Figure 1.
Operational modal analysis (OMA) of time-invariant structure based on locality preserving projections (LPP) algorithm.
Figure 1.
Operational modal analysis (OMA) of time-invariant structure based on locality preserving projections (LPP) algorithm.
Figure 2.
The process of moving the moving window.
Figure 2.
The process of moving the moving window.
Figure 3.
The moving window principle processing with time.
Figure 3.
The moving window principle processing with time.
Figure 4.
OMA of linear time-varying structure based on moving window locality preserving projections (MWLPP) algorithm.
Figure 4.
OMA of linear time-varying structure based on moving window locality preserving projections (MWLPP) algorithm.
Figure 5.
The mass slow-time-varying 3-DOF (degree of freedom) structure with external force.
Figure 5.
The mass slow-time-varying 3-DOF (degree of freedom) structure with external force.
Figure 6.
The Gaussian white noise and three object displacement response signal graphs. (a) Gaussian white noise; (b) The first object displacement response signal; (c) The second object displacement response signal; (d) The third object displacement response signal.
Figure 6.
The Gaussian white noise and three object displacement response signal graphs. (a) Gaussian white noise; (b) The first object displacement response signal; (c) The second object displacement response signal; (d) The third object displacement response signal.
Figure 7.
The real natural frequency of the mass slow-time-varying 3-DOF structure.
Figure 7.
The real natural frequency of the mass slow-time-varying 3-DOF structure.
Figure 8.
The natural frequency identified by MWLPP of the mass slow-time-varying 3-DOF structure when (a) t = 50.025 s, (b) t = 1200 s and (c) t = 1974.375 s.
Figure 8.
The natural frequency identified by MWLPP of the mass slow-time-varying 3-DOF structure when (a) t = 50.025 s, (b) t = 1200 s and (c) t = 1974.375 s.
Figure 9.
The density slow-time-varying cantilever beam structure with external force.
Figure 9.
The density slow-time-varying cantilever beam structure with external force.
Figure 10.
The Gaussian white noise (a) and 1st (b), 20th (c) and 40th (d) elements displacement response signal.
Figure 10.
The Gaussian white noise (a) and 1st (b), 20th (c) and 40th (d) elements displacement response signal.
Figure 11.
The real natural frequency of the density slow-time-varying cantilever beam structure.
Figure 11.
The real natural frequency of the density slow-time-varying cantilever beam structure.
Figure 12.
The modal shapes identified by MWLPP of four moments including (a) t = 100 s, (b) t = 650 s, (c) t = 1500 s and (d) t = 1974.375s in the mass slow-time-varying 3-DOF structure.
Figure 12.
The modal shapes identified by MWLPP of four moments including (a) t = 100 s, (b) t = 650 s, (c) t = 1500 s and (d) t = 1974.375s in the mass slow-time-varying 3-DOF structure.
Figure 13.
The natural frequencies identified by the MWLPP in the mass slow-time-varying 3-DOF structure.
Figure 13.
The natural frequencies identified by the MWLPP in the mass slow-time-varying 3-DOF structure.
Figure 14.
The MAC value of modal shapes identified by the MWLPP in the mass slow-time-varying 3-DOF structure.
Figure 14.
The MAC value of modal shapes identified by the MWLPP in the mass slow-time-varying 3-DOF structure.
Figure 15.
The change with time of damping ratio of slow-time-varying 3-DOF structure identified by MWLPP.
Figure 15.
The change with time of damping ratio of slow-time-varying 3-DOF structure identified by MWLPP.
Figure 16.
The modal shapes identified by MWLPP at four moments including (a) t = 0.75 s, (b) t = 1.75 s, (c) t = 2.75 s and (d) t = 3.75 s of the density slow-time-varying cantilever beam structure.
Figure 16.
The modal shapes identified by MWLPP at four moments including (a) t = 0.75 s, (b) t = 1.75 s, (c) t = 2.75 s and (d) t = 3.75 s of the density slow-time-varying cantilever beam structure.
Figure 17.
The natural frequencies identified by the MWLPP in the density slow-time-varying cantilever beam structure.
Figure 17.
The natural frequencies identified by the MWLPP in the density slow-time-varying cantilever beam structure.
Figure 18.
The MAC (model assurance criterion) value of modal shapes identified by the MWLPP in the density slow-time-varying cantilever beam structure.
Figure 18.
The MAC (model assurance criterion) value of modal shapes identified by the MWLPP in the density slow-time-varying cantilever beam structure.
Figure 19.
The modal shapes identified by MWLPP of four moments including (a) t = 100 s, (b) t = 650 s, (c) t = 1500 s and (d) t = 1974.375 s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Figure 19.
The modal shapes identified by MWLPP of four moments including (a) t = 100 s, (b) t = 650 s, (c) t = 1500 s and (d) t = 1974.375 s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Figure 20.
The natural frequencies identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Figure 20.
The natural frequencies identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Figure 21.
The MAC value of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Figure 21.
The MAC value of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Table 1.
The real natural frequency of the mass slow-time-varying 3-DOF structure when t = 50.025 s, t = 1200 s and t = 1974.375 s.
Table 1.
The real natural frequency of the mass slow-time-varying 3-DOF structure when t = 50.025 s, t = 1200 s and t = 1974.375 s.
Real Natural Frequency (Hz) |
---|
Order | t = 50.025 s | t = 1200 s | t = 1974.375 s |
---|
1 | 2.24 | 2.29 | 2.31 |
2 | 6.28 | 7.02 | 7.27 |
3 | 9.07 | 10.56 | 12.26 |
Table 2.
The real natural frequency of the density slow-time-varying cantilever beam structure when t = 0.5 s, t = 2 s and t = 3.795 s.
Table 2.
The real natural frequency of the density slow-time-varying cantilever beam structure when t = 0.5 s, t = 2 s and t = 3.795 s.
Real Natural Frequency (Hz) |
---|
Order | t = 0.5 s | t = 2 s | t = 3.795 s |
---|
1 | 16.70 | 17.80 | 19.46 |
2 | 104.66 | 111.56 | 121.96 |
3 | 293.03 | 312.38 | 341.48 |
Table 3.
The MAC (model assurance criterion) value of four moments including t = 100 s, t = 650 s, t = 1500 s and t = 1974.375 s in the mass slow-time-varying 3-DOF structure.
Table 3.
The MAC (model assurance criterion) value of four moments including t = 100 s, t = 650 s, t = 1500 s and t = 1974.375 s in the mass slow-time-varying 3-DOF structure.
Order | t = 100 s | t = 650 s | t = 1500 s | t = 1974.375 s |
---|
1 | 1 | 1 | 0.9908 | 1 |
2 | 0.9999 | 0.9988 | 0.9746 | 0.7391 |
3 | 0.9998 | 1 | 0.9908 | 0.8691 |
Table 4.
The average error of MWLPP and MWPCA (moving window principal component analysis) in identifying natural frequencies in the mass slow-time-varying 3-DOF structure.
Table 4.
The average error of MWLPP and MWPCA (moving window principal component analysis) in identifying natural frequencies in the mass slow-time-varying 3-DOF structure.
Method | | | |
---|
MWLPP | 0.059% | 0.128% | 0.244% |
MWPCA | 0.059% | 0.129% | 2.45% |
Table 5.
The number of un-identified windows between MWLPP and MWPCA in the mass slow-time-varying 3-DOF structure.
Table 5.
The number of un-identified windows between MWLPP and MWPCA in the mass slow-time-varying 3-DOF structure.
Method | First Order Un-Identified
Number | Rate | Second Order Un-Identified Number | Rate | Third Order Un-Identified Number | Rate |
---|
MWLPP | 0 | 0 | 808 | 1.049% | 747 | 0.970% |
MWPCA | 0 | 0 | 746 | 0.969% | 1827 | 2.373% |
Table 6.
The of modal shapes identified by MWLPP and MWPCA in the mass slow-time-varying 3-DOF structure.
Table 6.
The of modal shapes identified by MWLPP and MWPCA in the mass slow-time-varying 3-DOF structure.
Method | | | |
---|
MWLPP | 0.9979 | 0.9344 | 0.9254 |
MWPCA | 0.9982 | 0.9009 | 0.9052 |
Table 7.
The MAC value at four moments t = 0.75 s, t = 1.75 s, t = 2.75 s and t = 3.75 s of the density slow-time-varying cantilever beam structure.
Table 7.
The MAC value at four moments t = 0.75 s, t = 1.75 s, t = 2.75 s and t = 3.75 s of the density slow-time-varying cantilever beam structure.
Order | t = 0.75 s | t = 1.75 s | t = 2.75 s | t = 3.75 s |
---|
1 | 1 | 0.9998 | 0.9999 | 0.9997 |
2 | 0.9981 | 0.9858 | 0.9990 | 0.9988 |
3 | 0.9951 | 0.9742 | 0.9975 | 0.9947 |
Table 8.
The average error of MWLPP and MWPCA in identifying natural frequencies in the density slow-time-varying cantilever beam structure.
Table 8.
The average error of MWLPP and MWPCA in identifying natural frequencies in the density slow-time-varying cantilever beam structure.
Method | | | |
---|
MWLPP | 3.234% | 0.469% | 0.159% |
MWPCA | 3.07% | 0.470% | 0.161% |
Table 9.
The number of un-identified windows between MWLPP and MWPCA in density slow-time-varying cantilever beam structure.
Table 9.
The number of un-identified windows between MWLPP and MWPCA in density slow-time-varying cantilever beam structure.
Method | First Order Un-Identified Number | Rate | Second Order Un-Identified Number | Rate | Third Order Un-Identified Number | Rate |
---|
MWLPP | 0 | 0 | 6 | 0.018% | 124 | 0.376% |
MWPCA | 213 | 0.64% | 1 | 0.003% | 66 | 0.2% |
Table 10.
The of modal shapes identified by MWLPP and MWPCA in the density slow-time-varying cantilever beam structure.
Table 10.
The of modal shapes identified by MWLPP and MWPCA in the density slow-time-varying cantilever beam structure.
Method | | | |
---|
MWLPP | 0.9990 | 0.9971 | 0.9789 |
MWPCA | 0.9834 | 0.9773 | 0.8996 |
Table 11.
The MAC value of four moments including t = 100 s, t = 650 s, t = 1500 s and t = 1974.375 s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Table 11.
The MAC value of four moments including t = 100 s, t = 650 s, t = 1500 s and t = 1974.375 s in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Order | t = 100 s | t = 650 s | t = 1500 s | t = 1974.375 s |
---|
1 | 0.9995 | 1 | 0.9938 | 1 |
2 | 0.9988 | 0.9997 | 0.9707 | 0.8243 |
3 | 0.9970 | 1 | 0.9818 | 0.8731 |
Table 12.
The average error of MWLPP in identifying natural frequencies in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Table 12.
The average error of MWLPP in identifying natural frequencies in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Method | | | |
---|
MWLPP (with 10% white Gaussian noise) | 0.059% | 0.127% | 0.245% |
MWLPP(without white Gaussian noise) | 0.059% | 0.129% | 2.45% |
Table 13.
The number of un-identified windows by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Table 13.
The number of un-identified windows by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Method | First Order Un-Identified
Number | Rate | Second Order Un-Identified
Number | Rate | Third Order Un-Identified
Number | Rate |
---|
MWLPP
(with 10% white Gaussian noise) | 0 | 0 | 674 | 0.876% | 1062 | 1.380% |
MWLPP
(without white Gaussian noise) | 0 | 0 | 808 | 1.049% | 747 | 0.970% |
Table 14.
The of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Table 14.
The of modal shapes identified by MWLPP in the mass slow-time-varying 3-DOF structure with 10% white Gaussian noise.
Method | | | |
---|
MWLPP
(with 10% white Gaussian noise) | 0.9981 | 0.9260 | 0.9219 |
MWLPP
(without white Gaussian noise) | 0.9979 | 0.9344 | 0.9254 |