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Aerodynamic damping properties of two transmission towers estimated by combining several identification methods
J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
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Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Aerodynamic damping properties of two transmission towers estimated by combining several identification methods Mayumi Takeuchi a,n, Junji Maeda a, Nobuyuki Ishida b a b
Faculty of Human Environment Studies, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Japan Steel Tower Co. Ltd.,1-7-1 Kitahama, Wakamatsu-ku, Kitakyushu 808-0023, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 February 2008 Received in revised form 31 August 2010 Accepted 3 September 2010 Available online 29 September 2010
Based on full-scale measurement data, we report on several aerodynamic damping properties of two transmission towers under conditions of strong winds. In estimating the damping ratio from data that contain many components of close frequencies of tower and conductor vibrations, we cannot estimate the damping ratio of each component if we use an estimation method available for only a single-degreeof-freedom system. Therefore, we combined several identification methods and are introducing a new method of estimating damping properties. This method is applicable to the response record of a multidegree-of-freedom system such as the coupled structure of a transmission tower-and-conductors. Using the new estimation procedure, we were able to extract the component of every vibration mode of the towers from a measured time history and estimate the accurate damping ratios individually; and, our results revealed that the wind speed dependency of the aerodynamic damping property of the coupled tower-and-conductors system had wide differing characteristics depending on the vibration mode. Moreover, using a three-dimensional nonlinear analysis, we discuss, in detail, the effects of the progressive deformation of a transmission–conductor–plane due to an increase in wind speed on the development of the aerodynamic damping of the coupled tower-and-conductors system. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Aerodynamic damping 3D behaviors of a coupled tower-andconductors system Normal decomposition method Multi-degree-of-freedom RD technique
1. Introduction A transmission tower is a mixed structure connected by conductors with very different levels of rigidity; hence, the behaviors of the conductors greatly affect those of the tower (Maeda et al., 1999; Okamura et al., 2003). Because the structural damping force of the transmission tower is very small, the tower is extremely sensitive to aerodynamic damping forces acting on the conductors. These damping forces play an important role in the behavior of conductors under conditions of strong wind (Davenport, 1988; Momomura et al., 1997). The authors reported elsewhere that the results from the site measurement data for a transmission tower show the presence of strong aerodynamic damping force acting on the tower, and they illustrated its wind speed dependency (Maeda et al., 2003). In addition, the authors discussed analytical results using a coupled model for a towerand-conductors system and indicated that both the geometrical nonlinearity of the conductors’ stiffness and the presence of additional aerodynamic damping forces on the conductors
n
Corresponding author. Tel.: + 81 92 642 3367; fax: + 81 92 642 4394. E-mail addresses: [email protected] (M. Takeuchi), [email protected] (J. Maeda), [email protected] (N. Ishida). 0167-6105/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2010.09.001
significantly affect the tower’s behaviors under conditions of strong winds (Maeda et al., 2003; Fujimura et al., 2007). We know that the accuracy of a damping ratio estimated from measured response data tends to vary more widely than other vibration parameters, such as natural frequency. Earlier reports (Haviland, 1976; Davenport, 1983) have shown that the coefficient of variance of the damping ratio estimated from measured response data sometimes reaches 70%. Also, because there are many damping estimation methods and the applicable limit of each method differs, the accuracy of the estimated damping ratio depends on the estimation method used. In addition, the measured response data for the tower includes many components of close frequencies because the transmission tower is a mixed structure connected by conductors with different levels of rigidity, as mentioned previously. Thus, particular attention should be paid to the decomposition of the vibration mode components. The damping estimation methods for the measured response data of the structures are classified into two methods: those using a time domain response function and those using a frequency domain response function. The former is known as the Autocorrelation Method and the Random Decrement Technique (Tamura and Suganuma, 1996) and the latter is known as the Half Power Method and the Hilbert Transform Method (Agneni and Balis, 1988). However, these methods work theoretically only for a singledegree-of-freedom system. As mentioned above, the tower’s response data often includes many components of close frequencies;
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
hence, we cannot simply apply these methods for it. In recent years, an applicable method for the response data of a multi-degree-offreedom system was introduced. The method uses the WaveletLogarithmic Decrement (Ruzzene et al., 1997; Lamarque et al., 2000; Hans et al., 2000), the Frequency Domain Decomposition (Brincker et al., 2000, 2001), the Multi-Degree-Of-Freedom Random Decrement Technique (Tamura et al., 2002) and the Normal Decomposition Method (Fujimura et al., 2007). In particular, the latter three are applicable to response data containing several components of close frequencies. Therefore we expect that these methods will contribute to improving the accuracy of the damping ratio estimation of the transmission tower-and-conductors. Also, the coupled model for the tower-and-conductors system, which the authors have used previously, has only one-degree-offreedom in the direction of the wind. However the flatness of the conductor-plane is lost with an increase in wind speed and the angle of the plane against the wind direction changes. Thus, we have to show the properties of aerodynamic damping forces using a three-degree-of-freedom model. In this paper, the authors apply the general identification methods and the new methods to the measured response data for two transmission towers and compare the damping ratios estimated using each method. After discussing the estimations, the authors discuss in greater detail the aerodynamic damping properties of a three-dimensional coupled model of a tower-andconductors system, focusing on the wind speed-associated change of the deformation shape of the conductor-plane.
Photo 1. Tower A.
Table 1 Structural specifications of Tower A. Voltage Height of tower Line angle Insulator Conductor Ground wire Span Left side Right side Natural frequency 1st tower-mode 2nd tower-mode
873
500 kV 214.5 m 0 (deg.) Suspension type KTACSR/Est450 4 OPGW490 1463 m 361 m 0.5 Hz 1.3 Hz
2. Outline of observed towers and measurement systems V1 X Y U1 U2
2.1. Outline of measurement of tower A
K1, K2(214.5m)
Conductors
U3
One of the observed towers (Tower A) was a 214.5 m tall 500 kV suspension type transmission tower, as shown in Photo 1. The structural specifications and natural frequencies of Tower A are shown in Table 1. Eight ultrasonic anemometers and one windmill type anemometer were positioned on the tower for wind measurement, as shown in Fig. 1. Every anemometer was located several meters from a main post in a horizontal alignment so that the anemometers would not be affected by the tower. One target plate and four accelerometers for displacement and acceleration were positioned on/near the tower, as shown in Fig. 1. The reference (Maeda et al., 1999) details the displacement measurement system. For verification of the tower’s response characteristics, a total of 31 samples were selected from the measured wind, displacement and acceleration data acquired at a right angle to the conductors because the tower’s wind load becomes dominant in that direction. The data was gathered from 1993 to 1997, and each sample is 10 min long.
Wind
K3, K4(163.5m) U4
214.5m
Target plate U5
Windmill type anemometer: V1 Ultrasonic anemometer: U1~U8
U6
Accelerometer: K1~K4 K1, K3: In the direction of conductors
U7
K2, K4: At a right angle to conductors X Y Target plate for displacement measurement: X, Y X: In the direction of conductors
20m
U8
Y: At a right angle to conductors
34.4m
30 20 10 0 0
100
200
Time (s)
300
600
Acceleration of K2 (m /s2)
40
Displacement of Y (mm)
Wind speed of V1 (m/s)
Fig. 1. Position of measurement sensors on Tower A.
400
200
0 0
100
200
300
Time (s) Fig. 2. Time evolution of measurement data at the top of Tower A.
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0
100
200
Time (s)
300
10
2
101 10
Power spectrum / Variance (s)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Power spectrum / Variance
874
Displacement of Y
0
10-1 10-2 10
-3
10
-4
10
-5
0
1
2
3
4
10
2
101 10
Acceleration of K2
0
10-1 10-2 10
-3
10-4 10
-5
0
5
1
2
3
4
5
Frequency (Hz)
Frequency (Hz)
Fig. 3. Power spectra of displacement and acceleration during strong winds.
Table 2 Structural specifications of Tower B. Voltage Height of tower Line angle Insulator Conductor Ground wire Inside Outside Span Left side Right side Natural frequency 1st tower-mode
U1
500 kV 70 m 0 (deg.) Strain type ACSR610 4 AS100 OPAS100 217 m 528 m 1.2 Hz
K1, K2(70m)
Conductors
K3, K4 (58m)
70m
Wind
U2
K5, K6(24m)
Ultrasonic anemometer: U1,U2 Accelerometer: K1~K6 K1, K3, K5: In the direction of conductors K2, K4, K6: At a right angle to conductors
11m Fig. 4. Position of measurement sensors on Tower B.
Photo 2. Tower B.
Fig. 2 shows the time evolution of the wind, displacement and acceleration at the top of the tower when the mean wind speed of V1 was 32.1 m/s. The peak gust was 38.9 m/s. The maximum and average displacements found at a right angle to the conductors were 450.2 and 378.9 mm, respectively. And the maximum acceleration was 0.45 m/s2. Fig. 3 shows the power spectra of the displacement and acceleration. We can see that the wind energy component is remarkable in the lower frequency region of the displacement spectrum.
shown in Table 2. Two ultrasonic anemometers and six accelerometers were positioned on the tower for wind measurement and acceleration, as shown in Fig. 4. The data was gathered through long-term observation from 2003 to 2004. In this paper, we selected data recorded during a typhoon and a non-typhoon, and the evaluating time of the measured data is 10 min. Fig. 5 shows the time evolution of the wind and acceleration at the top of the tower when the mean wind speed of U1 was 29.1 m/s. The peak gust and maximum acceleration were 49.1 m/s and 0.68 m/s2, respectively. 3. Aerodynamic damping of the transmission towers based on measured data
2.2. Outline of measurement of tower B 3.1. Extracting the component of the tower’s dominant mode The other observed tower (Tower B) was a 500 kV strain-type tower, which is 70 m in height, as shown in Photo 2. The structural specifications and natural frequency of Tower B are
Before we estimate the damping ratio of a vibration mode, we have to extract the component of each mode from the measured
60 40 20 0 0
100
200
300 400 Time (s)
500
600
Acceleration of K2 (m/s2)
Wind speed of U1 (m/s)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
875
0.8 0.4 0.0 -0.4 -0.8 0
100
200
300 400 Time (s)
500
600
Fig. 5. Time evolution of measurement data at the top of Tower B.
data. Generally, the component of a tower-mode is extracted using a FFT band-pass filter (FFT). However, it is known that this filter has no standard for the selection of the band range of frequency, and so has some side-cut-effects. Thus, in this paper, the component of the tower-mode is extracted using the Normal Decomposition Method (NDM) (Fujimura et al., 2007). Because this method utilizes the orthogonality of the vibration mode shape, it is also very useful for recording the time history, which includes several components of close frequencies. The basic theory of NDM is described below. Here is the equation for the motion of multi-degree-offreedom € þ CxðtÞ _ þ KxðtÞ ¼ pðtÞ: MxðtÞ
ð1Þ
In the equation, x(t) is the displacement vector (mean component subtracted) and M, C, K and p(t) are the matrix of mass, damping, stiffness and fluctuating force vector, respectively. The damping matrix C is orthonormal to the natural vibration modes vector, and the displacement vector x(t) is written using a natural vibration mode matrix U and the generalized displacement vector q(t): xðtÞ ¼ UqðtÞ ¼ ½j1 j2 ji ½q1 q2 qi T :
ð2Þ
By substituting Eq. (2) in Eq. (1) and multiplying UT from the left, Eq. (1) can be rewritten as follows: € þC qðtÞ _ þ K qðtÞ ¼ p ðtÞ, M qðtÞ n
n
n
ð3Þ
n
where M , C , K and p (t) are the generalized mass matrix, the generalized damping matrix, the generalized stiffness matrix and the generalized fluctuating force vector, respectively. For example, the component of the restoring force in Eq. (1) is written as KxðtÞ ¼ KUqðtÞ:
ð4Þ
By multiplying both sides of Eq. (4) by UT from the left, we get the following:
UT KxðtÞ ¼ UT KUqðtÞ ¼ K qðtÞ:
ð5Þ
By deriving the second derivative of Eq. (5) with respect to t, we get the following: € € ¼ K qðtÞ: UT KxðtÞ
ð6Þ
Thus, a generalized acceleration of vibration mode q€ i ðtÞ, can be € using Eq. (7) as follows: derived from the measured time history xðtÞ € € ¼ K1 UT KxðtÞ: qðtÞ
ð7Þ
Fig. 6(a) shows the original acceleration data observed at the top of the Tower A under conditions of strong wind, indicating
power spectral density. We can see that the original data includes the components of the 1st and 2nd tower-modes containing many components of the conductors-mode around the frequency of each tower-mode. Fig. 6(b) and (c) show the time evolution and power spectral densities of acceleration both of which contain only the tower-modes extracted from the original acceleration data using NDM. However, both the 1st and 2nd tower-modes can be extracted explicitly, but several components of the conductorsmode remain near the 1st mode frequency of the tower. These components of the conductors-mode cannot be excluded because the tower parts of many of the tower-conductors-coupled-mode shapes around the 1st tower-mode component have almost the same shapes as the 1st tower-mode, as Okamura has reported (Okamura et al., 2003). And, NDM cannot extract only the 1st tower-mode component from the time history data of the transmission tower because the measured data contains only the tower-mode shape; hence, the conductors-mode shape cannot be considered in Eq. (7). 3.2. Estimating the aerodynamic damping ratio of the transmission towers The damping ratios of the 1st and 2nd tower-mode components were estimated from the extracted components of the vibration modes of the transmission tower using the Random Decrement technique (RD) (Tamura and Suganuma, 1996). However, as the natural vibration mode of the transmission tower is strongly affected by the fluctuating tension of the conductors, the tower’s measured response data often has many components of close frequencies, as mentioned previously (Fig. 6). If the data including several components of close frequencies is applied to RD, the RD evolution beats as a result, as shown in Fig. 7; hence, the envelope of the RD evolution cannot be fit to a free-vibration decrement curve. When the measured data related to the conductors-mode shape is lacking, we have to combine NDM with a decomposition method based on a mathematical method. In this case, the Multi-Degree-of-Freedom RD technique (MDOF-RD) (Tamura et al., 2002) is very useful for precisely estimating the damping ratio of each component. Also, the damping ratios of the measured components contains structural damping, which was estimated using vibration tests under calm conditions, hence the aerodynamic damping ratio is defined by subtracting the structural damping ratios (Tower A: 1st 0.30%, 2nd 0.49%, Tower B: 1st 0.86%; Fujimura et al., 2007) from the damping ratios of the components. Figs. 8 and 9 show the aerodynamic damping ratios of Towers A and B estimated from the measured acceleration data, which was measured orthogonally to the direction of the conductors under conditions of strong winds. In the figures, the values estimated by MDOF-RD, RD, NDM and FFT have been added. The analytical values shown in Fig. 8 and described below are the aerodynamic damping ratios of the component of the tower vibration mode of the coupled tower-and-conductors model. The
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Acceleration (m /s2)
0.4 0.2 0 -0.2 -0.4 0
100
200
300
400
500
600
Power spectrum ((m /s2)2 s)
876
0.15 1st tower-mode
0.10
Conductors-mode 2nd tower-mode
0.05 0 0.0
0.5
0.2 0.1 0 -0.1
108
112
116
120
Power spectrum ((m /s2)2 s)
Acceleration (m /s2)
0.3
104
1st tower-mode Conductors-mode
0.10 0.05 0 0.0
0.5
0
-0.1
112
116
120
Power spectrum ((m /s2)2 s)
Acceleration (m /s2)
0.1
108
1.0
1.5
2.0
Frequency (Hz)
0.2
104
2.0
0.15
Time (s)
-0.2 100
1.5
Frequency (Hz)
Time (s)
-0.2 -0.3 100
1.0
0.15 0.10 2nd tower-mode
0.05 0 0.0
0.5
1.0
1.5
2.0
Frequency (Hz)
Time (s)
Fig.6. Examples of extracted components of tower-mode of Tower A. (a) The original data at the top of Tower A. (b) The component of 1st tower-mode. (c) The component of 2nd tower-mode.
the variance of the estimated damping ratios by NDM or MDOFRD is remarkably smaller than by FFT or RD.
RD evolution (m/s2)
0.12 0.08 0.04
4. Aerodynamic damping properties of 3D-computational model
0 -0.04
4.1. Coupled model for the tower-and-conductors system
-0.08 -0.12 0
4
8
12
16
20
24
Time (s) Fig. 7. Example of the RD evolution beats.
coupled system of Tower A is shown in Fig. 10. As already reported in the references (Maeda et al., 2003; Fujimura et al., 2007), the aerodynamic damping ratios of the 1st and 2nd modes tend to increase with wind speed, and the aerodynamic damping ratios of the 2nd mode are smaller than those of the 1st mode. In the 1st tower-mode of Tower A, we focus on the differences between the damping estimation methods of MDOF-RD and RD. In the 2nd tower-mode, we focus on the differences between the decomposition methods of NDM and FFT. It should be noted that
A transmission tower is modeled to a coupled lumped-mass model which includes the insulators and the conductors for Tower A, as shown in Fig. 10, using FEM and the Mixed Formulation Method (MFM) for the tower and the conductors, respectively (Fujimura et al., 2007). In horizontal directions, each mass of the tower model has two-degree-of-freedom, one of which is in the direction of the conductors while the other is in a right angle to the conductors. Each mass point reaches its average displacement and then responds to a wind gust. The stiffness matrix of the tower model is estimated using linear FEM. A member of the tower is idealized to a beam element. The joints between the main posts are assumed to be rigid, and the others are assumed to be pin. However, a nonlinear deformation analysis is necessary when estimating the equilibrium positions of the conductors to the mean wind and dead loads. MFM using the potential energy function of a link mechanism is used,
10
1st tower-mode
Measured (MDOF-RD) Measured (RD)
8
Analytical
6 4 2 0 5
0
10
15
20
25
30
35
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
10
2nd tower-mode
877
Measured (MDOF-RD) Measured (RD)
8
Analytical
6 4 2 0 0
Mean wind speed of U1 (m /s)
5
10 15 20 25 Mean wind speed of U1 (m / s)
30
35
Aerodynamic damping ratio (%)
Fig. 8. Aerodynamic damping ratios estimated from the measured data (Tower A).
U, having more segments provides a more accurate mode shape.
10 1st tower-mode
Measured (MDOF-RD)
8 6 4 2 0 0
5
10 15 20 25 Mean wind speed of U1 (m /s )
30
35
Fig. 9. Aerodynamic damping ratios estimated from the measured data. (Tower B).
Observed tower
Each of the conductors of Tower A is divided so that a is 10 or over, judging by the result shown in Fig. 11, to estimate the aerodynamic damping with satisfactory accuracy. Each node of the divided conductors has three-degrees-of-freedom. The natural frequencies of the 1st and 2nd tower-modes of the coupled model are 0.5 and 1.4 Hz under calm conditions, respectively, and they are close to the measurement results. Additionally, the mode shapes of the coupled model fit the measurement results, as shown in Fig. 12. The coupled model is submitted to the wind profile of the power law of exponent 1/8, as shown in Fig. 13, at a right angle to the conductors. This wind profile is approximated using the measured data in cases of strong wind. 4.2. Definition of modal aerodynamic damping ratio In a buffeting response, the motion equation is MX€ ðtÞ þ Cs X_ ðtÞ þ KXðtÞ ¼ PðtÞ,
ð8Þ
where M, Cs, K, X(t) and P(t) are the matrix of mass, structural damping, stiffness, vector of displacement and wind force vector, respectively. As the tower is a lattice structure and the mass point area is small when compared with the scale of turbulence, its aerodynamic admittance is assumed to be constant. Therefore, the wind force P(t) is divided into the mean P, the fluctuating p(t) and _ the aerodynamic damping components axðtÞ as follows:
1463m
361m
Fig. 10. Coupled model for the tower-and-conductors system.
taking account of the displacement compatibility in the boundary condition (Magara et al., 1974). The stiffness matrix of the conductor model is estimated based on the small displacement from the equilibrium position using MFM. The matrices of the tower-andconductors comprise the stiffness matrix of the coupled model (Ozono et al., 1985, 1992). To make the coupled model, the conductors need to be divided into appropriate segments, as shown in Fig. 10. Fig. 11 shows the influence of the division number of the conductor model on the natural frequency and aerodynamic damping ratio of the 1st towermode of the coupled model that is assumed to be subjected to the wind profile of 40 m/s at the U1 of Tower A. The symbol a in the figure refers to the ratio of the division number of the conductor, nd, to the node number of the conductor-mode shape, nm. The figure indicates that the division number of the conductor has a greater effect on the estimated aerodynamic damping ratio than on the estimated natural frequency. Because every modal aerodynamic damping ratio is calculated using the natural vibration mode matrix
_ PðtÞ ¼ P þ pðtÞaxðtÞ P ¼ 0:5aU pðtÞ ¼ auðtÞ where a is the aerodynamic damping matrix: 2 3 0 6 7 0 6 7 6 7 6 7 rD1 A1 U 1 sym: 6 7 6 7 0 6 7 7, a¼6 6 7 0 6 7 6 7 rD2 A2 U 2 0 6 7 6 7 6 7 & 4 5 rDn An U n 0
ð9Þ
ð10Þ
where r, Di, Ai and U i refer to the air density, drag coefficient, area and mean wind speed at a right angle to the conductors of mass point i, respectively. The displacement vector X(t) has the mean X and fluctuating components x(t): KX ¼ P,
ð11Þ
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Natural frequency (Hz)
0.6
8 7 6 5 4 3 2 1 0
0.5 Natural frequency
0.4 0.3
Aerodynamic damping ratio
0.2 0.1 0 0
2
4
6
8
10
12
Conductor
Aerodynamic damping ratio (%)
878
2 4 Tower
1 3
5
Divided into 20 segments
14
(Ratio of the division number of the conductor to the node number of the conductor-mode shape)
= nd / nm = 20 / 5 = 4
(Definition of )
Fig. 11. Division number of conductors vs. modal values of the coupled model of Tower A.
aerodynamic damping ratio hai is defined using the natural vibration mode vector and the generalized matrices as follows (Caughey, 1960):
250
250
200
200
ai hai ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 Ki Mi
150
150
ai ¼ fi afi :
100
100
T
Analytical Measured
Analytical Measured 50
50
0
0 1st tower-mode
2nd tower-mode
Fig. 12. Comparison between the analytical and measurement results of the mode shapes of the 1st and 2nd tower-modes.
250 Measured Approximate
Height (m)
200
150
100
50
0 20
25
30
35
Mean wind speed (m/s) Fig. 13. Profile of mean wind speed.
€ þ ðCs þaÞxðtÞ _ þ KxðtÞ ¼ pðtÞ MxðtÞ
ð12Þ
Thus, the damping matrix C of a structure is written as: C ¼ Cs þa,
ð14Þ
4.3. Analytical aspect of aerodynamic damping Fig. 14 shows the modal aerodynamic damping ratios for each mode of natural frequency at the mean wind speed of U1, as shown in Fig. 1. The figure indicates the existence of two patterns in the plotted marks. One is a pattern of lower aerodynamic damping ratios. This pattern corresponds to the in-plane vibration mode of the conductors and is clearer in a region of lower wind speeds. The other is a pattern of higher aerodynamic damping ratios. This pattern corresponds to the out-of-plane mode of the conductors and approaches the pattern of the in-plane mode with an increase in wind speeds. In earlier reports, we used a model with one-degree–of-freedom in a right angle to the conductors and discovered that we cannot find the features of these patterns using only this model (Maeda et al., 2003). Fig. 15 shows comparisons between the estimated modal aerodynamic damping ratios using the model of degree of freedom of one and three when U U1 is 32.1 m/s. This is because the wind force adds not only to the out-of-plane direction but also to the in-plane direction when wind speed increases. The damping ratios of both patterns were found to be dependent on wind speed, and the lower mode has a larger damping ratio. This is in good agreement with the observed results. However, the damping ratios do not necessarily increase proportionally. Fig. 16 shows the aerodynamic damping ratios of the tower-modes selected from Fig. 14. It is thought that when wind speed increases and the angle of the conductor-plane is shifted to the direction opposite to that of the wind the development of the aerodynamic damping of the in-plane mode is affected, and the increase in the rigidity of the conductor resulting from its geographical nonlinearity should be kept in mind. The aerodynamic damping ratios of the tower-mode are smaller than those of the conductors-mode. And, the analytical values are in close agreement with the measurement results shown in Fig. 8. Therefore, it is estimated that the modeling of the coupled tower-and-conductors system in this paper is appropriate for use when estimating aerodynamic damping under conditions of strong wind.
5. Conclusions ð13Þ
The damping matrix C, which includes aerodynamic damping, is approximately orthonormal and hence the modal
Some of our findings concerning the aerodynamic damping properties of a transmission tower based on the wind response
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
20 Conductors-mode Tower-mode
16 12
UU1=10m/s
8 4 0 0
0.5
1
1.5
45 Conductors-mode Tower-mode
36 27
UU1=30m/s
18 9 0
2
0
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
40 Conductors-mode Tower-mode
24 UU1=20m/s
16
0.5
1
1.5
2
Natural frequency (Hz)
Natural frequency (Hz)
32
879
8 0
40 Conductors-mode Tower-mode
32 24
UU1=40m/s
16 8 0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Natural frequency (Hz)
Natural frequency (Hz)
Fig. 14. Modal aerodynamic damping ratios of the tower-mode of Tower A.
50 Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
50 Conductors-mode Tower-mode
40 30
1DOF 20 10 0
Conductors-mode Tower-mode
40 30
3DOF 20 10 0
0
0.5 1 1.5 Natural frequency (Hz)
2
0
0.5 1 1.5 Natural frequency (Hz)
2
Aerodynamic damping ratio (%)
Fig. 15. Comparison between modal aerodynamic damping ratios calculated using the model of degree of freedom of one and three when U U1 is 32.1 m/s.
5 4 1st tower-mode
3 2
2nd tower-mode
1 0 0
10
20
30
40
Mean wind speed of U1 (m /s)
(3) The coupled tower-and-conductors system has two patterns of aerodynamic damping forces corresponding to the vibration mode of the in-plane and out-of-plane modes of the conductors. (4) The aerodynamic damping ratios of the coupled model do not necessarily increase proportionally with an increase in wind speeds. (5) The analytical values of the aerodynamic damping ratios are in good agreement with the measurement results and hence it is estimated that the coupled tower-and-conductors model in this paper is appropriate for use when estimating aerodynamic damping under conditions of strong wind.
Fig. 16. Modal aerodynamic damping ratios of Tower A classified according to wind speed.
Acknowledgement measurement and the computational analysis of a coupled towerand-conductors model are summarized as follows: (1) The aerodynamic damping ratios of the tower-and-conductors are strongly dependent on wind speed. (2) A combination of the Normal Decomposition Method and the Multi-Degree-of-Freedom RD technique is very useful when the precise estimation of the damping ratio of a structure having close natural frequencies, such as a transmission tower, is desired.
This work was supported by Grant-in-Aid for JSPS Fellows, 21-4044, Japan Society for the Promotion of Science and Kyushu Electric Power Co., Inc.
References Agneni, A., Balis, C. L., 1988. Analytic signal in the damping coefficient estimation. In: Proceedings of the international conference on spacecraft and mechanical testing, pp. 133–139.
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Brincker, R., Zhang, L., Andersen, P., 2000. Modal identification from ambient response using Frequency Domain Decomposition. In: Proceedings of the 18th IMAC, pp. 625–630. Brincker, R., Ventura, C. E., Andersen, P., 2001. Damping estimation by frequency domain decomposition. In: Proceedings of the 19th IMAC, pp. 698–703. Caughey, T.K., 1960. Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics, 269–271. Davenport, A.G., 1983. The relationship of reliability to wind loading. Journal of Wind Engineering and Industrial Aerodynamics 13, 3–27. Davenport, A. G., 1988. The response of tension structures to turbulent wind: the role of aerodynamic damping. In: Proceedings of first international Oleg Kerensky memorial conference on tension structures, the Institution of Structural Engineers. London, pp. 20–22. Fujimura, M., Maeda, J., Morimoto, Y., Ishida, N., 2007. Aerodynamic damping properties of a transmission tower estimated using a new identification method. In: Proceedings of the 12th international conference on wind engineering, Preprints—vol. 1, pp. 1047–1054. Hans, S., Ibraim, E., Pernot, S., Boutin, C., Lamarque, C.H., 2000. Damping identification in multi-degree-of-freedom system via Wavelet-Logarithmic Decrement—part 2: study of a civil engineering building. Journal of Sound and Vibration 235 (3), 375–403. Haviland, R., 1976. A study of the uncertainties in the fundamental translational periods and damping values for real buildings. Evaluation of Seismic Safety of Buildings, Research Report 5 National Science Fundation. Lamarque, C.H., Pernot, S., Cuer, A., 2000. Damping identification in multi-degreeof-freedom system via Wavelet-Logarithmic Decrement—part 1: theory. Journal of Sound and Vibration 235 (3), 361–374. Maeda, J., Imamura, Y., Morimoto, Y., 1999. Wind response behavior of a power transmission tower using new displacement measurement. In: Proceedings of the 10th international conference on wind engineering, preprint—vol. 1, pp. 481–486.
Maeda, J., Morimoto, Y., Ishida, N., Tomokiyo, E., Imamura, Y., 2003. Aerodynamic damping properties of a high-voltage transmission tower. In: Proceedings of 5th international symposium on cable dynamics, pp. 255–262. Magara, E., Kunida, J., Kawamata, S., 1974. Analysis of cable nets in mixed formulation, Part 2 Rigoros solution of the geometrically nonlinear problems. Transactions of the Architectural Institution of Japan 218, 35–45 (in Japanese). Momomura, Y., Marukawa, H., Okamura, T., Hongo, E., Ohkuma, T., 1997. Full-scale measurements of wind-induced vibration of a transmission line system in a mountainous area. Journal of Wind Engineering and Industrial Aerodynamics 72, 241–252. Okamura, T., Ohkuma, T., Hongo, E., Okada, H., 2003. Wind response analysis of transmission tower in mountainous area. Journal of Wind Engineering and Industrial Aerodynamics 91, 53–63. Ozono, S., Maeda, J., Makino, M., 1985. Study on the coupling system of transmission lines and towers. Journal of Structural and Construction Engineering: Transactions of AIJ 353, 48–61 (in Japanese). Ozono, S., Maeda, J., 1992. In-plane dynamics interaction between a tower and conductors at lower frequencies. Engineering Structures 14, 210–216. Ruzzene, M., Fasana, A., Garibaldi, L., Piombo, B., 1997. Natural frequency and damping identification using Wavelet Transform. Mechanical System & Signal Processing 11 (2), 207–218. Tamura, Y., Suganuma, S., 1996. Evaluation of amplitude-dependent damping and natural frequency of buildings during strong winds. Journal of Wind Engineering and Industrial Aerodynamics 59, 115–130. Tamura, Y., Zhang, L., Yoshida, A., Nakata, S., Itoh, T., 2002. Ambient vibration tests and modal identification of structures by FDD and 2DOF-RD technique. In: Proceedings of the structural engineers world congress, T1-1-a-1.
Contents lists available at ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Aerodynamic damping properties of two transmission towers estimated by combining several identification methods Mayumi Takeuchi a,n, Junji Maeda a, Nobuyuki Ishida b a b
Faculty of Human Environment Studies, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Japan Steel Tower Co. Ltd.,1-7-1 Kitahama, Wakamatsu-ku, Kitakyushu 808-0023, Japan
a r t i c l e in f o
a b s t r a c t
Article history: Received 13 February 2008 Received in revised form 31 August 2010 Accepted 3 September 2010 Available online 29 September 2010
Based on full-scale measurement data, we report on several aerodynamic damping properties of two transmission towers under conditions of strong winds. In estimating the damping ratio from data that contain many components of close frequencies of tower and conductor vibrations, we cannot estimate the damping ratio of each component if we use an estimation method available for only a single-degreeof-freedom system. Therefore, we combined several identification methods and are introducing a new method of estimating damping properties. This method is applicable to the response record of a multidegree-of-freedom system such as the coupled structure of a transmission tower-and-conductors. Using the new estimation procedure, we were able to extract the component of every vibration mode of the towers from a measured time history and estimate the accurate damping ratios individually; and, our results revealed that the wind speed dependency of the aerodynamic damping property of the coupled tower-and-conductors system had wide differing characteristics depending on the vibration mode. Moreover, using a three-dimensional nonlinear analysis, we discuss, in detail, the effects of the progressive deformation of a transmission–conductor–plane due to an increase in wind speed on the development of the aerodynamic damping of the coupled tower-and-conductors system. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Aerodynamic damping 3D behaviors of a coupled tower-andconductors system Normal decomposition method Multi-degree-of-freedom RD technique
1. Introduction A transmission tower is a mixed structure connected by conductors with very different levels of rigidity; hence, the behaviors of the conductors greatly affect those of the tower (Maeda et al., 1999; Okamura et al., 2003). Because the structural damping force of the transmission tower is very small, the tower is extremely sensitive to aerodynamic damping forces acting on the conductors. These damping forces play an important role in the behavior of conductors under conditions of strong wind (Davenport, 1988; Momomura et al., 1997). The authors reported elsewhere that the results from the site measurement data for a transmission tower show the presence of strong aerodynamic damping force acting on the tower, and they illustrated its wind speed dependency (Maeda et al., 2003). In addition, the authors discussed analytical results using a coupled model for a towerand-conductors system and indicated that both the geometrical nonlinearity of the conductors’ stiffness and the presence of additional aerodynamic damping forces on the conductors
n
Corresponding author. Tel.: + 81 92 642 3367; fax: + 81 92 642 4394. E-mail addresses: [email protected] (M. Takeuchi), [email protected] (J. Maeda), [email protected] (N. Ishida). 0167-6105/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2010.09.001
significantly affect the tower’s behaviors under conditions of strong winds (Maeda et al., 2003; Fujimura et al., 2007). We know that the accuracy of a damping ratio estimated from measured response data tends to vary more widely than other vibration parameters, such as natural frequency. Earlier reports (Haviland, 1976; Davenport, 1983) have shown that the coefficient of variance of the damping ratio estimated from measured response data sometimes reaches 70%. Also, because there are many damping estimation methods and the applicable limit of each method differs, the accuracy of the estimated damping ratio depends on the estimation method used. In addition, the measured response data for the tower includes many components of close frequencies because the transmission tower is a mixed structure connected by conductors with different levels of rigidity, as mentioned previously. Thus, particular attention should be paid to the decomposition of the vibration mode components. The damping estimation methods for the measured response data of the structures are classified into two methods: those using a time domain response function and those using a frequency domain response function. The former is known as the Autocorrelation Method and the Random Decrement Technique (Tamura and Suganuma, 1996) and the latter is known as the Half Power Method and the Hilbert Transform Method (Agneni and Balis, 1988). However, these methods work theoretically only for a singledegree-of-freedom system. As mentioned above, the tower’s response data often includes many components of close frequencies;
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
hence, we cannot simply apply these methods for it. In recent years, an applicable method for the response data of a multi-degree-offreedom system was introduced. The method uses the WaveletLogarithmic Decrement (Ruzzene et al., 1997; Lamarque et al., 2000; Hans et al., 2000), the Frequency Domain Decomposition (Brincker et al., 2000, 2001), the Multi-Degree-Of-Freedom Random Decrement Technique (Tamura et al., 2002) and the Normal Decomposition Method (Fujimura et al., 2007). In particular, the latter three are applicable to response data containing several components of close frequencies. Therefore we expect that these methods will contribute to improving the accuracy of the damping ratio estimation of the transmission tower-and-conductors. Also, the coupled model for the tower-and-conductors system, which the authors have used previously, has only one-degree-offreedom in the direction of the wind. However the flatness of the conductor-plane is lost with an increase in wind speed and the angle of the plane against the wind direction changes. Thus, we have to show the properties of aerodynamic damping forces using a three-degree-of-freedom model. In this paper, the authors apply the general identification methods and the new methods to the measured response data for two transmission towers and compare the damping ratios estimated using each method. After discussing the estimations, the authors discuss in greater detail the aerodynamic damping properties of a three-dimensional coupled model of a tower-andconductors system, focusing on the wind speed-associated change of the deformation shape of the conductor-plane.
Photo 1. Tower A.
Table 1 Structural specifications of Tower A. Voltage Height of tower Line angle Insulator Conductor Ground wire Span Left side Right side Natural frequency 1st tower-mode 2nd tower-mode
873
500 kV 214.5 m 0 (deg.) Suspension type KTACSR/Est450 4 OPGW490 1463 m 361 m 0.5 Hz 1.3 Hz
2. Outline of observed towers and measurement systems V1 X Y U1 U2
2.1. Outline of measurement of tower A
K1, K2(214.5m)
Conductors
U3
One of the observed towers (Tower A) was a 214.5 m tall 500 kV suspension type transmission tower, as shown in Photo 1. The structural specifications and natural frequencies of Tower A are shown in Table 1. Eight ultrasonic anemometers and one windmill type anemometer were positioned on the tower for wind measurement, as shown in Fig. 1. Every anemometer was located several meters from a main post in a horizontal alignment so that the anemometers would not be affected by the tower. One target plate and four accelerometers for displacement and acceleration were positioned on/near the tower, as shown in Fig. 1. The reference (Maeda et al., 1999) details the displacement measurement system. For verification of the tower’s response characteristics, a total of 31 samples were selected from the measured wind, displacement and acceleration data acquired at a right angle to the conductors because the tower’s wind load becomes dominant in that direction. The data was gathered from 1993 to 1997, and each sample is 10 min long.
Wind
K3, K4(163.5m) U4
214.5m
Target plate U5
Windmill type anemometer: V1 Ultrasonic anemometer: U1~U8
U6
Accelerometer: K1~K4 K1, K3: In the direction of conductors
U7
K2, K4: At a right angle to conductors X Y Target plate for displacement measurement: X, Y X: In the direction of conductors
20m
U8
Y: At a right angle to conductors
34.4m
30 20 10 0 0
100
200
Time (s)
300
600
Acceleration of K2 (m /s2)
40
Displacement of Y (mm)
Wind speed of V1 (m/s)
Fig. 1. Position of measurement sensors on Tower A.
400
200
0 0
100
200
300
Time (s) Fig. 2. Time evolution of measurement data at the top of Tower A.
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 0
100
200
Time (s)
300
10
2
101 10
Power spectrum / Variance (s)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Power spectrum / Variance
874
Displacement of Y
0
10-1 10-2 10
-3
10
-4
10
-5
0
1
2
3
4
10
2
101 10
Acceleration of K2
0
10-1 10-2 10
-3
10-4 10
-5
0
5
1
2
3
4
5
Frequency (Hz)
Frequency (Hz)
Fig. 3. Power spectra of displacement and acceleration during strong winds.
Table 2 Structural specifications of Tower B. Voltage Height of tower Line angle Insulator Conductor Ground wire Inside Outside Span Left side Right side Natural frequency 1st tower-mode
U1
500 kV 70 m 0 (deg.) Strain type ACSR610 4 AS100 OPAS100 217 m 528 m 1.2 Hz
K1, K2(70m)
Conductors
K3, K4 (58m)
70m
Wind
U2
K5, K6(24m)
Ultrasonic anemometer: U1,U2 Accelerometer: K1~K6 K1, K3, K5: In the direction of conductors K2, K4, K6: At a right angle to conductors
11m Fig. 4. Position of measurement sensors on Tower B.
Photo 2. Tower B.
Fig. 2 shows the time evolution of the wind, displacement and acceleration at the top of the tower when the mean wind speed of V1 was 32.1 m/s. The peak gust was 38.9 m/s. The maximum and average displacements found at a right angle to the conductors were 450.2 and 378.9 mm, respectively. And the maximum acceleration was 0.45 m/s2. Fig. 3 shows the power spectra of the displacement and acceleration. We can see that the wind energy component is remarkable in the lower frequency region of the displacement spectrum.
shown in Table 2. Two ultrasonic anemometers and six accelerometers were positioned on the tower for wind measurement and acceleration, as shown in Fig. 4. The data was gathered through long-term observation from 2003 to 2004. In this paper, we selected data recorded during a typhoon and a non-typhoon, and the evaluating time of the measured data is 10 min. Fig. 5 shows the time evolution of the wind and acceleration at the top of the tower when the mean wind speed of U1 was 29.1 m/s. The peak gust and maximum acceleration were 49.1 m/s and 0.68 m/s2, respectively. 3. Aerodynamic damping of the transmission towers based on measured data
2.2. Outline of measurement of tower B 3.1. Extracting the component of the tower’s dominant mode The other observed tower (Tower B) was a 500 kV strain-type tower, which is 70 m in height, as shown in Photo 2. The structural specifications and natural frequency of Tower B are
Before we estimate the damping ratio of a vibration mode, we have to extract the component of each mode from the measured
60 40 20 0 0
100
200
300 400 Time (s)
500
600
Acceleration of K2 (m/s2)
Wind speed of U1 (m/s)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
875
0.8 0.4 0.0 -0.4 -0.8 0
100
200
300 400 Time (s)
500
600
Fig. 5. Time evolution of measurement data at the top of Tower B.
data. Generally, the component of a tower-mode is extracted using a FFT band-pass filter (FFT). However, it is known that this filter has no standard for the selection of the band range of frequency, and so has some side-cut-effects. Thus, in this paper, the component of the tower-mode is extracted using the Normal Decomposition Method (NDM) (Fujimura et al., 2007). Because this method utilizes the orthogonality of the vibration mode shape, it is also very useful for recording the time history, which includes several components of close frequencies. The basic theory of NDM is described below. Here is the equation for the motion of multi-degree-offreedom € þ CxðtÞ _ þ KxðtÞ ¼ pðtÞ: MxðtÞ
ð1Þ
In the equation, x(t) is the displacement vector (mean component subtracted) and M, C, K and p(t) are the matrix of mass, damping, stiffness and fluctuating force vector, respectively. The damping matrix C is orthonormal to the natural vibration modes vector, and the displacement vector x(t) is written using a natural vibration mode matrix U and the generalized displacement vector q(t): xðtÞ ¼ UqðtÞ ¼ ½j1 j2 ji ½q1 q2 qi T :
ð2Þ
By substituting Eq. (2) in Eq. (1) and multiplying UT from the left, Eq. (1) can be rewritten as follows: € þC qðtÞ _ þ K qðtÞ ¼ p ðtÞ, M qðtÞ n
n
n
ð3Þ
n
where M , C , K and p (t) are the generalized mass matrix, the generalized damping matrix, the generalized stiffness matrix and the generalized fluctuating force vector, respectively. For example, the component of the restoring force in Eq. (1) is written as KxðtÞ ¼ KUqðtÞ:
ð4Þ
By multiplying both sides of Eq. (4) by UT from the left, we get the following:
UT KxðtÞ ¼ UT KUqðtÞ ¼ K qðtÞ:
ð5Þ
By deriving the second derivative of Eq. (5) with respect to t, we get the following: € € ¼ K qðtÞ: UT KxðtÞ
ð6Þ
Thus, a generalized acceleration of vibration mode q€ i ðtÞ, can be € using Eq. (7) as follows: derived from the measured time history xðtÞ € € ¼ K1 UT KxðtÞ: qðtÞ
ð7Þ
Fig. 6(a) shows the original acceleration data observed at the top of the Tower A under conditions of strong wind, indicating
power spectral density. We can see that the original data includes the components of the 1st and 2nd tower-modes containing many components of the conductors-mode around the frequency of each tower-mode. Fig. 6(b) and (c) show the time evolution and power spectral densities of acceleration both of which contain only the tower-modes extracted from the original acceleration data using NDM. However, both the 1st and 2nd tower-modes can be extracted explicitly, but several components of the conductorsmode remain near the 1st mode frequency of the tower. These components of the conductors-mode cannot be excluded because the tower parts of many of the tower-conductors-coupled-mode shapes around the 1st tower-mode component have almost the same shapes as the 1st tower-mode, as Okamura has reported (Okamura et al., 2003). And, NDM cannot extract only the 1st tower-mode component from the time history data of the transmission tower because the measured data contains only the tower-mode shape; hence, the conductors-mode shape cannot be considered in Eq. (7). 3.2. Estimating the aerodynamic damping ratio of the transmission towers The damping ratios of the 1st and 2nd tower-mode components were estimated from the extracted components of the vibration modes of the transmission tower using the Random Decrement technique (RD) (Tamura and Suganuma, 1996). However, as the natural vibration mode of the transmission tower is strongly affected by the fluctuating tension of the conductors, the tower’s measured response data often has many components of close frequencies, as mentioned previously (Fig. 6). If the data including several components of close frequencies is applied to RD, the RD evolution beats as a result, as shown in Fig. 7; hence, the envelope of the RD evolution cannot be fit to a free-vibration decrement curve. When the measured data related to the conductors-mode shape is lacking, we have to combine NDM with a decomposition method based on a mathematical method. In this case, the Multi-Degree-of-Freedom RD technique (MDOF-RD) (Tamura et al., 2002) is very useful for precisely estimating the damping ratio of each component. Also, the damping ratios of the measured components contains structural damping, which was estimated using vibration tests under calm conditions, hence the aerodynamic damping ratio is defined by subtracting the structural damping ratios (Tower A: 1st 0.30%, 2nd 0.49%, Tower B: 1st 0.86%; Fujimura et al., 2007) from the damping ratios of the components. Figs. 8 and 9 show the aerodynamic damping ratios of Towers A and B estimated from the measured acceleration data, which was measured orthogonally to the direction of the conductors under conditions of strong winds. In the figures, the values estimated by MDOF-RD, RD, NDM and FFT have been added. The analytical values shown in Fig. 8 and described below are the aerodynamic damping ratios of the component of the tower vibration mode of the coupled tower-and-conductors model. The
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Acceleration (m /s2)
0.4 0.2 0 -0.2 -0.4 0
100
200
300
400
500
600
Power spectrum ((m /s2)2 s)
876
0.15 1st tower-mode
0.10
Conductors-mode 2nd tower-mode
0.05 0 0.0
0.5
0.2 0.1 0 -0.1
108
112
116
120
Power spectrum ((m /s2)2 s)
Acceleration (m /s2)
0.3
104
1st tower-mode Conductors-mode
0.10 0.05 0 0.0
0.5
0
-0.1
112
116
120
Power spectrum ((m /s2)2 s)
Acceleration (m /s2)
0.1
108
1.0
1.5
2.0
Frequency (Hz)
0.2
104
2.0
0.15
Time (s)
-0.2 100
1.5
Frequency (Hz)
Time (s)
-0.2 -0.3 100
1.0
0.15 0.10 2nd tower-mode
0.05 0 0.0
0.5
1.0
1.5
2.0
Frequency (Hz)
Time (s)
Fig.6. Examples of extracted components of tower-mode of Tower A. (a) The original data at the top of Tower A. (b) The component of 1st tower-mode. (c) The component of 2nd tower-mode.
the variance of the estimated damping ratios by NDM or MDOFRD is remarkably smaller than by FFT or RD.
RD evolution (m/s2)
0.12 0.08 0.04
4. Aerodynamic damping properties of 3D-computational model
0 -0.04
4.1. Coupled model for the tower-and-conductors system
-0.08 -0.12 0
4
8
12
16
20
24
Time (s) Fig. 7. Example of the RD evolution beats.
coupled system of Tower A is shown in Fig. 10. As already reported in the references (Maeda et al., 2003; Fujimura et al., 2007), the aerodynamic damping ratios of the 1st and 2nd modes tend to increase with wind speed, and the aerodynamic damping ratios of the 2nd mode are smaller than those of the 1st mode. In the 1st tower-mode of Tower A, we focus on the differences between the damping estimation methods of MDOF-RD and RD. In the 2nd tower-mode, we focus on the differences between the decomposition methods of NDM and FFT. It should be noted that
A transmission tower is modeled to a coupled lumped-mass model which includes the insulators and the conductors for Tower A, as shown in Fig. 10, using FEM and the Mixed Formulation Method (MFM) for the tower and the conductors, respectively (Fujimura et al., 2007). In horizontal directions, each mass of the tower model has two-degree-of-freedom, one of which is in the direction of the conductors while the other is in a right angle to the conductors. Each mass point reaches its average displacement and then responds to a wind gust. The stiffness matrix of the tower model is estimated using linear FEM. A member of the tower is idealized to a beam element. The joints between the main posts are assumed to be rigid, and the others are assumed to be pin. However, a nonlinear deformation analysis is necessary when estimating the equilibrium positions of the conductors to the mean wind and dead loads. MFM using the potential energy function of a link mechanism is used,
10
1st tower-mode
Measured (MDOF-RD) Measured (RD)
8
Analytical
6 4 2 0 5
0
10
15
20
25
30
35
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
10
2nd tower-mode
877
Measured (MDOF-RD) Measured (RD)
8
Analytical
6 4 2 0 0
Mean wind speed of U1 (m /s)
5
10 15 20 25 Mean wind speed of U1 (m / s)
30
35
Aerodynamic damping ratio (%)
Fig. 8. Aerodynamic damping ratios estimated from the measured data (Tower A).
U, having more segments provides a more accurate mode shape.
10 1st tower-mode
Measured (MDOF-RD)
8 6 4 2 0 0
5
10 15 20 25 Mean wind speed of U1 (m /s )
30
35
Fig. 9. Aerodynamic damping ratios estimated from the measured data. (Tower B).
Observed tower
Each of the conductors of Tower A is divided so that a is 10 or over, judging by the result shown in Fig. 11, to estimate the aerodynamic damping with satisfactory accuracy. Each node of the divided conductors has three-degrees-of-freedom. The natural frequencies of the 1st and 2nd tower-modes of the coupled model are 0.5 and 1.4 Hz under calm conditions, respectively, and they are close to the measurement results. Additionally, the mode shapes of the coupled model fit the measurement results, as shown in Fig. 12. The coupled model is submitted to the wind profile of the power law of exponent 1/8, as shown in Fig. 13, at a right angle to the conductors. This wind profile is approximated using the measured data in cases of strong wind. 4.2. Definition of modal aerodynamic damping ratio In a buffeting response, the motion equation is MX€ ðtÞ þ Cs X_ ðtÞ þ KXðtÞ ¼ PðtÞ,
ð8Þ
where M, Cs, K, X(t) and P(t) are the matrix of mass, structural damping, stiffness, vector of displacement and wind force vector, respectively. As the tower is a lattice structure and the mass point area is small when compared with the scale of turbulence, its aerodynamic admittance is assumed to be constant. Therefore, the wind force P(t) is divided into the mean P, the fluctuating p(t) and _ the aerodynamic damping components axðtÞ as follows:
1463m
361m
Fig. 10. Coupled model for the tower-and-conductors system.
taking account of the displacement compatibility in the boundary condition (Magara et al., 1974). The stiffness matrix of the conductor model is estimated based on the small displacement from the equilibrium position using MFM. The matrices of the tower-andconductors comprise the stiffness matrix of the coupled model (Ozono et al., 1985, 1992). To make the coupled model, the conductors need to be divided into appropriate segments, as shown in Fig. 10. Fig. 11 shows the influence of the division number of the conductor model on the natural frequency and aerodynamic damping ratio of the 1st towermode of the coupled model that is assumed to be subjected to the wind profile of 40 m/s at the U1 of Tower A. The symbol a in the figure refers to the ratio of the division number of the conductor, nd, to the node number of the conductor-mode shape, nm. The figure indicates that the division number of the conductor has a greater effect on the estimated aerodynamic damping ratio than on the estimated natural frequency. Because every modal aerodynamic damping ratio is calculated using the natural vibration mode matrix
_ PðtÞ ¼ P þ pðtÞaxðtÞ P ¼ 0:5aU pðtÞ ¼ auðtÞ where a is the aerodynamic damping matrix: 2 3 0 6 7 0 6 7 6 7 6 7 rD1 A1 U 1 sym: 6 7 6 7 0 6 7 7, a¼6 6 7 0 6 7 6 7 rD2 A2 U 2 0 6 7 6 7 6 7 & 4 5 rDn An U n 0
ð9Þ
ð10Þ
where r, Di, Ai and U i refer to the air density, drag coefficient, area and mean wind speed at a right angle to the conductors of mass point i, respectively. The displacement vector X(t) has the mean X and fluctuating components x(t): KX ¼ P,
ð11Þ
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Natural frequency (Hz)
0.6
8 7 6 5 4 3 2 1 0
0.5 Natural frequency
0.4 0.3
Aerodynamic damping ratio
0.2 0.1 0 0
2
4
6
8
10
12
Conductor
Aerodynamic damping ratio (%)
878
2 4 Tower
1 3
5
Divided into 20 segments
14
(Ratio of the division number of the conductor to the node number of the conductor-mode shape)
= nd / nm = 20 / 5 = 4
(Definition of )
Fig. 11. Division number of conductors vs. modal values of the coupled model of Tower A.
aerodynamic damping ratio hai is defined using the natural vibration mode vector and the generalized matrices as follows (Caughey, 1960):
250
250
200
200
ai hai ¼ pffiffiffiffiffiffiffiffiffiffiffiffi 2 Ki Mi
150
150
ai ¼ fi afi :
100
100
T
Analytical Measured
Analytical Measured 50
50
0
0 1st tower-mode
2nd tower-mode
Fig. 12. Comparison between the analytical and measurement results of the mode shapes of the 1st and 2nd tower-modes.
250 Measured Approximate
Height (m)
200
150
100
50
0 20
25
30
35
Mean wind speed (m/s) Fig. 13. Profile of mean wind speed.
€ þ ðCs þaÞxðtÞ _ þ KxðtÞ ¼ pðtÞ MxðtÞ
ð12Þ
Thus, the damping matrix C of a structure is written as: C ¼ Cs þa,
ð14Þ
4.3. Analytical aspect of aerodynamic damping Fig. 14 shows the modal aerodynamic damping ratios for each mode of natural frequency at the mean wind speed of U1, as shown in Fig. 1. The figure indicates the existence of two patterns in the plotted marks. One is a pattern of lower aerodynamic damping ratios. This pattern corresponds to the in-plane vibration mode of the conductors and is clearer in a region of lower wind speeds. The other is a pattern of higher aerodynamic damping ratios. This pattern corresponds to the out-of-plane mode of the conductors and approaches the pattern of the in-plane mode with an increase in wind speeds. In earlier reports, we used a model with one-degree–of-freedom in a right angle to the conductors and discovered that we cannot find the features of these patterns using only this model (Maeda et al., 2003). Fig. 15 shows comparisons between the estimated modal aerodynamic damping ratios using the model of degree of freedom of one and three when U U1 is 32.1 m/s. This is because the wind force adds not only to the out-of-plane direction but also to the in-plane direction when wind speed increases. The damping ratios of both patterns were found to be dependent on wind speed, and the lower mode has a larger damping ratio. This is in good agreement with the observed results. However, the damping ratios do not necessarily increase proportionally. Fig. 16 shows the aerodynamic damping ratios of the tower-modes selected from Fig. 14. It is thought that when wind speed increases and the angle of the conductor-plane is shifted to the direction opposite to that of the wind the development of the aerodynamic damping of the in-plane mode is affected, and the increase in the rigidity of the conductor resulting from its geographical nonlinearity should be kept in mind. The aerodynamic damping ratios of the tower-mode are smaller than those of the conductors-mode. And, the analytical values are in close agreement with the measurement results shown in Fig. 8. Therefore, it is estimated that the modeling of the coupled tower-and-conductors system in this paper is appropriate for use when estimating aerodynamic damping under conditions of strong wind.
5. Conclusions ð13Þ
The damping matrix C, which includes aerodynamic damping, is approximately orthonormal and hence the modal
Some of our findings concerning the aerodynamic damping properties of a transmission tower based on the wind response
M. Takeuchi et al. / J. Wind Eng. Ind. Aerodyn. 98 (2010) 872–880
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
20 Conductors-mode Tower-mode
16 12
UU1=10m/s
8 4 0 0
0.5
1
1.5
45 Conductors-mode Tower-mode
36 27
UU1=30m/s
18 9 0
2
0
Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
40 Conductors-mode Tower-mode
24 UU1=20m/s
16
0.5
1
1.5
2
Natural frequency (Hz)
Natural frequency (Hz)
32
879
8 0
40 Conductors-mode Tower-mode
32 24
UU1=40m/s
16 8 0
0
0.5
1
1.5
2
0
0.5
1
1.5
2
Natural frequency (Hz)
Natural frequency (Hz)
Fig. 14. Modal aerodynamic damping ratios of the tower-mode of Tower A.
50 Aerodynamic damping ratio (%)
Aerodynamic damping ratio (%)
50 Conductors-mode Tower-mode
40 30
1DOF 20 10 0
Conductors-mode Tower-mode
40 30
3DOF 20 10 0
0
0.5 1 1.5 Natural frequency (Hz)
2
0
0.5 1 1.5 Natural frequency (Hz)
2
Aerodynamic damping ratio (%)
Fig. 15. Comparison between modal aerodynamic damping ratios calculated using the model of degree of freedom of one and three when U U1 is 32.1 m/s.
5 4 1st tower-mode
3 2
2nd tower-mode
1 0 0
10
20
30
40
Mean wind speed of U1 (m /s)
(3) The coupled tower-and-conductors system has two patterns of aerodynamic damping forces corresponding to the vibration mode of the in-plane and out-of-plane modes of the conductors. (4) The aerodynamic damping ratios of the coupled model do not necessarily increase proportionally with an increase in wind speeds. (5) The analytical values of the aerodynamic damping ratios are in good agreement with the measurement results and hence it is estimated that the coupled tower-and-conductors model in this paper is appropriate for use when estimating aerodynamic damping under conditions of strong wind.
Fig. 16. Modal aerodynamic damping ratios of Tower A classified according to wind speed.
Acknowledgement measurement and the computational analysis of a coupled towerand-conductors model are summarized as follows: (1) The aerodynamic damping ratios of the tower-and-conductors are strongly dependent on wind speed. (2) A combination of the Normal Decomposition Method and the Multi-Degree-of-Freedom RD technique is very useful when the precise estimation of the damping ratio of a structure having close natural frequencies, such as a transmission tower, is desired.
This work was supported by Grant-in-Aid for JSPS Fellows, 21-4044, Japan Society for the Promotion of Science and Kyushu Electric Power Co., Inc.
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