WIND LOAOS ON FLAT PLATE PHOTOVOLTAIZ ARRAY FIELDS

(NOWSTEADY WINDS)

3PL Contract No. 954833

Low Cost Solar Arrqy Pro jec t Engi n e e d ng Area

Phase I V F ina l Report

August, 1981

Ronald D, M i 1 l e r Donald K. Zinmnerman

"The JPL Lou Cost Solar Array Pro ject i s sponsored by the U.S. Department 0: Energy and forms p a r t o f the Solar Photo-

v o l t a i c Conversion Program t o i n i t i a t e a major e f f o r t toward the development o f lo*-cost so la r arrays. This work was per- f o n e d f o r the Jet Propulsion Laboratory, C a l i f o r n i a I n s t i t u t e o f TechnolGgy by agrement between NASA and DOE."

Prepared f o r

J e t Propulsion Laboratory 4800 Oak Grove Dr ive

Pasaderia, Ca l i fo rn ia 91103

By the

k e i n g Engineering End Construction Lonipany ( A O i v i s i or, o f The Boei ng Cotnpal?y)

Seat t. 1 e, Washi ngt on

"This repor t was prepared as an account o f work sponsored by the United

States Government. Nei ther the United States nor the United States Department o f Energy, nor any o f t h e i r employees, nor any o f t h e i r contractors, subcontractors, o r t h e i r employees makes any warranty, express

o r implied, o r assumes any legal l i a b i l i t y o r respons ib i l i t y f o r the accuracy, completeness or useful ness of any i n f o m t i o n , apparatus, product

o r process disclosed, o r represents tha t i t s use w u l d not i n f r i n g e p r i v a t e l y owned r ights."

ii

Acknowledgements

Management and engineering on t h i s p ro jec t was provided by the Solar Systems organizat ion of the Boeing Engineering and Construction Compdny ( a D iv i s ion of

The Boeing Company). O r . Ronald Ross, W r . Robert Weaver, and Hr. Donald Woore

Manager and technical monitors , respect ively, i n the Engineeriny area of the

Lou Cost Salar Array Pro ject , J e t Propul s ion Laboratory, Ca l i f o rn ia I n s t i t u t e of Technology, cont r ibuted many he lp fu l suggestions during the proyrdin. W r .

Weaver served as technical monitor f o r the aerodynamics inves t iga t ion and Mr .

Moore f o r the s t ruc tu ra l response analysis. Mr. Abrahun Wilson, a lso of JPL,

de ta i led the JPL array s t ruc tu ra l design used i n t h i s analysis. tunnel t e s t was performed by the F l u i d Dynamics Laboratory s t c C f o f Colorado

S t a t e Univers i ty , For t Col l ins , Colorado.

t e s t i n g was ilr. Jack E. Cermak, thz Pro jec t D i rec to r was O r . Jon A. Peterkd,

and Pr inc ipa l Iwest !gators were N. Hosoya and D r . Hike Poreh.

The wind

The responsible Manayer f o r the

iii

Table o f Contents

1 .o Summary

2.0 Introduct ion

2.1 Objectives

2.2 Discussion and Background

2.3 Stl;dy Requirements

2.4 Report Organi zat i on

3.0 Technical Approach, Results and Discussion

4.0 Conclusions

5 .O Array Design Guide1 ines for Wind Turbulence

6.0 New Techno1 ogy

7.0 References

Appendix A Appendix B Appendix C

Paye 1

9

21

23

29

31

A 1 B1 c1

V

Figures

No. Page

1-1

1-2

1-3

3-1 3-2

3-3

3 -4

3-5

3-6

3-7

3 -8

5-1

5-2

5-3

Configdration I - Dynamic Analysis Results f o r 5 t h Array Within an Array F i e l d

Configuration I 1 - Dynamic Analysis Results f o r 5 t h Array Wi th in an Array F i e l d

Flow Chart t o Calculate Array Design Wind Loads due t o the Non-Steady Por t ion o f the Yind Array Configuration Character is t ics Typical Cross Spectrum between Pressure laps on an Array i n a 90 mph Wind Configuration I - Dynamic Analysis Results f o r 1 s t Array Unprotected from the Wind Configuration I - Dynamic Analysis Results f o r 1 s t Array

Protected from the Wind by a Fence Configuration I - Dynamic Analysis Results f o r 5 th Array Within an Array F i e l d

Configuration I 1 - Dynamic Analysis Results f o r 1 s t Array Unprotected from the Wind

Configuration 11 - Dynamic Analysis Results f o r 1 s t Array Protected from the Wind by a Fence Configuration I 1 - Dynamic Analysis Results f o r 5th Array Within an Array F i e l d Flow Chart t o Calculate Array Design Wind Loads due t o the Non-Steady Por t ion o f the Wind Wind Tunnel Test RMS Pressure Coeff ic ients on the 5 t h Array i n an Array F i e l d

Example Design Wind Loads on I n t e r i o r Arrays i n an Array F i e l d Using the Design Guidelines

2

3

4

10 11

14

15

16

17

18

19

24

26

26

Tab1 es

No. Pa ye

5 - 1 Results f o r <in I n t e r i o r 4rrdy Using the Design Guidelines

v i i

27

~ E C E C I N G FPLP k:~fi{WK NCT FILMED

1.0 SUMMARY

This report presents the results of a combined experimental (wind tunnel test results) and theoretical analysis u t i 1 iring randan harmonic analysis techniques t o predict the dynamic response and the structural dynamic loads of f l a t plate photovoltaic arrays due t o wind turbulence. Guidelines for use i n predicting the turbulent portion of the wind loading on future similar arrays using the results of this study are presented.

The dynamic response and the loads dynziiiic magnification factor o f the two array configurations (a four post array and a two post array) are similar. Figures 1-1 and 1-2 are typical magnification factors for the two array configurations located i n Identical positions w i t h i n the array field. The figures show the magnification factors a t a mid chord and outer chord location on :he array illustrated and a t four points on the chord. test experimental rms pressure coafficient t h a t the magnification factors are based on is a1 so shown on each figure. dynamic inagnif cat ion factor occur a t a mid chord location on an array and near the trail ng edge. A technique employing these magnification factors and the wind tunne tes t rms fluctuating pressure coefficients t o calculate design pressure loads due t o wind turbulence i s presented i n Figure 1-3.

The wind t u n n e l

In general, the largest response and

1

cr

t 0 .c 0

c E YI C 0 V

- c". M .

2 0 C

9 d

?

'9

s

cu.

n V \ m

0 L 0 c u rc 0

E 0 U V m L LL

v

c C 0

c, Q V cc

.r

.r

0 L 0 s u

L al U 3 0

F i g U t e I-1. Configuration I - Dynamic Analysis Results for 5th A m y with an Array Field

2

0

-0 E

E

u C al u y. cc W 0 V

al L f v) ul aJ L

.c

c

n

a 2 c m 4 E c L W c5 X w

W

h

5 L Y

L C Y 0 9 LL

E 0

u ca U

r

c cc c E PI

r"

P

f

0 t 0 0

I / ' A

' I I U

0

Figue 1-2. Configuration It - Dynmic Analysis Results for 5th Array within an Array Field

3

Step A (Decisigcs)

Des i gn conf igura t ion matches ? Conf igurat ion I (See

11 Figure 3-1)

Design conf igura t ion 1 s t s t ruc tu ra l na tura l

(3 )

Rms unsteady pressure

( 4 )

t.0 exceed 68.27% = l a 95.452 = 2 0

Oesign wind speed a t ,zlev.ition o f 10 meters

7 Step B

Obtain magn i f i ca t ion f a c t o r (M.F.) from appropr iate f igures 3-3 t o 3-8 using decisions 1 and 2

Step c + M u l t i p l y M.F. by rms unsteady pressure c o e f f i c i e n t from decis ion 3 = design pressure c o e f f i c i e n t wi th p r o b a b i l i t y no t t o exceed 68.27% I II Step D

-1 ~

M u l t i p l y Step C by leve l of .Y robab i l f t y no t t o exceed from decis ion 4.

Step E -w ! M u l t i p l y Step D by dynamic I wind pressure from decis ion

array surface due t o

Figur 3. Flow Chart ro Celculate A m y Design Wind 1 oads due to the N o n - S W y Portion of the Wind

4

2.u IKTRODUCTION

This repor t sumnarites a combined theore t ica l experimental analysis o f the s t ruc tu ra l dynamic loading on long, f l a t p l a t e photovol ta ic arrays r e s u l t i n g

from exposure t o the non-steady p o r t i o n o f the wind enli iroment. This repor t i s an extension t o the analyses reported i n DOE/JPL 954&33-79/2 and -81/3.

study was performed under contract number 954833 t o the J e t Propulsion Laboratory as par t C f the Engineering Area Task o f the Low Cost Solar Array

(LSA) Project. This p ro jec t i s being managed by JPL for t h e Department o f Energy, D i v i s i o n o f Solar Technology.

The

2.1 3bject ives

The cost associated w i t h the desfgn and const ruct ion o f so lar photovol ta ic arrays t o produce e l e c t r i c energy from sunl ight i s an important f a c t o r i n the acceptance and use o f so lar energy. The load due t o wind on an array and on

i t s support s t ruc tu re s t rongly inf luences the design and u l t i m a t e l y t h e cost o f

the photokol ta ic panels, panel and array support s t ructure and foundation o f the array. It i s , therefore, essentfal t o determine the t r u e maximum wind load tna t the array w i l l experience dur ing i t s l i f e t i m e i n order t o minimize the

s t ruc tu re costs. The ob jec t ive o f t h i s study was t o establ ish wind load guide- l i n e s on f l a t p l a t e photovol ta ic arrays f o r t h a t por t ion o f the wind

environment t h a t i s considered non-steady (turbulence) . 2.2 Discussion and Background

Three factors a f f e c t the amount o f wind loading on a body: the f l : v f i e l d i n which the body i s placed, the aerodynamic charac ter is t i cs o f the body i t c e l f ,

and the dynamic response o f the body due t o the wind loading. A l t t . ugh the s t ruc tu ra l loads r e s u l t i n g from t h i s l a t t e r f a c t o r are not t o t a l l y composed o f aerodynamic forces ( they a1 so include i n e r t i a forces) , these s t ruc tu ra l loads do r e s u l t from the wlnd loading, or more precisely, the f luc tua t ions i n wind

1 oadi ng.

The f l o w f i e l d o f the type t h a t would be found around arrays i n an array f i e l d has three aspects: obstacles, 2) atmospheric gusts, and 3 ) turbulence. The steady s ta te f low

1) the steady s ta te f low before i t encounters any

5

i s characterized by of a shear layer adjacent t o the ground whose shearing

effects dccrease w i t h e leva t ion above +he ground u n t i l a utijform f l ow i s attained. Gusting i s the r e s u l t of v e l o c l t y va r ia t i ons and changes i n the

d i r e c t i o n o f the p reva i l i ng wind due t o atmo&& i c i n s t a b i l i t f e s . may be caused by several factors. Gusting can cause turbulence when adjacent.

volumes o f a i r are moving a t d i f f e r e n t ve loc i t ies , thus producing a shearing e f fec t . Terrain roughness causes turbulence because o f shearing effects. An

obstacle i n the path o f the f l ow can also create turbulence by upsett ing the

flow and causing eddies and vor t i ces t o form. I n addit ion, the shape o f the

body w i l l a f fect the cha rac te r i s t i cs o f the turbulence. Turbulent flow i s

h igh l y complex, w i t h varying frequencies and i n t e n s i t i e s occurring .n a random manner .

Turbulence

The e f f e c t on the array forces due t o the f low f i e l d i s a func t ion o f the a i r f l o w Character ist ics and the r e s u l t i n g pressure d i s t r i b u t i o n over the array. When the f low i s turbulent, the pressure d i s t r i b u t i o n and, consequently, the

forces exerted on the array are nonuniform i n frequency and in tens i t y . These forces may cause v ib ra t ions i n the s t ruc tu re r e s u l t i n g i n addi t ional s t ruc tu ra l dynamic forces oa the array. Since turbulence var ies i n frequency and int.ensity, the resu l t t ng loading w i l l a lso vary as a func t ion o f the frequency and in tens i ty .

2.3 Study Requirements

The requirements of t h i s study involve analysts and test . They are:

1. Wind tunnel t e s t and data reduction i n the form o f auto and cross spectrums.

2. Theoretical s t ruc tu ra l dynamic modeling o f arrays. 3 . Dynamic analysis resul ts.

4. Establishment o f design guidel ines f o r est imating unsteady wind loads on photovoltaic arrays.

6

?he fol lwing is a summary of the statement o f work for Phase IV.

1 . Array V1b:ation Characteristics i ) A structural finite element computer model will be developed in

sufficient detail t o adequately define the structure for- use i n the calculation of structural mode shapes o f a photovoltzic array. The t i l t angle o f the array will match one I;f the t i l t dngles o f the arrays used i n the Phase I11 study (reported !n refertnce 1).

i i ) Structural vibratiorl characteristics of the array will be calculated t o produce mode shapes and ge ' -alized mass and stiffness d a t a for the array model developed i n i ) .

2. Array Structural Dynamic Load

i ) The generaltzed mass and stiffness d a t a acd the v i b r a t i o n mode shapes will be combined w i t h the steady state aerodynamics from one array f ie ld configuration chosen from the Phase I11 test d a t a t o generate the structural dynamic equations of motion.

i t ) The Phase I11 wind tunnel pressure time history d a t a will be ana1,red for the array configuration chosen ir! 2 (1) t o yield auto- and cross- spectral density finctions and will be used i n calculating the gweralized forcing functions fo r tne equations of niotion.

3. Structural Dynamic Loads Structura? dynamic responses and loads for the arrays will be calculated from the equations of motion and load equations deileSJped i n 2. The resulting aerodynanic forces from the dynamic analysis will be compared to the steady s ta te aerodynamic forces t o indicate the level of increase t o be expected from t h ? dynamic forces.

2.4 Report Organ iza t ion

The remainder of t h i s report presents the results of tho dynamic analysis and i t s formulation. Section 3.0 presents the basic technical approach, detai ls the results and discusses the pertinent findings. Conclusions are preselted i n Section 4.0. wind turbulence i n Section 5.G.

Thc use of the results as design gu(de1ines are presented for New Technoloqy and Referewes are outlined t n

7

Sections 6.0 and 7.0 respectively. Appendlx A pnrwts the analysts procedwe; Appe,ndix B, the wind tunncl rrarurad IU d d t r p r u s u m coefficients; id Appendix C, the ColorIldo State Unlversjty test nport for the test and test data reduction pertlnent to the dynr ic analysts.

a

3.0 TECHNICAL APPROACH, RESULTS AWD OISCUSSION

Becausf.: the pressure f l uc tua t i on on the arrays caused by the wind turbulence i s randan i n nature, random harmonic analysis techniques can be appl ied t o d e t e n t n e the dynwric response o f the arruys. The technical approach was t o

develop theore t lca l mathematical models o f arrays and solve f o r the response

of the arrays using dnd tunnel measured pressure f luc tua t ions on the arrays as the forcint: function. Pressure .measurements were recorded slmul taneously a t several l c t a t i o n s on r i g i d arrays dur ing a wind tunnel t e s t conducted a t

Colorado State Universi ty. These pressure data were then reduced t o y i e l d auto and cross spectra a t and between each pressure tap. The t e s t procedures

and resu l t s o f the w!nd tunnel t e s t i s presented i n Appendix C.

The s t ruc tu ra l dynamic response o f an array i s dependent on the s t ruc tu ra l charac ter is t i cs o f the array as wel l as the fo rc ing furrction. For t h i s study, tuo t yp i ca l arrays (Figure 3-1) were modeled theo re t i ca l l y u t i l i z i n g the modal approach; the nmde shapes are shown i n Appendix A. was a four post a r ray designed by JPL (see Ref. 5 f o r the de ta i led design conf igurat ion) and the second conf igurat ion was a two post a r r a y (support posts a t the center o f each end o f the array). the dynamic response o f the arrays were calculated and included the equation o f motion and pressure load equations on the arrays. procedure i s de ta i led i n Appendix A. fo rc ing funct ions (auto and cross spectra*) on these arrays were simulated from the wind tunnel t e s t f o r the wind condi t ions on the f i r s t a r ray of dn

a r r a y f i e l d exposed t o the wind, the f i r s t a r r a y protected by a fence, and the f i f t h a r ray w i th in an array f i e ld . The wind was considered t o be head-on t o the arrays.

The f i r s t conf igurat ion

The theoret ica l equations f s r

The theore t ica l The f l uc tua t i ng wind induced pressure

The wind tunnel t e s t resu l t s consis t ing o f auto and cross spectra o f the a r r a y pressure coe f f i c i en ts are o f s i g n i f i c a n t importance t o the dynamic response of

the arrays. Figure 3-2 shows a t yp i ca l normalized cross spectrun f o r the 5 th array located i n an array f i e l d w i t h a t i lt angle o f 35" and a 90 mph wind.

O f importance i n the cross spectrun i s tha t the phase angle i n a l l cases were

*The cross-spectrum re1 ates pressure versus frequency a t tw d i f f e r e n t po ints

( i and j); the auto-spectrum i s s i m i l a r except a t a s ing le po in t ( i = j ) . tenns dre used i n t h e analysis as shown i n Appendix A. d e c c r i p i ion o f the auto and c r o s s spectrum can be found i n References 3 and 4.

These A de ta i led technical

9

Conf lgurot ion I

b-1 2 0 f t t-4

Weight = 977 lbs F i r s t Plate Bending cbda = 10 Hz

conf igurot ion 11:

p-' 12 f t <-q

Weight - 933 l b s Pitch Mode = 3 H t

F i r s t Plate Bandfng Mode = 10 H t

zero. That i s , the pressure a t each pressure tap i s i n phase w i th each other pressure tap on the array. m e importance o f t h i s feature i s that only

symnetric modes o f v i b r a t i o n about the center line o f the a r r v can be exci ted by the wind turbulencc. Another important aspect o f the spectra Is t h a t for a

n inety mph wind the porrer i n the wind as Ind icated by the arrqy pressure

c o e f f i c i e n t spectra becomes i ns ign i f i can t (<1% o f naxiaun) a t frequencies

greater than 5 Hz.

.01

r l t i l i z i n j randm hdrmonic an;iysis techniques and the propert ies o f auto and

cross spectrums, 3 s t ruc tura l dynamic analysis was performed f o r the two array

conf igurat ions. The r e s u l t s presented as magnif icat ion fac to rs defined as the theoret ica l RMS dynamic load div ided by the experiiliental RMS pressure load i s shown i n f iyures 3-3 t o 3-5 f o r conf igurat ion I and f igures 3-6 t o 3-8 f o r

7 1

c+n f igu ra t i on 11. Figures 3-3 t o 3-5 show the ef fect o f locatio:! o f

configuratii,: I arrays i n an array f i e l d (1s t array wittrout a fence, 1s t a r r a y

protected by a fence, and 5 t h array w i t h i n an arrqy f i e l d , respect ively) . Each f i gu re a lso shows the e f f e c t of t he f i r s t modal s t ruc tu ra l frequency on the magni f icat ion f a c t o r and the e f f e c t o f l oca t i on on the array [ a i d chord L' ,d outer chord locat ions). .ocated near the t r a i l i n g edge o f t he array (downwind edge) and a t the mid chord locat ion. The magni f icat ion f a c t o r decreases w i th an increase i n the

C i r s t mode s t ruc tu ra l frequency. For the cond i t ion where the f i r s t d e frequency i s between one and two cysles/sec, t he magni f icat ion fac to rs change rap id ly . This i s caused by the combination of the very f l e x i b l e s t ructures 3s ind icated by the l o w natura l frequency) and the aerodynmtcs i n te rac t i ng

13 reduce the s t a b i l i t y o f the structure. A t frequencies above 3 i tr , the

s t ruc tu re becanes s t i f f enough SO t h a t t he i n t e r a c t i o n w i t h the aerodynamics

d.3es not present a probl%. The magni f icat ion fac to rs f o r the 5 th array and the 1 s t array protected by a fence tend t o be s l i g h t l y l a r g e r than the 1 s t

array that 1s unprotected. This i s expzcted since the fence an4 the uy#!nd a r r a y s w i l l increase the \eve! o f turbu'!ence I n the wind compared t o the wind f lowing over the 1 s t array.

I n general, t he la rges t magni f icat ion fac to rs are

The resu l t s f o r conf igurat ion I 1 ar rays ( f t gu res 3-6 t o 3-8) are s i m i l a r but

o f reduced i 'dgt i tude compared t o the r e s u l t s for Conf igurat ion I arrays. The ri!dbced magnitude i s a r e s u l t o f the l a rge r weight per u n i t surface area tha t i s required f o r s t i t i c load considerations by conf igurat ioa I 1 caapared t o

co i f i gu ra t i on I. As a r e s u l t o f the higher weight, l ess dynamic response

oc,w-s for the same 1s t node s t ruc tu ra l frequency. Also, t he magnif icat ton fac t l ) rs f o r the uu:er chord and the mid chord locat ions are very s im i la r i n shapt! and mab. 'tude f o r conf igurat ion 11.

Yn ' i ~ e r a l , the s t ruc tu ra l conf igurat ion (weight, s t i f f n e s s and shape) w i l l a f o t c t the dynamic response o f the structure.

stt"fness, the less the dynamic response for a given fo rc ing function. The wei:,ht and s t i r f ness o f the array considered as conf igurat ion I was about as l i g h t and r ? e x i b l e as possible and ye t maintain s t ruc tu ra l i n t e g r i t y f o r the

:t.;+i.. loads. The naninal f i r s t s t ruc tu ra l natura l frequency o f t h i s a r r a y t h a t would be exc i ted by the wind was j u s t s l i g h t below 10 Hz.

The la rger the weight and

For both

12

conf igurat ions, the s t ruc tu re whose 1 s t wind exc i ted s t ruc tu ra l frequency i s

below 5 Hz i s s u f f i c i e n t l y f l e x i b l e tha t I t i s questionable whether i t could wlthstand the s t a t i c loads w i t h s u f f i c i e n t confidence t o be a v iab le

structure. Because f o r conf igurat ion I t o have a lowest natura l frequency o f 5 HI requi res the s t ruc tu re t o be approximately 4 times as heavy or 4 times as f l e x i b l e as the nominal s t ruc tu ra l configuration, t he r e s u l t s for the

conf igur, t ion w i t h a 5 Hz frequency should be very conservative. If 5 Hz i s considered as the lower l i m i t f o r a var iab le structure, t he maximum magni f icat ion fac to r f o r conf igurat ion I i s 1.3 and conf igurat ion I 1 i s 1.08.

13

1=

c a L c, a t

* a

c cc c c: f,

x"

0 L 0 c L,

0

L c

e-

0 L 0 &E u L QI Y

d

F i g m 34. C o n ~ m t i o n I - Dynunic Ana'* Results for 1st Arny Procrchdfram ttm Windby8 Fence

15

,, a L I 4 c i$iq u

w 3

n

e l

,? d

' ? n u \ v) Y

c j

..I N . 1

c u a

Figt~n 3-5.

U

n . \ r L 0 Y V 1 LL

Y

C 0 c

U I: .c-

c I I 46

a 0 c U L a? Y 3 0

I 1 sl- . 9 (\r

m d U c u s .

cf I

Conf@rstion I - Dynamic Andysis R w l t s for 5th Array within an Array Fhld

2, U E Q) q L u- c m L a * < z

16

Y

"32,

cr c Q,

U

b b 0) 0 0

c

.r

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c, (0 0

c C m

.r

c

.r

2

P 0 c u L aJ 2

Q 0 8 0 c

cr - . LL

X

0 c

Fiwm 3-6. Canfigurothn II - Dynamic Analpsis Results for 1st Army Unprommd from the Wind

17

a

0 C aJ u Y- cc

u Q) L 3 cn cn Q) L a. v) X oe:

.r

.r

8 .)

c rp c)

c li k X

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a e V

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0 0

/ ./

/ \ ' I / \ / I

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i I /

J f I

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FWm 3-7. Conf@mtion I1 - Dynvnk And@$ Rewrits for ts t Array Pmthcard from the Wind by I Fence

18

m L 0 L: u 0 t .c

b&? U L L

4 Y

E

2 0 c u L a c, a

/ / '/

/ /

I (tl o v

0 c

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Figum 3-8. Configurstion lI - Dynamic Analysis Results for 5th Army within 817 Army F idd

19

4.0 CONCLUSIONS

Although the s t ructura l conf igurat ion af fects the dynamic response due t o wind turbulence, the general trends o f the t w o configurations studied are s imi lar . The mid chord locat ion on an array has the largest response and the largest magnif icat ion factors on !IC, h configurations. The outer chord magnif icat ion fac to r o f conf igurat ion I i s considerably lower than the mid chord factor because of the corner constraints, whereas f o r conf igurat ion 11, the outer chord magnif icat ion fac to r i s only s l i g h t l y reduced compared t o the mid chord. The largest dynamic response occ!rrs riear thz downwind edgs o f the array and decreases as the 1 s t s t ructura l modal frequency incredses. ”’ .;, t q minimize the dynamic loads due t o turbulence requires the maximum p r ~1 s t i f fness for the structure. A dynamic magnif icat ion factor o f 1.3 ap t o be a conservative fac to r f o r any type o f f l a t p la te photovoltaic array strucfure.

It should be noted tha t f o r the types of structures used i n t h i s analysis, i f the 1s t wind excited s t ructura l frequency i s below 5 Hz the structure i s s u f f i c i e n t l y f l e x i b l e tha t i t i s questionable whether i t could withstand the s t a t i c loads wi th s u f f i c i e n t confidence t o be a v iab le structure. conf igurat ion I t o have a lowest natural frequency o f 5 Hz requirzs the structure t o be approximately 4 times as heavy o r 4 times 8s f l e x i b l e as the nominal s t ructura l configuration, thus, the resu? t s f o r the conf igurat ion w i th a 5 Hz frequency should be very Conservative, lower l i m i t f o r a var iable structure, the maximn magnif icat ion factor f o r conf igurat ion I i s 1.3 and conf igurat ion I 1 i s 1.08.

For

If 5 Ht i s considered as the

5.0 ARRAY DESIGN GUIDELINES FOR WIND TURBULENCE

Based on the resu l ts o f the boundary layer wind tunnel t e s t and the resu t h i s dy3amic analysis, the following design guidelines are given f o r determining wind loading on photovoltaic f l a t p la te arrays f o r the turbu

t S o f

ent por t ion o f the wind. These guidellnes are v a l i d fer arrays a t l eas t 2 chord lengths from the edge and f o r a wind ve loc i ty p r o f i l e t ha t approximates a !/7 power law. dynamic sense, and because o f the conservatism i n t h i s analysis, the wind designs loads calculated using these gu'del i r e s would probably be a1 so adequate f o r the side edge arrays.

However, because o f the edge ccnstraints o f the arrays i n a

F i !re 5-1 presents guidelines t o ca lcu late design pressure loads on array surfaces due t o the non-steady poption o t the wind. (The desiprl guidelines f o r thc steady s tate por t ion 07 the wind t h a t needs t o be coqbined w i th the turbulent pot iAon f o r the t o t a l design load was documented i n reference 1.) This procedure requires a number o f decisions which include.; 1) the choice o f c o n f i p r a t i o n I o r 11 must be representative o f the design under consideration; 2) the choice o f ths rms unsteady pressure coef f i c ien ts f o r the appropiate design array t!!t i ng le and array loca t ion from the wind tunnel resu l ts presected i n Appendix B; 3) the leve l o f p w b a b i l i t y not t o exceed a given level (i.e., la = 68.27%, 2 u = 95.45%, 3 a = 99.73%); 4) the design configuration f i r s t s t ructura l natural frequency tha t can be excited by the wind turbulence; 5) the design f ree stream bind speed. The appropriate magnification fact p q from f igure 3-3 t o 3-8 mul t i p l i ed by the rms unstesdy presswe coef f i c ien t chosen from Appendix B i s the design rms pressure coef f i c ien t f o r wind turbulence wi th a 68.3% probab i l i t y o f not exceeding. For a higher p robab i l i t y such 2s 95.5%. t h i s design rms pressure coef f i c ien t would be mul t ip l ied by 2.0. p roc ed u re.

The fol lowing i s an example ii,,piying t h i s

Assume tha t the design i s s im i la r t o configuration I , has a t i l t angle o f 35" and the f i r s t s t ructura l frr ,uency normal t v the array I S 5 Hz. F o r any i n t e r i o r arrays, the rms unsteady pressure coef f i c ien t f o r t h i s condit ion extracted from Appendix B i s rpshown i n f igure 5-2. The magnification fac to r (M.F.) f o r 5 Hz a t the mid span lcca t ion ( f i gu re 3-5) and these M.F.

mu l t ip l ied by the r m s ~ C p from f igure 5-1 i s shown i n tab le 6.1 as a function o f chord position. Assurning tha t a leve l o f p robab i l i t y not t o exceea 95.5%

23

Step A (Decisions)

Design configuration matches ? Configuration 1 (See

I 1 Figure 3-1)

( 3 )

t o exceed 68.27% = 1 0 95.45% = 2cr

Design k ind speed a t

decisions 1 and 2

Step c

unsteady pressure coe f f i c i en t from decision 3 = design pressure coeff icient w i th probabi 1 i ty not t o exceed 68.275 I

I I

Step D + Mult ip ly Step C by leve l o f p robab i l i t y not to exceed from decision 4.

f I

Mul t ip ly Step 0 by dynamic I wind pressure from decision 5 = design pressure load on array surface due t o non-steady port ion of wind -

Fijpre 5- 1. Flow Chart +o Ca/culat.t 4rray p. the NonSteedy Ponicb,; . fh

24

::*. .r due ro

o f the design load due t o turbulence i s required, t he ns aCp times the M.F.

I s increased by a f ac to r o f 2. The design de l ta pressure c o e f f i c i e n t f o r the

turbulent por t ion o f the wind i s shown i n tab le 5-1 and i n f i gu re 5-3. pressure due t o a 90 z$ wind i s also shown i n tab le 5-1 and f i gu re 5-3.

The

The design load fo r the f r o n t array ( e i t h e r protected o r unprotected by a fence) i s calcuiated I n the same manner using the appropriate rms de l ta pressure coe f f i c i en t and magnif icat ion factors.

s impi f led by u t i l i z i n g the highest M.F. (M.F. - 1.3 f o r modal frequencies > E tk) f o r a l l conditions.

conf igurat ion natural frequency as wel l as not requ i r ing the determination o f

the M.F. along the chord from f igures 3-3 t o 3-8.

Thi s procedure can be

This would e l iminate the need t o determine the design

25

T i l t Angle = 35’

ACP

/ / /

.2 . 4 .6 .8 1 .o Fraction o f Chord (S/C)

8.2

6.15

P (PSf)

4 . 1

2. os

.4

. 3

P /

/

. 2 . 4 - 6 .8 1 .o Fract ion o f Chord (S/C)

F@um 5 4 Exampk l?asqn Wind L &s on Interior A m ys in an Ana y Frild Using the Desw Guidelines

26

Steps

Figure 5-1

Step A

Step A

Step B

Step C

Step 0

Step E

Rms ACp (see Figure 5-2)

q (psf) (90 mph wind)

H.F. (see Figure 3-51

Design A C p ( 2 0 ) (M.F.) (rmsACp)

Design Pressures ( P S f )

S/C -1 96(T.E

-104

20.5

1.19

.1238

.2475

5.07

.74

.095

1.27

.1207

-241 3

4.95

.26

-133

-

1.17

. i 556

.3112

6.38

06(L.E 3 ,2022

T.E. = T r a i l i n g Edge

L.i - Leading Edge

27

6.0 NEU TECANOLOGY

No reportable i t e m s o f new technology have been i d e n t i f i e d by b e i n g during the contract o f t h i s work.

23

7.0 REFERENCES

1. "Wind Loads on Fla t P l a t e Photovoltaic Array Fields", D C t l J P L - No. 954833-8113, Boei ng Engi neeri ng and Construct1 on Company, Apri 1 , 1981.

2. Miller, R. D.; and Graham, M. L.: "Random Harmonic Analysis Program, L221 (TEV156) Volume I ; Engi neeri ng and Usage". NASA CR-2857, 1979.

3. Bendat, J. S. and Piersol , A. G. : "Randan Data: Analysis and Measurement Procedures", (1971), John Wiley and Sons, Inc.

4. Crandall, S. H., "Random Vibrations", (1958), MIT and John Wiley and Sons, Inc.

5. Wilson, A.H. : " L w Cost Solar Array Structure Development," DOE/JPL - 1012-53, J e t Propulsion Laboratory, Pasadena, Cal i fornia , June 15, 1981.

31

Appendix A - Analysis Procedure

A mathematical model o f the photovol ta ic array included the important s t ruc tu ra l , mass and aerodynamic charac ter is t i cs necessary t o sirnul a te the

dynamic response o f the array t o the turbulence generated by the upwind arrays

and tha t contained i n the f r e e stream wind.

The s t ruc tu ra l model included the e l a s t i c charac ter is t i cs o f the complete

array. A f i n i t e element program NASTRAN was used t o model the s t ructure and t o ca lcu late modes o f v ibrat ion. The mode shapes f o r the two conf igurat ions t h a t can be exci ted by aerodynamic turbulence are shown i n Figure A-1. General i r e d aerodynamic forces representing the aerodynamics on the array were

computed using the steady s ta te pressure d i s t r i b u t i o n s measured i n Phase 111

(reference 1) and combined w i t h the modes o f v i b r a t i o n t o produce the fo l lowing equations o f motion f o r use i n a Boeing developed Ranurn Harmonic:; analysis computer program (reference 2).

M i ] generalized s t ruc tu ra l s t i f f n e s s and s t ruc tu ra l damping matr ix generalized mass m a t r i x

M4 M31 aerodynamic s t i f f n e s s matr ix Ms] aerodynamic damping matr ix

q 1 [ C]

l a p 1

vector of general ized coordinates matr ix of generalized forces associated w i t h u n i t pressures a t each o f the e x c i t a t i c n poiots vector o f e x c i t a t i o n prcssures

A corresponding set o f load equations using the force summation method was developed i n the fom,

Where 1 L i s the load vectcr and t h e [ & ] matrices are d s defined f o r the response equations, except t h a t they represent generalized load coef f i c ien ts .

A1

The auto and cross power spectral dens i t ies of the pressures measured i n the wind tunnel was used t o exc i te the array mathematical d e l . The loads output

spectrum was calculated u t i l i z i n g the fo l lowing equation:

where

X i (b, denotes conjugate

$ i j

t rans fer funct ion f o r system response t o e x c i t a t i o n i

i = j auto power spectral densi ty f o r pressure i i Z j cross power spectral densi ty between pressures i and j

In tegra t ion of the output spectrum produces the rms load response

Because the pressure taps on the wind tunnel model i n the Phase I 1 1 study were located on only one spanwise stat ion, other spanwise s tat ions were assumed t o

function o f u n i t y spanwi se

each spanwise s t a t i o n i s a However, the assumption o f pressure taps i s conservat

The assumpt V a l i d assumpt a c o r r e l a t i o n

ve.

have an i d e n t i c a l auto- and cross-spectrum chordwise and a cor re la t ion

on o f the same chordwise spectr

on since the process i s random. funct ion o f u n i t y between spanw

m a t

se

A 2

I [ , ii/ i

Configurat ion I I I / . I

F

I ;

! / Conf i gura t i on I 1 ;

f

Frequency Ratio: Mode 2/Mode 1 = 1.6

Fipm A- 1. A m M o d s sharpss ORIGINAL PAGE IS OF POOR QUALITY

A3

Configuration 2 I ,/ I

/

Frequency Ratio: Mode 3 / W e 1 = 3 . 3

FrPm, A-?. A m y Made shQy3BI lContirwd)

A4

Configuration I1 i

Frequency Ratio; Mode 2/Mode 1 = 3.6

A5

Appendix B

Wind Tunnel Test RM5 Unsteady Del ta Pressure Coef f i c ien ts

RMS pressure coe f f i c i en ts shown i n Figures B-1 t o B-3 of the f l uc tua t i ng pressures were obtained from the wind tunnel t e s t r e s u l t s reported i n reference 1. These coe f f i c i en ts were obtained fo r th ree tilt angles ( Z O O , 35" and 6 0 ° ) , and f o r t he f i r s t ar ray o f an array f i e l d unprotected and protected

by a fence, the f i f t h ar ray w i t h i n an array f i e l d , and f o r wind d i rec t i ons from the f r o n t and rear. The pressure coef f ic ients were ca lcu lated based on a wind reference v e l o c i t y a t 10 meters and a 1/7 power law wind ve loc i t y p ro f i 1 e.

, _ *.

61

T i l t Angle = 35'

.1

I -

Tilt Angle = 60'

- . .... - - --- .I -_ 1

T i l t Angle = 60'

-- / / /

.2 . 4 .6 .8 1 .o Fraction of CSord (S/C!

RMS Pressure Coefficients an the First Army of an Unprotected Array Field

82

T i 1 t Angle = 20’

AcP . I

/ / /

T i l t Angle = 35’ .2 I

A c P . l 1

Ti l t Angle = 35’

A%:: 1

I .2x 1 I 7

/h T i l t Ang:e = 6r”

I-+ p/ A%:: I -/----’ --

T i l t Angle = 6G0

ACP . 1

/ / / 1 .O .2 .4 .6 .8

Frac t i on o f Chord ( S / C f

Fi@m B-2. 8MS Pressure Coefficients on th First Array of an Army Fhld Protected by a Fen-

83

T i l t Angle = 20'

*CP

T i l t Angle = 20'

T i l t Angle = 35' - 2 I

Acp - J l I a . 8

T i l t Angle = 35'

T i l t Angle = 65* - * I Acp t

T i l t Angle = 60'

A C P q .I/, .2 . 4 .6 .9 1 .O

Fraction o f Chord ( S / c )

wirhin an Array Field Figure 8-3. RMS P m r e Coefficimts on the Fifth Array

A / ' /

e/- P -- A

/ / /

B4

APPENDIX C

UIND LOADS ON FLAT PLATE PHOTOVOLTAIC ARRAY FIELDS

(Spectra Analysis)

PHASE I V - FINAL REPORT

WIND TUNNEL TEST REPORT

Prepared f o r J e t Propul siofr Laboratory under Contract 954833

Boeing Engineering and Construction Company

( A D iv is ion of The Boeing Company)

Seat t le . Washington 98124

WIND PRESSURES AND FORCES ON FLAT-PLATE PHOTOVOLTAIC SOLAR ARRAYS--

CROSS-SPECTRAL ANALYSIS

N . Hosoya,* J . A. Peterka,** and J. E. Cennak***

f o r

Boeing Engineering and Construction Company P.O. Box 3707

Seatt le , Washington 98124

Fluid Dynamics and Diffusion Laboratory C i v i l Engineering Department

Colorado State University Fort Collins, Colorado 80523

Project 5-36022

June 1981

*Graduate Research Assistant **Associate Professor

***Professor-in-Charge, Fluid Mechanics and Wind Engineering Program CEH80- 8 lNfI-JAP-JEC57

TABLE OF CONTENTS

Section

1

2

LlST OF FIGURES . . . . . . . . . . . . . . . . . . LlST OF TABLES . . . . . . . . . . . . . . . . . . LlST OF SYMBOLS . . . . . . . . . . . . . . . . . . PREFACE . . . . . . . . . . . . . . . . . . . . . . PRESSURE TAPS AND ARRAY (‘ONFIGURATIONS . . . . . . 1.1 Pressure Tdps . . . . . . . . . . . . . . . . 1.2 Array Configurations . . . . . . . . . . . DATA ACQUISITION . . . . . . . . . . . . . . . . . 2.1 Pressure and Normal Force Coeff ic ien ts . . . . 2.2 Pitching Moment Coeff ic ien ts . . . . . . . . . 2.3 Spectral Analysis . . . . . . . . . . . . . .

2.3.1 Matrix Arrangement . . . . . . . . . . 2.3.2 Correlat ion Functions . . . . . . . . . 1 . 3 . 3 Power Spectral Dens i ty Functions . . .

3 RESULTS AND DISCUSSIONS . . . . . . . . . . . . . . 3.1 Pressure and Normal Force Coeff ic ien ts . . . . 3 . 1 Power Spectral Analysis . . . . . . . . . . .

J CONCLUSIONS . . . . . . . . . . . . . . . . . . . . REFERENCES . . . . . . . . . . . . . . . . . . . . FIGURES . . . . . . . . . . . . . . . . . . . . . . TABLES . . . . . . . . . . . . . . . . . . . . . .

i i i

i v

v

v i

1 7 c

2 3 3

J

- ..,

6

3

J .i

iii

LIST OF FIGURES

Pape

10

11

Figure

1 Position of Pressure Taps on Instrumented Model . . . Schematic Description of Army Field . . . . . . . Time Trace of t he Normal Force and Pitching Moment Coefficients . . . . . . . . . . . . . . . . .

2

3 12

13 Auto-Spectra f o r F i r s t Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 3 5 O b 4

Auto-Spectra for F i r s t Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 60°, 5

14

Auto-Spectra for F i r s t Array a t with Fence . . . . . . . . . . . . . . . . . . . . . a = 14S0, 6

15

Auto-Spectra for F i r s t Array a t with Fence . . . . . . . . . . . . . . . . . . . . . a = 160°, f

16

Auto-Spectra for F i f t h Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 20°, S

3

1 0

11

12

15

14

15

17

Autc-Spectra f o r F i f t h Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 35",

18

Cross-Spectra for First Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 3S0,

19

Cross-Spectra for First Array a t without Fence . . . . . . . . . . . . . . . . . . . . a = 60° ,

23

Cross-Spectra for First Array a t with Fence . . . . . . . . . . . . . . . . . . . . . a = l .So ,

27

Cross-Spectra f o r First Array a t w i t h Fence . . . . . . . . . . . . . . . . . . . . . 3 = 160°,

31

Cross-Spectra f o r F i f t h Array a t without F e n c e . . . . . . . . . . . . . . . . . . . . a * 20°,

35

Cross-Spectra f o r F i f t h Array at without Fence . . . . . . . . . . . . . . . . . . . . u = 3S0,

39

i v

LIST OF TABLES

'Yi iDlC _c_

1 List of Array Configurations Testcd . . . . . . . . . - Time-Averaged Pressure Coefficients . . . . . . . . .

Comparison of Integration of Auto-Spectra . . . . . . 3

V

LIST OF SYMBOLS

Symbol

c

FN

ti - t M

N

N"

P

OP

L

Definition

chord length of solar array (4 in =del, 8 ft full-scale)

pitching moment ceefficient

normal force coefficient

instantaneous pressure coefficient

fluctuation of pressure coefficient

mean pressure coefficient

root-mean-square of pressure coefficient

eccentricity of center of pressure from mid-chord

normal force per unit area

fence height

pitching moment per mit ared about center of chord

frequency N-c normalized frequency = - 'ref

local pressure on collector

pressure difference across co1:ector

reference dynamic pressure at 10 m full-scale

correlation function

reference wind velocity (approx. 43 fps model) at 10 m height

fence distance

angle of attack

time lag

power spectral function

normal ized power spectral function

v i

PREFACE

Simultaneous pressure measurements on flat plate photovoltaic solar

arrays were conducted in a simulated atmospheric boundary layer charac-

terized by a 1/7th power law mean velocity distribution.

describes random properties of fluctuating local pressures on solar

ari2j.s by cross-spectral analysis.

include the auto- and cross-spectra for fluctuating pressures for several

typical array configurations.

This report

The random properties to be analyzed

The essential experimental configurations, including facilities,

wind-tunnel models, instrumentation and the flow simulation technique,

have been described in a preceding report [SI "Wind Pressures and Forces

on Flat-Plate Photovoltaic Solar Arrays," Colorado State University

Report CER80-81NH-JAP-MP-JEC13.

presented in this supplementary report required some additional arrange-

ments for instrumentation and data acquisition. These arrangements will

be described in Sections 1 and 2 of this report.

The simultaneous pressure measurements

v i i

1 .O PRESSURE TAPS AND ARRAY CONFIGURATIONS

1.1 Pressure Taps

The simultaneous pressure measurements were obtained at four

pressure taps along the chord on the upstream surface and four pressure

taps on the downstream surface of the solar array. These pressure taps

tested were taps 1, 4, 7 and 10 on the upstream surface, and taps 11,

* 14, 17 and 20 on the downstream surface. The location of each pressure

.tap is shown in Figure 1 which duplicates Figure 10 of the preceding

report.

1.2 A m y Configurations

All array configurations tested for this phase of the study are

listed in Table 1.

cates Figure 11 in the preceding report.

rations are chosen to be presented and discussed in this report.

this study, the ground clearance and spacing of solar a&ys were 0.25 c

and 2.0 7 respectively, and were not varied.

for all array configurations.

are seen in Figure 2..

Array locations are shom'in Figure 2 which dupli-

Several typical a m y configu-

For

The wind direction was Oo

The definitions of these test parameters

2.0 DATA ACQUISITION

The mean wind velocity outside the simulated turbulent boundary

layer in the Meteorological Wind Tunnel, shown in Figure 1 of the main

report, was approximately 50 fps as was used for the preceding tests

(giving a Uref L 43 fps). The outputs from the 8 pressure transducers

were recorded simultaneously on digital magnetic tape for 35 seconds at

500 sampl.es oer second using a data acquisition system based on a

Hewlett-Packard System 1000 minicomputer.

by the same computer.

The data wer. then analyzed

Pressure, normal force and pitching m e - t

C 2

coefficients presented in this report

dynamic pressure at 10 m elevation in

were referenced

the prototype.

densities of local peak pressures were evaluated with

coefficients.

2.1 Pressure and Nom1 Force Coefficients

to the m u n

The power spectral

these pressure

Pressure and normal force coefficients are respectively defined in

Sections 3.2 and 3.3 of

apply to this report:

c = - P P %ef

C

the preceding report. The s u e definitions

2.2 Pitching Moaent Coefficients

Pitching moment coefficients, CH1 about the mid-chord o f the solar

array ate defined by

M

%t?f ' 5 4 % where M is the calculated pitching mment per unit surface area about

the mid-chord of the array,

approaching flow at 10 Q full-scale, and c

is the reference dynamic pressure of .-

is the chord length of the

array. In order to calculate M, the eccentricity of the center of

pressure, e, was obtained numerically by a curve fitting to the pressure

distribution using linear interpolation schsaes along the chord of the

array. Thus

M * - FN.e

where FN is the normal force-

'pitching moment lifting windward edge up is defined t o be positive.

c 3

2.3 Spectral Analysis

2.3.1 Matrix Arrangement

A set of cross-power spectral densities of local pressure

fluctuations on the solar array were calculated. For simplicity of

notatior., the pressure tnp numbers 1,4,7,10,11,14,17 and 20 can be

renamed as 1,2,3,...,8 in order. Any pressure tap number used in

this report will be, hereafter, referred to as the renamed one.

Matrices for the correlation and power spectral densities are

written as follows.

auto-correlation , if i = j

cross-correlation, if i # j [ R i j O ] = I

auto-spectrum , if i = j

cross-spectrum , if i # j

where T is time lag, N is frequency, and i , j refcr to the pressure

t3p numbers.

2 . 3 . 2 Correlation Functions

For this analysis, auto- and cross-correlation functions are defined

where C ( t ) and Cpmeank are instantaneous and time-averaged values

of pressure coefficient with respect t o tap number k. Pli

Defining a

fluctuating component of pressure coefficient, C' (t), as pk

C' (t) = c (t) - c pk pk pmeank

c 4

the auto-correlation function, at T - 0, becomes for m e tap

Rii(0) = C' (t) C' (t) Pi Pi

C2 PrmS i

in which an overbar denotes a time averaging.

2.3.3 Power Spectral Density Functions

Power spectral density functions are defined by

OD

a . .(N) = 4 I R. .(T) exp(-j2nNr)d~ 1J 1J

0

where j appearing in the exponential function

l?:e integral property of the suto-spectral

m 1 Oii(N)dN = Rii(0) = C2 i 0 %IS

(7)

refers to j2 = -1.

function requires

(8)

Because of this property, the power spectral density can be normalized

@?.IN) I? = 'C with units [ k] P,S

i j P,S

It is also common practice to nonnalize the frequency N by

Nc 'ref

N* = -

( 9 )

is the reference wind velocity. 'ref where

The cross-spectral analysis was performed digitally by a Fomier

Transform subroutine using standard techniques such as those in

references [l] and [3 ] . Transforms were perfonned on 8 time segments,

each 2048 samples in length, for each pressure record (recorded at

500 samples per second).

and appropriate averaging across da ta points i n frequency and time seg-

ments was performed t o reduce normalized standard e r r o r o - t he spectrum.

Because of memory l imi ta t ions , t he cross-spectra were only ca lcu la ted t o

a maximum frequency of 125 Hz (N*= 1.0).

a l l the energy i n the f luc tua t ing pressures.

of the cross-spectra reached a maximum of 11 percent a t the lower

frequencies.

Transforms were combined t o form cross-spectra

This frequency retained virtual11

Normalized standard e r r o r

3.0 RESULTS AND DISCUSSIONS

3.1 Pressure and Normal Force Coeff ic ients

Examples of time t r aces of noma1 force and pi tching moment

coe f f i c i en t s a r e shown i n Figure 3 obtained with the f i r s t so l a r array

a t

coe f f i c i en t s a r e tabulated i n Table 2 f o r s i x typ ica l a r ray configurat ions,

including P.uns 21331, 21344, 21346, 21352 and 21368. These values agree

well with those found i n the preceding report [SI.

3.2 Power Spectral Analysis

a = 35", without fence. The time-averaged mean and rms pressure

The auto-spectra l o r each pressure t ap a r e show. i n Figures 4

through 9 f o r the s i x cases l i s t e d in Table 2 .

spectra f o r the individual pressure taps and f o r both surfaces of the

so l a r array, the power spectra show t h a t t h e energy is sh i f t ed t o higher

frequencies f o r the f i r s t array with fence and the f i f t h array, consis tent

with t h e smaller sca les of turbulence experted behind the fence and within

the array f i e l d . The r ea r s ide of t h e f i r s t array a l s o shows t h i s e f f e c t .

'The area uncier t h e auto-spectra over the ava i lab le frequency range is

compared t o the measured variance i n f luc tua t ing pressure i n Table 3.

Figures 4 through 9 c lea r ly show t h a t the energy content i n the power

Comparing t h e auto-

C6

spectra is concentrated at N* c 0.1. At N* -=. 0.3, 9 spike in the

power spectra is seer1 for both the upstream and downstream surf: cos.

This spike in the power spectra is due to vibration of the wind-tunnel

model. Moreover, for some cases, another spiire at N* f~ 0.4 was obtained

which is a second mode of vibration of the model.

model the full-scale structural stiffness or damping so that these

response spikes do not indicate full-scale response.

No attempt was made to

Both the real and imaginary parts of the auto- and the cross-9pectra

were calculated.

The imaginarv rlart was essentially zero for all cases. The implication

of this finding is that the phase angles of the various frequencies are

uncorrelated.

fluctuations in turbulent Sowiary layer f3 ows.

Only the real part is shown in Figures 4 through 9.

This is a typical result in velocity or pressure

Figures 10 through 15 show cross-spectral plots for all combinations

of taps for each of the six cases listed in Table 2.

each array were grouped on plots so that similarly-shap- * -irves would

appear on the same plot. The plots reveal tht same increase in energy

at higher frequencies for arrays behind the fence and withiri ';.he array

field as was observed for the auto-sgsctra.

frequency of vibration shows in the plots.

Cross-spec:ra for

Also, the model natural

4.0 CONCLUSIONS

On the basis of the data presented in previous sections, the

following conclusions can be drawn.

1. Auto-spectra of local pressure fluctuations characteristically

f a l l rapidly with increasing frequency.

2 . Auto-spectra show higher energy at the high^ frqiencies

where the pressure tap is within the array field cr uk,L-rid a wind fence.

c 7

3. Cross-spectra shouxl prcperties similar to those in 1 and 2

above.

4. Cross-specrra between taps in separated zones were quite

similar . 5 . Ihe imaginary pa;ts of both auto- and cross-spectra were

essentiai :y zero.

C8

REFERENCES

1.

2.

3.

4.

5 .

Akins; R. E . and J. Densities and Corrciations Using Digital FFT Techniques," (1975), Colorado Sta t e University Report CER75-76REA-JAP13, Fort Collins.

. Peterka, "Cmputatlm of Power Spectral

Bearran, P. W . , "Wind L a d s on Structures i n Turbulent Flow," (1971), Ptoc. Modern Design of Wind Sensi t ive Structures , Paper 4, pp. 57-64.

Bendat, 2 . S. and A. G. Piersol , "Rucdom Data: Analysis and Measurement Procedures," j1971), 3 0 h hlilsy and Sons 11%:.

C e m a k , i. E., "Aerodynamics of Buildings," (1976), Annwl Review of Fluid Mechanics, Vol. 8, pp. 75-106.

Hosoya, N., J . A. Peterka, M. Poreh, and J. E. C e m k , tlUbrdi Forces and Pressures on Fiat-Pls te Photovoltsic Solar Arrays," (1980), Fluid Mechanics and Uind Engineering Prograa, Colorado S ta t e Unirersi.ty Report CERSO-8lEiH-JAP-WP-JErX3, Fort Coi i inr , Sept sm).,cr.

c 9

FIGURES

c 10

Figure 1. Position of Pressure Taps on Instrumented Model

c 11

ARRAY I ARRAY 2 ARRAY 3

WIND -

~ FENCE ”‘ .-PRESSURE TAPS FOR EDGE ’ AND CORNER STUDIES

... .. .. ..

. ._ - CENTER

PRES

..... ....

IRE TAPS

Figure 2 . Schematic Uescription of Array Field

c 12

1

cw( I

.8

.6

1 1

t 1 1 1 1

9.8 9.5 18.0 18.5 11.0

m 1 1 1 1

-. 1 LL 9.0 9.5 18.8 18.5 11.8

Figure 3. Time Trace of the Normal Force and Pitching Moment Coefficients

C 13

r; t a

x m k L4 U U v) k

m k u U a a 0 u 1 4

T

C 14

cn

c

I- L c

L

t-

k 0 rw ;d k Y

I 0

c 1s

A U')

v) v

u) t -

h P

w W

\ .

v, P

<ZH/ I> WtU133dS O l n V Q3ZIlVWklON

L

c c

I

I

i

Q, u E 0) U.

k 0 w (d k c,

I 0 e, 3 4

9

al

C 16

ll

<ZH/ I> WnU33dS OlnV a3ZIlVWtfON x Q k k -2

- . .. _ I .

n Q>

C 17

ai U

n P-

r- W

n

W -1 c3 z U

0 c u .r(

3

0 0 F\1

0 rr

C 18

n fn E n ln v) U

d W .J a z a

v; * >- a rr! Q: a

.? 1

1 v n Pl)

vi m

t i

c, d x d 4 k 4

5 cu *+-I LL

k 0 cu d k c, u Q, a

c 19

r CI h

t 1

t 1

0 v) M II

e lu

9 k k 4 +J

VI k 4 L

lu k c, 0 Q

I R I

! ? ! V I VI 0 k U

c 20

c J CZH/I) WnU133dS SSOU3 C3ZIlVWUON

- 0 ln M

1 U

f

<ZW/I) WnC1133dS SSOW OJZflVWtlON

c 21

1

I

c 22

ARRAY# I , ANGLE - 35, <4,5),<4,6)

Ftpurc 10. CrosJ-Spectra for Fir3t Array at ( colic ludod)

a * 3S0, without Fence

C 23

C 24

I . . - I . . . I . . I I . . . I . . . - f f E -. g - ig

t 1

.

c 2s

n QD

F-'

3 v

. CD v n

4

a w d 3 c

n OD ; v . n h

+' v n 0 i

i

U

n In

v

9 I

I . - * 1 . . . 1 . . . L . . . 1 . . . 1

6 iil. E 9

tJ (d

x a k k U c, m k -l+

L L

k 0 cu

I . . . 1 . . . I . . . c

1 H .. !! - ta - r! CZW/I) WnUl33dS SSOW 03ZIlVWdON

C 26

A R R A Y I I, ANGLE - 60. (2,5),~5,6).(5,7).(5.8)

l0.BBBBr 1 , v . , . . . * , I . " 1 . . . .

.eie .I@ I .me 10.888 NORflALIXD FREWENCY

Figure 11. Cross-Spectra for First Array at (concluded)

il = 60° , without Fence

C 27

2 f

1 t i

Q, u c 0 LL

c e, .r(

L . 0 rr, d

II

U

u m

0 ru ld k .> a Q) a m I m m 0 k c)

It I

C 29

1 II

d

rd k k d e, v) k *rl L

k 0 cu rd k

C 30

1.m - n N E

2 .lm-

W

I:

t- u W h UJ

UJ v) 0

u w N H -1 U x aL 0 Z

a n .elm -

.eel0 -

Figure 12. Cross-Spectra f o r First Array at (concluded)

a = 145", with Fence

c 31

t 4

c 32

n (D

s; G U

I I

n Q,

4 CI. a 2

h 0

u-4

:- I . - . 1 . . . I . . . I

(B F f 4 ' g f-'

c 33

c 34

ARRAY+ I , ANGLE - 169, C1,2>,<1,3),C1,4~,C2.4)

i*.mr ---- v) v) 0 8

Figure 13. Cross-Spectra for First Array at (concluded)

a = 160°, w i t h Fence

c 35

- 0 C N

1 II

c i

C36

A ‘0

c

n W

i

c 37

Figure 14. Cross-Spectra f o r Fifth Array at [ c onc 1 uded)

a = 20'. without Fence

c 39

n IC

v 4 . n h

n h

(u v n h

L

v

vi m 8

W d 4

t 1

C 40

n u)

vi m

I

u) (3

c * U p: rY <

F 0 w K)

%l i

k 0 w

1 . - . I . . - 1 . . . 1 . . I . . . - a - E! '9 . ig

c 41

H

C 42

J

d i e .!ee I .BBB 18.888 NWIALIZED FREOJENCY

Figure 15. Cross-Spectra for Fifth Array at (concluded)

a = 3S0, without Fence

c 43

TABLES

c 44

Table 1. List of Array Configurations Tested

a Fence* *

Array

21323 120 1 0.75 2.5 30

21325 145 5 -- -- -- 21527 145 2

21 331 145 1 0.75 2.5 30

21344 160 1 0.75 2.5 30

21348 20 1

21350 20 1 0.75 2.5 30

-- -- -- 21 356 35 1

21358 35 1 0.75 2.5 30

-- -- -- 21368 60 1

2 1370 60 1 0.75 2.5 30 -

*edge study (see Section 2.3 of the preceding report for the definition) **see Figure 2 for definition of parameters

c 45

Table 2a. Time-.tveraged Pyessure Coefficients

R u n 21331* Rur 21344* R u n 21346

145 1 160 1 20 5 a Array a Array a Array

Tap (original) L e a n ‘ ~ m s ‘bean ‘ ~ m s h e a n cPrms

1 (1) -.083 .OS1 -.lo5 .OS2 .020 .os1

2 (4) -.Oh8 -057 -.O% .061 .os5 .os0

3 (7) -.088 .078 -.133 ,072 .075 .069

4 (10) - . lo7 .068 -.lo9 .060 .077 .113

5 (11) -,I34 -040 -.096 .040 -.196 .210

6 (14) -.136 .039 -.096 .a38 -.116 -121

7 (171 -.I53 -042 -.lii .049 -.lo3 .078

8 (20) -. 131 .044 - . l o 7 .045 -.OS4 .072 - -.. - *with fence

C 46

Table 2b. Time-Averaged Pressure Coefficients

Run 21352 Run 21356 Run 21368

35 5 35 1 60 1 a Array a Array a Array

Tap (original) Cpmea* cP*s Cpmean cPrms %mean Cpms

1 (1) -.013 .060 .087 .OS4 ,223 .089

.376 ,095 2 (4) .001 .062 .292 .067

3 ( 7 ) -.004 -081 .436 .1C7 .482 .148

4 (10) -.002 .120 .451 .178 .279 .177

-.140 .147 -.300 .081 -,417 .066

6 (14) -.IO4 .lo9 -.290 .073 -.398 .063

7 (17) -.126 ,089 -.346 ,071 - e 4 7 2 .063

c 47

Table 3. Comparison of Integration of Auto-Spectra ~ ~

array = 1, a = 35OJ without fence

pressure taps

0.990

1.069

1.165

1 .c59

1.004

0.971

0.986

il. 963

array = 5 , a = 35", without fence

( l J '1 1.035

( 2 J 'f 1.016

(3, 5; 1.070

( 4 J 4, 0.986

(5, 5 ) ! .ob1

( 6 J 6, 1-06'

( 7 , 7) 0.991

(8, 8) 0.972

*by theory, this quantity should be identically equal to 1 --

(see Equation 8)