Effect of dynamic wind force on structures using spectral approaches with complex modal analysis Q.-H.Nguyen 1* , F. Legeron 2 , T.-P. Le 3 . 1 * PhD candidate, Department of Civil Engineering, University of Sherbrooke, QC, Canada, [email protected]2 Professor of Civil Engineering, University of Sherbrooke, QC, J1K2R1, Canada, [email protected]3 L’Institut Français de Mécanique Avancée (IFMA), France, Thien-Phu.Le@ ifma.fr ABSTRACT The dynamic wind loads are considered in terms of equivalent static wind load based on gust- effect factor in most of the current design practices. In ASCE 7-05, the equivalent static wind load is based on the gust-effect factor f G which corresponds with the displacement at the highest point of the structure for flexible or dynamically sensitive structures. Gust-effect factors are calculated directly from background and resonance response by spectral method using their real modal properties. The main focus of the present investigation is on the influence of non- proportional viscoelastic damping matrix (for independent-frequency case system) and symmetrical stiffness and damping matrices (frequency-dependent case system) on the spectral calculation. A new transfer function is introduced in complex modal analysis or spectral stochastic response analysis. The case study is clearly demonstrated with numerical calculations showing the accuracy and high performance in this method. INTRODUCTION The response of linear elastic flexible system under the dynamic wind load (DWL) has been studied in the past in order to determine the peak dynamic response (PDR) r g r r = ^ , where, r is the mean or time average response, g is the statistical peak factor and r is the standard deviation of the response (SDOR), [1], [2,3,4], [5]. The peak dynamic response can be calculated directly from DWL or indirectly from the equivalent static wind load (ESWL) [6]. In both cases, the damping matrix of the structure is one of the most difficult and important factor to be considered because of the nature of non-proportional viscous property and its significant influence on PDR. The free vibration of multi-degree-of-freedom (MDOF) with non-proportional damping matrix was developed mathematically by [7]. An improved approach was expressed by using a new coordination )}} ( { )}; ( {{ = )} ( { t y t y t W in the governing equations of motion. The coupled modal problem becomes uncoupled due to the appearance of complex eigenvectors and eigenvalues of the system. Another mathematical methods were presented by [8] and [9]. One could find that these last methods, in general, are difficult to apply on the problem of MDOF system under ambient excitation. The linear vibration system under random excitation is considered by using the approximate normal mode method presented by [10]. The coupled equation of motion becomes uncoupled based on the utilization of the fictitious damping ratio ik . In effect, this approximation is acceptable when the eigenvalues are well separated.
Transcript
Effect of dynamic wind force on structuresusing spectral approaches with complex modal analysis
Q.-H.Nguyen1*, F. Legeron2, T.-P. Le 3.
1 * PhD candidate, Department of Civil Engineering, University of Sherbrooke, QC, Canada,[email protected]
2Professor of Civil Engineering, University of Sherbrooke, QC, J1K2R1, Canada,[email protected]
3 L’Institut Français de Mécanique Avancée (IFMA), France, Thien-Phu.Le@ ifma.fr
ABSTRACT
The dynamic wind loads are considered in terms of equivalent static wind load based on gust-effect factor in most of the current design practices. In ASCE 7-05, the equivalent static windload is based on the gust-effect factor fG which corresponds with the displacement at thehighest point of the structure for flexible or dynamically sensitive structures. Gust-effect factorsare calculated directly from background and resonance response by spectral method using theirreal modal properties. The main focus of the present investigation is on the influence of non-proportional viscoelastic damping matrix (for independent-frequency case system) andsymmetrical stiffness and damping matrices (frequency-dependent case system) on the spectralcalculation. A new transfer function is introduced in complex modal analysis or spectralstochastic response analysis. The case study is clearly demonstrated with numerical calculationsshowing the accuracy and high performance in this method.
INTRODUCTION
The response of linear elastic flexible system under the dynamic wind load (DWL) has been
studied in the past in order to determine the peak dynamic response (PDR) rgrr =^
, where,r is the mean or time average response, g is the statistical peak factor and r is the standarddeviation of the response (SDOR), [1], [2,3,4], [5]. The peak dynamic response can be calculateddirectly from DWL or indirectly from the equivalent static wind load (ESWL) [6]. In both cases,the damping matrix of the structure is one of the most difficult and important factor to beconsidered because of the nature of non-proportional viscous property and its significantinfluence on PDR.The free vibration of multi-degree-of-freedom (MDOF) with non-proportional damping matrixwas developed mathematically by [7]. An improved approach was expressed by using a newcoordination )}}({)};({{=)}({ tytytW in the governing equations of motion. The coupled modalproblem becomes uncoupled due to the appearance of complex eigenvectors and eigenvalues ofthe system. Another mathematical methods were presented by [8] and [9]. One could find thatthese last methods, in general, are difficult to apply on the problem of MDOF system underambient excitation.The linear vibration system under random excitation is considered by using the approximatenormal mode method presented by [10]. The coupled equation of motion becomes uncoupledbased on the utilization of the fictitious damping ratio ik . In effect, this approximation isacceptable when the eigenvalues are well separated.
The vibration of multi-degree-of-freedom under wind loading is analyzed deeply. It is found thatthe numerical work becomes very cumbersome when the dynamic effects are considered in timedomain, [11], [12]. The problem demands even more computational effort when the aeroelasticwind force is accounted for [13]. The estimated response using spectral method is preferred inthis case [14], [15].In the present investigation, the spectral method is summarized and the development of thismethod for a non-proportional damping system under dynamic wind force is presented. A newtransfer function )( iH is introduced into the power spectral density (PSD) response function.This method is based on spectral method presented in [16]. At the end of the paper, somenumerical examples are presented to illustrate the effectiveness of the new approach in structuredesign.
THEORETICAL BACKGROUND
Under quasi-stationary flow wind load, the equation of motion of N-degrees of freedom (N-DOF) damped system with viscous damping can be written in matrix form as:
)}({=)}(){()}(){()}({ tftytyty b KCM (1)where )}();...;({=)}({ 1 tytyty N is the Lagrangian displacement vector; )(, KM and )(C are,respectively, the mass matrix, the stiffness matrix and the damping matrix of the structure. Thevector )}();...;({=)}({ 1 tftftf Nb is the aerodynamic wind force vector, presented by its cross
power spectral density (XPSD) matrix NmjPP NNcr
mjfNNff 1,...,=,,)]([=)]([=)(,
P . The
function )(,cr
mjfP is referred to as the sidedone XPSD of )(tf between the two points j and
m of the structure. These factors were discussed in detail in [14] and [17]. In the frequencydomain, Eq. (1) could be written as: )}({=)}({)()(2 bFYi KCM (2)where )}({ Y and )}({ bF are the fourier transforms of )}({ ty and )}({ tfb . Therefore, thedisplacement vector is:
)}(){(=)}({)()(=)}({ 12 bb FFiY HKCM (3)
Its XPSD function is:)()()(=)( * ΗPHP fy (4)
where "*" denotes the conjugate operation. The standard deviation of displacement for the k thdegree of freedom could be calculated directly by:
dPkyky )(=
0
(5)
This method is easy to use, but it places a high demand on computational effort because of theoperation )(=)()( 12 HKCM
i at each discrete frequency. For a frequency-independent system, two other spectral methods, which, correspond to the proportional or non-proportional nature of damping matrix C are preferred. The development of a spectral methodfor a frequency-dependent system is presented in following subsection.
SPECTRAL METHOD FOR A FREQUENCY-INDEPENDENT SYSTEM
In this subsection, it is supposed that the stiffness matrix and the damping matrix are bothsymmetrical independent on the frequency . Eq. (1) becomes:
)}({=)}({)}({)}({ tftytyty bKCM (6)
System with proportional damping matrix
The damping matrix KMC km = is assumed to be Rayleigh damping. Eq. (6) is usuallysolved by the principal transformation law:
)(=)(1=
tZty jkj
N
jk (7)
where };...;;{=}{ 21 Njjjj is the structural eigenvector of mode jth and)}();...;({=)}({ 1 tZtZtZ N is the vector of the principal coordinates representing the image of
)}({ ty in the structural principal space. By substitution of Eq. (7) into Eq. (6), the coupled Eq.(6) becomes uncoupled due to the proportional nature of the damping matrix:
NjM
tFtZtZtZ
j
jjjjjjj 1,...,=,
)(=)()(2)( 2 (8)
where j and j are the natural frequency and the damping ratio of the j th mode, respectively.
)}({}{=)( tftF bT
jj and }{}{= jT
jjM M are the generalized load and modal mass for j thmode, respectively.For a response )(trk for the k th degree of freedom, Eq. (7) becomes )(=)(
1=tZbtr jkj
N
jk .
Values kjb are obtained by standard methods of analysis [16]. The power spectral density (PSD))(
krP and the variance 2
kr of the response )(trk are calculated directly by:
dPihihbbdPmjFmjkmkj
N
m
N
jkrkr
)()()(=)(=,
1=1=00
2
(9)
where )( ih j and )( ihm are transfer functions:
])/()/(2[11=)( 2
jjjjj iK
ih
(10)
and )(,
mjFP is the XPSD between two generalized loads )(tFj and )(tFm :
}){(}{=)(, mf
TjmjFP P (11)
System with non-proportional damping matrixThe approximate method for using non-proportional damping matrix ignores the off-diagonalcoupling coefficients of the modal damping matrix Cc T= then solve the resultinguncoupled Eq. (8) as a typical mode superposition analysis or by spectrum method presentedabove [16]. However, there is no mathematical theory or sufficient experimental evidenceshowing why damping in a physical system should be described by proportional damping [19].The results presented in examples at the end of the paper indicate that this approximation is notapplicable in some cases. In addition, [20] presented an available method for identifying asystem with non-proportional damping matrix. Therefore, the development of spectrum method
using directly non-proportional damping matrix is necessary. Considering the general situationwhere the damping matrix C is not necessarily Rayleigh damping KMC km . Supposing
thattjs
j ety }{=)}({ and a new state vector.12)}({
)}({=)}({
Ntyty
tW
is:
tjsNJ
tjs
jj
j ees
tW .12}{=}{
}{=)}({
(12)
Eq. (6) becomes:
{0}
)}({=)}({)}({
][][
tftWtW b
BA
M00K
0MMC
(13)
It clearly indicated that ),( Njj ss are N pairs of conjugate eigenvalue which are solutions of thecharacteristic polynomial 0=]][][[ BAsdet and }){},({ NJJ are N pairs of conjugateassociated eigenvectors having N2 components [7]. }{ J is calculated from equation
{0}=}]]{[][[ Jj BAs and its form is Njs
Njj
jJ 1,..,2=,
}{}{
=}{.12
.
Make .12)}({=)}({ NtZtW Θ , where .12)}({ NtZ is called a new general coordinate vector and thefull matrix NNjm .22][= Θ contains NN.22 complex components. Eq. (13) is now:
{0}
)}({=)}(]{[)}(]{[
tftZbdiagtZadiag bT
JJ Θ (14)
where ΘΘ ][=][ .22 Aadiag TNNJ and ΘΘ ][=][ .22 Bbdiag T
NNJ . It is easy to show that ),( Njj aa
and ),( Njj bb are N2 pairs of conjugate. The solutions for N2 differential Eq. (14) order one :
.12.12 )()(=)}({
Njjt
N dthFtZ (15)
where )(th j is the impulsive - response function and )}({}{=)( fF Tjj is the complex modal
force, Nj 1,...,2= . Take a general equation at mode k , Nk 1,...,2= :
deFiHdthFtZ tikkkk
tk )()(
21=)()(=)(
(16)
Here dtethiH tikk
)(=)( and dtetFF tikk
)(=)( are Fourier transforms of )(thk and
)(tFk . Note that the vector turbulent wind loads )}({ tfb contains N stationary gaussianprocesses having a zero mean value, the fourier series representation could be used:
tnink
nnk
nk eFtFtF )(=)(=)(
=
^
=
(17)
The modal displacement for the k th mode is:
tnink
knknnk
nk eF
biatZtZ
)(1=)(
^=)(
==
(18)
or:
deFbia
tZ tik
kkk )(1
21=)(
(19)
From Eq. (16) and Eq. (19), one has:
dtethbia
iH tik
kkk
)(=1=)( (20)
and:
0<0,
0,1=)(
21=)(
t
teadeHth
tkakb
k
tikk
(21)
Keeping in view that ka and kb are functions of Θ , ][A and ][B which contain all theproperties of the system, one can estimate the response using directly the impulsive-responsefunction )(thk in the complex mode superposition analysis, or indirectly the transfer function
)( iHk in the spectral stochastic response analysis.
Complex mode superposition analysis
Replacing Eq. (21) into Eq. (15), the general coordinate vector .12)}({ NtY of the system is:
.12
)(
.121)(=)}({
N
tjajb
jj
t
N dea
FtZ
(22)
So the displacement vector .1)}({ Nty is:
.12.2.1 )}({=)}({ Nuh
NNN tZty Θ (23)
where, NNjmuh
NN .2.2 ][= Θ is the upper-half of the full complex xeigenmatri Θ . The other
response vector )(}{=)}({ 2
1=.1 tZBtr jkjN
jNk of the structure is calculated similarly by:
.12.2.1 )}({=)}({ NNNjN tZBtr (24)
where kjB is obtained by standard methods of analysis in using a vector elastic force)}({=)}({ tytf S K :
.12.2 )}({=)}({ Nuh
NNS tZtf ΘK (25)
Spectral stochastic response analysis
In [16], the PSD of the response was analyzed mathematically for a proportional dampingsystem. The following paragraph presents the development of this method for the non-proportional damping system. The results presented in numerical exemples indicate thereliability of this expansion.From Eq. (15) and Eq. (23), the equation of displacement for the k th degree of freedom is:
dthFtZty jj
t
kj
N
jjkj
N
jk )()(=)(=)(
2
1=
2
1=
(26)
The PSD of the displacement )(tyk is defined from its autocorrelation function:
diexpRPky
T
TTkyRky )()(lim21=)(
21=)( )( F (27)
where )()( kyRF is the Fourier transform of the autocorrelation function )(
kyR for )(tyk :
)()(=)(
2
1=
2
1= tZtZER mjkmkj
N
m
N
jky (28)
where ))(( txE denotes the mean value of the discrete random variable )(tx . Replacing )(tZ j
and )( tZm of Eq. (15) into Eq. (28) and noting that only )( jjF and )( mmF change acrossthe ensemble, one could obtain:
mjjmmjFmmjjkmkj
N
m
N
jky ddRhhR )()()(=)(,00
2
1=
2
1= (29)
where jj t = et mm t = . Replacing )(kyR from Eq. (34) into Eq. (32) and changing
variable jm = , Eq. (27) becomes:
)()()(=)(,
2
1=
2
1=
mjFmjkmkj
N
m
N
jky PiHiHP (30)
with:
}){(}{=
)()(lim21=)(
,
mfT
j
mPjPjmT
jmTTmjF
P
diexpRP
(31)
The transfer functions )( iH j and )( iHm are calculated by Eq. (20) and )( iH j and
)( iH j are conjugate )1=)((jj
j biaiH
. The variance of the displacement is:
dPRkykyky )(=(0)=
0
2
(32)
For the other response )(trk at the k th degree of freedom of the structure, its PSD becomes:
)()()(=)(,
2
1=
2
1=
mjFmjkmkj
N
m
N
jkr
PiHiHBBP (33)
and its variance dPkrkr
)(= 02
.
SPECTRAL METHOD FOR A FREQUENCY-DEPENDENT SYSTEM
In this paragraph, a more general case is considered with the symmetrical stiffness matrix and thedamping matrix both dependent on the frequency .From Eq. (1), the new system under turbulent wind load could be presented as:
{0}
)}({=)}({
)()}({
)(
)])]
tftWtW b
[B(ω[A(ω
M00K
0MMC
(34)
where the state vectortjs
NJtjs
jj
j ees
tW)(
.12
)()}({=
)}(){()}({
=)}({
. The complex
eigenvalues )(js and eigenvectors )(J are calculated from equations
0=)]([)]()[( BAsdet j and {0}=)}({)]([)]()[( Jj BAs , NJj 1,...,2=, .
Take .12)}(){(=)}({ NtZtW Θ where the full matrix NNjm .22)]([=)( Θ contains NN.22complex frequency-dependent components. Eq. (34) is now:
{0}
)}({)(=)}()]{([)}()]{([
tftZbdiagtZadiag bT
JJ Θ (35)
where )()]()[(=)]([ ΘΘ Aadiag TJ and )()]()[(=)]([ ΘΘ Bbdiag T
J . By using thefourier transform, Eq. (35) becomes:
dtetfbia
FiHZ tib
Tk
t
kkkkk
)}({)}({
)()(1=)()(=)( (36)
. The displacement )(tyk for the k th degree of freedom is:
deFiHtZty tijjkj
N
jjkj
N
jk )()(
21)(=)()(=)(
2
1=
2
1=
(37)
Using the same approach presented in subsection above, the PSD of the displacement )(tyk is:
)()()()()(=)(,
2
1=
2
1=
mjFmjkmkj
N
m
N
jky PiHiHP (38)
with)}(){()}({=)(
, mf
TjmjFP P (39)
Therefore, the variance of the displacement )(tyk is dPkyky )(= 0
2 .
NUMERICAL EXAMPLES
Two numerical examples are presented in this study. The turbulent wind velocities )(tv whichcreate the load on each mass is generated from PSD )(vP [14]:
2*5/3
)(.501)(
2002=)( v
zVzzV
zPv
(40)
here, smzV z /25=|)( 10= is mean wind speed at level )(mz above ground and )/1.892(=* smv isthe shear friction velocity. The XPSD of the wind velocity is considered in two cases: linearXPSD )(=)(
, vmj
cr
mjv PP where )(=)( ,, tvtv mjmj are the wind velocities at two masses j
and m (example 01) and non-linear XPSD (example 02):)()()(=)(
,,
mvjvmjvcr
mjv PPP (41)
with:
mj
mjrv
zyx
r
mjv
VV
rrC
21
)(2exp=)(
22,,
,
(42)
where rvC is the exponential decay coefficient of the turbulence component v in the directionyxr ,= and z , respectively. In the second example, 10.5=yvC and 0== zvxv CC .
It is assumed that the time series of wind force applied on j th mass )()( tAvtf jj where A is aconstant, in )/( mNs . The PSD of wind force is )(=)( 2
jvjf PAP and its XPSD is
)]([=)]([=)]([,
2
, cr
mjvcr
mjff PAPP .
EXAMPLE 01In order to evaluate the presented method, the 2-DOF asymmetrical vibration system shown inFig. 1 is considered as a first numerical example. The system has the mass matrix
1001
10= 5M and the stiffness matrix
212
221141750=kkk
kkkK . By changing the values
1k and 2k , two natural frequencies of the system are well separated or close. The case(1.5;0.1)=);( 21 kk is presented here.
The damping matrix is
22
221510=ccccc
C . By changing the values 1c and 2c of this
matrix, one can obtain a proportional or non-proportional damping matrix C . Several pairs);( 21 cc are considered.
Step-by-step solution approachThe Newmark constant-average-acceleration method is adopted. The algorithm presented in [21]with (0.5;0.25)=);( and 32768== tNT . 0,125 s4096= is used. The five loading vectors
)}(0.5);({=)}({ 11 tftftR are generated from the PSD function of Eq. (40). The mean of fivestandard deviations of displacement of second degree of freedom is calculated from the last /2Nvalues of each time series:
5/2
)(
=
22/2
5
1=
2
T
dtty jj
T
T
jNMy
(43)
Integral approachUsing Eq. (5), the SDOR of the displacement y2(t) is:
dPyIny )(= (2;2)
02 (44)
where
1212 )(=)( KCiMKCiMP fy P (45)
Quasi-proportional approach
When the damping matrix C is non-proportional, the modal damping matrix Cc T= is non-diagonal. Nevertheless, an approximate solution may be obtained by ignoring the off-diagonalcoupling coefficients of matrix and then solving the resulting uncoupled equations as a typicalmode superposition analysis. This approach is named quasi-proportional (QP) approach in thisstudy. The SDOR can be calculated from equation:
dPihihmjFmjmj
mj
QPy )()()(=
,022
2
1=
2
1=2
(46)
where, }){(}{=)(, mf
TjmjFP P
Approximation quasi-proportional approach
By ignoring the off-diagonal coupling coefficients of the modal damping matrix Cc T= , forlightly damped systems with well separated modal frequencies, the SDOR can be calculated byanother quasi-proportional approximation (QPA) approach:
2121=
)(4
)(= 2
22
2
1=02
22
2
1=2
RRBGBG
PK
dPK jjF
j
j
j
j
jjF
j
j
j
QPAy
(47)
Non-proportional approach
The standard deviation of displacement )(2 ty is:
dPiHiHmjFmjmj
mj
NPy )()()(=
,022
4
1=
4
1=2
(48)
where }){(}{=)(, mf
TjmjFP P , and }{ j , }{ m are j th, m th column vectors of the upper half
of the eigenvectors matrix uh2.4 .
Standard deviation comparison
In this case, (1.5;0.1)=);( 21 kk ; 1=A and the wind forces )(2=)( 21 tftf apply on two masses
1m and 2m , respectively. So its XPSD is
25.05.05.01
)(=)]([=)]([ 2 vvf PPAP .
The stiffness matrix is
6.11.01.06.1
141750=K . The two natural frequencies of the undamped
system are )/(5)(1.458;1.5=);( 21 srad .When (0.5;0.1)=;( 21 cc , the upper half of the eigenvectors matrix Θ of the system is:
ii
ii
ii
iiuh
0.390.250.390.25
0.400.280.400.29
0.390.250.390.25
0.400.280.400.29
=2.4Θ .
The standard deviation of displacement )(2 ty is calculated by five approaches presented above.Fig. 2 presents this response in time series and its PSD by four spectral approaches. Thecomparison of these results are shown in Table 1.
Figure 1: 2-DOF viscous linear vibration system
Figure 2: Displacement y2(t) when (k1; k2)=(1,5; 0,1) and (c1; c2)=(0,5; 0,1)
EXAMPLE 02The 4-DOF vibration system sketched in Fig. 4 is considered as a second numerical example.The damping matrix )(C , the stiffness matrix )(K and the mass matrix M are:
000000000000
10= 21
21
21
215
cccc
C ,
44
4433
3322
221
000
000)(
141750=
kkkkkk
kkkkkkk
K
Figure 3: 4-DOF viscous linear vibration system
and
1000010000100001
10= 5M . By changing the value , several responses are compared.
The XPSD of wind force which applies on four masses is)]()[(=)]([=)]([=)]([
,
22
,
mjvvcr
mvjvcr
mjff PAPAPP where )(vP is presented in Eq. (40)
and )(,
mjv is calculated by Eq. (42):
25
||2
10.5=)(
,
mj
mjv
yyexp
.
The interesting response in this example is the displacement )(4 ty for the 4 th mass. Its standarddeviation is calculated by two similar spectral approaches presented in example 01 .
Integral approachUsing Eq. (5), the SDOR of the displacement y4(t) is:
dPyIny )(= (4;4)
04
(49)
where
12
*
)()()(=)(
)()()(=)(
KCMH
HPHP
ify (50)
Non-proportional approachThe standard deviation of the displacement y4(t) is:
dPiHiHmjFmjmj
mj
NPy )()()()()(=
,044
8
1=
8
1=4 (51)
where )}()]{([)}({=)(,
mfT
jmjF PP , and )}({ j , )}({ m are j th, m th column vectors of
the upper half of the eigenvectors matrix )(2.4 uhΘ .
Standard deviation comparison
In this example, (4;3;2;1)=);;;( 4321 kkkk ; 2=1c and 1=A . The standard deviation of thedisplacement )(4 ty is calculated by two approaches presented above. Fig. 4 presents its PSD bytwo spectral methods in the case 0.1= . The comparison of these results is shown in Table 2.
Figure 4: PSD of y4(t) for (4;3;2;1)=);;;( 4321 kkkk ; c1=2 and β=0,1
Table 2: Standard deviation comparison )(4 ty
The differences between the two spectral approaches are very small. It means that thedevelopment of the method presented in subsection above is reliable.
CONCLUSION AND RECOMMENDATION
For a viscous linear elastic of MDOF system under dynamic wind loads, some conclusions arepresented here:
The development of spectral methods using a non-proportional damping matrix ofthe system (independent frequency or not) is performed in this study. Thedifference between the responses of the approachalproportionnon and
approachstepbystep or approachintegral is very low. The calculation ofany results by spectral method is faster and more reliable than by dynamictransient methods.
The errors between quasi-proportional and non-proportional or step-by-stepapproaches are considerable when the natural frequencies of the system are
closed. The displacement calculated from its background and resonantcomponents is always over-estimated in this example.
The utilization of spectral method with quasi-proportional damping matrix is notrecommendable. It may be acceptable when the natural frequencies of the systemare well separated and the structure is lightly damped.
The both aerodynamic and aeroelastic wind force should be analyzed byapproachalproportionnon with frequency dependent properties of the system.
The expansion of this approach for the asymmetrical frequency dependent systemmuch be studied in the next research.
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