Marmolejo et al 2019c - Revision history

Rimni at 14:17, 2 December 2020

2020-12-02T14:17:41Z

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Rimni: /* 3. Numerical simulation (closed modal frequencies) */

2020-12-02T13:59:15Z

‎3. Numerical simulation (closed modal frequencies)

Rimni at 13:57, 2 December 2020

2020-12-02T13:57:53Z

Rimni at 13:52, 2 December 2020

2020-12-02T13:52:47Z

Rimni: /* 3. Numerical simulation (closed modal frequencies) */

2020-12-02T13:50:42Z

‎3. Numerical simulation (closed modal frequencies)

Rimni at 13:34, 2 December 2020

2020-12-02T13:34:51Z

Rimni at 11:31, 21 January 2020

2020-01-21T11:31:15Z

Rimni at 11:29, 21 January 2020

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Rimni at 11:28, 21 January 2020

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Rimni at 11:25, 21 January 2020

2020-01-21T11:25:52Z

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 |-style="text-align:center"
-! MODE !!  Theoretical frequencies<br/>(Hz) !!  colspan="3"  |Identification by wavelets !! Frequency error<br/>(%)  !! Damping error<br/>(%)
+! style="vertical-align: top;"| MODE !!  Theoretical frequencies<br/>(Hz) !!  colspan="3" style="vertical-align: top;" |Identification by wavelets !! Frequency error<br/>(%)  !! Damping error<br/>(%)
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−	! MODE !! Theoretical frequencies !! colspan="3" \|Identification by wavelets !! Frequency error !! Damping error	+	! MODE !! Theoretical frequencies<br/>(Hz) !! colspan="3" \|Identification by wavelets !! Frequency error<br/>(%) !! Damping error<br/>(%)
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−	~~\| style="text-align: center;vertical-align: top;"\| (Hz)~~	+
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−	~~\|style="text-align: center;vertical-align: top;"\|(%)~~	+
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 |-style="text-align:center"
-! MODE !!  !! Theoretical frequencies !!  colspan="3"  |Identification by wavelets !! Frequency error !! Damping error
+! MODE !!  Theoretical frequencies !!  colspan="3"  |Identification by wavelets !! Frequency error !! Damping error
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 ==3. Numerical simulation (closed modal frequencies)==
-The numerical example is a simple supported beam which is added with <math display="inline">8000\, kg/m</math> symmetrically distributed masses (Figure <span id='cite-Figure_2'></span>[[#Figure_2|2]]). For this  beam <math display="inline">L=</math><math>60\, m</math>, <math display="inline">E=60\, GPa</math>, <math display="inline">I=</math><math>6.75\, {m}^{4}</math>, <math display="inline">{m}_{l}=150\, \frac{kN}{\frac{g}{m}}</math>, and <math display="inline">\zeta =</math><math>0.05</math>. The first five natural frequencies are: <math display="inline">0.194</math>, <math display="inline">0.435</math>, <math display="inline">2.511</math>, <math display="inline">2.961</math>,  and <math display="inline">6.491\, Hz</math>.
+The numerical example is a simple supported beam which is added with <math display="inline">8000\, kg/m</math> symmetrically distributed masses (<span id='cite-Figure_2'></span>[[#img-2|Figure 2]]). For this  beam <math display="inline">L=</math><math>60\, m</math>, <math display="inline">E=60\, GPa</math>, <math display="inline">I=</math><math>6.75\, {m}^{4}</math>, <math display="inline">{m}_{l}=150\, \frac{kN}{\frac{g}{m}}</math>, and <math display="inline">\zeta =</math><math>0.05</math>. The first five natural frequencies are: <math display="inline">0.194</math>, <math display="inline">0.435</math>, <math display="inline">2.511</math>, <math display="inline">2.961</math>,  and <math display="inline">6.491\, Hz</math>.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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 Stationary sensor <math display="inline">19.5\, m</math> away from the left support. Mobile sensor velocity  <math display="inline">\, 0.0167\frac{m}{s}</math>. Gaussian white noise acceleration excitation input at the beam supports. Response signals conditioned using Hilbert Transform and sub-sampling. Morlet wavelet used for the modal identification. 256 frequency bins for this wavelet; Wavelength approximately equal to the square root of length of the signals (half-length the Morlet analyzing wavelet scale); lower and the upper normalized frequency bounds of the analyzed signal, <math display="inline">{f}_{min}=</math><math>0.01</math> and <math display="inline">{f}_{max}=0.50</math>.
-Figures <span id='cite-Figure_3'></span>[[#Figure_3|3]] and <span id='cite-Figure_4'></span>[[#Figure_4|4]] show the stationary sensor recorded signal time history along with its scalogram, and the mobile sensor recorded signal time history traveling at 0.0167 m/s along with its scalogram, respectively.
+Figures <span id='cite-Figure_3'></span>[[#img-3|3]] and <span id='cite-Figure_4'></span>[[#img-4|4]] show the stationary sensor recorded signal time history along with its scalogram, and the mobile sensor recorded signal time history traveling at 0.0167 m/s along with its scalogram, respectively.
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-Modal frequencies obtained by selecting peaks (Figure <span id='cite-Figure_5'></span>[[#Figure_5|5]]); damping ratios obtained using the stationary sensor signal scalogram [1]; mode shapes obtained using the response signals wavelet transforms via Eqs.(6) and (7); comparison made between the identified and the exact mode shapes; in this example 59 modal coordinates were identified.
+Modal frequencies obtained by selecting peaks (<span id='cite-Figure_5'></span>[[#img-5|Figure 5]]); damping ratios obtained using the stationary sensor signal scalogram [1]; mode shapes obtained using the response signals wavelet transforms via Eqs.(6) and (7); comparison made between the identified and the exact mode shapes; in this example 59 modal coordinates were identified.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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-Comparative results performed from the numerical simulation and the theoretical example; identified mode shapes <math display="inline">{\phi }_{j}</math> compared to the exact mode shapes <math display="inline">{\phi }_{k}</math> using the Modal Assurance Criterion (MAC) [31] (Figure <span id='cite-Figure_6'></span>[[#Figure_6|6]]); damping ratio identified using technique identification by Le [1]; The Time-Frequency Toolbox software (TFTB) used for the wavelet signals processing [29,32]. Table <span id='cite-Table_1'></span>[[#Table_1|1]] summarizes the results of the modal parameter identification. As shown, the errors in natural frequencies are less than 1.6%, the maximum modal damping error is 10.0 %, and MAC values are all very close to 1.0.
+Comparative results performed from the numerical simulation and the theoretical example; identified mode shapes <math display="inline">{\phi }_{j}</math> compared to the exact mode shapes <math display="inline">{\phi }_{k}</math> using the Modal Assurance Criterion (MAC) [31] (<span id='cite-Figure_6'></span>[[#img-6|Figure 6]]); damping ratio identified using technique identification by Le [1]; The Time-Frequency Toolbox software (TFTB) used for the wavelet signals processing [29,32].  <span id='cite-Table_1'></span>[[#Table_1|Table 1]] summarizes the results of the modal parameter identification. As shown, the errors in natural frequencies are less than 1.6%, the maximum modal damping error is 10.0 %, and MAC values are all very close to 1.0.
-In Figure <span id='cite-Figure_7'></span>[[#Figure_7|7]], the <math>X</math> axis is the mobile sensor’s velocity. The Y axis is the variations in the Identified Modal Coordinates (IMC) per beam’s length unit. Two modes are considered correlated when the MAC value is at least  equal to 0.95, corresponding to 18 degrees angle [31]. For instance, if the speed of the mobile sensor is 0.024 m/s, its correspondent IMC is 2 (Identified Modal Coordinates per meter).
+In <span id='cite-Figure_7'></span>[[#img-7|Figure 7]], the <math>X</math> axis is the mobile sensor’s velocity. The Y axis is the variations in the Identified Modal Coordinates (IMC) per beam’s length unit. Two modes are considered correlated when the MAC value is at least  equal to 0.95, corresponding to 18 degrees angle [31]. For instance, if the speed of the mobile sensor is 0.024 m/s, its correspondent IMC is 2 (Identified Modal Coordinates per meter).
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
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 |}
+<div id='Table_1'></div>
 <div class="center" style="font-size: 75%;">'''Table 1'''. Numerical simulation results summary
 </div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;width:80%;font-size:85%;"
+{| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
-|- style="border-top: 1pt solid black;"
+|-style="text-align:center"
-| style="text-align: center;vertical-align: top;"|'''MODE'''
+! MODE !!  !! Theoretical frequencies !!  colspan="3"  |Identification by wavelets !! Frequency error !! Damping error
-|  style="text-align: center;vertical-align: top;"|'''Theoretical frequencies'''
+|-
-| colspan="3"  style="text-align: center;vertical-align: top;"|'''Identification by wavelets'''
-| style="text-align: center;vertical-align: top;"|'''Frequency error'''
-|  style="text-align: center;vertical-align: top;"|'''Damping error'''
-|-style="border-bottom: 1pt solid black;"
+|
 | style="text-align: center;vertical-align: top;"| (Hz)
@@ Line 305: / Line 306: @@
+|
+|
-|-style="border-top: 1pt solid black;"
+|-
 |  style="text-align: center;vertical-align: top;"|1
 |  style="text-align: center;vertical-align: top;"|0.194
@@ Line 338: / Line 339: @@
 |  style="text-align: center;vertical-align: top;"|6.000
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|5
+|  style="text-align: center;vertical-align: top;"|5
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|6.491
+|  style="text-align: center;vertical-align: top;"|6.491
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|6.413
+|  style="text-align: center;vertical-align: top;"|6.413
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.052
+|  style="text-align: center;vertical-align: top;"|0.052
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0.999
+|  style="text-align: center;vertical-align: top;"|0.999
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1.202
+|  style="text-align: center;vertical-align: top;"|1.202
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|4.000
+|  style="text-align: center;vertical-align: top;"|4.000
 |}
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
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-The noise influence was evaluated too. As per observable in Figure <span id='cite-Figure_8'></span>[[#Figure_8|8]], signal-to-noise ratio, results for values greater than <math display="inline">15\, dB</math> lie very close to 1 (MAC value). Such good identification results were obtained from the simulated ambient contaminated response data.
+The noise influence was evaluated too. As per observable in Figure <span id='cite-Figure_8'></span>[[#img-8|Figure 8]], signal-to-noise ratio, results for values greater than <math display="inline">15\, dB</math> lie very close to 1 (MAC value). Such good identification results were obtained from the simulated ambient contaminated response data.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-

@@ Line 43: / Line 43: @@
 Structural Health Monitoring (SHM) is an essential tool to determine the safety, usability and maintainability of critical structures such as bridges, buildings or dams for example. Structural modal parameters, that is natural frequencies, mode shape vectors, and damping ratios, are the most commonly used condition indexes for vibration based SHM and damage detection. Currently, a modal model of a structure can be derived by an Operational Modal Analysis (OMA) showing significant advantages over procedures requiring known excitation inputs; low cost, broadband excitation, and the continuous use of the structures in real-life operational conditions [1–4].
-In a traditional modal identification method (Figure 1<span id='cite-Figure_1a'></span>[[#Figure_1a|(a)]]), a commonly large number of stationary sensors are distributed in pre-defined locations along the structure. The modal coordinates are identified over such discrete points. To improve the spatial resolution additional gear, additional sensor setups as well as experienced staff for proper configuration would be required, a process which is further challenged by the physical constraints of a structure.
+In a traditional modal identification method (<span id='cite-Figure_1a'></span>[[#img-1|Figure 1(a)]]), a commonly large number of stationary sensors are distributed in pre-defined locations along the structure. The modal coordinates are identified over such discrete points. To improve the spatial resolution additional gear, additional sensor setups as well as experienced staff for proper configuration would be required, a process which is further challenged by the physical constraints of a structure.
-In mobile sensor setups (Figure 1<span id='cite-Figure_1b'></span>[[#Figure_1b|(b)]]), sensors are mounted on mobile platforms to move along the structure as desired. A mobile sensing scheme offers clear improvements where a fixed sensor scheme falls short. With mobile sensing, few sensors, almost basic connectivity and lesser setup testing time can provide more spatial information than a dense fixed sensors network [5–8].
+In mobile sensor setups (<span id='cite-Figure_1b'></span>[[#img-1|Figure 1(b)]]), sensors are mounted on mobile platforms to move along the structure as desired. A mobile sensing scheme offers clear improvements where a fixed sensor scheme falls short. With mobile sensing, few sensors, almost basic connectivity and lesser setup testing time can provide more spatial information than a dense fixed sensors network [5–8].
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-

@@ Line 524: / Line 524: @@
 |  style="text-align: center;vertical-align: top;padding:5px;"|'''Frequency diff.'''
 |  style="text-align: center;vertical-align: top;padding:5px;"|'''Damping diff.'''
-|- style="border-bottom: 1pt solid black;"
+|- style="border-bottom: 1pt solid black;border-top: 1pt solid black;"
+|
 |  style="text-align: center;vertical-align: top;padding-bottom:5px;"|'''f (Hz)'''

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 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:75%;"
 |-
-|  style="border-top: 1pt solid black;text-align: left;vertical-align:top;" |Identified Element
+|  style="border-top: 1pt solid black;text-align: left;vertical-align:top;" |'''Identified Element'''
-|  style="border-top: 1pt solid black;text-align: center;"|Mobile Platform Hz.
+|  style="border-top: 1pt solid black;text-align: center;"|'''Mobile Platform''' Hz.
-|  style="border-top: 1pt solid black;text-align: center;"|Mobile Sensor Hz.
+|  style="border-top: 1pt solid black;text-align: center;"|'''Mobile Sensor''' Hz.
-|  style="border-top: 1pt solid black;text-align: center;"|Stationary Sensor Hz.
+|  style="border-top: 1pt solid black;text-align: center;"|'''Stationary Sensor''' Hz.
-|  style="border-top: 1pt solid black;text-align: center;"|Stationary- Mobile Sensors Hz.
+|  style="border-top: 1pt solid black;text-align: center;"|'''Stationary- Mobile Sensors''' Hz.
 |-style="border-bottom: 1pt solid black;"
+|

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 |}
-is defined by putting all these ratios together at each time <math display="inline">b</math>. Eq. 28 is an estimation of the time-evolution of the complex mode shape <math display="inline">{U}_{j}</math> [23]. The Real Part of <math display="inline">{U}_{j}(b)</math> corresponds to the <math display="inline">j</math>th mode shape <math display="inline">{\varnothing }_{j}(b)</math>  [1,22,26,28], which is:
+is defined by putting all these ratios together at each time <math display="inline">b</math>. Eq. (28) is an estimation of the time-evolution of the complex mode shape <math display="inline">{U}_{j}</math> [23]. The Real Part of <math display="inline">{U}_{j}(b)</math> corresponds to the <math display="inline">j</math>th mode shape <math display="inline">{\varnothing }_{j}(b)</math>  [1,22,26,28], which is:
 {| class="formulaSCP" style="width: 100%; text-align: center;"