US 2024/0353276 A1 · CABLE FORCE IDENTIFICATION ALGORITHM CONSIDERING SEMI-RIGID CONSTRAINTS ON BOTH ENDS

Abstract

The present invention discloses a cable force identification algorithm considering semi-rigid constraints on both ends. The present invention only needs the basic parameters such as length and mass per unit length of a cable and then adopts a modal identification algorithm such as stochastic subspace method and frequency domain decomposition method to process signals collected by the acceleration sensor, and thus obtains the corresponding first natural frequency and mode shapes, and the cable force can be solved without obtaining any other data in advance. The present invention simplifies the cable to an equivalent single-degree-of-freedom system, modifies the first natural frequency of the cable through the mode shape of modal identification, avoids the cable force identification error caused by the change of the boundary mechanical characteristics of the cable, improves the efficiency and accuracy of cable force monitoring, has good application prospect in real-time monitoring of cable force of cable-stayed structure.

A cable force identification algorithm considering semi-rigid constraints on both ends, comprising the following steps:

step 1:
arranging an acceleration sensor vertically at the mid-span and both ends of a semi-rigid constrained cable respectively, and collecting vibration signals of the cable under environmental excitation or artificial excitation;
step 2:
processing the vibration signals collected in step 1 by the modal identification algorithm, and identifying the first natural frequency f1 and mode shapes at the mid-span and two endpoints of the semi-rigid constrained cable;
step 3:
establishing a semi-rigid constrained cable model which is mainly composed of a cable, and a transverse support spring and an axial support spring on the left and right ends, simplifying the semi-rigid constrained cable model to an equivalent single-degree-of-freedom model, and calculating the generalized mass M* and the overall stiffness K* of the semi-rigid constrained cable;
(a) calculating the generalized mass M* of the semi-rigid constrained cable
the first-order mode shape of hinged cables on both ends is
ϕ
0
⁢
sin
⁢
π
⁢
x
l
,
the mode shapes of transverse support springs on both ends are respectively ϕ1 and ϕ2, and the first-order mode shape of the semi-rigid constrained cable is the superposition of the first-order mode shape of the hinged cable and the mode shape of the transverse support spring and can be calculated by the following formula:
φ
1
(
x
)
=
a
⁢
x
+
b
+
ϕ
0
⁢
sin
⁢
π
⁢
x
l
(
26
)
a
=
ϕ
2
-
ϕ
1
l
(
27
)
b
=
ϕ
1
(
28
)
ϕ
0
=
φ
⁡
(
l
2
)
-
ϕ
1
+
ϕ
2
2
(
29
)
wherein x represents the horizontal coordinate along the length of the semi-rigid constrained cable;
ϕ0 represents the maximum mode shape value of the hinged cable;
l represents the length of the semi-rigid constrained cable:
φ
⁡
(
l
2
)
represents the mode shape value at the midpoint of the semi-rigid constrained cable;
and ϕ0, ϕ1 and ϕ2 are normalized:
the generalized mass M* of the semi-rigid constrained cable is:
M
*
=
∫
0
l
m
¯
⁢
φ
1
(
x
)
2
⁢
dx
=
m
¯
⁢
l
⁡
(
1
3
⁢
a
2
⁢
l
2
+
b
2
+
1
2
⁢
ϕ
0
2
+
abl
+
2
⁢
a
⁢
ϕ
0
⁢
l
π
+
4
π
⁢
b
⁢
ϕ
0
)
(
30
)
wherein m represents the mass per unit length of the semi-rigid constrained cable:
(b) calculating the overall stiffness K* of the equivalent single-degree-of-freedom model for the semi-rigid constrained cable
the equivalent single-degree-of-freedom model of the semi-rigid constrained cable is mainly composed of a spring with the equivalent lumped mass point of m0* and the stiffness coefficient of k0* and transverse support springs on the left and right ends;
and the overall stiffness K* of the equivalent single-degree-of-freedom model for the semi-rigid constrained cable is calculated by the following formula:
K
*
=
1
1
k
0
*
+
ϕ
1
4
⁢
ϕ
1
⁢
k
1
+
ϕ
2
4
⁢
ϕ
2
⁢
k
2
=
4
⁢
ϕ
1
⁢
k
1
⁢
k
0
*
4
⁢
ϕ
1
⁢
k
1
+
k
0
*
(
ϕ
1
+
ϕ
2
)
(
31
)
wherein k0* represents the generalized stiffness of the single-degree-of-freedom system of the hinged cable;
and k1 and k2 represent the stiffness of the transverse support springs on the left and right ends of the semi-rigid constrained cable respectively:
for the semi-rigid constrained cable on both ends, the first natural frequency is f1, the first-order natural circular frequency is ω1, and according to the fundamental vibration characteristics of the single-degree-of-freedom system, the relationship among K*, M′, f1 and ω1 is expressed by the following formula:
ω
1
=
K
*
M
*
=
2
⁢
π
⁢
f
1
(
32
)
step 4:
establishing the equivalent single-degree-of-freedom model of the hinged cables on both ends, which is composed of a spring with the equivalent lumped mass point of m0* and the stiffness coefficient of k0*, then calculating the generalized stiffness k0* of the hinged cables on both ends, and modifying the first natural frequency of the semi-rigid constrained cable;
the generalized stiffness k0* of the equivalent single-degree-of-freedom model of the hinged cables on both ends is:
k
0
*
=
1
⁢
6
⁢
M
*
⁢
ω
1
2
⁢
ϕ
1
⁢
fT
16
⁢
f
⁢
ϕ
1
⁢
T
-
M
*
⁢
ω
1
2
⁢
l
⁡
(
ϕ
1
+
ϕ
2
)
⁢
y
1
(
33
)
wherein f represents the mid-span sag of the hinged cable, which is the maximum displacement of the hinged cables on both ends;
T represents the measured cable force of the semi-rigid constrained cable;
and y1 represents the equivalent displacement of the transverse support spring of the semi-rigid constrained cable:
it can be known from the fundamental vibration characteristics of the single-degree-of-freedom system that the relationship among k0*, m0*, f0 and ω0 is as follows:
ω
0
=
k
0
*
m
0
*
=
2
⁢
π
⁢
f
0
(
34
)
the first generalized mass m0* of the hinged cables on both ends is
1
2
⁢
m
¯
⁢
l
,
and the first-order natural circular frequency ω0 and the first natural frequency f0 of the hinged cables on both ends are obtained respectively as follows through formula (9):
ω
0
=
k
0
*
m
0
*
=
{
3
⁢
2
⁢
M
*
⁢
ω
1
2
⁢
ϕ
1
⁢
fT
m
¯
⁢
l
[
16
⁢
f
⁢
ϕ
1
⁢
T
-
M
*
⁢
ω
1
2
⁢
l
⁡
(
ϕ
1
+
ϕ
2
)
⁢
y
1
]
}
1
2
(
35
)
f
0
=
ω
0
2
⁢
π
=
{
8
⁢
M
*
⁢
ω
1
2
⁢
ϕ
1
⁢
fT
π
2
⁢
m
¯
⁢
l
[
16
⁢
f
⁢
ϕ
1
⁢
T
-
M
*
⁢
ω
1
2
⁢
l
⁡
(
ϕ
1
+
ϕ
2
)
⁢
y
1
]
}
1
2
(
36
)
step 5:
substituting the modified first natural frequency into the cable force-frequency relation equation to solve the cable force:
T
=
M
*
⁢
ω
1
2
⁢
l
[
32
⁢
ϕ
1
⁢
f
+
π
2
(
ϕ
1
+
ϕ
2
)
⁢
y
1
]
1
⁢
6
⁢
π
2
⁢
ϕ
1
⁢
f
=
M
*
⁢
ω
1
2
⁢
l
[
32
⁢
ϕ
1
+
π
2
(
ϕ
1
+
ϕ
2
)
⁢
y
1
f
]
1
⁢
6
⁢
π
2
⁢
ϕ
1
(
37
)
wherein
y
1
f
=
ϕ
1
ϕ
0
.

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Cable force identification algorithm considering semi-rigid constraints on both ends

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Cable force identification algorithm considering semi-rigid constraints on both ends

Life of this patent

Abstract

Claims2