US 2020/0089733 A1 · SENSOR PLACEMENT METHOD FOR REDUCING UNCERTAINTY OF STRUCTURAL MODAL IDENTIFICATION

Abstract

Sensor placement for structural health monitoring and sensor placement method for reducing uncertainty of structural modal identification. Influences of structural model error and measurement noise on measured responses are separated. Structural stiffness variation is used as model error, and Gaussian noise is used as measurement noise. Monte Carlo method simulates a large number of possible cases, and structural mode shape matrices under each model error condition are obtained. Conditional information entropy index quantifies and calculates uncertainty of identified modal parameter results. Conditional information entropy index solves the problem of uncertain Fisher information matrix, which cannot be solved by traditional information entropy method. Optimal sensor placement corresponds to maximum conditional information entropy index value. The sensor placement method considers influences of structural model error and measurement noise on structural modal identification, which is helpful for improving accuracy of structural modal parameter identification.

A sensor placement method for reducing uncertainty of structural modal identification, wherein the steps are as follows:

establish the relationship between structural model error and measurement noise;
conditional information entropy based sensor placement method;
(1) establish the relationship between structural model error and measurement noise
step 1.1:
In structural health monitoring systems, prediction error between the measured and actual structural response is due to two causes:
model error and measurement noise, thereby establishing the following relationship:
y(t)=S(x(t,θ)+e(t,θ)) (1)
where:
y(t)∈ is a measured structural response of Ns degrees of freedom;
Nd is a number of total degrees of freedom of the structure;
S∈ is a selection matrix for the sensor locations;
θ∈ is a modal parameter vector to be identified;
e(t,θ)∈ is a prediction error between the measured and actual structural response, which can be expressed as:
e(t,θ)=emea(t,θ)+emod(t,θ) (2)
where:
emea(t,θ) is measurement noise;
emod(t,θ) is the part of the prediction error caused by the structural model error;
step 1.2:
define the form of the prediction error:
the measurement noise is assumed to be a zero-mean Gaussian noise with a covariance matrix of Σmea=diag(σ1, . . . , σNd), σ1=σ0;
the structural model error is represented by the stiffness variation of the structure which is shown as:
Δ


K
=
∑
j
=
1
N
e

β
j

K
j
(
3
)
where:
Ne represents a number of structural element stiffness matrices;
Kj is the jth element stiffness matrix;
βj is a perturbation coefficient of the jth element stiffness matrix;
change in the structural modal matrix is expressed as:
ΔΦ
i
=

[
∑
r
=
1
N
d
r
≠
i

-
Φ
r
T

K
1

Φ
i
λ
r
-
λ
i

Φ
r
,
…

,
∑
r
=
1
N
d
r
≠
i

-
Φ
r
T

K
N
e

Φ
i
λ
r
-
λ
i

Φ
r
]

[
β
1
,
…

,
β
N
e
]
T
=

E
i

β
(
4
)
where:
β is a perturbation coefficient vector of each element stiffness matrix, Ei is a sensitivity coefficient matrix of the ith mode;
ΔΦi is a change of the ith mode shape vector;
Φr is a rth mode shape vector;
λr and λl are the eigenvalues of the rth and ith mode respectively;
the superscript T indicates transposition;
the mode shape changes of each order of the structure are expressed as:
ΔΦ=[E1β,E2τ3, . . . ,ENmβ] (5)
where:
ΔΦ represents mode shape changes of each order mode of the structure;
the subscript Nm indicates a corresponding modal order;
step 1.3:
establish a measurement data expression that comprehensively considers the structural model error and measurement noise;
Eq. (1) is rewritten as:
y(t)=S((Φ+[E1β,E2β, . . . ,ENmβ])θ+emea(t,θ)) (6)
where:
Φ is a mode shape matrix calculated by the finite element model used in the structure;
as seen from Eq. (6), the prediction error between the measured response and the actual response is represented by the two parts caused by the model error and the measurement noise;
(2) conditional information entropy based sensor placement method
step 2.1:
use the probability density function to represent the uncertainty of the recognition result of modal coordinate parameters:
p

(
θ
|
Σ
mea
,
D
,
β
)
=
c

1
(
2

π


σ
0
)
NN
s

exp

[
-
NN
s
2

σ
0
2

J

(
θ
|
D
,
β
)
]

π

(
θ
|
β
)
(
7
)
J

(
θ
|
Σ
mea
,
D
,
β
)
=
1
NN
s

∑
k
=
1
N

[
y
k
-
Sx

(
θ
,
k
|
β
)
]
T

[
y
k
-
Sx

(
θ
,
k
|
β
)
]
(
8
)
where:
p(θ|Σmea, D,β) indicates conditional probability density function;
π(θ|β) is a prior distribution of modal coordinate parameters θ;
c is a constant, ensuring that the integral summation value of Eq. (7) is 1;
N is total number of samples;
k represents sampling time;
step 2.2:
according to Eq. (8), the Fisher information matrix is obtained:
Q(S,θ0|β)=(S(Φ+[E1β,E2β, . . . ,ENmβ]))T(S(Φ+[E1β,E2β, . . . ,ENmβ])) (9)
where:
Q(S, θ0|β) is the Fisher information matrix;
step 2.3:
derive the conditional information entropy for quantifying and calculating the uncertainty of the modal parameter identification;
h(S|Σmea,D,B)|∫β∈B−ln[det(Q(S,θ0|β))]π(β)dβ (10)
where:
h(S|Σmea, D, B) is conditional information entropy;
B is a range of the perturbation coefficients;
remove the negative sign to get a conditional information entropy index:
CIE(S)=∫β∈B ln [det(Q(S,θ0|β))]π(β)dβ (11)
step 2.4:
establish the finite element model of structure, determine the candidate sensor placement positions;
use Monte Carlo method to obtain the range of the perturbation coefficient B and the structural mode shape matrix in the corresponding situation;
the initial number of selected sensors is 0;
step 2.5:
whether to consider structural modal information redundancy:
do not consider, continue to the next step;
consider, jump to step 2.9;
step 2.6:
add one sensor location from the remaining candidate sensor positions to join the existing positions;
calculate the CIE(S) value, and the sensor location corresponding to the maximum value is selected;
step 2.7:
from the remaining candidate sensor positions, delete the selected sensor location;
check the remaining positions, if there is no remaining position, continue to the next step;
if there are remaining positions, return to step 2.6;
step 2.8:
get the final sensor placement and jump out of the loop;
step 2.9:
if there are sensor locations too close, they contain similar structural modal information, resulting in redundancy of acquired structural modal information;
introduce the concept of structural modal information redundancy as:
γ
p
,
q
=
1
-

Φ
p
-
Φ
q

F

Φ
p

F
+

Φ
q

F
(
12
)
where:
γp,q represents a redundancy coefficient between the pth node position and the qth node position in the finite element structure;
the subscript F indicates the Frobenius norm;
when the γp,q value is close to 1, it indicates that the modal information redundancy between the two positions is very large, containing almost the same displacement modal information, and these two positions do not need to exist at the same time such that you need to delete a position;
in actual operation, an appropriate redundancy threshold h is needed;
if the redundancy coefficient is greater than the redundancy threshold h, the corresponding sensor location will be deleted;
step 2.10:
add one sensor location from the remaining candidate sensor positions to join the existing position;
calculate the CIE(S) value, and the sensor location corresponding to the maximum value is selected;
step 2.11:
delete the selected sensor location from the remaining candidate sensor positions;
calculate the redundancy coefficients of the remaining positions and the existing sensor locations, and delete the positions from the remaining candidate sensor positions corresponding to the coefficients exceeding the redundancy threshold h;
step 2.12:
check the remaining positions, if there is no remaining position, continue to the next step;
if there are remaining positions, return to step 2.11;
step 2.13:
get the final sensor placement and jump out of the loop.

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Sensor placement method for reducing uncertainty of structural modal identification

Life of this patent

Sensor placement method for reducing uncertainty of structural modal identification

Life of this patent

Abstract

Claims1