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A study on the nonstationarity in wind and wind-induced response of tall buildings for adaptive active control
JOURNALOF
wi,.Im ,,em,,g ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 72 (1997) 213 224
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A study on the nonstationarity in wind and wind-induced response of tall buildings for adaptive active control R. A d h i k a r i a'*'l, H. Y a m a g u c h i b'2 ~'Department of AMES, Universi(v o/" Cal!lbrnia at San Diego, 9500 Gilman Drive, La Jolla, CA 92093 0411, USA b Department q[" Civil and Environmental Engineering, Saitama UniversiO', 255 Shimo Ohkuba, Urawa-Shi 338, Japan
Abstract
The nonstationarity in wind and its effects on the response of tall buildings are studied from the point of view of developing an adaptive active control scheme. The wavelet analysis of the wind and the structural response shows that the distribution of wind gusts with frequency close to the structural frequency could be one of the causes of nonstationarity in the structural response. It is also observed that the use of a longer averaging time results in a higher average peak factor for response and thus requiring one to develop a more robust control system, whereas a small averaging time may require frequent updating of the control system which is practically undesirable.
Keywords: Nonstationarity in wind; Wind-induced response; Wavelet analysis
I. Introduction
The wind-induced gust response of tall buildings causes significant discomfort to the building occupants [1,2]. An active control scheme, e.g. based on L Q control or H ~ control scheme, can be developed to maintain comfortable conditions in such buildings. So far, the design of such active control schemes are done by assuming a predefined fixed level of mean wind velocity and then considering wind as a stationary r a n d o m process. As the mean wind velocity during strong storms shows slowly varying tendency [3,4], a controller designed for a fixed mean wind velocity will not be an economical
* Corresponding author. ~Research Associate. Formerly at Department of Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkubo, Urawa-Shi 338, Japan. 2 Professor. 0167-6105/97/'$17.00 ( : 1997 Elsevier Science B.V. All rights reserved. PI! S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 6 1 -4
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and effective one. Therefore, it is quite rational to develop a control scheme which adapts itself to the changes in the mean wind velocity or the structural response. It is also to be remembered that if the fluctuating components of the wind velocity show considerable nonstationarity then the structural response may also be expected to be nonstationary and thereby directly governing the extent of robustness required in the control scheme. It is then necessary to know the relation between the mean wind velocity and its fluctuating components and also the effects of such fluctuating components on the structural response so that an appropriate time of adaptation as well as the extent of robustness required in the control algorithm can be assessed. Therefore. the main objective of this paper is lo investigate the nonstationarity associated with the wind and its effect on the structural response, mainly from the point of view of developing a robust adaptive active control scheme. As the averaging time used for the evaluation of the mean wind velocity, based on which adaptation scheme for the active control system is to be developed, will have some effects on decidmg the timing of adaptalion, the effects of changing the averaging time are also discussed in this paper.
2. Robust-adaptive control of wind-induced vibrations The mosl rational way of achieving an adaptive controller is to reflect the input characteristics in the control scheme. However, the realization of such adaptive controller requires the beforehand estimation of the input. The beforehand estimation of the wind force is practically not feasible as it is very difficult to evaluate the wind force acting on the structure. Therefore, instead of using the wind force, the controller can be made adaptive to the changes in the mean wind velocity which is measurable and can be predicted based on the retrospective observations made on the mean wind velocity [5]. The adaptiveness in the control scheme can be achieved by modeling the wind velocity as a piecewise stationary process, as shown in Fig. 1, and thereby incorporating a suitable provision for changing the control action with respect to the changes in the mean wind velocity of such piecewise stationary processes. The amount of fluctuations associated with such mean wind velocity and its effect on the structural response will govern the amount of robustness required in the control scheme. The L"
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R. Adhikari, H. Yamaguchi/J. Wind Eng. Ind. Aerodyn. 72 (1997) 213 224
215
expression for the control force of such a robust adaptive controller was derived in Ref. [6] as u = - (SB)-I[SAx
+ SHf(t)]
- sgn(~r(x))r/(t),
(1)
where x is the state vector and matrices A, B and H are represented for a dynamical system expressed in the state-space form, as (2)
£c = A x + B u + H f ( t ) .
In Eq. (1), a(x) is the so-called 'sliding surface,' f(t) is the wind force estimated by predicting the piecewise constant mean wind velocity, whereas r/(t) represents the bounds on the uncertainty in the external excitation and was shown to be given by ~l(t) >~ I S H ( f ( t ) M A X --f(t)RMs)l.
(3)
f(t)Mgx andf(t)aMs of Eq. (3) are to be evaluated based on the maximum and the RMS of the wind velocity, respectively, associated with the predicted mean wind velocity. It is to be noted that the f(t) term in Eq. (1) is related to the adaptive feature of the controller whereas q(t) is related to the robustness of the controller. A schematic diagram of the structural response using such robust-adaptive controller is shown in Fig. 2. It is seen in Fig. 2 that with an adaptive control scheme, it may be possible to control the structural response to a predefined level irrespective of the changes in the wind velocity. Therefore, in order to develop a rational guideline for control adaptation and the amount of robustness required in the control scheme, the prediction of the mean wind velocity or the structural response are indispensable. As the nonstationarity in the wind and the extent of its effects on the structural responses may govern the adaptation scheme and the possible magnitude of robustness required in the control scheme, the nonstationarity in the wind and its effects on the structural responses need to be investigated. In order to investigate the nonstationarity in the wind and its effects on the structural responses, the field data of wind velocity and response of a 100 meter tall buildings during two typhoons, viz. typhoon no. 9119 (typhoon Mireille) and typhoon no. 9117 (typhoon Kinna), were analysed. Time series of wind velocity and the acceleration responses were recorded by three-cup anemometers and accelerometers, Response (A) Uncontrolled\ ,~ Nonadaptive ~. /i constantgain control Adaptively " ~ / ] . /
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Fig. 2. SchematicU A diagram showing the performanceof the robust-adaptive controller in comparison to constant gain nonadaptivecontroller(U, is the triggeringvelocityfor the start of control, and U,,axis the maximum design wind velocityfor control).
210
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respectively, installed at the top storey of the building. The natural frequency and structural damping ratio of the building along both of the struclural axes were estimated to be 0.625 Hz and 1.0%, respectively. Because of very high wind speed of typhoon Mircille, instrument saturation occurred and. therefore, data recording had to be stopped during 90 1101nin fi:om the start of the data recording. Out of various averaging time intervals used. results of only 1 and 10rain averaging time are presented in most part of the prcscm paper whereas the effect of various averaging times are discussed in Section 5.
3. Characterislies of wind and wind-induced response 3./
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Fig. 3 shows the general characteristics of lhc observed wind. In case of typhoon Mireille, as is seen in part lil of Fig. 3a, a sudden change in tile wind direction from ENE to W N W is observed with the passage of the eye of the typhoon, whereas for typhoon Kinna. tile wind direction changes from the predominantly NNE to N N W with the passage of the typhoon. As the dynamic response of the structure is go~crned by the fluctuating components of the wind velocily [7]. the RMS and the maximunl of the wind velocity were examined and are shown in F'ig. 3. Part (ivl of Fig. 3a shows that the RMS of wind velocity for typhoon Mireille increases almost monotonically during 0 60 rain even though the increase in lhe mean wind velocity during that period is not significant. On the other hand. the RMS of the wind velocity is observed to be smaller during 150 200rain than during 0 lo 60 rain even though the mean wind velocity observed in the former case is larger than in the later case, lit) of Fig. 3a. Similar inconsistencies between mean wind velocity and R MS of the wind xelocity are observed in the case of typhoon Kinna also. Thus il is seen that a large mean wind velocity may not always result in large RMS ot" wind velocity. Also, (iii) of Fig. 3a indicates that the maximum wind velocity for typhoon Mireille shows considerable fluctuations during 0 60rain and the same is true for the maxinmrn wind velocity for Kinna during 120 180rain. (iii) of Fig. 3b. This behavior of the wind fluctuations can be related to the changes in the wind directiorls. Therelk)rc, while designing a robust adaptive controller, the uncertainty associated ~xith lhc fluctuating c o m p o n e n l s of wind velocity must also be considered by providing suitable magnitude of robustness to the control scheme.
It is also observed in Fig. 3 that the use of a smaller averaging time results in a rather large fluctuations in both the mean wind velocity and the RMS of the wind velocity, whereas the use of a larger averaging time results in rather smoothly wlrying tendency in both the mean wind velocity and the RMS of the wind velocity', but with higher RMS values. Therefore. if a controller with large robustness can be developed, then a large adaptation time can bc used which is practically desirable while the magnitude of the robustness required will depend on the extent of the effect of the wind fluctuations on the structural response.
R. Adhikari, H. Yamaguchi./~L Wind Eng. Ind. Aerodyn. 72 (1997) 213 224
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3.2. Characteristics of response Fig. 4 d e p i c t s the r e s p o n s e o f t h e b u i l d i n g w i t h respect to the s t o r m d u r a t i o n . Since r e s p o n s e in b o t h x- a n d y - d i r e c t i o n s for b o t h of the t y p h o o n s s h o w e d s i m i l a r c h a r a c t e r i s t i c s , o n l y the r e s p o n s e in t h e x - d i r e c t i o n are p l o t t e d in Fig. 4. It is o b s e r v e d in Fig. 4 t h a t t h e s t a t i o n a r i t y o r n o n s t a t i o n a r i t y in the m e a n w i n d v e l o c i t y is d i r e c t l y r e f l e c t e d in t h e s t r u c t u r a l r e s p o n s e . F u r t h e r m o r e , a careful i n s p e c t i o n of the r e s p o n s e for t y p h o o n M i r e i l l e Fig. 4a, also i n d i c a t e s t h a t t h e a c c e l e r a t i o n r e s p o n s e i n c r e a s e s
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rather monotonically from 0 to 60 mm of the typhoon duration even if the mean wind velocity remains ahnost constant during that time, {it) of Fig. 3a. This is because of the monotonic increase in the RMS the wind velocity as shown in {iv} of Fig. 3a. In case of typhoon Kinna also, it is observed in Fig. 4b that even if the mean wind velocity during 120 180rain is smaller than that during 40 120min, the structural response observed during 120 180rain is comparable to that during 40 120min. Therefore, the consideration of only the mean wind velocity for updating the control scheme may not be adequate and some other suitable parameter, e.g. the RMS or the maximum of the wind velocity, should also be identified. However, the effects of the RMS or the maximum of the wind velocity can be taken care by providing robustness to the controller, as given in Eq. [I) and Eq. (3}, and thereby keeping the adaptation scheme based only on the changes m the mean wind velocity. It is also seen in Fig. 4 that a smaller averaging time results in comparatively smaller RMS values of the structural response but with increased fluctuations as compared to the one obtained fiom a longer averaging time. As the robustness and the effectiveness of a controller have trade off tendency among themselves, the smaller RMS values are desirable from control point of view as the robustness required in the control scheme in this case would be smaller, whereas increased fluctuations are undesirable as it would require frequent adaptation in the control scheme which is practically undesirable. 3.3. (7~rre/ation hetween wind and wiml-induced response
When an adaptive controller is to be developed based on tile changes in thc predicted mean wind velocity, not only the relation between the mean wind velocity and the structural response is necessary, but the relation between the fluctuations in the wind and the Ituctuations in the structural response must also be considered in order to estimate the structural response for the predicted mean wind velocity so that an adaptation scheme for control can be developed accordingly and the magnitude of robustness required m the controller can be assessed. It has been shown in many literatures that the RMS of the response of a building or a tower can be conveniently represented as a function of the mean wind velocity with the RMS of the response being proportional to some power of the mean wind velocity [4]. Therefore, the
R. Adhikari, H. Yamaguchi/J. Wind Eng. Ind. Aerodyn. 72 (1997) 213-224
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relation between the fluctuations in the wind, RMS of wind velocity, and the RMS of the response accelerations were checked here and are shown in Fig. 5. In this case also, it was seen that the RMS of the response accelerations along both x- and y-axes showed similar characteristics with respect to the RMS of wind velocity for both of the typhoons. Therefore, only the RMS of the response accelerations along x-axis for both of the typhoons are plotted in Fig. 5. It is seen in Fig. 5a that the RMS of the response accelerations show an overall increasing tendency with increase in the RMS of the wind. In the case of typhoon Kinna, Fig. 5b, the relation between the fluctuating component of wind and the structural response is not very clear even though an increasing tendency in the RMS of response is identified with increase in the RMS of wind velocity.
4. Nonstationarity in wind and wind-induced response 4.1. Relation between response peak factors and mean wind velocity The relation between the mean wind velocity and the response peak factors are shown in Fig. 6. Here, the response peak factor is defined as the ratio of the maximum value of acceleration to its RMS values and is taken as a parameter to represent the fluctuations in the structural response. For comparison, the theoretical average peak factors for resonant response based on a narrow band stationary assumption made both on the wind turbulence as well as structure response are also plotted in Fig. 6. The theoretical values of the average peak factors were calculated by the following equation, originally derived by Davenport [8], Average peak factor = ~
+ 7 / ~ ,
(4)
where n is the natural frequency, T is the averaging time and 7 is the Euler's constant equal to 0.5772. As is seen in Fig. 6a, the peak factors for the case of typhoon Mireille show a decreasing tendency with increase in the mean wind velocity in the range upto
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20 25 m/s. B e y o n d this range, the increase in the m e a n w i n d v e l o c i t y d o e s not affect m u c h on the peak factors a l o n g tile y-axis, w h e r e a s the p e a k factors a l o n g the x - a x i s s h o w a d e c r e a s i n g t e n d e n c y with i n c r e a s e in the m e a n w i n d velocity. It is to be n o t e d here that in case of t y p h o o n Mireille. the v-axis r e p r e s e n t s close to the a l o n g - w i n d d i r e c t i o n a n d x-axis r e p r e s e n t s the a c r o s s wind d i r e c t i o n . H o w e v e r , in case of t y p h o o n K i n n a , as seen in Fig. 6b, the p e a k factors s h o w a d e c r e a s i n g t e n d e n c y with i n c r e a s e in the m e a n w i n d v e l o c i t y w h e n the a v e r a g i n g t i m e is 1 0 m i n , but w h e n the a v e r a g i n g t i m e is set to be I min, the p e a k factors are not affected by the increase in the m e a n w i n d velocity• It is o b s e r v e d in Fig. 6 that w h e n the a v e r a g i n g t i m e is set to be 1 0 m i n , very large values of p e a k factors, as c o m p a r e d to the t h e o r e t i c a l values of the p e a k factors, can o c c u r in the real case, thus h i g h l i g h t i n g the n o n s t a t i o n a r y c h a r a c t e r i s t i c s of the w i n d t u r b u l e n c e a n d the gust r e s p o n s e of the structure. T h e o c c u r r e n c e of such large p e a k factors is s o m e w h a t s t r a n g e a n d the p r o b a b l e c a u s e of the o c c u r r e n c e of such high p e a k factors is discussed next. 4.2. C a u s e q l n o n s l a t i o n a r i t 3' in s l r u c t u r a / r e s p o n s e
el wavelet a p p r o a c h
In o r d e r to locate the instant of o c c u r r e n c e of the large p e a k factors, the d i s t r i b u t i o n of the peak factors are p l o t t e d w i t h respect to the t i m e of t y p h o o n d u r a t i o n for t y p h o o n K i n n a a n d is s h o w n in Fig. 7. As is seen in Fig. 7, the peak factors for r e s p o n s e s in b o t h the x- a n d the ),-directions s h o w f l u c t u a t i n g t e n d e n c y a n d large values d u r i n g 120 180rain of the t y p h o o n record. It is to be n o t e d here that the m e a n w i n d v e l o c i t y o b s e r v e d d u r i n g this t i m e i n t e r w d was significantly s m a l l e r t h a n the m e a n w i n d v e l o c i t y o b s e r v e d d u r i n g 40 120 min. It s h o u l d also be n o t e d here that
221
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the time histories of wind velocity and structural response in both the x- and 3,-directions are examined and are shown in Fig. 8a Fig. 8c. Fig. 8a shows a sudden increase in the wind velocity at 130 rain. This increase in the wind velocity is followed by an increase in the structural response in both x- and ),-directions. However, it is also observed in Fig. 8 that even though the wind velocity at around 133 rain and at around 137 rain are larger than that at 130 min, the response corresponding to 133 or 137min is not larger than that at 130min. Also observed is the fact that even though the wind velocity at around 134 min is not extraordinarily large, the response in the ),-direction is considerable larger than at other time. In order to understand this characteristics of the structural response, the distribution of the wind gusts with frequency close to the structural natural frequency was investigated next. It was shown by Yamada and Ohkitani [9] that the kinetic energy contained in the turbulence field within a selected frequency band can be conveniently expressed in terms of the square of the wavelet coefficients. Also, since the first mode vibration of this building was observed to be dominant, the wavelet coefficients for wind velocity and the structuralresponse were calculated around the first natural frequency of thc building, and are shown in Fig. 8d-Fig. 8t". The analyzing wavelet chosen was Meyers wavelet [9] and the frequency band was chosen to be 0.208 0.833 Hz which includes only the first natural frequency of the structure. The occurrence of large values of wavelet coefficients for wind velocity around 130 and 134 rain, in Fig. 8d, indicates the presence of higher magnitude wind gusts with frequency close to the natural frequency of the structure. It is also seen that the wavelet coefficient forwind at 137 min is smaller than that at 130min, therefore, the structural response corresponding to 137min is observed to be smaller than that corresponding to 130rain, even though the wind velocity at 137 min is not smaller than at 130 min. The occurrence of higher values of wavelet coefficients for wind around 135 rain explains the existence of large structural responses around that time. Therefore, it can be concluded here that the structural responses arc caused not only by the large magnitude of wind velocity but also due to the occurrences of wind gusts having frequency close to the structural natural frequency. This is the reason why the peak factors observed were significantly larger even though the mean wind velocity was not observed to be significantly large.
4.3. E[fi, c.tx
(~/at~eraging time on the re.vl)On.se twak.litctop:s"
Fig. 9 shows the average and the maximum peak factors calculated for different averaging times. For comparison, the theoretical average peak factors obtained from Eq. (4), It is seen in Fig. 9 that the average peak factors increase with increase in the averaging time. This indicates that the use of a longer averaging time for adaptation of the control scheme would require the controller not only to be more robust but also to be designed to take care of large values of average peak factors. It is also seen from Fig. 9 that the actual average peak factors, evaluated from the field data, are always higher than the peak factors as derived by considering wind turbulence as
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~:
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Fig. 9. Variation in the response peak factor with averaging time. (a) T y p h o o n Mireille; (b) Typhoon Kinna. (Q, x-direction mean; O, x-direction max.; O, y-direction mean; O, y-direction max. peak factor; - - , peak factor based on stationary assumption).
a stationary random process. This highlights the nonstationary characteristics of the wind turbulence as well as the gust response of the building. Therefore, for the design of any control measure, the consideration of the wind turbulence or the gust response as a stationary random process may not lead to an efficient design.
5. Concluding remarks In the present paper, the nonstationarity in the wind and its effects on the structural response are discussed from the point of view of developing an adaptive active contreol scheme. Based on the analysis of the field data of the wind and the response of a tall building during two typhoons presented herein, the following conclusions could be drawn. (1) The fluctuating wind velocity component as well as the structural response show remarkable nonstationarity. The nonstationarity in the structural response could be due to various factors, e.g. nonstationarity in wind fluctuations due to the change in the mean wind velocity or wind direction as well as due to the concentration of gusts in wind with frequency close to the natural frequency of the structure (2) For an adaptive controller with adaptation scheme based on the changes in the mean wind velocity, the fluctuations in the structural response caused by the change in the wind direction or change in the wind fluctuations must be taken care by adding appropriate robustness to the control scheme (3) The response peak factors obtained are generally larger than the theoretical peak factors obtained by considering wind fluctuations and gust response as a stationary process (4) In general, the use of a longer averaging time for the evaluation of the mean wind velocity results in increased average peak factors. Thus the use of a longer adaptation time for an adaptive controller, with adaptation based on the changes in the mean wind velocity, would require the controller to be of larger capacity and to be more robust
224
R. .,tdhikari, tL }'~mmvuchi ,] ~lTnd En~. lmL
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Acknowledgements The authors are gratefill to Prof. Y. Fujino of University of Tokyo for his valuable suggestions and Mr. T. Wakahara and Mr. K. Shimada, Institute of Technology, Shimizu Corporation, for making the field data of the wind and wind-induced responses available for the present study.
References [I] [2] [3] [4] [511 [6] [7] [8] [9]
E.('.('.Choi, in:3rd Asia-PacilicSynlp. on Wind t!ng., Hong Kong, 1993, pp. 311 316. R.J. Hansen. ,I.W. Reed, E . H V.;illnl~llCkc. J. ,qlrtlcl. Fllg. AS('E 99 (ST7) (1973) 1589 1605. .I. Maeda. J. Wind Eng. 53fl992) 15 231m,lapancscl "f. ramura, K. Shimada. K. Plibi. M. Kawamura. 5' ttitomi, 121h Nalional Syrup. on Wind Eng., 1992, F'art l. pp. 107 112: Parl 2, pp. 113 1lS{in,lapanesel. M B. Priestlcy, Non-linear and Non-stationary Time Series Analysis, Academic Press. New York, 1988. R. Adhikari, H. Yamaguchi, 1st World Conf. on Sir. Conlrol. Los Angeles, vol. 2, 1994. pp. TA443 TA4-52. E. Simiu. R.H, Scallhin, Wind ~l]ccls Oil SIILIC[LII+eS;All Introduction It) Wind F,nginecring, Wiley, New York. 1986. H. Liu. Wind Engineering A Handbook 1iH-Sirticiural t~nginecrs, Preniice-HalL EnglewoodClifl~. NJ. 1991. M. Yamada, K. ()hkitani, Prog. I'hoor. Phy~, N0 14~11991)'799 815.
wi,.Im ,,em,,g ELSEVIER
Journal of Wind Engineering and Industrial Aerodynamics 72 (1997) 213 224
~ ~ i ~
A study on the nonstationarity in wind and wind-induced response of tall buildings for adaptive active control R. A d h i k a r i a'*'l, H. Y a m a g u c h i b'2 ~'Department of AMES, Universi(v o/" Cal!lbrnia at San Diego, 9500 Gilman Drive, La Jolla, CA 92093 0411, USA b Department q[" Civil and Environmental Engineering, Saitama UniversiO', 255 Shimo Ohkuba, Urawa-Shi 338, Japan
Abstract
The nonstationarity in wind and its effects on the response of tall buildings are studied from the point of view of developing an adaptive active control scheme. The wavelet analysis of the wind and the structural response shows that the distribution of wind gusts with frequency close to the structural frequency could be one of the causes of nonstationarity in the structural response. It is also observed that the use of a longer averaging time results in a higher average peak factor for response and thus requiring one to develop a more robust control system, whereas a small averaging time may require frequent updating of the control system which is practically undesirable.
Keywords: Nonstationarity in wind; Wind-induced response; Wavelet analysis
I. Introduction
The wind-induced gust response of tall buildings causes significant discomfort to the building occupants [1,2]. An active control scheme, e.g. based on L Q control or H ~ control scheme, can be developed to maintain comfortable conditions in such buildings. So far, the design of such active control schemes are done by assuming a predefined fixed level of mean wind velocity and then considering wind as a stationary r a n d o m process. As the mean wind velocity during strong storms shows slowly varying tendency [3,4], a controller designed for a fixed mean wind velocity will not be an economical
* Corresponding author. ~Research Associate. Formerly at Department of Civil and Environmental Engineering, Saitama University, 255 Shimo-Ohkubo, Urawa-Shi 338, Japan. 2 Professor. 0167-6105/97/'$17.00 ( : 1997 Elsevier Science B.V. All rights reserved. PI! S0 1 6 7 - 6 1 0 5 ( 9 7 ) 0 0 2 6 1 -4
214
R. Adhikari. I1. }~ a,v" ct " I ~l'imt t-Tnv Iml. A~'rodvn. 72 (1997) 213 224
and effective one. Therefore, it is quite rational to develop a control scheme which adapts itself to the changes in the mean wind velocity or the structural response. It is also to be remembered that if the fluctuating components of the wind velocity show considerable nonstationarity then the structural response may also be expected to be nonstationary and thereby directly governing the extent of robustness required in the control scheme. It is then necessary to know the relation between the mean wind velocity and its fluctuating components and also the effects of such fluctuating components on the structural response so that an appropriate time of adaptation as well as the extent of robustness required in the control algorithm can be assessed. Therefore. the main objective of this paper is lo investigate the nonstationarity associated with the wind and its effect on the structural response, mainly from the point of view of developing a robust adaptive active control scheme. As the averaging time used for the evaluation of the mean wind velocity, based on which adaptation scheme for the active control system is to be developed, will have some effects on decidmg the timing of adaptalion, the effects of changing the averaging time are also discussed in this paper.
2. Robust-adaptive control of wind-induced vibrations The mosl rational way of achieving an adaptive controller is to reflect the input characteristics in the control scheme. However, the realization of such adaptive controller requires the beforehand estimation of the input. The beforehand estimation of the wind force is practically not feasible as it is very difficult to evaluate the wind force acting on the structure. Therefore, instead of using the wind force, the controller can be made adaptive to the changes in the mean wind velocity which is measurable and can be predicted based on the retrospective observations made on the mean wind velocity [5]. The adaptiveness in the control scheme can be achieved by modeling the wind velocity as a piecewise stationary process, as shown in Fig. 1, and thereby incorporating a suitable provision for changing the control action with respect to the changes in the mean wind velocity of such piecewise stationary processes. The amount of fluctuations associated with such mean wind velocity and its effect on the structural response will govern the amount of robustness required in the control scheme. The L"
,,"b,
g
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R. Adhikari, H. Yamaguchi/J. Wind Eng. Ind. Aerodyn. 72 (1997) 213 224
215
expression for the control force of such a robust adaptive controller was derived in Ref. [6] as u = - (SB)-I[SAx
+ SHf(t)]
- sgn(~r(x))r/(t),
(1)
where x is the state vector and matrices A, B and H are represented for a dynamical system expressed in the state-space form, as (2)
£c = A x + B u + H f ( t ) .
In Eq. (1), a(x) is the so-called 'sliding surface,' f(t) is the wind force estimated by predicting the piecewise constant mean wind velocity, whereas r/(t) represents the bounds on the uncertainty in the external excitation and was shown to be given by ~l(t) >~ I S H ( f ( t ) M A X --f(t)RMs)l.
(3)
f(t)Mgx andf(t)aMs of Eq. (3) are to be evaluated based on the maximum and the RMS of the wind velocity, respectively, associated with the predicted mean wind velocity. It is to be noted that the f(t) term in Eq. (1) is related to the adaptive feature of the controller whereas q(t) is related to the robustness of the controller. A schematic diagram of the structural response using such robust-adaptive controller is shown in Fig. 2. It is seen in Fig. 2 that with an adaptive control scheme, it may be possible to control the structural response to a predefined level irrespective of the changes in the wind velocity. Therefore, in order to develop a rational guideline for control adaptation and the amount of robustness required in the control scheme, the prediction of the mean wind velocity or the structural response are indispensable. As the nonstationarity in the wind and the extent of its effects on the structural responses may govern the adaptation scheme and the possible magnitude of robustness required in the control scheme, the nonstationarity in the wind and its effects on the structural responses need to be investigated. In order to investigate the nonstationarity in the wind and its effects on the structural responses, the field data of wind velocity and response of a 100 meter tall buildings during two typhoons, viz. typhoon no. 9119 (typhoon Mireille) and typhoon no. 9117 (typhoon Kinna), were analysed. Time series of wind velocity and the acceleration responses were recorded by three-cup anemometers and accelerometers, Response (A) Uncontrolled\ ,~ Nonadaptive ~. /i constantgain control Adaptively " ~ / ] . /
"
~
'
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~
_
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(U)
Fig. 2. SchematicU A diagram showing the performanceof the robust-adaptive controller in comparison to constant gain nonadaptivecontroller(U, is the triggeringvelocityfor the start of control, and U,,axis the maximum design wind velocityfor control).
210
R -ldhikm'i, HI }~mmA,uchi ,/. llTlulEn~ Ind. -h'rodvn. 72 ¢1997) 213 224
respectively, installed at the top storey of the building. The natural frequency and structural damping ratio of the building along both of the struclural axes were estimated to be 0.625 Hz and 1.0%, respectively. Because of very high wind speed of typhoon Mircille, instrument saturation occurred and. therefore, data recording had to be stopped during 90 1101nin fi:om the start of the data recording. Out of various averaging time intervals used. results of only 1 and 10rain averaging time are presented in most part of the prcscm paper whereas the effect of various averaging times are discussed in Section 5.
3. Characterislies of wind and wind-induced response 3./
('ha/'aclcrLsH{~ ~/ w i n d
Fig. 3 shows the general characteristics of lhc observed wind. In case of typhoon Mireille, as is seen in part lil of Fig. 3a, a sudden change in tile wind direction from ENE to W N W is observed with the passage of the eye of the typhoon, whereas for typhoon Kinna. tile wind direction changes from the predominantly NNE to N N W with the passage of the typhoon. As the dynamic response of the structure is go~crned by the fluctuating components of the wind velocily [7]. the RMS and the maximunl of the wind velocity were examined and are shown in F'ig. 3. Part (ivl of Fig. 3a shows that the RMS of wind velocity for typhoon Mireille increases almost monotonically during 0 60 rain even though the increase in lhe mean wind velocity during that period is not significant. On the other hand. the RMS of the wind velocity is observed to be smaller during 150 200rain than during 0 lo 60 rain even though the mean wind velocity observed in the former case is larger than in the later case, lit) of Fig. 3a. Similar inconsistencies between mean wind velocity and R MS of the wind xelocity are observed in the case of typhoon Kinna also. Thus il is seen that a large mean wind velocity may not always result in large RMS ot" wind velocity. Also, (iii) of Fig. 3a indicates that the maximum wind velocity for typhoon Mireille shows considerable fluctuations during 0 60rain and the same is true for the maxinmrn wind velocity for Kinna during 120 180rain. (iii) of Fig. 3b. This behavior of the wind fluctuations can be related to the changes in the wind directiorls. Therelk)rc, while designing a robust adaptive controller, the uncertainty associated ~xith lhc fluctuating c o m p o n e n l s of wind velocity must also be considered by providing suitable magnitude of robustness to the control scheme.
It is also observed in Fig. 3 that the use of a smaller averaging time results in a rather large fluctuations in both the mean wind velocity and the RMS of the wind velocity, whereas the use of a larger averaging time results in rather smoothly wlrying tendency in both the mean wind velocity and the RMS of the wind velocity', but with higher RMS values. Therefore. if a controller with large robustness can be developed, then a large adaptation time can bc used which is practically desirable while the magnitude of the robustness required will depend on the extent of the effect of the wind fluctuations on the structural response.
R. Adhikari, H. Yamaguchi./~L Wind Eng. Ind. Aerodyn. 72 (1997) 213 224
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(b) Mireille; (b) T y p h o o n
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3.2. Characteristics of response Fig. 4 d e p i c t s the r e s p o n s e o f t h e b u i l d i n g w i t h respect to the s t o r m d u r a t i o n . Since r e s p o n s e in b o t h x- a n d y - d i r e c t i o n s for b o t h of the t y p h o o n s s h o w e d s i m i l a r c h a r a c t e r i s t i c s , o n l y the r e s p o n s e in t h e x - d i r e c t i o n are p l o t t e d in Fig. 4. It is o b s e r v e d in Fig. 4 t h a t t h e s t a t i o n a r i t y o r n o n s t a t i o n a r i t y in the m e a n w i n d v e l o c i t y is d i r e c t l y r e f l e c t e d in t h e s t r u c t u r a l r e s p o n s e . F u r t h e r m o r e , a careful i n s p e c t i o n of the r e s p o n s e for t y p h o o n M i r e i l l e Fig. 4a, also i n d i c a t e s t h a t t h e a c c e l e r a t i o n r e s p o n s e i n c r e a s e s
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rather monotonically from 0 to 60 mm of the typhoon duration even if the mean wind velocity remains ahnost constant during that time, {it) of Fig. 3a. This is because of the monotonic increase in the RMS the wind velocity as shown in {iv} of Fig. 3a. In case of typhoon Kinna also, it is observed in Fig. 4b that even if the mean wind velocity during 120 180rain is smaller than that during 40 120min, the structural response observed during 120 180rain is comparable to that during 40 120min. Therefore, the consideration of only the mean wind velocity for updating the control scheme may not be adequate and some other suitable parameter, e.g. the RMS or the maximum of the wind velocity, should also be identified. However, the effects of the RMS or the maximum of the wind velocity can be taken care by providing robustness to the controller, as given in Eq. [I) and Eq. (3}, and thereby keeping the adaptation scheme based only on the changes m the mean wind velocity. It is also seen in Fig. 4 that a smaller averaging time results in comparatively smaller RMS values of the structural response but with increased fluctuations as compared to the one obtained fiom a longer averaging time. As the robustness and the effectiveness of a controller have trade off tendency among themselves, the smaller RMS values are desirable from control point of view as the robustness required in the control scheme in this case would be smaller, whereas increased fluctuations are undesirable as it would require frequent adaptation in the control scheme which is practically undesirable. 3.3. (7~rre/ation hetween wind and wiml-induced response
When an adaptive controller is to be developed based on tile changes in thc predicted mean wind velocity, not only the relation between the mean wind velocity and the structural response is necessary, but the relation between the fluctuations in the wind and the Ituctuations in the structural response must also be considered in order to estimate the structural response for the predicted mean wind velocity so that an adaptation scheme for control can be developed accordingly and the magnitude of robustness required m the controller can be assessed. It has been shown in many literatures that the RMS of the response of a building or a tower can be conveniently represented as a function of the mean wind velocity with the RMS of the response being proportional to some power of the mean wind velocity [4]. Therefore, the
R. Adhikari, H. Yamaguchi/J. Wind Eng. Ind. Aerodyn. 72 (1997) 213-224
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relation between the fluctuations in the wind, RMS of wind velocity, and the RMS of the response accelerations were checked here and are shown in Fig. 5. In this case also, it was seen that the RMS of the response accelerations along both x- and y-axes showed similar characteristics with respect to the RMS of wind velocity for both of the typhoons. Therefore, only the RMS of the response accelerations along x-axis for both of the typhoons are plotted in Fig. 5. It is seen in Fig. 5a that the RMS of the response accelerations show an overall increasing tendency with increase in the RMS of the wind. In the case of typhoon Kinna, Fig. 5b, the relation between the fluctuating component of wind and the structural response is not very clear even though an increasing tendency in the RMS of response is identified with increase in the RMS of wind velocity.
4. Nonstationarity in wind and wind-induced response 4.1. Relation between response peak factors and mean wind velocity The relation between the mean wind velocity and the response peak factors are shown in Fig. 6. Here, the response peak factor is defined as the ratio of the maximum value of acceleration to its RMS values and is taken as a parameter to represent the fluctuations in the structural response. For comparison, the theoretical average peak factors for resonant response based on a narrow band stationary assumption made both on the wind turbulence as well as structure response are also plotted in Fig. 6. The theoretical values of the average peak factors were calculated by the following equation, originally derived by Davenport [8], Average peak factor = ~
+ 7 / ~ ,
(4)
where n is the natural frequency, T is the averaging time and 7 is the Euler's constant equal to 0.5772. As is seen in Fig. 6a, the peak factors for the case of typhoon Mireille show a decreasing tendency with increase in the mean wind velocity in the range upto
220
R. Adhikari, H. )2mlaguchi,'d klTnd Eng. lml. Aerodvn. 72 ¢'199D 213 224
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a.,
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Fig. 6. Relation belween response peak lhclors and the mean wind velocity. (aJ Typhoon Mireille: {b) Typhoon Kinna. I@. 1 rain average; O, 10rain average: - -, I rain average and stationary assumption: • 10 rain average and stationary assumptionl.
20 25 m/s. B e y o n d this range, the increase in the m e a n w i n d v e l o c i t y d o e s not affect m u c h on the peak factors a l o n g tile y-axis, w h e r e a s the p e a k factors a l o n g the x - a x i s s h o w a d e c r e a s i n g t e n d e n c y with i n c r e a s e in the m e a n w i n d velocity. It is to be n o t e d here that in case of t y p h o o n Mireille. the v-axis r e p r e s e n t s close to the a l o n g - w i n d d i r e c t i o n a n d x-axis r e p r e s e n t s the a c r o s s wind d i r e c t i o n . H o w e v e r , in case of t y p h o o n K i n n a , as seen in Fig. 6b, the p e a k factors s h o w a d e c r e a s i n g t e n d e n c y with i n c r e a s e in the m e a n w i n d v e l o c i t y w h e n the a v e r a g i n g t i m e is 1 0 m i n , but w h e n the a v e r a g i n g t i m e is set to be I min, the p e a k factors are not affected by the increase in the m e a n w i n d velocity• It is o b s e r v e d in Fig. 6 that w h e n the a v e r a g i n g t i m e is set to be 1 0 m i n , very large values of p e a k factors, as c o m p a r e d to the t h e o r e t i c a l values of the p e a k factors, can o c c u r in the real case, thus h i g h l i g h t i n g the n o n s t a t i o n a r y c h a r a c t e r i s t i c s of the w i n d t u r b u l e n c e a n d the gust r e s p o n s e of the structure. T h e o c c u r r e n c e of such large p e a k factors is s o m e w h a t s t r a n g e a n d the p r o b a b l e c a u s e of the o c c u r r e n c e of such high p e a k factors is discussed next. 4.2. C a u s e q l n o n s l a t i o n a r i t 3' in s l r u c t u r a / r e s p o n s e
el wavelet a p p r o a c h
In o r d e r to locate the instant of o c c u r r e n c e of the large p e a k factors, the d i s t r i b u t i o n of the peak factors are p l o t t e d w i t h respect to the t i m e of t y p h o o n d u r a t i o n for t y p h o o n K i n n a a n d is s h o w n in Fig. 7. As is seen in Fig. 7, the peak factors for r e s p o n s e s in b o t h the x- a n d the ),-directions s h o w f l u c t u a t i n g t e n d e n c y a n d large values d u r i n g 120 180rain of the t y p h o o n record. It is to be n o t e d here that the m e a n w i n d v e l o c i t y o b s e r v e d d u r i n g this t i m e i n t e r w d was significantly s m a l l e r t h a n the m e a n w i n d v e l o c i t y o b s e r v e d d u r i n g 40 120 min. It s h o u l d also be n o t e d here that
221
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Fig. 8. Time histories and wavelet coefficients of wind velocity and structural response; (a) wind velocity; (b) acceleration in x-direction; (c) acceleration in y-direction; (d) wavelet coefficients of wind velocity; (e) wavelet coefficients of acceleration in x-direction and (13 wavelet coefficients of acceleration in y-direction. t h o u g h the R M S o f the w i n d v e l o c i t y d u r i n g 1 2 0 - 1 8 0 m i n w a s o b s e r v e d to be a l m o s t t w o t i m e s l a r g e r t h a n the R M S of w i n d v e l o c i t y d u r i n g 4 0 - 1 2 0 r a i n , the m a x i m u m of t h e w i n d v e l o c i t y w a s n o t o b s e r v e d to be l a r g e r t h a n t h a t d u r i n g 4 0 - 1 2 0 m i n (see Fig. 3b). In o r d e r to i n v e s t i g a t e the o c c u r r e n c e o f s u c h l a r g e v a l u e s of t h e p e a k f a c t o r s
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the time histories of wind velocity and structural response in both the x- and 3,-directions are examined and are shown in Fig. 8a Fig. 8c. Fig. 8a shows a sudden increase in the wind velocity at 130 rain. This increase in the wind velocity is followed by an increase in the structural response in both x- and ),-directions. However, it is also observed in Fig. 8 that even though the wind velocity at around 133 rain and at around 137 rain are larger than that at 130 min, the response corresponding to 133 or 137min is not larger than that at 130min. Also observed is the fact that even though the wind velocity at around 134 min is not extraordinarily large, the response in the ),-direction is considerable larger than at other time. In order to understand this characteristics of the structural response, the distribution of the wind gusts with frequency close to the structural natural frequency was investigated next. It was shown by Yamada and Ohkitani [9] that the kinetic energy contained in the turbulence field within a selected frequency band can be conveniently expressed in terms of the square of the wavelet coefficients. Also, since the first mode vibration of this building was observed to be dominant, the wavelet coefficients for wind velocity and the structuralresponse were calculated around the first natural frequency of thc building, and are shown in Fig. 8d-Fig. 8t". The analyzing wavelet chosen was Meyers wavelet [9] and the frequency band was chosen to be 0.208 0.833 Hz which includes only the first natural frequency of the structure. The occurrence of large values of wavelet coefficients for wind velocity around 130 and 134 rain, in Fig. 8d, indicates the presence of higher magnitude wind gusts with frequency close to the natural frequency of the structure. It is also seen that the wavelet coefficient forwind at 137 min is smaller than that at 130min, therefore, the structural response corresponding to 137min is observed to be smaller than that corresponding to 130rain, even though the wind velocity at 137 min is not smaller than at 130 min. The occurrence of higher values of wavelet coefficients for wind around 135 rain explains the existence of large structural responses around that time. Therefore, it can be concluded here that the structural responses arc caused not only by the large magnitude of wind velocity but also due to the occurrences of wind gusts having frequency close to the structural natural frequency. This is the reason why the peak factors observed were significantly larger even though the mean wind velocity was not observed to be significantly large.
4.3. E[fi, c.tx
(~/at~eraging time on the re.vl)On.se twak.litctop:s"
Fig. 9 shows the average and the maximum peak factors calculated for different averaging times. For comparison, the theoretical average peak factors obtained from Eq. (4), It is seen in Fig. 9 that the average peak factors increase with increase in the averaging time. This indicates that the use of a longer averaging time for adaptation of the control scheme would require the controller not only to be more robust but also to be designed to take care of large values of average peak factors. It is also seen from Fig. 9 that the actual average peak factors, evaluated from the field data, are always higher than the peak factors as derived by considering wind turbulence as
R. Adhikari, H. Yamaguchi/J. Wind Eng. Ind. Aerodyn. 72 (1997) 213 224 12[ ........
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, ....
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:
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................... .k ,~," .~
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$ ............. i ............
~.............. . ~:. . . ~. . ~....
~:
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0
5
10 15 20 Averaging time (rain)
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Fig. 9. Variation in the response peak factor with averaging time. (a) T y p h o o n Mireille; (b) Typhoon Kinna. (Q, x-direction mean; O, x-direction max.; O, y-direction mean; O, y-direction max. peak factor; - - , peak factor based on stationary assumption).
a stationary random process. This highlights the nonstationary characteristics of the wind turbulence as well as the gust response of the building. Therefore, for the design of any control measure, the consideration of the wind turbulence or the gust response as a stationary random process may not lead to an efficient design.
5. Concluding remarks In the present paper, the nonstationarity in the wind and its effects on the structural response are discussed from the point of view of developing an adaptive active contreol scheme. Based on the analysis of the field data of the wind and the response of a tall building during two typhoons presented herein, the following conclusions could be drawn. (1) The fluctuating wind velocity component as well as the structural response show remarkable nonstationarity. The nonstationarity in the structural response could be due to various factors, e.g. nonstationarity in wind fluctuations due to the change in the mean wind velocity or wind direction as well as due to the concentration of gusts in wind with frequency close to the natural frequency of the structure (2) For an adaptive controller with adaptation scheme based on the changes in the mean wind velocity, the fluctuations in the structural response caused by the change in the wind direction or change in the wind fluctuations must be taken care by adding appropriate robustness to the control scheme (3) The response peak factors obtained are generally larger than the theoretical peak factors obtained by considering wind fluctuations and gust response as a stationary process (4) In general, the use of a longer averaging time for the evaluation of the mean wind velocity results in increased average peak factors. Thus the use of a longer adaptation time for an adaptive controller, with adaptation based on the changes in the mean wind velocity, would require the controller to be of larger capacity and to be more robust
224
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Acknowledgements The authors are gratefill to Prof. Y. Fujino of University of Tokyo for his valuable suggestions and Mr. T. Wakahara and Mr. K. Shimada, Institute of Technology, Shimizu Corporation, for making the field data of the wind and wind-induced responses available for the present study.
References [I] [2] [3] [4] [511 [6] [7] [8] [9]
E.('.('.Choi, in:3rd Asia-PacilicSynlp. on Wind t!ng., Hong Kong, 1993, pp. 311 316. R.J. Hansen. ,I.W. Reed, E . H V.;illnl~llCkc. J. ,qlrtlcl. Fllg. AS('E 99 (ST7) (1973) 1589 1605. .I. Maeda. J. Wind Eng. 53fl992) 15 231m,lapancscl "f. ramura, K. Shimada. K. Plibi. M. Kawamura. 5' ttitomi, 121h Nalional Syrup. on Wind Eng., 1992, F'art l. pp. 107 112: Parl 2, pp. 113 1lS{in,lapanesel. M B. Priestlcy, Non-linear and Non-stationary Time Series Analysis, Academic Press. New York, 1988. R. Adhikari, H. Yamaguchi, 1st World Conf. on Sir. Conlrol. Los Angeles, vol. 2, 1994. pp. TA443 TA4-52. E. Simiu. R.H, Scallhin, Wind ~l]ccls Oil SIILIC[LII+eS;All Introduction It) Wind F,nginecring, Wiley, New York. 1986. H. Liu. Wind Engineering A Handbook 1iH-Sirticiural t~nginecrs, Preniice-HalL EnglewoodClifl~. NJ. 1991. M. Yamada, K. ()hkitani, Prog. I'hoor. Phy~, N0 14~11991)'799 815.