Real-time identification of time-varying bridge cable tension forces

Real-time identification of time-varying bridge cable tension forces based
on adaptive sparse time-frequency analysis
Yuequan Bao1,2, Zuoqiang Shi3, James L. Beck1, Hui Li2, Thomas Y. Hou1
1
Applied and Computational Mathematics, California Institute of Technology, Pasadena CA, 91125, USA
1
School of Civil Engineering, Harbin Institute of Technology, Harbin China, 150090
3
Mathematical Sciences Center, Tsinghua University, Beijing, China, 100084
Abstract:
For cable bridges, the cable tension force plays a crucial role in the construction, assessment and
long-term structural health monitoring (SHM). Cable tension forces vary in real time with the change
of the moving vehicle loads and environmental effects, and this continual variation in tension force
may cause fatigue damage of a cable. Traditional vibration-based cable tension force estimation
methods can only obtain the time-averaged cable tension force and not the instantaneous force. This
paper proposes a new approach to identify the time-varying cable tension forces of bridges based on
an adaptive sparse time-frequency analysis method. This is a newly developed method to estimate the
instantaneous frequency by looking for the sparsest time-frequency representation of the signal within
the largest possible time-frequency dictionary (i.e. set of expansion functions). In the proposed
approach, first, the time-varying modal frequencies are identified from acceleration measurements on
the cable, then, the real-time cable tension is obtained from the relation between this force and
frequencies. By utilizing the integer ratios of the higher order modal frequencies and fundamental
frequency of the cable, the effects of measurement noise on instantaneous frequencies of different
modes of the cable can be further reduced. A cable experiment is implemented to illustrate the ability
of the proposed approach. The results show that the time-varying cable tension forces can be well
1 estimated with small errors by the proposed time-frequency based approach, even in
case where
there is partial data loss in the measurements. .
Keywords: Structural health monitoring; time-varying cable tension identification; adaptive sparse
time-frequency; matching pursuit; data loss.
1 Introduction
Structural health monitoring (SHM) systems for the safety of structures have been widely investigated
and installed on many of civil infrastructure systems, such as long-span bridges, offshore structures,
large dams and other hydraulic engineering structures, nuclear power stations, tall buildings, large
spatial structures, and geotechnical engineering structures [1-4]. For large span bridges, such as
cable-stayed bridge and suspension bridge, the cables are a crucial element for overall structural safety
of the structure. The cable tension forces vary in real time because of the loads from moving vehicles
and other environmental effects, and this variation in cable tension forces may cause fatigue damage.
Therefore, estimation of the time-varying cable tension forces is important for the maintenance and
safety assessment of cable-based bridges. Methods for cable tension force estimation are either based
on inference from cable vibration signals or, less commonly, from special sensors on the cables.
The vibration-based methods for estimating cable tension forces calculate these forces from a
relation between the natural frequency of cable vibrations and the tension force in the cable. These
methods widely studied and are used in practice with the advantages of being inexpensive, convenient,
and nondestructive monitoring. Kim and Park [5] classified existing vibration-based methods into four
categories depending on whether the sag-extensibility and bending stiffness are taken into account or
2 not. They are the flat taut string theory that neglects both sag-extensibility and bending stiffness [6],
the modern cable theory that takes account of the sag-extensibility but still neglects bending stiffness
[7-10], the axially loaded beam model that considers the bending stiffness but neglects the
sag-extensibility [11], and the final category that takes account of both sag-extensibility and bending
stiffness using a practical formula [5, 12-16]. However, these vibration-based methods are usually not
able to estimate in real-time the time-varying cable tension force, but only the average cable tension
forces. Li et al. [17] proposed an extended Kalman filter based method to estimate the time-varying
cable tension force using the measured acceleration data and wind speed data on the bridge. This
method can identify the time-varying cable tension forces, but information about the wind speed data
is needed.
In additional to the vibration-based methods for cable tension force identification, direct
measurements using traditional force sensors and elasto-magnetic (EM) sensors have been also used
for cable tension force measurements in SHM of bridges. The traditional force sensor is used in a
series connection with the cable to measure the strain, either by a vibrating wire transducer, strain
gauge, hydraulic pressure sensor or Fiber Bragg Grating (FBG) sensor. The series connection means
that these sensors are not readily replaceable and they are difficult to calibrate under the high stress
states occurring in field applications. In addition, they tend to have unstable long-term performance.
These sensors are therefore not widely used for long-term monitoring of bridge cables. Another
sensing technique is the EM sensors, which have been used to measure static cable tensions. The
principle behind EM sensors is based on the variation under stress of the magnetic permeability in a
ferromagnetic material [18]. Field tests and applications of EM sensors for monitoring the cable
tension in some bridges have been reported [19-20].
3 For the identification of time-varying cable tension forces, one idea is to estimate the cable tension
force by identifying the time-varying natural frequencies of the cable through time-frequency analysis
of the cable vibration signal. Time-frequency analysis theories, including the short time Fourier
transform, Gabor transform, Wavelet transform and Hilbert-Huang transform, have been investigated
for non-linear signal analysis and non-linear structural damage identification in civil infrastructure
systems [21-25]. However, there are no reports about the successful applications of these
time-frequency analysis methods for real time identification of time-varying cable tension forces.
For the frequency varying through time over a small range and with noisy data, a high resolution
time-frequency analysis method is needed that is insensitive to noise. Recently, a new adaptive data
analysis method to study trends and instantaneous frequencies for nonlinear and non-stationary time
series data has been developed [26-28]. By combining the Empirical Mode Decomposition method
(EMD) [29] and Compressive Sensing (CS) theory [30-32], this method is able to look for the sparsest
time-frequency representation of the signal within the largest possible dictionary consisting of EMD
intrinsic mode functions. The advantages of this novel adaptive sparse time-frequency analysis method
are its high resolution in the time-frequency domain and its robustness to measurement noise. Here we
employ this adaptive sparse time-frequency analysis method to estimate in real-time the time-varying
cable tension force by using only the cable acceleration data.
2 Adaptive sparse time-frequency analysis
Typically, the time-frequency analysis method consists of two parts: a large dictionary of
time-frequency functions used to represent the signal and a decomposition method to decompose the
4 signal over the dictionary.
As an example of a dictionary, the Fourier transform, one of the most widely used frequency
analysis methods, is based on the dictionary of the well known Fourier harmonic basis functions:
sin2 kt, cos2 kt : k  0,1,2,
(1)
 
where we assume that time has been scaled for the signal to lie in [0, 1]. For any signal f  t  , t  0,1 ,
we have the following Fourier expansion:
M
f  t   a0   ak cos2k t  bk sin2k t 
(2)
k 1
where the coefficients ak , bk can be obtained by the Fourier integral. Once we get the Fourier
expansion, we can define the frequency for each component,
 t  , to be 2k , which is the derivative
of the phase function 2k t .
The Fourier series is a powerful tool which has been widely used in many different applications.
However, in many applications of time-frequency analysis, the Fourier series is not adequate because
the signal frequencies are time varying, whereas the frequencies given by the Fourier series are all
constants over the whole time span. For example, consider a simple chirp signal f  t   cos50t ,
2
t0,1 . Intuitively, the frequency should continuously increase as time grows, but this information
is not revealed by its Fourier coefficients as can be seen in Fig. 1.
5 Fig. 1. The chirp signal (left) and its Fourier coefficients (right).
In order to get a time-varying frequency, one natural idea is to enlarge the dictionary of the
time-frequency functions to incorporate functions with time-changing frequency. One such
generalization is to replace the Fourier basis by the so-called AM-FM signals, which can be written
a t  cos  t  . Where we require that a t  and the derivative of   t  ,   t  , are less oscillatory
than cos  t  . “Less oscillatory” means that over a few oscillations of cos  t  , the variation of
a t  and  t  is small so that that they can be well approximated by constants during these few
oscillations. Then in this short time interval, the signal is decomposed approximately over the Fourier
basis, which means that the frequency
 t 
can be defined to be the derivative of the phase function
  t     t 
(3)
We can therefore define informally the dictionary of AM-FM signals by:
D  a  t  cos  t  : a  t  ,   t  are less oscillatory thancos  t 
(4)
To give the rigorous definition of “less oscillatory”, we define a linear space V   ,
   k  

  k  
V    span 1,  cos   
,  sin   





   L  1k L   L  1k L 
where
(5)
 1 2 is a parameter to control the smoothness and L   1   0  2 is the number of
oscillations. A function, a t  , is said to be “less oscillatory” than cos  t  means that a t  V   .
Then, the dictionary D is well defined as following:
D  a  t  cos  t  : a  t  ,   t  V  
(6)
For arbitrary signal f  t  , we need to decompose it over above dictionary up to a given tolerance
threshold

6 M
f  t   ak cosk  t   r  t 
(7)
k 1
where r  t  is a small residual and r 2   .
For each component, its frequency can be defined as
k  t   k  t 
(8)
The remaining problem is how to do the decomposition. In Fourier analysis, since the basis is
orthogonal, the unique decomposition can be obtained by the Fourier transform. Unfortunately, the
dictionary D in Eq. (6) is highly redundant (the functions are not linearly independent) which means
that the decomposition is not unique. Taking the chirp signal in Fig. 1 as an example; obviously, both
f  t  cos  50 t 2  and its Fourier series are feasible decompositions using dictionary D, but the

decomposition cos 50 t
2
 is the one we want since this decomposition gives us an instantaneous
frequency. Compared with the Fourier series, a fundamental feature of this decomposition is that it is
very sparse. The whole signal is represented by only one component while there are about 100
components in Fourier series (see Fig. 1). For general time-frequency analysis of signals, we therefore
look for the sparsest decomposition among all feasible decompositions based on dictionary D in Eq.
(6), which is defined informally by the following optimization problem:
Minimize
 ak 1kM , k 1kM
M
M
Subject to：
f  t  -ak  t  cos k  t     , on 0,1
(9)
k 1
ak  t  cos k  t    D
The above optimization problem can be seen as a nonlinear l0-norm minimization problem, which
also arises in recent developments of compressive sensing (CS), where two common types of methods
are matching pursuit and basis pursuit. Correspondingly, there are also two different algorithms to
solve the optimization problem Eq. (9), which are based on the generalization of matching pursuit and
7 basis pursuit, respectively.
2.1 Algorithm for complete signal based on matching pursuit
If the signal is complete, the optimization problem Eq. (8) can be approximately solved by the
following nonlinear matching pursuit [27]:
Step 1: Let
r0  t   f  t  , k=1
(10)
Step 2: Solve the following nonlinear least-square problem:
 ak ,k  argmin rk1  acos 2 , subject to:
2
a, 
a cos  D
(11)
Step 3: Update the residual
k
rk  f  a j cos j
(12)
j 1
Step 4: if rk
2
  , stop, where  is a previously set residual threshold in Eq. (11). Otherwise, set
k  k  1 and go to Step 2.
In the above algorithm, the key part is to solve the nonlinear least-squares problem in Step 2. It is
found that this problem can be approximately solved by using Gauss-Newton type iteration along with
the Fast Fourier transform. By combining these techniques, an efficient method was proposed in [27].
Under some assumptions, the convergence of this method has been proved [29].
The performance of this algorithm depends on the assumption that the components of the
underlying decomposition are approximately orthogonal to each other. If some of the samples are
missing in the measured signal such that the discrete l2 norm can not approximate the continuous l2
norm, this orthogonality property is destroyed. Then, the decomposition given by this algorithm may
have large error. In order to deal with this kind of signals, we introduce another algorithm based on
basis pursuit.
8 2.2 Robust adaptive sparse time-frequency analysis for signals with missing samples
based on basis pursuit
In this algorithm, we need to assume that the number of components, M in Eq. (11) is known a prior.
Then, we solve following nonlinear l1 optimization problem to get the decomposition [30]:

min x 1 , subject to 
1 ,,M x  f  
(13)
  1 ,, M 
(14)
   cos  
(15)
x,1 ,,M
where
and  j , j  1,, K is defined as
and  is the basis of V   in Eq. (5)
   k  

  k  

cos
,
sin
 = 1，
  
  
   L  1k L   L  1kL 

 

(16)
After multiplying by cos   , the columns of  is the basis to represent the component of the
decomposition.
Once the optimization problem Eq. (13) is solved, the instantaneous frequencies are given as
j  t    j  t ，j  1,, M
(17)
Hou and Shi [30] proposed an iterative algorithm based on Augmented Lagrangian Multiplier
(ALM) and soft shrinkage to solve the optimization problem in Eq. (13). In this method, the phase
function  is fixed then Eq. (13) becomes a linear l1 optimization problem.
3 Real-time identification of time-varying tension in bridge cables
According to the widely used flat taut string theory that neglects both sag-extensibility and bending
stiffness [5], the cable force can be calculated by
9 2
f 
F =4mL2  n 
n
(18)
where fn denotes the nth natural frequency in Hz; F, m and L are the tension force, mass density, and
length of cable, respectively. Given the measured frequency and the mode number, the cable tension
can be calculated directly.
Consider time-varying cable tension, then Eq. (18) is expressed as
  t  
F  t  =4mL  n 
 2 n 
2
2
(19)
where F  t  and n  t  =2 f n  t  are the time-varying cable tension force and nth natural frequency,
respectively.
An important and useful feature of vibration of cable is that the modes frequencies have integer
multiples relations between the high order mode frequencies and the fundamental frequency,
n  t   n1  t  , if the sag-extensibility and bending stiffness are omitted. This feature means that we
only need to calculate one instantaneous frequency which significantly simply the algorithm.
3.1 Algorithm for signal with complete samples
For the complete signal, the algorithm is Section 1.1 can be simplified to be following:
min
 ak 1kK , 1
K
2
k 1
2
f  ak cos  k1  , subject to: ak V   , 1 V  
(20)
Here K is the number of modes used to calculate the instantaneous frequency which is a given positive
integer.
The above nonlinear least-squares problem is solved by a Gauss-Newton type iteration method which
is given in Algorithm 1. The explanation of this algorithm can be found in [27], where Hou and Shi
also proposed a fast algorithm based FFT to solve the least-squares problem Eq. (18).
10 Algorithm 1
Input: Initial guess of phase function 10   0 and parameters   0 ,  ,  0 .
Output: Phase function 1
1. while    do
2.
3.
while 1n 1  1n
  0 do
2
Solve the following least-squares problem:
a
n 1
k
, bkn 1   Argmin f    ak cos 1n  bk sin 1n 
K
ak ,bk
k 1
Subject to: ak  V 
n
1
4.
,b
k
 V 
n
1

2
(21)
2
Update 1n :
 k  PV

1n ;
d 
t
 b n 1   
  arc tan  kn 1    ,  k    k  s  ds
0
  dt

 ak   
 
(22)
and

where PV
 ; 
n
1
n 1
1
 K

     k  k ,  k   a  b     ak2  bk2  
k 1
 k 1

K
2
k
n
1
2
k
1
(23)
is the projection operator to the space V 1n ;  and V 1n ;  is the space
defined in Eq. (5) with    .
5.
end while
6.
    
7. end while
3.2 Algorithm for the signals with missing samples
For the signals with missing samples, the algorithm is Section 2.2 can be also be simplified by using
the relation n  t   n1  t  . The degree of freedom of the optimization problem Eq. (13) can be
11 reduced to following,
 x f
min x 1 , subject to 

2

(24)
where
   ,,  


 K1  
 1
(25)
   cos  j    , j  1,, K

j1
1

(26)
and  j1 , j  1,, K is defined as
This problem is solved by an iterative algorithm which is stated in Algorithm 2. The l1 optimization
problem Eq. (24) is solved by Augmented Lagrangian Multiplier method (ALM) which is also
accelerated by FFT due to the fact that V   is spaned by the Fourier basis in  -coordinate [30].
Algorithm 2
Input: Initial guess of phase function 10   0 and parameters   0 ,  ,  0 .
Output: Phase function 1
1. while    do
2.
3.
while 1n 1  1n
2
  0 do
Solve the following l1 optimization problem:
 a
n 1
k


, bkn 1  argmin x 1  y
x ,y
1

(25)
Subject to:   n  x+  n  y  f
1
1

2
where
  n    n ,  ,  K n  ,   n     n ,  ,  K n 
 1
1
1
 1  
 1  
 1
4.
Update 1n :
 k  PV

1n ;
d 
t
 b n 1   
  arc tan  kn 1    ,  k    k  s  ds
0
  dt

 ak   
 
12 (26)
(27)
and

5.
 j 
n
1


 cos  j1n    n , 
1
 j 
n
1


 sin  j1n    n , j  1, , K
1
(28)
Update 1n in the same way as that in Algorithm 1.
6.
end while
7.
    
8. end while
After we obtain the solution   t  , the fundamental modal frequency is 1  t   1  t  . Then, the
time-varying cable tension force F  t  is estimated as
   t  
F  t  =4mL 

 2 
2
2
(29)
4 Cable Experiments
4.1 Experimental set-up
An experiment with a model cable of 1403 cm length (see Figure 2) was carried out by Li et al. [17]
and it is employed here to illustrate the proposed approach. The vibration of the cable is excited by
two 550-kW blower fans to simulate wind. A force sensor that is installed between the left anchorage
of the cable and the sliding bearing is used to measure the time-varying cable tensions. A threaded rod
is installed in a series connection with the cable to adjust the cable tension in real time, as shown in
Figure 2(b); the threaded rod is operated manually to generate the cable tension variation. Two
accelerometers are placed at 2.43 and 3.60m from the sliding bearing to measure the in-plane and
out-of-plane vibrations of the cable. The DSpace data acquisition system is used to record the
acceleration and cable tension force data with a sampling frequency of 200 Hz.
13 (a)
(b)
(d)
(c)
Fig. 2. Experimentaal setup: (a) experimental
e
l model; (b) tension adjusting device;; (c) data acq
quisition
system andd (d) blowerss and cable.
Three experimentaal cases are considered:
c
Case 11: the initial cable tension
n is 6500 N, the variation
n of cable force is 15% w
with duration
n of 30 s;
Case 22: the initial cable tension
n is 6500 N, the variation
n of cable force is 25% w
with duration
n of 25 s;
Case 3: the initiall cable tensio
on is 6500 N
N, the variation of cable force is from
m 5%, 10%, 15% and
20% witth duration of 30s, 25s, 30
0s and 30s, rrespectively.
4.2 Iden
ntification reesults
4.2.1 Ideentification results
r
for Ca
ase 1
For Casee 1, the meassured cable acceleration
a
ssignal and itts Fourier am
mplitude specctrum betweeen [2, 15]
Hz are sshown in Figg. 3. Fig. 3(b
b) clearly shoows the firstt five modal frequencies of the cablee are very
close to bbeing integeer multiples of
o 1, 2, 3, 4, aand 5 of the fundamentall frequency aaround 2.5 Hz..
H
14 400
1
Amplitude
Acceleration (m/s/s)
2
0
-1
200
-2
0
10
20
Time (s)
0
2
30
4
6
8
10
Frquency (Hz)
12
14
(a)
(b)
Fig. 3. Measured acceleration and the Fourier transform of the signal for Case 1: (a) acceleration data;
and (b) Fourier amplitude spectrum
Using the measured acceleration data and solving the optimization problem of Eq. (7), the identified
time-frequency results of Case 1 are shown in Fig. 4, which clearly shows the first five time-varying
frequencies.
14
Frequency (Hz)
12
10
8
6
4
2
0
5
10
15
20
Time (s)
25
30
35
40
Fig. 4. Identified first five time-varying frequencies for Case 1
Using each of these five time-varying frequencies to calculate the cable tension forces by Eq. (14),
the results are shown in Fig. 5, where the solid lines with (labelled ‘Measured’) are the cable tension
forces measured by the force sensor without de-noising in the experiments and the dashed lines
(labelled ‘Identified’) are the identified cable tension forces by the proposed approach. As shown in
Fig. 5, the identified time-varying cable tension forces are closed to the measured cable tension forces
15 To quantify the identification error, the relative error of the cable tension force is calculated by

F  t   T  t 
T t  2
2
(14)
100%
where  is the percentage identification error; F  t  is the identified cable tension force and T  t  is
the measured cable tension force at the sampled times. The identification errors for the cable tension
forces calculated using the identified first to fifth time-varying frequencies are  =4.26% , 3.62%,
3.27%, 4.36% and 3.43%, respectively. The time-varying cable forces estimated using each identified
mode show some differences, although in each case the identification errors are small.
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
(a)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
(b)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
(c)
16 25
30
35
40
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
(d)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
(e)
Fig. 5. Identified time-varying cable tension forces of Case 1 by: (a) the first time-varying frequency
with error   4.26% ; (b) the second time-varying frequency with error   3.62% ; (c) the third
time-varying frequency with error   3.27% ; (d) the fourth time-varying frequency with error
  4.36% ; and (e) the fifth time-varying frequency with   3.43% .
By imposing the constraints between the higher-order modal frequencies and the fundamental
frequency of the cable, the identification results obtained from the optimization Eq. (15) are shown in
Fig. 6, where the identified time-varying cable force for Case 1 is smooth and stable with the
corresponding identification error being   3.34% .
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
Fig. 6. Identified time-varying cable tension force for Case 1 by combining the first five time-varying
frequencies, with the identification error   3.34%
4.2.2 Identification results for Case 2
For the Case 2, the measured cable acceleration signal and its Fourier amplitude spectrum between [2,
15] Hz are shown in Fig. 7.
17 400
1
Amplitude
Acceleration (m/s/s)
2
0
-1
-2
0
10
20
Time (s)
30
200
0
2
40
4
6
8
10
Frquency (Hz)
12
14
(a)
(b)
Fig. 7. Measured acceleration and the Fourier transform of the signal for Case 2: (a) acceleration data;
(b) Fourier amplitude spectrum.
Using the measured acceleration data as shown in Fig. 7(a) and solving the optimization problem of
Eq. (7), leads to the identified time-frequency results of Case 2 that are shown in Fig. 8, which clearly
shows the first five time-varying frequencies.
14
Frequency (Hz)
12
10
8
6
4
2
0
5
10
15
20
25
Time (s)
30
35
40
45
Fig. 8. Identified first five time-varying frequencies for Case 2.
The time-varying cable tension force identification results of Case 2 are shown in Fig. 9, which
shows that the time-varying cable tension force can be well identified with small identification error.
Additionally, the identification results from combining the first five time-varying frequencies are also
more smooth and stable than the results identified from each signal frequency as shown in Fig. 10,
where the identification error is 3.58%.
18 Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
(a)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
(b)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
(c)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
(d)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
(e)
Fig. 9. Identified time-varying cable tension forces of Case 2 by: (a) the first time-varying frequency
with error   3.76% ; (b) the second time-varying frequency with error   3.62% ; (c) the third
time-varying frequency with error   3.53% ; (d) the fourth time-varying frequency with   4.25% ;
(e) the fifth, time-varying frequency with error   3.46% .
19 Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
Fig. 10. Identified time-varying cable tension force of Case 2 by combining first five time-varying
frequencies, with the identification error   3.58%
4.2.3 Identification results for Case 3
For the more complex scenario of Case 3, the measured cable acceleration signal and its Fourier
1500
2
Amplitude
Acceleration (m/s/s)
amplitude spectrum between [2,15] Hz are shown in Fig.11.
0
-2
0
50
100
150
Time (s)
200
1000
500
0
2
250
4
6
8
10
Frquency (Hz)
12
14
(a)
(b)
Fig. 11. Measured acceleration and the Fourier transform of the signal for Case 3: (a) acceleration
data; and (b) Fourier amplitude spectrum.
The identified time-frequency result for Case 3 are shown in Fig. 12, which shows the first five
time-varying frequencies are well identified. Using each of this five time-varying frequencies, the
calculated time-varying cable tension forces are shown in Fig. 13, which shows that the identification
results are close to the measured cable forces, with relative identification errors are   3.04% , 2.96%,
2.56%, 2.50% and 2.26% for each of the five time-varying frequencies. The identification result in Fig.
14 shows the improvement from combining the first five time-varying frequencies as in Eq. (15), with
the identification error being   1.95% .
20 14
Frequency (Hz)
12
10
8
6
4
2
0
50
100
150
Time (s)
200
250
300
Cable force (N)
Fig. 12. Identified first five time-varying frequencies of Case 3.
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
200
250
300
200
250
300
200
250
300
Cable force (N)
(a)
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
Cable force (N)
(b)
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
(c)
21 Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
200
250
300
200
250
300
Cable force (N)
(d)
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
Cable force (N)
(e)
Fig. 13. Identified time-varying cable tension forces of Case 3 by: (a) the first time-varying frequency
with error   3.04% ; (b) the second time-varying frequency with   2.96% ; (c) the third
time-varying frequency with error   2.56% ; (d) the fourth time-varying frequency with error
  2.50% ; and (e) the fifth time-varying frequency with error   2.26% .
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
200
250
300
Fig. 14. Identified time-varying cable tension force of Case 3 by combining the first five time-varying
frequencies with the identification error   1.95% .
4.3 Robustness of the approach to data loss
Wireless sensors and networks are widely studied and applied for SHM [35, 36]. The use of wireless
sensors for cable tension force estimation has also been reported [37]. For wireless sensors, data
packet loss a frequent problem
for numerous reasons, including: radio interference, e.g., other
devices operating on the same frequency; weather problems such as rain and lightning; poor
installation; antenna orientation; large transmission distances; radio wave obstructions; hardware
22 problems; etc. This problem has motivated the development of a method [38]. To consider the
potential of applying the proposed time-varying cable tension force identification method for wireless
sensors, its robustness for partial loss of experimental data is investigated. In this example, the
experimental cable acceleration data of Cases 1-3 are simulated with 10% random loss of the
transmitted data packets, each packet containing 5 data points. The acceleration data with 10% random
data loss for Cases 1-3 are shown in Fig. 15, which also shows the corresponding time segment from
2
Acceleration (m/s/s)
Acceleration (m/s/s)
10-12 seconds where the data loss occurs. As shown in Figs. 15 (b, d, f), the lost data are zero-padded.
1
0
-1
-2
0
10
20
Time (s)
2
0
-2
10
30
10.5
2
1
0
-1
-2
0
10
20
Time (s)
Acceleration (m/s/s)
Acceleration (m/s/s)
-2
150
Time (s)
12
11.5
12
-2
10.5
11
Time (s)
(d)
0
100
11.5
0
10
30
2
50
12
2
(c)
0
11.5
(b)
Acceleration (m/s/s)
Acceleration (m/s/s)
(a)
11
Time (s)
200
2
0
-2
10
250
10.5
11
Time (s)
(e)
(f)
Fig. 15. The acceleration data and its local magnification with 10% data loss for Cases 1-3.
By solving the optimization problem of Eq. (16), the identified results for the time-varying cable
tension forces for Cases 1-3 with 10%
loss of the original acceleration data are shown in Fig. 16,
which shows that the identified time-varying cable tension forces close to the measurements from the
23 force sensors. The corresponding identification errors for Cases 1-3 are   3.41% , 3.77% and 2.38%,
respectively, which are only slightly larger than the errors without data lost. The limited impact of the
random loss of 10% of the original signal packets is also apparent by the closeness of Fig. 16(a), (b)
and (c) with Fig. 6, 10 and 14, respectively, which further illustrates the robustness of the proposed
approach.
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
Time (s)
25
30
35
40
(a)
Cable force (N)
8000
Measured
Identified
7000
6000
5000
0
5
10
15
20
25
Time (s)
30
35
40
45
Cable force (N)
(b)
8000
Measured
Identified
7000
6000
5000
0
50
100
150
Time (s)
200
250
300
(c)
Fig. 16. Identification results with 10% loss of the acceleration data: (a) identification results for Case
1 with identification error   3.41% ; (b) identification results of Case 2 with identification error
  3.77% ; and (c) identification results for Case 3 with identification error   2.38% .
5 Conclusions
24 A time-varying cable tension force identification method based on adaptive sparse time-frequency
analysis was proposed in this paper. The first step is to estimate the time-varying modal frequencies
from a cable vibration signal by producing the sparsest decomposition of the signal using a large
time-frequency dictionary. Then second step is to obtain the time-varying cable tension force by the
relationship between the cable force and the cable natural frequency derived from the flat taut string
theory.
Model cable experiments under three scenarios are implemented to illustrate the capability of the
proposed approach. The results show that the time-varying modal frequencies of the cable can be well
identified and that the time-varying cable tension forces calculated from each time-varying frequency
separately are close to the force sensor measurements. By considering the integer ratios of the different
modal frequencies to the fundamental frequency of the cable, much better and more robust results can
be achieved. The relative identification errors of the time-varying cable tension forces for all three
experimental scenarios are less than 5%, which is an acceptable error for structural health monitoring
purposes.
The robustness of the proposed approach is demonstrated by an example where 10% of the data
packets are lost but good results with small identification errors are still obtained. This robustness
implies that the proposed approach has good potential to be used in conjunction with a wireless sensor
based SHM monitoring system that incorporates accelerometers on the cables.
In the proposed approach, only the acceleration signal from cable vibrations is needed, which is
readily measured in a real bridge. Therefore, field tests on a cable bridge are practical to further verify
the validity of the proposed method.
25 Acknowledgements
One of the authors (Yuequan Bao) acknowledges the support provided by the China Scholarship
Council while he was a Visiting Associate at the California Institute of Technology.This research was
also supported by grants from the National Basic Research Program of China (Grant
No.2013CB036305), the National Natural Science Foundation (Grant No. 51378154, 51161120359),
the National High Technology Research and Development Program of China (No. 2014AA110401)
and the Ministry of Science and Technology (Grant No. 2011BAK02B02), which supported the first
and fourth authors (Yuequan Bao and Hui Li). The research of Zuoqiang Shi was supported by
National Natural Science Foundation of China (Grant No. 11201257). The research of Thomas Hou
and Zuoqiang Shi was also in part supported by a NSF Grant DMS 1318377.
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