SYSTEM IDENTIFICATION: A STATISTICAL APPROACH
5.4. IDENTIFICATION OF MODAL PARAMETERS
where, 1( m) and
U1(m)
are the first singular value and first singular vector at the frequency ωm. While the first singular value reaches maximum within the narrow modal frequency range for the mth mode, the corresponding singular vector
U1(m)
becomes an estimate of the mth mode shape. It may be mentioned that the rank of the output PSD is expected to be nearly equal for multiple repeated modes for repeated modal frequencies. Thus the singular values acts as the complex mode indication function (CMIF) (Shih et al. 1988).After the estimation of mode shape, FSDD use enhanced PSD to evaluate natural frequency and damping ratio. Enhanced PSD in the vicinity of the mth mode is evaluated as
1
1
ˆ ( ) ( ) ˆ ( ) ( )
m
H
yy m yy m
eG i U G i U
(5.24)
The enhanced PSD is a scalar function of frequency and can be approximated as an SDOF system in the narrow frequency range around a modal frequency. An SDOF curve fitter can be utilized in the vicinity of a modal frequency for identification of natural frequency and damping ratio of that mode. It may be mentioned that in the case when all the measurements are not considered as reference, similar approach is applicable with FSDD.
5.4.1. Identification using NExT-ERA
Ambient data captured with both of the reference channels are used while employing NExT- ERA for modal identification. Such multiple-reference based NExT-ERA is also termed as MNExT-ERA (He et al. 2006) and expected to improve the performance of modal parameters identification. Impulse response is estimated in the form of the inverse Fourier transform of the cross power spectral density function. In the computation of cross power spectral density function, Hanning windows with 50% overlap is used in order to reduce the effects of spectral leakage. Finally, a suitable model-order as well as a suitable size of Hankel matrix is required to be found out for good performance of ERA. In this regard, Hankel matrices are formed with various ratios of row-numbers and column-numbers. Ratios are considered in a reasonably wide range from 0.1 to 10 and chosen as nearly {0.1, 0.3, 0.5, 1, 5, 10}. Ratios are chosen with higher density in lower part of the range 0.1–10, since better performances are observed in lower part of the range. It may be mentioned that an arbitrarily selected single data-set amongst the recorded data-sets (total 55 numbers) is considered for this analysis towards selection of suitable size of Hankel matrix. Other data-sets are assumed to perform in similar manners.
At first, an observation is made on the profile of singular values of the Hankel matrix to have an idea/suggestion regarding the system order. Singular values of Hankel matrices are evaluated based on three different ratios of row-numbers and column-numbers as {0.1, 5, 10}
covering a wide spectrum. Profiles of first 200 stronger singular values associated to these ratios are shown in the Fig. 5.3. Singular values are normalized to unity for even-handed observation for three different ratios of row-numbers and column-numbers. It is observed from Fig. 5.3 that demand of system order is indeed on lower side in the range of model orders up to 200. A reasonably conservative choice for system order is made as 100 for the subsequent analysis using NExT-ERA.
Fig. 5.3. Singular values (normalized to unity) of Hankel matrices associated to different ratios of the number of rows and the number of columns
Six different sizes of Hankel matrix based on those six ratios of row-numbers and column-numbers are considered to find out the suitable size of Hankel matrix for good performance of ERA. Performance of ERA is examined here with the help of stabilization diagram. A stabilization diagram displays the poles of a system at different model orders and differentiates physical poles from numerical poles. Physical poles occur at the same frequency at increasing model orders forming a vertical column of poles and hence show the trend of stabilization. Stabilization diagrams are commonly used to observe the presence of any mode. Superior stabilization provides higher confidence in ascertaining presence of any mode. Therefore, the stabilization diagram can be utilized for examining the suitable size of the Hankel matrix to have good performance in identification of modal parameters while employing ERA. The frequency is plotted along the abscissa while the model order is plotted along the ordinate in a stabilization diagram. Model order is considered in a reasonable range from 100 to 140. This range can be considered fairly sufficient since the lower-bound is selected in terms of a higher model order. A commonly used stabilization criteria are also considered here as: 1% for frequency, 5% for damping, and 2.5% for mode shape. Additional
0 20 40 60 80 100 120 140 160 180 200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Singular values in ascending order
Normalized values
Ratio=0.1 Ratio=5 Ratio=10
measures such as damping margin and modal phase collinearity (MPC) (Juang and Pappa 1985) are considered to avoid further the presence of numerical poles obtaining clearer stabilization. Both these measures can be considered appropriate based on the assumption of lower damping existing in the Saraighat Bridge. Damping margin is taken as 15% to avoid a pole having damping ratio more than 15%. On the other hand a pole having MPC less than 0.75, a value chosen based on the commonly considerations of MPC, are avoided. In this exercise, full Markov parameter sequences are used for all the cases associated to different sizes of Hankel matrices. The stabilization diagrams are shown in the Figs. 5.4(a–f). Meaning of the symbols used are: ‘f’ as pole with stable frequency, ‘d’ as pole with stable frequency as well as damping, ‘v’ as pole with stable frequency as well as mode shape, ‘s’ as pole with stable frequency as well as mode shape and damping and ‘+’ as new pole. Symbols are plotted aligning at left horizontally and at centre vertically. The background curve is considered as the first singular values of PSD matrices. The ratio of row-number and column- number as 0.5 is finally selected based on the observations on all the stabilization diagrams as shown in Figs. 5.4(a–f).
(a)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+++ ++++ ++ ++ ++ ++
s
svv ss ss ss vvss ++ vv ff
s
sss ss ss ss ss ss ff
s
svv ssvv ss ++++ ++ss
s
svv ++ss ss ss ffss ++ ss ++
s
svv vvss ss ss ++ss vv ff
s
sss ++ss ss ss ff ss ++ vv ++
s
sss ss++vv ss vvss ++ ss
s
sss++ ss ss ss ff vv ss ++ ss ++
s
svv vv ss ss ss ss ff vvff vv
s
sss ss ss ss ss ++ vvssss
s
sss ++ssvv ss ss ss ssss
s
sss++ ss ss ss ++ ss ssvv
s
svv ++ ss ss ss vv ss vv
s
svv ++ ss ss ss ss
s
sss ss ss ss ss ++ ++ ++
s
sss ++ss ss ss ss ss
s
sss vvss ss ss ss vv ++ ++
s
sss ssvv ss ++ ss ss vv ff
s
sssff ss ss ss ss ss vv ++ ss
s
sssvv ss ss ss vv ss ss
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 0.1
Frequency(Hz)
Model Order
(b)
(c)
(d)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ +++sss++sss+ ++ ++ss++ ++ss ++ ++ss ++ss ++ss +++ss+++ ++++ ss++ v
v ssss ++ vvss ss ss vvss ss ss ss++ vvvv ff v
v vvss ss ss ss ss ++ vv ff++++ ff++
v
v vvvv vv ++ vv ++vv ++++ ff ff ++
s
s ssssvv ++ ss s++s ss ss ff vv++ ++vvvv ++ff s
s ssssss ss vvffss ss vv ss vv vffv ss ssss ff ++ s
s ssssss vv ffssvv ss ss ss vvddss vv vvvv ff ff s
s ssssvv ++ ss++vv vv vv ss ss sffffs ss ssvv dd s
s vvvssssssv dd ssssvv ssss vv vv ss sss svvvvss ss ssvvss ++++vv v
v ssvv vv ssvv ss vv vv vvssssff vv vv v
v vvvssv ++ ssvv ++ vvss vvssssff vv vss svvssvs ++ v
v vvvv++ vv ff ++vv vv ++ vvff ++
s
s ssssssssssvv ss sss sssss vvssss vvss ss ssss sssssssvvffs ss sss sffvvssss++ffff ++ss v
v vvvv vv vv vv vvvv ++ vvff ++
s
s ssssssssssssssss ss svvsss ++ssss vv ssvv vvvssvff ss sss svvssss ssss ff vv v
v vv vv vv vv vvff vvvv ff
s
s ssssssvv ss ssss ss ss ss vv ssssvv vv vvvv ++vv ff s
s vvssssss ++vvff++ vvvv vv ss ssvvvv ss ssvv ff++ ++ s
s vvvvvv++ ++ vv ssvv vv vv sssvvs ss ssvv vv s
s ssssssss ss ss vvss ss ++ ss vvvvss++ ++ vvssss vv ++ s
s sssvvsss ss ss vvss vv vv ss ssssvvff ff vvssss ++ff ++
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 0.3
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+++++ ++ ++ ++++++ ++ ++ ++ ++ ++++ ++++++ ++++++ s
sssss ss vv vssvvv ss ss ss ss ssss ss ssss sddsvv s
s ss ss ss ss ssss ss ssss
v
vvssv++vv ff ff vv vv
s
sssvv vv ssssss ssss vv ssss ssss ssssssss ss sss ssssvvs vv ++++ v
vvv vv vv vv dd vv
+
+ ssss ss ss ss ssss ss ssss
v
v vvvvvvvvssssvv vvvv ++ +sssssss+vvsvvvvvv ssvvvvv++v++ ++vvff ssss ssss sssvvvsvff ssss svvvsvff ++ +
+ ss ++ ss ss ss ss++ ss ++
v
v vvvssssv++++ ++vvvv vvssss vvss ss ss vvvvssdd sssvvvsv ++++ v
v vv vv vv vv vvvv vv vv vvvv
s
s ssssvv vv vv vv vvss vv ss ss vv vssssv ++ ssssffvv ff ++ ++ s
sssss ss ss ss ssss ss ss ss ssssss ssss v
v vv ++ vvvv vv
s
s ssssvv vv vv vv ssss ss ss ss vvssss ss ss svvvvs ss s
sss svv s ss ss ssss ss vv ss vvvvvv ff ss svvvvs ss +
+ ++ ++
s
s sssvvvsssss sv vsffv ss svvffsss ssssssvv vvvv vvvv vvss vvvvvvvvssssssss ssvv ffssvvvvss ffss ssss ++ +
+ ssssss ss ss ssss ssss
v
v ssssss++ ++ vv vvvvss vvssss ss vv ssvv vss v ss vv vv v
v ssssss++vv vv vss svsss ss ss vv vv vffv vv ++ vv ++ v
v ssss++vv ss ss vv vvss vvvvv vvss sffsv ffsssvvs ++ v
v vv v++v vv vvvv vv vvvv vv vv
s
s sssvv vs++v ss ssss vvvv vv vv ffssvv ++ ++ssss vv
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 0.5
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+ + +
+++++ ++++++ ++ ++++++ ++++++++++ ++++++ s
sssssssssvv vv vssvvv++ vv ss svvffs ss ssss ff s
sssss ssssss ssssssssss ss ssss sdds vvssss ++ ++ s
sssss ssssss ssssssssss ss ssssvv ss ssvvff ++ ff++ s
sssss ssssss ssssvvvvss ss vv vssv ss ssffvv dd ff s
sssss ssssss ssssssssss ss++ ssssvv vv ffssss ff ++ s
sssss ssssss ssssssssss ss ss ssss ss ss svv vsv vv vv s
sssss ssssss ssssssssss ss ss ++ ssss ss ssffvvvv ss s
sssss ssssss ssssssssss ss ss vv ssssvv ssvvvvvv ++ ++ ++ s
sssss ssssss ssssssssss ssvv ++ vv vvssffff ss svv vsv vv vv ++ vv s
sssss ssssss ssssssssss ssvv vv ss ssssff ssssvvvv vv vv ss s
sssvvvvssss ssssssssss ssvv vv vv ssss ss ssss svvs ss ss ss s
sss sssvv vvvsv ssssssssssssssss ssss svvs vvss ssss ssss ssss sssss sssssss ss ss ss v
vvsssvs svvsvv vvssssvvss ss ss ss vv vv ssvv vv vv vvss vv v
v vv vv vv vv vvvff ffv vvvvvv vvff vv
+
+ vvvss svsvv ss ssssssss ss ss vv ss vvvv vsss++sv ssvvvvvv ff vv vv s
s ss ss ssss ss ss ss
v
vvvvvffff vv vvvvvv vv vv vv vvvv vvvvvv vv +
+ ssssssssss++ ++ss ssssssss ss ss ss ss ssssssss ssss ss ss s++++sss ss v
v ssssssssssvv vv ss ssssssss ss ss ss sssvvs sssvvsss ++ v
v ssssssssss svv vs v ss ssssssss ss ss vvss ss ss sffff ffs vvss ffff ff ++ ++ v
v vv vv vvvvvv ++vvvvvv vv ++ vv ++
s
s ssssssssss ss ss ss ssssssss ss ss ss ss vvvvss ss svvsss ssssssvv vv
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 1
Frequency(Hz)
Model Order
(e)
(f)
Fig. 5.4. Stabilization diagram based on NExT-ERA for six different ratios of row-numbers and column-numbers of Hankel matrix
5.4.2. Identification using SSI
In the identification of modal parameters using SSI, size of the output block Hankel matrix is commonly considered to be an important parameter affecting the performance in modal identification. Number of block-rows (i) is generally varied to construct Hankel matrices with different sizes. Six possible cases of numbers of block-rows are considered here as {10, 25,
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+ + + +++ +
+++++++++++++++ ssssss ++ ++++ ++ ++++ ++ ++ s
ssssssssvvsssss ssssss ss ssss ss ffffvv ss s
sssss svvvvvvsvv ssssss ss vvvvvvff vvvvvv ss s
sssssssssssssssssssvv ssssss ssss ff ffffvvvsssvsssss ffffsssssvvs ff v
v vv vvvvvv vv vv
s s s s s s s
sssssssssssssvv++ ssssss ss ssssssssvv sssvvs ffff s
sssssssssssssvvvv ssssss ss vvssss ss ssssss vvss s
sssssssssssssssssssssssssvvvv ssssss ssss sssssvv vs svs vvvvss ++ v
v vvvv ssssss ffffvvvv ++ff
s
sssssssssssssssvv ssssss ss ssss ss vvvvssss ffff s
sssssssssss sssssffsss ssssvv ssss ++sssssssss sss ssssssssssss ssssssss v
vvvvssssv vvvvvffddv vvss++vv ss ssssssvvff vvvsssvsssss v
vvvvv vvvff +ddv+ ssss vv vvvv s
sssvvvv vssvvssssvvv ++ssssssss ss ssssssvv sssvvsss ff ff s
sssssssss ssssssssss vvss ssssss ss vvvv vffffv ssssssss ssss s
sssssssss ssssssssss ss ssss ssssvv ssssssss s
s s s s s s
sssssssss ssssssssss ssssssss ss ssss svvs ssssssss ++ s
sssssssss ssssssssss vvss ssssss ss ffvvvv vvff++ ss svvvvsff ff s
ssssssssssssss sssssssssss vvssssss ssss vv vsssssss sssv vv sssssssssssssssssvv ds d v
vvv vvvvvdd +v+ ssssss ss vv vvv vvv vv s
ssssssssssvvvvsvvvv vv ssssssvvvv vv sssssssvv ffs
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 5
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+ + + + + +++ +
+++++++++++++ ssssssss ++ ++++ ++ ++++++++ ++ s
sssssssssssss ss ssss ss vvvvvv ++ ss s
s s s s s s s s
sssssssssssss ssssssss ss ssss ss ssss ff ss s
sssssssssssss ssssssss ss ffss svvs ssss ss ss s
sssssssssssss ssssssss ss ssssss ss ++vvvffv vv s
sssssssssssss ssssssss ss ssssss ss vvssss ss s
sssssssssssss ssssssss ss ssssss ss ++ffffvv ++ s
sssssssssssssssssssssss ssssssss ssss vv vssssssss svs ffss svvsssss vvss v
v vv ssss vv vvvv
s
ssvvvsssvvsssssvvsvssssss++++ vvssssssvvss ssss sssssssss ssvvs vvssssssssssssss ffss v
v ++vvvv vv
s s s s s s s s s
sssssssssssssss++vvvv ssssssss ss ssssss ss vvssssss vv s
sssssssss ssssssssvvvv ss ssssss ss vvvvss ss s
s s
s s s s s s s s
ssssssssssssssss sssssssssssvv vvssvv ss vvvsssvvsvffss ss sffvvvvvssvs ++ v
v ++++++ ssssss ss vv vv
s
sssssssssssssssvvvv ssss ss ssssvv ssssssss ++ s
sssssssss svvvvsff vvssssss ss vvvssvff ++ ssssssss s
sssssssssssssssvv ssssss ss vvvvvvvv vv vvvvvvvv s
sssssssssssssss ++ssssss ss ssssss svvs++++ ss svv ffffs ++++++ ++ s
sssssssssssssss vvssssss ssssssss ssvvvv ssvvvvffff ss s
sssssssssssssss vv ss vvvffv vvff ff vvvvvvff
Stabilization diagram (NExT-ERA)
Ratio (approx.) of numbers of rows and columns: 10
Frequency(Hz)
Model Order
50, 75, 100, 125} covering from lower number to a reasonably large number of block-rows.
In all the cases, all data points are considered to construct the Hankel matrices. Denoting the number of total data points and number of columns as ny and jmax respectively, jmax can be expressed as jmax = ny–2i+1 while considering all the data points. Consideration of all the data points with column-number as jmax is commonly observed to perform well compared to any other cases with column-number as j < jmax. It may be mentioned that an arbitrarily selected single data-set out of all the recorded data-sets (total 55 numbers) is considered in place of all the data-sets for this analysis for selection of suitable size of Hankel matrix.
First, an observation is made on the profile of singular values of the weighted projection matrix as in Eq. (5.16) to have an idea regarding the system order. The UPC algorithm is considered in this regard. Profiles of first 200 stronger singular values associated to the numbers of block rows are shown in the Fig. 5.5. Singular values are normalized to unity for even-handed observation among different numbers of block rows. Comparatively higher demand of system order than ERA is observed from the Fig. 5.5. A choice for system order is made as 100 for the subsequent analysis using SSI. Hence, a reasonable number of modes are expected to be identified as 50.
Fig. 5.5. Singular values (normalized to unity) of output block Hankel matrices associated to different number of block rows
0 20 40 60 80 100 120 140 160 180 200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Singular values in ascending order
Normalized values
i=10 i=25 i=50
0 20 40 60 80 100 120 140 160 180 200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Singular values in ascending order
Normalized values
i=75 i=100 i=125
The stabilization diagram is employed here to understand the performance of SSI associated to six different numbers of block rows as mentioned in preceding section. Three stabilization diagrams are constructed for each number of block rows using three different algorithms (UPC, PC and CVA). Total 18 numbers of stabilization diagrams are constructed and these are shown in Figs. 5.6(a–r). In the stabilization diagram, model orders are considered from 100 to 140 with lower-bound being fairly high. Meanings of the symbols used in these diagrams are similar to those used in stabilization diagram for NExT-ERA.
Damping margin of 15% and MPC margin of 0.75 are also considered here as in the case of NExT-ERA. It is observed from Figs. 5.6(a–r) that good performance is demonstrated while number of block rows is considered as 50 associated with CVA as well as PC algorithms and 75 associated with CVA algorithm. Finally the number of block rows as 50 associated to PC algorithm is chosen for efficient performance in modal identification.
(a)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++ ++ ++ s
s ++ss ss ssvv ss ss ssvv ss ffvv ssssvv s
s vvvv ss vvvv ss ssvv ss vvss vv s
s++++ ss vv ddss ++ ss ss ssff ss ss ss ss s
s ss++ ++ss ss ssss ss ss ssvv ss ss ss s
s ss ffss ss vvss vv ss ss ss++ ++ vv ss vv s
s ss vvvv vv dd ss ++ vv ssvv vv vv vv ss ++ ff s
s ss ss ss ss ss ss ss ss
v
v vv ++ vv ++ vv vv
s
s ss ss++ss ++ ++ss ff vvss ssss ss sss ss ss sssvv ffss ssss ++ ++ ff v
v vv ff ff vv ff vv
s
s ++ ffvv vv ss vvss ssvv ss ss ssssvv vv vv vv vv v
v++ vv vv ++ss vv vvss ss vv vss v ff ff ++ v
vff+ss+ ++vv vvss vvss ss vvss ssvv v
v vv vvvv vv vv vv
s
s ss ss ss
v
v vv ++vv ff vv vv vv
s
s vv vv++ vv ++ vv ss ss ss ss ff
s
s ++++ ffvv vvvv vv ff ss vvss ++ vv s
s dd ++ +ff+++ ssss vv ff ++ ff ss ssss++ dd ss ++ v
v ff ff vv vv vv ++ ff vv
s
s++++ ++ ++++ ss++ ++vv ss ss ff vv s
sff ff ff ++ss ssvv ++ ss ss ff
Stabilization diagram (SSI)
Algorithm:CVA. Number of block rows:10
Frequency(Hz)
Model Order
(b)
(c)
(d)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++ ++ ++ ++ ++ ++ ++++
s
s vv ++ vv ++ vv ss vv vv++ ++++ff s
s ++vv ++ ssvv ss sss sss ss sss sss svvs ++ v
v ffvv ++ ++ vv ff
s
s ++ vv ssss vv vv ssssvv ff ff v
v vvss vv vv ++ ssvv dd ffss v
v ++ ++ vv ff vv ff
s
s vv vv ssvv ++ff ++ vv ddff ++ v
v vv ++++ dd vv vv ++ vv
s
s ++ ++ dd ++ff ff vv ++
s
s ++ vv ++ ++ ++ vv ssvvff ++ ff s
s vv vv vv ff ff ++ vv++ ++ff
s
s vv ss dd vv ++ vvss vv vv
s
s ss++ ssff ss ss vv ++vv dd ff ff s
s ++ vv ss ss vv vv ff ffss ddvv s
s vv vv++ ++ss ddvv ff vv vv vv ++ ++
s
s vv vv ff ff ++ +ffff+ ++ ff
s
s vv ++ ff ff ss ++
v
v ++ vv++ ffvvff ff vvff ++ ffff++vv ++ +dd+vv +
+ +
+vv ++vv ++ ++ ff ff++ vv++ff ++ ++ vv +++ff+
Stabilization diagram (SSI) Algorithm:PC. Number of block rows:10
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ s
sff vv vv ss ss ssvvss ss vv vv ++ s
s++ vv++ ss ff vv ++ ++ ss ssvv ssvv ff++ s
s++ ss ss vv vv ff ss ss ss vv ff ++ s
s ss ss
v
vff vvff++ ff ff dd ++ ++ ++vv vv ++
s
s++vv ++ssdd vv ++dd ff ff ssvv vv s
s ssff ss ss ssss ffss ss ss sss sss ++ ++ dd v
v vddssv vvss vvss ss ff ss ss ff ffss ++ v
v vv ++ ++ ++vv ff++ ++ vv ++ ff
s
s ss ssff ff ssss vvss ++ ss sss sss ss vv ff v
v ++vv vv ++ ++ vv ++
s
s++ vvff ssss ss ++ ssssvv ++ ff ffss v
v vv vv ss++vv ff ++ ssvv ff ++vv ++
v
v vv++ vvss ffss vv vv ss ssvv dd vv vv v
v vv vv ff ss ss s++ +s+ ff ff ff ff
v
v ffvvss vv ff vvss ssff ++ ss vv vv++ v
v ++ vv vv vv
s
s++ ++ffff vv ++ff ss vv ss ss ++ ss s
s++ ++ss ++ss vv ff vvss ff vvss v
v ++++vv ++ vv dd ++
s
s +ff+vv ++ ++ ++vv ss ++ss ++
Stabilization diagram (SSI)
Algorithm:UPC. Number of block rows:10
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++
s
s ss ss ss ++ ++ vv ss ss ss ss sss sss ss ss ss dd ++ s
s ss ss ss ss ss++ vv ss ddvvvvss ss ss ss ff s
s ss ss ss ss vv++ ss ss ssvvvvss vvss ss ff ++ ++ s
s ss ss++ ss++ vv ss ssvvvvss ss ss ss ss ff
s
s ss ssvv ++ vv++ ss ss vvvvss ss ss ss ff s
s ss ssss ++vv ss ss vv vvvv ss ss ss ss s
s ss ss ss vvvv ss ssssvv vvss ss ss
s
s ss ssss ++ ++ vvvv ss vv vssv ss ss ++++ss ff s
s ss ss++ ff vvvv ss vvvvss ss ss ffss vv s
s ss ssss ++ ss ++vv ++ ss vvvvss ss ss ffffss dd s
s ss ss ss ss vvvv ++++ ss ss vvvvss vvss vvssss dd s
s ss ss++ ss ss ++ vv ++ff ss ss ++ff ffvv vv ss ss ssss ss dd s
s ss ssss ss vv vv ss++ vvss ss ss ssvvvv vv ss ss ssssss s
s ss ssss ++ ++vv ss ss ss vv vvv vvv ss ss vvssss s
s ss ssff dd ++ss ++ff ss ss ss svv vsv++ ss ss ssssss s
s ss ssvv ++++ ++ vv ++ ss ssvvvv vv ss ss ff ss s
s ss ssss ++ ff vv vv vv ssssvv vv ss ss ss sffs s
s ss ssss vv++++ ss ss++ vv ssffvv ss ss ffssss s
s ss ssss vvssff ss vv ss ssvvvv++ ss ss ssssss s
s ss ssss ssss ss ss ss ssff ss ss vvss ++
s
s ss ssss vvss ss ++ ss ++ ++ssvv ss ss ss ss ff
Stabilization diagram (SSI)
Algorithm:CVA. Number of block rows:25
Frequency(Hz)
Model Order
(e)
(f)
(g)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++++++ ++ ++ ++ ++ ++ ++ ++++++ ++ ++ ++++ s
s ss svvs ss ssss ss ssvvddss ss ss ++vv s
s ss ssff ssff vvvv ss ssvv ss ss vvss s
s ss ssss ssss vvvv ss ss ss++ ss ss ffff ff s
s ssvvff ssss++ ++ vv++ ++ ss ss ss ss ss ss ss s
s ss ssss ss ssss vv vv vv ss ss ssffvv vv ++ ++ vvss ss s
s ss ssss ss ssss ss ss ss ss ++ ss ss sffs ssff vv ssvv ++++vv s
s ss ssss ss ss ss ssvv ss ss ssssss ss ff dd vv vv ffss s
s vv vvssff ffss ss ss ssvv ++ ss ss ssvv ff vv vv ss s
s ss ssss ssss ss ss ssvv ss ss ssvv vffvff ss vv++++ ss s
s ss svvsff ssssvv vv vv vv vv vv ss s++s ss dd ssff ++ss ++ s
s ss ssss ssss vv ss ss ss ss svvs ss ss ffdd ff ff s
s ss svvs ss ss vv vv ss ssss ssdd ++ vv vv ++ s
s ssvvff ss vv vv ++ ss ss sffvvs ss ++ vvss ++ s
s ss ssssssss ssss vv vv ssss ssss ssss ss svvsss vv vss ssv ++ v
v vv ++ vv++ ddvvvvvv ++ ff++ ++
s
s ssvvff ssff ss ++ vv ++ dd++ ++ ss vv vssv vv ss ss ss ssff ++ ++ s
s ss ssss ssss ++ ++ ++ddss ss ss ++vv vvvv ++ ssvv ff++ s
s ss ssss ss ++ ss vvss vv vv ffvvff ss ss ++vv ++ s
s ss ssssff vv vv ++ ff ++ ss ssff vv vv ff ff s
s ss ssssdd ss vv vv ss ++ ss vv vvss dd s
s ssvvvvdd ss ++ ++ vv vv ss++ff ss ++ss vvss++ vv
Stabilization diagram (SSI) Algorithm:PC. Number of block rows:25
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ s
s vvss ++ ss vv ++ ff ff vv ss ssvv vvss vv ff
s
s ssvvff ++ vv ++ ss vv ss ss ss ssvv vv ss ss ++dd s
s ss ss sdds ss ss ss ss ss ss ssvvffss vvss ss ++ ++ s
s ss ss++ ss++ ++ ss++ ss ss ssssss ss ss ss ss vv s
s ssss ss ss ss ss ssss v
v ff vv vv vv ++ ff vv vv vv vv
s
s ss ssssss ss dd vv++ ss ss ss ss ssssvv vv vv ff s
s ss ssss ss ss ++vv ss ss ss ssssvv vv ss ss ff ++ s
s ss ssvv ss vvvv ff ss vvvvss ++ ss ss s
s ss ss ss ss ++vv vv ss vv ++ vvss ++ ++
s
s ss sss sss svvs ssss ss sss sss ssss ssss vvss ss ++ss ++ v
v ++ ++ ++vvddvv
s
s ss ss ss ss ++++ vv ss ss ss ++ ss ssvv vv ss s
s ss ss ss ss ff vv ss ++ ss ss vv ++ vvss vv s
s ss ss ss ss ss ss v
v ++ vv vv++ ff ++vvff++ ++ vv vv++++vv s
s ss ssss ss ++ff ++ ++++ vv ss vv dd ffvv vvss ++ s
s ss sss sssvv ss vv ++ ssss++ ssss ss ff vvffss++ vvss ssss++ ++ v
v vv vv ++++ vv vv vvff ff ff vv++
s
s ss ssvv vv ffff vv vv ff vv ssff ++ffssvv ss ss s
s ssss svv vsv vv vv vv ssss vv ffss ++ vvvvss ++ ss ss v
v ++ vv vv++ vv vv ++ vv vffv ++ ++
s
s ss ssss ++ff ++ ++ ++++ vv vv vv ssff vvffss ss
Stabilization diagram (SSI)
Algorithm:UPC. Number of block rows:25
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ ++ ++++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ s
s ss ssssss vv vvss vvss ss ss ss ss ss ss ss ssss s
s ss svvvsv vv ssvv ss ss vv ssff ss ss ss ssss ++ ++ s
s ss ssssvv ss ssvv ss vv ss ss ss ss ss ss ssss s
s ss ssssss ss ssss ss vv vv ss ss ss ss ss ssss ++ s
s ss ssssvv vv ssss ++ss ++ ff ++ ssvvss ss ss ssss vv s
s ss ssssvv ss ss ss ssss ss vv ss ss ss ss ss ssss vvff s
s ss ssssss vv ss sffs vvvv ss ss ssvv ss ss ss vvss vvff s
s ss ssss ss ssffvv ++vv ss ss ssvv ss ss ss vvss ++ vv s
s ss ssss vv ssffvv ssss vv ss ++ ++ ssvvss ss ss ssss ss dd s
s ss ssss vv vv vvvv ssvv ss ss ss ss ssvvss ss ss ssss ss ss s
s ss ssss vv vv vv vvvv vv ss sffs ss ss ss ss ss ss ssss ss ++vv s
s ss ssss ss ssvv vvvv ss ss vv vv ss ss ss ss ss vvss++ ss dd s
s ss ssss ssvv vv vvvv vv vv ssvv vffv ss ss ssssss ss vv s
s ss ssss ++++ssvv vvvv ss ssffvvss ssvv ssss ss ffss s
s ss svvs vvvvssvv ++ vvvv vv ssvv vv ssvv ssss vv s
s ss ssss ++vvss ss ssss ss ss ss ss ssvv ssss ss s
s ss ssss ssvv vv vvvv ss ss ss ss ss sffs ssss ss s
s ss svvs ++ss ssff++ ++ ssss ss ++ ssvvss vv vffv ssss vv s
s ss ssss vvvvvv vv ssss ss ss ffss ss ss ssffvv vvss ss s
s ss svvs ssvv++ss ss ++ ssss vv ss ffssvv vv ssvvvv ssss ss s
s vv vvss vv vv vv ++ vvss ss ss ss ssssvv ss ssss ssss ++ ss
Stabilization diagram (SSI)
Algorithm:CVA. Number of block rows:50
Frequency(Hz)
Model Order
(h)
(i)
(j)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ s
s ss vv ssssss ++ ss sss sss ssss ss ss sss ss ss sss ss sss sss vv vssv v
vff vv vv vv ff
+
+ ss s++ +s+ ++ss ss ss ss ss ss ss ss ss ss ss ss ++ v
v sssssvv vs svs ss ssvvss ss ss ss ss ss ss ss ss ss ss ff ++ v
v vv vv ++
s
s ssssss ss vv ssvv ssffvv ss ++ ss ss ss ss ss ss ff ffvv ss s
s ssssss ss ss ssvv vv++vv ss ss ssss ss ss ss++ ffvv ++ s
s ssssss ss ss ssss vvvvvv ss ss ssss s++s ss ssvv ss s
s ssss ss ss ss ss ss ss ssss ss ss ss v
v vv ++vv ++ vv vv ff
s
s ss ssss ssss ss ssssvv ssvvvv ssss vvss ssvvvv vv ssssssss v
v vvvv vv ++ ++vv vv vv ff ffdd vv vvvv ++ ++
s
s ssssssssssssssvv ss ssssss vv vssv ssss ++ ss vvss ssss ssss svv vvs ffff v
v vv vv vv++ vv vv ff ff vv ffvv ++ ++
s
s ssssssvv vv ++ssvv ++ vvvv ++ ff vv vv v+++v+ss ssvv vv ffff s
s ssssssss ss ssffss ss ssss ss ss ss vvvvvvvvss ssvv vv s
s ssssss ss ssvvss vv vvvv vv ss ss vv vss svs ss ssssvv ++ s
s ssvvvv vv ssddvv ++ vv ++ ++ff ++ ss vvssssss ss ssss ss ++ vv s
s ssssss ss ssvvffss vv ss vv vv ff ss ssssssss ss vv vddv ++ s
s ssssss ss ssvvss vv ss ss ss ss ss svvsss ss ss ssffff ++ s
s ssssss ss ssssss ss ss ss vv ss ssffvvss vv vddv ff ++ s
s ssssss ss ssssss ss ss ss ss ss ssssvv vv ++ vv vvss s
s ssssss ss ss ssvv ss ss vv ++ss ss ssvv vvss ss ssvv
Stabilization diagram (SSI) Algorithm:PC. Number of block rows:50
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++++ ++ ++++ ++++ ++ ++ ++++++++ ++ ++ ++ ++ s
svvff ++vv ssvv ++ ++vv vv ssss ss ss ss ss ssff ff s
svv ff vv vv ssss vv ssvv vvvv
+
+ ss ss++ ss++ss++ ++ ++ ++ ssss ss ss ss++ v
v ss ssvv ssvvssvv ++ vv vss svs ++ss ++ss sssss++s ss s++s ss ssvv vvss ++ v
v ss sssvvs vvvvvvss vvss ss ss ss ss ssssss ss ss ss ssvvvv ff vv ff ff ++ ++
s
s ss ssvvvssssv vv vvss vv ss ssss ssss vv ssssssvv vv ff ssssvv vv ++ ++ss v
v vv ++ vv vvvv vvss ss svvs ssvv ssvv vv vvss ff ssffvv ff ss v
v vv vvdd ++ ++++vvff vv vv vv ff dd ++ s
s ss ssss vv vvssvv vv ss ss ss ss vvvv vvss ff vv ++ ++ ff ff s
s vv vvdd vv vv vssv vv ss ss ss vv vv vss svs ff ss ss ssff vv s
s vv vvff ++ ++ ++vvvv ss ss ss vv vv ssffvv vv ss ss ssvvvv vv s
s ss ssss ss ssvvvvvv vv ssvv ss ss ssvv vvss ss ++vv vv s
s ss ssss ss ssffvvvv vv ss ss ss ss ssss ss ss ss ss ++++ ++ dd s
s ss ssss ss ssvvssss ++ vv vv ++dd ss ffvv++ ++dd ssvv ssss ff vv s
s ssss ss ss ss ssss
v
v vvvv ++ vv++ ++ ff vv ff ff ddff vv vv ff s
s ss ssss vv ss ssvv vv vv ++vv vv ss++dd ffffvv vvss++ ff s
s ss ssss ss ss ssss ss ss ssvv vv vv ff ++ff ++ ssvvff ++ ++ s
s ss ssss ss vv vv vvvv ++vv ++++ vv ffvv vv ss ssvv vv ++ vv vv s
s ss ssss ss vv vvvvss ++ +vv+ vv vv ++ffdd ff ssssvv s
s ss sss svvss vv ss ss svvsss vvvv vv ssvv ssvv +ss sss s+ss ss ss ssss v
v vv ++ vv vv vv ++ ++++ vv ++ ++ ff ff ff++ vv vv ++ ++ ++
Stabilization diagram (SSI)
Algorithm:UPC. Number of block rows:50
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ s
s ssss ss ssssvv ++ss ss++vvss ss ss ss s
s ssss vv vvssss ++ vvvv ss sss sss vvss ss s
s ssss ss ssvvvv vvss ++ ss svvs ss vvss ss ++ s
s ss++ ss vvssvv vv ++ ss vv vvss ss ss ss ss ff s
s ssss vv ss ssvv ++vv ss ss vv vvvvvv vvss ss vv s
s ssss vv vvssss ss ss vv ss svvs ss ss ss ss ++ ++ s
s ssvv ++ vv vvvv ss++ vv vv ss sss sss vvss ss ++ s
s ssvv ss vvvv vvvv ss ss ss svvs ss vvss ss ++ s
s ssss ss ssvv ssss ss vv ss svvs ss ss ss ++ ++ s
s ssvv ++ vv ssvv vvvv vv ss vvss ss vvss ss ssff s
s ssss vv vv ssss ssvv ++ vv vv vv ss ss ss ss ss s
s ssss vv ss ssvv ssvv ss ss vv vv ss ss ss s
s ssss vv ++ vvvv ss++ ++ ss vv vv vv ss ss ss ++ ++ s
s ssss ss ss ssvv vvvv vv vv vv ss ss ss ss ss ss s
s vvvv vv vv ssvv vvvv vv vv vv vv v++v ss ss ss ++ s
s vvvv vv vv ssvv ssvv vv ss vv vv vssv ss ss ss s
s vvvv vv vv ssvv ssvv ss ss ss ss ssvv ss ss ss s
s ssss ss ss vvss vvss vv vv vv vv vv vv ss ss ss ++ s
s vvvv vv ss ssvv ssss ss ss ss ss ssvv vvss ss vv s
s ssss vv vv vvvv vvvv ss ss ssvv vv ss ss ss ss s
s vvss vv ss vvvv vvss ss ss ss ssvv ss ss ss vv ++
Stabilization diagram (SSI)
Algorithm:CVA. Number of block rows:75
Frequency(Hz)
Model Order
(k)
(l)
(m)
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+++ ++++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++++ ++ s
s ssss ss ss ss ss ss ss
v
v vssv vv ss ssvv ss ddffss ss ssvvvv vvss v
v ++ vvvvvv vv vvvv vvvv ++
s
s ssssssss ssssssssss ss sss sssvv ssss vv vss sv vs svs ssssssssssss ssss ss ++ v
v vv vv
s
s ssss ss ssvv vv++ff++ ss ss ss ssssss ss s
s ssss ss vvvv vvvv ss ++ ++ ++ vvss ss ssssss ss ++ s
s ssss ss vvss ssss ss vv vv ss ss ss ss ss svvs vv ++ ++ ++ s
s ssss ss ssss vvss ss ss ss ss ss ss ss svvs vv vv ss s
s ssss ss ssss ssss ss ss ss ss ss ss ssssvv vv vv ++ s
s ssss ss ssss vvss vv ss ++ ss ss ss ss ss svvs ss ss s
s ssss vv vvss vvss vv ss ss ss ss ss ss ssssss ss vv ++ s
s ssss ss ssss ssss ss ss ss ss ss ss ss ssssvv vvff s
s ssss ss ssss ssss vv ss ss ss ss ss ss ssssvv ffff ++ s
s ssss ss vvss ssss vv vv ss ss vv vffv vffv ssvvvv vvff ff ss s
s ssss ss vvss ssss ss ss ss ss ss svvs ssss ssssvv vvvv +
+ ss ssss ss ssss ssss ss ss ss ss ss ss ssss ssssss ++ v
v vvvv ++ ++
s
s ssss ss ss ss ss ssss ssssss ss v
v vvvv ffvv vv vv vv
+
+vv vv ++ vvvv vvss vv ss ss ssvv vvss ssssvv ffssss ff +
+ vv v++v ++ ++ vv ssss ss ss ss vv vv vv ssvvvv vvvv ++vv++ +
+ ss ss ++ ++ ssss ss ss ss ++ss ssss v
v vv vv++++vv vv vvvv ++ vv vv
s
s ss ss ss ssssssss ssss ss ss ss vv vvss vv ssssss vv
Stabilization diagram (SSI) Algorithm:PC. Number of block rows:75
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+++ ++++ ++ ++ ++++ ++++ ++ ++++ ++ ++ ++++++ s
sssssvv vv ss sssvvs ss ss ss vv ss ss s
s svvsss vv vv vvvvvv ss ss ss ++ ++ ss ss ++ s
sssssss vv ss svvvvs ss ssvv ++vv++ ++ vvvvdd ++ ++ s
sssssss vv vv vvssss vvss ss vv vvss ssvvvv vv++vv ++ s
s svvsss vv ss ssssss vv vvss ff ssvv ssvvvv vvss ss vv++ s
sssssss vv ss sssvvs vvssvv++ ++ff vv v++v vvvvvv vvss vv s
s svv vsv vv vv vvssss ++vv ++ ssvv ss svvs ss svvs ss s
sss ssss ss ss ss ssss vv vv vv vvss ssssss ssvv vv ++ ++ s
sssssss ss ss svvsss vvss vv ff vv vv ssvvvv ++ss vv s
sss ssss ss ss svvvvs vvss ss vv v++v vv ssvv ss s
sssssss ss ss ss ssss ss ss ss ffvvvv vv ssvv ff ++ s
sss ssss ss ss ss ssss ss ss ss vv vssv ssff ffvv ++++ff s
sssssss ss ss sssvvs ss ss ss vvss ss vv vffffv vvss ++ s
svvssss vv ss ss ssss ss ss ss ssss ss vv vffv ffvv ss ++ s
sssddvv vv ss ssssss ss ss ss vvvv vv vv vffv dd ff vv +
+ ssss sss++s ss ssssss ss ss ss ++ss ss++ ss v
v ssss ssvv vvss ss ssssss ss ss ss ss vvvv vv vv vffv sssdd++s vv v
v ssss sssvvssvvs ss ss ssssss ss ss ss ss vvss ++ssvvvv vv ff ffvv vv ++ss vv v
v ssss ssss ss ss ssssss ss ss ss ss ss vvvv vv ssvv vffv ff ffss ++ v
v ssss sssvvsss ss ss ssssss ss ss vv vv vssv ss sffssvvs ddff ++ v
v ssss ssss ss ss ssss vv vv ss vv vvssvv ffvv dss d v
v vv vv vv vv vv vv++ vv vffvvv ++vv
Stabilization diagram (SSI)
Algorithm:UPC. Number of block rows:75
Frequency(Hz)
Model Order
0 2 4 6 8 10 12
100 105 110 115 120 125 130 135 140
+
+ ++ss++ss++ss ++ss+++ss+ss ss++ ++ ++ss ++ ++ss ++ v
v vv vv ++
s
s ssvvss ssssss ++ ss vv vv vvss ++ s
s ss++ssss++ vvssss vv vv ss ss ss ss ++ ++
s
s vvvv vssvvv vv vvvv ss ss ss vvss vv vv s
s ss svvvvv vvsss s++++vs vv vvv vvvvvvv ssvv ssss+++vvff+ssvv vvss ssss ssss vv s
s svv vs svs vv vvvv++ ss ++ vv vvvvss ss ss ss ++ s
s svvsssss vv vssv ++ss ++ ss ss ss svvs ss ss ss vv ++ s
s svvvvsss vv vff vvv++ vvss vv ss ssvvvv ss ss ss ss s
s svv vsssv ss svv vsvvv vvss ss ++ vv vss svs vvss ss ss s
s svvss sss ss ssffvv vvss vv ss ss ssssss ss ss ss vv +
+ s
s ss sss svvssss ss vv vss sv vs sssssvs vvssss ssss vv ssss svv vs svs ss sss sss ss ssss v
v vv vvvv vv vv ++ ff ++
s
s ss ssss ss ss ss ss
v
v vv vvvv++vv vv ++ vv vv vv vvvvvv vv vv
s
s ss svv vsv vv++vv v++v vvvv ff vv++ss ss ss ss s++svv ss ss ss s
s vvvss sv vsssssssv++vvsssss ss sssssssss vvffss ss vv ffvv vvss ssss ss sss sss ssss ++ v
v vv vv++ vv vvvv vv++ vv vvff ++ s
s ssssssssvv++vv vvv vvvss vvss vvss++ vvvvss ss ss ss++ ss s
s ssss svvvvs++ ss svv vs svs ss ss vv vv ++ vv ffss s++s ss ss ss ss s
s vvvv v++ +ssv+ vvvv vssvvv ss ss ss ss ++ vv ss ssssssss ss ss ss ss
Stabilization diagram (SSI)
Algorithm:CVA. Number of block rows:100
Frequency(Hz)
Model Order