JSV-D-07-00741 Title

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COVERING LETTER

Title of paper: Experimental Identification of damping in higher modes of slosh using Hilbert-Huang Analysis.

Authors: Kaushal J Shah (Corresponding Author) Prasanna S Gandhi

Contact Address: c/o Prof. Prasanna S Gandhi

Department of Mechanical Engineering Indian Institute of Technology, Bombay Powai, Mumbai, India

Pin Code : 400 076

Telephone Number: Mobile: (0091) 98941 61202

Fax : (009122) 2572 6875 [c/o Prof Prasanna S Gandhi]

E-mail Address: [email protected]

Cover Letter

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E xp e r i m e nt al I d e nt i ﬁ c at i o n o f d am p i n g i n h i gh e r m o d e s o f s l o s h u s i n g H i l b e r t - H u an g

A n al ys i s .

K aus hal J S hah

^a,

∗

, P ras anna S G andhi

^b^,¹

aNa tio na l Institute o f Tec hno logy, T iruc hira ppa l li, De pa rtme nt o f Mec ha nica l Enginee ring, T iruc hira ppa l li 620 0 1 5, Ta mil Na du, India

bIndia n Institute o f Tec hno logy Bo mba y, De pa rtme nt o f Mec ha nica l Enginee ring, Po wa i, Mumba i 4 0 0 0 76, Ma ha ra stra , India

A b s t r ac t

S los hing is t he os cillat ory b ehavior of liquid s in res p ons e t o s ome excit at ion. T his can b e a p rob lem in liquid cargo aut omob iles , aircraft s , s p acecraft s , liquid met al hand ling, and p ackaging of various liquid s . To enhance p erformance and / or s ave cos t in t hes e ap p licat ions , b et t er id ent iﬁcat ion of s imp le mo d els and mo d el p aramet ers in t hes e ap p licat ions is inevit ab le. Typ ical s uch mo d els in t he lit erat ure are d evelop ed us ing eit her mult ip le comp lex p end ula or mult ip le s p ring mas s d amp er analogy. T hough es t imat ion of p aramet ers for ﬁrs t mo d e vib rat ion of liquid has b een es t ab lis hed , exp eriment al es t imat ion of d amp ing in higher mo d es has not b een d one s o far, mainly b ecaus e t he liquid vib rat ions at higher frequencies give mix mo d al res p ons e. We p rop os e in t his p ap er an exp eriment al met ho d t o es t imat e p aramet ers , es p ecially d amp ing in higher s los h mo d es . We us e a recent ly d evelop ed t echnique of emp irical mo d e d ecomp os it ion along wit h H ilb ert analys is t o s ep arat e out t he various mo d es in t ime d omain. Furt her, nonlinear leas t s quare minimiz at ion is us ed t o Main text

Click here to download Manuscript: slosh_damping_JSV.tex

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es t imat e s ucces s fully d amp ing in higher mo d es . M at ch of d amp ing of t he firs t mo d e es t imat ed us ing p revious met ho d s and us ing higher mo d es d at a wit h t he p rop os ed met ho d , verifies t he effect ivenes s of met ho d . Es t imat ion of d amp ing in t his manner for higher mo d es is carried out for keros ene and wat er at various fill fract ions in a cylind rical t ank. Res ult s s how t hat higher mo d e d amp ing rat io is great er t han lower mo d e d amp ing and b ot h s how a s at urat ion t rend as fill fract ion increas es . Als o con- t ainer d imens ions and s urface t ens ion effect s p lay an imp ort ant role in d et ermining t he magnit ud e of d amp ing.

Ke y wo rds: S los h, D amp ing, H ilb ert -H uang Analys is , EM D

1 N o m e n c l at u r e

D D iamet er of t ank (m)

E M D E mpirical M o de D ecomp osit ion I M F I nt rinsic M o de Funct ion

R Radius of t ank (m) f Frequency (H z)

g Accelerat ion due t o gravity (m/ s²) h H eight of ﬂuid in t ank (m)

h

D F ill Fract ion

∗ S t ud net B . Tech, N at ional Ins t it ut e of Technology T iruchirap p alli, D ep art ment of M echanical Engineering.

Ema il a ddre sse s: kaus hal 2 6 @ gmai l . c o m( K aus hal J S hah) , gandhi @ i i t b. ac . i n ( P ras anna S G and hi) .

1 As s is t ant P rofes s or, D ep art ment of M echanical Engineering, Ind ian Ins t it ut e of Technology, B omb ay, Powai.

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ν K inemat ic Viscosity (m²/ s) ) σ S urface Tension ( J/ m²) ρ D ensity (k g/ m³)

ζ D amping Rat io

2 I nt r o d u c t i o n

S loshing is t he oscillat ory b ehavior of liquids in resp onse t o some excit at ions.

S loshing can b e problem in liquid cargo aut omobiles, aircraft s, spacecraft s, liquid met al handling, and packaging of various liquids. S loshing of liquids in packaging indust ry can lead t o improp er sealing of cont ainers t hereby reducing shell life of t he pro duct [ 1 ] . I n aut omobiles, sloshing is induced in fuel t anks due t o sudden accelerat ion, braking or t urning. T his can lead t o inst ability and can hamp er p erformance and cont rol of t he vehicle [ 2 ] . I n large liquid cargo cont ainers such as t ankers and ships, sloshing can cause t he vehicle t o roll over leading t o accident s [ 3 ] . H owever, compared t o t he ab ove ment ioned areas of applicat ion, t he impact of sloshing in t he aerospace applicat ion is very severe. S loshing of prop ellant s in part ial filled fuel t anks in space ro ck- et s is a p ot ent ial source of dist urbance which may b e crit ical t o t he st ability and st ruct ural int egrity of t he vehicle. S loshing is induced by t ank mot ions result ing from guidance and cont rol syst em commands or from changes in t he vehicle accelerat ion, such as t hose o ccurring during engine cut off or when vehicle encount ers wind shears. S mall excit at ion amplit udes can lead t o de- velopment of slosh forces and moment s of large magnit udes, t he nat ure and int ensity of t hese forces dep end on t he fill fract ion, geomet ry, and lo cat ion of

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t he cont ainer, and on t he prop ert ies of t he prop ellant . As t he prop ellant is consumed, t he fill fract ion changes wit h t ime, hence t he sloshing forces and moment s also change cont inuously. At one inst ance in t he p owered phase of t he flight , t he prop ellant ’ s oscillat ory frequency may nearly coincide wit h ei- t her t he fundament al elast ic b o dy b ending frequency or t he dynamic cont rol frequency of t he vehicle. T his could cause a failure of st ruct ural comp onent s wit hin t he vehicle or excessive deviat ion of t he vehicle from is preplanned flight pat h [ 4] .

T herefore, mo deling of slosh phenomenon is very imp ort ant in order t o pre- dict t he b ehavior of prop ellant at different st ages of t he flight . D evelopment of an accurat e mo del can help in space mission simulat ion and formulat ion of effect ive cont rol st rat egies for robust and successful launch of vehicles wit hout failure. T ill dat e, mult iple complex p endula or mult iple spring mass analogies have b een used t o mo del slosh [ 4] . I n b ot h t he mo dels/ analogies ment ioned ab ove, t he knowledge of damping rat io is very imp ort ant t o obt ain an accurat e mo del. H ence t he est imat ion of damping b ecomes very imp ort ant . Available lit erat ure [ 4–9 ] shows t hat ext ensive work has b een done in t he area of comput ing first mo de damping. H ence, an accurat e mo del t o represent first mo de of slosh is available. H owever, t o t he b est of our knowledge est imat ion and crit ical st udy of higher mo de damping b ehavior of liquids has not b een done so far for t he reasons ment ioned b elow.

T he primary fo cus of t his pap er is t he det erminat ion of higher mo de damping in order t o mo del higher mo des of slosh accurat ely. E st imat ion of damping in higher mo des b ecomes diﬃcult since exp eriment s show t hat excit ing higher mo des of sloshing wit hout cont ribut ion from lower mo des is not p ossible. T herefore, in order t o det ermine higher mo de damping, it is necessary t o

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separat e out t he mo dal resp onse dat a in t ime domain and t hen analyz e t hem individually. To achieve t his, we have employed a recent ly develop ed t echnique of empirical mo de decomp osit ion ( E M D ) t o separat e out t he various mo des int o int rinsic mo de funct ions ( I M F ) . Applicat ion of t he H ilb ert t ransform on t he int rinsic mo de funct ions ( I M F ) helps us t o det ermine t he inst ant aneous frequency. Furt her pro cessing of t he dat a separat ed in t ime domain gives us t he required higher mo de damping. D amping in higher mo des, est imat ed by t his way, is analyz ed for wat er and kerosene sloshing in cylindrical t ank.

T his pap er is organiz ed as follows: S ect ion 3 present s t he basics of numerical t echnique implement ed t o est imat e t he damping rat io. S ect ion 4 illust rat es t he exp eriment al pro cedure and dat a analysis of a typical exp eriment . S ect ion 5 present s and discusses t he result s of slosh damping in higher mo des for liquid sloshing in cylindrical t ank. T he exp eriment s are p erformed wit h two different liquids wit h different viscosity and surface t ension. F inally, S ect ion 6 concludes t he research findings.

3 P r o p o s e d N u m e r i c al Te ch n i qu e

We prop ose in t his sect ion a numerical t echnique t o est imat e t he higher mo de damping. T his sect ion is divided int o two part s. T he ﬁrst part fo cuses on t he H ilb ert H uang Analysis and highlight s it s imp ort ant feat ures. T he second sect ion shows how exp onent ial curve ﬁt t ing is used t o det ermine t he damping rat io from a given exp eriment al dat a.

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3. 1 Hi lb e rt Hua n g A n a lys i s

From ext ensive set of exp eriment s carried out at various frequencies and slosh amplit ude we know t hat higher mo des cannot exist indep endent ly of t he lower mo des. When we excit e t he liquid at higher mo de, various lower mo de frequencies get excit ed, making t he signal non- st at ionary in nat ure. B ecause of t he p eculiar non- st at ionary nat ure of t he slosh signal, ext ract ing accurat ely higher mo de damping using convent ional frequency domain t o ols ( Fourier and Wavelet Transform) is seldom p ossible. Fourier series, as we know, can analyz e only st at ionary and linear dat a accurat ely and hence cannot b e used in t his case. Wavelet on t he ot her hand can b e used t o analyz e non- st at ionary dat a but , wavelet t ransform is based on t he H eisenb erg U ncert ainty P rinci- ple, hence we cannot det ermine t he exact t ime- frequency represent at ion of a signal. T he wavelet t ransform has a go o d t ime and p o or frequency resolu- t ion at high frequencies, and go o d frequency and p o or t ime resolut ion at low frequencies [ 1 1 ] . T his drawback can lead t o loss of essent ial informat ion. I n order t o overcome t his drawback, we opt ed for H ilb ert H uang Analysis b ecause t he various prop ert ies are derived by diﬀerent iat ion rat her t han convolut ion.

T herefore, it is not limit ed by t he uncert ainty principle and can b e applied t o non- st at ionary dat a t o obt ain precise t ime- frequency- amplit ude represen- t at ion of a signal for feat ure ext ract ion. T his sect ion present s our prop osed met ho d based on H ilb ert H uang Analysis [ 1 0 ] for est imat ing t he higher mo des damping in slosh.

T he ob ject ive of H ilb ert H uang Analysis is t o generat e t he H ilb ert S p ec- t rum. T his is p erformed in two st eps. F irst , by E mpirical M o de D ecomp o- sit ion( E M D ) , t he t ime- series input dat a is decomp osed int o a set of funct ions

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known as I nt rinsic M o de Funct ions( I M F s) . S econdly, applicat ion of H ilb ert Transform on t hese funct ions is used t o generat e a frequency- t ime plot called t he H ilb ert S p ect rum. T he following sect ions provide a descript ion on E M D and t he prop osed E M D algorit hm used for mo dal separat ion.

3. 1 . 1 Em pi ri ca l Mo de Deco m po s i ti o n

E M D is used t o decomp ose a t ime- series dat a int o a set of comp onent s, known as I M F s. For a signal t o b e considered as an I M F a t ime- series dat a has t o sat isfy t he following two condit ions:

( 1 ) I n t he whole dat a span, t he numb er of ext rema and t he numb er of z ero crossings must b e eit her equal or diﬀer at most by one.

( 2 ) T he mean value of t he envelop e deﬁned by t he lo cal maxima and t he envelop e deﬁned by t he lo cal minima is z ero.

A pract ical pro cedure, known as sift ing pro cess [ 1 0 ] , is employed t o decomp ose a t ime- series dat a int o I M F s. An imp ort ant feat ure of t he sift ing pro cess is t o ﬁnd appropriat e t imescales t hat may reveal imp ort ant informat ion emb edded in t he original signal.

3. 1 . 2 Si fti n g A lgo ri thm

G iven a signal x(t) , t he int rinsic mo de funct ions are obt ained using t he algorit hm describ ed b elow.

( 1 ) I dent ify all t he maxima and minima in x(t) .

( 2 ) I nt erp olat e all t he maxima and minima t o form t he upp er envelop e ema x(t) and lower envelop e emi n(t) using cubic spline algorit hm.

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( 3 ) C omput e t he average m(t) = (ema x(t) + emi n(t) ) / 2 .

( 4) S ubt ract m(t) from t he signal t o obt ain t he residual funct ion: r(t) = x(t) − m(t) .

( 5 ) I t erat e on t he residual r(t) unt il it sat isﬁes t he st opping crit erion men- t ioned lat er in sect ion 3 . 1 . 3 . T his ensures t hat t he pro cessed signal is sat isfying t he deﬁnit ion of I M F ment ioned ab ove ( see 3 . 1 . 1 )

( 6 ) O nce we obt ain an I M F , deﬁned in sect ion 3 . 1 . 1 , we remove it from t he signal: x(t) = x(t) − I M F(t) and rep eat t he pro cedure t o obt ain ot her I M F s.

As more and more numb er of I M F s are generat ed, t he numb er of ext rema decreases as we go from one residual t o t he next t hus guarant eeing t hat t he complet e decomp osit ion is achieved in a ﬁnit e numb er of st eps. At t he end of t he algorit hm we have:

x(t) =

�n

j= 1

I M Fj(t) +rn(t) ( 1 )

where rn(t) is t he residue.

3. 1 . 3 Sto ppi n g Cri te ri a

S everal researchers [ 1 0 , 1 2 , 1 3 ] have develop ed furt her t he E M D t echnique wit h numerous ways of st opping t he pro cess. T he choice of st opping crit erion plays a very imp ort ant role in signal analysis. T he sift ing algorit hm ment ioned in S ec- t ion 3 . 1 . 2 cont ribut es t o two fact ors. F irst ly, it eliminat es riding waves. Riding waves in t he signal play an imp ort ant role as t hey cont ain informat ion regarding inst ant aneous frequency of t he signal. S econdly, it smo ot hens out uneven amplit udes. S mo ot hening of amplit ude can lead t o a signal of const ant ampli-

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t ude and varying frequency. T herefore, siﬁt ing causes over- decomp osit ion of t he signal if not st opp ed at appropriat e st ep wit h correct st opping crit erion.

O ver- decomp osit ion can lead t o loss of frequency informat ion. H ence we will not b e able t o obt ain accurat e t ime- frequency plot s. S imilarly smo ot hening of amplit udes can lead t o loss of informat ion regarding t he amplit ude t rends of various high frequency dat a emb edded in t he signal. I n order t o prevent loss of informat ion due t o eit her over- decomp osit ion of t he signal or smo ot hening of amplit udes, we need t he correct st opping crit eria. We found for our case t he following st opping crit eria work b est t o prevent loss of informat ion. Also t hese crit erions make t he algorit hm fast er and more eﬃcient .

( 1 ) N umb er of ext rema and t he numb er of z ero crossings must b e eit her equal or diﬀer at most by one. ( Refer [ 1 0 ] )

( 2 ) C onﬁdence L imit which is deﬁned as t he consecut ive numb er of sift ings, where in t he numb ers of z ero- crossing and ext rema are t he same. T he value should range from 4 t o 8 . ( Refer [ 1 2 ] )

( 3 ) T hreshold which is deﬁned as s(t) = mo d(m(t)/ a(t) ) , where a(t) = (ema x(t) − emi n(t) ) / 2 should b e less t han 0 . 0 5 . ( Refer [ 1 3 ] )

3. 1 . 4 Hi lb e rt Spec trum

O nce I M F s have b een obt ained as result of t he sift ing pro cess ( Refer 3 . 1 . 2 ) , it is p ossible t o generat e t he H ilb ert S p ect rum ( T ime versus frequency plot ) t hat represent s t he variat ion of frequency of t he I M F s wit h resp ect t o t ime.

T he not ion of frequency for each I M F is obt ained by employing t he concept of analyt ic signals [ 1 0 ] .

An analyt ic signal is a complex signal wit h one- sided sp ect rum t hat preserves

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all informat ion cont ained in t he original signal. A very simple way of est imat - ing an analyt ical signal is by employing t he H ilb ert Transform. T he real part of an analyt ical signal is t he original input t ime- series, whereas it s complex comp onent is t he H ilb ert Transform of t hat signal.

G iven an analyt ic signal, z ( t ) , deﬁned as:

z(t) = x(t) +i y(t) = a(t)e^{j θ}^(t) ( 2 )

where x(t) is t he input t ime- series and y(t) t he H ilb ert Transform ofx(t) . T he inst ant aneous at t ribut es of z(t) can b e deﬁned as follows:

a(t) =

�

x(t)² + y(t)² ( 3 )

θ(t) = a r cta n

�

y(t) x(t)

�

( 4)

f(t) = 1 2π

dθ dt

( 5 )

where a(t) is t he inst ant aneous amplit ude, θ(t) is t he inst ant aneous phase and f(t) is t he inst ant aneous frequency.

Wit h t he deﬁnit ion of inst ant aneous at t ribut es given ab ove, t he H ilb ert S p ec- t rum is generat ed as follows:

( 1 ) E st imat e I M F s from t he input signal from sift ing pro cess. ( Refer sect ion 3 . 1 . 2 )

( 2 ) E st imat e t he inst ant aneous at t ribut es of each I M F deﬁned by E quat ions ( 3 ) , ( 4) , ( 5 ) .

( 3 ) G enerat e t he H ilb ert S p ect rum by plot t ing inst ant aneous frequency versus t ime.

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From t he H ilb ert S p ect rum generat ed, we can det ermine t he various mo des exist ing in t he signal by analyz ing t he diﬀerent frequencies t hat exist s in t he sp ect rum. Also since we are able t o infer t he t ime durat ion of each mo de and it s inst ant aneous amplit ude, we can det ermine t he damping of various mo des as well. T he applicat ion of t his t echnique, how t o int erpret t he H ilb ert S p ect rum and est imat ion of higher mo de damping will b e illust rat ed in t he S ect ion 4 wit h t he help of an example.

3. 2 Lea s t Sq ua re Expo n e n ti a l Curve Fi tti n g

B y ﬁt t ing an exp onent ially decaying envelop e ( t o I M F s in t his case) , we can det ermine t he damping rat io of any signal having damp ed nat ure. T he lo cal maximas of t he signal are used as dat a p oint s in order t o get t he envelop e. For a t ime decaying signal, t he exp onent ially decaying curve can b e represent ed as:

y = Ae⁻^{2π f ζ t} ( 6 )

I n order t o obt ain b est fit exp onent ial curve, we use least square minimiz at ion t echnique. I n t his t echnique we minimiz e, t he sum of t he squares of t he offset of t he p oint s from t he fit t ed exp onent ial curve, given by

�n

i= 1

|yf i tte d− ye x pe r i me n ta l|². ( 7 )

T he b est ﬁt paramet er values are given by

a =

�

n

i= 1(x²iyi)

�

n

i= 1(yil n yi) −

�

n

i= 1(xiyi)

�

n

i= 1(xiyil n yi)

�

n i= 1 yi

�

n

i= 1(x²iyi) − (

�

n

i= 1 xiyi)²

, ( 8 )

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and

b =

�

n i= 1 yi

�

n

i= 1(xiyil n yi) −

�

n

i= 1(xiyi)

�

n

i= 1(yil n yi)

�

n i= 1 yi

�

n

i= 1(x²iyi) − (

�

n

i= 1 xiyi)²

, ( 9 )

where, B = −2πf t and A=e^a corresp onding t o E quat ion ( 6 ) . From t he H ilb ert sp ect rum, we can det ermine t he frequency(f) of t he signal. H ence, by using t his informat ion one can comput e t he value of damping rat io (ζ) from t he const ant B .

4 N u m e r i c al I m p l e m e nt at i o n

T his sect ion develops t he pro cedure t o obt ain t he dat a in order t o est imat e damping in higher mo des and furt her present s t he analysis of corresp onding exp eriment al result s. T he case under considerat ion is a cylindrical t ank wit h wat er of mass 7 kg as sloshing liquid. T he exp eriment is p erformed on a sp ecial designed exp eriment al rig, for more det ails regarding t he set up, refer S ect ion 7 . 1 .

4 . 1 Expe ri m e n ta l Pro cedure

( 1 ) C alculat e t he t heoret ical nat ural frequency of slosh p ert aining t o t he liquid[ 4] . T he exp eriment is conduct ed at a frequency lower t han t he t heoret ical frequency by 1 0 - 1 5 % . T his is done in order t o avoid splashing, which o ccurs when t he t ank is excit ed at t he nat ural frequencies of t he liquid.

( 2 ) P rovide t he cylindrical t ank wit h a sinusoidal excit at ion at frequency calculat ed in ( 1 ) in t ranslat ion. T his is achieved by using a reference t ra-

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ject ory given by X0s i n( 2πf t) , where X0 is t he amplit ude of excit at ion and f is t he frequency of excit at ion. T he t ra ject ory t racking is t hen exe- cut ed by a P I D cont roller op erat ing on error b etween act ual and reference p osit ion of t he t ank. T he P I D gains are adjust ed in such a way t hat t he t racking error is minimiz ed. For example, in our exp eriment set up ( Refer S ect ion 7 . 1 ) P I D gains of 3 5 , 1 5 and 0 . 8 resp ect ively pro duce maximum t racking error of 0 . 0 6 5 mm.

( 3 ) Wait t ill t he init ial t ransient mo des dies out and st eady st at e sloshing is achieved. O nce t his is achieved conduct a ” Q uick S t op” at z ero velo city of t he sinusoidal mot ion in order t o prevent jerking of t he cylindrical t ank. Q uick st op enables liquid t o have damp ed oscillat ion at it s nat ural mo dal frequency ( close t o t he excit at ion frequency) . F igure 2 show a typical quickst op dat a obt ained from t he load cell.

( 4) Allow t he liquid t o undergo nat ural damp ed oscillat ion and capt ure t he forces and moment s dat a using a load cell mount ed underneat h t he cylindrical t ank. ( Refer S ect ion 7 . 1 )

( 5 ) Perform H ilb ert H uang Analysis ( Refer S ect ion 3 . 1 ) on t he dat a obt ained from t he load cell t o ext ract various det ails from t he signal.

4 . 2 A n a lys i s o f ﬁrs t m o de da ta

To illust rat e t he exp eriment al pro cedure t o get t he dat a and furt her t he numerical pro cedure t o pro cess it , we p erform analysis of t he first mo de dat a in t his sect ion. To obt ain first mo de dat a, t he cylindrical t ank is excit ed at 1 . 9 H z which is 1 3 % less t han t he t heoret ical first mo de nat ural frequency of wat er of mass 7 kg [ 4] . T he force F x dat a is recorded by t he six- axis load

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sensor mount ed underneat h t he cylindrical t ank ( Refer 7 . 1 ) . O n p erforming H ilb ert - H uang Analysis on F x signal as describ ed in S ect ion 3 . 1 on t he dat a, we obt ain t he various I M F s as shown in F igure 3 and t he H ilb ert S p ect rum as shown in F igure 4. We can observe t hat t he first I M F ( F igure 3 ) shows a decaying t rend similar t o t hat of t he input signal. H ence, we can comput e t he damping rat io of t he first mo de by applying least square met ho d as describ ed in S ect ion 3 . 2 . T he discrepancy in values of damping rat io as obt ained from t he original signal and t hat from t he first I M F is less t han ±5 % . From t he H ilb ert S p ect rum ( F igure 4) we can observe t hat t he first mo de nat ural frequency corresp onding t o 2 . 1 9 3 H z [ 4] exist s t hroughout t he dat a. T he damping rat io comput ed from t he first I M F is ζ = 0.0 0 5 2 8 .

4 . 3 A n a lys i s o f hi ghe r m o de da ta

I n order t o obt ain higher mo de dat a, t he t ank must b e oscillat ed at a frequency great er t han t he ﬁrst mo de nat ural frequency. I n t he case considered here, t he exp eriment has b een conduct ed by excit ing t he cylindrical t ank at 3 . 4H z . F igure 5 shows t he force dat a recorded by t he load cell in t he direct ion of t ranslat ion. We can clearly see t hat t he signal is mix mo dal in nat ure. H igher mo des or frequency signal is sup erimp osed over t he base signal init ially and as t ime progress t he higher frequency signal disapp ears.

O n p erforming t he H ilb ert H uang Analysis ( Refer S ect ion 3 . 1 ) on t he dat a, we obt ained various I M F s as shown in F igure 6 and H ilb ert sp ect rum as shown in F igure 7 . From H ilb ert S p ect rum ( F igure 7 ) we clearly not ice t hat at a part icular inst ance of t ime, various mo de frequencies or mo des exist simult aneously.

For example, b etween t ime durat ion 0 . 5 sec t o 2 sec, we see t hat t hree mo de

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frequencies exist s simult aneously corresp onding t o 2 . 2 H z, 4. 7 H z and 9 . 8 H z . T his shows t hat higher mo des cannot exist indep endent ly and are always ac- companied by lower mo des. We can also observe t hat higher mo de frequencies such as 4. 7 H z ( from 0 t o 2 seconds) and 9 . 8 H z ( from 1 t o 1 0 seconds) die out fast er t han t he ﬁrst mo de nat ural frequency ( 2 . 2 H z ) .

T he observat ions ment ioned in t he ab ove paragraph are also reflect ed in t he I M F s obt ained as shown in F igure 6 . We see t hat t he first I M F shows a signal of frequency 9 . 8 H z from t ime 0 seconds t o 1 2 seconds approximat ely and t hen t he signal is predominat ely first mo de frequency ( 2 . 2 H z ) . S imilarly, second I M F shows a signal corresp onding t o 4. 7 H z exist ing for t he first 2 seconds.

From 2 t o 1 2 seconds, it s predominant ly first mo de frequency and t he rest is t ransient dat a. F inally t he t hird I M F shows first mo de dat a from 0 t o 2 seconds and t he rest b eing t ransient in nat ure. C ollect ively from all I M F s, we see t hat first mo de frequency ( 2 . 2 H z ) exist s t hroughout t he signal and t he higher mo des damp out fast er. Also we see t hat t he first I M F isolat es in t ime domain t he highest frequency at and t he ot her frequencies are obt ained in t he remaining I M F s in t he decreasing order of magnit ude of frequency.

T he I M F s obt ained also show how t he amplit ude of t hese mo des vary wit h resp ect t o t ime. H ence t he I M F s can b e used t o comput e t he damping rat io of higher as well as lower mo des. From t he ﬁrst I M F we can comput e t he higher mo de damping rat io by implement ing t he least square met ho d ( Refer S ect ion 3 . 2 ) as shown in F igure 8 . S imilarly, from t he second I M F we can comput e t he damping rat io corresp onding t o t he ﬁrst mo de using t he same t echnique.

T he damping rat io of t he ﬁrst mo de was comput ed t o b e ζ = 0.0 0 3 6 2 and t he higher mo de t o b e ζ = 0.0 0 8 6 7 .

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5 Re s u l t s an d D i s c u s s i o n

T he exp eriment al pro cedure ment ioned in S ect ion 4 was carried out for various cases. T wo different liquids, namely kerosene and wat er were used for exp eriment al purp oses. Table 1 t abulat es t he imp ort ant prop ert ies p ert aining t o t he two liquids. F irst t he exp eriment s was p erformed by varying t he mass of wat er from 4 t o 9 kg. T he mass of t he liquid is represent ed as fill fract ion wit h is a dimensionless numb er. F ill fract ion is t he rat io of height of t he liquid in t he t ank t o t he diamet er of t he t ank ( h/ D ) . S imilarly, t he exp eriment s were p erformed wit h kerosene, by maint aining t he fill fract ion t o b e t he same as t hat obt ained for different masses of wat er.

T he damping rat io obt ained for each set of exp eriment s is plot t ed versus t heir resp ect ive ﬁll fract ion. F igures 9 t o 1 2 illust rat e t he result s obt ained on ex- p eriment at ion for kerosene and wat er.

5. 1 Lo we r m o de da m pi n g a n a lys i s

I n t he lit erat ure available t o dat e, many empirical formulae have b een develop ed t o det ermine t he damping rat io of ﬁrst mo de dat a. [ 4, 5 , 7 , 8 ] . T he two imp ort ant formulae used are ment ioned in References[ 4, 7 , 8 ] . B ot h t he formulae develop ed show t he same t rend wit h a small variat ion in numerical values.

T he empirical formula used in t his pap er for comparison purp ose is [ 7 , 4]

ζ = .7 9ν⁴

1

2R

−3

4 g

−1 4

�

1 + 0.3 1 8 s i nh( 1.8 4_R^h)

�

1 +

1 − _R^h cos h( 1.8 4_R^h)

� �

. ( 1 0 )

Values obt ained from t his empirical formula have b een plot t ed along wit h t he exp eriment al values b ot h for kerosene as well as wat er. From F igures 9 and 1 0

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we not ice t hat t here is a slight deviat ion of t he exp eriment al values from t he empirical values t hough t hey b ot h share a similar t rend. T he deviat ion of wat er is observed t o b e great er t han t hat of kerosene. T his can b e explained as follows. I n t he empirical formula ment ioned in E quat ion ( 1 0 ) , t here is no paramet er t hat considers surface t ension effect s of t he liquid. S ince t he t ank under considerat ion is relat ively smaller in siz e, surface t ension can significant ly affect damping rat io. D amping fact or almost doubles when t he surface t ension increases by a fact or of t hree approximat ely. [ 4, 1 5 ] . From Table 1 we can see t hat t he surface t ension of kerosene is less t han wat er by a fact or of t hree approximat ely, hence kerosene shows less deviat ion from t he empirical formula where as wat er show a much higher deviat ion. T he damping rat io obt ained exp eriment ally for wat er is approximat ely twice t hat of value given by empirical relat ion.

We can also observe t hat t he damping rat io of first mo de obt ained from t he higher mo de excit at ion dat a is in excellent agreement wit h t he same obt ained from first mo de excit at ion for kerosene. H owever t here is discrepancy in similar dat a for wat er. T his can again b e at t ribut ed t o t he surface t ension effect . I f we t ake a closer lo ok at t he prop ert ies of t he two liquids under considerat ion ( Refer Table 1 ) , t he kinemat ic viscosity of kerosene is great er t han t hat of wa- t er. From E quat ion 1 0 t he damping rat io is direct ly prop ort ional t o kinemat ic viscosity, hence t he damping rat io of kerosene must b e great er t han t hat of wat er. B ut we observe t he exact opp osit e. T his can again b e at t ribut ed t o surface t ension effect s.

We also observe t hat t he exp eriment al values show a sat urat ing t rend which is very similar t o t hat shown by t he empirical values. Anot her imp ort ant feat ure t hat we not ice is t hat t here is a dip in values of damping rat io as we progress

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from _D^h < 1 t o _D^h > 1 . T his is shown by t he t heoret ical formula as well.

T his happ ens b ecause when _D^h < 1 , t he cont ainer b ot t om int eract s wit h t he slosh mass and hence plays an imp ort ant role in arrest ing slosh. T he degree t o which t he cont ainer b ot t om aﬀect s damping is dep endant on t he shap e of t he cont ainer. [ 1 4]

5. 2 Hi ghe r m o de da m pi n g a n a lys i s

F igures 1 1 and 1 2 show t he damping rat io obt ained from higher mo de dat a along wit h t he lower mo de dat a. O n each dat a p oint we show error bar range of values obt ained when a part icular exp eriment was rep eat ed several t imes.

I n addit ion t o noise in t he measurement , t he error bar can b e at t ribut ed t o t he change in environment al condit ions under which different exp eriment s are carried out . S ince our exp eriment al condit ions are very sensit ive t o surface t ension and kinemat ic viscosity, which are again sensit ive t o t emp erat ure changes, a small change in t emp erat ure can b e reflect ed easily in t he magnit ude of t he damping rat io obt ained. From t hese figures we can observe t hat t he higher mo de damping is great er t han lower mo de damping. T his confirms t he fact t hat higher mo des damp out fast er t han t he lower mo des. T his is can also we observed from t he H ilb ert S p ect rum ( Refer F igure 7 as an example) .

We also observe t hat like lower mo de damping rat ios, higher mo de damping also shows a sat urat ion t rend, so we can generaliz e t hat damping, higher or lower mo de sat urat es as t he fill fract ion increases. O n comparing higher mo de damping wit h lower mo de, we not ice t hat higher mo de damping do es not show a dip t rend as shown by first mo de dat a. T his is b ecause t he higher mo de slosh mass is very less compared t o t hat of first mo de also it lies ab ove t he cent er

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of gravity at low liquid dept hs ( According t o t he Pendulum M o del[ 4] ) . D ue t o t his fact or, t he higher mo de slosh mass do es not int eract wit h t he cont ainer b ot t om hence it do es not display a dip feat ure in it s damping rat io.

Also we can see t hat higher mo de damping rat io of wat er is great er t han t hat of kerosene in spit e of kinemat ic viscosity of kerosene b eing great er t han t hat of wat er. T his can b e explained on similar lines as how surface t ension aﬀect s lower mo de damping.

6 C o n c l u s i o n

Wit h t he prop osed met ho d based on H ilb ert H uang Analysis we carry out accurat e est imat ion of higher mo de damping in liquid slosh. T he t echnique is based on t ime domain decomp osit ion of dat a in various I M F s from which t he paramet ers p ert aining t o individual higher mo de can b e est imat ed. We confirm t he accuracy of t he implement ed numerical t echnique by comparing t he empirical and exp eriment al result s of first mo de slosh. We find t hat t he damping rat io is significant ly dep endant of t he dimensions or t he t ank under considerat ion and t he fill fract ion of t he liquid. As t he diamet er of t he t ank and fill fract ion of t he liquid reduces, surface t ension effect s and cont ainer b ot t om int eract ions come int o play resp ect ively. T hese two paramet ers affect t he magnit ude of damping rat ios obt ained in b ot h higher and lower mo de.

T he pap er also compares t he damping t rends in lower and higher mo des.

B ot h lower as well as higher mo de damping show a sat urat ion t rend similar t o t he exist ing empirical formulae. Also higher mo de damping rat io is great er t han t hat of lower mo de, validat ing t he fact t hat higher mo des damp out fast er. T he t echnique present ed is fairly general and can b e applied in variety

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of ot her cases say for example est imat ion of ot her paramet ers such as higher mo de mass and p endulum hinge lo cat ion of p endulae analogy, damping in higher mo des for t anks wit h baﬄes and so on.

7 A p p e n d i x

7. 1 Expe ri m e n ta l Se tup.

T he exp eriment al set up for measurement of slosh induced paramet er has a cylindrical t ank made of acrylic having int ernal diamet er of 1 9 0 mm and height 45 0 mm and two ball screw typ e linear st ages: one used t o provide t ranslat ion excit at ion and t he ot her for pit ching typ e excit at ion of t he t ank. E xp eriment s p erformed in t he following pap er, require only t ranslat ion excit at ion of t he t ank. P hot ograph of t he set up is shown in F igure 1 3 . t ranslat ion mot ion is obt ained by convert ing rot ary mot ion t o linear mot ion. B rushless D C mot ors wit h built in enco ders are used t o provide rot ary mot ion. S ix axis load cell is used t o capt ure moment s and forces coming due t o sloshing in all t hree direct ions. T he load cell is assembled in t he set up just b elow t he t ank so t hat forces due t o frict ion can b e avoided. S ensor signals have b een capt ured and act uat ors cont rolled using S I M U L I N K and dS PAC E D AQ card int erface.

E xp eriment al set up is designed so t hat higher mo de slosh can also b e carried out . T he mot or is cont rolled using dS PAC E D AQ card. T his card can t ake feedback from t he enco der, which is used t o cont rol t he mot ion of t he mot or in sinusoidal form. Wit h t he help of S I M U L I N K mo del and P I D cont rol, mot or can b e cont rolled t o give any desired mot ion. T his rig is also designed t o conduct exp eriment s at higher mo de frequencies. [ 1 6 , 1 7 ]

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Re fe r e n c e s

[ 1 ] M at t ias G rund elius and B o B ernhard s s on, C ont rol of L iquid S los h in an Ind us t rial Packaging M achine. Proceedings o f the 1 999 IEEE Inte rna tio na l Co nfe re nce o n Co ntro l A pplica tio ns Ko ha la Coa st- Isla nd o f Ha wa i’ i, Ha wa i’ i, U SA . A ugust 22- 27, 1 999.

[ 2 ] S t efan aus d er Wies che, C omp ut at ional s los h d ynamics : t heory and ind us t rial ap p licat ion. Co mputa tio na l Mec ha nic s 30 (20 0 3), 374 387, Springe r- Ve rla g 20 0 3, DO I 1 0 . 1 0 0 7/s0 0 4 66- 0 0 3- 0 4 1 3- 8.

[ 3 ] Yonghwan K im, Exp eriment al and N umerical Analys es of S los hing F lows . D ep art ment of N aval Archit ect ure and O cean Engineering, S eoul N at ional U nivers ity

[ 4] Ab rams on, H . N, T he D ynamic B ehavior of L iquid s in M oving C ont ainers . NA SA Spec ia l Pub lica tio n SP- 1 0 6, W a shingto n, D. C. (1 966).

[ 5 ] M iles , J ohn W, O n T he S los hing O f L iquid s In a C ylind rical Tank. Re po rt Numbe r A M6- 5, G M- T - 1 8, T he Ra mo - W oo lridge Co rpo ra tio n, G uided Missile Re sea rc h Divisio n, A pril 1 956.

[ 6 ] K . M . C as e and W. C . Parkins on, D amp ing of s urface waves in an incomp res s ib le liquid . Jo urna l Fluid Mec ha nic s, Pa ge s 1 72- 1 84 , Vo lume 2, Pa rt 2, Ma rc h 1 957.

[ 7 ] Franklin T . D o d ge, T he N ew D ynamic b ehavior of L iquid s in moving cont ainers . S out hwes t Res earch Ins t it ut e S an Ant onio, Texas .

[ 8 ] D avid G . S t ep hens , H . Wayne L eonard and Tom W. Ferry, J r. , Inves t igat ion of t he d amp ing of liquid s in right circular cylind rical t ank, includ ing t ime-variant liquid d ep t h. NA SA Tec hnica l No te s D- 1 367.

[ 9 ] Raouf A. Ib rahim, L iquid S los hing D ynamics - T heory and Ap p licat ions . Ca mb ridge U nive rsity Pre ss.

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[ 1 0 ] N . E. H uang, Z . S hen, S . R. L ong, M . C . Wu, H . H . S hih, Q . Z heng, N . -C . Yen, C . C . Tung, and H . H . L iu, T he emp irical mo d e d ecomp os it ion and t he Hilb ert s p ect rum for nonlinear and non-s t at ionary t ime s eries analys is . Proceedings o f the Ro ya l Soc ie ty Lo ndo n, A , 4 54 , 90 3- 995, 1 998.

[ 1 1 ] Rob i Polikar, T he Wavelet Tut orial http: //use rs. ro wa n. edu/ po lika r

[ 1 2 ] N . E. H uang, M . -L . C . Wu, S . R. L ong, S . S . P. S hen, W. Q u, P. G lo ers en, and K . L . Fan, A conﬁd ence limit for t he emp irical mo d e d ecomp os it ion and H ilb ert s p ect ral analys is . Proceedings o f the Ro ya l Soc ie ty Lo ndo n, Se r. A 4 59, pa ge s 231 7234 5, 20 0 3.

[ 1 3 ] G ab riel Rilling, Pat rick F land rin and Paulo G oncalves, O n Emp irical M o d e D ecomp os it ion And it s Algorit hms . La bo ra to ire de Physiq ue (U MR CNR S 5672), Ecole N ormale S up erieure d e L yon 46 , allee d It alie 6 9 3 6 4 L yon C ed ex 0 7 , France.

[ 1 4] K eyur J os hi, M o d eling and Analys is of F luid S los h und er Trans lat ion and P it ching Excit at ion. Ma ste r’ s the sis, IIT Bo mba y, June 20 0 6.

[ 1 5 ] M ikis hev, G . N . and D oroz hkin, An Exp eriment al Inves t igat ion of Free O s cillat ions of a L iquid in C ont ainers . ( in Rus s ian) . Iz v. Akad , Nauk S S S R, t d . Tekh. N auk, M ekh. Trans lat ed int o Englis h by D . K ana, S out hwes t Res earch Ins t it ut e, J une 3 0 , 1 9 6 3 .

[ 1 6 ] P. S . G and hi, M ohan J at ind er, N . Anant hkris hnan, and J os hi K eyur, D evelop ment of 2 D O F act uat ion s los h rig: A novel mechat ronic s ys t em. In IEEE Inte rna tio na l Co nfe re nce o n Industria l Tec hno logy, IEEE, Dece mbe r 20 0 6.

[ 1 7 ] J at ind er M ohan, D es ign and d evelop ment of s los h rig for launch vehicle fuel t anks , Ma ste r’ s the sis, IIT Bo mba y, June 20 0 4.

[ 1 8 ] R. T . Rat o, M . D . O rt igueira1 and A. G . B at is t a, An Imp roved Emp irical M o d e D ecomp os it ion.

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[ 1 9 ] C hris t op her D . B lakely, A Fas t Emp irical M o d e D ecomp os it ion Technique for N ons t at ionary N onlinear T ime S eries , Ce nte r fo r Sc ie ntiﬁc Co mputa tio n a nd Ma the ma tica l Mode ling, U nivers ity of M aryland , C ollege Park M D 2 0 7 40 U S A

[ 2 0 ] WeiWang, J unfeng L i, T ians huWang, D amp ing comp ut at ion of liquid s los hing wit h s mall amp lit ud e in rigid cont ainer us ing F EM , Ac ta Mec h Sinica (20 0 6) 22: 9398 DO I 1 0 . 1 0 0 7/s1 0 4 0 9- 0 0 5- 0 0 81 - 3.

8 L i s t o f t ab l e s an d ﬁ gu r e s

Table 1 P rop ert ies of liquids used in exp eriment at ion.

F igure 1 I llust rat ion of S ift ing algorit hm.

F igure 2 Force dat a showing p oint of quickst op.

F igure 3 T he ﬁrst t hree I M F s obt ained on p erforming E M D on ﬁrst mo de dat a.

F igure 4 H ilb ert sp ect rum of ﬁrst mo de dat a formed from t he ﬁrst I M F . F igure 5 Force dat a obt ained from t he load cell at higher mo de excit at ion.

F igure 6 T he ﬁrst t hree I M F s obt ained on p erforming E M D on higher mo de dat a.

F igure 7 H ilb ert sp ect rum of higher mo de dat a formed from t he ﬁrst t hree I M F s.

F igure 8 L east square exp onent ial curve ﬁt t ing t o det ermine damping rat io of higher mo de from t he ﬁrst I M F .

F igure 9 D amping rat io versus ﬁll fract ion for ﬁrst mo de dat a for kerosene.

E rror margin has not b een plot t ed for clarity of diagram. M aximum error is 8 . 3 % . F igure 1 0 D amping rat io versus ﬁll fract ion for ﬁrst mo de dat a for wat er.

E rror margin has not b een plot t ed for clarity of diagram. M aximum error is 7 % . F igure 1 1 D amping rat io versus ﬁll fract ion for higher mo de dat a for kerosene.

F igure 1 2 D amping rat io versus ﬁll fract ion for higher mo de dat a for wat er.

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F igure 1 3 P hot ograph of slosh rig int erfaced wit h P C .

9 Tab l e s

Tab le 1

P rop ert ies of liquid s us ed in exp eriment at ion.

F luid D ens ityρ K inemat ic Vis cos ity ν S urface Tens ion σ k g/ m³ 1 0⁻⁶(m²/ s) 1 0⁻³(J/ m²)

Wat er 1 0 0 0 1 . 1 0 . 0 7 2

K eros ene 8 1 0 2 . 4 0 . 0 2 8

1 0 F i gu r e s

0 2 4 6 8 10

−2

−1 0 1 2

Time (sec)

Amplitude(N or Nm

) −4

−2 0 2 4

residue signal maxima minima e_max(t) m(t) e_min(t)

F ig. 1 . Illus t rat ion of t he S ift ing algorit hm.

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0 5 10 15 20 25 30 35 40

−6

−4

−2 0 2 4 6

Time (sec)

Amplitude (N)

Point of Quickstop

F ig. 2 . Force d at a s howing p oint of quicks t op .

−2 0 2

−1 0 1

Amplitude (N)

0 5 10 15 20 25 30

−0.5 0 0.5

Time (sec)

IMF 2 IMF 1 Input Signal

IMF 3

F ig. 3 . T he ﬁrs t t hree IM F s ob t ained on p erforming EM D on ﬁrs t mo d e d at a.

0 5 10 15 20 25 30

0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency (Hz)

Time (sec)

From IMF 1

F ig. 4. H ilb ert s p ect rum of ﬁrs t mo d e d at a formed from t he ﬁrst IM F .

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0 5 10 15 20

−2

−1.5

−1

−0.5 0 0.5 1 1.5 2

Time (sec)

Amplitude (N)

F ig. 5 . Force d at a ob t ained from t he load cell at higher mo d e excit at ion.

−2 0 2

−1 0 1

Amplitude (N or Nm)

−1 0 1

0 5 10 15 20

−1 0 1

Time (sec)

IMF 3 IMF 1

IMF 2 Input signal

F ig. 6 . T he ﬁrs t t hree IM F s ob t ained on p erforming EM D on higher mo d e d at a.

0 5 10 15 20

0 5 10 15

Frequency (Hz)

Time (sec)

IMF 1 IMF 2 IMF 3

F ig. 7 . Hilb ert s p ect rum of higher mo d e d at a formed from t he ﬁrs t t hree IM F s .

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0 1 2 3 4 5 6 7 8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6

Time (sec)

Amplitude(N or Nm)

IMF 2 Fitted curve

F ig. 8 . L eas t s quare exp onent ial curve ﬁt t ing t o d et ermine d amp ing rat io of higher mo d e from t he ﬁrs t IM F .

0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10⁻³

Fill Fraction (h/D)

Damping Ratio (ζ)

From higher mode excitation From lower mode excitation Empirical values

F ig. 9 . D amp ing rat io vers us ﬁll fract ion for ﬁrs t mo d e d at a for keros ene. Error margin has not b een p lot t ed for clarity of d iagram. M aximum error is 8 . 3 % .

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0.6 0.8 1 1.2 1.4 1.6 1.8 0

1 2 3 4 5 6 7 8x 10⁻³

From higher mode excitation From lower mode excitation Empirical values

F ig. 1 0 . D amp ing rat io vers us ﬁll fract ion for ﬁrs t mo d e d at a for wat er. Error margin has not b een p lot t ed for clarity of d iagram. M aximum error is 7 % .

0.6 0.8 1 1.2 1.4 1.6 1.8

0 1 2 3 4 5 6 7 8x 10⁻³

Fill Fraction (h/D)

Damping Ratio (ζ)

Error margin lower mode Lower mode damping Error margin higher mode Higher mode damping

F ig. 1 1 . D amp ing rat io vers us ﬁll fract ion for higher mo d e d at a for keros ene.

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0.6 0.8 1 1.2 1.4 1.6 1.8 0

0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016

Fill Fraction (h/D)

Damping Ratio (ζ)

Error margin lower mode Lower mode damping Error margin higher mode Higher mode damping

F ig. 1 2 . D amp ing rat io vers us ﬁll fract ion for higher mo d e d at a for wat er.

F ig. 1 3 . P hot ograp h of s los h rig int erfaced wit h P C