In order to identify the modal parameters of time invariant three-dimensional engineering structures with damping and small nonlinearity, a novel isometric feature mapping (Isomap)-based three-dimensional operational modal analysis (OMA) method is proposed to extract nonlinear features in this paper. Using this Isomap-based OMA method, a low-dimensional embedding matrix is multiplied by a transformation matrix to obtain the original matrix. We find correspondence relationships between the low-dimensional embedding matrix and the modal coordinate response and between the transformation matrix and the modal shapes. From the low-dimensional embedding matrix, the natural frequencies can be determined using a Fourier transform and the damping ratios can be identified by the random decrement technique or natural excitation technique. The modal shapes can be estimated from the Moore–Penrose matrix inverse of the low-dimensional embedding matrix. We also discuss the effects of different parameters (i.e., number of neighbors and matrix assembly) on the results of modal parameter identification. The modal identification results from numerical simulations of the vibration response signals of a cylindrical shell under white noise excitation demonstrate that the proposed method can identify the modal shapes, natural frequencies, and ratios of three-dimensional structures in operational conditions only from the vibration response signals.
Operational modal analysis (OMA) has received widespread attention because it enables the identification of the modal parameters of a structure in its working condition using only the vibration response [
Dimensionality reduction techniques are an effective means of overcoming the curse of dimensionality. Current methods can be categorized as either linear or nonlinear dimensionality reduction [
Similarly, there are many nonlinear dimensionality reduction methods, such as locally linear embedding (LLE) [
Based on Isomap algorithm, this paper proposes a three-dimensional OMA method for complex three-dimensional continuum structures.
The primary contributions of this paper can be summarized as follows: An Isomap-based OMA method is proposed for the identification of modal shapes, modal natural frequencies, and modal ratios of three-dimensional structures We identify the correspondence between the low-dimensional embedding matrix and the modal response matrix and between the transformation matrix and the modal shapes We conduct a theoretical analysis of the characteristics of the Isomap-based OMA method We analyze the effects of different parameters (e.g., matrix assembly method, number of neighbors, and dimensionality reduction method) on the algorithm We design numerical simulations of the vibration response signals of a cylindrical shell to verify the effectiveness of our algorithm
The remainder of this paper is organized as follows. In Section
According to dynamic structural vibration theory, the dynamic differential equation of a
The free and random vibrations of weakly damped systems have a displacement response that can be expressed as follows in modal coordinates:
The modal responses
Taking advantage of single-DOF (SDOF) identification approach such as FFT, it is easy to recognize the
After
OMA is to identify modal shape matrix
Isomap is a very popular manifold learning algorithm. Unlike PCA, which is designed only for linear dimensionality reduction, Isomap can solve nonlinear dimensionality reduction problems [ Build a neighborhood graph G. For each sample point Estimate the geodesic distances. Call the shortest path algorithm to calculate the distance between any two points. This is also called the geodesic distance, that is, the distance to the adjacent point is the Euclidean distance, and that to the nonadjacent point is the shortest path distance. The specific calculation process for the distance matrix Call the classical multidimensional scaling (MDS) algorithm to obtain the matrix
Connection of a sample to its neighbors in the Isomap algorithm.
The following can be obtained from the steps of MDS. The aim of the MDS algorithm is to obtain the representation of the sample in the
Setting
From the following equations
In summary, the classical MDS algorithm can be split into the following steps: Using ( Using ( Perform eigenvalue decomposition on Construct
We now introduce the Isomap algorithm; an example of using Isomap for nonlinear dimensionality reduction is presented in Figure
An example of using Isomap for nonlinear dimensionality reduction: (a) original three-dimension data before dimensionality reduction; (b) two-dimension data after dimensionality reduction by Isomap.
PCA and classical MDS can be effectively applied to Euclidean structures but fail to extract nonlinear features [
The Isomap algorithm can obtain low-dimensional representations
Comparing (
Physical interpretation of Isomap-based OMA.
The OMA of “three-dimensional” structure is a more complex problem, because it requires the assembly of the modal parameter in three directions to calculate the “three-dimensional” modal parameter.
Therefore, we introduce two three-dimensional matrix assembly methods in this section, namely, least-squares matrix substitution (LSMS) and direct matrix assembly (DMA). After the data matrix has been assembled, we use the Isomap algorithm to find the modal parameters.
Continuum-structure mechanical systems can be divided into
In (
In the LSMS-based Isomap method, a single-dimensional displacement response signal is first decomposed. To reduce the influence of the Gaussian-distributed signal measurement noise, the main and biggest structure’s dynamic response dimension is selected in practical engineering cases. Substituting
Process of LSMS-based Isomap for three-dimensional structure OMA.
From (
The modal coordinate response and modal shape of the overall structure are solved in one pass:
Using the Moore–Penrose matrix inverse of (
The process of the DMA-based Isomap method for three-dimensional OMA is described in Figure
Process of DMA-based Isomap for three-dimensional structure OMA.
Isomap-based OMA has the following characteristics: The order of the identified modal parameters is different from the theoretical value. The Isomap algorithm is an improved version of classical MDS, and the order of recognition follows the contribution of the principal components, running from small to large. The amplitude information of the modal shapes is lost. According to ( Some modal information may be missing. When the contribution of an independent component is small, it is difficult to identify its modal parameters.
The LSMS method selects the maximum vibration response data of the three-dimensional structure to calculate the modal response matrix, which is used to determine the other modal parameters. However, the modal coordinate response of the direction of maximum vibration response is not equal to the global modal coordinate response. Although we obtain this matrix at a small cost, the resulting three-dimensional modal shape must have a large error.
The DMA-based method directly obtains the global modal shape and overall modal coordinate response of three-dimensional structures and then uses Isomap to identify the modal parameters. Compared with LSMS, DMA avoids the need for matrix inversion operations and is more robust. Furthermore, this method offers greater accuracy because matrix inversion errors and ill-posed problems [
Performance comparison of two assembly methods in Isomap.
Matrix assembly methods | LSMS | DMA |
---|---|---|
Matrix inversion error | ✓ | × |
Ill-posed problems | ✓ | × |
Robustness | Sensitive to measurement noise | Insensitive to measurement noise |
The neighbor value K refers to the number of data points to which a given point is connected. Each point is connected to the nearest K points to form a graph G, and then some algorithm (e.g., Dijkstra’s algorithm and Floyd’s algorithm) is used to calculate the shortest path.
If K is too small, the connectivity graph may not be formed, which will affect the calculation of the shortest distance. At the same time, the negative effects of noisy points will be amplified. A sufficiently large value of K can reduce the difference between the path length and the true geodesic distance. If K is too large, the computation time may become unbearable and underfitting may occur.
Both MDS and PCA are linear dimensionality reduction techniques. The PCA method looks for the low-dimensional embedding of the data points which best preserves their variance, as measured in the high-dimensional input space. The classical MDS method finds an embedding that preserves the interpoint distances. We use the Euclidean distance in the proposed method, and the results of PCA are consistent with those from MDS [
Isomap is a nonlinear dimensionality reduction technique. When there are nonlinear features in the three-dimensional structures, the extraction performance of linear methods is poor. The linear and nonlinear relationships between the data also affect the algorithm results.
We conducted simulations to study a cylindrical shell with a complex three-dimensional structure. The cylindrical shell is simply supported at both ends, and a certain number of vibration sensors are positioned on its surface to record the vibration response in three directions, with a vibration exciter used to simulate the working conditions. Figure
Simulation process.
The cylindrical shell has a thickness of 0.005 m, radius of 0.1825 m, length of 0.37 m, and elasticity modulus of 205 GPa. The material has Poisson’s ratio of 0.3, density of 7850 kg/m3, and mode damping ratios of 0.03, 0.05, and 0.1.
The cylindrical shell was considered to be a uniform axial distribution of 38 circles, and each circle had 115 uniformly distributed observation points, giving a total of
Response signals in three directions. (a) Response in
To evaluate the effect of identification using the proposed method for three-dimensional structures, the mode shapes and natural frequencies were calculated using FEA. These were considered the real modal parameters for comparison with the identified modal parameters. The modal assurance criterion (MAC) reflects the effectiveness of the modal identification given by the proposed method. MAC is defined as
From the response signals in the three directions shown in Figure
The modal shapes and natural frequencies calculated by the FEA method with a damping ratio of 0.03 were considered to be the real values. Figure
Real modal shapes calculated by FEA.
Embedding dimension of 6 and number of neighbors
To enable a better comparison, we rotated the coordinates and obtained the results shown in Figure
Real modal shapes after rotating the coordinates.
The modal shapes and natural frequencies identified by the LSMS-based Isomap algorithm are shown in Figures
Modal shapes identified by LSMS-based Isomap.
Modal frequencies identified by LSMS-based Isomap.
Modal shapes identified by DMA-based Isomap.
Modal frequencies identified by DMA-based Isomap.
Comparison of natural frequencies with different assembly methods.
Orders | Frequency calculated by FEA (frequency/Hz) | Orders of Isomap components | Identified by LSMS-based Isomap (frequency/Hz) | Relative error (%) | Identified by DMA-based Isomap (frequency/Hz) | Relative error (%) |
---|---|---|---|---|---|---|
1 | 1054.9 | 2 | 1059 | 0.387 | 1059 | 0.387 |
2 | 1145.7 | 1 | 1145 | −0.061 | 1149 | 0.288 |
3 | 1239.6 | 3 | 1237 | −0.210 | 1237 | −0.210 |
4 | 1441.9 | 4 | 1441 | −0.062 | 1441 | −0.062 |
5 | 1740.0 | 6 | — | — | — | — |
7 | 1871.7 | 5 | 1874 | 0.123 | 1874 | 0.123 |
The symbol “--” indicates that the result was not recognized or too small.
Comparison of MAC of modal shapes with different assembly methods.
Order of real modal shape | Order of identified modal shape | ||
---|---|---|---|
1 | 2 | 0.8783 | 0.6564 |
2 | 1 | 0.6837 | 0.6180 |
3 | 3 | 0.5634 | 0.7817 |
4 | 4 | 0.4238 | 0.5241 |
5 | 6 | 0.0014 | 0.0049 |
7 | 5 | 0.6495 | 0.8932 |
Comparison of modal shapes with different methods.
Method | Modal shapes when | |||||
---|---|---|---|---|---|---|
1th order | 2th order | 3th order | 4th order | 5th order | 7th order | |
FEA | ||||||
LSMS | — | |||||
DMA | — |
Comparison of modal ratios with different assembly methods.
Order of real modal ratios | Order of identified modal ratios | Identified by the LSMS-based Isomap | Identified by DMA-based Isomap |
---|---|---|---|
1 | 2 | 0.0204 | 0.0286 |
2 | 1 | 0.0187 | 0.0196 |
3 | 3 | 0.0270 | 0.0210 |
4 | 4 | 0.0216 | 0.0214 |
5 | 6 | — | 0.0678 |
7 | 5 | 0.0282 | 0.0277 |
The modal shape and frequency identified by the LSMS-based Isomap algorithm with an embedding dimension of 5 are shown in Figures
Modal shapes identified when the embedding dimension is 5.
Modal frequencies identified when the embedding dimension is 5.
Tables
Comparison of MAC at different dimensions.
FEA method | LSMS method | |||
---|---|---|---|---|
Real order | Identify order | Order | ||
1 | 2 | 0.8783 | 2 | 0.8783 |
2 | 1 | 0.6837 | 1 | 0.6837 |
3 | 3 | 0.5634 | 3 | 0.5634 |
4 | 4 | 0.4238 | 4 | 0.4238 |
5 | 6 | — | 6 | 0.0014 |
7 | 5 | 0.6495 | 5 | 0.6495 |
Comparison of modal frequencies at different dimensions.
FEA method | LSMS method | ||||
---|---|---|---|---|---|
Real order | Real frequencies | Identify order | Frequencies | Order | Frequencies |
1 | 1054.9 | 2 | 1059 | 2 | 1059 |
2 | 1145.7 | 1 | 1145 | 1 | 1145 |
3 | 1239.6 | 3 | 1237 | 3 | 1237 |
4 | 1441.9 | 4 | 1441 | 4 | 1441 |
5 | 1740.0 | 6 | — | 6 | — |
7 | 1871.7 | 5 | 1874 | 5 | 1874 |
Tables
Comparison of MAC for different neighbor numbers K.
Real order | 1 | 2 | 3 | 4 | 5 | 7 | |
---|---|---|---|---|---|---|---|
Identify order | 2 | 1 | 3 | 4 | 6 | 5 | |
LSMS | 0.8850 | 0.6863 | 0.5591 | 0.4179 | 0.0003 | 0.6544 | |
DMA | 0.6609 | 0.6198 | 0.7790 | 0.5221 | 0.0002 | 0.8877 | |
LSMS | 0.8783 | 0.6837 | 0.5634 | 0.4238 | 0.0014 | 0.6495 | |
DMA | 0.6564 | 0.6180 | 0.7817 | 0.5241 | 0.0049 | 0.8932 | |
LSMS | 0.8800 | 0.6851 | 0.5632 | 0.4261 | 0.0050 | 0.6534 | |
DMA | 0.6599 | 0.6208 | 0.7820 | 0.5260 | 0.0042 | 0.8981 | |
LSMS | 0.8798 | 0.6851 | 0.5635 | 0.4262 | 0.0519 | 0.6532 | |
DMA | 0.6594 | 0.6206 | 0.7824 | 0.5263 | 0.0213 | 0.8990 | |
LSMS | 0.8796 | 0.6851 | 0.5636 | 0.4265 | 0.1423 | 0.6543 | |
DMA | 0.6585 | 0.6201 | 0.7827 | 0.5271 | 0.0724 | 0.9007 |
Comparison of modal frequencies for different neighbor numbers K.
Real order | 1 | 2 | 3 | 4 | 5 | 7 | |
---|---|---|---|---|---|---|---|
Identify order | 2 | 1 | 3 | 4 | 6 | 5 | |
LSMS | 1052 | 1146 | 1237 | 1441 | — | 1874 | |
DMA | 1059 | 1145 | 1243 | 1441 | — | 1874 | |
LSMS | 1059 | 1145 | 1237 | 1441 | — | 1874 | |
DMA | 1059 | 1149 | 1237 | 1441 | — | 1874 | |
LSMS | 1059 | 1145 | 1237 | 1443 | — | 1868 | |
DMA | 1059 | 1149 | 1237 | 1443 | — | 1868 | |
LSMS | 1059 | 1145 | 1237 | 1443 | — | 1868 | |
DMA | 1059 | 1149 | 1237 | 1443 | — | 1868 | |
LSMS | 1059 | 1146 | 1237 | 1443 | — | 1868 | |
DMA | 1059 | 1149 | 1237 | 1443 | — | 1868 | |
FEA | 1054.9 | 1145.7 | 1239.6 | 1441.9 | 1740.0 | 1871.7 |
Comparison of modal ratios for different neighbor numbers K.
Real order | 1 | 2 | 3 | 4 | 5 | 7 | |
---|---|---|---|---|---|---|---|
Identify order | 2 | 1 | 3 | 4 | 6 | 5 | |
LSMS | 0.0201 | 0.0186 | 0.0272 | 0.0216 | 0.0846 | 0.0294 | |
DMA | 0.0286 | 0.0195 | 0.0211 | 0.0216 | 0.0668 | 0.0284 | |
LSMS | 0.0204 | 0.0187 | 0.0270 | 0.0216 | 0.1140 | 0.0282 | |
DMA | 0.0286 | 0.0196 | 0.0210 | 0.0214 | 0.0678 | 0.0277 | |
LSMS | 0.0204 | 0.0187 | 0.0270 | 0.0213 | 0.0702 | 0.0275 | |
DMA | 0.0285 | 0.0196 | 0.0210 | 0.0212 | 0.0317 | 0.0275 | |
LSMS | 0.0204 | 0.0187 | 0.0271 | 0.0212 | 0.0560 | 0.0274 | |
DMA | 0.0285 | 0.0196 | 0.0210 | 0.0211 | 0.0364 | 0.0274 | |
LSMS | 0.0204 | 0.0188 | 0.0271 | 0.0212 | 0.0592 | 0.0274 | |
DMA | 0.0286 | 0.0196 | 0.0210 | 0.0211 | 0.0359 | 0.0274 |
For the modal ratio, we apply the random decrement technique (RDT) [
Example of the modal ratio solution process.
We compare the modal damping ratio identification methods (RDT and NExT) using different dimensionality reduction methods (PCA [
RDT and NExT results with a real modal damping ratio of 0.1.
Real order | Identified order | Real modal ratio = 0.1 | |||||||
---|---|---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||||
LSMS | DMA | LSMS | DMA | ||||||
RDT | NExT | RDT | NExT | RDT | NExT | RDT | NExT | ||
1 | 2 | 0.0532 | 0.0567 | 0.0587 | 0.0624 | 0.0510 | 0.0567 | 0.0605 | 0.0627 |
2 | 1 | 0.1155 | 0.1241 | 0.1045 | 0.1139 | 0.1136 | 0.1245 | 0.1074 | 0.1139 |
3 | 3 | 0.0668 | 0.0639 | 0.0650 | 0.0605 | 0.0682 | 0.0645 | 0.0653 | 0.0612 |
4 | 4 | 0.0716 | 0.0847 | 0.0869 | 0.0967 | 0.0770 | 0.0837 | 0.0956 | 0.0971 |
5 | 6 | 0.0742 | 0.0758 | 0.0748 | 0.0759 | — | — | — | — |
7 | 5 | 0.1079 | 0.1192 | 0.1308 | 0.1351 | 0.1369 | 0.1388 | 0.1473 | 0.1420 |
RDT and NExT results with a real modal damping ratio of 0.05.
Real order | Identified order | Real modal ratio = 0.05 | |||||||
---|---|---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||||
LSMS | DMA | LSMS | DMA | ||||||
RDT | NExT | RDT | NExT | RDT | NExT | RDT | NExT | ||
1 | 2 | 0.0312 | 0.0331 | 0.0312 | 0.0454 | 0.0297 | 0.0331 | 0.0373 | 0.0442 |
2 | 1 | 0.0716 | 0.0854 | 0.0585 | 0.0679 | 0.0825 | 0.0861 | 0.0549 | 0.0673 |
3 | 3 | 0.0388 | 0.0410 | 0.0543 | 0.0611 | 0.0391 | 0.0410 | 0.0524 | 0.0597 |
4 | 4 | 0.0274 | 0.0367 | 0.0625 | 0.0780 | 0.0298 | 0.0368 | 0.0619 | 0.0821 |
5 | 6 | 0.0434 | 0.0450 | 0.0361 | 0.0450 | — | — | 0.0364 | 0.0471 |
7 | 5 | 0.0492 | 0.0485 | 0.0393 | 0.0453 | 0.0499 | 0.0496 | 0.0416 | 0.0462 |
RDT and NExT results with a real modal damping ratio of 0.03.
Real order | Identified order | Real modal ratio = 0.03 | |||||||
---|---|---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||||
LSMS | DMA | LSMS | DMA | ||||||
RDT | NExT | RDT | NExT | RDT | NExT | RDT | NExT | ||
1 | 2 | 0.0185 | 0.0204 | 0.0213 | 0.0286 | 0.0181 | 0.0204 | 0.0206 | 0.0286 |
2 | 1 | 0.0183 | 0.0188 | 0.0160 | 0.0197 | 0.0167 | 0.0187 | 0.0163 | 0.0196 |
3 | 3 | 0.0230 | 0.0271 | 0.0212 | 0.0209 | 0.0230 | 0.0270 | 0.0212 | 0.0210 |
4 | 4 | 0.0168 | 0.0212 | 0.0149 | 0.0210 | 0.0160 | 0.0216 | 0.0163 | 0.0214 |
5 | 6 | 0.0270 | 0.0267 | 0.0269 | 0.0267 | — | — | 0.0550 | 0.0678 |
7 | 5 | 0.0275 | 0.0274 | 0.0290 | 0.0274 | 0.0298 | 0.0282 | 0.0297 | 0.0277 |
PCA and Isomap by LSMS with a real modal damping ratio of 0.03.
Real order | Identified order | LSMS | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.8785 | 1058 | 0.294 | 0.8783 | 1059 | 0.389 |
2 | 1 | 0.6843 | 1145 | −0.061 | 0.6837 | 1145 | −0.061 |
3 | 3 | 0.5639 | 1237 | −0.210 | 0.5634 | 1237 | −0.210 |
4 | 4 | 0.4274 | 1442 | 0.007 | 0.4238 | 1441 | −0.062 |
5 | 6 | 0.6282 | 1737 | −0.172 | 0.0014 | — | — |
7 | 5 | 0.6564 | 1872 | 0.016 | 0.6495 | 1874 | 0.123 |
PCA and Isomap by LSMS with a real modal damping ratio of 0.05.
Real order | Identified order | LSMS | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.6311 | 1059 | 0.389 | 0.6316 | 1065 | 0.957 |
2 | 1 | 0.3026 | 1143 | −0.236 | 0.3019 | 1143 | −0.236 |
3 | 3 | 0.1082 | 1232 | −0.613 | 0.1088 | 1229 | −0.855 |
4 | 4 | 0.0801 | 1437 | −0.340 | 0.0788 | 1441 | −0.062 |
5 | 6 | 0.0037 | 1747 | 0.402 | 0.0227 | — | — |
7 | 5 | 0.0219 | 1868 | −0.198 | 0.0211 | 1879 | 0.390 |
PCA and Isomap by LSMS with a real modal damping ratio of 0.1.
Real order | Identified order | LSMS | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.7248 | 1073 | 1.716 | 0.7256 | 1080 | 2.379 |
2 | 1 | 0.4890 | 1140 | −0.498 | 0.4887 | 1137 | −0.759 |
3 | 3 | 0.3430 | 1199 | −3.275 | 0.3379 | 1178 | −4.969 |
4 | 4 | 0.2435 | 1414 | −1.935 | 0.2434 | 1391 | −3.530 |
5 | 6 | 0.6293 | 1747 | 0.402 | 0.0002 | — | — |
7 | 5 | 0.2520 | 1854 | −0.946 | 0.2710 | 1800 | −3.831 |
PCA and Isomap by DMA with a real modal damping ratio of 0.03.
Real order | Identified order | DMA | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.6554 | 1059 | 0.389 | 0.6564 | 1059 | 0.389 |
2 | 1 | 0.6180 | 1151 | 0.463 | 0.6180 | 1149 | 0.288 |
3 | 3 | 0.7833 | 1237 | −0.210 | 0.7817 | 1237 | −0.210 |
4 | 4 | 0.5284 | 1443 | 0.076 | 0.5241 | 1441 | −0.062 |
5 | 6 | 0.7289 | 1738 | −0.115 | 0.0049 | — | — |
7 | 5 | 0.9038 | 1873 | 0.069 | 0.8932 | 1874 | 0.123 |
PCA and Isomap by DMA with a real modal damping ratio of 0.05.
Real order | Identified order | DMA | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.5395 | 1052 | −0.275 | 0.5614 | 1052 | −0.275 |
2 | 1 | 0.1402 | 1143 | −0.236 | 0.1521 | 1140 | −0.498 |
3 | 3 | 0.1928 | 1225 | −1.178 | 0.1770 | 1237 | −0.210 |
4 | 4 | 0.1213 | 1434 | −0.548 | 0.1245 | 1441 | −0.062 |
5 | 6 | 0.0005 | 1877 | 7.874 | 0.0003 | 1889 | 8.563 |
7 | 5 | 0.0041 | 1879 | 0.390 | 0.0044 | 1890 | 0.978 |
PCA and Isomap by DMA with a real modal damping ratio of 0.1.
Real order | Identified order | DMA | |||||
---|---|---|---|---|---|---|---|
PCA (MDS) | Isomap | ||||||
MAC | Frequencies | Relative error (%) | MAC | Frequencies | Relative error (%) | ||
1 | 2 | 0.5232 | 1081 | 1.716 | 0.5193 | 1081 | 2.474 |
2 | 1 | 0.4812 | 1044 | −0.498 | 0.4804 | 1138 | −0.672 |
3 | 3 | 0.2107 | 1185 | −4.405 | 0.2033 | 1176 | −5.131 |
4 | 4 | 0.1703 | 1406 | −2.490 | 0.1788 | 1392 | −3.461 |
5 | 6 | 0.7279 | 1747 | 0.402 | 0.0003 | — | — |
7 | 5 | 0.4739 | 1854 | −0.946 | 0.5082 | 1889 | 0.924 |
From Figures From Tables From Figures Different K values have little effect on the recognition results (see Tables Tables Isomap-based OMA can effectively identify the three-dimensional modal parameters (see Tables Tables
In this paper, we have described the application of the Isomap algorithm to OMA to identify modal shapes, modal natural frequencies, and modal ratios of three-dimensional structures. Promising results were obtained from simulations of a cylindrical shell emitting nonlinear response patterns. We also compared the influence of various parameters. The simulation results show that the parameters of the Isomap algorithm (number of neighbors K and dimension
In future work, we will attempt to improve the accuracy of our method at higher damping ratios. The method discussed in this paper uses the most basic Isomap algorithm. Improved versions of Isomap may achieve better results. Furthermore, the experiment with the actual structure and the problem of finding the missing mode are worth studying.
All the data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was financially supported by the National Natural Science Foundation of China (Grant no. 51305142) and the Promotion Program for Young and Middle-Aged Teachers in Science and Technology Research at Huaqiao University under Grant ZQN-PY212.