AEROELASTIC GALLOPING OF TALL STRUCTURES IN SIMULATED WINDS by PETER P. SULLIVAN B.S., Clrad State University Frt Cllins, Clrad A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department f Mechanical Engineering) We accept this thesis as cnfirming t the required standard THE UNIVERSITY OF BRITISH COLUMBIA JUNE, 1977 (c) Peter P. Sullivan, 1977
In presenting this thesis in partial fulfillment f the requirements fr an advanced degree at the University f British Clumbia, I agree that the Library shall make it freely available fr reference and study. I further agree that permissin fr extensive cpying f this thesis fr schlarly purpses may be granted by the Head f my Department r by his representatives. It is understd that cpying r publicatin f this thesis fr financial gain shall nt be allwed withut my written permissin. Department f Mechanical Engineering The University f British Clumbia 2075 Wesbrk Place Vancuver, Canada V6T 1W5 Date
i i ABSTRACT This thesis studies the effects f mdel aspect rati n the static frces and gallping vibratins f bluff shapes when expsed t a turbulent bundary layer similar t the atmsphere. Previus investigatins have analyzed the gallping scillatins f finite prismatic bdies expsed t a turbulent shear flw n the basis f the quasisteady thery and the assumptin f an average lateral frce. Herein cnsideratin is given t the variatin f lateral frces with height and the gallping scillatins f tw finite square twers are predicted. The turbulent bundary layer was grwn ver a lng fetch f rughness and at the lcatin f the static and dynamic tests was 28" deep and had prperties similar t a suburban r frested full scale expsure. The gemetric scale f the mdels fund frm an analysis f velcity spectra was abut 1/500. Fr the height t width ratis examined, aspect rati had little effect n the average static frces fr small angles f attack. The lcal static frces, measured frm the pressure distributin, had a wide variatin ver the height f the mdel. Fr the finite sectins examined the respnse predicted frm the lcal frces gave higher amplitudes fr the same reduced velcity as cmpared, t the respnse fund frm the average frces. The results f the dynamic tests agreed with the gallping respnse predicted frm the lcal sectinal frces indicating that the three-dimensinal effects are imprtant in the cnsideratin f the gallping phenmenn. The measurements f velcity spectra in the wake f the rigid
iii 28" mdel indicate that the Struhal shedding frequency varies alng the span f the mdel. Similar velcity spectra behind the gallping 28" mdel did nt exhibit a discernible peak at the statinary value f the Struhal number.
PAGES iv, v DO NOTEXIST.
vi TABLE OF CONTENTS Page ABSTRACT... ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE x i i i viii ix xi CHAPTER 1 INTRODUCTION. 1.1 Backgrund 1.2 Purpse 2 THEORY 2.1 Quasi-steady assumptin 2.2 Energy apprach 3 DESCRIPTION OF EXPERIMENTS AND APPARATUS 3.1 Outline f experiments cnducted 3.2 Wind tunnel 3.3 Velcity measurements 3.4 Static mdels 3.5 Frce measurements 3.6 Pressure measurements 3.7 Elastic mdels and munting 3.8 Deflectin measurements and calibratin 3.9 Damping measurements 3.10 Frequency measurements and density calculatin 4 RESULTS AND DISCUSSION.. 4.1 Velcity measurements 4.2 Average frce measurements 4.2.1 Drag cefficient 4.2.2 Lift cefficient 4.2.3 Lateral frce cefficient
vii CHAPTER Page 4.3 Lcal lateral frce cefficient 4.4 Theretical gallping respnse 4.4.1 Respnse using average lateral frce cefficients 4.4.2 Respnse using lcal lateral frce cefficients 4.5 Respnse measurements 4.6 Wake measurements 5 SOME EFFECTS OF ASPECT RATIO ON THE FORCE COEFFICIENTS.. 60 5.1 ' First experiment 5.2 Secnd experiment 5 CONCLUSIONS 68 6.1 Full scale interpretatin f mdel results 6.2 Cnclusins BIBLIOGRAPHY..... APPENDIX 1. APPENDIX 2
viii LIST OF TABLES TABLE Page I Summary f mdel prperties fr the dynamic tests.... 29 II 69 III Average aerdynamic cnstants fr the 28" and 20" mdels 77 IV Matrix f aerdynamic cnstants fr the 28" mdel.... 77 V Matrix f aerdynamic cnstants fr the 20" mdel.... 78
ix LIST OF FIGURES FIGURE Page 1 Finite vertical structure vibrating in a turbulent bundary layer flw 6 2 Elastic 20" mdel and upstream surface rughness.... 15 3 Test mdels (left t right 20" and 28" elastic mdels and variable height static mdel) 17 4 Dynamic balance and elastic test mdel 20 5 Typical calibratin curve fr accelermeter utput versus mdel deflectin 24 6 Typical decay plt fr damping calibratin 26 7 Elastic mdel and munting rig fr dynamic tests.... 28 8 Variatin f velcity with height in the bundary layer. 31 9 Variatin f turbulence intensity with height in the bundary layer 32 10 Pwer spectrum f the lngitudinal velcity cmpnent.. 34 11 Variatin f average drag cefficient with angle f attack fr fur aspect ratis.. 36 12 Variatin f average lift cefficient with angle f attack fr fur aspect ratis 38 13 Variatin f average lateral frce cefficient with tan a fr fur aspect ratis. 40 14 Effect f turbulence intensity n C fr square sectin ref. (4) Y 41 15 Variatin f lcal lateral frce cefficient alng the span f the 28" mdel 43 16 Variatin f lcal lateral frce cefficient alng the span f the 20" mdel 44 17 Gallping respnse calculated frm the average lateral frce cefficients 47
X FIGURE Page 18 Cmparisn f theretical and experimental gallping amplitudes 49 19 Cmparisn f theretical and experimental gallping amplitudes 50 20 Effect f mde shape n the theretical gallping respnse cmputed frm the lcal lateral frce cefficients... 52 21 Pwer spectrum f the lngitudinal velcity fluctuatins in the wake f the rigidly munted 28" mdel 55 22 Pwer spectrum f the lngitudinal velcity fluctuatins in the wake f the gallping 28" mdel 58 23 Variatin f average lift cefficient with angle f attack fr three mdels f high aspect rati 61 24 Effect f aspect rati n the drag cefficient f square prisms in smth flw. 64 25 Effect f aspect rati n the lift cefficient f square prisms in smth flw...... 65 26 Effect f aspect rati n the lateral frce cefficient f square prisms in smth flw 66
xi NOMENCLATURE tip amplitude f the mdel - dimensinless amplitude = a/h - dimensinless amplitude = a n/3 - average aerdynamic cnstants - matrix f aerdynamic cnstants incrprating lcal changes - lngitudinal dimensin f the mdel - numerical cefficients - viscus structural damping - functin invlving the velcity prfile and mde shape f the structure average drag cefficient - lcal lateral frce cefficient - average lateral frce cefficient - average l i f t cefficient - pressure cefficient n the upper surface f the mdel pressure cefficient n the lwer surface f the mdel - distance frm the base f the mdel t the pint f rtatin - ttal drag frce - functin invlving the velcity prfile, mde shape and aerdynamic cnstants A.. - natural frequency f the mdel lcal lateral frce - ttal lateral frce - lateral dimensin f the mdel inertia f the rtating assembly abut the axis f rtatin spring, stiffness length f the mdel - ttal l i f t frce - turbulent length scale f the lngitudinal velcity cmpnent frequency f velcity fluctuatin
xii P - pressure n the lwer surface f the mdel PJJ - pressure n the upper surface f the mdel R g - Reynlds number S - Struhal number = nh/v S(n) - pwer spectral density f the lngitudinal velcity cmpnent t - time V, - reduced velcity at the height f the mdel = /u) n h U - dimensinless velcity = V^/fS. V - lcal mean velcity - mean velcity at the height f the mdel V. rel - relative mean velcity seen by the vibrating mdel v(z) - functin describing the velcity variatin with height v 1 - ttal RMS velcity fluctuatin f the lngitudinal velcity cmpnent W - ttal wrk dne by the damping frces x - alng-wind directin y - crss-wind directin y - velcity f vibrating mdel y n - mde shape f the structure z - vertical directin z_ - distance frm the pint f rtatin t the pint f attachs ment f the springs a - angle f attack Q 3 - fractin f critical damping = ct 6 - bundary layer thickness Y - pwer law expnent n - mass parameter fr a square sectin = pmp m 9 - angular rtatin f the mdel p - density f the fluid - density f the mdel u>_ - natural circular frequency
ACKNOWLEDGEMENTS The authr wishes t thank Dr. G.V. Parkinsn whse amiable attitude made this wrk a pleasant experience. Sincere thanks must g t Bb Strachan whse expertise in cmputer science made the acquisitin f large amunts f data a bearable experience. The authr is mst grateful t the numerus graduate students wh assisted in this wrk.
1 CHAPTER 1 INTRODUCTION 1.1 Backgrund An elastlcally munted structure may vibrate when expsed t a fluid flw. The causes f such mtin may be randm buffeting by turbulence r they can be cherent instabilities arising frm the interactin between the structure and the wind. The latter instabilities are usually scillatry and are caused by the separating shear layers frm an aerdynamically bluff shape. One such instability results frm the tw separated shear layers which are unstable and rll up t frm discrete vrtices which result in an scillatry pressure lading n the afterbdy f the structure. When this peridic lading ccurs at a frequency clse t the natural frequency f the structure a resnant vibratin can ccur. The resnant vibratins are termed vrtex-induced and are characterized by displacements f the rder f the width f the structure. They can nly ccur ver a discrete wind speed range defined by the Struhal number. A secnd class f scillatry instabilities is termed gallping. Gallping is typically a lw-frequency high amplitude mtin in a single uncupled mde f vibratin in a plane perpendicular t the wind directin. Self-excited scillatins f the gallping type are caused by the aerdynamic instability f the crss-sectin f the bdy s that the mtin generates frces which increase the initial amplitudes. A cntinuus increase in steady-state amplitudes with increasing wind speed is characteristic f a gallping phenmenn.
2 There have been numerus studies made f gallping instability. Smith (1) investigated extensively the gallping mechanism f a twdimensinal square prism in smth flw. Using Parkinsn's (2) quasisteady assumptin f frces excellent agreement was fund between experiment and thery fr the square sectin. Later Santsham (3) under similar test cnditins t Smith's shwed that the same quasi-steady apprach culd be applied t the 2/1 rectangle under the cnditin that the nset velcity fr gallping is much higher than the velcity at which vrtex-induced resnance ccurs. Laneville (4) investigated the effects f turbulence intensity and scale n the nature f gallping scillatins. This study f tw-dimensinal rectangular cylinders shws the quite surprising result that an increasing turbulence intensity can cmpletely change the stability characteristics f a sectin. Higher levels f turbulence made thse sectins which behave as sft scillatrs in smth flw mre stable in a turbulent stream. An ppsite trend was bserved fr thse sectins which are stable at rest in smth flw, i.e. they became mre unstable with an increased level f turbulence. The scale f the turbulence, within the range tested, shwed n marked influence n the stability characteristics. Nvak in a series f papers (5,6,7,8,9) has examined the gallping scillatins f lng prismatic bdies, typical f a tall structure, when expsed t atmspheric and grid-generated turbulence. Nvak (5) first extended the quasi-steady apprach t cntinuus elastic systems, expsed t a turbulent shear flw, n the basis f an energy cnsideratin. Later studies (6) investigated the effects f turbulence
3 n the general character f gallping scillatins. The cnclusin was that turbulence generally reduces the amplitudes f scillatin but has n severe effect n the nset f the scillatin fr a square sectin. Other studies (7, 8) have shwn that turbulence can change the stability characteristics f prismatic bdies, and that gallping scillatins can arise with sectins which frmally d nt bey Den Hartg's stability criterin. Nvak als prpsed a universal respnse curve which wuld permit the predictin f gallping characteristics frm a single dynamic test f a particular bluff bdy. 1.2 Purpse Investigatins int the aerelastic gallping f structures is imprtant because strng lateral self-excited scillatins can develp at a certain wind speed as a result f the lateral frce cmpnent. The nset velcity is usually high but the cnstantly decreasing specific weight, damping and stiffness f tall buildings, typical f mdern practice, enhance the pssibility f this aerelastic instability. The tendency t gallping at velcities lwer than the nset velcity prduces a negative damping which reduces the inherent psitive structural damping and results in an increased respnse t lateral wind gusts. The purpse f this study is t investigate the gallping behavir f a finite vertical structure f square crss-sectin which has mechanical prperties similar t a tall building, and which is situated in a turbulent flw representative f the atmsphere. The effects f building aspect rati (i.e. the rati f height t width f the structure) n the gallping characteristics are t be examined als. Previusly Nvak (6, 7, 8) assumed an average frce cefficient was
applicable and cmputed the gallping respnses using the quasi-steady thery. Herein cnsideratin is given t the fact that the lateral frce cefficients are variable with height in a bundary layer and a cmparisn is made between the respnses predicted by an "average" and a variable frce cefficient. In additin sme f the simpler.aspects f the cmplex prblems f flw arund a three-dimensinal bluff bdy are cnsidered.
5 CHAPTER 2 THEORY 2.1 Quasi-steady assumptin Fr the mathematical descriptin f the scillatins due t the aerdynamic instability f bluff cylinders the quasi-steady apprach is assumed t be valid. The quasi-steady thery assumes that the frces experienced by the vibrating cylinder are the same frces exerted n a static mdel which is at an angle f attack equal t the apparent angle f attack seen by the vibrating cylinder, as shwn by Fig. 1. Under tw-dimensinal, smth flw cnditins the quasi-steady assumptin leads t a weakly nn-linear differential equatin which can be slved fr bth the steady and transient amplitudes f vibratin (2). In sme practical cnsideratins gallping can ccur with finite vertical structures which are expsed t a sheared turbulent bundary layer flw. Nvak (5, 6, 7) has examined such systems and using the quasi-steady average frces has slved fr the amplitudes f steady vibratin n the basis f an energy balance. In the present experiments cnditins were three-dimensinal, as in Fig. 1, and a similar energy apprach was used in the slutin f the prblem. In a bundary layer there is a velcity gradient between the wall and the free stream. This velcity variatin can be expressed by the simple relatin V(z) = V v(z)... 1 where V. is the velcity at the reference pint, here the tp f the structure, and v(z) is a functin describing the velcity prfile.
7 Dealing with atmspheric bundary layers it is mst cnvenient t cnsider v(z) t be a pwer law prfile whse shape depends n the rughness gemetry (11, 12). Turbulent flw past a bluff bdy implies that there are fluctuating cmpnents f velcity and frce. Since the time fr develping a steady gallping scillatin is hundreds f cycles (1) the verall effect f the velcity fluctuatins is small and can be accurately ignred. T dispse f the lateral fluctuating frce cmpnents is mre difficult but is accurate if the nset velcity is much higher than the velcity at which vrtex resnance ccurs (3, 6). Therefre treating nly time average values the lcal mean frce in the y-directin is given in cefficient frm by the expressin F y (z,a) = C fy (z,a) hv 2 (z) At present there is n adequate thery which gives as a functin f the angle f attack, a, r as a functin f the vertical dimensin, z. The lateral frce cefficient can nly be determined experimentally thrugh frce r pressure measurements, as in Appendix 1. The necessary empiricism is then intrduced by assuming the lateral frce cefficient can be represented by a plynmial curve fit f the data in the general frm C r (z,a) = E A. (z) tan a 1 2 fy ^ i where nw the aerdynamic cnstants, A^, are variables with height. Fr symmetrical prismatic sectins is an dd functin f
8 tan a and as such even pwers f tan a shuld vanish. Preserving the even pwers hwever, results in a smther apprximating functin and is accmplished by prperly cnsidering the abslute value f tan a (5). Intrducing the abslute value signs in all even pwered terms, and using the quasi-steady implicatin that '> tan a = y(z) V(z) results in the general equatin fr the lateral frce cefficient as C f (z,a) = -Z A, (fikl) 2 *" 1 + I A, (z)(m) 2S l ^ 4 l 3 fy r = 1 2r-l v '\V(z)/ g = s l 2s v MV(z)/ y(z) The abve expressin represents a frce cefficient which is analgus t the frce cefficient btained fr tw-dimensinal smth flw cnditins. The main difference is that the expressin in equatin 3 is sme cmplex functin f the vertical dimensin. 2.2 Energy apprach The nly net exchange f energy between the mechanical and aerdynamic frces, ver a perid f vibratin, is that due t the dissipative frces. Therefre cnsidering a structure with idealized viscus damping, c, the ttal dissipative frce acting ver a differential length dz is F(z,y)dz = (C fy (z,y) hv 2 (z) - cy(z))dz...\ 4 Steady vibratins exist when the ttal wrk dne by the damping frces, ver a perid f vibratin, is identically zer. Thus the equatin determining the steady amplitudes f vibratin is
i 2TT/U) W = 0 =j j F(z,y)dz ydt ~ ' Nting the similarity between the three-dimensinal system and the tw-dimensinal single degree f freedm system, Nvak (5) assumed that the structure wuld have a respnse similar t that f a free vibratin. The assumed mtin which is accurate t the first apprximatin is given by y(z, t) = ay n (z) cs u^t Here a is the amplitude at the reference pint, the tip f the structure, y (z) is the nrmalized mde shape and t is the natural circular frey n v r n quency. Previusly Nvak (6, 7, 8) applied an average lateral frce cefficient ver the height f the structure and then cmputed the steady gallping amplitudes using the abve equatins. If this tw-dimensinal assumptin f frces is accurate the general algebraic equatin describing the amplitudes f steady vibratin, resulting frm the integratin f equatin 5 is: 11 " 1 1 = Z A.B.C./a* U i=l \U where nw U and a* are dimensinless wind velcity and amplitude given by U = n V & ; a* = na n Here A_^ are average cefficients fund frm equatin 2, B^ are numerical cefficients, fr dd i = r are
10 1«3:5... r B = 2 'r 2.4*6... (r+1) and even i = s are 4 2r4-6... s s ir 1-3-5... (s+1) and C are cefficients describing the vibratin mde and wind prfile given by 2-i 5 v < z > y n^z) C. = 1 A 2,.dz.i+1, dz If a tw-dimensinal average frce is nt accurate an alternate lateral frce cefficient can be cnsidered. T d s requires that the average aerdynamic cnstants, A^, in equatin 2 be replaced by a functin which expresses their dependence n the vertical dimensin, z. The plynmial expressin used here was A.(z) = E A..[ )^. 8 If equatin 8 is substituted fr A^ in all previus equatins the expressin describing the steady gallping amplitudes is = E B II u i=l i D iu/ where all terms are as befre except the cefficients D ± replace A^ and are given by
11 D. = _ )dz By examining equatin 7 r 9 it is bserved that the gallping respnse predicted is universally valid. Thus, gallping scillatins f all elastic systems having the same crss-sectin, height and mde shape shuld cllapse nt a single universal respnse curve when expsed t the same wind prfile fr all mass and damping cnfiguratins. This fact shuld enable the direct determinatin f the universal respnse curve fr a particular structure by measuring the gallping characteristics f a single arbitrary elastic mdel in the wind tunnel. The amplitudes f statinary scillatin can be fund frm equatin 7 r 9 but sme f these amplitudes may be unstable. Parkinsn (2) and Nvak (7) have examined in detail the stability f gallping amplitudes. Fr the analysis f stability, the first derivative f equatin 5 is needed and in the sense f rbital stability amplitude a g is stable when dw * da a. s < 0... 10 and with the ppsite sign the mtin is unstable. Perfrming the differentiatin in equatin 10 leads t the general equatin predicting the stability f a particular amplitude which is i=l i+l\_ /a*\ i _ 1 _ 1 u < 0 11,...»... i i - l
12 Once again the cefficients A^C^ can be replaced by cefficients fr thse cases in which the lateral frce cefficient is nt cnstant with height. In summary, using the quasi-steady assumptin f frces the gallping respnse f a vertical structure can be predicted. In thse cases where an average lateral frce cefficient is an adequate apprximatin the gallping respnse is that given by equatin 7. Fr threedimensinal situatins where the lateral frce cefficient varies with height the gallping scillatin is predicted by equatin 9. T analyze the stability f the steady amplitudes the relatinship given by equati 11 shuld be used.
13 CHAPTER 3 DESCRIPTION OF EXPERIMENTS 3.1 Outline f experiments cnducted The purpse f this study is t investigate the gallping behavir f a finite vertical structure expsed t a simulated atmspheric bundary layer flw. T this end experiments were cnducted as fllws: a) Velcity measurements Measurements were made f the bundary layer's velcity prfile, turbulence intensity and spectral distributin f energy. This data was used t define a characteristic length scale s that mdel data can be cmpared t full scale infrmatin. b) Frce and pressure measurements Once the prperties f the turbulent flw had been sufficiently characterized their effects n the static behavir f finite square cylinders were investigated. Frce measurements were made n square cylinders f fur different aspect ratis at varius angles f attack. Later pressure measurements were cnducted n tw prisms t btain lcal lateral frce infrmatin. c) Respnse measurements The dynamical behavir f tw elastically munted square twers, placed in the same turbulent flw as fr the frce measurements, was investigated fr varius cmbinatins f mdel damping and frequency. The variatin f tip amplitude with wind speed was recrded and cmpared with the theretical predictins, which utilized the frce and pressure readings made
14 in 2). d) Wake measurements Additinal infrmatin abut the gallping behavir f the square twers was btained by examining the spectra i f velcity fluctuatins in the wake behind the rigidly and elastically munted 28" mdel. 3.2 Wind tunnel All experiments were cnducted in the industrial aerdynamics wind tunnel at U.B.C. The wind tunnel is an pen circuit, blwer type tunnel 8' wide and initially 5.17' high with a test sectin 80' lng. The area cntractin rati is 4:1 and a cnstant speed, variable pitch fan blws air thrugh the test sectin at speeds between 7 and 70 ft/s. The test sectin rf can be adjusted t maintain ambient pressure in the tunnel. Pressure taps lcated at 8' intervals alng the back wall f the tunnel were cnnected t a multitube manmeter t accurately set the pressure gradient t zer. The velcity prfile and turbulence characteristics f the bundary layer are determined by thie rughness f the flr cvering. The turbulent bundary layer fr this study was created by cvering the entire test sectin with rughness elements 1.5" high, 0.75" wide and 0.041" thick, 6" apart in staggered rws. A view f the rughness can be seen in Fig. 2. 3.3 Velcity measurements The imprtant prperties f the turbulent bundary layer were measured with a single ht-wire with linearized respnse. The. ht-wire system used was a DISA type 55D01 anemmeter. The signal frm the anemmeter was fed int a DISA linearizer, type
Figure 2. Elastic 20" mdel and upstream surface rughness
16 55D10, and frequencies higher than lokhz were eliminated with a DISA 55D25 filter. Using the linearized ht-wire signal mean and RMS measurements were made f the bundary layer's prfile and turbulence intensity at three different wind speeds. In rder t determine the scales f the turbulence the spectrum f the lngitudinal velcity cmpnent was analyzed. The spectrum was measured with a Bruel and Kjaer, type 1614, 1/3 ctave band filter. Digital readut was accmplished by using a Schlumberger Time Dmain Analyzer with real time averaging. Spectra f the lngitudinal velcity cmpnent were cmputed at several different heights in the bundary layer. 3.4 Static mdels The frce and pressure measurements were cnducted n a 2" by 2" square plexiglass twer. The twer was cmpsed f 4" tall segments which culd be assembled t frm a twer f the desired aspect rati. The lcal frces were determined by the integratin f the lcal pressures, and t this end tw 4" lng sectins were built and fitted with rws f pressure taps. Each rw cnsisted f seven pressure prts f 0.025" in diameter. A view f the mdel can be seen in Fig. 3. 3.5 Frce measurements One f the necessary inputs t the gallping thery f Chapter 2 is the determinatin f the lateral frces. It was mst cnvenient t measure the lift and drag frces and then cmpute an average lateral frce via equatin 3 in Appendix 1. T measure the lift and drag, an Aerlab pyramidal strain gauge balance was emplyed. The balance is designed t supprt a mdel in the
17 Figure 3. Test mdels (left t right 20" and 28" elastic mdels and variable height static mdel)
18 wind tunnel and vary its angle f yaw ver a 360 range with a precisin f 0.1. Links separate the individual frce r mment cmpnents s that each can be measured individually. Since the mdels were munted vertically in the wind tunnel the lift frce was measured with the side frce unit with angles f attack being replaced by angles f yaw. The electrical signals cming frm the drag and side frce lad cells were then fed int a PDP 11/10 minicmputer t be digitized. The sample perid at each angle f attack was apprximately 40 secnds, and three runs were made at each aspect rati. Average drag and lift cefficients were then cmputed. The reference dynamic pressure was measured frm a pitt static tube with an inclined Lambrecht manmeter. The reference pitt tube was lcated 45" abve the tunnel flr and left f the tunnel centerline. This dynamic pressure was measured t be 6% higher than the dynamic pressure measured at a height f 28" abve the tunnel centerline. Thus dynamic pressures measured frm the reference pitt at 45" were reduced by 6% t yield the actual dynamic head at the height f 28", i.e. the lcatin in the wind tunnel where the mdels were tested. 3.6 Pressure measurements T btain the lcal lateral frces acting n the mdel the: pressures at a particular height were measured and then integrated. Since measuring and recrding pressures at 14 taps fr 30 : mre angles f attack at several different lcatins required a large number f readings the pressure measuring system was autmated. The system used was quite effective in btaining and string the large quantity f pressure data accumulated.
19 Pressure taps acrss a face f the mdel were cnnected t a Scanivalve multlprt scanner whereupn the pressure signal was cnverted int a vltage by a Barcel, type 511, pressure transducer. The electrical signal was further amplified and cnditined by a Datametrics Electric Manmeter, type 1018B, and was then input int the PDP 11/10 minicmputer t be digitized and stred. The multiprt pressure scanner was driven by an electrical impulse frm the cmputer, s after each 30 secnd sample the scanner was advanced and a new tap was sampled and cnverted int pressure cefficient frm. The pressure cefficients calculated were nn-dimensinalized by the dynamic pressure at the particular height f the rw f pressure taps. Once all the pressure cefficients had been calculated a cubic spline curve fit and then a Simpsn's rule integratin rutine were used t perfrm the calculatin f the lcal lateral frce cefficient given by equatin 9 in Appendix 1. 3.7 Elastic mdels and munting T verify the theretically predicted gallping respnses dynamic measurements were cnducted n tw elastically munted square twers. A mdel tgether with its munting is shwn in Fig. 4. The basic mdel was attached t a vertical 1/2" diameter hllw steel rd which in turn was fixed t a 1" diameter thin walled aluminium tube which was supprted by tw cylindrical air cre bearings. The 1/2" steel rd after passing thrugh the aluminium tube was flexibly cnnected t rigid steel legs by tw hrizntal helical tensin springs. The air cre bearings prviding the mdel supprt were similar t thse designed by Smith (1). T prevent any mtin frm ccurring in the alng-wind directin the air bearings were drilled and fitted with end
Figure 4. Dynamic balance and elastic test mdel 20
21 plates. The mdel was thus capable f rigid bdy rtatin in a single degree f freedm in a plane perpendicular t the wind directin. The air supply t the bearings and end plates came directly, via a flexible hse, frm the cmpressed air line available in the lab and was kept cnstant thrughut a test. The tw mdels tested were each built f 1/2" thick balsa wd and measured 2" by 2" and were 20 and 28 inches tall. A varnish was used t prtect the surfaces and crners f the mdels and this resulted in a smth exterir finish. In additin t the damping frces already inherent in the pivting system, eddy current damping was als emplyed. The dissipative frces due t the eddy currents are almst entirely equivalent t viscus damping which was desirable in this analysis. T this end a thin aluminium disk was attached t the bttm f the steel rd t prvide eddy current damping as it mved between the ples f a G electrmagnet as shwn in Figs. 4 and 7. The current pwering the electrmagnet was prvided by a D.C. pwer supply and was cntrlled by a variable resistance. After each test any residual magnetism was remved frm the magnet by switching the current ver t a slwly decreasing A.C. supply. The helical tensin springs were made by cutting the required number f cils frm a knwn spring and then were calibrated by a simple lad deflectin test. The tw springs had a cmbined stiffness f 14.5 lb/in. Changes equivalent t changes in mdel density were prduced by varying the vertical distance frm the pint f rtatin t the pint f
22 attachment f the springs. Varying this distance effected a change in frequency and a cnsequent change in average 'effective density' f the mdel. The cmplete air bearing mdel supprt system was built n a rigid steel frame apprximately 21" high, which in turn was supprted under the wind tunnel by a heavy table. After the mdel was aligned t zer angle f attack the steel frame was firmly clamped t the supprt table. The mdel was fixed"t the air bearing system thrugh a hle in the tunnel flr, and an inch space separated the tunnel flr and the air bearing system t reduce pssible effects due t tunnel vibratins. The entire system exhibited n perceptible mtin at even the highest amplitudes f mdel vibratin. 3.8 Deflectin measurements and calibratin The amplitudes f steady vibratin due t the mechanism f gallping are knwn t be large and ccur at a frequency clse t the natural frequency f free vibratin. These characteristics, the large amplitudes and the frequency f vibratin, dictated the type f deflectin measuring instrumentatin. Strain gauges are ften used in dynamic systems f the type described here (10) but nn-linearity in the strain gauges due t the large amplitudes culd be a prblem. The deflectin f the mdel was instead measured by a Bruel and Kjaer, type 4332, accelermeter munted inside and quite near the tp f the mdel. The accelermeter used was fairly large, it had a mass f 30 gtams, but als had a high sensitivity abut 46 mv/g and a flat frequency respnse t abut 1 Hz. Since the desired quantity was displacement nt acceleratin the high impedance utput
23 frm the accelermeter was fed int a Bruel and Kjaer 2625 preamplifier where the signal was integrated and amplified. The resulting lw impedance signal had a D.C. ffset f 13 vlts which was blcked by a capacitr befre being fed int a Krhn Hite lw pass filter which remved signals abve 160 Hz. The filtered utput was then displayed n an scillscpe, pltted n a Hneywell visicrder scillgraph and digitized by the PDP 11/10 cmputer. RMS data were measured and 20 secnd samples were taken ver a ne minute perid after the flw had stabilized in the wind tunnel. Lnger sampling perids were used if the amplitudes fluctuated a great deal. A slight disadvantage in using the accelermeter t measure the mdel displacement was that the calibratin f the mdel fr deflectin had t be dne in a dynamic test. T determine the displacement versus vltage characteristics f the accelermeter a very thin wire was attached t the tip f the mdel and fixed t the side f the wind tunnel. The resulting hrizntal deflectin was then measured with a pair f vernier calipers, using a rigid stand munted next t the mdel as a reference pint. was sharply cut. The visicrder was set t a knwn speed and the wire The riginal displacement was taken as the peak f the first scillatin cycle f the scillgraph trace. Knwing the displacement versus vltage respnse f the visicrder the mdel deflectin culd then be cnverted int a vltage. Calibratin f the mdel fr deflectin was perfrmed befre each test and cnducted at five different initial displacements. The cnstant btained by pltting vltage versus displacement, Fig. 5, was used as input t the cmputer prgram which
A- Peak utput vltage (vlts) Figure 5. Typical calibratin curve fr accelermeter utput versus mdel deflectin S3
25 cnverted the electrical signals int RMS displacements. 3.9 Damping measurements The damping f the mdel fr a particular current in the electrmagnet was btained by plucking the mdel in the crss wind directin and recrding the utput nt the visicrder scillgraph. The decay curve was repeated fr three different initial displacements and was measured and pltted n a semi-lg graph as shwn in Fig. 6. The lg decrement used was the average fr the three trials. Decay traces were taken befre each dynamic test and the lg decrement was crrespndingly calculated. This prcedure incrrectly includes the still-air aerdynamic damping f the mdel itself, but this is relatively small and is partly cmpensated fr by the higher values f nnaerdynamic damping actually ccurring during gallping. Later, after perfrming the damping calibratins it was fund that the percentage f critical damping already present in the pivting system due t frictin was a significant amunt. Depending n the frequency the fractin f critical damping, C/C Q1 ;> due t the pivting system was between 0.0045 and 0.008. Since the nset velcity is directly prprtinal t the damping present tests were ften perfrmed with the electrmagnet nt present, s as t keep the wind speed within a reasnable range. The damping in the pivting system was then assumed t be entirely viscus and was calculated in the same manner as previusly utlined.. 3.10 Frequency measurements and density calculatin The mdel frequencies were calculated frm the scillgraph traces resulting frm the
26
27 damping calibratin. A 1 cycle per secnd triangular wave frm a functin generatr served as the time base fr the calculatin f the frequency. The frequency measurements were repeatable, but were checked befre each dynamic test. One f the necessary inputs t the gallping thery which was cmputed directly frm the frequency measurements was the determinatin f the average density, p m> f the mdel. T determine the effective density f the mdel the mment f inertia f the pivting system had t be calculated. The inertia f the rtating assembly, i.e. mdel, steel rd, aluminium shaft, springs damping plate and accelermeter, abut the hrizntal axis f rtatin was btained frm the equatin f free vibratin. Given the gemetry in Fig. 7, if the mment f inertia abut the axis, 0, is I the equatin f free vibratin is given by 2 16 + C0 + 2kz 6 = 0 s and the mdel inertia can be calculated directly frm the expressin 1 = 2kz 2 s 2 (2nfr where 2k = ttal spring stiffness f = frequency z = vertical distance between the pint f g * rtatin and the springs 6 = angular rtatin Nw fr a rectangular prism rtating abut the axis 0 the inertia is knwn and the effective density can be cmputed frm
Figure 7. Elastic mdel and munting rig fr dynamic tests
29 m = where h d length f the mdel lateral dimensin f the mdel distance frm the base f the mdel t the pint f rtatin, here d = 2.4" V = vlume f the mdel m The dynamic tests were cnducted n tw square twers fr varius cmbinatins f frequency and damping. A summary f the mdel prperties fr each particular test are presented in Table 1. (See nmenclature fr definitin f the symbls.) TABLE.I Height f zs Cm B/n (in) (c/s) (in) (lb/ft 3 ) xlo" 3 xlo -3 28 3.79 5.00 11.4 6.6 1.250 5.28 28 3.58 5.00 12.7 7.3 1.150 6.35 28 3.79 5.00 11.4 7.9 1.320 5.98 28 4.41-6.00 12.1 8.0 1.210 6.61 20 4.65 4.31 14.1 4.5 0.950 4.74 20 4.65 4.31 14.1 5.2 0.942 5.52 20 4.72 4.31 13.7 7.3 0.913 8.00 20 5.38 5.00 14.1 7.4 0.913 8.11
30 CHAPTER 4 l RESULTS AND DISCUSSION 4.1 Velcity measurements In a neutrally stable atmspheric bundary layer the prperties f the mean flw are knwn t be almst entirely dependent n the rughness f the surface (11,12). Simply put this means that the rugher the surface, the greater the drag frce at the surface, turbulence intensity, the Reynlds stresses, the gradient height and the retardatin at the surface. In (11, 12) Davenprt has catalgued sme f the prperties f typical atmspheric bundary layers and their crrespnding surface rughness. The intent here was t make velcity measurements f the mdel bundary layer's prperties and cmpare these t the full scale infrmatin in (11, 12). The variatin f the mean velcity and the RMS turbulence intensity with height are shwn respectively by Figs. 8 and 9. Frm Fig. 8 it can be seen that fr the particular rughness used the bundary layer thickness, 6, is apprximately 28" and the prfile expnent, y> in the equatin is 0.264. The pwer law expnent was btained by pltting lg ( V /V2g») versus lg ( z /6). The velcity prfile and the turbulence intensity were measured fr three different gradient wind speeds and bth prperties were fund t be reasnably invariant with R g.
32 4 1 1 1 1.0 t AT 28" 27.8 ft/s 41.0 ft/s 0.8 _L AT* & 45.0 ft/s A / 0.6 J. A T A T 0.4 -L A T A * «>T A 0.2 4- A AT* am 0.5 4-4- 0.10 0.15 v«/v 4-0.20 Figure 9. Variatin f turbulence intensity with height in the bundary layer
33 An imprtant characteristic f a turbulent bundary layer is the distributin f energy with frequency. Spectral measurements f the lngitudinal velcity cmpnent were made at several heights in the bundary layer at a gradient wind speed f 37 ft/s. The spectra measured at 2/3 f the mdel height fr the 28 y and the 20" mdels, Fig. 10, are given as dimensinless pwer versus nn-dimen&inal frequency. T determine the turbulent length scale, L, the measured spectra were cmpared with Vn Karman's theretical distributin f energy, i.e. X T. n L 4 v n S(n)...2,-.5/6 1 + 7 0, ( ^ "J where n = the frequency L = the length scale V = the mean lcal velcity v 1 = the ttal RMS velcity fluctuatin Matching the measured spectra t the theretical curve was dne ver the mderate frequency range.* Fr the spectra taken at 18.67" the characteristic length scale was 1.03' and at 13.33" the scale was fund t be 0.942'. The pwer law expnent, the distributin f turbulence intensity and the spectra all scale t what Davenprt calls a suburban r frested expsure. Fr the 28" mdel crrespnding full scale data has a typical eddy length f 560' and fr the 20" mdel the atmspheric scale f turbulence is 475'. Taking the gemetric scale as the rati f full scale turbulence t the scale f wind tunnel turbulence implies that fr t;he 28" mdel the scale is 1/540 and fr the 20" mdel the scale is 1/500. *(This crrespnds t dimensinless frequencies, in Fig. 10, between 1.0 and 1.8.)
1 h H 1 Height abve 18.67" flr > 0.20 -L CO 0 13.33" Thery Vn Karman V 28-37.0 ft/s n 0.10 CD 0.08 P. CO CO <a 0.06 rh e H CO S 0.04 6 H 0.02 4-1 h + 0.2 1.2 1.4 0.4 0.6 0.8 1.0 n Dimensinless frequency X L v /V Figure 10. Pwer spectrum f the lngitudinal velcity cmpnent 1.6 1.8
35 Thus bth mdels represent tall twers expsed t a "suburban" wind. 4.2 Average frce measurements 4.2.1 Drag cefficient By examining Fig. 11 it is seen that fr angles f attack between 0 and 30 degrees the effect f lwering the mdel's aspect rati results in a lwer average drag cefficient. Fr small angles f attack (a - 13 ), the drag cefficients tend t becme independent f wind rientatin as the aspect rati decreases. When cmpared t Laneville's (4) tw-dimensinal results the drag cefficients measured here are lwer and the minimum drag ccurs at a slightly higher angle f attack. Vickery in (13) measured the drag cefficients, at zer angle f attack, f several finite square bluff shapes expsed t a flw f 10% turbulence intensity. His results indicate that fr a decrease in aspect rati frm 15 t 2 a reductin in the mean drag cefficient culd be as much as 30%. In Vickery's data a marked feature f the variatin f C D with aspect rati was the attainment f a maximum value f drag cefficient at a finite value f aspect rati. Fr the range f height t width ratis examined here the drag cefficients were always fund t increase with increasing aspect rati. The reductin in drag fr the shapes tested is mst likely the end prduct f tw mechanisms, ne being the increased flw ver the tip f the mdel and tw the verall higher level f turbulence intensity appraching a lwer aspect r"ati mdel. Bth mechanisms serve t reduce
t 1 t 1.6 +- u e <u H rt <4-l 1.4 + <"1. 2 60 cd u <D Ml. 0 cd u cu > < A A,A-A" D A Q A J A D ^ A 0 A Q Q * 2 a A F * A A A Mdel 28" 20" 16" 12" height R = 40,000 e 0.8 + 1 1 1 1 f 1 1 5.0 10.0 15.0 20.0 25.0 30.0 * 35.0 Angle f attack a( ) Figure 11. Variatin f average drag cefficient with angle f attack fr fur aspect ratis ON
37 the drag cefficients by increasing the base pressure behind the mdel. 4.2.:2 Lift cefficient As shwn by Fig. 12 the effect f decreasing the mdel aspect rati is t prgressively reduce the maximum negative lift, leaving the initial trend at small angles f attack unchanged. The invariance f lift cefficient at small angles f attack fr different turbulence intensities was als bserved by Laneville. Cmparisn f the tw-dimensinal and three-dimensinal lift cefficient curves reveals that in the latter situatin the slpes at the rigin are nt nearly as steep indicating that the square sectin is mre stable in a three-dimensinal flw. The invariance f the lift curve slpes fr small angles fr aspect ratis between 14 and 6 was unexpected. The first appearance f a tip flw seems t cause a discrete jump in the initial slpe f the lift cefficient curve with further increases in mdel threedimensinality having negligible effect n the initial trends f the lift curve slpe. A decrease in aspect rati des appear t reduce the angle at which the maximum lift ccurs. This result is difficult t islate frm Fig. 12. since a reductin in aspect rati crrespnds t an increased turbulence intensity which is knwn t reduce the angle at which the maximum negative lift ccurs. Mre experiments will have t be dne t explain the unexpected behavir f the lift curve slpe with mdel aspect rati. 4.2.3 Lateral frce cefficient The average lateral frce cefficients calculated frm the lift and drag measurements, as in Appendix 1, are
Average lift cefficient I I I I Mi O l-l C H r < O n H- Mi 0) rt c H" H P M 0> O 01 T3 Mi r rt < r n n P> w > rt OQ 3 I- 1 H- r TO O M (A h- 1 r H" Ml rt Ml O p O rt r rt r MI O Ml H* O O H» r 0 rt a < ' r rt af O am > pa > a > i> a a > > > 3 cw M r Ml (U rt rt t O 5^ OJ On (D t cn O II = s = O r sr r H- OP rt 8
39 pltted in Fig. 13 fr several different mdel aspect ratis. As a cmparisn the lateral frce cefficients frm (4), btained under twdimensinal istrpic turbulent flw cnditins are pltted in Fig. 14. Cmparing the tw-dimensinal and the three-dimensinal results reveals that in the latter situatin the initial slpe at the rigin is much reduced frm the tw-dimensinal case. The reduced slpe f the lateral frce cefficient in the three-dimensinal situatin is indicative f the lwer values f lift cefficient measured. As can be seen thugh C^ fr a finite square sectin even in a turbulent shear flw exhibits a psitive slpe fr small angles indicating the sectin is unstable. The reduced slpe is evidence that the finite square sectins tested shuld all gallp but at a higher reduced velcity than under tw-dimensinal cnditins. ~ Frm an examinatin f Fig. 13 it is bserved that fr small angles f attack the lateral frce cefficient is almst cmpletely independent f the particular mdel's aspect rati. The invariance f the lateral frce cefficient is due t the measured lift cefficients which had little dependence n the mdel's height t width rati fr small angles f attack. In general decreased aspect rati and increased turbulence intensity nly serve t reduce the value f the maximum lateral frce and the angle at which it ccurs. If the assumptin f an average lateral frce is adequate the characteristics f these lateral frce cefficients shuld be reflected in the mdel's gallping respnse. Finite square twers having an aspect rati between 14 and 6 shuld all have reasnably the same nset velcity but gallp with
Average lateral frce cefficient i 'Fy Ul Ul a > m > D n m i.. II Hit) (D i t O P O «OA It Ni ON O t 0. 00 fd cr OQ i 1
I Lateral frce cefficient.. '. O H* N> W * O Ui c w Mi Mi (B O rt f rt C M e r- 1 rs 3 tt> O H* 3 rt fl> 3 Ui 03 H H* rt 3 a 3 O O r ^< O 0 CO DJ c H r CO n rt 3 i-i O Ul O n> Ml \ 4>
42 lwer amplitudes as the mdel aspect rati decreases. 4.3 Lcal lateral frce cefficient In a turbulent bundary layer the lcal drag and lift cefficients can be expected t vary alng the height f the mdel. Naturally the lcal lateral frce cefficient will reflect these changes and will be dependent upn the vertical dimensin, in sme manner. T ascertain the variatin f lcal lateral frce with z, lcal pressures were measured and integrated as in sectin 3.6. As shwn by Figs. 15 and 16 the dependence f the lcal lateral frce upn the dimensinless height, 11, fr bth the 28" and the 20" mdels is quite striking. The lateral frce cefficient fr the 28" mdel basically ges thrugh three regins. Over the bttm third f the mdel the lateral frce cefficient is negative indicating the flw is well reattached and accrding t Den Hartg's criterin is stable fr all wind speeds. The frce cefficient is negative ver this regin due mst likely t the high levels f turbulence and the cmplex manner in which the fluid separates in a sheared flw. In the middle sectin f the mdel the lateral frce cefficient curves again have psitive slpes at the rigin and are crrespndingly unstable. The slpe at the rigin, the maximum lateral frce and the angle at which it ccurs all gradually increase as the vertical dimensin appraches 70% f the mdel height. Near this pint the lateral frce cefficient curve has a slpe and a maximum value bth apprximately 2/3 f the crrespnding values btained under tw-dimensinal cnditins f 9% turbulence intensity, (Fig. 14). The variatin f the lateral frce cefficient
I Lpcal lateral frce i cefficient c H* OQ C H (D H O < P )-( H* P rt H- 0 ' a I- 1 Hi > > H- 1 O P r-> I- 1 P rt 1 ' Ul (D ' H P h- 1 H Hi P 3 H P r r O r Hi Hi O H- r H* Ui r 3 rt P I 1 O 3 00 rt 3* r OJ id P O LO O 3 f O IJO Ul t> > 4 < > O N r II ON VO J r VO c VO ON ro Ul CO rt 3* r r a r
00 c (D I 1 ON O < ft) H H> 6) rt 3 Hi O Lcal lateral frce cefficient i + < C fy< 2 > H LO t- 1 > M ft) re (D n ft) r- 1 O Ul r+l O O M H O (0 0 O O ft) S3 P < ef H> H- O H- (T> Ul 0 rt ft) O 3 OQ rr CT fd CO TJ ft) 0 O MN Ul rt cr (D Kl O < O N ID "^v. II ON VD J t 00 ON ON S3 S O fd
45 with angle f attack at this lcatin hwever, is distinctly different than under tw-dimensinal cnditins as the maximum frce ccurs at a higher angle f attack and the frce is psitive ver a brader range f angles. After 70% f the mdel height has been reached the slpe f the frce cefficient curve and the maximum value bth begin t fall ff as the influence f the mdel's tip cmes int cnsideratin. Fr the 20" mdel the lcal lateral frce cefficients btained at three dimensinless heights have basically the same variatin with height and angle f attack as the lcal lateral frce cefficients measured n the 28" mdel. The majr difference between the measured cefficients fr the 28" and the 20" mdel is that fr the latter the lateral frce cefficients all attain their maxima at lwer angles f attack. Thugh the frce cefficients n the 20" mdel were nt measured at exactly the same dimensinless height as fr the 28" mdel, it can be seen that fr crrespnding heights the initial slpes f the lateral frce cefficient curves are nearly identical. In summary the measurements f lcal lateral frce cefficients have shwn that there is a definite variatin f lateral frce alng the span f the mdel and that the frces n tw mdels f different aspect ratis are distributed in the same way. The majr effect f aspect rati and turbulence intensity is t reduce the maximum mean lateral frce and the angle at which it ccurs. Since the lateral frce cefficient exhibits distinctive changes depending n the height it is t be expected that the gallping respnse utilizing the lcal lateral frce cefficients will be different than the gallping respnse fund *(Althugh nt shwn it has been verified that the pressure measurements can be integrated t btain the average frce cefficients in Fig. 13.)
46 frm the average lateral frce cefficients. 4.4 Theretical gallping respnse 4.4.1 Respnse using average lateral frce cefficients The gallping respnse f the mdel twers was predicted using the thery utlined in Chapter 2. The aerdynamic cnstants, A^, in equatin 7 were btained frm a curve fit f the experimental average lateral frce cefficient data. The curve fit is a fifth rder plynmial including the even terms and was perfrmed with a least squares rthgnal plynmial cmputer rutine (14). The average aerdynamic cnstants, A_^, fr the 28" mdel and the 20" mdel can be fund in Appendix 2. In Fig. 17 the variatin f tip amplitude with tip wind speed is pltted fr mdels f three different aspect ratis. The respnse curves are universally valid; that is, fr all cnfiguratins f mdel frequency, damping and density the variatin f tip amplitude with tip wind speed is given by these curves (5). The theretical respnses are calculated assuming a rigid bdy rtatin abut the hrizntal axis s that the nrmalized bending mde crrespnds t y n (z) = z /i The velcity variatin is taken as a pwer law prfile with the expnent equal t 0.264. By examining Fig. 17 it, can be seen that fr a given wind speed as the mdel height decreases s des the amplitude f scillatin. The nset velcity fr all three mdels is basically the same and is greater than in the tw-dimensinal cases (1,4). The invariance f nset velcity