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2 Eperimental dnamic analsis o a steel mast ecited b wind load Bart Peeters Department o Ciil Engineering, Katholieke Uniersiteit Leuen, Belgium Guido De Roeck Department o Ciil Engineering, Katholieke Uniersiteit Leuen, Belgium ABSTRACT: This paper discusses the eperimental determination o the modal parameters o a steel transmitter mast. Wind is the most important loading condition in the design o such a structure. The mast should sustain the load without ecessie stresses but moreoer in order to preent malunctioning o the antennae, the rotation at the top has to be limited. Because o the time dependent character o the wind, the response is inherent dnamic and susceptible to resonance eects. The dnamic magniication at resonance is largel dependent on the damping ratios o the lower ibration modes. Thereore a ibration eperiment was perormed on a steel transmitter mast in order to determine these damping ratios. Since it is impossible to measure eactl the dnamic wind load, onl response measurements were recorded. A noel implementation o the stochastic subspace identiication method is used to etract the modal parameters rom this output-onl data. INTRODUCTION In the design process o a steel transmitter mast, the damping ratios o the lower modes are important actors. The wind turbulence spectrum has a peak alue at a er low requenc around.4 H (Balendra 993). All eigenrequencies o the considered structure are situated at the descending part o the turbulence power spectrum, and thus in act onl the ew lower modes o ibration are important or determining the structure s response to dnamic wind load. The structure under consideration is a steel rame structure with antennae attached to the top. In order to preent malunctioning o the antennae, the rotation at the top has to be limited to E. Onl once in ears, this alue ma be eceeded. The dnamic response (and thus the rotation angle) o a structure reaches its maimum at resonance, where the amplitude is inersel proportional to the damping ratio. So the damping is directl related to the maimum rotation angle. A high damping ratio means that the amount o steel needed to meet the speciication o limited rotation can be reduced. The onl wa to determine the true damping ratios is b perorming a ibration test on the structure. Such a test does not onl ield the damping ratios, but also the eigenrequencies and the mode shapes at the sensor locations. This allows to alidate and eentuall update a inite element model o the structure. A practical wa to ecite the mast is using the wind. Howeer this ecitation cannot be measured eactl and the traditional modal parameter estimation methods are ecluded. Classicall input and output measurements are irst conerted into requenc response unctions or impulse response unctions and the modal parameters are then etracted rom these unctions. When using ambient ecitation (e.g. traic on bridges, waes on oshore platorms, wind on towers,...) the detailed knowledge o the input is replaced b a white noise assumption. Onl output measurements are aailable to determine the modal parameters. The most widel used method in ciil engineering to determine the eigenrequencies o a structure based on output-onl measurements is the rather simple peakpicking method, where the eigenrequencies are simpl determined as the peaks o the power spectra. One practical implementation o this method was realied b Felber (993). The major adantage o the method is its speed. Disadantages are the subjectie selection o eigenrequencies, the lack o accurate damping estimates and the determination o operational delection shapes in stead o mode shapes, since no modal model is itted to the data. Thereore we are looking or more adanced methods. The stochastic subspace identiication method (SSI) is such a method (Van Oerschee & De Moor 996). It identiies a stochastic state space model directl rom the output-onl measurements without the need to conert the time signals to correlations or spectra. The state space model is a er general model that is also suitable or our purposes: it can describe a linear

3 ibrating structure ecited b white noise. Subspace identiication does not inole an nonlinear computations and is thereore much aster and more robust than the prediction error method (Ljung 987) that tries to identi an ARMAV-model. The ke step o SSI is the projection o the row space o the uture outputs into the row space o the past outputs. In this paper a noel implementation o SSI is used, the socalled reerence based stochastic subspace identiication: SSI/re (Peeters & De Roeck 999). The idea is now to take instead o all past outputs onl a set o well-chosen reerence sensors. This reduces the dimensions o the inoled matrices and also the computation time, without haing a negatie inluence on the qualit o the estimated modal parameters. DATA ACQUISITION. Structure Figure represents the bottom and top o the mast. A tpical cross section is gien in Figure. The mast has a triangular cross section consisting o 3 circular hollow section proiles o which the section and the thickness decrease rom bottom to top. The 3 main tubes are connected with smaller tubes orming the diagonal and horiontal members o the truss structure. The structure is composed o segments o 6 m, reaching a height o 3 m. At the top in the centroid o the section an additional tube rises aboe the truss structure resulting in a total height o 38 m. A ladder is attached to one side o the triangle. Together with the diagonals, this ladder is disturbing somewhat the smmetr o the structure. Further, the mast is ounded on a thick concrete slab supported b three piles. A irst test was carried out on 4 Februar 997 (Peeters & De Roeck 998). The obtained damping ratios were er low:.-.%. Howeer at that time the transmitter equipment (the antennae) was not et placed. Thereore a new test was perormed on 6 March 998. The sectorial antennae or a cellular phone network, situated at a height o about 33 m, are epected to hae an important inluence on the dnamics o the structure. The additional mass (+%) is considerable and it is located at a place where large displacements occur.. Measurements The measurement grid or the dnamic test consisted o 3 points: eer 6 m, rom to 3 m, 3 horiontal accelerations were measured. Their measurement direction are denoted on Figure b H, H, H3. Assuming that the triangular cross section remains undeormed during the test, the 3 measured accelerations are suicient to describe the complete horiontal moement o the considered section. At ground leel ( m) also 3 ertical accelerations were Figure. Bottom and top o mast. Figure. Tpical cross section o mast with accelerometer positions (H, H, H3). measured in order to hae a complete description o all displacement components o the oundation. Another dierence as opposed to the irst test was that supplementar sensors were installed on the central tube at 33 m. These sensors measuring in both horiontal directions allow a better characteriation o the mode shapes. Due to the limited number o acquisition channels and high sensitiit accelerometers, the measurement grid o 3 sensor positions was split up in 4 setups. In output-onl modal analsis where the input orce remains unknown and ma ar between the setups, the dierent measurement series can onl be linked i there are some sensors in common. The three sensors at 3 m are suited as reerences since it is not epected that these are situated at a node o an mode shape.

4 PSD [(m/sec ) /H] [H] PSD [(m/sec ) /H] [H] Figure 3. Comparison o power spectral densities o the signal measured at 3 m in -direction (Fig. ). Top: irst test, ; bottom: second test, The cut-o requenc o the anti-aliasing ilter was set at H. The data was sampled at H. A total o 37 samples was acquired or each channel, resulting in a measurement time o about min or each setup. Figure 3 represents the power spectral densities o the signals measured at the same location in both tests. Comparing these spectra alread reeals that the dnamic behaiour o the mast changed quite drasticall due to the added eccentric mass o the antennae. 3 SYSTEM IDENTIFICATION 3. Reerence based stochastic subspace identiication It is beond the scope o this paper to eplain in ull detail the stochastic subspace identiication method or the reerence based ariant. The interested reader is reerred to literature: Van Oerschee & De Moor (996) and Peeters & De Roeck (999). Here onl the main ideas behind the method are gien. The method assumes that the dnamic behaiour o a structure ecited b white noise can be described b a stochastic state space model (). A justiication o this statement can or instance be ound in Peeters & De Roeck (998). k+ = A k + w k k = C k + k () k ß l are the measurements o l outputs at discrete time instant k. k ß n is the state ector. The number o elements o the state space ector n is twice the number o ecited modes o the structure. The modes occur in comple conjugated pairs. w k ß n is the process noise due to disturbances and modelling inaccuracies; k ß l is the measurement noise due to sensor inaccurac. The are both unmeasurable ector signals assumed to be ero mean, white and stationar. Aß n n is the state matri, describing the dnamics o the sstem as characteried b its eigenalues. Cß l n is the output matri, translating the internal state o the sstem into obserations. When appling subspace identiication or the estimation o the modal parameters ollowing tools rom linear algebra are successiel applied: QRactoriation, singular alue decomposition (SVD), least squares and eigenalue decomposition. One starts with the output measurements. A data reduction is obtained b QR. SVD is used to reject the noise, assumed to be represented b the higher singular alues. The state space matrices are ound b least squares and the modal parameters (eigenrequencies, damping ratio s and mode shapes) are obtained rom the state matrices b an eigenalue decomposition. The ke step o stochastic subspace is the projection o the row space o the uture outputs into the row space o the past outputs, computed using QR. The idea o the reerence based stochastic subspace method is to take instead o all past outputs onl a set o wellchosen reerence sensors. This reduces the dimensions o the inoled matrices and also the computation time. In modal analsis applications oten a lot o sensors are used. The hae a certain spatial distribution oer the structure, leading to signals o dierent qualit. Some sensors are located at nodal points o a mode shape and others ma be located at points close to ied boundaries: e.g. or most ciil engineering structures it is generall impossible to obtain a ree-ree setup. Since the number o reerences is limited, their qualit is important: all

5 model order [ ] 3 d dd dd d d dd d d d d d d d model order [ ] 3 d d d d [H].... [H] Figure 4. Stabilisation diagrams: requenc s. model order. The criteria are % or requencies, % or damping and % or ectors. Let: complete requenc range; right: oomed around irst two (closel spaced) modes. r, stable pole;., stable req. and ector;.d, stable req. and damping;., stable req. The sum o the power spectral densities o all channels is also represented on the graph. Table. Estimated eigenrequencies and damping ratios: aerages, and standard deiations are represented in the table: and Mode number ( ) [H].487 (.3).49 (.).93 (.3).976 (.) (.) 7. (.) (.7) ( ) [%]. (.).4 (.3). (.).8 (.7).4 (.3).3 (.7).9 (.8) ( ) [H].7 (.).79 (.).93 (.4).6 (.).7 (.) (.3) 4.68 (.4) ( ) [%]. (.).7 (.).7 (.).3 (.).7 (.). (.). (.), based on 8 samples. Both tests modes must be present in the data measured b the reerences. I the best sensors are selected as reerences, no loss o identiication qualit is epected. On the contrar, because the lower qualit sensors are partiall omitted, the identiication results ma be more accurate. The reason wh the projection is not limited to the reerence uture outputs too, is that in this case one would obtain mode shapes amplitudes at the reerence sensors onl, whereas one is interested in the mode shapes at all measured locations. 3. Identiication in practice The stochastic subspace identiication method and some preprocessing and isualiation tools were implemented as a graphical user interace (Laquière et al. 998) or Matlab (Mathworks 998). Beore identiication the data was decimated with actor 8: it was iltered through a digital low-pass ilter with a cuto requenc o H and resampled at. H. This operation reduces the number o data points to 384 and makes the identiication more accurate in the considered requenc range - H. The higher modes (- H) can be identiied in a separate identiication procedure without decimating and using onl a limited time rame. In theor, the model order n (twice the number o identiied modes) equals the number o nonero singular alues ater appling SVD as one o the intermediate subspace steps (section 3.). Howeer in practice due to noise (modelling inaccuracies, measurement noise and computational noise) the higher singular alues are not eactl ero, and the model order can or instance be ound b looking at the maimal distance between two successie singular alues. Further, i the aim is to obtain modal parameters, it is better to tr man dierent model orders. Eer order ields a state space model () and corresponding modal parameters. All these modal parameters can be plotted in stabilisation diagrams allowing to distinguish the spurious modes rom the phsical ones. Figure 4 shows such a stabilisation diagram. When ooming around.7 H, it is clear that the subspace algorithm ound modes. With its closel spaced modes, the mast is a tpical eample where the peak-picking method (section ) would ail. The spectra onl show one peak around.7 H, whereas there are modes.

6 =.7H, =.%. 3. =.9H, =.7%.. =.7H, =.%. 7. =4.63H, =.%. 9. =6.8H, =.% =6.9H, =.%.. =8.9H, =.%. 3. =9.6H, =.%. 7. =.8H, =.3%. 8. =3.H, =.3%. Figure. Some representatie mode shapes, obtained b appling the reerence based stochastic subspace method to the output data. 4 IDENTIFICATION RESULTS Rather than tring to ind one order and related state space model where all modes are stable, dierent orders are selected to determine the modal parameters. There are 4 setups and eer setup was measured twice. So, there are 8 estimates or eer eigenrequenc and damping ratio. The mean alues and standard deiations are represented in Table. Also the results o the irst test (997--4) are represented in Table. Unortunatel there is no statistical inormation present or mode shapes, since 4 setups ield onl mode shape estimate. The uncertainties on the eigenrequencies are etremel low. Due to the increased mass the eigenrequencies are lower in the second test. As usual, the damping

7 CONCLUSIONS PSD [(m/sec ) /H] [H] Figure 6. Comparison o spectrum estimates. º, stochastic subspace;, Welch s. ratios are more uncertain. Howeer, it seems that placing the antennae at the top had a positie inluence on the damping ratios, in the sense that the are somewhat higher or the lower modes. But still the are er low. Because the antennae are in an eccentric position the bending-torsion behaiour changed and it was diicult to pair the mode shapes beore and ater placing the antennae. Some representatie mode shapes are shown in Figure. For instance the third mode is a new mode which was not present in the irst test. From the stochastic state space model () an analtical epression or the power and cross spectra between the outputs can be obtained. These spectra can be compared with spectra rom a nonparametric identiication method: e.g. Welch s aeraged periodogram method based on appling discrete Fourier transorms to the data. In Figure 6 such a comparison is made, indicating that subspace identiication was able to identi a model that closel its the measurements. From the identiied model () it is also possible to compute the contributions o each mode to the total response. In Figure 7 the iltered measurements and the contributions o the seen modes present in the considered requenc range are represented. A ibration eperiment was perormed on a steel transmitter mast to estimate the modal parameters. The eigenrequencies, damping ratios and mode shapes were successull etracted rom the acceleration measurements using the reerence based stochastic subspace method. Connecting the sectorial antennae at the top had a drastic eect on the dnamical behaiour o the mast. The obtained damping ratios are er low and een lower than the damping ratios or steel structures that can be ound in design codes. REFERENCES Balendra, T Vibration o buildings to wind and earthquake loads. London: Springer-Verlag. Felber, A.J Deelopment o a hbrid bridge ealuation sstem. PhD thesis, Uniersit o British Columbia, Canada. Laquiere, A., B. Van den Branden, B. Peeters & G. De Roeck 998. Introduction to MACEC, modal analsis o ciil engineering constructions, a toolbo or use with Matlab. Leuen: Department o Ciil Engineering, K.U.Leuen. Ljung, L Sstem identiication: theor or the user. Englewood Clis, New Jerse: Prentice Hall. Mathworks, The 998. Using Matlab, ersion.. Peeters, B. & G. De Roeck 998. Stochastic subspace sstem identiication o a steel transmitter mast. In Proceedings o IMAC 6, the International Modal Analsis Conerence, Santa Barbara, USA, 3- Februar 998: Bethel: SEM. Peeters, B. & G. De Roeck 999. Reerence based stochastic subspace identiication in ciil engineering. In Proceedings o the nd International Conerence on Identiication in Engineering Sstems, Swansea, UK, 9-3 March 999. Swansea: Uniersit o Wales. Van Oerschee, P. & B. De Moor 996. Subspace identiication or linear sstems: theor - implementation - applications. Dordrecht: Kluwer Academic Publishers..*total mode mode mode 3 mode 4 mode mode 6 mode t [sec] Figure 7. Modal contributions to the total response. The top chart is the measured data; the contributions rom the irst 7 modes are ordered rom top to bottom. The amplitudes o the measured data hae been multiplied b..