TACKING SIMULATION OF SAILING YACHTS WITH NEW MODEL OF AERODYNAMIC FORCE VARIATION DURING TACKING MANEUVER

Transcription

1 Journal of ailoat Technolog, rticle -., The ociet of Naval rchitects and Marine Engineers. TCKING IMULTION OF ILING CHT WITH NEW MODEL OF ERODNMIC FORCE RITION DURING TCKING MNEUER utaka Masuama Department of Mechanical Engineering, Kanaawa Institute of Technolog, JPN Toichi Fukasawa Department of Marine stem Engineering, Osaka Prefecture Universit, JPN Manuscript received pril 8, ; revision received Jul 3, ; accepted Octoer 8,. stract: mathematical model for the tacking maneuver of a sailing acht is presented as an extension of research the same authors. The authors have proposed the equations of motion for the tacking maneuver expressed in the horiontal od axis sstem. The calculation method was applied to a 34-foot sailing cruiser and the simulated result showed good agreement with the measured data from full-scale tests; however, the modeling of aerodnamic force variation during tacking was insufficient due to lack of information aout the sail forces. In this report, the authors performed full-scale measurement of sail forces during tacking maneuvers using a sail dnamometer oat Fujin. The Fujin is a 34-foot sailing cruiser which has a measurement sstem to otain simultaneousl sail forces, sail shapes, and oat attitude. Based on the results of full-scale measurements, a new model of aerodnamic force variation for the tacking maneuver was proposed. The equations of motion were also simplified to more easil perform the numerical simulation. Using this calculation method, the tacking simulations were performed and compared with the measured data from three full-scale oats. The simulated results showed good agreement with the measured data. This simulation method provides an effective means for assessment of tacking performance of general sailing achts. Kewords: aerodnamics, hdrodnamics, maneuvering, motions, performance assessment, performance prediction, sails. NOMENCLTURE B readth at design waterline D design draft including fin keel F n Froude numer GM metacentric height of oat I,, moments of inertia of oat aout x -, - and -axis in general od axis sstem J,, added moments of inertia aout x -, - and -axis in general od axis sstem K, N moments aout x- and -axis in horiontal od axis sstem L length on design waterline l R distance etween quarter-chord point of rudder and C.G. of oat m mass of oat m x,, added masses of oat along x -, - and -axis in general od axis sstem sail area actual total sail area Copright NME

2 Journal of ailoat Technolog, rticle -. U, velocit components of oat along x- and -axis in horiontal od axis sstem U apparent wind speed W B oat velocit x CG x-coordinate of C.G. of oat x GCE x-coordinate of geometric centre of effort of sail X, force components along x- and -axis in horiontal od axis sstem GCE -coordinate of geometric centre of effort of sail α R effective attack angle of rudder β leewa angle γ apparent wind angle W γ R decreasing ratio of inflow angle for rudder δ rudder angle ρ densit of water ρ a densit of air φ heel angle or roll angle heading angle Coordinate sstem G x,, coordinate of horiontal od axis sstem G x,, coordinate of general od axis sstem o d x d, d, d coordinate of sail dnamometer axis sstem INTRODUCTION Tacking of a sailing acht is a quick maneuvering motion accompanied large rolling angle changes in a short period of time. To anale this tpe of large amplitude motion, a mathematical model for the simulation was presented the same authors Masuama et. al. 993; 995. In these previous reports the authors emploed equations of motion expressed the horiontal od axis sstem introduced Hamamoto et. al. 988; 99. In this coordinate sstem, the maneuvering motion of the oat and aero/hdrodnamic forces acting on it can e expressed easil. Both added mass and added moment of inertia, which are referenced to the od axes fixed on the oat, can e otained using the coordinate transformation. The calculation method was applied to a 34-foot sailing cruiser and a -foot Japanese traditional tall ship Naniwa-maru Masuama et. al. 3. The simulated results showed good agreement with the measured data otained during full-scale tests. Keuning et al. 5 extended our model to e ale to appl the calculation method to a large variet of achts using their extensive dataase of Delft stematic acht Hull eries DH. In this previous research, however, the modelling of aerodnamic force variation during tacking was insufficient due to a lack of information aout the sail forces. To clarif the sail force variation during tacking, the authors 8 previousl measured the forces using a sail dnamometer oat Fujin and proposed a new model of aerodnamic force variation. In this paper, which is an extension of the previous report, the equations of motion are simplified to more easil execute the tacking simulation, and the numerical simulations are performed and compared with the measured data for three fullscale oats, Fujin, Fair, and ea Dragon. Copright NME

3 Journal of ailoat Technolog, rticle -. DIRECT MEUREMENT OF IL FORCE UING FUJIN ail Dnamometer Boat Fujin The sail performance was measured using the sail dnamometer oat Fujin. The Fujin was originall uilt for conducting tests on sails the authors for the Japanese merica s Cup entr in 994. The design of the Fujin is ased on the R-.3m class, which is an International Measurement stem IM ocean racer designed amaha Motor Co. Ltd. In this oat, load cells and CCD cameras were installed to simultaneousl measure the sail forces and shapes. t the same time, the sailing conditions of the oat e.g., oat speed, heel angle, wind speed, and wind angle are measured. Figure shows the sail plan and general arrangement of the dnamometer frame of the Fujin as well as the coordinate sstem of the sail dnamometer. It should e noted that the origin of the sail dnamometer sstem is not on the Center of Gravit C.G. of the oat ut located at the aft face of the mast x d -direction and the height of the deck level d -direction. The measurement sstem installed in the Fujin and the relationships etween sail shapes and performance were reported Masuama and Fukasawa 997a; 997. Recentl, the measured sail shapes were reanaled and taulated for use as input data for the numerical calculation. This sail shape dataase, in association with the results of latest RN-ased CFD, was also reported Masuama et al. 7; 9. tead ail Performance for Upwind Condition The aerodnamic coefficients and the coordinates of the center of effort of the sails for the close-hauled condition are defined as: X x d CE X ρau N d,, d d d d ρau K d CE, d, where X d and d are the force components along the x d and d axes of the sail dnamometer sstem respectivel, and K d and N d are the moments around the x d and d axes. x CE and CE are the x d and d coordinates of the center of effort of the sails CE. It should e noted that the x d and d axes are fixed on the oat and therefore inclined with pitch and heel angles. This means the measured side force d is not in the horiontal plane ut is normal to the mast. Figure shows the measured aerodnamic coefficients and the coordinates of CE as a function of apparent wind angle γ for the mainsail and 3% ji configuration shown in Figure. The aerodnamic forces acting on the mast and rigging are included in the measured X d and d forces. The solid smols indicate the results of staroard tack and the open smols indicate the port tack. In Figure a the side force coefficient ' d is expressed as positive for oth port and staroard tacks. The measurements var widel, especiall in the ' d data, ecause the are ased on measurements taken with the sails trimmed in different was. Copright NME 3

4 Journal of ailoat Technolog, rticle -. Figure. ail plan and arrangement of sail dnamometer frame of Fujin. Copright NME 4

5 Journal of ailoat Technolog, rticle -. Figure. ail performance variation for mainsail and 3% ji configuration as a function of apparent wind angle γ. The apparent wind speed U and apparent wind angle γ were measured an anemometer attached to the ow unit. The ow unit, which was attached to the ow pulpit, includes a post that can rotate freel to maintain its vertical attitude when the oat heels in order to measure the wind data in the horiontal plane. ince the measured side force d acts normal to the mast, the values of d do not var with heel angle like the horiontal plane component, ut are affected decreases in oth the effective attack angle and the dnamic pressure on the sails as shown in ppendix. This decreasing ratio is approximated a function of cos φ. However, the heel angles during measurement were less than degrees, and therefore the difference with the upright condition was less than 6%. For this reason, the measured data in Figure are indicated without heel angle correction. Under this assumption, the asic sail force curves used in the tacking simulation in the upright condition, X' and ', are shown as solid and dotted curves, which refer to the present results and as well as other wind tunnel test results Masuama and Tatano 98. Copright NME 5

6 Journal of ailoat Technolog, rticle -. Figure shows the variation of the CE coordinates of the sails. The x d - and d - coordinates of the geometric center of effort x GCE and GCE are.63 m aft and 4.8 m aove the origin of sail dnamometer sstem, which are indicated dashed lines in the figure. The geometric center of effort is calculated as the center of actual sail area of the mainsail and ji mast area is not included. The measured coordinates of x CE are near x GCE and move forward with increasing pparent Wind ngle W toward the origin. However, the coordinate of x CE coincides with the x GCE for the closest angle to the wind. In the experiments with the Fujin using other sail configurations, such as the mainsail with a 75% ji and mainsail alone, the coordinates of x CE also coincided with the x GCE of each sail plan in the upwind condition. Therefore, for the tacking simulation, the coordinate of x GCE is used as the point of application of d force for various sail configurations. There is a wide scatter in the experimental values for CE. This is thought to e ecause the measured heel moment contains a large component from the mass of the dnamometer frame and rigging 659kg. This moment should e sutracted from the measurement, taking into account the measured heel angle. If there is a slight error in the position of the center of gravit of the dnamometer frame, or in the measured heel angle, the error in the calculated moment will e large. ome ias in the values of CE etween port and staroard tacks might e caused the slight discrepanc in alignment etween the x d axis of the dnamometer frame and the center line of the hull. However, although the measured data are scattered, the coordinates of CE coincides with GCE for the tacking simulation. Using these results, the heel moment K d and aw moment N d for the tacking simulation are otained multipling the d force the coordinates of G GCE and x G GCE, respectivel, where the superscript of G means the coordinate from the center of gravit of the oat. ail Force ariation During Tacking Figure 3 shows two examples of the measured data in the time domain for X' d, ' d, K' d and N' d during tacking for seconds, eginning five seconds efore the initiation of tacking and ending 5 seconds after. Figure 3a shows tacking from staroard to port tack, and Figure 3 shows tacking from port to staroard tack. The scatter in the data at the crossing points of the curves is caused the crew action on the dnamometer frame in releasing and trimming the ji sheet. In the measured data, the inertia forces and moments due to the mass of the dnamometer frame are included. These effects appear clearl at the eginning and end of the tacking maneuver ut are not as significant during the middle stage. Hence the measured data are modified onl sutracting the forces and moments due to the gravitational force acting on the dnamometer frame ased on the measured heel angle at ever moment. The evaluation of the effect of the inertia forces and moments on the measured data is discussed in ppendix. Figure 4 shows the variation of sail force coefficients during tacking as a function of the heading angle of the oat, where º means heading in the true wind direction. During tacking, the ji sheet was released just efore the ji was ackwinded on the new tack in order to minimie luffing of the ji and loss of wind power. The curves show the results of tacking cases from staroard to port tack. It should e noted again that forces and moments are shown using the sail dnamometer coordinate sstem. The variations start from the close-hauled condition of staroard tack until the oat is on port tack i.e., from -45º to 45º. The corresponding W, from γ 3º to -3º, are also indicated in the second ascissa in the figure. Figure 4a shows the variation of X' d. When the oat Copright NME 6

7 Journal of ailoat Technolog, rticle -. heads directl into the wind, X' d ecomes aout -., i.e., drag force coefficient. Figures 4 to 4d show the forces and moments ecome ero not at º, ut around º, which indicates a dela in the variation of forces and moments compared to the change of heading angle. This could e caused the sail filling with wind due to the awing motion from the former tack to º on the new tack when the ji sheet was released. Figure 3. Examples of measured sail force coefficients in the time domain during tacking. Figure 4. ariation of sail force coefficients during tacking as a function of heading angle of oat tacking from staroard to port tack. Copright NME 7

8 Journal of ailoat Technolog, rticle -. Figure 5 shows the same variation for the case of port tack to staroard tack. In this case, ' d, K' d and N' d ecome ero at around -º, and the variation of forces and moments are almost smmetrical to Figure 4. Therefore, the ias in the ero crossing point of the forces and moments in the tacking maneuver is smmetrical. Figure 5. ariation of sail force coefficients during tacking as a function of heading angle of oat tacking from port to staroard tack. Model of ail Force ariation for Tacking imulation The equations of motion for tacking simulation are expressed in the horiontal od axis sstem, which is discussed in the next section. In this axis sstem, the sail forces used in the calculation are expressed in horiontal components, and the moments are expressed around the C.G. of the oat, rather than the origin of the sail dnamometer sstem. These components are defined as X,, K and N as: Copright NME 8

9 Journal of ailoat Technolog, rticle -. X X ' cos ρau ' cos ρau G GCE K ' cos ρau N ' x G GCE X ' G GCE 3 sin cos ρau 3 The derivation of these formulas, including effect of heel angle, φ, are descried in ppendix. The asic sail performance curves of X's and 's in Figure show the stead state values and do not express the dnamic variation due to tacking. Therefore, the model of sail force variation for the tacking simulation is defined as old lines in Figure 6, referring to the measured data in Figures 4 and 5. Figure 6a shows the case of tacking from staroard to port tack. The ascissa indicates apparent wind angle γ. In the model, the asic sail performance curves of X's and 's are divided into three stages. tage is the range of γ that is greater than º. In this region, the coefficients var with γ according to the asic curves. tage B is the range of γ º to -º. In this region, the coefficients are assumed to var linearl along the lines determined from the results of Figures 4a and 4. tage C is the range of γ -º to -3º. In this region, the asic pattern of the coefficients is expressed as asic performance curves. However, it ma take several seconds to recover to the asic curves due to the dela in trimming the sails for the new tack. Therefore, the coefficients are assumed to increase from the lowest values to the asic curve values with time. The recover time was chosen from 5 to seconds taking the simulated heel angle corresponding to the measured heel angle. Figure 6 shows the case of tacking from port to staroard tack, where the variation pattern proceeds in the opposite direction. In the numerical calculation, the rolling effects on the sail were considered as a variation of apparent wind angle, γ, and speed, U, due to the motion at the center of effort of the sail. The sail forces and moments expressed in equation are used for the equations of motion in the following section. Copright NME 9

10 Journal of ailoat Technolog, rticle -. X','. taroard tack X' B.5 C..5 Port t ack ' Increasing with elapsed time X' γ [deg] ' a tacking from staroard to port tack taroard tack C X','..5 B Port tack ' X'..5 X' γ Increasing with [deg] -.5 elapsed time -. ' tacking from port to staroard tack Figure 6. Model of sail force variation for tacking simulation. Copright NME

11 Journal of ailoat Technolog, rticle -. EQUTION OF MOTION FOR TCKING IMULTION To express the large amplitude motion such as a tacking maneuver of a sailing acht, we used motion equations expressed in the horiontal od axis sstem introduced Hamamoto et al. 988; 99. It should e noted that this coordinate sstem is different than the sstem for the sail dnamometer. The origin of the coordinate sstem is on the C.G. of the oat as shown in Figure 7. The x-axis lies along the centerline of the oat on the still-water plane and is positive forward. The -axis is positive to staroard in the still water plane, and the -axis is positive downwards. In this coordinate sstem, the maneuvering motion of the oat and aero/hdro-dnamic forces acting on it can e expressed in the horiontal plane even though the oat heels. Both added mass and added moment of inertia, which are referenced to the od axes fixed to the oat, can e otained a coordinate transformation. The authors previousl presented the equations of motion expressed in four simultaneous differential equations ut excluded pitching and heaving motions Masuama et al. 993; 995. The distance etween center of gravit and center of added mass of the oat, x G, was taken into account. For a sailing acht with a fin keel, the value of x G is small ecause the fin keel has a large mass and lateral projected area, which affect oth the centers of gravit and added mass. Therefore, to create an easier simulation procedure, it is assumed that the center of added mass of the hull coincides with the center of gravit of the oat i.e., x G. In this case, the forces and moments expressed in the general od axis sstem are formulated from the Euler-Lam equations of motion Lam, 93. These equations are transformed into those expressed in the horiontal od axis sstem, assuming the pitch angle, θ, is equal to ero, as shown in the ppendix 3. The equations of motion expressed in the horiontal od axis sstem for the motions of surge, swa, roll and aw are derived as follows. The left sides are formulas -3 in ppendix 3 and the right sides are fluid dnamic forces acting on the hull and sail with reference to the horiontal od axes. Figure 7. Definition of coordinate sstem and forces and moments, -ve is indicated direction. Copright NME

12 Journal of ailoat Technolog, rticle -. Copright NME surge: R H x X X X X X m m m U m m sin cos 3 swa: R H x m m U m m m m m cos sin sin cos 4 roll: { } sin cos sin mggm K K K K J I J I J I R H 5 aw: { } { } R H N N N N J I J I J I J I cos sin cos sin 6 These equations are the same as equations through 4 in reference of Masuama et al. 995, except the term x G was eliminated. In equation 3, X is the hull resistance in the upright condition, which is calculated from model tests or the DH dataase Keuning and onnenerg 999. The previous traditional expressions Masuama et al., 995 are adopted to descrie the hdrodnamic forces acting on the hull. The stead forces on the canoe od and fin keel are descried using hdrodnamic derivatives as follows: D L N N N N N N D L K K K K K K D L D L X X X X B H B H B H B H ρ ρ ρ ρ 7 where, ' is defined as: β β sin sin B B

13 Journal of ailoat Technolog, rticle -. Equations 7, in which higher order terms are eliminated, are a simplification of equations 5 in reference of Masuama et al The hdrodnamic forces and moments on the rudder are expressed as: X R K N R R R C C C C Xδ δ Kδ Nδ sinα R sinδ ρ B L D sinα R cosδ cos ρ B L D sinα R cosδ ρ B L D sinα R cosδ cos ρ B L D 8 where C Xδ to C Nδ are coefficients determined rudder angle tests. The effective attack angle of the rudder, α R, is given α R δ γ R β l R tan 9 U where γ R is the decreasing ratio of inflow angle, which is caused mainl the downwash from the fin keel. The third term indicates the inflow angle due to the turning motion of the oat, where the l R is the horiontal distance etween the quarter-chord of rudder and the C.G. of the oat. For tacking maneuvers, the turning radius is relativel small and hence the inflow angle at the rudder position ecomes greater than 3 degrees. This means that the rudder is outside of the downwash of the fin keel and that a decrease in inflow angle might not occur. Therefore γ R is not multiplied here the third term of equation 9. HDRODNMIC FORCE CTING ON HULL Principal Dimensions of Boats for Tacking imulation Tacking simulation was performed for three oats, Fujin, Fair and ea Dragon. The Fair is a.4 m LO sailing cruiser designed Masuama. The principal dimensions of the Fair and a comparison of its tacking performance etween measured and simulated tests have een previousl descried Masuama et al. 993; 995. However, the sail plan and shape of the fin keel were modified after the previous experiments. Therefore, the measurement of tacking maneuvring was repeated the authors. The ea Dragon is a 6.4 m LO small sailing cruiser uilt in Japan. Using this oat, the authors also conducted man rolling and tacking tests Masuama et al. 8a. Figure 8 shows the sail plan and principal dimensions of Fair and ea Dragon. Copright NME 3

14 Journal of ailoat Technolog, rticle -. a Fair ea Dragon Figure 8. ail plans of Fair and ea Dragon. Hdrodnamic Derivatives and Coefficients The hdrodnamic derivatives of the hull and rudder were otained from model tests. In order to appl the derivatives for Equations 7 and 8, olique towing tests and rudder angle tests were performed. The analsis procedure from the measured data to the hdrodnamic derivatives is descried in a previous paper Masuama et al The results of the rudder angle tests and olique towing tests for the Fujin are shown in Figures 9 and, respectivel. Because of the restrictions of the experimental facilit, heel moment, K, was not measured. Hence, the K moment around the LWL level was evaluated multipling the force 43% of the fin keel draft according to Keuning et al. 3. Tale shows the hdrodnamic derivatives of the hull and coefficients of the rudder that were otained. The calculated results from equations 7 and 8 using these hdrodnamic derivatives and coefficients are indicated as curves. It should e noted that the N moment is expressed around midships. The decreasing ratio of inflow angle, γ R, was also measured rudder angle tests under olique towing conditions for the Fujin, modified Fair, and ea Dragon. The results showed agreement with the data of the original Fair Masuama et al Copright NME 4

15 Journal of ailoat Technolog, rticle -. XR'. R'.5-4 δ 4 [deg] δ -4 4 [deg] NR'.4-4 δ 4 [deg] β β 5 β β Figure 9. ariation of hdrodnamic coefficients of rudder with rudder angle δ for Fujin. XH'.4 H'. - β [deg] β - [deg] NH'.4 - β [deg] φ φ - φ - φ Figure. ariation of hdrodnamic coefficients of hull with leewa angle β for Fujin without rudder forces. Copright NME 5

16 Journal of ailoat Technolog, rticle -. Hdrodnamic derivatives of the hull due to awing motion, such as X,, and N calculated using the following equations from Masuama et al. 995:, are X m Cm ρl D where C m is assumed to e.3 Hasegawa 98. π d m ρbl D a h h ρb F c 4 L h h d 3 m d m N.54 ρbl D a ρb F c L L where, h h h w In equations and, the first term is the contriution from the canoe od, which is expressed an empirical formula Inoue et al. 98. The second term is the contriution from the fin keel, which is calculated lifting surface theor Etkin 97. Here, F, a, and c are the lateral projected area, lift curve slope, and mean chord length of the fin keel, respectivel. The terms h, h, and h w are the distances from the C.G. of the oat to the leading edge of the fin keel, the axis of rotation for vanishing lift force, and the aerodnamic center of the fin keel, expressed the percentage of the mean chord length, c. The contriution from the rudder is alread considered in the third term of equation 9. The derivative due to rolling,, was calculated considering the change in attack angle of oth the fin keel and the rudder caused the rolling angular velocit as: D ρ Ba c d 3 where c is chord length of the fin keel or rudder at coordinate. The contriution of the canoe od was neglected. The damping coefficient for rolling, K, was otained a rolling test of the full-scale oat. logarithmic decrement, σ, and coefficient, α, are defined as: Copright NME 6

17 Journal of ailoat Technolog, rticle -. t, α t T T σ σ ln 4 where φ is roll angle and T is roll period. From the rolling test of Fair and ea Dragon with mainsail, the value of α was evaluated as.4 and.53, respectivel Masuama et al. 995 and 8a. The damping effect of the sails was ver large, so these tests were also performed under running conditions to clarif the effect of velocit without sails. In this case the value of α increased linearl with the oat velocit. From these results, the value of α for these oats is formulated using Froude numer as: α α.4f n 5 Using the coefficient α, the damping coefficient for rolling, K, is given K α I J 6 where the value of I J are also otained from the rolling test using following relation: T I J mg GM 7 π The hdrodnamic derivatives and coefficients of these oats are shown in Tale. Copright NME 7

18 Journal of ailoat Technolog, rticle -. Tale. Principal dimensions, hdrodnamic derivatives, added masses and added moments of inertia of Fujin, Fair and ea Dragon. Boat Name Fujin Fair ea Dragon Class R.3 KIT 34 LO [m] LWL L [m] BMX [m] BWL B [m] Draft Canoe od [m] Draft Fin keel D [m] Displacement* [kg] GM [m] x CG from Midship [m] l R from C.G. [m] ail areamainsail [m ] ail areaji [m ] x G GCE from C.G. [m] G GCE from C.G. [m] m x [kg] M Hull, ail [kg] 3, 8 4, 8 74, m [kg] 4 36 I [kg m ] I [kg m ] I [kg m ] J Hull, ail [kg m ] 7, 8 86, 8 9, J [kg m ] J [kg m ] X X φφ X X φ φφ φ φ K K φ K φφ K φ K K φ N N φ N φφ N φ N N C Xδ C δ C Kδ C Nδ * Including crew weight Copright NME 8

19 Journal of ailoat Technolog, rticle -. dded Masses and dded Moment of Inertias dded mass of the hull along x-axis, m x, was assumed to e the same as the value of a spheroid Newman 977. ince the frequenc of swaing and awing motion is low, m and J are calculated with the doule model expressed the Lewis form coefficient, C and C 3, as: m π ρ L D C x dx 8 J π ρ L x D C x dx 9 where, C 3C C C C 3 x 3 alues of m and J are also calculated as: m π 8 ρ L B C x dx J π 8 ρ L x B C x dx where, C 3C C C C 3 x 3 The value of J, including mainsail, was otained from the rolling test with mainsail using Equation 7. The added masses of the fin keel, rudder and sail along the -axis were calculated assuming the were ellipsoid planes with the same lateral areas. The added moment of inertias of the fin keel and rudder around the -axis were evaluated as: Copright NME 9

20 Journal of ailoat Technolog, rticle -. J k, r m k, r x C. G. k, r x, where C. G. k r means the distance from the C.G. of the oat to the center of lateral area of the fin keel or rudder. dded masses and moments of inertias of the fin keel and rudder were included in the values of m and J of the hull. The values of these added masses and moments are also shown in Tale with inertia forces and moments of these oats. COMPRION BETWEEN MEURED ND IMULTED REULT The Runge-Kutta method was emploed to calculate the equations of motion. The rolling and awing motions were calculated around the C.G. of the oat. Input data for the simulation consisted of the true wind velocit and the measured time histor of the rudder angle during the tacking maneuver at increments of. seconds. Results of Fujin Figure shows the comparison etween measured and simulated results of Fujin. Figure shows tacking from staroard to port tack, and shows tacking from port to staroard tack. The sail force variations in Figures 3a and 3 correspond to these cases, respectivel. The indicated results were recorded for 35 seconds, eginning 5 seconds efore the start of tacking. Figure a shows the oat trajectories. olid circles indicate the positions of measured C.G. of the oat at each second, while open circles indicate the simulated positions. The illustrations of the small oat smol indicate the heading angle ever three seconds. The wind lows from the right side of the figure and the grid spacing is taken as 5 meters. Figure shows the time histories of rudder angle δ, heading angle, heel angle φ, and oat velocit B. The solid lines are measured data and the dotted lines are simulated data. In Figures and, the patterns of rudder angle variation can e considered as standard for tacking maneuvers. s shown, tacking with a awing motion of 9 degrees is completed in 7 to 8 seconds. The oat velocit decreases aout 3%, and the oat takes aout 5 seconds to recover to the previous velocit after the awing motion is completed. The measured time histories of and φ indicate the dela of the ero crossing point of φ compared with. This might e caused the sail filling with wind due to the awing motion until around º on the opposite tack as shown in Figures 4 and 5. The simulated time histories show a slight dela when compared to the measured data. In particular, the dela of the simulated heel angle is relativel large. This might e caused the over-estimation of the damping coefficient for rolling, K φ. For this point further investigation might e necessar. However, the simulated results of velocit decrement show agreement with the measured results. This suggests that the model of sail force variation proposed in this report is adequate for the tacking simulation. In Figures a and a, although the simulated trajectories show slightl larger turning radiuses than the measured trajectories, the simulated results show agreement with the measured values overall. Results of Fair Figure shows the comparison etween measured and simulated results of Fair. The contents of these figures are identical to Figure. In these cases, the rudder angle Copright NME

21 Journal of ailoat Technolog, rticle -. variations in the first stage are relativel small. These cause a dela in the awing motion of the oat. Hence it takes more than seconds to complete the tacking maneuver. On the other hand, the simulated results show a prompt response to the rudder angle variation. Therefore the simulated time histories var slightl earlier compared with the measured histories. B the same reasoning, the simulated trajectories in Figures a and a show smaller turning radiuses than the measured trajectories. Results of ea Dragon Figure 3 shows the comparison etween measured and simulated results of ea Dragon. This oat has a relativel small fin keel and old, inefficient sails. The closest measured angle to the wind is therefore large and the tacking angle ecomes aout degrees. The simulated trajectories show agreement with the measured trajectories, ut there is a slightl smaller minimum angle to the wind than the measured values due to the etter sail coefficients used in the calculation. Unfortunatel, the measured and simulated time histories of the heel angle do not agree well ecause this oat is severel affected crew movements on the deck due to the light displacement. The patterns of rudder angle variation are almost identical to Figures and. However, the simulated time histories of var relativel earlier than the measured histories. This also might e caused the light displacement of the oat. Overall, although the timing of oat motion indicated in the simulated time histories shows a slight discrepanc, the tendenc and amount of variation of the oat motion indicate good agreement with the measured data, including the decrement of oat velocit. measured simulated measured simulated UT5.7m/s start of tacking UT5.4m/s start of tacking WIND WIND WIND WIND 5m 5m [deg] 7 δ φ start of tacking δ a Trajector of oat φ B elapsed time Boat attitude parameters [m/s] [sec] From staroard to port tack B [deg] 7 δ φ start of tacking φ δ B a Trajector of oat : Heading ngle φ : Heel ngle δ : Rudder ngle B: Boat elocit [m/s] elapsed time [sec] Boat attitude parameters From port to staroard tack. Figure. Measured and simulated results of tacking maneuver of Fujin. 4 3 B Copright NME

22 Journal of ailoat Technolog, rticle -. measured simulated measured simulated UT4.8m/s start of tacking UT4.9m/s start of tacking WIND WIND WIND WIND 5m 5m [deg] 7 δ φ start of tacking δ a Trajector of oat [m/s] elapsed time [sec] Boat attitude parameters From staroard to port tack. φ B 4 3 B [deg] 7 δ φ φ B start of tacking δ a Trajector of oat : Heading ngle φ : Heel ngle δ : Rudder ngle B: Boat elocit [m/s] elapsed time [sec] Boat attitude parameters From port to staroard tack. Figure. Measured and simulated results of tacking maneuver of Fair. 4 3 B measured simulated measured simulated UT5.6m/s start of tacking UT5.5m/s start of tacking WIND WIND WIND WIND 5m 5m [deg] 7 35 δ φ -35 start of tacking δ a Trajector of oat [m/s] elapsed time [sec] Boat attitude parameters φ From staroard to port tack. B 4 3 B [deg] 7 δ φ φ δ B start of tacking a Trajector of oat : Heading ngle φ : Heel ngle δ : Rudder ngle B: Boat elocit [m/s] elapsed time [sec] Boat attitude parameters From port to staroard tack. Figure 3. Measured and simulated results of tacking maneuver of ea Dragon. φ 4 3 B Copright NME

23 Journal of ailoat Technolog, rticle -. CONCLUION In this report, the sail force variations during tacking were measured using a sail dnamometer oat Fujin. Based on the results a new model of aerodnamic force variation for tacking maneuver was proposed. Then the equations of motion were simplified to more easil execute the numerical simulation eliminating the term x G. Using these equations of motion expressed in the horiontal od axis sstem, tacking simulations were performed and compared with the measured data for three full-scale oats Fujin, Fair, and ea Dragon. lthough the timing of oat motion indicated in the simulated time histories shows a slight discrepanc, the tendenc and amount of variation of the oat motion indicates good agreement with the measured data, particularl the simulated decrement of oat velocit. This shows that the proposed model of sail force variation is adequate for the tacking simulation. This simulation method provides an effective means for assessment of tacking performance of general sailing achts. cknowledgements The authors wish to thank amaha Motor Co. Ltd. for permitting the description of the principal dimensions and specifications of the Fujin; Mr. David. Helgerson for his comments on this article; Mr. H. Mitsui, the harormaster of the namiu Ba eminar House of Kanaawa Institute of Technolog, for his assistance with the sea trials; and the graduate and undergraduate students of the Kanaawa Institute of Technolog who helped with the sea trials, particularl H. rakawa and T. Onishi, who performed the sea trials of modified Fair and ea Dragon. This research is partl supported the Grant-in-id for cientific Research of the Ministr of Education, cience, ports and Culture in Japan. References Campell, IMC Optimisation of a sailing rig using wind tunnel data. Proceedings of the 3th Chesapeake ailing acht mposium, nnapolis, MD, Etkin, B. 97. Dnamics of tmospheric Flight. John Wile ons, New ork. Hamamoto, M. and kioshi, T tud on ship motions and capsiing in following seas st report. Journal of the ociet of Naval rchitects of Japan, 47, Hamamoto, M. 99. new coordinate sstem and the equations descriing maneuvering motion of a ship in waves. Proceedings of workshop on prediction of ship maneuverailit, West-Japan ociet of Naval rchitects, Fukuoka, Japan, Hasegawa, K. 98. On a performance criterion of autopilot navigation. Journal of the Kansai ociet of Naval rchitects,78, Inoue,., Hirano, M. and Kijima, K. 98. Hdrodnamic derivatives on ship maneuvering. International hipuilding Progress, 8 3, -5. Kerwin, J. E velocit prediction program for ocean racing achts revised to Feruar, 978. H. Irving Pratt Ocean Race Handicapping Project, MIT Report No Copright NME 3

24 Journal of ailoat Technolog, rticle -. Keuning, J.. and onnenerg, U. B pproximation of the calm water resistance on a sailing acht ased on the delft sstematic acht hull series. Proceedings of the 4th Chesapeake ailing acht mposium, nanpolis, MD, 8-. Keuning, J.. and ermeulen, K. J. 3. The aw alance of sailing achts upright and heeled. Proceedings of the 6th Chesapeake ailing acht mposium, nnapolis, MD, -7. Keuning, J.., ermeulen, K. J., and de Ridder, E. J. 5. generic mathematical model for the maneuvering and tacking of a sailing acht. Proceedings of the 7th Chesapeake ailing acht mposium, nnapolis, MD, Lam, H. 93. Hdrodnamics. Camridge Universit Press, Camridge, UK. Masuama,., and Tatano, H. 98. Hdrodnamic analsis on aailing part 4: wind tunnel experiments on acht sails. Journal of the Kansai ociet of Naval rchitects, 85, 7-5. Masuama,., Nakamura, I., Tatano, H., and Takagi, K "Dnamic performance of sailing cruiser full-scale sea tests." Proceedings of the th Chesapeake ailing acht mposium, nnapolis, MD, Masuama,., Fukasawa, T., and asagawa, H Tacking simulation of sailing achts numerical integration of equations of motion and application of neural network technique. Proceedings of the th Chesapeake ailing acht mposium, nnapolis, MD, 7-3. Masuama,., and Fukasawa, T. 997a. Full scale measurement of sail force and the validation of numerical calculation method. Proceedings of the 3th Chesapeake ailing acht mposium, nnapolis, MD, Masuama,., Fukasawa, T., and Kitasaki, T Investigations on sail forces full scale measurement and numerical calculation part : stead sailing performance. Journal of the ociet of Naval rchitects of Japan, 8, -3. Masuama,., Nomoto, K., and akurai,. 3. Numerical simulation of maneuvering of Naniwa-maru, a full-scale reconstruction of sailing trader of Japanese heritage. Proceedings of the 6th Chesapeake ailing acht mposium, nnapolis, MD, Masuama,., Tahara,., Fukasawa, T. and Maeda, N. 7. Dataase of sail shapes vs. sail performance and validation of numerical calculation for upwind condition. Proceedings of the 8th Chesapeake ailing acht mposium, nnapolis, MD, -3. Masuama,., Fukasawa, T., and Onishi, T. 8a. Dnamic stailit and possiilit of capsiing of small light sailing cruiser due to wind. Proceedings of the International Conference on Innovation in High Performance ailing achts, Lorent, France, Masuama,., and Fukasawa, T. 8. Tacking simulation of sailing achts with new model of aerodnamic force variation. Proceedings of the Third High Performance acht Design Conference, uckland, NZ, Masuama,., Tahara,., Fukasawa, T., and Maeda, N. 9. Dataase of sail shapes versus sail performance and validation of numerical calculation for the upwind condition. Journal of Marine cience and Technolog, 4, Newman, J. N Marine Hdrodnamics. MIT Press, Camridge, M. Copright NME 4

25 Journal of ailoat Technolog, rticle -. PPENDIX Cancellation of Inertia Forces and Moments due to Mass of the Dnamometer Frame Inertia forces and moments due to mass of the dnamometer frame are cancelled with the following procedure. Using the finite difference scheme, the angular velocit k and angular acceleration k are otained from the time histor of roll angle as: k k k - h k k k k - h where h is time step interval and φ k is roll angle at k-th step. In order to confirm this cancelling procedure, a rolling test of the Fujin was performed. Figure - a shows the time histories of the measured roll angle, and calculated angular velocit and angular acceleration formulas - and -. The measured time step interval is. second h. sec. Figure - shows the time histories of the d forces. The result without cancellations is shown in solid line, the result with cancellation of the gravit force alone is dotted line, and the result with cancellation of oth gravit force and inertia force is the alternating long and short dash line. Here, the inertia force inertia is calculated as: inertia [ N ] rdm M 3-3 DG where DG is the distance etween C.G. of the oat and C.G. of the dnamometer frame.98 m and M is mass of the frame 659 kg. Figure - c shows the same time histories of the K d moments. ince the K d moment is expressed around the origin of the sail dnamometer coordinate sstem, the inertia moment K inertia is calculated as: K inertia r dm [ N m] r r dm rdm R M 874 DG M -4 where is the distance etween C.G. of the oat and the origin of sail dnamometer coordinate sstem. m, and R is the radius of gration of the dnamometer frame around x d axis 3.96 m. The cancelled results of d force and K d moment show successful elimination of the contriution of the mass of the dnamometer frame due to oth the gravitational force and the inertial force on the measured data. Therefore, this cancelling procedure is considered to e adequate. Copright NME 5

26 Journal of ailoat Technolog, rticle -. φ, φ ', φ '' 3 - φ [deg] φ ' [deg/sec] φ '' [deg/sec ] elapsed time [sec] a Rolling angle, angular velocit and angular acceleration d Force [N] 3 - d without cancellation d with cancellation of gravit force d with cancellation of garvit force and inertia force elapsed time [sec] d Force Kd Moment [N-m] Kd without cancellation Kd with cancellation of gravit force moment Kd with cancellation of garvit force moment and inertia moment elapsed time [sec] c Kd Moment Figure -. Time histories of rolling test of Fujin showing cancelling procedure of inertia forces and moments due to mass of dnamometer frame. To determine the effect of inertial force on the data of a tacking maneuver, the same procedure was applied to the measured data in Figure 3. Figure - shows the results of cancellation on the d force and K d moment. The solid lines are the same as those in Figure 3, which show the cancelled effect of gravit, and the dotted lines represent the results of cancelled inertial forces. The cancelled results appear clearl at the starting and finishing stages of the tacking maneuver, ut are not as significant at the middle stage. Therefore the measured data are shown onl sutracting the forces and moments due to the gravit force acting on the dnamometer frame. Copright NME 6

27 Journal of ailoat Technolog, rticle -..5 start of tacking.5 start of tacking.. 'd 'd, K'sd K'd Canceled gravit force effect alone Canceled gravit -. force and inertia 'd force effect elapsed time [sec] a Tacking from staroard tack to port tack 'd, K'd.5. K'd -.5 Canceled gravit force effect alone Canceled gravit -. force and inertia force effect elapsed time [sec] Tacking from port tack to staroard tack Figure -. Results of cancellation of inertia force on d and inertia moment on K d from the measured time histories during tacking. PPENDIX ariation of ail Force Coefficient with Heel ngle The equations of motion for tacking simulation are expressed in the horiontal od axis sstem. The sail forces and moments should e expressed in their horiontal components. In the upright condition, the aerodnamic coefficients X' and ' are expressed using lift coefficient L' and drag coefficient D' as: X L L sinγ D cosγ D cosγ sinγ -5 where the suscript of means value at the upright condition. In the heeled condition, the effect of heel on the aerodnamic forces is produced the reduction of oth the apparent wind angle and apparent wind speed as given Kerwin 978 and Campell 997. The apparent wind angle in the heeled condition γ φ is expressed as follows using apparent wind angle γ and apparent wind speed U : γ tan U sin γ cos cos γ U tan tan γ cos -6 The apparent wind speed in the heeled condition U φ is also expressed as: U cosγ U sinγ cos U sinγ sin U -7 Copright NME 7

28 Journal of ailoat Technolog, rticle -. For the close-hauled condition, the sail ma not stall due to the small attack angle. Therefore, the lift force will decrease proportionall to the reduction of oth the apparent wind angle and the dnamic pressure of flow i.e., the square of the apparent wind speed. Hence the decreasing ratio of lift force the heel angle φ can e descried as: γ γ U U tan tanγ cos γ { sinγ sin } -8 The vector of lift force inclines with heel angle and rotates in the normal plane to the apparent wind axis. ince the angle etween the apparent wind axis and the oat center line heeling axis is γ, the rotating angle of the lift force vector φ in the normal plane to the apparent wind axis is given : cosγ sin sin -9 Therefore, the decreasing ratio of horiontal component of the lift force is expressed as: γ γ U U tan cos tanγ cos γ { sin γ sin } cos sin cosγ sin { } - Expanding equation - in a power series and assuming that γ is small, results in γ γ U 3 U cos cos cos cos sin cos - Equation - can further e expanded in terms of φ, and results in γ γ U U cos - Equation - is incidentall equal to the first two terms of the power series for the cos φ function. Hence the curve of cos φ was compared with the calculated results of equation - for three γ cases Figure -3. The calculated results show agreement with the curve of cos φ in spite of the large γ. Therefore, we adopted the formula of cos φ to express the decreasing ratio of the horiontal component of the lift force in place of equation -. Copright NME 8

29 Journal of ailoat Technolog, rticle -.. Decreasing ratio of lift force cos φ γ eq - γ 4 eq - γ 6 eq Heel angle φ [deg] Figure -3. Comparison etween the curve of cos φ and the calculated results of equation -. Finall, when the lift coefficient represents the variation of the lift force including the contriution of dnamic pressure of apparent wind speed, the horiontal component of lift coefficient in the heeled condition L' is descried as: L L cos -3 The main part of the drag is caused the induced drag, which is in proportion to the square of the lift force. The reduction of lift force expressed equation -8 is also approximated cos φ. The vector of the drag force is in line with the apparent wind axis and does not incline the heel angle. Therefore the horiontal component of the drag coefficient D' is descried as: D D cos -4 From these results, the aerodnamic coefficients in the horiontal components X' and ' are then expressed as follows using the coefficients at the upright condition L' and D' : Copright NME 9

30 Journal of ailoat Technolog, rticle -. X L sin γ D cosγ L X cos cos L cosγ D sin γ L cos sin γ D cos cosγ D cos cosγ cos sin γ -5 The moment K is generated mainl the force. However, K is affected the component normal to the mast. Hence, G cos G cos GCE GCE K -6 where G GCE is -coordinate of the geometric center of effort of the sail from the C.G. of the oat and negative upwards. The moment N is also generated mainl the force. However, it is well known that the N is also affected the heel angle φ due to the application point of the thrust force X moving outoard to lee side. Therefore N' can e written, including the effect of X', as: G G ' x GCE GCE N ' X sin cos -7 where x G GCE is x-coordinate of the geometric center of effort of the sail from the C.G. of the oat. PPENDIX 3 Transformation of Forces and Moments from General Bod xis stem to Horiontal Bod xis stem Let the velocities and angular velocities expressed in the general od axis sstem G-x,,, which is fixed on the oat axes, e u, v, w and p, q, r. The forces and moments expressed in this axis sstem are formulated from the Euler-Lam s equations of motion Lam 93 as: Copright NME 3

31 Journal of ailoat Technolog, rticle -. X Z K M N m m u x m m rv m m qw m m v m m pw m mx ru m m w m mx qu m m pv I J p m m wv { I J I J } I J q m mx uw { I J I J } I J r m m uv I J I J x qr rp { }pq -8 where X to N on left side are forces and moments expressed in the general od axis sstem. For the sake of simplicit, it is assumed that the center of added mass of the hull coincides with the C.G. of the oat, and the principal axes of inertia of the added mass also coincide with those of the oat. Then we willl consider the horiontal od axis sstem G-x,,, which originates on the C.G. of the oat. The x-axis lies along the center line of the oat on the still-water plane and is positive forward. The -axis is positive to staroard in the still-water plane. The - axis is positive down as shown in Figure -4. The Euler angles,, θ, φ, are used for the translation from the earth fixed axis sstem into the general od axis sstem. To express the tacking motion of the oat in calm water, we assume the pitch angle, θ, is equal to ero. In this case, the angles around x-, -, -axis of the horiontal od axis sstem, Ψ, Θ, Φ, are descried as: Ψ, Θ θ, Φ The x -axis also coincides with the x-axis. Therefore, the components of the vector expressed in the horiontal od axis sstem are transformed into those expressed in the general od axis sstem as: x cos sin x sin cos -9 Copright NME 3