A PURELY EMPIRICAL APPROACH

CHAPTER III CHAPTER III

4.2 A PURELY EMPIRICAL APPROACH

C HAPTER IV ANALYSIS OF THE MODEL

0

0 _{I I}

.,;.

B o. 13

..c a 0 E er}-

u

I-z ⁰ w CJ

1--1 0.t

u

1-1

LL LL w 0

oo u,_;-

a: (!)<!fl(!)

1-1

I- _~

a: 0

WO ^(!)

z ·-

1 - 1 0

(!;

(!) (!)

o C...)

...

I I I

'4. 00 5.00 6.00

a

CJ

r:g

z ·- wa

1--1

u

1--1

LLO LL J')

w ci_

0 1

u

Q a: a a:: CJ

o,_;_

I

a

i.I)

REDUCED VELOCIT'i'.

I I

B = 0. 13

7.00

Vr

-

I ^{I I}

4. 00 5.00 6. 00 7.CO

REDUCED VELOC !Tl, V r

Fig. 4.2.la

0 C)

.,;. ^{I I}

B == 0.25

a ..r::.O

E er}-

u

"

I -zo w CJ 1-1 N · -

u

LL

LL Wa ~(!;

oi:...> (£6

u,_;- _CCJ

a:

1-1

I - ^(!)

a::o ^(!) z ·-WO

1 - 1 0 ~

8

^{al) (!) CJ(!)}

0 0

~

14. co s'. co 6.00 ^I

REDUCED VELOCITi.

0 0

r:g

z ·- wo

t-.t

u

LL o

LL Lt>

We)_

0 1

u

Q a: o a: CJ

o ·

1-

0 Ul

I I

B 0.25

7 '.

cc

vr

14~.-c-o~~--,s'.~c-o~~~6~'.-c-o~~-7--r.cc

REDUCED VEUKITI, 'vr

Actual Experimental Data Points of Cnh and Cdh Coefficients [61]

0

0 ..;.-+---'-'---'''---+

r-

.

z o

we:_

1 - i N

u

LL LL

Wo oo u,...;- a:

t-a: 0 WU z ·-

1-i 0

0 CJ

s a.so

(!) (!)

!-+---..--,---~.---+

4. co 5.00 6. co 7. co

0 u

~o

I-0

z ·- wo u

1-i

I.LO LL '.f>

we)_

0 1

u

t.'J cr:o

a: ^CJ

0 ,...;_

I

0 i.f)

REDUCED VELQC ITI. V ^r

(!) (!)

I ^I

B O.SO

!-+---..--,---~.---+

4.00 5.00 6. CO 7.CO

REDUCED VELOCITY. Vr

Fig. 4. 2. lb r-

0 u

..;.

z o w ^CJ

1--f N- u

1-i

LL LL

Wo

D CJ

u,...;-

0 u

(!) (!)

I I

B 0. 75

~-+---.---~ 1

4. co s'. co s'. co ^7. cc

0 u

. 'o i - u

z ·- wo u

1-i

LL.0 LL f l

we)_

0 1 u

0 JI

REDUCED VELQC!Ti', \ r

I I

8 0. /5

1-+---....-,---~1---+

4. co 5. co 6.00 7. co REDUCED VELCCITI',

vr

Actual Experimental Data Points of Cmh and Cdh Coefficients [61]

solved simultaneously for Band Q, given the other parameters Qn,

n, s,

and S. However, since B is known only at four discrete points, those equations are solved instead for Qn and Q, given B,

n, s,

and S. From Eqs. (3.2.16) and (3.2.17), one obtains two different expressions for

(4.2.1)

which does not contain the coefficient Cdh' and

n (4.2.2)

2(21TS)²

which does not contain the coefficient Cmh• Through Eqs. (4.2.1) and (4.2.2), Qn is evaluated as a function of Q for each of the four values of B and at each experimental data point for Cmh and Cdh given in Figs.

4.2.la and 4.2.lb. Qn is plotted as a function of Q in Figs. 4.2.2a to 4.2.2g, for values of parameters S,

n,

and

s

chosen as follows:

Table 4.2.1

Values of Parameters for Purely Empirical Approach

Case Figure

s

n

s

1 4.2.2a 0.20 .00514 .00103

2 4.2.2b 0.20 .00514 .00145

3 4.2.2c 0.20 .00514 .00181

4 4.2.2d 0.20 .05 .01

5 4.2.2e 0.20

.so

.10

6 4.2.2f 0.20 .20 .OS

7 4.2.2g 0.20 .20

.oos

In cases 1 to 3, the parameters were chosen so to enable direct comparison with Feng's experiments [9]. By maintaining the ratio

n/s

constant in cases 4 and 5, the effect of n on the solutions could be verified. Finally, the manner in which

s

affects the solution is shown in cases 6 and 7.

For each of the four B values, Eq. (4.2.1) and (4.2.2) may at most intersect at two points. Just to aid visual interpretations of those intersections, interpolation curves (dashed lines) were also drawn in Fig. 4.2.2g. The intersection points obtained in the case considered in Fig. 4.2.2g form the amplitude and frequency response curves plotted in Fig. 4.2.3. These curves, though rather sketchy, show the same quali- tative behavior encountered experimentally [9, 27, 46] and are the first

·indications that indeed forced vibration data can be used to generate the response of the induced vibration case.

4.2.1 Observations on the Available Data

Based on the results of Fig. 4.2.2 it is observed that:

1) What appeared to be an extensive set of Cmh(Q,B) and Cdh(~6,B)

coefficient data for the purpose of response prediction is, in fact, not so. As indicated by Figs. 4.2.1, there are only four values for the variable B that yielded meaningful results. Even so, the considerable scatter of data points makes defining a smooth interpolation curve difficult.

2) In past experiments, the total force acting on the cylinder was measured and then decomposed into its orthogonal compon- ents, as given by Eq. (3.2.2). In this manner, each experi- ment, for a certain amplitude B and reduced velocity Vr, yielded only one value for each forcing coefficient Cmh

0 0

CN CN

~oj

^Uo z . ^{B s .,, =} '{ = ^{= c 0. 13 0.20 0.} 0. ^C0514 C0103

w- ^[3

:::J

0 w ^{I G}

8

cf3

G G GEi GG G

G G

q B "' 0. 25 ' s = 0.20

>-o ¹⁾ = 0.00514

Uo '{"' O.C0103 ~[31338

z .

rE G G

w- [3 [3 [3 G

:::J

89

G (38

w G

a: LL o CJ

G a::o _GG G

LLo

B G

0 0 I

Cl o \J1

N

.

^{I I I} w

c ^{CN I}

q ^{B =a.so} 8 GG Gg q B = 0. ·75

s = 0.20 _{G (38}

' s = 0 . . w

>-CJ .,, = 0.00514

138 G >-o ¹¹ = 0.00514

U o

r

⁼ o. 00103 [JllBB3 Gct:E raE ^{G U o}

r

= o. 0010s 13

z ·- ^{t- :z: .} _{B tm} ^{[3 [3[3 IB3}

w- [3 ^{G G B} G w- ^[3

:::J 8 :::J

0 0 _G G

w ^{GG G} w ^G

a: 0 a::o _@

LL a LL a

_J ci- G % - G G _I-

G _ J O G

a: cr. G

a::: a:::

:::J :::J

~-o ~a

a: ⁰ a: CJ

z~ z~

I ^0.70 o:ao ^{o'. qo}

l:

^{co 1: l 0 l. 20 I 0.70 ' 0.80 ' a.go I l. ' co l. ' 10 1. 20}

FREOUENCI etF V IBRRTI ON.

n

FREQUENCY etF V IBRRTI {jN,

n

Fig. 4.2.2a

Solution of Eq. (4.2.1) 8 , and Eq . ( 4 . 2 . 2) 0

C'.i

q c B "' 0. 13

s = 0.20

) - C) "it = ^{0. C0514}

U o _{z ·-} t= ^0.00145

[3 [3 GB ~ G

w- G

::J

C3 w ₈

a:: C) 8 8

LL o G _JO ·-

a: GG

a::

::J

~-C)

a:o

z~

I c. '70 o'. 80 o'. go i '.co i '. l 0

FREQUENCY C!F V IBRRTI CIN.

n

0 Cl

CN

q

) - 0 UC.l

z ·-

w-

::J C3 w er.a

LL c.>

_J 0- cr: a::

::J r-a

a: ^CJ

B = ^0.50 s = 0.20

11 "' 0. COS14

'{ = 0.00145

G GG

G ~

138GGl3f:2

IB ~Gt=lE rnE 13 138 G

Ge

G G

G G

...

'-

l. 20

z~ ,-t--~~~-.-~~~-.-~~~--..~~~~...-~~~~

C.'70 0.80 a.go ^{l.CO 1.10} 1. 20

FREQUENCY C!F VIBRRTICIN •

.U

Fig. 4.2.2b

q CN B = 0.25 s = o. :.rn

.

) - 0 ?t = O.C0514

U u t = 0. 00145 _~GtEG

z .

w- GG G ^{[13 [3}

::J @G:

C3 ⁸

w ^{GG 8} _{@ '9}

a:: 8

LL o ^G

0 138

_JC)

G G

a: a::

::J

~-a

a: C.l z~

I I I I I

C.70 0.80 0.90 l. co

FREQUENCI C!F V IBRRTI ClN.

0 0 CN

q ^{B c: 0. '75} s = 0. :.w

) - 0 1! = 0.00514

U o t = 0.00145 _{G GGS3}

z . _{B Im}

w- G

::J 0

~o~

^{G GG}

LL u @

G G G

_JO

cr: ¹³

a::

::J

1~0 a:o

z~

I I . I I I

c. '70 0.80 a.go 1. co

FREQUENCY ClF V IBRRTI CJN.

Solution of Eq. (4.2.1) !3 , and Eq . ( 4. 2 . 2) 0

I-

I

l. 10 1. 20

n

!

\J1

+--I

13 I-

I

1. 10 1. 20

n

CN

q

>-o

UCJ z ·- w ...

:::J 0 w a::o

LL CJ _JO

a: cc

:::J

cc i-O CJ

B = 0. 13

s ^{"= 0.20}

1t "' 0. COS14

'{ = 0. C0181

t3 G GEi GG G G ~

G G G

G GG

c N

~ j

>-o

~~ w ...

::J 0 w a::o

LL CJ I _J o-1 ^G

cc a:

::J i-O cco

B = 0.25

s = 0.20

"I = 0. C0514

· r

^{= O.C0181}

13 [3 [3 (]3 [3 c:ElJEE!BGl3 CE G

GG G

@GJ3

GG

3

@~

G G

z...: ,--~~~--~~~-.-~~~---~~---.,~..--~~~~ z...: ,--~~~--y-~~~~~~~-.-~~~--~~~-i-

o. 70 0. 80 0. 90 1. co 1. 10 1. 20 c. '10 0. 80 0. 90 1. co 1. l 0 1. 20

FREQUENCY ~F VIBRRTIClN . .0, FREQUENCY ClF VIBRRTIClN . .0,

0 0

CJ u

N ^{I I I I}

c ^CN

q B = 0.50 q ^B = 0. ·75

s:: 0.20 s = 0.20

>-o 1t = O.C0514 ^)-- 11 <= 0. COSl 4

UCJ '{ = 0. COlSl ~G[,ffi ^{aE(3 G} ug

· r

^{= O.C0181 G}

z ·- ^.... z . _{El [HI} 13 [31393

w ... _G G G %8 ^{w ... [3}

::J G8 ::J

0 ^{G G 8} 0

w _G ^G

~ ~

a::a ₈₈ ^{(? G GG}

LL CJ LL o

G (t CJ

_J ci- ^{G ~} G ...

_J CJ G G

cc cr: ^G

a: a:

::J ::J

i-O I-CJ

cco a:u

z...: z...:

I c. 10 o'. eo o.qo ^I 1. ^I co 1. ' .1.0 l. 20 ^I c. ^I 10 0.80 ^I 0.90 ^I 1. ^I co l. 10 ^I 1. 20

FREQUENCY ClF V IBRRTI ClN,

n

FREQUENCY CJF V IBRAT IClN.

n

Fir;. 4.2.2c

Solution of Eq. (4.2.1) B , and Eq . ( 4 . 2 . 2) O

I Vl Vl I

cN

~OJ

^{U o} _{z .}

w-::J G w

a:::o I.Lo

_ J O

a: a:

::J l--0

a:o

B = ^{0. 13} s = 0.20

1) = 0. cs

'{ = O.Cl

G

~ G

G G Gii [3[3 C3

G G G

G

GG

CN

q

..

>--o UCJ z w· ....

::J G w

a:::o LL CJ _ J O

a: I G

a:

::J l--0

a:o

B "" 0. 25

s = 0.20

1) = 0. cs

'{'"" 0. Cl

[3 13

GG

~13i:EG I-

rE 13 G G G

138 G & ~

G G

G

z~ 1--~~~---~~~--~~~--~~~--~~--+- z~ 1-+-~~~--.--~~~--.--~~~-..-~~~-.-~~~-+

c. '70 0. 80 0. 90 1. co 1. 10 1. .20 C. lO 0. 80 0. 90 1. CO 1. l 0 l. 20

0 CJ CN

c:

^>--o U o

j

z . w-

::J G w a::o

LL CJ

FREQUENCY CIF VIBRATICIN.

n

B = 0.50

s ^c 0.20

., =a.cs

'{ "' 0. Cl

[3 GG

GG eG eg

eC3

GG G

9 [JllSJl33 GB°E il!:E l3 G G

G8 G

0 C)

c N

~OJ

^{U o} _{z .}

w-

::J G w er. I.Lo

CJ

l--0 l--0

cr.u a:o

FREQUENCY CIF VI BRRT I CIN,

n

B :.: 0. 15

s = 0.20

., =a.cs '{ = 0. Cl

G eIBl ^GGGlB'.3

G

I-

G G

G

El

G G G

f]

^{G% G [ (} G

z~ 1 I I I I z~ ,-.-~~~-p-~~~-r~~~--.~~~~..-~~~~

c. -;o o. ⁸⁰ o. qo ^1. co ^1. io ^{i. 20} c. ¹⁰ o. ^t10 o. qo i. co i. io ^{i. 20} FREQUENCY CIF VIBRATICIN.

n

^{FREQUENCY Cl~} VIBRRTICIN.

n

Fig. 4.2.2d

Solution of Eq. (4.2.1) ^{!!I ,} and Eq . ( 4 . 2 . 2) 0

I

\Jl

°"

I

cN

:OJ

^Llo _{z .}

w ...

:::J 0 w a:o I.Lu

_JO

a: a:

:::J f- 0

a:u

B = 0. 13

s = 0.20

'If = 0. 50

'{ = c. 10 G

G G [3

I

~~G

G

G 8

8

~

GG

z.,....; 1+-~~~-.-~~~...-~~~r-~~~.--~~--+-

c. 10 0.80 0.90 l.CO 1. 10 1. 20

FREQUENCY CIF V IBRRTI CIN, il

u CJ

N ^{I I I I}

q c 8 = 0.50

G GG Gg

~ s ⁼ 0.20 _{G (38 c:P3G}

>-CJ ^'It = 0. 50

G 8 ~cf'GdE

Llo '{= 0.10

G 11#3

z ·- ^f-

w ...

13 G G 8

:::J [3 <?

w 0 88 ^G

a:o

LL o

·- G % ^-

_JO G

cr a:

:::J i - 0

a: ^CJ z...:

I

0:80 o:qc 1: CG 1: i 0

C.'70 1. 20

FREQUENCY Cl F V IBRRT IClN,

n

Fig. 4.2.2e

I I I

cN

:OJ

^{8 s 11 "= = 0.20 o.} 0. !:>O ²⁵

UCJ '{ = 0. 10 :z: .

w-

[3 ~ [3

~

G~GG

⁸

:::J

0 I ^{C3 G}

w

cr:: o I GG

LL CJ

[3 t33e~G 89

G G

_JC)

CT I G G

a:

:::J 1--C)

z.,....; a:u l~~~~-y-~~~...-~~~-.--~~~-.-~~~~

c. 70 0. 80 0. 90 l. co ^1. 10 1. 20

FREQUENCY CIF VIBRRTICIN. il

C) u

q CN B '= 0. 15

~ s "'0. :.w _[3

)-- 1l ~ 0. !:>O _C3Gi5J3

ug

_{'{ =}

0. 10 ~ ^G

z . w· ....

G B

:::J

0 _G G

w ^G

~C) u ^El

G 8

_JO G

cr _G

a:

::i f - U

a:u z...:

I I I I I -- T

c. '70 0.80 o.qo 1. co 1. 10 1. 20

FREQUENCY CIF ^V IBRRT IClN,

n

Solution of Eq. (4.2.1) 8, and Eq. (4.2.2) O

I

\J1 -..j

I

cN cN

q _{B "' 0. 13} q _{B == 0.25}

s ⁼⁼ 0.20 s = 0.20

>-o ^{'?) == 0. 20} >-o ^'?) = 0. 20

~131338

U o z ·-

r

= a. cs G~ ~ ^{13 .... U o :z .}

r

⁼⁼ a. cs

w- GG w ...

l33 IJ G ~GG⁸

:'.:) [3 :'.:) C3 G ^[3

0 0 _{GG G}

f5 9

w _G G G w

a::o 0::0 G

G GG ^G

LL CJ LL o

_J 0- ^.... _ J O I-

a: ^GG a: ^{G G}

a: a:

:'.:) ::::J

f-O 1--Cl

a: CJ a: CJ

z~ :z ~~

I I

o:go ^{I I I}

c. '70 0.80 1. co 1. 10 1. 20 c. ·10 0.80 C.90 1. co l. 10 1. 20

FREQUENCY CJF V IBRRTI CJN,

n

^FREQUENCY CJF V IBRRTI CJN.

n

0 u o

u

I Vl N ro

c ^{CN I}

~OJ

^{U o} _{z .} ^{B s}

_r

^{'?) = "' ==} == ^{0.50 0. o ..} o. cs ^{20 w} w ...

:'.:) 0 w a: LL o

CJ

8G eG eg

[3 G C3

fJJ G «133GBE ~ ^G

G G

% ^G

~OJ

^{U o} _{:z .} ^{s B}

_r

^{'?) = = =} == ^{0. '/5 0.20 0. 20} o. os w ...

:'.:)

w 0

cr:o

LLu

e@ ^{13 1313~}

[3

[3

G G

G

f] ^G

^'e

^{8 [}

^{t" (} ~

f-O f-O

a:u cru

z~ _r I I I I z~ !-+-~~~--.-~~~--~~~~~~~~--~~~~

c. ·10 o. ao o. 90 i. co ^i. io i. 20 c. 70 o. eo o. qo l. co ^i. io i. 20

G ~ 8 G G G

G

FREQUENCY CJF VIBRRTICJN,

n

^FREQUENCY CJF VIBRRl'ICJN.

n

Fig. 4.2.2f

Solution of Eq. (4.2.1) G , and Eq . ( 4 . 2 . 2) 0

cN o•

~OJ

^UCJ _zci

w ...

::J a w a:

LL o _CJ --1 ci- a: a:

::Jo I-CJ

a: .

I

B = 0. 13

s == 0.20

,, = 0.200

'{=a.cos

I I

,,.~,

/~

\

e- _,,,,,. G ~

--

^~

- G - - ~-e-eB-GGf3-.

f--

00

'

z~ .---~~~---~~--....-~~~-r-~~~--.-~~~-r

c. 70 o. 80 0. 90 l. co 1. l 0 l. 20

0 CJ Q I

CN

~OJ

^UCJ _zci

w--

::J a w a:

LL o CJ _ J O

cr: a::

::Jo 1-U

FREQUENCY OF VIBRRl.ION,

n

I

B = 0.50

s = o. 20 ,, = 0.200

t == 0. cos

G GG

8^G/~..._G

L G8 \

e,c.3

^{G \}

I \ G

cfo ~\

--B - - G~~--ID!IEl338~-G-

!-

G\

a:o z.1---~----~~~--~~----~~----~~~---+-

c. 10 o.eo o.qo i.co i. 10 1. 20

FREQUENCY OF VIBRAl.ION,

n

Fig. 4. 2. 2r;

d~ I

^{B = 0.}

2~

^{I I I}

I

~o uu

zo

w ...

::J CJ

w a::o

LLu _ J O

cr: a:

::Jo I-CJ

a: .

s = 0.20

.., == 0. 200

'{=a.cos

_,(f'

G_.~ G

,,,,...&B..-e ~w

_, G \

-~--& - ID~ ^- -<±13Bf.!i3GG-133.£3 -

,,

,,,,... \

G""' G

\

z~ .~---.---.,---...---~---r

c. ^"10 o. 80 0. 90 l. co 1. 10 1. 20

0 CJ

FREQUENCY OF VIBRATION,

n

d~ I

^{B r- c.}

1~

^{I I I}

I

~o uo z ci_

w ...

::J CJ w a:

LLg _ J O

cr: a:

::Jo I-CJ

s = 0.20

"'I,= o. :we

~ = 0. cos

8 ... ~

I \

-43 - - - e-[ID-t-8-GG~ ~

-

^13-

G/ '

/ G\

a: ci

z~---.---..---r---..---r-

c. 10 a.ea o. qo i.co 1. 10 1.20

FREQUENCY OF VIBRAl.ION,

n

Solution of Eq. (4.2.1) 8 , and Eq. (4.2.2) 0

I Vl

\D I

lJ.j 0 .fl

en o Z o 0 ·-

Q_.,...

cn

I

't) 0. 200

~ - 0. C05

Fig. 4.2.3a

Frequency Response for Case 7

I

\

I I I

(!) approxl!nate 1nter£ect .ton£

or

Eq. U. 2. J) and Eq. U. 2. 2) interpol at ion .~o.'"' re.re.ne.nce

(I) I .

lJ.j

a: \

\(!)

w~

0 ·-

=:JO I -

I (')/

I

I

_J /(')

Q_

\

(!)' ...

'"""' ^..._

---

...

___________

^~

2: o I (I)

a:~-+-~~~~--~~~~-.-~~~~-..~~~~--.~~~~~,--~~~~~ _{0 I I I I I} 0. 60 0. 80 1. 00 1. 20 1. 40 1. 60

NORMAL I ZED VELOC I Tl,. ws/ wn

Fig. 4.2.3b

Amplitude Response for Case 7

1. 80

and Cdh• Thus, the resulting Cmh and Cdh values when plotted should align under their corresponding Vr value.

But close examination of the actual experimental data points [61] shows that there appears to be a shift between the Cmh and Cdh coefficients obtained within each experimental run.

3) From Figs. 4.2.2, it is noted that if the experimental points Cmh and Cdh were aligned under the same Vr value, then

instead of plotting the results from Eq. (4.2.1) and (4.2.2) separately, one could have plotted the difference between these two equations. After all, the sole purpose of the procedure is to determine the simultaneous solution of Eqs.

(3.2.16) and (3.2.17) in terms of Q and Qn•

4) Upon examining Figs. 4.2.2b and 4.2.2c, it appears that either there is something wrong with the data, or else,

there is a new characteristic of the vortex induced vibra- tion phenomenon never before encountered or reported experi- mentally. Fig. 4.2.2b shows that Eqs. (3.2.16) and (3.2.17) barely have a solution for B 0.13, have no solution for B 0.25 and B

=

0.75 and have some sort of a solution for B 0.50. In terms of amplitude response, this would trans- late into having a plot similar to that in Fig. 2.2.6 but with an added closed loop of solutions above those· shown for values B

=

0.50 and no solutions in between. A similar reasoning applies to Fig. 4.2.2c. Recently, Staubli [70]

found some analytical results similar to the aforementioned.

Based on other sets of experimental data [3], however,

it still seems that there must be something wrong with the present data set and that in fact, Figs. 4.2.2b and 4.2.2c should not exhibit any solution for B

=

0.50.

5) It is observed that the maximumamplitudesof vibration obtained by the present approach are substantially smaller than those obtained experimentally by Feng [9], as can be seen from Table 4.2.2.

Table 4.2.2

Comparison Between Maximum Amplitudes of Vibration Obtained Experimentally and from Purely Empirical Approach

Bmax

n l;; Feng¹ Model²

.00514 .00103 .524 - ^.25

.00514 .00145 .396 - ^.13

.00514 .00181 .204

<

^.13

.00514 .00257 .146

<

^.13

.00514 .00324 .082

<

^.13

1 Feng's results [9] as digitized by Hall [29].

2 Purely empirical approach.

6) It is noted that the maximum amplitude of vibration occurs for Vr

=

5.00; that is, where Cdh(Vr,B) is a minimum.

This also corresponds to the point where Cmh(Vr,B)'

=

O.

It will be shown later that this feature will allow mixing the present experimental data with other sets of experimental observations.

In spite of all the possible problems related to the

presently available experimental data, it is believed

that these data still carry the basic information necessary to predict, at least qualitatively, the behavior of the induced vibration of cylinders. A method for accomplishing this is presented.

4.3 . AN ANALYTICAL-EMPIRICAL APPROACH

It is clear from the foregoing that some appropriate interpolation of the data will be required in order to produce reasonable model response curves. Sarpkaya, in presenting the experimental data, also included a smoothed version, reproduced in Figs. 2.2.2 and 2.2.3. However, no mention was made as to how this version was obtained from the raw data.

As is obvious at this point, the surfaces Cmh(Q,B) and Cdh(Q,B) must be well defined and continuous in both Q and B in order to ensure a contin- uous solution for the system of Eqs. (3.2.16) and (3.2.17). If there were enough data points in both Band Vr (or Q), and if these data had a relatively smooth behavior, a numerical interpolation could be employed and the purely empirical approach described in the previous section applied. But this is not the case for the available data and may still not be, even when a more complete experimental set of data becomes avail- able. One must, therefore, resort to some analytical interpolation scheme.

Ideally, the interpolation expressions should be chosen to reflect the very nature of the vortex induced vibration phenomenon. A least square fit in two dimensions could then be applied to the experimental data so as to select the constants appearing in the interpolation

expressions. But if the nature of the fluid-structure interaction was known, there would be no need for an approximate model.

4.3.1 Fitting of Experimental Data

Considering the observations made with respect to the available experimental data, the situation is far from ideal. Instead of attemp- ting to interpolate the actual data points through two dimensional

surfaces, which could prove fruitless, an alternate interpolation scheme is adopted based on the smoothed data of Figs. 2.2.2 and 2.2.3.

In this procedure, an expression for the curves in the Vr direction is chosen, in order to retain the characteristics deemed most important, but no attempt is made to make a best fit of the actual data points.

In particular, analysis of Cd1CVr,B) curves presented in Fig. 2.2.3, for the range where Eq. (3.2.18) is satisfied, shows that

1) all curves have a first zero crossing practically at about Vr

=

4.80;

2) all curves reach a minimum at about the same point, Vr

=

5.00;

3) there is a second zero crossing for Vr

>

5.00, that is depen- dent on B.

Accordingly, an expression for Cdl (Vr,B) is chosen as follows:

x

[d(x-a)² + bx] , x~a (4.3.la)

( x-a \ ²

x [ 1 - c-a ") ] _{, x a}

>

[d(x-a)² + bx]

(4.3.lb)

where

x

= vr -

4.80 and the other parameters are defined as

a

=

0.2 considered constant. x

=

a is where the minimum occurs in the local coordinate system.

b b(B) , function of the amplitude and an approximate value for the inverse of the minimum.

c

=

c(B) , function of the amplitude, x

=

c where the second zero cros- sing occurs in the local coordinate system.

d

=

0.1 , controls the broadness of curves.

The relationship between these parameters and the Cal curve is shown in Fig. 4.3.1.

A similar examination of the behaviour of the Cml(Vr,B) curves, presented in Fig. 2.2.3, shows that

1) all curves have a zero crossing practically at Vr

=

5.15;

2) the slope of the curves at the zero crossing point is depen- dent on B;

3) the curves tend to have a practically constant behavior for values of Vr slightly greater than 5.5.

The expression for Cm1<Vr,B) is chosen as

(4.3.2a)

Cm1<Vr,B)

- e (V r - V r₀ )

(4.3.2b) where the parameters are defined as

Vro , zero crossing of the curves

e

=

e( B) f f ( B)

slope of the Cmh curves at Vr

=

Vro

assumed as an asymptotic value that Cmh would tend to, for relatively large values of Vr•

The relationship between these parameters and the Cml curve is shown in Fig. 4.3.2.

a

-J

c

Fig. 4.3.1 Parameters for definition of Ca1<Vr,B) surfaces

ton,B=e

f

Fig. 4.3.2. Parameters for definition of Cml (Vr,B) surfaces

-67-

The Cd1CVr,B) and Cml(Vr,B) curves, being smoother than the Cdh(Vr,B) and Cmh<vr,B) curves, have been chosen for interpolation. lbwever,

one set of coefficients can easily be recovered from the other by using the following equations:

21T 3B

Cmh(Vr,B)

= - -

Cml (VpB) v r 2

(4.3.3)

32TIB²

cdh<vr,B)

=---

Cd1CVr,B) 3V r ²

(4.3.4)

It should be stated that the above expressions have been arrived at after a relatively extensive examination of other possible expressions.

Among all the possibilities considered, it is felt that Eqs. (4.3.1) and (4.3.2) are the ones that best match the smoothed data of Fig. 2.2.3.

As has been indicated, several of the parameters appearing in Eq.

(4.3.1) and (4.3.2) were defined as functions of B so the complete two- dimensional surfaces for Cm1CVr,B) and Cd1(Vr,B) surfaces can be gener- ated. These functions were chosen, bearing in mind the simplest possible expressions and considering only the present set · of experimental data to the extent possible.

Specification of the parameter b.

As can be seen from Table 4.2.2, the present experimental data

yield maximum amplitudes of vibration far below the values experimentally found by Feng [9]. Since the prediction of amplitude is one of the most important aspects of the capability of any model, it is felt that some improvement should be made in this area. Experimental data¹exactly simi-

1 As of this writing Staubli [70] has not published his complete work.

lar to the present set have not, as of yet, been published • However, as mentioned before, the maximum amplitude of vibration occurs at Vr

=

^5.00,

corresponding to minimum values of Cdh (Vr,B) and close to zero values of Cmh (Vr,B). Data corresponding to forces acting at peak response on flexibly mounted cylinders are used to supplement Sarpkaya's data, based upon the assumption that both phenomena are similar. Mixing of the results from forced cylinder experiments with those obtained from flex- ibly mounted cylinder experiments is, admittedly, not a very desirable procedure. Nevertheless, in doing so, the prediction capability of the model is so greatly enhanced that this by far off sets any of the proce- dure's undesirable effects.

Evaluating Eq. (4.3.1) for Vr

=

5.0 and using Eq. (4.3.4), yields

b(B)

-

^--~-32TIB² 75

1 (4.3.5)

Values for Cdh(5.0,B) from Blevins [9] are mixed with corresponding values from Sarpkaya and a smooth curve, given by the expression that follows, drawn through the experimental points.

5.0,B) -l.375B² + l.483B + 0.200 (4.3.6)

This curve is shown, along with the experimental points, in Fig. 4.3.3.

Note that the empirical relationship gives results which are larger than those obtained by Sarpkaya, for B

=

0.13 and B

=

0.25, and gives a smaller coefficient for B

=

^0.50.

For all the other parameters, function of B, appearing in Eqs.

(4.3.1) and (4.3.2) no similar measurements have been found. Consequently,

,...a CJ

OJ

•-t---_.... ____________________

__...._~~~~--~----~--+-

-

D

..

lf)

II i.n

Lr---

> .

~a

...c

"'O

u

I a

Lf)

1 - 0

z w

u

t-1 U1

LL~ LL o

w

D

u

CJ 0

ITO

[!]

(!) (!) (!)

[!) ~rom 5arpi:aya £"6'JJ

· (!) ~rom Blevins CSJ

prtJsen t mode 1

(!) (!)

(!)(!) (!)

(!) (!)

(!) (!)

~'=b-.-0-0---...---1--.50

0.25 0. 50 0. 75 1.

co

1. 25