Experimental verification of selected methods ...

Experimental verification of selected methods sensitivity to damage size and location J. Iwaniec1, P. Kurowski1 AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics Department of Robotics and Mechatronics Mickiewicz Alley 30, 30-059 Krakow, Poland email: [email protected]

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Abstract The main emphasis of the paper is put on the experimental verification and comparison of classical modal analysis techniques and recurrence plots sensitivity to damage size and location. Identification experiments were carried out for the laboratory object subjected to random and chirp excitations, respectively. In the course of carried out experiments, the process of damage propagation was simulated by the successive drilling into one of the object elements. Measured time histories of system responses were analyzed with the application of the classical modal analysis, recurrence plots (RP), cross recurrence plots (CRP) and joint recurrence plots (JRP) methods. Obtained results proved that the RP, CRP and JRP methods are much more sensitive to changes in dynamical system properties resulting from damage initialization and propagation than classical modal analysis methods and can be successfully applied to damage detection and tracking changes in the system natural frequencies.

1

Introduction

Recently, demands for reliable performance of mechanical systems, durability, safety of operation, comfort of exploitation, optimization of mass and dimensions have been increasing. For economic reasons, shorter design time, cost reduction, longer exploitation period, minimization of necessary inspections and repairs, have been required. Therefore development of reliable methods for early damage detection as well as research into sensitivity of already existing methods to damage appearance and propagation are of key importance. Nowadays, for the purposes of system identification and state diagnosing the methods of modal analysis are commonly used [1, 2, 3, 4]. Although the application of modal analysis methods to damage detection [5, 6, 7] provides satisfactory results, it should be considered that appearance of system unserviceabilities (such as cracks, clearances, etc.) invokes system nonlinear properties. Therefore, in the paper, application of the recurrence plots method [8, 9, 10] to damage detection is proposed. The method is dedicated to investigation into changes in dynamic behaviour of mechanical systems during their normal work. The main advantage of method application consists in the possibility of detecting transitions to chaotic behaviour, variability of parameters in time, cyclic processes, etc. on the basis of measured, relatively short, time histories of system dynamic responses. Moreover, interpretation of information contained in the recurrence plots can be supported by the estimation of the measures provided by the Recurrence Quantification Analysis (RQA). The main emphasis of the paper is put on the experimental verification and comparison of the efficiency of classical modal analysis techniques and recurrence plots. Identification experiments were carried out for the laboratory object (the steel frame) subjected to random and chirp excitations, respectively. In order to simulate the process of damage propagation, in the course of carried out identification experiments, the object was subjected to the successive drilling into one of its elements. Measured system responses were

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analysed with the application of the classical modal analysis, recurrence plots (RP), cross recurrence plots (CRP) and joint recurrence plots (JRP) methods. The structure of the paper is as follows. Section 2 concerns algorithms of the applied methods and considerations concerning their applicability to model-based damage detection. Identification experiment is described in the Section 3. There are also presented the results of damage detection (steel frame modelbased diagnostic) obtained with the application of the classical modal analysis methods and RP, CRP and JRP methods. Finally, the paper is concluded in section 4.

2

Applied methods

The concept of recurrences was introduced by Henri Poincaré [8] in 1890. Even though Poincaré's discovery has aroused great interest, it gained practical importance more than 70 years later due to the development of efficient computer systems [9]. In 1987, based on the Poincaré's achievements, Eckmann, Oliffson and Ruelle formulated the recurrence plots method (RP) [10]. Originally the method was used for the purposes of visualisation of system trajectories, especially in the higher dimensional phase spaces. The following development of the Recurrence Quantification Analysis (RQA) [11, 12, 13], providing the measures supporting interpretation of information contained in recurrence plots, has consolidated the method as a tool for nonlinear data analysis. Nowadays recurrence plots and their quantification – RQA are used for the purposes of detection of qualitative changes in the behaviour of dynamical systems [14]. Discussed methods find applications in numerous fields of research, such as technique [15], astrophysics [16, 17], biology [18], chemistry [19], geology [20], cardiology [21], neuroscience [22] and economy [23]. The recurrence plots method and its extensions, making it possible to analyse dependencies between two different systems by comparing their states (cross recurrence plots, CRP) or to compare different systems by looking for time instants when both of them recur simultaneously (joint recurrence plots, JRP), are briefly discussed in Sections 2.1 ÷ 2.3.

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Recurrence plot (RP)

The main idea of the RP method, introduced by Eckmann, Oliffson and Ruelle [10], consists in revealing all the times when the phase space trajectory of the considered dynamical system visits roughly the same area in the phase space. Such recurrence of a state x at time i at a different time j is marked within a twodimensional squared matrix [R], called recurrence matrix, with ones and zeros:

[R ] = Θ (ε − {x } − {x } ), i, j

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(1)

where N is the number of considered states xi, εi denotes a threshold distance, || || a norm and Θ( ) the Heaviside function. Graphical representation of such a matrix is known in the literature under the term 'recurrence plot' (RP). In other words, the recurrence plot represents graphically the matrix [Ri,j] given as:

[Ri, j ] = 01 :: 

{xi } ≈ {x j } , {xi } ≠ {x j }

i , j = 1, K , N

(2)

where {xi} ≈ {xj} are the points belonging to the neighbourhood of radius ε (defined according to the applied norm). In order to estimate recurrence plot (for the considered dynamical system) it is necessary to specify values of three parameters: threshold ε, time delay τ and embedding dimension m. Values of these parameters should be selected carefully, since they have significantly influence on the informative content of the estimated recurrence plot. For instance, if the value of ε is too small, there may be almost no recurrence points and the information concerning recurrence of the considered system states can be lost. On the

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contrary, if ε is chosen too large, almost every point is a neighbour of every other point, which leads to appearance of spurious structures visible on the obtained recurrence plot. In the literature various criterions for selection of threshold ε have been proposed. However, the value of ε corresponding to the assumed RP point density is usually selected. Appropriate time delay τ can be determined with the application of the auto-correlation or mutual information function [24, 26]. The mutual information function (MI) method, on the contrary to the linear autocorrelation function approach, takes into account also nonlinear correlations: MI = −∑ pij (τ ) ln ij

pij (τ )

(3)

pi p j

where pi denotes probability of finding the value of system time history in the ith interval, pij denotes probability that observation in a given time instant t belongs to the ith interval and observation in (t + τ) time instant to the jth interval. In practical applications, as the ‘reasonable’ value of τ, the value corresponding to the first minimum of the mutual information function is assumed. For the purposes of the smallest sufficient embedding dimension m estimation, the false nearest neighbours method [25] is frequently used. The method determines neighbours of each point of the considered system phase space trajectory for a given dimension of phase space mn and checks whether these points are still the closest neighbours in the mn+1 dimensional phase space. For the properly selected dimension of the phase space, the number of false nearest neighbours is close to zero. In the original definition of the method [10] the L2 norm is used and the neighbourhood radius is selected in such a way that it contains a fixed amount of states. For such a neighbourhood, the radius εi changes for each xi (i = 1, …, N) and Ri,j ≠ Rj,i, which leads to asymmetrical recurrence diagrams. This criterion can be adjusted in such a way that the recurrence point density has a fixed predetermined value. Such neighbourhood criterion is called fixed amount of nearest neighbours (FAN) and plays important role in estimation cross recurrence plots (CRPs), which is discussed in the further part of the paper. Recurrence Quantification Analysis is a method of nonlinear data analysis which quantifies the number and duration of recurrences of a dynamical system represented by its state space trajectory. Measures based on diagonal structures are able to find chaos-order transitions while measures based on vertical (horizontal) structures are able to find chaos-chaos transitions [13, 22]. Definitions of the most popular RQA measures, such as the Recurrence Rate (RR), Determinism (DET), Laminarity (LAM), Averaged diagonal line length (L), Trapping Time (TT), Longest diagonal line (Lmax), Longest vertical line (Vmax), Divergence (DIV), Entropy (ENTR) can be found in [13, 11, 12, 14].

2.2

Cross recurrence plot (CRP)

The cross recurrence plots (CRP) method was introduced in order to analyse relations between two systems by comparing their states [9]. However, the analysed data should represent the same observable (should come form the same or comparable process). The method can be treated as a generalization of the linear cross-correlation function. If two dynamical systems represented by the {xi} and {xj} trajectories in the d-dimensional phase space are considered, the cross recurrence matrix is defined as follows [9]:

[CR

x, y i, j

(ε )] = Θ (ε − {xi } − {x j } ),

i = 1, K , N

j = 1, K , M

(4)

It should be stressed that both considered systems are represented in the same phase space. While using experimental data, it may happen that embedding dimensions estimated for both time series are not equal. In such a case, the higher value of embedding parameter should be selected. Before computing the CRP diagram, the components of {xi} and {xj} should be normalised or (alternatively) the FAN criterion should be used.

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Joint recurrence plot (JRP)

As it was stated before, CRP method can not be used for the purposes of comparing two physically different time histories. On the contrary to the CRP method, the JRP method is dedicated to comparing different dynamical systems by considering the recurrence of their trajectories in their respective phase spaces separately and looking for time instances when both of them recur simultaneously [9]. For two dynamical systems represented by the {xi} and {xj} trajectories, the joint recurrence matrix is defined as:

[JRP (ε x, y i, j

x

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)]

,ε y = Θ ε x −

{xi } − {x j } )Θ (ε y − {yi } − {y j } ),

i, j = 1, K , N .

(5)

While considering practical engineering applications, it can be stated the JRPs are more suitable than CRPs for research into two interacting system that influence each other or adapt to each other [9]. On the other hand, the CRPs are more appropriate than JRPs to look for relationship between dynamic behaviour of the same system subjected to different processes / external influences.

2.4

RP, CRP and JRP methods applicability to damage detection

In order to prove and illustrate the applicability of the RP, CRP and JRP methods to system diagnostics a number of analytical data was analysed. First, the properties of system moving in a circular path with uniform speed were analysed. System dynamic motion equation can be written as: y = A ⋅ sin ωt

(6)

The following motion parameters were assumed: A = 1.5, ω = 0.5 [1/s], t = 0, 0.1, …, 64 [s]. Vibration displacements time history and phase space trajectory of the considered system are presented in the Fig. 1. 1.5

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Figure 1: Vibration displacements time history (a) and phase space trajectory of the considered system Optimal value of time delay τ was established by looking for the first minimum of the mutual information function (MI) while minimal dimensions of the phase space necessary to represent the trajectory of the considered system were determined with the application of the false nearest neighbours method. It was estimated that τ = 29 (Fig. 2a) and m = 2 (Fig. 2b), respectively. Recurrence (RP), cross recurrence (CRP) and joint recurrence (JRP) plots determined for the considered system are presented in the Fig. 3. Determined diagrams are identical. Diagonal lines are smooth and continuous. Vertical distances between successive lines of the diagram are equal, reflecting the regularity (repeatability) of system dynamic properties in time.

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Figure 2: Optimal value of time delay τ = 29 determined with the use of the mutual information function (MI) method (a), embedding dimensions (m = 2) estimated by means of the false nearest neighbours method (b) CRP(y, y) τ = 30, m = 2, ε = 0.1, FAN

RP(y) τ = 30, m = 2, ε = 0.1, FAN

JRP(y, y) τ = 30, m = 2, ε = 0.1, FAN

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Figure 3: Diagrams a) RP(y), b) CRP(y, y), c) JRP(y, y) determined for the system moving in a circular path with uniform speed 1 0.8 0.6

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Figure 4: Vibration displacements time histories of the considered systems ( y1,  y2) Next, two systems moving in a circular path with uniform speed were analysed: y1 = A ⋅ sin ωt = A ⋅ sin (2π ⋅ f ⋅ t )

(7)

y 2 = A ⋅ sin[2π ⋅ ( f + ∆f ) ⋅ t ]

(8)

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The following parameters were assumed: A = 1, f = 5 [Hz], ∆f = 0.5 [Hz], t = 0, 0.0006, …, 10 [s]. Vibration displacements time histories of the considered system are presented in the Fig. 4. RP(y1) τ = 80, m = 3, ε = 0.1, FAN

JRP(y1, y2) τ = 80, m = 3, ε = 0.1, FAN

CRP(y1, y2) τ = 80, m = 3, ε = 0.1, FAN

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Figure 5: Diagrams a) RP(y1), b) CRP(y1, y2), c) JRP(y1, y2) determined for the system moving in a circular path with uniform speed Recurrence plot determined on the basis of the signal y1, cross recurrence and joint recurrence plots determined for the signals y1 and y2 are presented in the Fig. 5. The difference between diagrams RP(y1) and CRP(y1, y2) is hardly visible. In case of joint recurrence plot JRP(y1, y2) it is clearly visible that the trajectories of considered systems initially recur simultaneously in their respective phase spaces (joint recurrence is observed). Later in time, joint recurrence decays, which is illustrated by white areas in the upper left and lower right corner of the estimated JRP diagram. Finally, the response of mechanical system y12 (Fig. 6) represented by the sum of harmonic components was considered:  y12 = y1 + y 2   y1 = sin( 2π ⋅ f1 ⋅ t )  y = sin( 2π ⋅ f ⋅ t ) 2  2

(9)

where t = 0, 0.0006, ..., 10], f1 = 5 [Hz], f2 = 7 [Hz]. 2

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Figure 6: Vibration displacements time histories: a) y12, b) y12 (−) and y12' Then, frequency of the second component y2 was increased by ∆f2 = 0.5 [Hz]. Modified signal was denoted by y2 ' and the sum of signals y1 and y2 ' was calculated:

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 y12 ' = y1 + y2 '   y2 ' = sin(2π ⋅ [ f 2 + ∆f 2 ]⋅ t ).

(10)

Time histories of signals y12 and y12' are presented in the Fig. 6 while determined RP(y12), CRP(y12, y12') and JRP(y12, y12') diagrams are shown in the Fig. 7. RP(y12) m = 4, τ =60, ε = 0.1, FAN

CRP(y12, y12') m = 4, τ =60, ε = 0.1, FAN

JRP(y12, y12') m = 4, τ =60, ε = 0.1, FAN

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Figure 7: Diagrams a) RP(y12), b) CRP(y12, y12'), c) JRP(y12, y12') Both CRP and JRP methods proved to be sensitive to changes in the frequencies of signal harmonic components (even by small values, such as ∆f = 0.5 [Hz]). It is especially important in system diagnostics since at the early stage of damage propagation changes in the system natural frequencies are often observed.

3 3.1

Results of steel frame damage detection Identification experiment

At the stage of modal experiment planning it was assumed that energy necessary to excite the tested system would be delivered in a controlled manner using electro-dynamic exciter. During the experiments the Robotron exciter suspended on a dedicated stand was used. The excitation was provided to the system through the intermediate element (stinger). The point of excitation application is indicated in the Fig. 8. During the carried out experiments the system was fixed to the steel base (Fig. 8). The network of measurement points consisted of 8 points. While creating the frame geometrical model, the Cartesian coordinate system with the origin in the point 8 was selected (Fig. 9a). The following measurement directions were assumed: − ox - along construction base, − oy - along the shorter side of the steel base, − oz - along the longer side of frame, from bottom to top. In the course of the identification experiment, the controlled damage (in the form of hole) was introduced (Fig. 9b). The hole diameter was gradually increased - initially the structure was undamaged. In the consecutive measurement sessions the hole diameter equaled 2, 3, 4, 5, 6, 7, 8, 10 [mm]. In each measurement session the network of measurement points remained unchanged. The following equipment was used: − − −

28-channeled analyzer of dynamical signals piezoelectric accelerometers impedance heads

SCADAS III PCB 356A16 type PCB 288D01 type

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electro-dynamic exciter power amplifier software module TestLab Spectral Testing

Robotron ModalShop 2050E02 LMS Company

Figure 8: The point of excitation application in the course of modal experiments 500 excitation

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Figure 9: Network of measurement points and assumed coordinate system (a), introduced damage (b) The most important features of the performed experiments are as follows: − − − − − − − −

Two types of tests were carried out: with the application of: − random excitation (white noise) - for the purposes of modal analysis, − chirp excitations - treated as a basis for RP, CRP and JRP diagrams estimation. Exciting force acted on the system in the ox direction of the assumed coordinate system, point of force application is indicated in the Fig. 9. In each measurement session characteristics of exciting forces were registered. System responses (in the form of vibration accelerations time histories) were measured in each measurement point in three, mutually perpendicular, directions. On the basis of measured signals, transfer functions [m/s2/N] between excitation and system responses were estimated. In the course of identification experiments, the coherence functions between exciting forces and system responses were monitored. Measurement frequency ranged from 1 Hz to 400 Hz. Frequency resolution of carried out measurements was set at 0.25 Hz.

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Identification experiment consisted of nine measurement sessions (tests) corresponding to the subsequent stages of the change in the hole diameter.

3.2

Results of classical modal analysis

Estimation of modal parameters was preceded by the analysis of data quality. Sample spectra of exciting forces amplitudes are presented in the Fig. 10a. It is visible that the amplitude of the exciting force was sufficiently uniform in the considered frequency band for all the tests. Selected characteristics of SUM indicator are shown in the Fig. 10. Local maxima of the SUM characteristics indicate the natural frequencies that have been mapped in the recorded data for the frequency range 1 ÷ 400 Hz. In case of the considered object, in each test, SUM indicator functions were determined on the basis of 25 characteristics. SUM indicators were used in the process of natural frequencies estimation. 11

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f1 = 8.29 Hz

f3 = 43.21 Hz

f4 = 91.03 Hz

f6 = 137.82 Hz

Figure 11: Selected identified mode shapes of the undamaged system

f10 = 398.64 Hz

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System modal parameters were estimated with the use of the Modal Analysis module of the Test.Lab software package. From the obtained results, with the application of the modal model consolidation procedures from the VIOMA software, the most representative mode shapes were selected. Results of modal parameters estimation with the application of the PolyMAX algorithm are gathered in the Table 1 and Table 2. Selected identified mode shapes are presented in the Fig. 11.

0 mm Natural No. Damp. freq. Hz 0% 1 8.29 0.17 2 33.08 5.29 3 43.21 1.52 4 91.03 1.77 5 118.73 5.04 6 137.82 1.87 7 168.33 0.10 8 270.53 1.57 9 289.00 1.06 10 398.64 0.32

2 mm Natural Damp. freq. Hz 0% 8.34 1.03 33.97 5.02 40.67 0.49 90.50 2.07 115.13 9.00 137.29 1.96 167.74 0.61 269.12 1.73 288.62 0.97 390.76 1.46

3 mm Natural Damp. freq. Hz 0% 8.31 0.52 34.62 3.71 91.20 1.74 125.54 7.71 138.21 1.86 167.62 0.05 269.77 1.67 289.00 0.96 390.27 0.52

4 mm Natural Damp. freq. Hz 0% 8.32 1.06 34.67 3.94 41.39 0.65 91.37 1.74 124.59 8.92 138.28 1.95 167.56 0.23 269.72 1.69 288.98 0.95 391.95 1.56

5 mm Natural Damp. freq. Hz 0% 8.30 0.51 34.57 5.11 41.29 0.59 91.53 1.66 125.72 7.02 138.69 2.14 269.73 1.64 289.01 1.00 393.17 1.87

Table 1: Results of modal parameters estimation for undamaged system (φ = 0 mm) and damaged system (φ = 2, 3, 4, 5 mm)

0 mm No. Natural Damp. freq. Hz 0% 1 8.29 0.17 2 33.08 5.29 3 43.21 1.52 4 91.03 1.77 5 118.73 5.04 6 137.82 1.87 7 168.33 0.10 8 270.53 1.57 9 289.00 1.06 10 398.64 0.32

6 mm Natural Damp. freq. Hz 0% 8.32 0.94 34.48 5.07 41.72 0.36 91.54 1.72 125.66 6.20 138.82 2.03 269.83 1.69 289.04 1.02 393.53 1.63

7 mm Natural Damp. freq. Hz 0% 35.03 3.73 91.07 1.98 123.73 7.43 138.55 2.09 167.88 0.12 270.06 1.68 289.24 0.93 390.73 1.38

8 mm Natural Damp. freq. Hz 0% 8.31 1.05 35.36 3.49 91.44 1.81 129.56 7.60 139.24 2.26 269.69 1.73 289.30 0.93 390.11 1.12

10 mm Natural Damp. freq. Hz 0% 8.32 0.78 34.65 3.18 91.33 1.75 129.41 7.57 138.70 2.25 267.62 1.83 289.09 0.95 387.95 2.13

Table 2: Results of modal parameters estimation for undamaged system (φ = 0 mm) and damaged system (φ = 6, 7, 8, 10 mm) While analysing the results of modal parameters estimation, carried out for the increasing size of introduced damage, no relation between values of estimated parameters and damage size was observed. In case of the considered system, even for damage amounting to 20% of frame element volume (hole of φ = 10 mm), modal analysis (PolyMAX algorithm) failed to detect presence of damage.

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Results obtained with the application of the RP, CRP and JRP methods

Time histories of the considered system dynamic responses to random excitation were analyzed with the application of the RP, CRP and JRP methods. In the carried out research, the FAN definition of the neighbourhood was used. The smallest sufficient embedding dimension m was estimated with the use of the false nearest neighbours algorithm [25]. Appropriate time delay τ was determined by searching for the first minimum of the mutual information function [24]. The phase space of the considered systems was reconstructed by delay embedding. All the necessary computations were performed with the use of functions implemented in the CRP Toolbox for MATLAB [26, 27, 28].

Figure 12: Diagrams a) RP(y), b) CRP(y, ydam), c) JRP(y, ydam) determined on the basis of system responses measured in the point 5 in the ox direction. Assumed parameters: m = 6, τ = 1, ε = 0.2, FAN criterion. Below the results obtained for the system with damage introduced (in the form of hole of diameter d = 10 [mm]) are discussed. Due to space limitations, only diagrams determined for the point 5 are presented (Fig. 12 ÷ Fig. 14). Table 2 contains results of the RQA analysis of system dynamic responses measured in points 2, 5 and 7 (Fig. 9a) in three directions. The following notation was used: − −

y - time history of the undamaged system dynamic response (measured in a given point), ydam - time history of the damaged system dynamic response (measured in a given point).

Figure 13: Diagrams a) RP(y), b) CRP(y, ydam), c) JRP(y, ydam) determined on the basis of system responses measured in the point 5 in the oy direction. Assumed parameters: m = 5, τ = 1, ε = 0.2, FAN criterion. Cross recurrence CRP(y, ydam) and joint recurrence JRP(y, ydam) plots determined for the signals y and ydam differ significantly from the recurrence diagram RP(y) estimated for the undamaged system. In order to asses qualitatively observed differences, the RQA analysis was carried out. Results of the RQA analysis of system dynamic responses measured in points 2, 5 and 7 in three directions are gathered in the Table 2.

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Figure 14: Diagrams a) RP(y), b) CRP(y, ydam), c) JRP(y, ydam) determined on the basis of system responses measured in the point 5 in the oz direction. Assumed parameters: m = 6, τ = 1, ε = 0.2, FAN criterion.

Type of analysis

Point

RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam) RP(y) CRP(y, ydam) JRP(y, ydam)

p2 (ox) p2 (ox) p2 (ox) p2 (oy) p2 (oy) p2 (oy) p2 (oz) p2 (oz) p2 (oz) p5 (ox) p5 (ox) p5 (ox) p5 (oy) p5 (oy) p5 (oy) p5 (oz) p5 (oz) p5 (oz) p7 (ox) p7 (ox) p7 (ox) p7 (oy) p7 (oy) p7 (oy) p7 (oz) p7 (oz) p7 (oz)

RQA measures RR 0.1998 0.2001 0.0406 0.1998 0.2000 0.0413 0.1998 0.2000 0.0408 0.1998 0.2000 0.0405 0.1997 0.1999 0.0406 0.1998 0.1999 0.0397 0.1998 0.2000 0.0402 0.1997 0.1999 0.0405 0.1998 0.2000 0.0418

DET L ENTR LAM TT T1 0.9330 6.2302 2.5407 0.3524 2.4940 5.0050 0.9312 6.1329 2.5226 0.3183 2.5050 4.9987 0.8291 3.9127 1.8577 0.0730 2.1439 24.6240 0.6833 8.3542 2.8602 0.1826 2.4793 5.0063 0.7032 8.5800 2.8868 0.1812 2.3918 5.0000 0.4854 5.1944 2.2535 0.0164 2.0267 24.2249 0.3932 5.5232 2.3525 0.5329 2.3927 5.0063 0.4131 5.6520 2.3832 0.5897 2.4015 5.0000 0.1715 3.7774 1.7368 0.2169 2.0835 24.5052 0.5002 5.9918 2.4713 0.1723 2.4934 5.0063 0.4478 5.3875 2.3369 0.2182 2.4031 5.0000 0.2100 3.6664 1.7046 0.0260 2.0902 24.6718 0.9255 9.3904 3.0059 0.2294 2.7053 5.0075 0.9259 8.7227 2.9438 0.2128 2.5668 5.0012 0.8409 5.3628 2.3207 0.0205 2.0249 24.6251 0.9415 6.8508 2.6742 0.2208 2.3545 5.0063 0.9202 6.0043 2.5006 0.2259 2.3678 5.0012 0.8462 4.1663 1.9626 0.0298 2.0214 25.1668 0.5996 3.3353 1.1024 0.2623 2.1227 5.0063 0.5999 3.2994 1.1113 0.2799 2.1103 5.0000 0.3101 2.5636 0.7295 0.0391 2.0056 24.8659 0.9323 10.1094 3.1266 0.2334 2.7360 5.0075 0.9407 10.5110 3.1924 0.2152 2.6496 5.0012 0.8709 6.1768 2.5277 0.0197 2.0547 24.6926 0.8363 4.6305 1.6139 0.1433 2.6222 5.0063 0.7897 4.0756 1.4902 0.1369 2.4007 5.0000 0.5808 2.9532 1.1119 0.0192 2.2120 23.9349

T2 0.4784 0.4391 25.6235 0.3836 0.3868 24.4279 0.5102 0.4958 27.6211 0.4308 0.4799 25.0111 0.3318 0.3452 24.8827 0.4358 0.4424 25.5512 0.4478 0.4475 25.3634 0.3193 0.2977 24.9448 0.3272 0.3958 24.1896

Table 3: Results of the RQA analysis (for selected measurement points) Carried out research revealed that values of the RR, DET, L, ENTR, LAM and TT measures estimated for JRP(y, ydam) diagrams are significantly lower than values of these measures estimated for the RP diagram of undamaged system. On the contrary, values of the T1 and T2 measures estimated for JRP(y, ydam) diagrams are significantly higher than values of these measures estimated for the RP diagram of undamaged system. While comparing RQA measures estimated for RP diagrams of undamaged system and CRP(y, ydam) diagrams, these relations are not so straightforward.

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Conclusions and final remarks

The paper concerns the issues of experimental verification of classical modal analysis techniques and recurrence plots sensitivity to damage initialization and propagation. The first group of methods is nowadays commonly used for linear system identification and model-based diagnostics, while the second is applied to analysis of nonlinear data from various fields of research, such as technique, astrophysics, biology, chemistry, geology, cardiology, neuroscience and economy. In the paper the authors discussed the idea of recurrence plots (RP), cross recurrence plots (CRP), joint recurrence plots (JRP) and Recurrence Quantification Analysis (RQA), briefly presented the most commonly used criterions and methods of embedding parameters (m, τ, ε) selection. In order to prove and illustrate the applicability of the RP, CRP and JRP methods to system diagnostics a number of analytical data was analysed. The research was carried out for the laboratory steel frame. In the course of carried out experiments, the process of damage propagation was simulated by the successive drilling into one of the object elements. For each size of damage two types of modal tests were performed: with random and chirp excitation, respectively. Measured time histories of system responses to chirp excitation were analyzed with the application of the classical modal analysis, while the analysis of system responses time histories to random excitation was carried out with the application of RP, CRP and JRP methods. The smallest sufficient embedding dimension m was estimated with the use of the false nearest neighbours algorithm while the appropriate time delay τ was determined by searching for the first minimum of the mutual information function. Results of modal analysis carried out for the increasing size of introduced damage revealed no relation between values of estimated parameters and size of introduced damage. Even for damage amounting to 20% of frame element volume, applied PolyMAX algorithm failed to detect presence of damage. JRP method proved to be more suitable for model-based diagnostics than the CRP method. Carried out research revealed that values of the RR, DET, L, ENTR, LAM and TT measures estimated for JRP(y, ydam) diagrams are significantly lower than values of these measures estimated for the RP diagram of undamaged system. On the contrary, the values of T1 and T2 measures estimated for JRP(y, ydam) diagrams are significantly higher than values of these measures estimated for the RP diagram of undamaged system. While comparing RQA measures estimated for RP diagrams of undamaged system and CRP(y, ydam) diagrams, these relations are not so straightforward. Obtained results proved that the RP, CRP and JRP methods are much more sensitive to changes in dynamical system properties resulting from damage initialization and propagation than classical modal analysis methods and can be successfully applied to damage detection and tracking changes in the system natural frequencies. However, the further research into sensitivity of the discussed methods has to be performed.

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