Experimental investigation of bi-stability in a vertically excited rectangular tank with finite liquid depth

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1 Developments n Martme Transportaton and Explotaton of Sea Resources Guedes Soares & López Peña (eds) 04 Taylor & Francs Group, London, ISBN Expermental nvestgaton of b-stablty n a vertcally excted rectangular tank wth fnte lqud depth N. Kaloumenos, A. Grammatkopoulos, C.C. Spandonds & K.J. Spyrou School of Naval Archtecture and Marne Engneerng, Natonal Techncal Unversty of Athens, Greece ABSTRACT: An expermental nvestgaton on the generaton and character of sloshng of a parametrcally excted lqud nsde a rectangular tank s reported. The focus s set on the phenomenon of lqud surface b-stablty,.e. when, for dentcal exctaton parameters values, sloshng may or may not be actvated, dependng on the ntal state of the surface when the exctaton s appled. From an earler theoretcal nvestgaton t had been conjectured that, on the plane of frequency versus ampltude of exctaton, the area of b-stablty s located adjacently to the prncpal resonance regon of the correspondng Matheu-type system. Ths was produced from a nonlnear mathematcal model focused on the frst free surface mode, derved by elmnatng hgher modes through an adaptve mode-orderng scheme. The comparson aganst expermental results presented n the current paper corroborates that the theoretcal predctons are far. The experments provde also evdence about the actvaton of hgher modes and the occurrence of phenomena of mode competton. InTroducton As parametrc sloshng s meant the moton of a lqud s free surface, trggered by an exctaton that acts perpendcularly to the undsturbed free surface. As ponted out many years ago, for vbratng structures contanng lquds such a phenomenon may sometmes ncur catastrophc consequences (Dodge 966). Shp motons along the gravty vector, thus parametrcally exctng the transported lquds, arse physcally n combnaton wth other rectlnear or angular shp motons. Nevertheless, t s very useful to understand fundamentally the varous ways n whch sloshng moton can appear, especally when ths happens qute unexpectedly. The semnal nvestgaton of Benjamn & Ursell (954) produced a frst glmpse of the stablty chart assocated wth the behavour of the free surface of a parametrcally excted lqud. It was derved from a lnear Matheu-type equaton. Whlst ths captures the regon of lnear nstablty, t does not suffce for predctng the ensung free surface elevatons, yeldng unrealstc nfnte wave ampltudes nsde the nstablty regon. In a seres of papers, Mles nvestgated parametrc sloshng on the bass of an averaged Lagrangan approach, usng weakly nonlnear models (see for example Mles 994). Extendng Mles approach, Decend (995) and Decent & Crag (995) found hysteress due to competton between the fnte-ampltude and the flat-surface solutons. Pernet et al. (009) carred out three-dmensonal smulatons for the full nonlnear vscous problem, up to relatvely hgh exctaton ampltude. They reproduced the remarkable square and hexagonal free surface patterns that had been expermentally captured earler by Ktyk et al. (005). Also, the ssue of mode competton had been nvestgated expermentally by Smonell & Gollub (989) and by Crak & Armtage (995) for shallow depth tanks. A comprehensve collecton of earler efforts concernng the nonlnear behavour of lquds carred n tanks of varous shapes and subjected to parametrc exctaton can be found n Ibrahm (005). Research on the problem of lqud sloshng n a rectangular vertcally excted tank was ntated recently n our group. A descrpton of our modellng approach, mplemented so far for -D cases, s presented n Spandonds & Spyrou (0 & 0). It s based on the adaptve multmodal analyss ntroduced by Faltnsen & Tmokha (00, 009), appled earler for drectly excted sloshng. Couplng of the model wth a contnuaton algorthm of nonlnear dynamcal systems corroborated that a b-stablty regon exsts, located entrely nsde the doman where, from a lnear perspectve, the surface should appear as quescent. Ths analyss revealed further that, n lne wth well-known behavour of nonlnear parametrcally excted systems, the lower boundary of the b-stablty regon s defned by the locus of the foldng ponts of the curve of lmt-cycle ampltude. The ntrgung feature of ths regon, yet a common one for those famlar wth strongly nonlnear behavour, 9

2 s that t s ntal-condtons-dependent;.e. one may obtan a stable wavy surface or a flat surface dependng on how much dsturbed the free surface had been when the harmonc exctaton was frstly appled. In the presented work the man objectve was the expermental reproducton of parametrc sloshng. The experments were conducted usng a shakng table faclty dscussed n a later secton. Attenton has been pad on actvatng wave modes only along the longer tank sde. Even though complex exctaton scenaros could easly be examned, at ths stage was assumed harmonc vertcal forcng wth small to moderate ampltude. Exctaton frequences were kept around the frequency of prncpal resonance of the lowest mode (n reference to the longer sde). Results have been compared aganst the relevant generc lnear stablty chart as well as aganst predctons obtaned by the modal analyss method (that s brefly outlned n the followng secton). The appled mode orderng scheme leads to a fundamental core model comprsed of a sngle, Matheu type, nd order ordnary dfferental equaton, presentng two thrd-order nonlneartes: one due to a product of elevaton wth the square of elevaton velocty; and another due to a product of elevaton squared wth acceleraton. SUMMARY OF THEORETICAL RESULT The heght-to-length rato of the consdered rectangular tank s hl= 04.. The hydrodynamc problem has been formulated n terms of Laplace s equaton for the velocty potental Φ(y, z, t) appled throughout the flud volume Q(t). The well-known boundary condtons are enforced on the free surface Σ(t) and on the tank surface S(t). Faltnsen & Tmokha (00 & 00) converted the governng equatons nto an nfntedmensonal system of ODEs, postulatng Fourer seres representatons for descrbng surface s elevaton ζ(y, t) and the velocty potental Φ(y, z, t), as ndcated below: ζ( yt, ) = β ( t) f ( y) = Φ( yzt,,) = R () t ϕ ( y, z) = πz ( ) cosh π ϕ( yz, ) = cos l y+ l l πh cosh l () () (3) y l f ( y ) = cos π +, (4) l The axes orgn s fxed at the mddle of the undsturbed surface. The y axs ponts to the rght along the length of the tank and the z axs s vertcal, pontng upwards. β (t) corresponds to the tme-dependant free surface elevaton that occurs due to the th natural mode. Keepng nonlnear terms up to thrd-order, applyng a mode orderng scheme based on β = Ο(ε /3 ), β µ = O(ε), µ > fnally retanng β only, ncorporatng dampng and assumng perpendcular harmonc exctaton n 3 = n 3α cos(σt), the nfnte system of ODEs s reduced for purely vertcal exctaton to the followng smple equaton: β ζ σβ σ σ + + n a σ β cos( t) g ββ β + d ( + β ) = 0 (5) where: d = π h h 4 l tan h π tan h π l (6) β stands n ths case for the tme-dependant free surface elevaton at y = -l/;.e. at the left tank wall. σ µ s µth natural frequency and δ s Kronecker s symbol; whle d, t are functons of lqud s heghtto-tank-length-rato. It s apparent that omsson of the nonlnear terms leads to a Matheu-type system. The mathematcal model was coupled to the computatonal algorthm MATCONT (Doodge et al. 003). Ths algorthm s much more than an ODE solver, allowng to trace effcently steadystate solutons of Eq. as one vares ether the frequency rato or the exctaton ampltude (or even both smultaneously). Stablty analyss yelded that the forcng-versus-frequency parameters plane s dvded nto three areas. Area A s host to quescent steady solutons:.e. every vertcal external exctaton leads nvarably, no matter what the ntal state of the free surface was, to a flat lqud surface. Area B s the classcal area of nstablty where the exctaton generates free surface oscllaton. Typcal parametrc oscllatons assocated wth ths area have frequency about half the forcng frequency. However, other much more complex responses can be found here too. Area C s the one where b-stable behavour s exhbted. The same external exctaton leads ether to a quescent surface or to a wavy one, dependng on the ntal condton (determned through the ntal values of β and j ). 9

The motons can reach ampltudes of 30 o and 0.5 m. In Table are collected the key features of the shakng table.

3 3 EXPERIMENTAL SETUP In Fgure s shown the vbraton testng faclty ( shakng table ) of NTUA s School of Naval Archtecture and Marne Engneerng. It s based on a table platform that s able to perform Sx Degree Of Freedom (6-DOF) low frequency motons accordng to approprate nput tme hstores, emulatng motons performed by shps n waves. The motons can reach ampltudes of 30 o and 0.5 m. In Table are collected the key features of the shakng table. A Labvew nterface program reads or generates desred tme hstores and computes the approprate actuator motons. To translate the platform s calbraton/measurement to real moton, real tme PID loops are used that control the sx actuators. Moreover a table-mounted Mnature Atttude Headng Reference System (AHRS) wth GPS (Mcrostran 3DM-GX3-35) reports to the man PC the exact poston of the table at any tme. Fgure. The shakng-table of the School of Naval Archtecture and Marne Engneerng, Natonal Techncal Unversty of Athens. Flow vsualzaton s possble by the use of a hgh speed vdeo system. Ths system conssts of two hgh speed vdeo cameras (Trouble-Shooter HR color) wth a resoluton of pxels and a maxmum speed of 6,000 mages per second that cooperate n a master/slave mode. A specalzed software program (MIDAS 4.0) s used for eventcapture camera control, synchronzaton wth data sources, and automated montorng. The use of two PCD-300B sensor nterfaces workng n a master/slave mode enables the man PC to perform up to 6 dfferent stress and force, pressure, acceleraton and dsplacement measurements through the use of stran gages and transducers, respectvely. DCS 00 A dynamc data acquston software enables easy, nteractve settng of measurng condtons and sensor nformaton as well as montorng of measurng data on numerc and varous graph wndows. The tank that was used s shown n Fgure. It was ntally desgned for cargo shft nvestgatons (IMO 0). The tank s made from noncoloured Perspex of 0 mm thckness to permt drect observaton. Its sze s 0.6 m 0.4 m 0.4 m (wdth length heght). To keep the scenaro close to the one that had been nvestgated by the theoretcal approach (that s, to have fnte lqud depth wth h/l = 0.4), we partly flled the tank wth water up to 0.4 m heght. The geometry of the tank and the heght of the lqud yeld the natural frequences of the system, whch depend ether on tank s length or tank s wdth (as a matter of fact n heght-to-length and heghtto-wdth rato). In Tables and 3 are presented the natural frequences correspondng to the frst few modes, for longtudnal and transverse waves. In general, complex wave formaton can be observed, featurng combned longtudnal and transverse oscllatons (one example s shown n Fg. 3). Table. Shakng-table propertes. Table sze 0 cm 0 cm Degrees of freedom Sx (6) Max. payload tons Max. dsplacement n heave* 30 cm Max. velocty n heave* 50 cm/s Max. dspl. n sway/surge* 30 cm Max. velocty n sway/surge* 30 cm/s Max. dspl. In n roll/ptch/yaw* 30 deg Max. velocty n roll/ptch/yaw* 5 deg/s Frequency range 0 8 Hz Max acceleraton, any axs 0.3 g *Reduced range when complex movements are performed. Fgure. The orthogonal tank used for the experments. It s made of Perspex of 0 mm thckness. 93

Table. Natural frequences accordng to length of tank. Odd modes Even modes st 6.608 nd 0.0693 3rd.4069 4th 4.3333 5th 6.057 6th 7.5553 7th 8.969 8th 0.7 Table 3.

The exctaton was always harmonc and along the vertcal axs. A dense grd of exctaton ampltudes and frequences was examned.

4 Table. Natural frequences accordng to length of tank. Odd modes Even modes st nd rd th th th th th 0.7 Table 3. Natural frequences accordng to wdth of tank. Odd mode Even mode st nd rd th th th th th 4.8 condton. The exctaton was always harmonc and along the vertcal axs. A dense grd of exctaton ampltudes and frequences was examned. The frequency range was from 0 to 5 rad/s and the ampltude range was from 0 to 0.05 m. In order to verfy the results, each experment was repeated twce. 4. Zero ntal condtons Suffcent tme was allowed between consecutve runs n order for the free surface to calm down and thus acheve practcally a zero ntal condton for the ensung run. The free surface dynamc response was labelled as stable f at the end of the run t appeared to be calm, lke at the begnnng of the run. It was unstable f, towards the end of the run, a non-decayng wavy surface was prevalent. In Fgure 4 are shown tme nstances of lqud s surface moton for two dfferent exctatons, exhbtng two dfferent wave patterns: that are ether the frst (up) or the second (down) antsymmetrc modes (modes that correspond to the 3rd and st natural frequency or else to 3rd mode s fundamental and st mode s prncpal resonance, respectvely). Fgure 3. Combned wave n longtudnal and n transverse drecton. The tank s excted along the gravty vector. The tank presents a slght tlt around the transverse axs. However, by restrctng the range of exctaton frequency around 3 rad/s (the frequency whereabout the vertex of the prncpal resonance regon for the correspondng D tank should be expected) and keepng the exctaton ampltude relatvely low, the actve modes assocated wth tank length can be relatvely safely nvestgated, thus practcally reducng the 3D to a D problem (nevertheless, some nterference should stll be expected snce the st, nd and 3rd natural frequency of the transverse waves (see Table 3), are close to the borders of the nvestgated frequency range). 4 EXPERIMENTAL RESULTS Over 00 runs were performed, separated nto two sets, accordng to the ntal free surface Fgure 4. Examples of obtaned wave forms: (upper) st ant-symmetrc mode correspondng to the st natural frequency (exctaton frequency:.9 rad/s, ampltude:.9 cm); (lower) nd ant-symmetrc mode correspondng to the 3rd natural frequency (exctaton frequency:.3 rad/s, ampltude:. cm). 94

In Fgure 5 s summarsed the regon of nstablty (located to the nteror of the sold lne), as obtaned from the campagn of runs wth zero ntal condtons.

5 Fgure 5. Instablty chart (sold lne). A comparson s shown aganst lnear theory predctons for the prncpal parametrc nstablty of the st mode (dashed lne) and the fundamental nstablty of the 3rd (dotted lne). In Fgure 5 s summarsed the regon of nstablty (located to the nteror of the sold lne), as obtaned from the campagn of runs wth zero ntal condtons. On the graph are supermposed theoretcal predctons for the correspondng lnear Matheu-type model of the st (dotted lne) and 3rd (dashed lne) natural mode. The coloured regon corresponds to unstable free surface. Theoretcal predctons appear well located aganst the measurements. Insde the nstablty area was observed mode competton. It appears that an nternal border exsts (not shown). To ts left, every exctaton yelds the pattern of the 3rd mode (full wave oscllaton); whereas to ts rght, s realsed the st mode (half wave oscllaton). However, around the border these tests had to be kept on for longer tme, untl the steady pattern emerged. 4. Non-zero ntal condtons In the second seres of tests, the same ranges of frequency and ampltude were examned. However, shortly before each run, a short volent roll exctaton was appled to the tank, so that the free surface acqures some knd of oscllatory pattern when the parametrc exctaton was appled. Thus a non-zero ntal condton was acheved. The obtaned pcture of the stablty regon presented notable dfferences compared to that of the frst seres. Two were the key fndngs of ths nvestgaton: a. The nstablty area s much wder compared to that obtaned from the frst seres (Fg. 6). Ths supples concrete expermental evdence about the exstence of an area where ntal condtons affect crucally lqud s dynamc response. In Fgure 7 s shown a comparson of two qualtatvely dfferent steady-state free surface Fgure 6. Comparson of expermental results for the two seres of tests. Results of the st seres (zero ntal condtons) are ndcated by the dashed lne whle those of the nd (non-zero ntal condtons) are ndcated wth the sold lne. Area A corresponds to a flat surface steadystate and area B to a wavy one. In area C are hosted the ntal-condton-dependent cases. Fgure 7. Lqud response obtaned for ntally horzontal free surface (upper); and for ntally dsturbed free surface (lower). Exctaton frequency and ampltude were.7 rad/s and.5 cm respectvely. responses obtaned nsde the b-stablty area (area C). In the upper pcture the surface retans the calm water characterstcs despte the vertcal oscllaton of the tank. In the lower pcture, the surface oscllates accordng to the st mode. b. A second notable fndng s that, nsde the b-stablty area C the free surface follows qute dfferent patterns under slghtly dfferent exctaton frequency and ampltude values. 95

Fgure 9. Comparson between numercal (sold lne) and expermental (dotted lne) results. Instablty chart ndcatng the three dfferent areas. Fgure 8. Free surface oscllatons n dfferent modes.

6 Fgure 9. Comparson between numercal (sold lne) and expermental (dotted lne) results. Instablty chart ndcatng the three dfferent areas. Fgure 8. Free surface oscllatons n dfferent modes. In the upper pcture (obtaned wth exctaton frequency. rad/s and ampltude. cm) s captured the 3rd ant-symmetrc mode (5th natural frequency). In the lower pcture, s captured the 4th ant-symmetrc mode (7th natural frequency). The exctaton frequency was 4.39 rad/s and the ampltude was.7 cm. In contrast to what happens n area B where purely fundamental and prmary resonance of 3rd and st mode respectvely were observed (as well as evdence of ther competton appeared), n area C addtonal patterns appeared. Specfcally, for exctaton frequency between and. rad/s and for ampltude hgher than cm the free surface followed sometmes the 5th natural mode (3rd ant-symmetrc mode) as shown n Fgure 8a. Even hgher order natural modes appeared. In Fgure 8b s shown a tme nstance where the 7th natural mode (4th antsymmetrc mode) has appeared. These phenomena possbly have to do wth hgher resonances of these modes (hgher than fundamental). It s noted that double perod phenomena are also expected to occur nsde the nstablty area, for hgh enough exctaton ampltudes (Ibrahm 005). 5 EXPERIMENTAL VS NUMERICAL RESULTS Comparson between the fndngs from the experments and from the predctons of our mathematcal model s presented n Fgure 9. The predctons of the numercal method are n far agreement wth the expermental results. One dfference s that the nstablty area obtaned from the experments s wder. However, the numercal model had been focussed only on the prncpal resonance of the frst mode and furthermore, t was restrcted to non-lnear terms up to the 3rd order. Another dfference s that the b-stablty area appears wder too n the case of the experments. Ths can be understood snce hgher order natural modes (nd, 5th and 7th natural modes) were actvated n the tests whch however were not accounted n the mathematcal model. It should be noted that more complex models nvolvng these modes are currently under nvestgaton. 6 CONCLUSIONS By expermental technques -D lqud sloshng n a rectangular, vertcally excted tank has been nvestgated, wth focus on valdatng the predcton of a b-stablty area n parameters plane. The results obtaned from the experments were compared aganst numercal results obtaned from modal analyss. The nvestgaton was lmted to a specfc fnte lqud depth, correspondng to a tank-heght-to-depth rato of 0.4. The experments ndeed confrm the exstence of an area of bstablty of parametrcally excted sloshng. Insde ths area, one may obtan a stable wave or a flat surface, dependng on the free surface s state when the exctaton was frstly appled. The theoretcal predctons are n good qualtatve agreement wth the real free surface dynamcs, despte the assumpton of a sngle domnant mode and the excluson of the hgher order non-lnear terms. A next step towards confrmng the capablty of the approach to produce more realstc results 96

7 wll be the nvestgaton of the dynamc behavour assocated wth the mmedately hgher order nonlnear model and new valdaton of these numercal predctons by further seres of experments. REFERENCES Benjamn T.B. & Ursell F., 954. The stablty of the plane free surface of a lqud n a vertcal perodc moton, Proceedngs of the Royal Socety A5: Crak A.D.D. & Armtage J, 995. Faraday exctaton, hysteress and wave nstablty n a narrow rectangular wave tank. Flud Dynamcs Research 5: Decent S.P., 995. Hysteress and mode competton n Faraday waves. Ph.D. thess, Unversty of St Andrews, Scotland. Decent S.P. & Crak A.D.D., 995. Hysteress n Faraday resonance. Journal of Flud Mechancs 93: Dodge F.T Vertcal Exctaton of Propellant Tanks. In the book: The Dynamcs of Lquds n Movng Contaners. NASA Report SP 06. Dhooge A., Govaerts, W., Kuznetsov, Y.A., Mestrom,W., Ret, A.M., Sautos, B., 003. MATCONT and CL_MATCONT: Contnuaton Toolboxes for MAT- LAB, Gent (Belgum) and Utrecht (Netherlands) Unverstes. Faltnsen, O.M. & Tmokha, A.N., 00. Adaptve multmodal approach to nonlnear sloshng n a rectangular tank, Journal of Flud Mechancs, 43: Faltnsen, O.M., Tmokha, A.N., 009. Sloshng, ISBN: , Cambrdge Unversty Press, New York. Ibrahm, R.A., 005. Lqud Sloshng Dynamcs, ISBN: , Cambrdge Unversty Press, New York. Internatonal Martme Organzaton (IMO), 0. Sold Bulk Cargoes Code [IMSB Code 68(85)], IMO publshng, ISBN: Ktyk A.V., Embs JMekhonoshn., V.V., Wagner C., 005. Spatotemporal characterzaton of nterfacal Faraday waves by means of a lght absorpton technque, Physcal Revew E 7, Mles J.W., 994. Faraday waves: rolls versus squares, Journal of Flud Mechancs 69: Pérnet N., Jurc D., Tuckerman L.S., 009. Numercal smulaton of Faraday waves, Journal of Flud Mechancs 635, 6. Smonel F & Gollup J.P., 989. Surface wave mode nteractons: effects of symetry and degeneracy, Journal of Flud Mechancs 99: Spandonds, C & Spyrou K.J., 0. Parametrc Sloshng n A D Rectangular Tank wth fnte lqud depth. Proceedngs, 4th Internatonal Congress of the Internatonal Martme Assocaton of the Medterranean (IMAM), Genova, September: Spandonds, C.C. & Spyrou K.J., 0. B-stablty n a vertcally excted rectangular tank wth fnte lqud depth, Ocean Systems Engneerng (3):

REFRACTION. a. To study the refraction of light from plane surfaces. b. To determine the index of refraction for Acrylic and Water.

$REFRACTION. a. To study the refraction of light from plane surfaces. b. To determine the index of refraction for Acrylic and Water.$ Purpose Theory REFRACTION a. To study the refracton of lght from plane surfaces. b. To determne the ndex of refracton for Acrylc and Water. When a ray of lght passes from one medum nto another one of dfferent