Proceedings of the ASME nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France

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1 Proceedngs of the ASME nd Internatonal Conference on Ocean, Offshore and Arctc Engneerng OMAE June 9-,, Nantes, France OMAE-6 ROLL DAMPING COEFFICIENTS ASSESSMENT AND COMPARISON FOR ROUND BILGE AND HARD CHINE HULLFORMS Ermna Begovc Unversty of Naples Federco II Naples, Italy Carlo Bertorello Unversty of Naples Federco II Naples, Italy Jasna Prpc Orsc Faculty of Engneerng Unversty of Rjeka Rjeka, Croata ABSTRACT The sustanable desgn of small passenger vessels and large sze pleasure craft ndcates new trends: lower speed, reduced fuel consumpton and better seakeepng n all range of veloctes. Small ampltudes of roll moton are consdered one of the most mportant comfort parameter. These trends call for revewng some commonly used concepts and desgner practce. Fundamental parameters as metacentrc heght and mdsecton form affect comfort as well as shp safety n counteractng way. Round blge, due to good seakeepng characterstcs and to large amount of avalable data from systematc seres s the most common hull form choce when crusng speed mples dsplacement or sem dsplacement mode. Hard chne hull forms are generally chosen when relatve speed leads to some hydrodynamc lft, although recently they have been consdered for low relatve speed. They allow smplfed and cheaper constructon and among ther nterestng features s hgher roll dampng. Ths appears very attractve for craft that are generally ftted wth stablzng systems and ask for the most reduced roll motons at any and zero speed. Ths paper presents the results of expermental tests relatve to roll decay of two shp models of the same dmensons, dsplacement and GM values: they have the same warped bottom hull form developed at Unversty of Naples, the frst s hard chne and the second has round blge from transom to mdshp. Both of them are sutable for dsplacement and semdsplacement craft operatng at medum-low relatve speed. The tests have been carred out at Unversty of Naples towng tank, n calm water, at Fn rangng from. to.. Natural roll frequences and lnear and non lnear dampng coeffcents are reported for all tested speeds allowng a far comparson of the roll characterstcs due to the varaton n hull form. INTRODUCTION Increased length recently acheved by pleasure craft, the demand for superor comfort and recent ncrease n fuel cost lead to consder low relatve speed as very nterestng feature for sustanable desgn of pleasure boats and ths remans only possble choce for max and mega yachts. Smlar consderatons can be done for small passenger vessels although transport effcency mples careful consderaton of speed reducton. Wthn ths frame, round blge dsplacement hull forms are generally used. They have been studed extensvely for dfferent type of vessels and as systematc seres (Seres NPL, D Seres, NOVA); plenty of data are avalable about ther hydrodynamc performances both n calm water and n rough sea. Hard chne hull forms, n some cases wth chne developed only n the after part of the hull (Systematc Seres SKLAD), have been consdered recently for low relatve speed motor yachts and trawlers. They allow smplfed constructon and farng processes, but ther ntroducton s mostly due to the practcal evdence of hgher roll and ptch dampng. Unfortunately there are lttle nformaton about resstance and behavor of such hard chne warped hull forms at low relatve speed. As regard roll dampng, publshed tests on hard chne hull forms have been performed at hgh or very hgh relatve speed when hydrodynamc forces are predomnant. Copyrght by ASME

2 Roll moton and roll nduced acceleratons have been always consdered n small craft desgn wthn safety matters. Very stff craft as tugs and fshng vessels need to be carefully consdered to ther hgh GM and consequent hgh acceleraton values. At present, the pleasure craft larger sze that means longer offshore passages and open water moorng has focused desgner nterest on roll moton as one of the most nfluencng factors on comfort aboard. Unfortunately roll dampng optmzaton counteracts optmal values of other fundamental parameters as GM or appendxes wet surface. It s very dffcult or even mpossble to use roll behavor as a prmary desgn crtera and the usual practce looks for compromse solutons or consders stablzng devces. Besde the wdely used actve fns, zero speed actve fns and gyroscopc stablzers are also avalable and wdely used. The wave nduced roll motons at low and zero speed, the hull form dampng ntrnsc characterstcs and the further nfluence of dedcated devces based on hydrodynamc nteractons or gyroscopc effects need to be evaluated to provde adequate desgn references. Some general consderatons on roll dampng of modern yachts are presented n Gallarde () where the contrbutons of hull form, blge keels, passve fns and actve fns n roll dampng at anchor were reported based on the expermental campagns carred on at MARIN, gvng approxmately % as the percentage contrbute for hull dampng, % for blge keels, % for passve fns and 6% for actve fns. Theoretcal-emprcal determnaton of roll dampng s n most cases based on the method proposed by Ikeda et al. (97 a, b) for typcal cargo shps wth vertcal sdes and horzontal bottom wth C B varyng from.6 to.. The method conssts n breakng down the roll dampng nto several components accordng to the physcs of ther appearance and summng them to fnd out the equvalent roll dampng coeffcent. Basc method consders fve components partcpatng n shp total roll dampng, namely: frcton, wave, eddy, lft and blge keels. Wave makng, frcton and lft roll dampng are lnear components proportonal to the roll angular velocty. Eddy makng and blge keel components are non lnear. In Ikeda (97 a), the authors have found on the bass of seres of experments that eddy makng component for a naked hull s proportonal to the square of roll frequency and to roll ampltude.e. eddy makng at zero speed depends on hull form only. Ths component decreases rapdly wth forward speed and then t reduces to non-lnear correcton for the lft force. The frst mprovements by Ikeda, Hmeno and Tanaka (97b) extended the method to the forward speed case, later n Tanaka and Ikeda (9) the method was modfed for small craft hard chne hull wth a large skeg. Ths approach was revewed by Ikeda () reportng brefly the mprovements made to the orgnal method and the applcaton of mproved components predcton to assess optmum blge keel sze and locaton. Ikeda (99) appled hs method to hard chne hull forms typcal for hgh relatve speed and planng regme. The dsagreement of predcted and measured results suggested the exstence of other roll dampng components. In Ikeda and Katayama reported the roll predcton method for hgh speed planng craft where the experenced addtonal dampng s related to the vertcal lft force. The proposed method determnes the vertcal lft from Savtsky (96) sem-emprcal procedure, calculates the contrbuton n roll dampng due to ths force and adds t to the orgnal Ikeda method. The method s valdated at very hgh Froude number (F n.6) for monohedral hull of degree deadrse angle at dfferent runnng trms. It also concludes, as expected, that at hgh forward speed the domnant effect s due to the vertcal lft. Durng 9-tes some expermental estmatons of roll dampng for monohedral hulls wth, and degree deadrse were reported n Ward Brown and Klosnsk Brown (99 a, b). The free oscllaton roll tests were made at one dsplacement and covered varatons n speed (C V.,.,.), trm (, and 6 degrees), and yaw. Measured quanttes ncluded tme hstores of the roll extncton, from whch the frequency and logarthmc decrement of the roll moton were determned. The hydrodynamc effects of added nerta and dampng n roll are deduced and emprcal expressons for these quanttes are obtaned n terms of craft geometry and operatng condtons. The authors expressed added nerta as functon of the wetted length and shown that t s ndependent from speed, trm and yaw angle. The dampng ncreases wth yaw angle, speed, and wetted length, but t s otherwse ndependent of trm. Obtaned emprcal expressons ft expermental results wthn % precson and authors appled them to predct the response of a ft planng craft at speeds up to 6 knots. Although hard chne planng hull forms have been subject of studes and some results exst, there s complete lack of data on slow speed behavor. Furthermore, exstng data are always relatve to monohedral hull forms only whle actual desgn trends are focused on warped hard chne hull forms. Ths work s focused on the determnaton of roll dampng due to the dfference n round blge and hard chne of warped hull forms n dsplacement and sem-dsplacement regme. Hard or rounded blge chne, extended from transom to md-shp, has been chosen as the geometrcal nfluencng factor. ROLL DAMPING EXPERIMENTAL ASSESSMENT To determne roll dampng expermentally the usual practce s to perform a so-called roll decay test, n whch roll moton s nduced by releasng the hull after applyng a transversal nclnaton. The subsequent roll moton s measured and fnally the roll dampng s computed from these measurements. All methodologes of roll dampng analyss are startng from the one-degree of freedom equaton for roll moton n the form: ( I + A ) & φ + B ( & total φ ) + C () where I and A are moment of nerta and added moment of nerta, C s roll restorng coeffcent, B -total stands for nonlnear dampng coeffcent ncludng vscous effects and φ s roll angle. Copyrght by ASME

3 Dampng coeffcent B -total can be expressed as a seres expanson of φ & and φ & n the form: B & & & & total () B + B φ + B whch s nonlnear representaton. Usng ITTC Recommended Procedures and Gudelnes () nomenclature, the equaton () can be rewrtten n a more common form by dvdng for the acceleraton coeffcents as: & φ + & φ + β & φ & φ + γ & φ + ω () Where B I + M B β I + M () B γ I + M ω I C + M π T Usng Fourer expansons for non lnear term: & φ & φ ω & φ () π It s possble to obtaned lnearzed roll equaton n the form: && φ + & φ + β ω & φ + γ ω & φ + ω π then lnear equvalent coeffcent can be expressed as: && φ B total eq ( I + A ) + & eq φ + ω and ( ) + ω β + ω γ (6) eq π To better understand cubc coeffcent effect the Authors performed dfferent analyss for model speed zero and m/s. The valdty of dfferent models was checked through Runge Kutta soluton of ordnary dfferental equaton and measured tme seres for small and large ntal ampltudes. The procedure has four prncpal steps. From the typcal roll decay curve (Fg. ) the maxmum (t, φ ) and mnmum (t +, φ + ) are found for each decrement and mean oscllaton ampltudes are calculated as: φ + φ+ φ MEAN (7) t+ t 6 Tme (s) Roll Decay (deg) φ φ + Fgure. Typcal roll decay curve. Equvalent dampng coeffcent eq s then defned as: φ eq, log ln + ω β φ MEAN t+ t + π () φ φ+ eq, Ln _ + ω MEAN β (9) t t π eq, Ln _ + φ φ+ t t + + ω π + ω MEAN MEAN γ β () Accordng to Lewandovsky the cubc term s emprcal adjustment of quadratc term wth no smple physcal justfcaton, Chakrabart (99) consders only quadratc term and ITTC reports that the coeffcents B -, B - are consdered constant durng a steady perodc oscllaton and that they may depend on the roll ampltude φ a and on encounter frequency ω E. but doesn t consder or explan the behavour of B - connected wth cubc term. Bulan (9) consders cubc coeffcent relevant for the bare round blged hull at zero speed and moreover t allows proper natural frequency determnaton. φ Typcal dagram of ln t + t φ+ ampltude φ MEAN- s gven n Fg.. as functon of mean Copyrght by ASME

4 ln(φ -φ + )/(t + -t ) y.9x +. R².. φ MEAN (deg) Fgure. Typcal logarthmc decrement. And the damped frequency ω ω + (), eq, eq, For each performed decay at each speed the analyss of roll decay conssts n:. assessment of lnear and logarthmc decrement dampng accordng to the procedure 7 -. assessment of frequency of oscllaton ω ( φ) for each sngle decrement. mergng the data of each decrement performed at one forward speed V and fnal calculaton of coeffcents It should be noted that mergng of all decrements for consdered speed ncreases the precson of fttng coeffcents assessment. 7 6 (φ -φ + )/( t + -t ) y.7x +.x R².97 ROUND BILGE VS HARD CHINE Two models were bult; the only dfference among them s hard chne (Fg. ) vs. round blge (Fg. ). The models prncpal characterstcs are gven n Table. φ MEAN (deg) Fgure. Typcal lnear decrement. When consderng logarthmc decrement the roll dampng dependency on ampltude s captured as the frst order polynomal to the equvalent roll dampng scatter and coeffcents and β are obtaned drectly from the fttng curve shown n Fg. wth the frst order polynomal y ax+b. Fgure. Model W Hard Chne hull form. b β a () If lnear decrement model s used then the coeffcents of polynomal y ax+bx +cx are, β and γ respectvely: a π β b ω γ c ω π The natural roll frequency of sem-perod s defned as:, π t t ω () + Fgure. Model R Round blge hull form. Fgure 6. Hard chne - Round blge secton comparson. Copyrght by ASME

5 Table. Models prncpal characterstcs. Hard Chne Round Blge L OA (m).9.9 L WL (m).6.6 B (m).. B WL (m).7.7 B (m).. T (m).. Trm (deg) Δ (kg) LCG (m) from.6.6 transom VCG (m) from BL.7.6 GM T (m).7. CB.. C WP.7.7 k -ar (m).66.6 k -ar (m).96. Prevously to tank tests LCG, VCG, k and k of both models were measured usng the nertal balance (shown n Fg. ). Roll and ptch tme hstores were measured by accelerometer Cross Bow CXLGP-R-AL at samplng frequency of Hz to determne the perod of motons on the bass of oscllatons repeatng the measure three tmes. The rad of nerta are then calculated subtractng the part due to the nertal balance structure. Obtaned k -ar and k -ar values are. B WL and.6 L OA respectvely. Shp/model scale rato s λ, that means. m L OA n full scale. Ths s standard value for a passenger ferry, or for a t full load dsplacement max yacht. Fgure. The equpment used to measure mass nertal characterstcs of the models. EXPERIMENTAL ROLL DECAY SET UP Several roll decay tests n calm water have been carred out at zero speed, and at model veloctes of.,.,.7,.,. and.9 m/s correspondng to shp speeds of,, 7, 9, and 7 knots n full scale. The models have been towed by elastc ropes fxed n the proxmty of waterlne, as shown n Fg. 6. The ropes are fxed to the alumnum bars connected to the towng carrage. Intal heelng of the model s manually nduced and the nclnaton angle s controlled n real tme by nertal platform. The nclnaton angles vared from to degrees n order to obtan suffcent data base to determne both lnear and nonlnear dampng coeffcents. The roll tme hstory has been recorded by nertal platform Xsense at frequency of Hz. Yaw was controlled by nertal platform but t was not observed durng the tests. IMO MSC./Crc. (6) recommends at least four tests wth dfferent ntal angles and f the roll dampng s very hgh, the number of tests should be ncreased to obtan suffcent number of peaks of the roll angle. Recordng of the roll tme hstory should start before the release of the model to confrm that no angular velocty s gven when releasng. Recordng should contnue untl the model has reached rollng angles smaller than.. As the consdered hull forms have very hgh dampng, for each speed at least decays were performed. Fgure 7. Perspectve lne drawng of Hard Chne model (only n ths pcture secton numbers from bow). Fgure 9. Expermental set up Copyrght by ASME

6 RESULTS Each decay was analyzed accordng to the procedure descrbed and for each speed, all decays are merged and dagrams are plotted. One complete procedure for hard chne at zero speed s gven for large angle test. recorded roll decays 6 (φ -φ + )/(t + -t ) lnear decay for HC at zero speed y.x +.9x R².97 y -.6x +.7x +.7x R².9 Roll [deg] Roll [deg] Tme [s] recorded roll decays Tme [s] Fgure. Completed and analysed roll tme hstory φ MEAN (deg) 6 6 Fgure. Determnaton of dampng coeffcents lnear decrement From the trend lne equatons, the coeffcents, β and γ are determned for three cases. To have an dea about the precson of dfferent methods, tme seres of damped oscllatons s smulated by Runge Kutta method. In Fgures -7 examples are gven for zero speed and for model speed of m/s. Roll seres Vs Tme log ln ln expermental Mergng all the performed tests, dampng coeffcent are obtaned from dagrams gven at Fg. and. Ampltude (deg).9..7 ln(φ /φ + )/(t + -t ) logarthmc_decay for HC at zero speed tme Fgure. Hard Chne hull at zero speed φ MEAN (deg). 6 6 Fgure. Determnaton of dampng coeffcents logarthmc decrement Ampltude (deg) - - Roll seres Vs Tme tme Fgure. Round blge hull at zero speed log ln ln expermental 6 Copyrght by ASME

7 Roll seres Vs Tme log ln ln expermental focused on warped hull forms n semdsplacement regme as they are most nterestng for the prevously reported desgn trends. Ampltude (deg) - Table. Summary results for hard chne hull, model scale LOGARITHMIC DECREMENT HC model tme Fgure. Hard chne hull at v m/s Ampltude (deg) Roll seres Vs Tme log ln ln expermental V β EQ ω Τ LINEAR MODEL HC model tme Fgure 6. Round blge hull at v m/s It can be observed the lnear model wth cubc member ddn t brng any mprovement, furthermore n all cases t gves the worst predcton. Therefore the fnal results reported n Tables and gvng the coeffcents, β and EQ for logarthmc and lnear analyss wth lnear and quadratc dampng. All the values are gven for model values. Fnally to report the obtaned values to the shp the followng relatons are vald: T λ T SHIP MODEL ω ω MODEL SHIP SHIP ω λ MODEL MODEL ω SHIP MODEL These results are gven n Dagrams as a functon of shp speed. It can be noted that hard chne hull has slghtly hgher roll perod up to speed of knots. The dfference n perods s about.-. seconds, or more n general s about -7% n both cases. At the full scale speed of 7 kn, natural perod s dentcal for both hulls. Ths result can be explaned through two observatons: the frst s that the consdered speed corresponds to F n. where some hydrodynamc lft can be expected and therefore the vscous effects due to the varaton n hull-forms does not nfluence the roll perod but dampng only. Secondly, tryng to connect ths results wth Ikeda s work, at speeds hgher than F n., the eddy makng dampng has been transformed nto dampng due to the lft. It s mpossble to compare results from Ikeda s method and those presented n ths work. That s because the relatve speed consdered are dfferent as well as hull forms. Ths paper s λ V β EQ Table. Summary results for round blge hull, model scale LOGARITHMIC DECREMENT RB model V β EQ ω Τ LINEAR MODEL RB model V β EQ Copyrght by ASME

8 (/s) β HC_lnear dampng HC_quadratc dampng.6 RB_lnear dampng ω (rad/s) HC_natural frequency RB_natural frequency. v (kn). 6 6 Fgure 7. Natural roll frequences as functon of shp speed T (s) HC_roll perod RB_roll perod. v (kn). 6 6 Fgure. Natural roll perods as functon of shp speed EQ (/s) RB_equvalent dampng HC-equvalent dampng v (kn). 6 6 Fgure 9. Equvalent roll dampng coeffcent EQ.. RB_quadratc dampng. v (kn). 6 6 Fgure. Lnear and quadratc dampng coeffcents In. Fg. equvalent roll dampng EQ s gven and Fg. 9, lnear and quadratc dampng parts and β coeffcents are gven as functon of shp speed. It can be seen the beneft of hard chne n quadratc dampng coeffcent β n all speed range and especally up to v kn (Fn.). Equvalent lnear dampng coeffcent EQ s lower for hard chne up to the shp speed of approxmately knots (F n.), then t becomes sgnfcantly hgher. Ths was n some way unexpected result; we expected to observe n all speed range the superor dampng of hard chne. A possble explanaton for ths s n the determnaton of dampng coeffcents methodology. As t can be seen from dagrams and of Runge Kutta the dampng obtaned does not reproduce the measured values at zero speed, especally for hard chne hull. None of the models: logarthmc decrement: lnear + quadratc; lnear decrement: lnear + quadratc, lnear decrement: lnear + quadratc + cubc was able to descrbe the effects of vortces created by hard chne. 6 COMMENTS AND CONCLUSIONS Ths work s focused on determnaton of roll dampng of warped hull form at low relatve speed for whch lttle data are avalable. The effect of hard chne and round blge on roll dampng has been nvestgated for two models of same characterstcs and same nertal propertes n dsplacement and sem-dsplacement regme. It was expected to experence the dfference between two forms n nonlnear roll dampng only, whch s ampltude dependent and therefore roll decay tests were performed for angles from to degrees at zero speed up to F n.. Roll decays have been analyzed consderng logarthmc and lnear decrements. Obtaned results are natural roll frequences and perods, equvalent lnear dampng EQ and lnear and quadratc β dampng coeffcents as functons of shp speed. Obtaned dfferences between two models are about 7 % n term of roll perod up to the F n ~.. Then the dfference n perod becomes almost zero, ndcatng the predomnant effect of hydrodynamc forces over vscous effects due to the hull form. The results confrm hgh dampng propertes of warped hull form n dsplacement and semdsplacement regme. The fnal choce between round blge or hard chne wll probably depend Copyrght by ASME

9 on dfferent factors such as shp purpose, servce speed, shpyard technology, etc. At very low relatve speed both forms have smlar dampng behavor. The supposed better dampng due to the hard chne seems not apprecable. At Fn greater than.7 hard chne has shown clearly superor behavor. It has to be ponted out that ths fact s due to the flow separaton at chne and developng of hydrodynamc lft and not due to the vortces generated by hull form. The dampng coeffcents assessment represent a useful tool to predct the contrbute of stablzng devces. Ths s especally true for recently used gyroscopc stablzers whose dampng performance s drectly added to ntrnsc hull form dampng. NOMENCLATURE A added moment of nerta (kgm ) B maxmum breadth (m) B WL breadth at water lne (m) B total dampng coeffcent (kgm /s) C roll restorng coeffcent (Nm) C V - Froude number based on shp beam, F n Froude number based on shp length, C V F n V g B V g L GM metacentrc heght (m) I roll moment of nerta (kgm ) k roll radus of nerta (m) k ptch radus of nerta (m) L OA length over all (m) L WL length on water lne (m) LCG longtudnal center of gravty, measured from transom stern (m) T draft (m) T natural roll perod (s) v shp speed (kn) VCG vertcal center of gravty, measured from BL (m) Δ dsplacement (kg) λ - shp model scale - lnear dampng coeffcent (/s) EQ - equvalent lnear dampng coeffcent ω - natural roll frequency (rad/s) β - quadratc dampng coeffcent (/rad) γ - cubc dampng coeffcent (/rad ) ACKNOWLEDGMENTS Ths work has been fnancally supported by Unversty of Naples Federco II wthn the frame of - research program. Assessment of Intact Stablty Weather Crteron On Expermental Bass, Project HYD-III-CEH-, Rev..- Fnal-, November 9, [] Chakrabart S.K., (99), Offshore Structure Modellng, Advances Seres n Ocean Engneerng, Vol. 9, Feb. 99, Worls Scentfc Pubblshng Co [] Gallarde G., Toxopeus S., Verwoest T., Hoojmans P. (), Hydrodynamcs of Large Motor Yachts: Past Experence and Future Developments, [] Ikeda Y. (), Predcton Methods of Roll Dampng of Shps and Ther Applcaton to Determne Optmum Stablzaton Devces, Marne Technology, Vol., No., pp.9-9 [] Ikeda Y., Katayama T. (), Roll Dampng Predcton Method for a Hgh Speed Planng Craft, Proceedngs od the 7th Internatonal Conference on Stablty of Shp and Ocean Vehcles, STAB, Vol. B, pp.- [6] Ikeda Y., Umeda N., (99), A Predcton Method of Roll Dampng a Hard-Chne Boat at Zero Forward Speed, Jour. of the Kansa Soc. of Naval Arch. Japan, No., 99, pp.7 6. [7] Ikeda Y., Hmeno Y., Tanaka N. (97a), Components of Roll Dampng of Shp at Forward Speed, Journal of Socety of Naval Archtecture Japan, Vol., pp. - [] Ikeda Y., Hmeno Y., Tanaka N. (97b), A Predcton Method for Shp Roll Dampng, Report of Department of Naval Archtecture, Unversty of Osaka, Japan, No. [9] Lewandovsky E.M. (), The Dynamcs of Marne Craft: Maneuverng and Seakeepng, vol., Copyrght by World Scentfc Publshng Co. Pte. Ltd. [] Savtsky D. (96), Hydrodynamc Desgn of Planng Hulls, Marne Tecnology, Vol., No., 96, pp.7-9 [] Tanaka N., Ikeda Y., (9), Study on Roll Characterstcs of Small Fshng Vessels (part ) effect of skeg and hardchne on roll dampng, Journal of Kansa Socety of Naval Archtects, 96, -7 (n Japanese) [] P. Ward Brown and Walter E. Klosnsk. (99), Expermental Determnaton of the Added Inerta and Dampng of Planng Boats n Roll, Report No. CG-D-- 9, Davdson Laboratory, Stevens Insttute of Technology, 99 [] P. Ward Brown and Walter E. Klosnsk. (99), Expermental Determnaton of the Added Inerta and Dampng of a Degree Deadrse Planng Boat n Roll, Report No. CG-D--9, Davdson Laboratory, Stevens Insttute of Technology, 99 [] 6th ITTC Specalst Commttee on Stablty n Waves () - ITTC Recommended Procedures REFERENCES [] Bulan G., Francescutto A., Fucle F. (9), Determnaton of Relevant Parameters for the Alternatve 9 Copyrght by ASME

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