On optimization of the observation station based on the effective independence method

2.1. Effective independence method

The most widely used algorithm is EI (Effective Independence method), proposed by Kammer [10], from which many other algorithms have evolved. Its core idea is to remove those observation stations that contribute the least to the target linearly -independent mode to make the determinant of the Fisher matrix [11] $Q={\mathrm{\Phi }}_s^T{\rm M}{\mathrm{\Phi }}_s$ that corresponds to the remaining ones get the maximum value. In the matrix, ${\mathrm{\Phi }}_s$ represents the inherent mode of former S-order, thus ensuring that the resultant modes of the remaining observation stations are the best evaluated.

Matrix $E$ of EI goes as follows:

where $\mathrm{\Psi }$ represents feature vector of matrix Fisher and $\lambda$ represents the corresponding feature value while ${\left\{I\right\}}_i$ equals to the total of coefficients. $E$ represents the linearly-independent contribution of a given observation station to a target modal vibration mode.

2.2. Modal kinetic energy method and strain energy method

The sensor arrangement of modal kinetic energy method and modal strain energy method [12, 13] is similar to that of effective independence method. The main difference lying in that in the case of the former, the observation station is set where the modal kinetic and strain energy reach the highest, rather than being decided by the maximum determinant of Fisher matrix. To sum up, the principle obeyed in applying the two methods to select the optimal observation station is that the kinetic and strain energy must peak wherein.

The modal kinetic and strain energy can be expressed as follows:

where $\mathrm{\Phi }\in R^{n\times p}$ represents modal matrix and $P$ target modal value. The greatest strength of the modal kinetic energy method and modal strain energy method is that they work well even in adverse environment.

2.3. Mode assurance code

Values of inherent modal vibration modes of structures form a set of orthogonal vectors on nodes. But it is impossible that the number of observation stations is equivalent to that of nodes. Instead, the number of nodes is far larger. That is even truer for large and complex structures. Jointed by measurement accuracy and the effects of noise, the orthogonality of target modal vibration modes measured cannot be ensured. Therefore, in selecting observation stations, it is necessary to retain the completeness of dynamic characteristics of the structural models and to ensure broad intersect angle between modes to its best.MAC (Mode assurance code) matrix [3] gives a fair orthogonal evaluation on the modal vectors and can be expressed as follows:

In the equation, ${\mathrm{\Phi }}_i$ and ${\mathrm{\Phi }}_j$ are respectively $i$-order and $j$-order modal vectors. Whether the two corresponding mode shapes are orthogonal can be judged by examining the non-diagonal element of MAC matrix of each mode within the freedom degree of measuring. When MAC is less than 0.25, the two vectors are generally believed to be mutually orthogonal; when MAC is greater than 0.9, the two vectors are believed to be relevant.

3.1. Reduction improvement

Suppose that the stiffness matrix of a system with n degree of freedom is $K$, mass matrix $M$ and characteristic pair ${\lambda }_j$ and ${\varphi }_j$ ($j=1-n$), the vibration mode can be expressed as follows:

where ${\varphi }_j^m$ represents the column vector derived from the measuring coordinate of a vibration mode and ${\varphi }_j^s$ the column vector of a reduced coordinate. Correspondingly, the characteristic equation of the structural vibration is expressed as:

From the second part of Eq. (6), we can infer:

To unfold Eq. (7) according to Neumann series, we may get:

where “$o$” represents an extremely small high order. To omit the high order of Eq. (8), we may arrive at:

From the first part of Eq. (6) we can also infer:

For the finite model of most structures, the mass matrix is usually diagonal matrix. Hence, we come to:

To substitute Eq. (10) into Eq. (9), we may get:

To substitute Eqs. (11) and (12) into Eq. (9), we may get:

From Eq. (13), we may infer:

Therefore, the improved reduced equation is expressed as:

where the transformation matrix $T$ is:

To substitute Eqs. (15) and (13) into Eq. (5), we may arrive at:

If both sides of Eq. (17) time $T^T$ we may arrive at:

Suppose that:

We may get:

To substitute Eq. (9) into Eq. (20), we may get:

Therefore, the reduced characteristic equation is expressed as:

3.2. Weight coefficient

For large-scale structures, due to highly distinct vibration distribution of modes, the contribution of high-order modes to the modal strain energy cannot be ignored. In order to cover the contribution to the strain energy of all modes and improve the accuracy of the result, we come up with a weight coefficient reflecting the contribution of high-order modes to amend the matrix of modal vibration modes. The weigh coefficient is calculated under:

where ${\omega }_i$ represents the frequency of $i$-order and $c_i$ weight coefficient of $i$-order.

3.3. Optimization method

The selection of an observation station when applying mode kinetic energy method and made strain energy is decided by the total energy of all target modes at the station, which will lead to information loss of some target modes due to uneven energy distribution of orders in large and complex structures. To tackle this dilemma, guided by the two methods mentioned here, we select observation station according to the average modal kinetic and strain energy coefficients of each station.

3.3.2. Effective independence-average modal strain energy coefficient method

From the Eq. (3), we may infer the modal strain energy coefficient of the $i$th freedom degree at $n$th order, namely:

30

${\zeta }_{in}=\frac{MS{E}_{in}}{\sum _{j=0}^{N}MS{E}_{jn}},0\le {\zeta }_{in}\le 1,\sum _{i=1}^m{\zeta }_{in}=1.$

Then, the average modal strain energy coefficient is:

31

${\zeta }_i=\sum _{n=1}^N\frac{{\xi }_{in}}{N}.$

To combine Eq. (2) and (31), we may get the effective independence method based on average strain energy coefficient as follows:

32

$E^{\mathrm{\text{'}}\mathrm{\text{'}}}={\left[\mathrm{\Phi }\mathrm{\Psi }\right]}^2{\lambda }^{-1}{\left\{I\right\}}_i\zeta .$

After the iterative calculation of $E^{\text{'}}$and $E^{\text{'}\text{'}}$, we may choose the observation station that corresponds to the maximum of $E^{\text{'}}$ and $E^{\text{'}\text{'}}$ until that the maximum value of non-diagonal elements of MAC matrix that corresponds to all the selected stations is eligible. In order to avoid concentrated stations, we need to check that whether the distances between this station and all other selected one are longer than $D_i$ and $D_i^{\text{'}}$, which are already smallest. Herein:

33

$D_i=D\mathrm{l}\mathrm{g}\frac{{\xi }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\mathrm{l}\mathrm{g}{\xi }_i},$

34

$D_i^{\mathrm{\text{'}}}=D\mathrm{l}\mathrm{g}\frac{{\zeta }_{\mathrm{m}\mathrm{i}\mathrm{n}}}{\mathrm{l}\mathrm{g}{\zeta }_i}.$

There-into, $D$ is a constant, representing the smallest distance appointed. That the areas with greater structural vibration are equipped with more sensors and vice versa will not only ensure the completeness and independence of the target modes but also avoid concentrated stations.

Fig. 1The plane model

4.1. About the model

In conducting the experiment of observation station arrangement optimization, a GARTEUR plane model was applied. As is shown in Fig. 1, the model has 2640 cells and 5106 nodes, hence, 30636 freedom degrees. Modal vibration of the plane is mainly $X$, $Y$ and $Z$ -directed, so the 3 rotational freedom degrees can be designated as ancillary ones, which makes the follow-up optimization calculating much easier. Material parameters of plane models: the elastic modulus is 7 E10 Pa, Poisson’s ratio of 0.3 and material density 2800 Kg/m³.

To take the modes of first 15 orders as target vibration modes and their inherent frequency is shown in Table 1.

4.2. Observation station arrangement optimization and comparison

a) Sensor arrangement of effective independence method for Guyan reduction based on modal kinetic/strained coefficients and the respectively improved ones and effective independence-average modal kinetic/strained energy coefficient methods are compared as is shown in Fig. 2.

1) Fig. 2(a) Guyan-reduction average modal kinetic energy coefficient effective-independence method and Fig. 2(d) Guyan-reduction average modal strained energy coefficient effective-independence method give rise to relatively concentrated sensor placement. In other words, these sensors are unevenly distributed and some of them are even sitting together. In this way, some modal information will be obtained repeatedly while some other important modal information will be lost. We can see that the resultant observation stations of these two methods entail many sensors. As a result, high economic costs will be caused during collection tests of modal data and the experiment will be complicated, thereby being not conducive to the modal data collection. In a word, both of the two are undesirable.

2) Fig. 2(a) improved-reduction average modal kinetic energy coefficient effective-independence method and Fig. 2(d) Improved-reduction average modal strained energy coefficient effective-independence method, however, will give rise to relatively disperse sensor placement. Sensors are evenly distributed and the number of observation stations are smaller. Consequently, economic costs caused during collection tests of modal data will be lowered and the experiment will turn easier. Compared with the methods Fig. 2(a) and Fig. 2(b), these two methods not only largely reduce the number of observation stations but also bring about evener and more reasonable sensor placement. It is fair to say that to improve reduction, to some extent, renders the model more concise and faithful. Thus, these two discussed here are favorable.

3) Fig. 2(c) effective independence-average modal kinetic energy coefficient method and Fig. 2(f) effective independence-average modal strained energy coefficient method proposed in this paper actually bring about rather even senor placement and a fairly small quantity of observation stations. Hence, the modal information will not be lost and the economic costs will be reduced by a large margin. Moreover, the experiment will become even easier. Among Fig. 2(a), (d), (b), (e), (c) and (f), the latter two methods turn out to be the optimal ones in that they allow the fewest observation stations, best sensor placement, lowest economic costs of tests and easiest experiment. It can be concluded hereby that weight coefficients play a positive role in the remedying of high-order targeted modes of vibration. In summary, the two sensor placement optimization schemes put forward in this paper are the best, both in terms of the number and positions of sensors coupled by the fact that they best retain the real structures under test and complete dynamic characteristics.

b) Column graphs of MAC matrix obtained from the six mentioned optimization algorithm are shown in Fig. 3.

Fig. 2Sensor arrangement of varied methods

a) Guyan-reduction average kinetic energy coefficient effective-independence method (Number of needed observation stations: 82)

b) Improved-reduction average kinetic energy coefficient effective-independence method (Number of needed observation stations: 44)

c) Effective independence-average modal kinetic energy coefficient method (Number of needed observation stations: 34)

d) Guyan-reduction average modal strain energy coefficient effective-independence method (Number of needed observation stations: 79)

e) Improved-reduction average modal strain energy coefficient effective-independence method (Number of needed observation stations: 42)

f) Effective independence -average modal strain energy coefficient method (Number of needed observation stations: 33)

The maximum value of non-diagonal element of MAC matrix is an important index to judge whether target modal vibration modes are relevant or not. When MAC is less than 0.25, the two vectors are generally believed to be mutually orthogonal; when MAC is greater than 0.9, the two vectors are believed to be relevant. Judging the Fig. 3, one can see that: 1) part of the non-diagonal element values of MAC matrix derived from methods Fig. 3(a) and (d) are somewhat great, indicating that the orthogonality between the targeted modes of vibration is unsatisfactory and that Fig. 3(a) and (d) are not ideal schemes. 2) The non-diagonal element values of MAC matrix derived from methods Fig. 3(b) and (e) are relatively small and a lot smaller than those from Fig. 3(a) and (d). Thus, the orthogonality between the targeted modes of vibration of Fig. 3(b) and (e) are good and better than that of Fig. 3(a) and (d), implying that Fig. 3(b) and (e) are better schemes than Fig. 3(a) and (f). Methods Fig. 3(c) and (f) proposed in this paper results in quite small non-diagonal element values of MAC matrix that are almost zero. Right here, one may conclude that the resultant orthogonality is much preferable than Fig. 3(a), (d), (b) and (e) and that weight coefficients are beneficial to the remedying of high-order targeted modes of vibration. Therefore, Fig. 3(c) and (f) are the soundest sensor placement optimization schemes.

By comparing the results of the six different sensor arrangement scheme, we may infer that the effective independence – average modal kinetic/strain energy coefficient methods not only maintain the maximum linear- independence of target modes and genuine and complete dynamic characteristics if the observed modes but also avoid the loss of information of high-order modes at the low-energy, being the most reliable and effective one among the six optimization algorithms.

Fig. 3Column diagram of MAC matrix in applying varied methods

a) Guyan-reduction average kinetic energy coefficient effective-independence method

b) Improved-reduction average kinetic energy coefficient effective-independence method

c) Effective independence-average modal kinetic energy coefficient method

d) Guyan-reduction average modal strain energy coefficient effective-independence method

e) Improved-reduction average modal strain energy coefficient effective-independence method

f) Effective independence -average modal strain energy coefficient method

5.1. Effective independence – average modal kinetic energy coefficient method

The experiment is to test the resultant scheme of the effective independence-average modal kinetic energy coefficient method proposed in this paper. The optimized placement of observation stations hereby is the optimized placement Fig. 2(c) among all methods in Fig. 2.

a) Coordinate –based locations of sensor placement (Table 2).

b) Frequency response function logarithmic curve shown in Fig. 6.

c) Comparison and contrast between experiment frequency and simulation frequency (Table 3).

d) The comparison and contrast of MAC matrix orthogonal graphs is shown in Fig. 7.

e) The comparison and contrast of the vibration modes of the first experimented 8 orders and simulation vibration modes is shown in Fig. 8.

Fig. 6Frequency response function logarithmic curve

Fig. 7Comparison and contrast of MAC matrix orthogonal graphs

a) Experimented MAC matrix orthogonal graph

b) Simulated MAC matrix orthogonal graph

Fig. 8The comparison and contrast of the vibration modes of the first experimental 8 orders and simulation vibration modes

a) Experimented vibration mode of the 1st order

b) Simulated vibration mode of the 1st order

c) Experimented vibration mode of the 2nd order

d) Simulated vibration mode of the 2nd order

e) Experimented vibration mode of the 3rd order

f) Simulated vibration mode of the 3rd order

g) Experimented vibration mode of the 4th order

h) Simulated vibration mode of the 4th order

i) Experimented vibration mode of the 5th order

j) Simulated vibration mode of the 5th order

k) Experimented vibration mode of the 6th order

l) Simulated vibration mode of the 6th order

m) Experimented vibration mode of the 7th order

n) Simulated vibration mode of the 7th order

o) Experimented vibration mode of the 8th order

p) Simulated vibration mode of the 8th order

5.2. Effective independence – average modal kinetic energy coefficient method

The experiment is to test the resultant scheme of effective independence-average modal strained energy coefficient method proposed in this paper. The optimized placement of observation stations hereby is the optimized placement Fig. 2(f) among all methods in Fig. 2.

a) Coordinate – based locations of sensor placement (Table 4).

b) Frequency response function logarithmic curve shown in Fig. 9.

c) Comparison and contrast between experiment frequency and simulation frequency (Table 5).

d) The comparison and contrast of MAC matrix orthogonal graphs is shown in Fig. 10.

e) The comparison and contrast of the vibration modes of the first experimented 8 orders and simulation vibration modes is shown in Fig. 11.

Fig. 9Frequency response function logarithmic curve

By examining the experiments and simulation tests of the two methods proposed in the paper, we may see: differences between the experiment frequency and simulation experiment of both methods are small and simulation frequencies, which are less than five percent; MAC orthogonal graphs of each experiment and simulation test are quite similar and each vibration modes are orthogonally convergent; modal vibration modes of each experiment and simulation test are in agreement. Then, the two hybrid optimization methods proposed here may serve good guidance to the observation station optimization of large-scale structure se and are of great practical value.

Fig. 10Comparison and contrast of MAC matrix orthogonal graphs

a) Experimented MAC matrix orthogonal graph

b) Simulated MAC matrix orthogonal

Fig. 11The comparison and contrast of the vibration modes of the first experimented 8 orders and simulation vibration modes

a) Experimented vibration mode of the 1st order

b) Simulated vibration mode of the 1st order

c) Experimented vibration mode of the 2nd order

d) Simulated vibration mode of the 2nd order

e) Experimented vibration mode of the 3rd order

f) Simulated vibration mode of the 3rd order

g) Experimented vibration mode of the 4th order

h) Simulated vibration mode of the 4th order

i) Experimented vibration mode of the 5th order

j) Simulated vibration mode of the 5th order

k) Experimented vibration mode of the 6th order

l) Simulated vibration mode of the 6th order

m) Experimented vibration mode of the 7th order

n) Simulated vibration mode of the 7th order

o) Experimented vibration mode of the 8th order

p) Simulated vibration mode of the 8th order