8. moving force identification with generalized orthogonal function expansion

8/12/2019 8. Moving Force Identification With Generalized Orthogonal Function Expansion

1/25


2/25

210 M o v i n g L o a d s D y n a m i c A n a l y si s a n d I d e n t if i c a t io n Te c h n i qu e s

wherei(t) is the set of functions. There are many series expansions, such as polynomialfunctions, Fourier series and orthogonal functions, such as the Legendre polyno-mials, Chebyshev polynomials, Bessel functions and B-spline functions. Generalizedorthogonal functions and wavelets will be discussed in the following sections.

8.2.2 G e n e r a l i z e d O r t h o g o n a l F u n c t i o n

The Chebyshev polynomial has been used widely in numerical analysis. This sectionintroduces the generalized orthogonal function from the first kind of Chebyshev poly-nomialTn(x). It is a polynomial inx of degreen, defined by the relation (Mason andHandscomb, 2003):

Tn(x)=cos n whenx=cos (8.2)If the range of the variable x is in the interval [1,1], then the range of the cor-

responding variable can be taken as [0,]. These ranges are traversed in oppositedirections, sincex= 1 corresponds to=andx= 1 corresponds to= 0. We havethe fundamental recurrence relation from Equation (8.2):

Tn(x)=2xTn1(x) Tn2(x), (n=2, 3, . . .) (8.3)

with the initial conditions:

T0(x)=1, T1(x)=x (8.4)

All the polynomials{Tn(x)} can be generated recursively from Equation (8.3). Theorthonormal polynomials can be obtained as follows, with scaling of the polynomials:

1T0(x),

2Ti(x), i=1, 2, . . . (8.5)

For an independent variabletin a general range [0,T], we can map the independentvariabletto the variablex with the transformation

x= 2tT

1 (8.6)

and this leads to a shifted Chebyshev polynomial (of the first kind)Tn(t) of degreenin variabletin the interval [0,T] which is a generalized orthogonal function given as:

T1= 1

T2=

2

2

Tt 1

Downloadedby

[ISTANBULTEKNIK

UNIVERSITESI]at06:5212Ja

nuary2014


3/25

M o v i n g F o rc e I d e n t i f ic a t i o n 211

T3=

2

2

2

Tt 1

2 1

Tj+1=2

2

Tt 1

Tj Tj1

(8.7)

8.2.3 W a v e l e t D e c o n v o l u t io n

The Daubechies wavelets and associated scaling functions j,k(t) are obtained bytranslation and dilation of functions(t) and(t) respectively (Law et al., 2008).

J,k(t)=2J/2(2Jt k) J, kZ (8.8)

J,k(t)=2J/2(2Jt k) J,kZ (8.9)

whereJis the resolution. The scaling function(t) and wavelet function(t) can bederived from the dilation equation as:

(t)=k

ak(2t k) (8.10)

(t)

= k(

1)ka1

k(2t

k) (8.11)

where ak, a1k are the filter coefficients and they are fixed for specific wavelet orscaling function basis. It is noted that only a finite number ofak, a1k are nonzero forcompactly supported wavelets.

The scaling function(t) and wavelet function(t) have the following properties:

(t)dt=1 (8.12)

(tj)(t k)dt=j,k, j, kZ (8.13)

tm(t)dt=0, (m=0, 1, . . . ,L/2 1) (8.14)

where m denotes the number of vanish moments and L is the order of Daubechieswavelet withL= 2m.

The translation of the scaling and wavelet functions on each fixed scale forms theorthogonal subspaces:

VJ= {2J/2(2Jt k), JZ} (8.15)

Downloadedby

[ISTANBULTEKNIK


nuary2014


4/25


WJ= {2J/2(2Jt k), JZ} (8.16)

such thatVJforms a sequence of embedded subspaces:

{0}, . . . , V1V0V1, . . . , L2(R) and VJ+1=VJWJ (8.17)

where is the operator for the addition of two subspaces. At a certain resolutionJ, theapproximation of a function f(t) in L2(R) space using J,k(t) as basis can be denoted as:

PJ(f)=k

J,kJ,k(t), J, kZ (8.18)

wherePJ(f) is the approximation off(t) andJ,k is the approximation coefficient.LetQJ(f) be the detail of the function usingJ,k(t) as basis at the same levelJ, and

QJ(f)=k

J,kJ,k(t), J,kZ (8.19)

whereJ,kis the detail coefficient. The approximationPJ+1(f) of the next level (J+ 1)of resolution is given by:

PJ+1(f)=PJ(f) +QJ(f) (8.20)This forms the basis of multi-resolution analysis associated with waveletapproximation.

The Wavelet-Galerkin approximation to the signalf(t) at a certain resolutionJcanbe expressed as:

h(t)=k

J,k2J/2(2Jt k), J, kZ (8.21)

from Equations (8.8) and (8.18). Substitutingy= 2Jtinto Equation (8.21), we obtain:

h(y)=k

J,k(y k); J,k=2J/2J,k, J, kZ (8.22)

Ify takes up only integer values, the approximation is discretized at all dyadic pointswitht= 2Jyas:

h(i)=h(iy)=hi, (i=0, 1, 2, . . . ,NT) (8.23)whereNTis the number of time instances. Equation (8.22) can be rewritten as:

hi=k

kik=k

ikk (8.24)

Downloadedby

[ISTANBULTEKNIK


nuary2014


5/25


withk=(k). In matrix form this becomes:

h1h2

h3............

hNT1

=

0 0 0 L2 2 11 0 0

0

3 2

2 1 0 0 4 3... ...

L2 L3 L4 0 00 L2 L3 0 0... ...0 0 0 L3 1 0

12

3............

NT1

(8.25)

The periodic boundary condition has been included in Equation (8.25) for the finitedomain analysis denoted as:

1=NT12=NT3

...

L+2=NTL+2

and

NT= 0NT+1=1

...

NT+L2=L2

(8.26)

8.3 M o ving F o rce I dent ifica t io n8.3.1 B e a m M o d e l

8.3.1.1 G e n e r a l i z e d O r t h o g o n a l F u n c t i o n E x p a n s i o n

The strain in the Euler-Bernoulli beam at a pointx and timetcan be written as:

(x, t)= h2w(x, t)

x2 (8.27)

whereh is the distance between the lower surface and the neutral plane of bending ofthe beam. Substituting the transverse displacementw(x, t) in Equation (2.7) of Chapter2 into Equation (8.27), and assuming there areNmodes in the responses, we have:

(x, t)=Q (8.28)

where

= {h1(x),h2(x), . . . , hN(x)}; Q= {q1(t),q2(t), . . . ,qN(t)}T.

and i(x) is the second derivative ofi(x).The strain is then approximated by a generalized orthogonal functionT(t) as shown

in Equation (8.5) as:

(x, t)=Nfi=1

Ti(t)Ci(x) (8.29)

Downloadedby

[ISTANBULTEKNIK


nuary2014


6/25


where{Ti(t),i= 1,2, . . . ,Nf} are the generalized orthogonal functions;{Ci(x), i= 1,2, . . . ,Nf} is the vector of coefficients in the expanded expression. Note that the waveletform of the orthogonal function in Equation (8.21) can also be used. The strains attheNs measuring points can be expressed as:

=CT (8.30)

where

T= {T0(t),T1(t), . . . ,TNf(t)}T;= {(x1, t), (x2, t), . . . , (xNs , t)}T;

C=

C10(x1) C11(x1) C1Nf(x1)

C20(x2) C21(x2) C2Nf(x2)...

......

...

CNs0(xNs) CNs1(xNs) CNsNf(xNs)

and{x1,x2, . . . , xNs} is the vector of the location of the strain measurements. By theleast-squares method, the coefficient matrix can be obtained as:

C=TT(TTT)1 (8.31)

Substitute Equation (8.28) into Equation (8.30), we have:

Q=(T)1TCT (8.32)

where

=

h1(x1) h2(x1) hN(x1)

h1(x2) h2(x2) hN(x2)

......

......

h1(xNs) h2(xNs) hN(xNs)

and it can be obtained from Equation (2.4) in Chapter 2.

8.3.1.2 M o v i n g F o r c e I d e n t i f i c a t i o n T h e o r y

The vector of generalized coordinates obtained from Equation (8.32) can be substitutedinto Equation (2.8) for the beam, and rewrite it in matrix form to become:

IQ+ CdQ+KQ=BP (8.33)where

Cd=diag(2ii);K=diag(2i)

Downloadedby

[ISTANBULTEKNIK


nuary2014


7/25


B=

1(x1(t))/M1 1(x2(t))/M1 1(xNp(t))/M12(x1(t))/M2 2(x2(t))/M2 2(xNp(t))/M2

......

......

N(x1(t))/MN N(x2(t))/MN N(xNp(t))/MN

The required QandQcan be obtained by directly differentiating Equation (8.32) tohave:

Q=(T)1TCTQ=(T)1TCT

The moving forces obtained from Equation (8.33) using a straight forward least-

squares solution would be unbound. Let the left-hand-side of Equation (8.33) berepresented by U.Regularization technique is used to solve the ill-posed problem inthe form of minimizing the function:

J(P,)= BP U2 +P2 (8.34)

where is the non-negative regularization parameter.The success of solving Equation (8.34) lies in how to determine the regularization

parameter. Two methods are used in this chapter. If the true forces are known, the

parameter can be determined by minimizing the error between the true forces and thepredicting values as:

S= PP (8.35)

In the practical case when the true forces are not known, the method of generalizedcross-validation (GCV) is used to determine the optimal regularization parameter. TheGCV function to be minimized in this work is defined by (Golub et al., 1979):

g()= B

P

U

2

2{trace[IB((BTB+I)1BT)1]}2 (8.36)

wherePis the vector of estimated forces.

8.3.2 P l a t e M o d e l

The displacement w(xs, ys, t) at location (xs, ys) and at time t is rewritten in matrixform from Equation (3.16) in Chapter 3 as:

w(xs, ys, t)=WsQ (s=1, 2, . . . ,Ns) (8.37)where Ns is the number of measuring points, and Q is a matrix of qij(t) fromEquation (3.16).

Ws= {W11(xs, ys),W12(xs, ys), . . . ,W1n(xs, ys),W21(xs, ys), . . . ,Wmn(xs, ys)}

Downloadedby

[ISTANBULTEKNIK


nuary2014


8/25


The modal strains inx-direction can be written as:

Wij(xs, ys)= zti

a 2

sini

axsYij(ys), (i=1, 2, . . . ,m; j=1, 2, . . . ,n).

whereztis the distance from the measuring point at the outer surface to the neutralsurface of bending. ForNs measuring points:

wns=WnsQ (8.38)

where

wns=

[w(x1, y1, t),w(x2, y2, t), . . . ,w(xNs , yNs , t)]T

Wns=

W11(x1, y1) W12(x1, y1) Wmn(x1, y1)W11(x2, y2) W12(x2, y2) Wmn(x2, y2)

...... ...

W11(xNs , yNs ) W12(xNs , yNs ) Wmn(xNs , yNs )

Nsmn

The modal displacement can be obtained from Equation (8.38) by least-squaresmethod as:

Q=(WTnsWns)1WTnswns (8.39)

Since the displacements or strains are measured, the velocities and accelerations canbe obtained by dynamic programming filter (Trujillo and Busby, 1983) or orthogonalpolynomial method described in Section 8.2, and the modal velocities and accelera-tions are calculated by the least-squares method from Equation (8.39). They are thensubstituted into Equation (3.17) for the plate to form the matrix equation:

B=SP (8.40)where

S=

2sinax1(t)

Y11(y1(t))

hab

0Y211(y)dy

2sinax2(t)

Y11(y2(t))

hab

0Y211(y)dy

2sin

axNp (t)

Y11(yNp (t))

hab

0Y211(y)dy

2sinax1(t)

Y12(

yl(t))

hab0

Y2

12

(y)dy

2sinax2(t)

Y12(y2(t))

hab0

Y2

12

(y)dy

2sinaxNp (t)

Y12(yNp (t))

hab0

Y2

12

(y)dy

..

....

..

.

2sinm

ax1(t)

Ymn(yl(t))

hab

0Y2mn(y)dy

2sinm

a x2(t)

Ymn(y2(t))

hab

0Y2mn(y)dy

2sin

ma

xNp (t)Ymn(yNp (t))

hab

0Y2mn(y)dy

Downloadedby

[ISTANBULTEKNIK


nuary2014


9/25


B=

q11(t) + 21111q11(t) +211q11(t)q12(t) + 21212q12(t) +212q12(t)

...

qmn(t) + 2mnmnqmn(t) +2mnqmn(t)

; P= p1(t),p2(t), . . . ,pNp (t)T

(8.41)

The moving load Pcan be obtained by the straightforward least-squares methodfrom Equation (8.40). But the solutions are frequently unstable in the sense that smallnoises in the responses would result in large changes in the predicted moving force.Regularization technique is utilized to improve the conditioning. The load identifica-tion is formulated as a nonlinear least-squares problem.

minJ(P,)=(B SP,R(B SP)) +(P,P) (8.42)

where is an optimal regularization parameter or a vector. R is a weight matrixand it can be determined from the measured information (Santantamarina and Fratta,1998). Generalized cross-validation method (Golub et al., 1979) and L-Curve method(Hansen, 1992) are then used to determine the optimal regularization parameter inthis study.

8.4 Applica t io ns

8.4.1 I d e n t i f i c a t i o n w i t h a B e a m M o d e l

The method, described in previous sections, is illustrated in the following simulationstudies. The effect of discarding some of the information contained in the measuredresponses on the error of identification is studied.

8.4.1.1 S i n g l e - S p a n B e a m

A single span simply supported beam is studied with two varying forces moving ontop at a constant spacing of 4.27 m.

f1(t)=9.9152 104[1 + 0.1 sin(10t) + 0.05 sin(40t)] N;f2(t)=9.9152 104[1 0.1 sin(10t) + 0.05 sin(50t)] N.

(8.43)

The parameters of the beam are as follow: EI= 2.5 1010 Nm2, A= 5000 kg/m,L= 30m, h= 1 m. The first eight natural frequencies of the beam are 3.9, 15.61, 35.13,62.48, 97.58, 140.51, 191.25 and 249.8 Hz, and they are used in the computation ofthe analytical mode shapes from Equation (2.4) in Chapter 2. The forces are movingat a speed of 30 m/s. Random noise is added to the calculated strains to simulate the

polluted measurement and 1, 5 and 10 percent noise levels are studied with:

=calculated+EpNiose var(calculated) (8.44)

where is the vector of strains;EP is the noise level; Niose is a standard normal dis-tribution vector with zero mean and unit standard deviation;calculatedis the vector of

Downloadedby

[ISTANBULTEKNIK


nuary2014


10/25


Table 8.1 Error of Identification for single span beam

Number of Noise levelmode shapes

1% 5% 10%

First Second First Second First Secondforce force force force force force

2 31.51 31.53 45.11 45.05 174.27 112.503 11.56 12.00 13.04 13.24 16.31 16.084 6.78 6.18 7.05 6.95 8.47 8.695 5.16 3.62 5.07 3.89 5.46 4.806 3.90 3.10 4.02 3.28 4.14 3.667 3.43 2.99 3.45 3.13 3.66 3.44

8 3.15 2.86 3.19 3.01 3.42 3.299 9.52 8.96 9.48 8.94 9.48 8.9410 18.02 17.30 18.51 17.99 18.42 17.88

calculated strains; var(calculated) is the standard deviation ofcalculated. The errors in theidentified forces are calculated as:

Error=PPTruePTrue

100% (8.45)

Table 8.1 shows the errors of identification from using different number of modeshapes in the identification. The time step is 0.001 s in the calculation. The strainconsists of responses from the first eight mode shapes polluted with 5 percent noiselevel. Ten measuring points are available in the identification and they are evenlydistributed along the beam length. The different combination of number of modeshapes used in the identification and the number of measuring points are studied.Figure 8.1 shows the identified results using three and six mode shapes. The followingobservations were made:

1. Results in Table 8.1 show that the errors in the identified forces are insensitive to

the noise level in the responses. This is because orthogonal functions have beenused to approximate the strains in the identification, and this approximationsuppresses the errors due to high frequency measurement noise.

2. When the number of mode shapes used in the identification is the same asthe number of mode shapes in the responses, i.e. eight mode shapes, theerrors of identification are the smallest. The errors become large when the numberof mode shapes used in identification is either larger or smaller than the numberof mode shapes in the responses. This indicates that the pairing of the number ofmode shapes in both the responses and the identified forces has a large effect on

the errors in the identification. The correct pairing can be determined from aninspection of the frequency content in the measured responses.3. Figure 8.1 shows that there are large discrepancies in the identified forces near the

beginning and the end of the moving forces when only three modes are used in theidentification. These discrepancies are much less when six modes are used. This isbecause of the sudden appearance and disappearance of the forces at these points,

Downloadedby

[ISTANBULTEKNIK


nuary2014


11/25


15 104 The first axle force

10

5

5

N

0

0 0.2 0.4 0.8 1.20.6 1

15 104 The second axle force

Time (s)

10

0

N

5

0 0.2 0.4 0.8 1.20.6 1

Figure 8.1 Identified results with different number of mode shapes ( True loads; - - from 3modes; from 6 modes)

which can be represented by an equivalent impulsive force. These impulsive forcesexcite the beam with a broad-band vibration that covers a large number of modalfrequencies. Therefore, more mode shapes should be used in the identification totake advantage of the information of the forces at higher modal frequencies in theresponses at the beginning and the end of the time histories.

8.4.1.2 T w o - S p a n C o n t i n u o u s B e a mTable 8.2 shows the errors in the identified moving forces on a two-span continuousbeam with different numbers of mode shapes and numbers of measuring points. Theparameters of the beam are the same as for the single-span beam except that each spanmeasures 30 m long. The first eight natural frequencies of the beam are 3.9, 6.1, 15.61,19.75, 35.12, 41.22, 62.43 and 70.48 Hz. Figure 8.2 shows the identified forces fromusing strains polluted with 5 percent noise level at six measuring points. Inspection ofthe results in Table 8.2 and Figure 8.2 gives the following observations:

1. Results in Table 8.2 show that the errors increase as the noise level in the responseincrease. The errors are more than twice of that under similar conditions for thesingle-span beam. Therefore, moving load identification in a multi-span beamwould be less accurate than that in a single-span beam.

2. When the number of mode shapes used in the identification equals to that in theresponses as shown in the first two rows and the lower part of Table 8.2, the

Downloadedby

[ISTANBULTEKNIK


nuary2014


12/25

Table 8.2 Identified Errors for Two-span Beam

No. of mode No. of mode No. of Noise levelshape in shapes in measuring

responses identification points 1% 5% 10%N1 N2 Ns

First Second First Second First Secondforce force force force force force

10 10 14 7.85 9.07 18.48 19.44 27.64 28.9910 10 10 7.78 8.98 16.67 17.98 26.28 27.4210 9 10 8.90 10.61 15.95 17.23 24.24 25.1210 8 10 11.54 13.71 23.05 24.12 31.37 32.5910 7 10 13.81 16.19 17.02 19.01 21.99 23.7010 6 10 17.05 19.55 20.76 22.99 26.61 28.54

6 6 6 15.99 18.15 20.15 21.91 26.26 27.786 6 8 16.06 18.21 20.93 22.69 27.74 29.246 6 10 15.98 18.15 19.98 21.77 26.11 27.696 6 12 15.90 18.08 18.90 20.73 24.02 25.656 6 14 15.85 18.04 18.19 20.06 22.47 24.14

20

104 The first axle force

15

10

5

N

0

0 0.5 1 2.51.5 2

5

20 104 The second axle force

Time (s)

15

5

0

N

5

0 0.5 1 2.51.5 2

10

Figure 8.2 Identified forces on continuous beam from different number of mode shapes ( Trueloads; - - from with 3 modes; from with 6 modes)

Downloadedby

[ISTANBULTEKNIK


nuary2014


13/25


13.715m

0.2m

2.743m

4.865m

24.325m

Figure 8.3 A typical single span bridge deck

errors in the identified forces varies only slightly with more measuring points.

The number of the measuring points is best selected to be equal to the number ofmode shapes.3. Results from the upper part of Table 8.2 also show that the errors would be

smallest when the number of mode shapes in the identification is the same as thatin the responses. This confirms the observation made in the case of the singlespan beam.

4. The identified forces in Figure 8.2 have large fluctuations close to the intermediatesupport at 1.0 s. This is due to the presence of the small responses generated atthis time instance with subsequently a small signal to noise ratio in the measureddata.

8.4.2 I d e n t i f i c a t i o n w i t h a P l a t e M o d e l

A simply supported prototype bridge composed of five I-section steel girders and aconcrete deck as shown in Figure 8.3. It is noted that the model is similar to the oneused by Fafard and Mallikarjuna (1993) in their study of bridgevehicle interaction.It is also similar to the continuous bridge deck shown in Figure 4.12 of this book. It iswide enough to accommodate four lanes of traffic. The parameters of the bridge deckare listed as follow:a= 24.325 m,b= 13.715 m,h= 0.2m,Ex= 4.1682 1010 N/m2

Ey= 2.9733 1010

N/m2

, = 3000 kg/m3

, xy= 0.3. For the steel I-beam: webthickness= 0.01111 m, web height = 1.490 m, flange width = 0.405 m, flangethickness= 0.018 m. For the diaphragms, the distance between two diaphragms is4.865 m, cross-sectional area= 0.001548 m2,Iy= 0.707 106 m4,Iz= 2 106 m4,J= 1.2 107 m4.The rigidities in thex-direction of the equivalent orthotropic platecan be calculated according to Bakht and Jaeger (1985), as Dx= 2.415 109 Nm,

Downloadedby

[ISTANBULTEKNIK


nuary2014


14/25


Table 8.3 Natural Frequencies of the equivalent orthotropic plate (Hz)

n 1 2 3 4 5 6 7 8m

1 4.96* 6.31 10.01 16.07 24.81 36.46 51.08 68.702 19.84* 21.26 25.41 32.17 41.51 53.49 68.22 85.823 44.65* 46.07 50.32 57.33 67.06 79.48 94.60 112.494 79.37* 80.80 85.07 92.17 102.06 102.50 114.74 115.585 124.01* 124.02 125.42 125.44 125.46 129.40 129.43 129.51

* Longitudinal bending modes.

Dy= 2.1813 107 Nm, Dxy= 2.2195 108 Nm. The natural frequencies of the bridgedeck are listed in Table 8.3. It should be noted that this structure is similar to, but not

the same as, that in Chapter 4.A two-axle vehicle model is used in the simulation. The axle spacing and wheel

spacing are 4.26 m and 1.829 m respectively. The four wheel loads are listed as follows:

P1(t)=3134. (1 + 0.1 sin(10t) 0.1 sin(20t) + 0.05 sin(40t))kgP2(t)=6166. (1 0.1 sin(10t) 0.1 sin(20t) + 0.05 sin(40t))kgP3(t)=3134. (1 + 0.1 sin(10t) + 0.1 sin(20t) + 0.05 sin(40t))kgP4(t)=6166. (1 + 0.1 sin(10t) + 0.1 sin(20t) + 0.05 sin(40t))kg

(8.46)

whereP1andP3are the front wheels andP2andP4are the rear wheels withP2andP4afterP1 andP3 respectively. The total vehicle load is 18.6 Tonnes and the proportionof axle loads follows the pattern of vehicle type H20-44 from AASHTO (2002). Thevehicle moving speed is 20 m/s, and the time step of analysis is 0.001 s in the simulation.White noise is added to the calculated displacements or strains to simulate the pollutedmeasurements.

8.4.2.1 S t u d y o n t he N o i s e E ff e c t

The vehicle is moving along the centerline of the deck. The measured responses from

25 modes (m= 5, n= 5) in Table 8.3 are used in the calculation, and the numberof modes in the identification Equation (8.40) is the same as that in the responses.According to discussions in Chapter 6, the number of measuring points should not beless than the number of vibration modes in the measured information. And therefore25 measuring points are selected evenly distributed on the five I-beams. The identifiedindividual wheel loads from using displacement responses with 1 percent and 5 percentnoise levels are shown in Figure 8.4. The following observations are made:

1. The beginning or end of the identified results is under-estimated when there is

noise in the responses. This is due to the small responses at the beginning or theend of the time duration, and the fact that the regularization parameter has beenoptimized over the total time duration of the event.

2. Errors in the identified results increase with the noise level. Hence when the noiselevel is high, a data treatment process (such as filtering or smoothing) should beused to reduce the noise in the responses before the computation.

Downloadedby

[ISTANBULTEKNIK


nuary2014


15/25

M o v i n g F o rc e I d e n t i f ic a t i on 223

5

(a) Wheel load1

Time (s)

104

4

3

N

2

1

00 0.5 1.51

8

(b) Wheel load2

Time (s)

104

6

4N

2

00 0.5 1.51

5

(c) Wheel load3

Time (s)

10

4

4

3

N

2

1

00 0.5 1.51

8

(d) Wheel load4

Time (s)

10

4

6

4N

2

00 0.5 1.51

Figure 8.4 Identified results with different noise levels ( true load; - - - 1% noise; 5% noise)

8.4.2.2 I d e n t i f i c a t i o n w i t h I n c o m p l e t e M o d a l I n f o r m a t i o n

In practice, the vibration modes selected for identification are not the same as that inthe responses. In general the lower modes of the structure are dominating the measuredresponses, and they are used in the identification. The moving loads are identified againwith this incomplete modal information with fewer modes in the identification than

those in the responses. The parameters of the system used in the simulation are thesame as those in the last study. The vehicle is moving along the centerline of the bridgedeck. Table 8.4 shows the errors in the identified results, and Figure 8.5 shows theidentified results using 20 modes (m= 4,n= 5) or (m= 5,n= 4) with 1 percent noisein the responses. The following observations are made from the results:

1. Whenm 4,n 4, an acceptable result can be obtained with most of the errorsless than 10 percent at 1 percent noise level. This is because the natural frequenciesof these modes (shown in Table 8.3) have covered most of the excitation frequency

range of the car as shown in Equation (8.46). In practice, the frequency rangerequired in the identification can be obtained from the spectrum of the responses.2. The more vibration modes used in the identification, the fewer errors are found in

the identified results(Table 8.4). However, large errors still exist at the beginningand the end of the load time histories as seen in Figure 8.5. This is due to the factthat impulses are generated by the moving loads at the beginning and the end of

Downloadedby

[ISTANBULTEKNIK


nuary2014


16/25


Table 8.4 Errors (percent) in the identified loads from different measured information

Noise Level 1% 5% 10%

m n Total Axle-1 Axle-2 Axle-1 Axle-2 Axle-1 Axle-2mode no.

5 5 25 6.97 6.00 17.90 14.46 28.68 22.166.51 5.98 15.58 13.38 24.77 20.28

5 4 20 9.05 7.74 17.95 14.80 28.07 21.949.23 6.60 16.50 12.78 26.31 20.41

5 3 15 30.56 18.62 32.25 20.32 36.11 23.9931.27 18.29 32.64 19.75 36.02 22.91

4 5 20 7.06 6.10 18.01 14.59 28.79 22.336.59 6.06 15.69 13.51 24.89 20.44

4 4 16 9.13 7.82 18.05 14.93 28.17 22.109.34 6.68 16.63 12.91 26.44 20.574 3 12 30.86 18.82 32.53 20.51 36.32 24.15

31.57 18.43 32.90 19.88 36.23 23.043 5 15 10.82 9.83 23.71 21.93 34.64 32.22

9.90 9.37 20.68 20.12 30.20 29.673 4 12 12.57 12.03 23.23 21.97 33.66 31.59

14.67 10.29 23.12 20.03 32.58 30.453 3 9 34.69 23.48 36.19 25.91 40.10 32.10

38.30 23.86 38.86 25.21 40.69 30.022 5 10 22.91 18.95 35.41 37.72 44.28 49.50

19.71 17.13 32.53 35.17 40.23 46.76

Note:The errors in table correspond to each wheel load as wheel1|wheel2wheel3|wheel4

the time duration, and a lot of higher modes of the structure are excited whichare not covered by the selected vibration modes in the identification.

3. There are large errors in the identified results withm= 5,n= 3. This shows thatthe torsional modes are also very important in the moving load identification on

bridge decks even if the vehicle is moving along the centerline.

8.4.2.3 E f f e c t s o f T r a v e l P a t h E c c e n t r i c i t y

There are four lanes on the bridge deck. Normally the vehicle is not moving exactlyalong the centerline. Table 8.5 shows the errors in the identified results with the carmoving at different eccentricities and using different number of vibration modes inthe identification. The identified results for different eccentricities with 25 modes(m= 5,n= 5) are shown in Figure 8.6. The parameters are the same as for the abovestudies, and the responses are calculated with 25 modes (m

=5,n

=5). The following

intermediate conclusions can be drawn from Table 8.5 and Figure 8.6.

1. Whenm 3,n= 5, an acceptable result can be obtained with most of the errorsless than 10 percent at 1 percent noise level. This shows that the method proposedin the chapter is also effective to identify the eccentric moving loads on the bridgedeck.

Downloadedby

[ISTANBULTEKNIK


nuary2014


17/25


18/25


5

(a) Wheel load1

Time (s)

104

4

3

N

2

1

00 0.5 1.51

8

(b) Wheel load2

Time (s)

104

6

4N

2

00 0.5 1.51

5

(c) Wheel load3

Time (s)

10

4

4

3

N2

1

00 0.5 1.51

8

(d) Wheel load4

Time (s)

10

4

6

4N

2

00 0.5 1.51

Figure 8.6 Identified results for different eccentricities (1 percent noise) ( true load; -.-.-e = 0;- - -e = 1/8b; e = 3/8b)

2. In the cases withn


19/25


Table 8.6 The correlation coefficients between measured and reconstructedresponses at 5/8L

Case Number of Measuring locations Correlation

Mode Shapes Coefficient

A 3 1/4L,1/2L,3/4L 0.9809B 4 1/8L,1/4L,1/2L,3/4L 0.9470C 5 1/8L,1/4L,1/2L,3/4L,7/8L 0.9752D 3 1/8L,1/4L,3/8L,1/2L,3/4L,7/8L 0.9853E 4 1/8L,1/4L,3/8L,1/2L,3/4L,7/8L 0.9837F 5 1/8L,1/4L,3/8L,1/2L,3/4L,7/8L 0.9822G 6 1/8L,1/4L,3/8L,1/2L,3/4L,7/8L 0.9716

at 5/8L obtained from the identified forces with different number of mode shapes inthe identification. The number of measuring points is taken equal to the number ofmode shapes in the identification. Figure 8.7 shows that the identified forces fromCases (A) and (G) of the study using 3 and 6 sensors respectively. The combined forceis also presented in Figure 8.7(c). The following observations are made:

1. Table 8.6 shows that the correlation coefficients are all larger than 0.9 for differentcombination of modes and measuring points. It shows that the method basedon generalized orthogonal function is effective to identify the moving forces inpractice.

2. There is a low frequency component in the identified individual forces inFigure 8.7. This is the pitching motion of the moving car.

3. The identified forces from using six modes are closer to the static forces at thebeginning and the end of the time histories than those obtained from usingthree modes. This gives experimental evidence that more mode shapes in thecomputation should be used to identify the moving forces near these locations.

8.5.2 P l a t e M o d e l

8.5.2.1 E x p e r i m e n t a l S e t - u p

The model vehiclebridge system fabricated in the laboratory as described in Section6.3.4.2 is used for this study. The same model car as described in Section 6.3.4.2 isused in the experiment. Details on the experimental setup are referred to Figure 6.28.The layout plan of the sensors is reproduced in Figure 8.8 for easy reference in thisstudy.

Twenty-five strain gauges were located at the bottom of the ribs to measure the strainof the bridge deck as shown in Figure 8.8. Six B&K model 4370 accelerometers were

placed at the bottom of beams 4 and 5 at 1/4, 1/2 and 3/4 span for the accelerationmeasurements.p1(t) andp3(t) are the left and right wheel loads at the front lookingin the direction of the traveling path;p2(t) andp4(t) are the left and right wheel loadsat the back followingp1(t) andp3(t).

The rigidities of the equivalent orthotropic plate are calculated as Dx= 7.3677 104 Nm, Dy= 4.2696 103 Nm and Dk= 8.6018 103 Nm. The first ten measured

Downloadedby

[ISTANBULTEKNIK


nuary2014


20/25


21/25


610 305 305

Nine Photoelectric Sensors in Equal Spacing

Rail 3

Rail 2

Rail 1

(a) Top face of the bridge deck

(b) Bottom face of the bridge deck

(c) Section A-A

(Dimensions are in millimetres)

12.5

Rail

Gauges

Photoelectric

Sensors

6.35

25

A

A

610 610

122

2441 6 11 16

2 7 12 17

3 8 13 18

4 9 14 19

5 10 15 20

152 305 152

244

610

1220

Figure 8.8 Layout of the bridge deck

wheel loads, axle loads and the combined load from different vibration modes usedin the identification with different number of measuring points when the model carmoves along the centerline. Table 8.7 shows the correlation coefficients between thereconstructed and measured strains at 3/8a of each beam for different moving paths

of the car. The following observations are made from the Figures and Table 8.7.

1. The method based on generalized orthogonal function is effective to identifyindividual moving wheel loads and acceptable results can be obtained.

2. The correlation coefficients between the reconstructed and measured strains onthe beams adjacent to the moving path of the car are larger than 0.8. This shows

Downloadedby

[ISTANBULTEKNIK


nuary2014


22/25


100

50

50

0N

0 1 2 3

(a) Wheel load1

4 5 6

100

50

50

0N

0 1 2 3

(c) Wheel load3

4 5 6

200

100

100

0N

0 1 2 3

(b) Wheel load2

4 5 6

200

100

100

0N

0 1 2 3

(d) Wheel load4

Time (s)4 5 6

Figure 8.9 Identified wheel loads for different combinations of measured information ( staticloads; - -m = 3, n = 2(9); ---m = 3, n = 3(15); m = 3, n = 2(15))

that the method is effective to identify the wheel loads moving with or withoutan eccentricity.

3. When the distance between the measuring point and the path of moving car islarge, the correlation coefficient is small, as seen from Beam #1 fore= 3/8b. Thisis because of the small responses at the measuring points, and the reconstructedresponse is very sensitive to error in the identified loads.

4. The identified loads from the case with 15 vibration modes (m= 3, n= 3) isalways smaller in all the results shown in Figures 8.98.11. The reason is due toan unequal number of modes used in the responses and in the identification, andit will be discussed in next section.

8.5.2.3 E f f e ct o f U n e q u a l N u m b er o f M o d e s i n t h e R e s p o n se a n d

i n t h e I d e n t if i c a t io n

Figures 8.98.11 show that the identified loads from (m= 3 and n= 3) is less thanthe loads identified from (m= 3 andn= 2). This difference cannot be the result of

Downloadedby

[ISTANBULTEKNIK


nuary2014


23/25

150

100

50

50

0

N

0 1 2 3

The front axle load

s

s

4 5 6

300

200

100

100

0

N

0 1 2 3

The back axle load

4 5 6

Figure 8.10 Identified axle loads for different combinations of measured information ( staticloads; - - m = 3, n = 2(9);--- m = 3, n = 3(15); m = 3, n = 2(15))

300

250

200

150

N

100

150

50

0

0 1 2 3

s

4 5 6

Figure 8.11 Identified total loads from using different modes ( static loads; - - m = 3,n = 2(9);- - -m = 3, n = 3(15); m = 3, n = 2(15))

Downloadedby

[ISTANBULTEKNIK


nuary2014


24/25


Table 8.7Correlation coefficient between reconstructed and measured strains at3/8a

Eccentricity Modes Beam 1 Beam 2 Beam 3 Beam 4 Beam 5

0 m = 3; n = 4(15) 0.783 0.897 0.931 0.909 0.799(Rail 3) m = 3; n = 3(15) 0.935 0.944 0.931 0.951 0.941

m = 3; n = 2(15) 0.901 0.935 0.929 0.944 0.922m = 3; n = 2(9) 0.932 0.949 0.933 0.953 0.947

3/8b m = 3; n = 4(15) 0.112 0.772 0.897 0.939 0.936(Rail 1) m = 3; n = 3(15) 0.166 0.793 0.915 0.951 0.948

m = 3; n = 2(15) 0.039 0.813 0.914 0.948 0.947m = 3; n = 2(9) 0.044 0.692 0.839 0.849 0.837

1/8b m = 3; n = 4(15) 0.550 0.794 0.848 0.790 0.758(Rail 2) m

=3; n

=3(15) 0.859 0.937 0.945 0.949 0.974

m = 3; n = 2(15) 0.896 0.947 0.953 0.948 0.966m = 3; n = 2(9) 0.880 0.922 0.920 0.929 0.945

Note: (15) denotes 15 measuring points located evenly on the five beams; (9) denotes ninemeasuring points located evenly on the three beams near the moving path of the car.

any calibration error. An inspection of Equations (8.38) to (8.41) gives the followingreasons for the existence of this error.

Equation (8.38) is valid for both the measured responses and for the identification.LetNR=mRnR andNI=mInIbe the number of the modes in the responses andin identification respectively. Equation (8.38) can be rewritten as follows:

wns=WNsNRQNR (8.47)where

WNsNR=

W1(x1, y1) W2(x1, y1) WNR (x1, y1)W1(x2, y2) W2(x2, y2) WNR (x2, y2)

...... ...

W1(xNs , yNs)W2(xNs , yNs) WNR (xNs , yNs)

NsNR

QNR= {q1(t),q2(t), . . . ,qNR (t)}TWe have two possible cases:Case (a): WhenNI


25/25


The terms in last bracket in Equation (8.48) represents the responses from the lowersets ofNImodes in identification. But in practice, the total measured responses areused instead, leading to an over-estimation of the forces when substitutingQNIand itsderivatives into Equation (3.17) in Chapter 3.

Case (b): WhenNR

8. moving force identification with generalized orthogonal function expansion

Documents