Again, increasing the number of recorded data will lead to more accurate results in lower frequency modes. 5. Experimental results and discussions 5.1. Test setup and results Tests are conducted on two structures, a plate and a pulley ring both of which are tested under a free-free boundary condition and are suspended by simple strings. A 34 channel NI PXIe 1082 data acquisition system along with PCB piezoelectric accelerometers (with 100 mV/g sensitivity and 10 kHz frequency range) and a PCB force sensor (with 1124 mV/kN sensitivity and 4.448 kN measurement range) are used to measure the response of the structure. The first test consisted of a 300 mm 305 mm rectangular steel plate which is studied in modal laboratory condition. The thickness of the plate is 3 mm and since the sides of the structure only slightly differ (5 mm), the plate will have closely spaced modes. In order to confirm this assumption, a simple finite element model of the plate is created and the approximate natural frequencies and mode shapes of the structure are identified. It is revealed that this structure does possess pairs of closely Table 6 Modal parameters of the tested pulley ring. Method
EMA (Experiment)
NExT-ERA (Present Work)
EMA (Polymax) (Experiment)
Modal parameter
u (Hz)
u (Hz)
z (%)
NExT-ERA (Present Work)
z (%)
Mode 1 Mode 2 Mode 3
1549.59 1635.52 1645.14
1549.31 1633.09 1643.39
0.05 0.19 0.15
0.03 0.17 0.13
124
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
spaced modes or repeated mode shapes (Fig. 4). Now, an FRF based roving hammer test is performed to measure the actual modal properties of the plate structure. Schematic of this test is depicted in Fig. 5a. Sixteen points on the plate are marked and hammer impacts are acted through these sixteen locations of the plate. For the EMA test a simple hammer, to which the force sensor is attached, is used to excite the structure with a steel tip. Modal View software is also utilized to analyze the collected data in this test. The damping ratios of the plate are extracted by Polymax method. Based on the results attained by EMA hammer test (Table 5) the MOF for the close modes of the structure is about 0.17, so it can be deduced that the modes are moderately close. In addition, to perform the OMA NExT-ERA test as the structure is quite light attaching sixteen concurrent accelerometers would not be appropriate since the added mass will definitely change the modal properties of the structure. Therefore, fifteen separate experiments were carried out by using only two accelerometers as shown in Fig. 5 (b). The first accelerometer is fixed at point one as a reference point, whilst the other is shifted through the other fifteen locations in each experiment. For each experiment, the structure is excited by a couple of random hammer impulses and the response is recorded. Table 5 is also listed the results from this data.
Fig. 7. (a) Pulley ring dimensions, (b) OMA test setup for pulley ring.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
125
Fig. 8. An example of recorded data for NExT-ERA procedure.
In the second test a steel pulley ring with more closely repeated modes is tested, so that the accuracy of the method can be studied in a more critical situation. A finite element model of the ring is analyzed initially to ensure that the repeated modes do indeed exist. Next, in order to assess the results of NExT/ERA method, an FRF based hammer test is executed to identify the close modes of the structure. Fig. 6 shows the coherence of measured data for the EMA test which somehow ensures the validity of the results in the vicinity of natural frequencies. According to the results obtained by hammer test (Table 6), the MOF value is calculated about 0.32 which demonstrates a greater extent of closeness in comparison with the plate model. The modal properties are extracted using the response data at eight locations of the ring. Next, the OMA test is performed at seven steps using two accelerometers in each experiment as before. Since the structure (Fig. 7a) is quite light, during the tests the movement of the location of accelerometer sensor causes a change in the modal properties of the structure in each experiment. In order to reduce the effect of sensor movement, six weights (Fig. 7b) with the same weight as accelerometer sensor, are added to the pulley ring to maintain the symmetry of the structure. Each experiment is repeated three times to obtain average data. This number for averaging is achieved by experience and is a commonlyused number during modal analysis tests. In each of the experiments, the ring is excited by use of several random hammer impulses while the structure response is being recorded (Fig. 8). 5.2. Discussions on test results for the plate Using fifteen pairs of time history acceleration data which are extracted by a sampling frequency of 2500 Hz and a recording length of about 10 s, the average natural frequencies and damping ratios for the plate are estimated by employing the NExT-ERA method. Table 5 represents these results as well as those obtained by simple hammer test. Based on these results, it can be concluded that the accuracy of NExT-ERA in calculating the closely spaced modes of the plate is reasonable. Also, the OMA time domain method predicts the modal damping ratios with an acceptable accuracy.
Fig. 9. MAC diagram for mode shapes identified by NExT-ERA vs. simple hammer test of the plate.
126
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
Fig. 10. AutoMAC diagram for mode shapes of the plate identified by simple hammer test.
Fig. 11. AutoMAC diagram for mode shapes of the plate identified by NExT-ERA.
Fig. 12. AutoMAC diagram for mode shapes of the pulley ring identified by EMA hammer test.
The MAC diagram for the mode shapes identified by NExT-ERA vs. mode shapes identified by hammer test is also plotted in Fig. 9. Reference [28] notes that computing viable MAC values for axis-symmetric structures with repeated modes, is associated with major challenges. In these cases, usually one of the identified repeated mode shapes is a combination of two true mode shapes. Indeed each of the identified repeated mode shapes may be rotated with respect to the corresponding analytical mode shapes. In other words the repeated modes may exhibit a coupling with respect to each other. Accordingly,
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
127
Fig. 13. AutoMAC diagram for mode shapes of the pulley ring identified by NExT-ERA.
Table 7 Modal parameters of the tested pulley ring by various time domain methods. Method
NExT-ERA
NExT-PRCE
SSI
NExT-ERA
NExT-PRCE
SSI
Modal parameter
u (Hz)
u (Hz)
u (Hz)
z (%)
z (%)
z (%)
Mode 1 Mode 2 Mode 3
1549.31 1633.09 1643.39
1549.24 1633.00 1642.33
1548.88 1633.08 1642.75
0.03 0.17 0.13
0.03 0.17 0.23
0.03 0.18 0.17
Fig. 9 indicates that the mode shapes of the repeated modes are not reasonably consistent with those extracted by hammer test. Also plotting the AutoMAC diagrams for the mode shapes calculated by each method (as shown in Fig. 10 and Fig. 11) shows that each set of mode shapes is well correlated within itself. Therefore both methods showed great robustness to identify well-correlated mode shapes for the plate. 5.3. Discussions on test results for the pulley ring Average modal parameters extracted by NExT-ERA with 10 s recording time and 5000 Hz sampling frequency are shown in Table 6. It can be seen that NExT-ERA has reasonable accuracy in identifying the close modes of the structure and their related parameters. For both methods the AutoMAC diagrams are plotted in Fig. 12 and Fig. 13. It can be deduced that the repeated
Fig. 14. AutoMAC diagram for mode shapes of the pulley ring identified by (a) SSI (b) NExT-PRCE (c) NExT-ERA.
128
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
mode shapes identified by NExT-ERA are in much better correlation in comparison with those identified by the hammer test. In fact, the AutoMAC diagram for the EMA mode shapes has serious problems. Yet, as it was depicted in Fig. 6 the measured data for the EMA test entailed adequate quality, therefore the reliability of the performed test cannot be blamed for the detected deficiency in the AutoMAC diagram. It is possible that increasing the output coordinates of the ring may result in extracting mode shapes with better correlation. However, it seems that the EMA method shows some shortcomings when used in calculating the finely-correlated mode shapes when the natural frequencies are too close, whilst the NExT-ERA method avoids such problems. Moreover, in order to assess the accuracy of NExT-ERA with other leading time domain algorithms in modal identification, the results obtained by the method are compared with PRCE and EITD in combination with NExT, as well as Stochastic Subspace Identification (SSI) for the pulley ring (Table 7). Fig. 14 also depicts the AutoMAC diagrams for these methods. In order to use NExT-EITD method, the response from all coordinates should be recorded at the same time, or similar excitation should be performed for each step. Since only two sets of accelerometer sensors are available for data recording the experiment is conducted at several steps, therefore NExT-EITD cannot extract valuable modal parameters in this case and excludes during comparison. However, from Table 7 and Fig. 14 it can be deduced that other time domain methods are also capable to identify closely-spaced modes with reasonable accuracy. 6. Conclusions In this paper the accuracy of the time domain OMA method, NExT-ERA, is studied for several cases. Firstly, the method showed very good accuracy for a simulated 5-DOFs mass-spring system. It was however concluded that a sufficiently long recording time is also required to extract the damping ratios with acceptable accuracy. The effectiveness of the method when implemented in the finite element models of the beam and plate was extremely good. In these cases, the method shows a great capability in identifying the modal properties related to the closely-spaced modes as well. Under the test condition, the natural frequencies and the mode shapes for the repeated modes of the structure were extracted with extremely good accuracy by NExT-ERA. The damping coefficients obtained by the method were also close to those identified by the FRF based impact test. In the case of the pulley ring, which possessed a pair of natural frequencies with greater extent of closeness, the mode shapes identified by the hammer test lacked adequate correlation. Yet, the mode shapes identified by NExT-ERA were in much better correlation, which proves NExT-ERA as a more robust method compare to EMA method for identifying mode shapes of structures, especially in case of close modes. A comparison study is also conducted between NExT-ERA with other reliable time domain methods in OMA. The comparison shows that all other considered OMA methods identify close modes of the structure in a reasonable manner. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
R.J. Allemang, A.W. Phillips, Spatial information in autonomous modal parameter estimation, J. Shock Vib. 404132 (2014). S. Tan, Q. Wang, Z. Wu, Effects of damping ratio and critical coupling strength on pogo instability, J. Spacecr. Rockets 53 (2016) 370e379. K. D'Souza, B.I. Epureanu, Noise rejection for two time-based multi-output modal analysis techniques, J. Sound Vib. 330 (2011) 1045e1051. R.M. Lin, Development of a new and effective modal identification method e mathematical formulations and numerical simulations, J. Vib. Control 17 (2010) 741e758. R. Brincker, L. Zhang, P. Andersen, Modal identification of output-only systems using frequency domain decomposition, Smart Mater. Struct. 10 (2001) 441e445. S.K. Au, F.L. Zhang, Y.C. Ni, Bayesian operational modal analysis: theory, computation, practice, Comput. Struct. 126 (2013) 3e14. W.J. Yan, L.S. Katafygiotis, A two-stage fast Bayesian spectral density approach for ambient modal analysis, Mech. Syst. Signal Process. 55 (2016) 139e155. L. Zhang, T. Wang, Y. Tamura, A frequency-spatial domain decomposition (FSDD) technique for operational modal analysis, Mech. Syst. Signal Process. 24 (2010) 1227e1239. H. Vold, J. Kundrat, G. Rocklin, R. Russell, A multi-input modal estimation algorithm for mini-computers, SAE Tech. Pap. 820194 (1982). S.R. Ibrahim, Large modal survey testing using the ibrahim time domain identification technique, AIAA J. Spacecr. Rockets 19 (1982) 459e465. J.N. Juang, R. Pappa, An Eigensystem realization algorithm for modal parameter identification and model reduction, Journal of Guidance, Control Dyn. 8 (1985) 620e627. G.H. James, T.G. Carne, J.P. Lauffer, A.R. Nard, The natural excitation technique (NExT) for modal parameter extraction from operating structures, Modal Anal.: Int. J. Anal. Exp. Modal Anal. 10 (1995) 260e277. J.M. Caicedo, S.J. Dyke, E.A. Johnson, Natural excitation technique and eigensystem realization algorithm for phase I of the IASC-ASCE benchmark problem: simulated data, J. Eng. Mech. 130 (2004) 49e60. J.M. Caicedo, Practical guidelines for the natural excitation technique (NExT) and the Eigensystem realization algorithm (ERA) for modal identification using ambient vibration, Exp. Tech. 35 (2011) 52e58. K.A. Petsounis, S.D. Fassois, Parametric time domain methods for the identification of vibrating structures e a critical comparison and assessment, Mech. Syst. Signal Process. 15 (2001) 1031e1060. S.J. Hu, W.L. Yang, F.S. Liu, H.J. Li, Fundamental comparison of time-domain experimental modal analysis methods based on high- and first-order matrix models, J. Sound Vib. 333 (2014) 6869e6884. D.M. Siringoringo, Y. Fujino, System identification of suspension bridge from ambient vibration response, Eng. Struct. 30 (2008) 462e477. D.Y. Chiang, C.S. Lin, Identification of modal parameters from ambient vibration data using eigensystem realization algorithm with correlation technique, J. Mech. Sci. Technol. 24 (12) (2010) 2377e2382. G. Zhang, J. Ma, Z. Chen, R. Wang, Automated eigensystem realization algorithm for operational modal analysis, J. Sound Vib. 333 (15) (2014) 3550e3563. D.M. Siringoringo, Y. Fujino, Experimental study of laser Doppler vibrometer and ambient vibration for vibration-based damage detection, Eng. Struct. 28 (2006) 1803e1815. A. Agneni, L.B. Crema, G. Coppotelli, Output-only analysis of structures with closely spaced poles, J. Mech. Syst. Signal Process. 24 (5) (2010) 1240e1249.
S.A. Hosseini Kordkheili et al. / Journal of Sound and Vibration 412 (2018) 116e129
129
[22] C. Rainieri, G. Fabbrocino, Performance assessment of selected OMA techniques for dynamic identification of geometrical systems and closely spaced structural modes, J. Theor. Appl. Mech. 49 (3) (2011) 825e839. [23] A. Malekjafarian, R. Brincker, M.R. Ashory, M.M. Khatibi, Identification of closely spaced modes using Ibrahim Time Domain method, in: Proceeding of 4th IOMAC, 2012. [24] G. Chen, Z. Wang, A signal decomposition theorem with Hilbert transform and its application to narrowband time series with closely spaced frequency components, Mech. Syst. Signal Process. 28 (2012) 258e279. [25] Y. Guo, A. Kareem, System identification through nonstationary data using timeefrequency blind source separation, J. Soundand Vib. 371 (2016) 110e131. [26] T. Wang, O. Celik, F.N. Catbas, L.M. Zhang, A frequency and spatial domain decomposition method for operational strain modal analysis and its application, Eng. Struct. 114 (2016) 104e112. [27] N.M.M. Maia, Extraction of Valid Modal Properties from Measured Data in Structural Vibrations, PhD thesis, Imperial College, Department of Mechanical Engineering, London SW7, UK, 1988. [28] R. Brincker, M.L. Aenlle, Mode shape sensitivity of two closely spaced eigenvalues, J. Sound Vib. 334 (2015) 377e387.