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Nonlinear vibrations of guy cable systems
0045-7949/85 $3.00+ .@I D 1985PergamonPress Ltd.
Compurers & Strucrures Vol. 21,No. 112, PP. 33-50,1985 Printedin theU.S.A.
NONLINEAR
VIBRATIONS
OF GUY CABLE SYSTEMS
Institut de Recherches
HENRI PASTORELand GASTON BEAULIEU d’Hydro-QuCbec (IREQ), 1800 Mont&e Ste-Julie, Varennes, Canada, JOL 2P0
Qukbec,
Abstract-A nonlinear static and dynamic analysis has been performed with ADINA for the cable system of a 50-kW vertical axis wind turbine. The nonlinear static deformation and tension distribution are compared to the results of a nonlinear catenary cable model. The first natural frequencies and mode shapes, computed with ADINA, are in close agreement with experimental values obtained with the Ibrahim Time Domain (ITD) modal analysis method. The aerodynamic forces acting on the rotor of a vertical axis wind turbine contain many harmonics. These excitation forces located in a rotating coordinates system are transformed into the fixed coordinates of the guy cable system and a solution for the forced nonlinear vibration is obtained with ADINA. The results are compared to the experimental values obtained from field tests on IREQ-50-kW wind turbine.
tal results
1. INTRODUCTION
obtained
from field tests on IREQ-50-kW
wind turbines. axis wind turbines are supported by a system of guy cables, comprising three or more equally spaced cables. The pretension force applied to these cables is generally from 12 to 20% of their normal breaking strength. Due to the aerodynamic loads acting on the blades of the rotating turbine, the cables experience large fluctuations in their instantaneous tension. As much as 100% increase and 75% decrease in tension have been measured on some experimental vertical axis wind turbines. These large variations in tension excite very strongly the vibration modes of the guy cable system which behaves in a nonlinear manner. Beaulieu has studied the nonlinear static response of these cable systems, first with a parabolic cable model[ll and then with a catenary model[2]. A nonlinear dynamic simulation model has also been developed by Robert and Beaulieu[3] and used to study the behavior of a large 4-MW wind turbine[4]. A preprocessor to generate the finite element model of a guy cable system for ADINA has been developed by Pastorel[S]. This preprocessor has been used to generate the guy cable system used in the present study. A nonlinear static and dynamic analysis has been performed with ADINA for the cable system of the IREQ-50-kW vertical axis wind turbine. The nonlinear static deformation and tension distribution are compared to the results of a nonlinear catenary model. A free vibration analysis is then performed and the first three natural frequencies and mode shapes of one of the cables are compared with experimental values obtained with the Ibrahim Time Domain (ITD) modal analysis method. After an explanation on the method of computation for the excitation forces, the results of the nonlinear forced vibration solution are presented. The dynamic simulation results are then compared with experimen-
Vertical
33
2. DESCRIPTIONOF THE GUY CABLE SYSTEMAND ITS FINITE ELEMENT MODEL 2.1 The wind turbine guy cable system
The IREQ-50-kW wind turbine system is shown in Fig. 1. Three guy cables of 22 mm in diameter and 40.3 m long are used to maintain the wind turbine upright. These guy cables have been instrumented. Each one is equipped with a tension cell at the lower end. The tension on the top is also measured on one of the cables. In addition, this cable is equipped with five groups of accelerometers. Each group has two accelerometers mounted at 90” to each other; one measuring the vertical acceleration, the other the horizontal acceleration. The top of the wind turbine central column is also equipped with two accelerometers; one oriented in the direction of the instrumented cable, the other at 90” to the first one. These load cells and accelerometers are connected to a data acquisition system capable of sampling each channel many times per second. Some of the experimental results obtained are presented later, for comparison purpose with the analytical results from ADINA. 2.2 The finite element model
The wind turbine is composed of a fixed structure, the guy cables and supporting tower, and of a rotating structure, the rotor. The purpose of this paper is to study the guy cable system, i.e., the nonrotating structure. Therefore the guy cables are represented in details while a very simplified model is used for the rotor. The complete finite element model uses simple two-node truss elements. Each guy cable is described by 12 nonlinear elements. The rotor, being pinned at both ends is described by one truss ele-
H. PASTORELANDG.
34
.2 6.f
TZH
BEAULIEU
T4.H : Cable #4 Top Tension A3.H : Accelerometer # 3 Horizontal A3.V : Accelerometer # 3 Vertical AS.4: A~~e~rorne~r Top #I -E : Anemo~ter
I r0P
I lose
Fig. 1. IREQ-50 kW wind turbines
Table I. Guy cables and rotor parameters GUY CABLES PARAMETERS Young modulus
12.8 E4 MPa
Diameter
22.0
milimeters
Length
40.3
meters
Linear weight
1.97
kg/m
Inclination angle
35.0
degrees
ment along the X-axis. Its inertia around the Y- and Z-axis are taken into account by putting equivalent masses at the top of the truss element. The axis system and the nodal points used to model the guy cable system and rotor are shown in Fig. 2. The three guy cabies are identical except for their linear weight which is different due to the instrumentation wires which run on them. The equivalent masses representing the five groups of accelerometers are also included for the instrumented cable. The basic parameters used for the cable and rotor are presented in Table 1.
WIND TURBINE ROTOR PARAMETERS Total
rotor
mass
3.81
metric
tor,s
Rotor height
18.0
In-plane
17.2 E4 kg-m**2
inertia
Out-of-plane
inertia
meters
16.5 E4 kg-m*2
3. NONLINEAR
3. I Catenary
able
STATIC ANALYSIS
model
A catenary cable model[2] has been developed to analyze the nonlinear static response of cable system. The model is based on a hyperbolic cosine function describing the catenary shape of a cable.
35
Nonlinear vibrations of guy cable systems
‘4”
Guy cable 2
Guy cable
FE nodes o Accelerometers on the instrumented guy cable
J
l
N
Fig. 2. FE idealization of the guy cable system.
This approach was first used by O’Brien and Francis[6] and then by Irvine[7]. The nonlinear static deformation and tension distribution of one of the cable has been computed with this catenary model and compared to the nonlinear static results from ADINA. In this model, the differential equation representing a perfectly flexible cable is:
Tti&+w!!!=o hdx2
dx
.
(1)
By integrating with the boundary conditions y = 0 when x = 0 and y = h when x = d, we obtain:
dy -= dx
- sinh
E- 91=-
sinh (qx - +)
[
(2)
where T,, = w= 4= d= h=
horizontal component of cable tension linear weight of cable w/Th = catenary parameter distance between anchor points height between anchor points
+ =
The nonlinear comes:
static shape of the cable then be-
y = .! [cash $I - cash (qx - +J)] . 4
(3)
36
H.
PASWREI. AND
G. BEAULIFLJ
With this catenary model, it is easy to obtain the tension distribution, T, along the cable:
own weight, and their behavior under an applied load at the top of the central column. The purpose of the nonlinear static solution is to obtain an equilibrium where the cable sag and nominal tension correspond to the physical system being studied. In order to converge toward this where equilibrium. one must specify an initial value of strain for each of the cable elements. Many sucds cessive runs are required to converge to the desired ;il;: = cash (qx - 4, . (5) solution except if an accurate value is computed for the initia1 strain. In the case of an inclined cable, the initial strain can be computed using a tensile Therefore, force equal to the desired nominal tension minus T = T,, cash tqs - 4) (6) the component of weight along the cable axis. Using this initial strain value, the desired equilibrium soOne can also obtain the elastic deformation of the lution is normally achieved with two ADINA nonlinear static runs. For each run we used four time cable and the stretched and unstretched length[2]. The nonlinear static solution of ADINA is com- step solutions with equilibrium iterations and stiffpared, in Fig. 3, with the static deformation and ness reformation for each time step. Figure 3 gives the results of a nonlinear static tension distribution computed with eqns (3) and (6). The tension dist~bution is identical for the two run for one cable having a nominal tension of 29.25 models and only minor differences exist in the dis- kN. The upper curve shows the sag of the cable and placements. The catenary model appears to be the lower curve the distribution of tension from the slightly less rigid (being a continuous model) than bottom to the top of the cable. The behavior of the whole system resulting from the finite element model. the application of a large horizontal load to the top of the column has also heen analyzed with ADINA. 3.2 ~o~[i~~a~ static a~al~.~~.~with AUlNA Two types of static analysis were done on the Such large quasi-static load can occur in reality durstructure: the deflection of the cables under their ing stormy conditions. Figure 4 describes this case. In the upper part are presented curves giving the displacement of the top of the column as a function 0 : i : j of the applied load for various values of the nominal tension in the cables. These curves show a fairly quasi-linear behavior of the global system except for very high loads. In the lower part of Fig. 4, the variation of the tension in the slack and tight cables are presented as a function of the top displacement. These curves make clear the nonlinear behavior of the cables even for small displacements of the top, especially for the slack cable. To perform this computation we used, after restart from the static equilibrium, eight steps of nonlinear solution with stiffness reformation and equilibri~lm iteration in each -16 i ; , ; / 1 0 0.1 0.2 0.3 0.a 0.5 o.'s0.7 0.8 0.9 : time step.
------I
I
I
I
NODES The nonlinear static solution results with ADINA compare very well with the model solution results. The convergence is very good when twelve two-node truss are used to represent a segment of cable. i(. FREE
0
0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.0 0.9
I
NOJES Fig. 3. Nonlinear static: comparison eatenary method and ADINA.
VIBRATION
obtained catenary achieved elements
ANALYSIS
The method used to extract the eigenvahtes and the eigenvectors is the standard subspace iteration method. We used the standard tolerances in ADINA for the eigenvalues. A large number of multiple roots and closely spaced roots are present in the cable system. Thus, a large number of iterations
37
Nonlinear vibrations of guy cable systems 35
,
I
70
60
6
4 2 9 i lbp displacement in cm
6
30 j 5 *s
s
____..___...._____ i _____...
_+
9
1 .......
__.j . . . ... .. .. ...i
0 0
2
4 6 Frequency
8
10
Fig. 5. Subspace iteration method.
01 0
;
1
;
2
i
;
I
3
4
5
Topdisplacement in cm Fig. 4. Nonlinear static ADINA: IREQJO kW wind tur-
bine, three guy cable system.
is required for the free vibration analysis. In fact, the number of iterations required increased exponentially with the frequency, as can be seen in Fig. 5. A better strategy would have been to use a subspace iteration with a shift around 5 Hz. This was not absolutely necessary for the simple structure presented in this paper but if a more complex structure is analysed, the strategy with a shift is recommended. We have used it successfully for analysing a six-cable supporting system for a megawatt size wind turbine. 4.2 Results
Up to five families of modes have been computed with ADINA. The frequencies for each of the modes are listed in Table 2. A family of modes is defined as one group of frequencies involving the three guy cables, each one vibrating either in the vertical plane (vertical mode) or at right angle to the vertical plane (sway mode). All the possible combination of the first vertical and sway modes constitute the first family of modes shown in Fig. 6. Similarly, the combination of the second vertical and sway modes form the second family, and so on. From the results obtained with ADINA, only the first family of modes exhibits a strong coupling between the cables. For the other higher families, the
three cables are uncoupled and vibrate almost independently from each other. The guy cable system and simple rotor model display two other modes involving the structure as a whole. These are the two tilt modes shown on Fig. 7. These two modes are important because they couple very strongly with the rotating turbine and can in certain cases lead to instability. It is therefore of primary imTable 2. Frequencies of cable system computed with ADINA
Frequency
Families of modes
In Hertz 1.48 1.49 1.50 1.51 1.52 1.53 1.20 1.21 1.22 1.23 t.24*Ml**
1 30.0 kN 19.5 kN
2.932.94 2.982.86 2.992.99 2.392.392.42 2.422.43 2.43~ 4.344.35 4.404.414.424.43 3.533.583.58 3.583.593.60**
3 30.0kN 19.5 kN
5.835.865.725.755.775.76 4.614.614.674.68 4.694.?0*
4 30.0kN 19.5 kN
6.916.927.017.027.047.05 5.695.615.685.705.715.72~ #4 #4 #6 #6 #2 #2 snayvertsrayvertsrayvert
description tilt30.0 kN 19.5kN
6.00 6.24 5.94 6.19* w rrequency and mode shape experimentaly verified l
frequency esperimentaly
verified
38
H.
PASTOREI. AND G.
BEAUJ.JEL
4.3 Modul Mode t-1
i2Hz
6
6
Modej-2 121Hz
2
6
6
4
2
6
6
i
4 2
2
4
Modef-4 123Hz 6
2
2 6
4
6
6
6
4
hdei-6 1,31Hz
4
Fig.
6.
First family of mode.
po~ance to identify them in a free vibration analysis. Another observation that is made from the solution of these modes is that their frequency is only slightly modified with change in value of the nominal tension of the guy cable. Their frequency is mostly controlled by the rotor mass and lateral stiffness of the guy cable system which remains relatively constant (for small displacement) with guy cable tension variation as seen on Fig. 4. Furthermore, a study of Fig. 4 reveals that for low guy cable tension, the system behaves like a softening spring as far as the tilt modes are concerned.
identification (ITD) tec~hnique
wirh the Ihruhim
Time Domrrir,
The IREQ-50 kW wind turbine is used lo verify the ADINA model of the guy cable system. The frequencies and mode shapes have been measured for the instrumented cable (cable #4). Two different types of test are presented. In test #l. a cable was attached at the top of the wind turbine, in line with the instrumented cable. and a force was applied to it. A helicopter hook was used as a quick-release mechanism. All the accelerometers and Ioad cells described in section 2. I were sampled at 60 pointsis for 17 s. In test #7. a weight of 30 kg was suspended close to the group #I of accelerometers. Once again, the load was removed using the quick-release mechanism and instrumentation channels sampled. The cable tension was different for the two tests (done at several weeks interval). The recorded time traces were transferred to IREQ main computer for analysis with the IT‘D technique@]. A modal identification was performed using the ITD technique. Up to eight frequencies have been extracted from the experimental data. With five groups of accelerometers along the span of the cable, only the first three mode shapes can be measured with sufficient accuracy. The first three frequencies and normalized vertical mode shapes are presented in Fig. 8 for test #i. The quality of the mode shapes produced demonstrates the usefulness of the ITD technique. In order to check the accuracy of the ADINA model and predictions. a comparison was done with the experimental results. Figures 9. 10 and 1I dis-
6.8 Hz
1
4
6 5.95Hz Fig. 7. Tilt modes.
-1.5-i 0
/ 0.2
, 0.4
Accelerometers
IV-i 0.6
0.8
1
location
Fig. 8. Ex~riment~ vib~~i~n of a guy cable: IKE@50 kW wind turbine; vertical modes. test # 1.
39
Nonlinear vibrations of guy cable systems 2
Legend
Legend
r---l
o
o Mods #I
j1.3OHzj ITD#i I.5 ..'------+ '...... q Mode #I1'03 Hz1 ITD#z _____ _ _'_________-
:FF i
Mode #3 (3.70HzjITD#I
1.5
-.L.-.-.
1
0.5
.______,
.;>.
j
.____...
>.
q
:\.
. . . . . . . .
.!
07
:
I
.
-0.5 .. --..-..--.--. 1
ci
0.i
0.k
Accelerometers
0.‘6
0.8
1
Location
6
0.2 0.4 Accelerometers
0.'6 o.‘a Location
i
Fig. 9. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison ITD ADINA.
F’lg. 11. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison ITD ADINA.
play the first three modes of vibration for the two experimental tests compared with the ADINA results. Only small deviations in mode shapes and frequencies can be noticed and the results demonstrate that the analytical predictions are indeed very close to the experimental results. The effects of the excitation method can be seen between test #l and #2. The mode shapes are modified, especially the third mode in Fig. 11, when a transverse force is
used to excite the cable. The along-the-span excitation appears to produce better results as far as mode shape is concerned. The frequencies obtained with ADINA are compared with the experimental frequencies in Table 3. The error varies from 1.0% to 5.7%. 5.
NONLINEAR
FORCED VIBRATIONS
5.1 The excitation forces
The aerodynamic forces acting on the guy cable system of a vertical axis wind turbine are of two types: distributed aerodynamic loads acting on the Table 3. Comparison and precision of frequencies i
:
1 0 Mode #2 12.43 XZ[ -t-.-.-~
ADINA i
ADINA EXP. 1.24
6
0.Z 0.k Accelerometers
Ok Oi Location ITD ADINA.
ADINA 1.26
ERROR 1.6 %
1.30
1.31
1.0 %
2.50
2.43
2.0 %
3.70
3.60
2.8 %
4.58
4.70
2.6 %
5.41
5.72
5.7 %
6.00
5.95
1.0 %
6.34
6.25
1.4 %
i
Fig. 10. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison
FREQUENCIES vs EXPERlMENTAb
40
H. PASTORELAND G. BEAULIEU
cables themselves, and loads coming from the turbine while it is extracting power. The first ones are negligible when compared to the latter ones. Rotor forces result from the integration of the aerodynamic loads acting on the blades of the rotor. They are computed by aerodynamic codes prior to their input into ADINA. The aerodynamic code used here is based on the double multiple stream tube theory including the Ma& model of dynamic stall for vertical axis wind turbinesl9, 101. The compu-
0
1
2
3 F&juency
Fig. 12. North-south
tation of the loads is done in three steps: (1) The first step computes for any section of the blade and for various positions of the rotor, the induced velocity and angle of attack. (2) The second step computes the normal and targential forces at different sections of the blades, using data for the aerodynamic profile, which come from wind tunnel tests or from mathematical models. These data are corrected using empirical models to take into ac-
4
5
8 ti
7
8
9
10
Hertz
force MADMD: IREQ-SO kW wind turbine: top of central column.
41
Nonlinear vibrations of guy cable systems
count the dynamic stall effects on the blades at high angle of attack. (3) The third step integrates the normal forces along the blades and splits them between the top and bottom of the turbine, taking into account the variation of the wind velocity between the bottom and the top of the rotor due to the atmospheric boundary layer. The integrated forces acting on the top of the turbine are then transformed from the rotating ref-
erence frame of the turbine to the fixed reference frame of the supporting system (Figs. 12 and 13). These transformed forces are input to ADINA at node 37 (Fig. 2). In this approach, the rotor is considered as being rigid. 5.2 Nonlinear forced vibration with ADINA
The case presented here is a simulation of forced excitation while the wind turbine is rotating at 80 rpm in a mean wind speed of 12.25 m/s. The aer-
7 I
1.2 ,
012346678
Fkquency Fig. 13. East-west
in Hertz
9
io
force MADMD: IREQ-50 kW wind turbine; top of central column.
42
H. PASTOREI.AND G. BEAL:LW
odynamic load computations have been performed every five degrees of rotor angular displacement. The time step for the forced vibration analysis was chosen such that a new series of loads was applied at each time step. This time step was suf~ciently small, permitting many points per cycle of vibration, even for the higher harmonics considered in the analysis. The method of integration is the standard Newmark method based on the displacement criterion for convergence. For the whole analysis.
we used stiffness reformation and equilibrium iteration at each time step. As can be seen in Fig. 14. the simulation has been done for a long time (about 13 s) which consists of25 revotutions of the wind turbine i 1800time steps). This long simulation was done in order to compare ADINA results with experimental ones in the time domain and also in the frequency domain. The upper part of Fig. 15 presents a zoom of the acceleration between 4 and 6 s and the lower part
..I I
I i ..
I i
I
I
--I
I
I -I 3
Fig. 14. Nonlinear dynamic ADINA: IREQ-50 kW wind turbine; central node of guy #4,
43
Nonlinear vibrations of guy cable systems
1
-41 4
i
4.25
1
i
4.50
1
i
1
Time
in
Fkquency Fig. 15. Vertical acceleration
’ ii
4.75
1
i
5.25
1
j
5.50
1
i
5.75
f
’ A
seconds
in Hertz
ADINA: IREQ-50 kW wind turbine; central node of guy #4.
the spectrum of the acceleration for the complete simulation. The main harmonics contained in the acceleration response can be seen on the spectrum. Because these curves represent the transverse vertical acceleration of the central mode of the span we notice that only the odd families of modes do appear on the spectrum. The aerodynamic loads excitation frequencies at 2 per revolution (2P) and 4 per revolution (4P) can be seen at 2.67 and 5.34 Hz, respectively. The two tilt modes are clearly seen
around 6 Hz and seem to be well excited. Figure 16 is split in the same way as Fig. 15 and shows the displacement between 4 and 6 s and its spectrum. The spectrum analysis shows that the displacement is essentially composed of the first family of modes, the second spike being the 2P excitation. It is clear from this curve that the first mode of the guy cable is excited, even if the main excitation force has a frequency of 2.66 Hz. The explanation for this is given in Fig. 17 where the tension is plotted. We
44
H. PASTOREL. AND G. BEAULIEU
4
4.25
4.50
4.75
Time 1000
5.25
5 in
5.50
5.?5
ii
secxmds
I _ _...... +i.. *_._-.i ..........j.. ...._._ :_.__--__-_:_.. :... .._i.,._.____. ..._.._,.. ......_.:.. ..._.._._.....,. ........... . ......_. P.._ ......1 j_:--.-; _......j ....._...:... .. ...i ........_.:A
0
1
2
+/ ~uency
Fig. 16. Vertical displacement
5
6
7
in
&%tz
0
9
10
ADINA: IREQ-SO kW wind turbine: central node of guy #4.
can see that the tension has a strong 2P component. The frequency of the first mode of vibration is approximately half of this 2P component; thus, we are in a situation of parametric excitation of the cable[il, 121. We can also see the tilt modes contribute significantly to the dynamic tension. 5.3 Nonlinear forced vibration-Comparison
4
with
experimental results
The resuhs of the nonlinear forced vibration obtained with ADINA are compared with experimen-
tal results from the IREQ-50-kW wind turbine operating at 80 rpm in a 12.25 m/s mean wind speed. In Fig. 18, the acceleration of the central node point of cable #4 is seen to vary from -2 to +3.5 in ADINA, compared with -3.5 to +3.3 on the experimental CUNC The agreement between the analytical and experimental curves is quite good if we take into account the variations in the excitation forces produced by the gusts of wind which are not taken into account in the analytical approach. The gusts of wind produce transient peak loads which
Nonlinear vibrations of guy cable systems
2.25 m/s 1 !nom 29 .5 k1
4
4.25
4.50
4.75
Time
5 in
5.25
5.50
5.75
seconds
012345678
s
10
Frequency in Hertz Fig. 17. Dynamic tension ADINA: IREQ-50 kW wind turbine; top of guy #4.
can be seen when one analyses in detail a large amount of experimental data. The analytical and experimental frequency content of the acceleration time traces are shown in Fig. 19. We can see that the excitation force at 2P is the main harmonic, both in the analytical and experimental spectrum. The guy cable frequencies are visible on both curves. The analytical and experimental amplitudes are not the same for all frequencies, mainly for the tilt modes, around 6 Hz, which are much more excited in the analytical
model than in reality. The fact that no damping has been introduced in the finite element model and that the second harmonic of the excitation forces (4P) is smaller on the experimental spectrum, accounts, at least partially, for these differences. In Figs. 20 and 21 the dynamic tension time traces and spectrum are seen to compare relatively well. The overall tension variations are in close agreement, the tension varying between -26 and +34 kN. The frequency content is about the same with differences in amplitude for the frequencies.
H. PASTOREI. ANDG. BEAULIE~
4
4.2
4.4
4.6 Time
4.0
5
5.2
5.4
5.6
5.8
5
seconds
in
4 3 2 1 0 -1 -2
-3
-44 0
;Y
02
\i
0.4 0.6 0.8 Time
Fig. 18. Vertical acceleration:
in
1
1.2 1.4 1.6 1.8
41 2
seconds
IREQ-50 kW wind turbine; central node of guy #4.
Nonlinear vibrations of guy cable systems
0.1
0.01
Fig. 19. Vertical acceleration: IREQ-50 kW wind turbine; central node of guy #4.
47
H.
PASTOREI. AND G. BEAULIEL~
I RPM
25 4s LOEn
29.5 kN
...____..~.....
_______.+.. .1.
Ii ~
r’ ........i........ f
!
. . __... .....f ~
~
4
4.2 4.4 4.6 4.0
02
0.4
0.6 Thne
5.2 5.4 5.6 5.8
6
in seconds
Time
0
5
0.0
1
1.2
1.4
1.6
1.8
in seconds
Fig. 20. Dynamic tension: IREQ-50 kW wind turbine; top of guy #4.
2
Nonlinear vibrations of guy cable systems
0
1
2
3
4
5
Ikequen~
8
7
9
8
in Hertz
i, Frequency
10
lb
in Hertz
Fig. 21. Dynamic tension: IREQ-50 kW wind turbine; top of guy #4.
Once again, actual forces due to gusts of wind and damping effects have not been included so far in the analytical model. 6. GENERAL CONCLUSIONS
From the results of this work, we can see that ADINA is a good and versatile tool for the study of vibrating cables. The idealisation of the cable with two-node nonlinear truss elements appears to
be adequate,
and there is no need for using higherorder elements. The comparison between the catenary model and the ADINA model emphasizes this point. The good agreement observed between the experimental curves and the dynamic response computed with ADINA is encouraging. However, some work remains to be done to include a damping model in the analysis and to input better aerodynamic forces to represent more accurately the field testing conditions.
H.
50
P.AS~OREL. AND G. BEA~JLIELJ
Acknon>ledgements-We wish to thank Bernard Masse who provided the aerodynamic loads, Bernard Saulnier. Yvon Vigneau, Dorien Marois, Luc Mattel for the experimental tests, and Francois Leonard. REFERENCES
1. G. Beaulieu, Analyse statique des haubans de I’Colienne des Iles-de-la-Madeleine. Rapport IREQ 2187 (Aug. 1980). 2. G. Beaulieu, Statique non-lintaire des cables: Application au systeme de haubanage des toliennes a axe vertical. Rapport IREQ-8RT 31256 (June 1984). 3. R. Robert and G. Beaulieu. Calcul de la dynamique des haubans des Coliennes a axe vertical avec un modele naraboliaue de cable. Raooort IREOdRT 30826 .. (28 June 198’4). 4. G. Beaulieu. Analyse statique et dynamique du syst&me de haubanage d’EOLE. Rapport IREQ-8RT 31286 (21 Sept. 1984). 5. H. Pastorel. Modele d’elements finis pour les haubans
6. 7.
8,
9.
10.
11, 12.
des toliennes a axe vertical. Rapport IREQ (in preparation). W. T. O’Brien and A. J. Francis, Cable movements under two dimensional loads. Journul of Strrrctur~d Division, Proceedings ofASCE, 89-123 (June 1964). H. M. Irvine, Cable Strrrctures. MIT Press. Cambridge. MA (1981). S. R. Ibrahim and E. C. Mikulcik. The exoerimental determination of vibration parameters from time responses. Vibration Bulletin (46) (Aug. 1976). B. Masse, Etude et comparaison des techniques de calcul aerodvnamiaue. (Proiet Eole. Etude #31 Ranport IREQ-2672 (Nov. 1982”). B. Masse, Description de deux programmes d’ordinateur pour le calcul des performances et des charges aerodynamiques pour les eoliennes a axe vertical. Rapport IREQ-2379. (July 1981). Bollotin, Nonconservative Problems oftheThrory c!f Elastic Stuhility. Mac Millan, New York (1963). N. Minorsky. Nonlinecw Oscillations. Chapter 20. Van Nostrand, New York (19621.
Compurers & Strucrures Vol. 21,No. 112, PP. 33-50,1985 Printedin theU.S.A.
NONLINEAR
VIBRATIONS
OF GUY CABLE SYSTEMS
Institut de Recherches
HENRI PASTORELand GASTON BEAULIEU d’Hydro-QuCbec (IREQ), 1800 Mont&e Ste-Julie, Varennes, Canada, JOL 2P0
Qukbec,
Abstract-A nonlinear static and dynamic analysis has been performed with ADINA for the cable system of a 50-kW vertical axis wind turbine. The nonlinear static deformation and tension distribution are compared to the results of a nonlinear catenary cable model. The first natural frequencies and mode shapes, computed with ADINA, are in close agreement with experimental values obtained with the Ibrahim Time Domain (ITD) modal analysis method. The aerodynamic forces acting on the rotor of a vertical axis wind turbine contain many harmonics. These excitation forces located in a rotating coordinates system are transformed into the fixed coordinates of the guy cable system and a solution for the forced nonlinear vibration is obtained with ADINA. The results are compared to the experimental values obtained from field tests on IREQ-50-kW wind turbine.
tal results
1. INTRODUCTION
obtained
from field tests on IREQ-50-kW
wind turbines. axis wind turbines are supported by a system of guy cables, comprising three or more equally spaced cables. The pretension force applied to these cables is generally from 12 to 20% of their normal breaking strength. Due to the aerodynamic loads acting on the blades of the rotating turbine, the cables experience large fluctuations in their instantaneous tension. As much as 100% increase and 75% decrease in tension have been measured on some experimental vertical axis wind turbines. These large variations in tension excite very strongly the vibration modes of the guy cable system which behaves in a nonlinear manner. Beaulieu has studied the nonlinear static response of these cable systems, first with a parabolic cable model[ll and then with a catenary model[2]. A nonlinear dynamic simulation model has also been developed by Robert and Beaulieu[3] and used to study the behavior of a large 4-MW wind turbine[4]. A preprocessor to generate the finite element model of a guy cable system for ADINA has been developed by Pastorel[S]. This preprocessor has been used to generate the guy cable system used in the present study. A nonlinear static and dynamic analysis has been performed with ADINA for the cable system of the IREQ-50-kW vertical axis wind turbine. The nonlinear static deformation and tension distribution are compared to the results of a nonlinear catenary model. A free vibration analysis is then performed and the first three natural frequencies and mode shapes of one of the cables are compared with experimental values obtained with the Ibrahim Time Domain (ITD) modal analysis method. After an explanation on the method of computation for the excitation forces, the results of the nonlinear forced vibration solution are presented. The dynamic simulation results are then compared with experimen-
Vertical
33
2. DESCRIPTIONOF THE GUY CABLE SYSTEMAND ITS FINITE ELEMENT MODEL 2.1 The wind turbine guy cable system
The IREQ-50-kW wind turbine system is shown in Fig. 1. Three guy cables of 22 mm in diameter and 40.3 m long are used to maintain the wind turbine upright. These guy cables have been instrumented. Each one is equipped with a tension cell at the lower end. The tension on the top is also measured on one of the cables. In addition, this cable is equipped with five groups of accelerometers. Each group has two accelerometers mounted at 90” to each other; one measuring the vertical acceleration, the other the horizontal acceleration. The top of the wind turbine central column is also equipped with two accelerometers; one oriented in the direction of the instrumented cable, the other at 90” to the first one. These load cells and accelerometers are connected to a data acquisition system capable of sampling each channel many times per second. Some of the experimental results obtained are presented later, for comparison purpose with the analytical results from ADINA. 2.2 The finite element model
The wind turbine is composed of a fixed structure, the guy cables and supporting tower, and of a rotating structure, the rotor. The purpose of this paper is to study the guy cable system, i.e., the nonrotating structure. Therefore the guy cables are represented in details while a very simplified model is used for the rotor. The complete finite element model uses simple two-node truss elements. Each guy cable is described by 12 nonlinear elements. The rotor, being pinned at both ends is described by one truss ele-
H. PASTORELANDG.
34
.2 6.f
TZH
BEAULIEU
T4.H : Cable #4 Top Tension A3.H : Accelerometer # 3 Horizontal A3.V : Accelerometer # 3 Vertical AS.4: A~~e~rorne~r Top #I -E : Anemo~ter
I r0P
I lose
Fig. 1. IREQ-50 kW wind turbines
Table I. Guy cables and rotor parameters GUY CABLES PARAMETERS Young modulus
12.8 E4 MPa
Diameter
22.0
milimeters
Length
40.3
meters
Linear weight
1.97
kg/m
Inclination angle
35.0
degrees
ment along the X-axis. Its inertia around the Y- and Z-axis are taken into account by putting equivalent masses at the top of the truss element. The axis system and the nodal points used to model the guy cable system and rotor are shown in Fig. 2. The three guy cabies are identical except for their linear weight which is different due to the instrumentation wires which run on them. The equivalent masses representing the five groups of accelerometers are also included for the instrumented cable. The basic parameters used for the cable and rotor are presented in Table 1.
WIND TURBINE ROTOR PARAMETERS Total
rotor
mass
3.81
metric
tor,s
Rotor height
18.0
In-plane
17.2 E4 kg-m**2
inertia
Out-of-plane
inertia
meters
16.5 E4 kg-m*2
3. NONLINEAR
3. I Catenary
able
STATIC ANALYSIS
model
A catenary cable model[2] has been developed to analyze the nonlinear static response of cable system. The model is based on a hyperbolic cosine function describing the catenary shape of a cable.
35
Nonlinear vibrations of guy cable systems
‘4”
Guy cable 2
Guy cable
FE nodes o Accelerometers on the instrumented guy cable
J
l
N
Fig. 2. FE idealization of the guy cable system.
This approach was first used by O’Brien and Francis[6] and then by Irvine[7]. The nonlinear static deformation and tension distribution of one of the cable has been computed with this catenary model and compared to the nonlinear static results from ADINA. In this model, the differential equation representing a perfectly flexible cable is:
Tti&+w!!!=o hdx2
dx
.
(1)
By integrating with the boundary conditions y = 0 when x = 0 and y = h when x = d, we obtain:
dy -= dx
- sinh
E- 91=-
sinh (qx - +)
[
(2)
where T,, = w= 4= d= h=
horizontal component of cable tension linear weight of cable w/Th = catenary parameter distance between anchor points height between anchor points
+ =
The nonlinear comes:
static shape of the cable then be-
y = .! [cash $I - cash (qx - +J)] . 4
(3)
36
H.
PASWREI. AND
G. BEAULIFLJ
With this catenary model, it is easy to obtain the tension distribution, T, along the cable:
own weight, and their behavior under an applied load at the top of the central column. The purpose of the nonlinear static solution is to obtain an equilibrium where the cable sag and nominal tension correspond to the physical system being studied. In order to converge toward this where equilibrium. one must specify an initial value of strain for each of the cable elements. Many sucds cessive runs are required to converge to the desired ;il;: = cash (qx - 4, . (5) solution except if an accurate value is computed for the initia1 strain. In the case of an inclined cable, the initial strain can be computed using a tensile Therefore, force equal to the desired nominal tension minus T = T,, cash tqs - 4) (6) the component of weight along the cable axis. Using this initial strain value, the desired equilibrium soOne can also obtain the elastic deformation of the lution is normally achieved with two ADINA nonlinear static runs. For each run we used four time cable and the stretched and unstretched length[2]. The nonlinear static solution of ADINA is com- step solutions with equilibrium iterations and stiffpared, in Fig. 3, with the static deformation and ness reformation for each time step. Figure 3 gives the results of a nonlinear static tension distribution computed with eqns (3) and (6). The tension dist~bution is identical for the two run for one cable having a nominal tension of 29.25 models and only minor differences exist in the dis- kN. The upper curve shows the sag of the cable and placements. The catenary model appears to be the lower curve the distribution of tension from the slightly less rigid (being a continuous model) than bottom to the top of the cable. The behavior of the whole system resulting from the finite element model. the application of a large horizontal load to the top of the column has also heen analyzed with ADINA. 3.2 ~o~[i~~a~ static a~al~.~~.~with AUlNA Two types of static analysis were done on the Such large quasi-static load can occur in reality durstructure: the deflection of the cables under their ing stormy conditions. Figure 4 describes this case. In the upper part are presented curves giving the displacement of the top of the column as a function 0 : i : j of the applied load for various values of the nominal tension in the cables. These curves show a fairly quasi-linear behavior of the global system except for very high loads. In the lower part of Fig. 4, the variation of the tension in the slack and tight cables are presented as a function of the top displacement. These curves make clear the nonlinear behavior of the cables even for small displacements of the top, especially for the slack cable. To perform this computation we used, after restart from the static equilibrium, eight steps of nonlinear solution with stiffness reformation and equilibri~lm iteration in each -16 i ; , ; / 1 0 0.1 0.2 0.3 0.a 0.5 o.'s0.7 0.8 0.9 : time step.
------I
I
I
I
NODES The nonlinear static solution results with ADINA compare very well with the model solution results. The convergence is very good when twelve two-node truss are used to represent a segment of cable. i(. FREE
0
0.1 0.2 0.3 0.4 0.5 0.5 0.7 0.0 0.9
I
NOJES Fig. 3. Nonlinear static: comparison eatenary method and ADINA.
VIBRATION
obtained catenary achieved elements
ANALYSIS
The method used to extract the eigenvahtes and the eigenvectors is the standard subspace iteration method. We used the standard tolerances in ADINA for the eigenvalues. A large number of multiple roots and closely spaced roots are present in the cable system. Thus, a large number of iterations
37
Nonlinear vibrations of guy cable systems 35
,
I
70
60
6
4 2 9 i lbp displacement in cm
6
30 j 5 *s
s
____..___...._____ i _____...
_+
9
1 .......
__.j . . . ... .. .. ...i
0 0
2
4 6 Frequency
8
10
Fig. 5. Subspace iteration method.
01 0
;
1
;
2
i
;
I
3
4
5
Topdisplacement in cm Fig. 4. Nonlinear static ADINA: IREQJO kW wind tur-
bine, three guy cable system.
is required for the free vibration analysis. In fact, the number of iterations required increased exponentially with the frequency, as can be seen in Fig. 5. A better strategy would have been to use a subspace iteration with a shift around 5 Hz. This was not absolutely necessary for the simple structure presented in this paper but if a more complex structure is analysed, the strategy with a shift is recommended. We have used it successfully for analysing a six-cable supporting system for a megawatt size wind turbine. 4.2 Results
Up to five families of modes have been computed with ADINA. The frequencies for each of the modes are listed in Table 2. A family of modes is defined as one group of frequencies involving the three guy cables, each one vibrating either in the vertical plane (vertical mode) or at right angle to the vertical plane (sway mode). All the possible combination of the first vertical and sway modes constitute the first family of modes shown in Fig. 6. Similarly, the combination of the second vertical and sway modes form the second family, and so on. From the results obtained with ADINA, only the first family of modes exhibits a strong coupling between the cables. For the other higher families, the
three cables are uncoupled and vibrate almost independently from each other. The guy cable system and simple rotor model display two other modes involving the structure as a whole. These are the two tilt modes shown on Fig. 7. These two modes are important because they couple very strongly with the rotating turbine and can in certain cases lead to instability. It is therefore of primary imTable 2. Frequencies of cable system computed with ADINA
Frequency
Families of modes
In Hertz 1.48 1.49 1.50 1.51 1.52 1.53 1.20 1.21 1.22 1.23 t.24*Ml**
1 30.0 kN 19.5 kN
2.932.94 2.982.86 2.992.99 2.392.392.42 2.422.43 2.43~ 4.344.35 4.404.414.424.43 3.533.583.58 3.583.593.60**
3 30.0kN 19.5 kN
5.835.865.725.755.775.76 4.614.614.674.68 4.694.?0*
4 30.0kN 19.5 kN
6.916.927.017.027.047.05 5.695.615.685.705.715.72~ #4 #4 #6 #6 #2 #2 snayvertsrayvertsrayvert
description tilt30.0 kN 19.5kN
6.00 6.24 5.94 6.19* w rrequency and mode shape experimentaly verified l
frequency esperimentaly
verified
38
H.
PASTOREI. AND G.
BEAUJ.JEL
4.3 Modul Mode t-1
i2Hz
6
6
Modej-2 121Hz
2
6
6
4
2
6
6
i
4 2
2
4
Modef-4 123Hz 6
2
2 6
4
6
6
6
4
hdei-6 1,31Hz
4
Fig.
6.
First family of mode.
po~ance to identify them in a free vibration analysis. Another observation that is made from the solution of these modes is that their frequency is only slightly modified with change in value of the nominal tension of the guy cable. Their frequency is mostly controlled by the rotor mass and lateral stiffness of the guy cable system which remains relatively constant (for small displacement) with guy cable tension variation as seen on Fig. 4. Furthermore, a study of Fig. 4 reveals that for low guy cable tension, the system behaves like a softening spring as far as the tilt modes are concerned.
identification (ITD) tec~hnique
wirh the Ihruhim
Time Domrrir,
The IREQ-50 kW wind turbine is used lo verify the ADINA model of the guy cable system. The frequencies and mode shapes have been measured for the instrumented cable (cable #4). Two different types of test are presented. In test #l. a cable was attached at the top of the wind turbine, in line with the instrumented cable. and a force was applied to it. A helicopter hook was used as a quick-release mechanism. All the accelerometers and Ioad cells described in section 2. I were sampled at 60 pointsis for 17 s. In test #7. a weight of 30 kg was suspended close to the group #I of accelerometers. Once again, the load was removed using the quick-release mechanism and instrumentation channels sampled. The cable tension was different for the two tests (done at several weeks interval). The recorded time traces were transferred to IREQ main computer for analysis with the IT‘D technique@]. A modal identification was performed using the ITD technique. Up to eight frequencies have been extracted from the experimental data. With five groups of accelerometers along the span of the cable, only the first three mode shapes can be measured with sufficient accuracy. The first three frequencies and normalized vertical mode shapes are presented in Fig. 8 for test #i. The quality of the mode shapes produced demonstrates the usefulness of the ITD technique. In order to check the accuracy of the ADINA model and predictions. a comparison was done with the experimental results. Figures 9. 10 and 1I dis-
6.8 Hz
1
4
6 5.95Hz Fig. 7. Tilt modes.
-1.5-i 0
/ 0.2
, 0.4
Accelerometers
IV-i 0.6
0.8
1
location
Fig. 8. Ex~riment~ vib~~i~n of a guy cable: IKE@50 kW wind turbine; vertical modes. test # 1.
39
Nonlinear vibrations of guy cable systems 2
Legend
Legend
r---l
o
o Mods #I
j1.3OHzj ITD#i I.5 ..'------+ '...... q Mode #I1'03 Hz1 ITD#z _____ _ _'_________-
:FF i
Mode #3 (3.70HzjITD#I
1.5
-.L.-.-.
1
0.5
.______,
.;>.
j
.____...
>.
q
:\.
. . . . . . . .
.!
07
:
I
.
-0.5 .. --..-..--.--. 1
ci
0.i
0.k
Accelerometers
0.‘6
0.8
1
Location
6
0.2 0.4 Accelerometers
0.'6 o.‘a Location
i
Fig. 9. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison ITD ADINA.
F’lg. 11. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison ITD ADINA.
play the first three modes of vibration for the two experimental tests compared with the ADINA results. Only small deviations in mode shapes and frequencies can be noticed and the results demonstrate that the analytical predictions are indeed very close to the experimental results. The effects of the excitation method can be seen between test #l and #2. The mode shapes are modified, especially the third mode in Fig. 11, when a transverse force is
used to excite the cable. The along-the-span excitation appears to produce better results as far as mode shape is concerned. The frequencies obtained with ADINA are compared with the experimental frequencies in Table 3. The error varies from 1.0% to 5.7%. 5.
NONLINEAR
FORCED VIBRATIONS
5.1 The excitation forces
The aerodynamic forces acting on the guy cable system of a vertical axis wind turbine are of two types: distributed aerodynamic loads acting on the Table 3. Comparison and precision of frequencies i
:
1 0 Mode #2 12.43 XZ[ -t-.-.-~
ADINA i
ADINA EXP. 1.24
6
0.Z 0.k Accelerometers
Ok Oi Location ITD ADINA.
ADINA 1.26
ERROR 1.6 %
1.30
1.31
1.0 %
2.50
2.43
2.0 %
3.70
3.60
2.8 %
4.58
4.70
2.6 %
5.41
5.72
5.7 %
6.00
5.95
1.0 %
6.34
6.25
1.4 %
i
Fig. 10. Vibration mode of guy cable: IREQ-50 kW wind turbine; comparison
FREQUENCIES vs EXPERlMENTAb
40
H. PASTORELAND G. BEAULIEU
cables themselves, and loads coming from the turbine while it is extracting power. The first ones are negligible when compared to the latter ones. Rotor forces result from the integration of the aerodynamic loads acting on the blades of the rotor. They are computed by aerodynamic codes prior to their input into ADINA. The aerodynamic code used here is based on the double multiple stream tube theory including the Ma& model of dynamic stall for vertical axis wind turbinesl9, 101. The compu-
0
1
2
3 F&juency
Fig. 12. North-south
tation of the loads is done in three steps: (1) The first step computes for any section of the blade and for various positions of the rotor, the induced velocity and angle of attack. (2) The second step computes the normal and targential forces at different sections of the blades, using data for the aerodynamic profile, which come from wind tunnel tests or from mathematical models. These data are corrected using empirical models to take into ac-
4
5
8 ti
7
8
9
10
Hertz
force MADMD: IREQ-SO kW wind turbine: top of central column.
41
Nonlinear vibrations of guy cable systems
count the dynamic stall effects on the blades at high angle of attack. (3) The third step integrates the normal forces along the blades and splits them between the top and bottom of the turbine, taking into account the variation of the wind velocity between the bottom and the top of the rotor due to the atmospheric boundary layer. The integrated forces acting on the top of the turbine are then transformed from the rotating ref-
erence frame of the turbine to the fixed reference frame of the supporting system (Figs. 12 and 13). These transformed forces are input to ADINA at node 37 (Fig. 2). In this approach, the rotor is considered as being rigid. 5.2 Nonlinear forced vibration with ADINA
The case presented here is a simulation of forced excitation while the wind turbine is rotating at 80 rpm in a mean wind speed of 12.25 m/s. The aer-
7 I
1.2 ,
012346678
Fkquency Fig. 13. East-west
in Hertz
9
io
force MADMD: IREQ-50 kW wind turbine; top of central column.
42
H. PASTOREI.AND G. BEAL:LW
odynamic load computations have been performed every five degrees of rotor angular displacement. The time step for the forced vibration analysis was chosen such that a new series of loads was applied at each time step. This time step was suf~ciently small, permitting many points per cycle of vibration, even for the higher harmonics considered in the analysis. The method of integration is the standard Newmark method based on the displacement criterion for convergence. For the whole analysis.
we used stiffness reformation and equilibrium iteration at each time step. As can be seen in Fig. 14. the simulation has been done for a long time (about 13 s) which consists of25 revotutions of the wind turbine i 1800time steps). This long simulation was done in order to compare ADINA results with experimental ones in the time domain and also in the frequency domain. The upper part of Fig. 15 presents a zoom of the acceleration between 4 and 6 s and the lower part
..I I
I i ..
I i
I
I
--I
I
I -I 3
Fig. 14. Nonlinear dynamic ADINA: IREQ-50 kW wind turbine; central node of guy #4,
43
Nonlinear vibrations of guy cable systems
1
-41 4
i
4.25
1
i
4.50
1
i
1
Time
in
Fkquency Fig. 15. Vertical acceleration
’ ii
4.75
1
i
5.25
1
j
5.50
1
i
5.75
f
’ A
seconds
in Hertz
ADINA: IREQ-50 kW wind turbine; central node of guy #4.
the spectrum of the acceleration for the complete simulation. The main harmonics contained in the acceleration response can be seen on the spectrum. Because these curves represent the transverse vertical acceleration of the central mode of the span we notice that only the odd families of modes do appear on the spectrum. The aerodynamic loads excitation frequencies at 2 per revolution (2P) and 4 per revolution (4P) can be seen at 2.67 and 5.34 Hz, respectively. The two tilt modes are clearly seen
around 6 Hz and seem to be well excited. Figure 16 is split in the same way as Fig. 15 and shows the displacement between 4 and 6 s and its spectrum. The spectrum analysis shows that the displacement is essentially composed of the first family of modes, the second spike being the 2P excitation. It is clear from this curve that the first mode of the guy cable is excited, even if the main excitation force has a frequency of 2.66 Hz. The explanation for this is given in Fig. 17 where the tension is plotted. We
44
H. PASTOREL. AND G. BEAULIEU
4
4.25
4.50
4.75
Time 1000
5.25
5 in
5.50
5.?5
ii
secxmds
I _ _...... +i.. *_._-.i ..........j.. ...._._ :_.__--__-_:_.. :... .._i.,._.____. ..._.._,.. ......_.:.. ..._.._._.....,. ........... . ......_. P.._ ......1 j_:--.-; _......j ....._...:... .. ...i ........_.:A
0
1
2
+/ ~uency
Fig. 16. Vertical displacement
5
6
7
in
&%tz
0
9
10
ADINA: IREQ-SO kW wind turbine: central node of guy #4.
can see that the tension has a strong 2P component. The frequency of the first mode of vibration is approximately half of this 2P component; thus, we are in a situation of parametric excitation of the cable[il, 121. We can also see the tilt modes contribute significantly to the dynamic tension. 5.3 Nonlinear forced vibration-Comparison
4
with
experimental results
The resuhs of the nonlinear forced vibration obtained with ADINA are compared with experimen-
tal results from the IREQ-50-kW wind turbine operating at 80 rpm in a 12.25 m/s mean wind speed. In Fig. 18, the acceleration of the central node point of cable #4 is seen to vary from -2 to +3.5 in ADINA, compared with -3.5 to +3.3 on the experimental CUNC The agreement between the analytical and experimental curves is quite good if we take into account the variations in the excitation forces produced by the gusts of wind which are not taken into account in the analytical approach. The gusts of wind produce transient peak loads which
Nonlinear vibrations of guy cable systems
2.25 m/s 1 !nom 29 .5 k1
4
4.25
4.50
4.75
Time
5 in
5.25
5.50
5.75
seconds
012345678
s
10
Frequency in Hertz Fig. 17. Dynamic tension ADINA: IREQ-50 kW wind turbine; top of guy #4.
can be seen when one analyses in detail a large amount of experimental data. The analytical and experimental frequency content of the acceleration time traces are shown in Fig. 19. We can see that the excitation force at 2P is the main harmonic, both in the analytical and experimental spectrum. The guy cable frequencies are visible on both curves. The analytical and experimental amplitudes are not the same for all frequencies, mainly for the tilt modes, around 6 Hz, which are much more excited in the analytical
model than in reality. The fact that no damping has been introduced in the finite element model and that the second harmonic of the excitation forces (4P) is smaller on the experimental spectrum, accounts, at least partially, for these differences. In Figs. 20 and 21 the dynamic tension time traces and spectrum are seen to compare relatively well. The overall tension variations are in close agreement, the tension varying between -26 and +34 kN. The frequency content is about the same with differences in amplitude for the frequencies.
H. PASTOREI. ANDG. BEAULIE~
4
4.2
4.4
4.6 Time
4.0
5
5.2
5.4
5.6
5.8
5
seconds
in
4 3 2 1 0 -1 -2
-3
-44 0
;Y
02
\i
0.4 0.6 0.8 Time
Fig. 18. Vertical acceleration:
in
1
1.2 1.4 1.6 1.8
41 2
seconds
IREQ-50 kW wind turbine; central node of guy #4.
Nonlinear vibrations of guy cable systems
0.1
0.01
Fig. 19. Vertical acceleration: IREQ-50 kW wind turbine; central node of guy #4.
47
H.
PASTOREI. AND G. BEAULIEL~
I RPM
25 4s LOEn
29.5 kN
...____..~.....
_______.+.. .1.
Ii ~
r’ ........i........ f
!
. . __... .....f ~
~
4
4.2 4.4 4.6 4.0
02
0.4
0.6 Thne
5.2 5.4 5.6 5.8
6
in seconds
Time
0
5
0.0
1
1.2
1.4
1.6
1.8
in seconds
Fig. 20. Dynamic tension: IREQ-50 kW wind turbine; top of guy #4.
2
Nonlinear vibrations of guy cable systems
0
1
2
3
4
5
Ikequen~
8
7
9
8
in Hertz
i, Frequency
10
lb
in Hertz
Fig. 21. Dynamic tension: IREQ-50 kW wind turbine; top of guy #4.
Once again, actual forces due to gusts of wind and damping effects have not been included so far in the analytical model. 6. GENERAL CONCLUSIONS
From the results of this work, we can see that ADINA is a good and versatile tool for the study of vibrating cables. The idealisation of the cable with two-node nonlinear truss elements appears to
be adequate,
and there is no need for using higherorder elements. The comparison between the catenary model and the ADINA model emphasizes this point. The good agreement observed between the experimental curves and the dynamic response computed with ADINA is encouraging. However, some work remains to be done to include a damping model in the analysis and to input better aerodynamic forces to represent more accurately the field testing conditions.
H.
50
P.AS~OREL. AND G. BEA~JLIELJ
Acknon>ledgements-We wish to thank Bernard Masse who provided the aerodynamic loads, Bernard Saulnier. Yvon Vigneau, Dorien Marois, Luc Mattel for the experimental tests, and Francois Leonard. REFERENCES
1. G. Beaulieu, Analyse statique des haubans de I’Colienne des Iles-de-la-Madeleine. Rapport IREQ 2187 (Aug. 1980). 2. G. Beaulieu, Statique non-lintaire des cables: Application au systeme de haubanage des toliennes a axe vertical. Rapport IREQ-8RT 31256 (June 1984). 3. R. Robert and G. Beaulieu. Calcul de la dynamique des haubans des Coliennes a axe vertical avec un modele naraboliaue de cable. Raooort IREOdRT 30826 .. (28 June 198’4). 4. G. Beaulieu. Analyse statique et dynamique du syst&me de haubanage d’EOLE. Rapport IREQ-8RT 31286 (21 Sept. 1984). 5. H. Pastorel. Modele d’elements finis pour les haubans
6. 7.
8,
9.
10.
11, 12.
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