MOTIONS OF A SAILING YACHT IN LARGE WAVES: AN OPENING SIMPLE INSTATIONARY MODELLING APPROACH

h Inernaional Symposium on Yach Design and Yach Consrucion MOTIONS OF A SAILING YACHT IN LARGE WAVES: AN OPENING SIMPLE INSTATIONARY MODELLING APPROACH Fabio Fossai, Sara Muggiasca Absrac. Due o he increasing demand of mehods and ools for he analysis of yach behaviour in a realisic environmen and in paricular he developmen of ime domain approaches able o simulae yach moion when under sail in a seaway, in recen years a number of Dynamic Velociy Predicion Programs have been developed. Up o now, while very ineresing resuls are available regarding dynamic effecs on hydrodynamic forces acing on he yach hull and appendages, he physics of unseady sail aerodynamics have received far less aenion. In his paper an opening simple model is presened wih he aim o reproduce unseady sail aerodynamics aking ino accoun hree dimensional effecs and unseady mainsail-jib ineracion. Under he assumpion ha cerain ime and lengh scales for he yach and is wave paern are shor compared wih he ime and lengh scales of he wave moion, he yach is modelled as a single poin mass consrained o move on a surface governed by he equaions of wave moion: normal and angenial equaions of vessel moion are derived and soluions are invesigaed for an arbirary wo-dimensional wave moion. INTRODUCTION One of he mos challenging asks in yach design modelling and simulaion is he developmen of mehods and ools for he analysis of yach behaviour in a realisic environmen. One paricularly difficul ask is developing ime domain approaches ha simulae yach moion, including manoeuvring and course-keeping, when under sail in a seaway. To overcome hese difficulies, modelling mus move from Velociy Predicion Programs (VPP) o Dynamic Velociy Predicion Programs. In recen years a number of Dynamic Velociy Predicion Programs have been developed. To he auhors knowledge, he work of Spens, De Sai and Brown [] is he firs aemp o apply classical ship manoeuvring heory o sailboas. Several years laer Lecher [] proposed an analysis of off wind seering characerisics based on he classic roo locus mehod for solving nonlinear differenial equaions. Professor Gerrisma [3] presened a paper in which he wave induced moions and manoeuvring characerisics of a ypical offshore racing yach have been analysed using a linear seering model wih sabiliy derivaives derived by Planar Moion Mechanism eperimens. No reamen of rig influence was considered. In a following paper by Gerrisma and Moeyes [4] several hull forms and sails influence have been included in he phenomena analysis. Deparmen of Mechanical Engineering - Poliecnico di Milano - ORC ITC Research Associae Deparmen of Mechanical Engineering, Poliecnico di Milano

In 995 Masuyama e al. [5] presened a model based on a force derivaive approach: a very good correlaion of numerical simulaion wih full scale resuls has been obained by he auhors using an eperimenal deerminaion of he hydrodynamic coefficiens. In paricular coefficiens describing he hull forces were deermined by applying regression on he resuls obained from dedicaed eperimens in he owing ank for he one paricular yach considered in his sudy. In he same paper a compleely differen approach has been also presened based on neural nework schemaizaion. The neural approach represens a very useful opion even if does no provide any insigh ino he physics of he sail in unseady flow. One of he mos generic model has been recenly proposed by Keuning e al. [7]: his model is based on he idea of use he eensive resuls and daabase as obained wihin he Delf Sysemaic Yach Hull Series in order o generae generally applicable approimaions for he coefficiens in he equaion of moions. The hydrodynamic model proposed by Keuning in [7] has been recenly used by Baisin e al. ([8]) who developed a dynamic VPP, as a firs feasibiliy sudy, as a plug-in of MSC ADAMS muli-body dynamics analysis ool. Afer his lieraure survey and analysis we can conclude ha while very ineresing resuls are available regarding dynamic effecs on hydrodynamic forces acing on he yach hull and appendages, he physics of unseady sail aerodynamics have received far less aenion and are consequenly less well undersood. Mos of he abovemenioned auhors use he seady sae approach based on force coefficiens empirically derived or measured from wind unnel eperimens as a funcion of he apparen wind angle. As an eample in [8] he aerodynamic forces are modelled following he IMS formulaion ([9], []), i.e. a seady sae approach, boh for he lif and drag of sails and for windage elemens. Some auhors like in [7], [], [] and [], include he induced velociies by he roll and he yaw moions ino he apparen wind angle and apparen wind speed epressions following he so called Quasi Seady Theory (QST) approach. On he oher side, in order o ake ino accoun unseady effecs also CFD simulaions have been proposed: for insance Jacquin e al. [6] presened an all inclusive VPP-CFD simulaions. These numerical mehods include all he basic feaures of unseady flow bu hey are sill in an early developmen sage and are also compuaionally very epensive. Very few eperimenal sudies have recenly addressed he unseadiness of sails aerodynamics: in 8 Gerhard e al. ([]) used unseady poenial flow heory o predic pressure disribuion on a wo dimensional slice of a mainsail carrying ou harmonic oscillaion boh perpendicular o and along he direcion of he inciden flow and heoreical predicion has been compared o wind unnel measuremens on an oscillaing D rigid mainsail slice model, leading o he conclusion ha he aerodynamic performance can only be prediced using an aerodynamic unseady model. The mainsail only analysis of Gerhard e al. in ([]) has subsequenly been eended o a wo sails configuraion using a couple of D cambered profiles. The main finding however is sill he same i.e. ha he aerodynamics can be prediced more accuraely wih an unseady model. More recenly Augier e al. ([8], [9]) presened a dedicaed insrumenaion sysem designed o perform full scale measuremens of loads in sanding and running rigging as well as yach moion and in ([5], [6]) some resuls of pressure measuremens a full scale are repored. In paricular Augier s papers are focused on he resuls obained on an upwind por ack run in head moderae swell and underline he imporance of an unseady model o represen he flow and he sress in he sails in real condiion. To he auhors knowledge, he only available eperimenal resuls specifically concerning unseady sail aerodynamics were presened by he auhors in [3], where unseady aerodynamics has been invesigaed on a 3D sailplan by means of dynamic wind unnel ess. Yach induced pich moion frequency and ampliude effecs on sails forces have been

invesigaed by means of forced moion oscillaion ess carried ou by means of a rigid sailplan using a carbon fibre manufacured rig scale model of a 48 IMS cruiser-racer sloop: dynamic ess have been performed wihin he ypical encouner frequency range and ypical pich ampliudes corresponding o bes VMG condiion for differen rue wind speed and relevan sea sae highlighing some fundamenal issues of he unseady sail aerodynamics. Preliminary obained resuls have been used in [4] o address developmen of numerical models which could be easily incorporaed ino a Dynamic Performance Predicion Program for he analysis of sailing yach behaviour in a realisic environmen allowing for reproduce yach sails aerodynamic coefficiens in he ime domain. To eend he scope of he invesigaion furher forced moion ess have been recenly carried ou on he same yach rig equipped by radiional sof sails [7] in order o provide an insigh ino he sails fleibiliy effecs on sails aerodynamics and a horough invesigaion of he sailplan unseady aerodynamics dependence on differen yach courses. In he presen paper he numerical approach oulined in [4] is embedded in a very simple ime domain model which aims o simulae yach moion when under sail in a seaway: his model should be considered jus an opening model in an aemp o reproduce yach pich frequency and ampliude effecs on sail aerodynamics aking ino accoun hree dimensional effecs and unseady mainsail-jib ineracion. The proposed model is based on he assumpion ha cerain ime and lengh scales for he yach and is wave paern are shor compared wih he ime and lengh scales of he wave moion. In paricular he yach is modelled as a single poin mass consrained o move on a surface governed by he equaions of wave moion: normal and angenial equaions of vessel moion are derived and soluions are invesigaed for an arbirary wo-dimensional wave moion. THE YACHT MODEL The proposed mahemaical model rely heavily on he assumpion ha he yach dimensions are so small in comparison o he scale of he waves ha her influence on he waves may be negleced and he ime scale of her oscillaions are shor compared wih he wave period. Thus he vessel is modelled as a poin mass which is consrained o move on he moving surface of he sea. The sea surface defined by he funcion f (, ) which represens an holonomic and rheonomous consrain and a any given ime i is a general cylinder whose generaors lie perpendicular o he rue wind incoming direcion (D wave moion hypohesis of classical wave mechanics heory). Curvaure of he earh is negleced so graviy is a uniform parallel field of magniude g. The environmen of he boa is shown in figure : he vessel runs sraigh on her course which forms a fied angle µ beween yach cenreline and rue wind direcion in a Caresian frame wih z ais verical and ais poining along he yach course.

z ξ B T c λ R mg µ TRUE WIND Yach environmen Yach course vs wind and waves direcion Figure The yach is locaed by he coordinaes: ( ) z f ( ( ), )) () The eernal wave moion is unaffeced o firs order by he presence of he vessel (due o he smallness of he yach) and does no need o be specified a he momen: his can be described by means of any heory of wo dimensional progressive waves including irregular sea environmen.. KINEMATICS Using eq.() o define he posiion of he yach, differeniaing we find he yach s velociy V B and acceleraion A B : Posiion: ( ) z f ( ( ), )) () Velociy: V Z d( ) V & ( ) d df ( ( ), )) f & + d f (3) Acceleraion: a Z dv d Z a dv d f && + f && ( ) & + f & + f (4) where a do means an ordinary derivaive wih respec o ime and he subscrips and signify parial derivaives.

ξ z z ξ V B A B & f & + f Yach velociy Yach acceleraion Figure The yach s velociy and her acceleraion can be resolved ino normal and angenial componens which are respecively defined by he following epressions (see Appendi ): ( + f ) V (5) n f ( ) + f + f f ( + f ) V & (6) n ( f + f f ) cosσ a & & + (7) ( + f ) f an a & + (8). DYNAMICS The normal and angenial boa equaion of moion can be considered according o he following epressions (fig.(3)): ma n m~ a B mg cosσ (9) T Ro mg sinσ () where: B is he buoyancy (normal o he free surface) T is he aerodynamic driving force (parallel o he surface) R is he oal hydrodynamic resisance (parallel o he surface) o g is he acceleraion due o graviy (verically downward) σ g ( f ) is he free surface slope m is he mass of he yach m ~ is he apparen mass for he longiudinal moion

ξ z B T R mg Figure 3 Firs consider he normal equaion of moion: he yach acceleraion normal componen is given in erms of () by eq.(7) and buoyancy resuls from he normal pressure gradien acing on he surfaces of he yach displaced volume. Defining wih ρw waer densiy and wih g (, ) he normal acceleraion on he free surface by Archimedes principle buoyancy is given by: B ρ w g(, ) () I should also be noed ha because of he wave moion he acual yach displaced volume is differen from he res displacemen which is relaed o he yach mass by he following relaionship: m ρ w (3) Thus, considering eq.(7) and eq.(3) he normal equaion of moion becomes: m ( f + f & + f ) cosσ ρ g(, ) ρ g cosσ & (4) w w Because of he wave moion a some posiions on he wave he vessel s normal acceleraion is greaer han ha of he surrounding waer and she displaces a larger volume ha a res while a some posiion he reverse will happen. Le s consider now he angenial equaion of moion: aking ino accoun he yach acceleraion angenial componen given by eq.(8) we have [ & ( + f ) + f an ] T Ro mg sinσ m~ & (5) In eq.(5) he mass is named m ~ o represen he longiudinal apparen mass of he yach which includes he added mass. The apparen mass depends on he form of he hull, hence on he acual yach displaced volume. For a ypical sreamlined form of a sailing yach hull he added mass value is epeced o fall in he range [%-%] of he yach mass. In conclusion eq. (5) is he general equaion of moion which is a second order differenial equaion for he unknown posiion of he yach () while eq.(4) can be used o evaluae he acual yach displaced volume.

As will be shown in he following, due o aero-hydrodynamics nonlinear characerisic numerical mehods of forward inegraion mus be used o find soluions from given iniial condiions. 3 THE AERODYNAMIC MODEL In order o evaluae aerodynamic hrus (figure 3) appearing in he yach equaion of moion (eq.5), he mahemaical model proposed by he auhors in [4] will be considered. This model rely on eperimenal daa which can been obained by means of dynamic wind unnel ess and allows for reproduce yach sails aerodynamic coefficiens in he ime domain. Deails of he eperimenal se-up and es mehodology used in he dynamic wind unnel ess can be found in ([3], [4]): for readers convenience only a brief summary of he main poins highlighed by eperimenal invesigaions will be provided in he following. 3. OUTLINE OF UNSTEADY SAIL AERODYNAMICS Figure 4 shows he unseady effecs in sail aerodynamics due o he yach pich moions in waves. As can be seen boh apparen wind speed V AW and apparen wind angle β AW corresponding o he seady sae condiions change due o pich induced velociy, resuling in a new apparen wind speed V ris and a new apparen wind angle β din which change in ime and depend on he paricular heigh considered. Clearly he problem would be similar when we should ake ino accoun he effecs of all he oher moions no considered in he presen paper. Figure 4: Dynamic effecs on wind riangle The saring poin was o solve he wind riangle using he effecive angle heory ([7]) on a plane perpendicular o he mas (figure 5) assuming he sailplan cenre of effor as he reference poin o define he insananeous apparen wind speed and he insananeous apparen wind angle.

Figure 5: Reference plane π for he effecive angle evaluaion When he yach is piched, he onse flow is no longer perpendicular o he leading edge of he sails and o accoun for his, he effecive angle concep was used. The effecive angle β eff is geomerically relaed o he apparen wind angle β AW as well as o he pich angle θ and can be calculaed from: β eff - anβ AW an cosϑ (6) Similarly he effecive wind speed V eff can be obained from β AW and θ as a fracion of he apparen wind speed V AW as follows: ( sin cos cos ) V V β + β ϑ eff AW AW AW (7) The combinaion of he dynamic velociy a he cenre of effor due o yach pich angular velociy wih he seady-sae effecive speed V eff leads o he dynamic wind speed V ris and dynamic apparen wind angle β din conceps as shown in Figure 6. Figure 6: dynamic wind riangle According o [4] a new represenaion of he aeroelasic effecs is hen proposed where aerodynamic forces are defined in dynamic condiions wih reference o he dynamic wind riangle concep and are presened as a sor of aerodynamic hyseresis loops, obained by

ploing he sailplan aerodynamic forces agains he dynamic insananeous apparen wind angle β din evaluaed a he insananeous cenre of effor heigh (Z CEH ) (figure 7). F Figure 7: insananeous cenre of effor (lef) and insananeous aerodynamic forces (righ) In paricular sail aerodynamic driving and heeling force coefficiens (C and C Y respecively) can be defined in he ime domain as a funcion of he dynamic apparen wind angle β din (i.e. he insananeous angle of aack) according o he following relaionship: C C Y F ρv F ρv Ris Y Ris S S (8) where F and F Y are respecively he measured aerodynamic driving and heeling forces componens, S is he sailplan oal area, ρ is he air densiy and V ris is he dynamic resulan wind speed evaluaed a he cenre of effor heigh. As an eample, in figure 8 (aken from [3]) unseady driving force coefficien C and heeling force coefficien C Y obained by means of pich forced moion wind unnel ess using a carbon fibre manufacured rig scale model of a 48 cruiser-racer sloop are ploed versus he dynamic apparen wind angle β din. In he same figure dashed lines represen seady sae aerocoeffs curves obained by means of ess wih he sails a res. β din β din Figure 8 Driving and heeling dynamic force coefficiens vs. β din

As can be seen he driving and heeling force coefficiens developed in dynamic condiions are differen for a given angle of incidence (dynamic apparen wind angle) depending on wheher ha angle is increasing or decreasing. This leads o an hyseresis loop, which denoes he presence of a phase shif beween he force and he insananeous angle of aack β din. If he force and he insananeous angle of aack were perfecly in phase, he hyseresis loop would appear as a single line equivalen o he saic seady-sae coefficien rend. When he force and he angle of aack are ou of phase, he plo appears as a loop. In paricular he direcion of roaion of he loop indicaes he relaive phase beween force and he insananeous angle of aack and herefore he energy sign: clockwise (+ circles) indicaes energy pumping, aniclockwise (- circles) indicaes energy dissipaion. The area inside he loop represens he amoun of energy ha can be dissipaed or pu ino he sysem by he aerodynamic forces depending on he sign of he phase shif. In our case he direcion of roaion of he driving force coefficien loop is clockwise and his means ha energy from he piching moion (he waves) is convered ino driving force (neverheless his addiional hrus force is of course a he cos of he damping force on he piching yach in waves). The loop ais has a differen slope wih respec o he seady-sae curve slope and his means ha dynamic effecs influence boh he equivalen damping and siffness characerisics of he aerodynamic forces field. Anoher very imporan poin concerning unseady effecs on sail aerodynamics highlighed by eperimenal invesigaions is he fundamenal role of he reduced velociy V R (or as an alernaive he reduced frequency f R ). As well known, wih reference o an oscillaing airfoil, reduced velociy V R (or as an alernaive he reduced frequency f R ) are defined as follows: V f R R V VT fc C fc C V VT where: V is he flow velociy C is he airfoil chord f is he airfoil oscillaion frequency T is he oscillaion period (/f) (9) Since he reduced frequency is direcly proporional o he airfoil oscillaion frequency, i is a parameer ha epresses he speed wih which he angle of incidence varies. I is also ineresing o observe ha he reduced velociy (eq.9) also epresses he relaion beween he airfoil oscillaion period and he ime (V/C) aken by a fluid paricle o cover he lengh of he chord, or he ime needed o cross he area ineresed by he wing secion. Reduced frequency, which is he reciprocal of reduced velociy (as we can deduce from eq. 9) obviously epresses he converse relaion. A his poin i is easy o undersand ha, he shorer he ime needed o cross he airfoil region is wih respec o he ime needed o complee an oscillaion (he period T), ha is for high values of reduced velociy or conversely for low values of reduced frequency, he more we can consider he condiions of he fluid-airfoil ineracions o be saic, and he more we can use he classic saic or seady sae formulaion for calculaing

he aerodynamic forces. On he oher hand, he smaller he reduced velociy (or he greaer he reduced frequency) he more he ineracion condiions will differ from hose of he saic sae and specific informaion on he forces developed in dynamic condiions would be needed. Concerning he reduced frequency compuaion, moving from he airfoil o he sails siuaion, wind unnel eperimens showed ha he sum of each sail chord measured in correspondence of he cenre of effor heigh (Z CEH ) can be considered in order o provide he sailplan reference chord value (figure 9). Figure 9: Sailplan chord lengh definiion As an eample figures (aken from [4]) show he driving and heeling force coefficiens obained by forced moion ess performed on he previous menioned cruiser-racer sloop sailplan a AWA wih a pich ampliude of wihin he eplored reduced velociy range (from V R 3 up o V R 4). C [-].55.5.45.4 V * 4.9 - A deg - V 3. m/s - freq. Hz V * 4.8 - A deg - V 3. m/s - freq.3 Hz V * 8.47 - A deg - V 3. m/s - freq.5 Hz V * 6.4 - A deg - V 3. m/s - freq.7 Hz V * 4.3 - A deg - V 3. m/s - freq. Hz V * 3.53 - A deg - V 3. m/s - freq. Hz saic CY -.3 -.4 -.5 -.6 V * 4.9 - A deg - V 3. m/s - freq. Hz V * 4.8 - A deg - V 3. m/s - freq.3 Hz V * 8.47 - A deg - V 3. m/s - freq.5 Hz V * 6.4 - A deg - V 3. m/s - freq.7 Hz V * 4.3 - A deg - V 3. m/s - freq. Hz V * 3.53 - A deg - V 3. m/s - freq. Hz saic.35 -.7 9.5.5.5.5 3 3.5 4 4.5 β din -.8 9.5.5.5.5 3 3.5 4 4.5 β din Driving force coefficien vs. β din a differen V R Heeling force coefficien vs. β din a differen V R Figure A quie large amoun of eperimenal ess recenly carried ou on boh rigid and sof sails ([7]) confirm ha reduced velociy play a fundamenal role in sails aerodynamics and ha yach piching moion has a srong and a non-rivial effec on he aerodynamic forces field. In paricular as he reduced velociy decreases, he enclosed area of he hysereic loop and he loop ais slope increase and his means ha unseady condiions lead o aerodynamic equivalen damping and siffening effecs.

3. SAIL AERODYNAMICS MODEL As previously menioned, in order o evaluae aerodynamic hrus (figure 3) appearing in he yach equaion of moion (eq.5), he mahemaical model proposed by he auhors in [4] will be considered. This model rely on eperimenal daa obained from dynamic wind unnel ess and moves from he basic idea o compare he measured aerodynamic forces wih he hysereic effecs measured on mechanical sysems. In fac i is possible o recognize ha he hyseresis loop of he aerodynamic forces highlighed by eperimenal invesigaions and menioned in he previous paragraph are similar o he ypical hyseresis loop of a non-linear and non-conservaive mechanical elemen leading o he possibiliy o develop a rheological model in he ime domain in order o reproduce he sailplan aeroelasic forces based on he variaion of he insananeous apparen wind angle. Generally speaking, rheological models, adoped o compue he aeroelasic forces induced by a variaion in he insananeous angle of aack, consis in a mechanical sysem whose componens are springs, dampers, coulomb fricion elemens, bump sops and oher mechanical pars ha are inerconneced and ha reac agains he imposed moion (figure lef) and he modelled sails aerodynamic coefficien is given by he force ransmied o he ground by he equivalen mechanical sysem. β, & din β din Figure : Typical rheological model componens (lef) and The proposed rheological model (righ) The inpu of he proposed model is he insananeous apparen wind angle β din and is ime derivaive dβ din /d, being his angle funcion of he incoming nominal (seady sae) apparen wind and of he yach moion, in paricular of he sailplan pich angular velociy. Moving on wih he aerodynamics-mechanics analogy, he deformaion of he mechanical sysem corresponds o he dynamic apparen wind angle in he sailplan, so ha he mechanical force due o he imposed srains corresponds o he aerodynamic force due o a change in he insananeous angle of aack on he sails. The major ask, in he rheological model parameer idenificaion, is o gran a good performance wih he capabiliy of reproduce boh he reduced velociy dependence and he yach moion ampliude dependence of he phenomenon. Several models o compue he aeroelasic forces induced by a variaion in he insananeous angle of aack have been considered connecing differen simple block like springs, dampers, coulomb fricion elemens, ec and, finally, he numerical model adoped consiss in a mechanical sysem whose componens are a linear spring wih one eremiy conneced o a linear dampers as shown in figure (righ). Sail aerodynamic driving force

coefficien is equal o he force ransmied o he ground by he equivalen mechanical sysem resuling by he muual ineracion beween he spring and he damper as a funcion of he insananeous apparen wind angle β din and is ime derivaive & β din. The dynamic behaviour of he proposed rheological sysem (fig. righ) is described by he following equaions: C k r & α rk k ( β α rk ) ( β α ) rk () where he sailplan driving force coefficien C is assumed o be he force ransmied o he ground, which depends no only on he model inpu which is he insananeous apparen wind angle β din and is ime derivaive & β din, bu also on he spring-damper connecing poin displacemen α rk and is ime derivaive α& rk which can be obained considering he following incremenal relaionship: d α rk ( + d) α rk ( ) + * k β ( ) αrk ( ) r ( ) () Tuning properly he spring siffness parameer k and damper damping parameer r allows for aking ino accoun he aerodynamic equivalen damping and siffness variaion wih he reduced velociy. For he presen work rheological model parameers have been idenified, wihin a suiable range of reduced velociies, using eperimenal daa provided by a quie large amoun of dynamic wind unnel ess carried ou a -7-3 apparen wind angles in order o comply wih he ypical upwind sailing condiions [7]. As an eample figure () summarizes driving force coefficiens obained wihin he eplored reduced velociy range wih a model pich ampliude of. As can be seen, aerodynamic hrus developed in dynamic condiions denoes he presence of a phase shif beween he force and he insananeous angle of aack β din leading o groups of hyseresis loops placed in correspondence of each mean apparen wind angle wihin he considered range. C [-].8.7.6.5.4.3.. 8 4 6 8 3 3 34 36 β din V * 4.9 - A deg - V.98 m/s - freq. Hz V * 3.95 - A deg - V.97 m/s - freq.3 Hz V * 8.38 - A deg - V.98 m/s - freq.5 Hz V * 5.98 - A deg - V.97 m/s - freq.7 Hz V * 4.8 - A deg - V.97 m/s - freq. Hz V * 3.49 - A deg - V.97 m/s - freq. Hz saic Figure

Using hese resuls he rheological model parameers may be idenified using a weighed leas square curve fiing algorihm ha minimizes he error beween he eperimenally measured hyseresis loops and hose produced by he numerical model. Following his approach, for each of he invesigaed apparen wind angle, rheological model parameers dependence on reduced velociy can be assessed and reproduced by means of piecewise-linear fiing funcions. Figure 3 shows he siffness and damping parameers rend vs he reduced velociy, which have been idenified using eperimenal resuls repored in fig.. k - siffness.5.4.3 deg 7deg 3deg 4 6 8 4 6. r - damping.5 4 6 8 4 6 V red Figure 3: Driving force coefficien-rheological model parameers vs. reduced velociy Le s consider now again he vessel running sraigh on her course and moving on he sea surface in a plane which forms a fied angle µ beween yach cenreline and he incoming waves: assuming he amospheric wind blowing wih speed W in he same direcion of he waves and defining he rue wind angle β TW, we can consider he angenial componen using he surface angle σ which will define he rue wind speed W TW in he π plane perpendicular o he mas (figure 4): V TW W W cosσ () ( + f ) λ π β TW c ξ B W TW σ T TRUE WIND R mg Yach course vs rue wind and waves direcion Figure 4 Effecive angle heory plane

According o effecive angle heory, aking ino accoun for he yach s velociy angenial componen we can define he dynamic resulan wind speed V ris and he dynamic insananeous apparen wind angle β din in he π plane using he following equaions: V β ris din ( V + W cos β ) + ( W sin β ) TW TW WTW sin βtw arcg V + WTW cos β TW TW TW (3) Finally he aerodynamic hrus (figure 4) appearing in he yach equaion of moion (eq.5) can be evaluaed by means of he following epression: T ρ as C ( β ) V din ris (4) where S is he sailplan oal area, ρ a is he air densiy and C is he unseady aerodynamic driving force coefficien which can be evaluaed a each ime sep by means of equaions () using he siffness and damping parameers chosen in correspondence of he reduced velociy obained combining he relevan wave frequency and wind speed values. 4 THE HYDRODYNAMIC MODEL As far as he oal hydrodynamic resisance R o which appears in he yach equaion of moion (eq.5) is concerned, in he presen paper he choice was o include a formulaion which would be as much simpler as possible bu able o include he effecs due o he acual yach displaced volume variaion under he effecive graviaional acceleraion g (, ). As usual he oal hydrodynamic resisance has been spli ino fricional and wavemaking componens as summarized in he following. Under he model assumpions highlighed in par. and aking ino accoun he angenial velociy V HO of a waer paricle on he free surface, he hull is ravelling a a speed: V V H O (5) a a displacemen in locally fla waer, under he effecive graviaional acceleraion g,. Considering: ( ) he Reynolds number: ( V V ) H O LWL Re (6) υ he Froude number: ( V V ) H O Fn (7) g(, )LWL

and assuming hese dimensionless quaniies wih he acual yach displaced volume are he parameers describing he curren geomery and flow condiions, he fricional resisance componen is evaluaed according o he following epression: R f ρ waw ( ) ( V VH O ) CRf (8) where: R f is he fricional resisance ρ w is he waer densiy A is he hull weed surface w C is he skin fricion coefficien rf The skin fricion coefficien has been evaluaed using he well-known ITTC 57 formula. The wavemaking resisance componen has been evaluaed according o he MIT-VPP original formulaion [5]: R w a BWL Cv ρ g a 5 w (9) Tc C + a v 3 where: R W is he wavemaking resisance componen Tc is he canoe body draf BWL is he heoreical waerline beam C v is he volumeric coefficien is he acual canoe body displaced volume g is he acceleraion of graviy ρ is he waer densiy w The values of he coefficiens a, a and a 3 are a funcion of he relaive speed are repored in [5] V and LWL 5 SOLUTION OF EQUATIONS OF MOTION In order o find soluions of he normal and angenial equaions of moion of he vessel moving on he free surface due o nonlineariies of aero-hydrodynamic force fields, numerical mehods of forward inegraion mus be used from given iniial condiions. As a consequence of nonlinear formulaions of he aerodynamic hrus and of he oal hydrodynamic resisance oulined in he previous paragraphs, he general equaion of moion (eq. (5)) is a nonlinear second order differenial equaion for he unknown posiion of he yach () : his equaion has been solved using a modified Newmark mehod including a frozen ime ieraions loop a each ime.

More in deails, due o he dependence of boh fricional and wavemaking resisance componens on he acual displaced volume moving from sep n o n+, he yach posiion velociy and acceleraion are evaluaed using a frozen ime ieraion loop where, a each ieraion, eq.(4) is used o calculae he yach displaced volume requesed o evaluae he abovemenioned force componens. 6 SOME RESULTS The proposed mahemaical model has been used o invesigae he dynamic response of a 48 cruiser-racer sloop whose main dimensions are shown in able. 6. REGULAR WAVES LOA (m) 4.5 BMA (m) 4.9 LWL (m) 3.5 BWL (m) 3.78 DISPL (kg) 43 SAIL AREA (m ) 4 AW (m ) 45.5 Tc (m).83 Table. Main yach dimensions. Analyses have been iniially carried ou considering yach sailing in regular waves and equaions (4) and (5) have been specialized aking ino accoun linear wave heory formulaion. In paricular he surface profile is described by he following equaion: ( k ω ) ξ ( ( ), )) ξ cos + (3) where ξ is he wave ampliude k is he wave number ω e is he encouner frequency e The encouner frequency is evaluaed by means of: ω ω k& cos µ (3) e where & is he yach speed and µ is he heading angle (fig.). The incoming wave direcion is assumed o be he same as he direcion of he rue wind, so he rue wind angle TWA can be relaed o he heading angle of equaion (5) according o: TWA 8 µ (3) True wind speed and rue wind angle values were chosen in such a way o mach apparen wind angle values where aerodynamic hrus coefficien hyseresis loops are available from dynamic wind unnel ess.

In paricular seady sae analyses have been performed a he beginning, considering only he longiudinal seady equaions of moion: T R (33) o wih reference o he following cases (able): CASE TWA (deg) TWS (knos) # 44.7 6 # 38.8 #3 6 Table leading o seady sae yach speed and apparen wind angle values repored in able 3. CASE Yach speed (knos) AWA (deg) # 6.6 # 9.4 6.5 #3 7.8 3. Table 3 Then regular waves have been considered and, in order o link wind inensiy o significan wave heigh and period properies, relaionships relaing o he Norh Alanic [8] were used (Figure 5). Figure 5. Wave significan heigh and wave period versus wind speed. More in deails able 4 shows he wave heigh and period values considered for analyses carried ou on yach sailing in regular waves.

CASE TWS (knos) Hs (m) T (s) # 6. 5.8 #. 6.8 #3.5 6 Table 4. Wave heigh and wave period values In he following, resuls obained wih reference o CASE # are shown. In paricular fig. 6 shows he regular wave surface encounered by he yach sailing under her way and figures 7 and 8 show respecively he angenial componen of he yach speed and he acual displaced volume ime hisories. Saring from iniial condiions which correspond o seady sae soluion obained as previously described, he yach velociy develops as dynamic oscillaions wih encouner period of 4.56 [s] which resuls ou of phase wih he wave profile, as well as he acual volume displacemen. Lile by lile ime is increasing yach speed oscillaions becomes saionary in he neighbourhood of a mean value which is abou 95% of he seady sae problem soluion (eq.33). ξ [m] - 4 6 8 Fig. 6: CASE #- wave profile ime hisory [m/s] 5 4 3 Tangenial boa speed 4 6 8 Fig. 7: CASE #- angenial yach speed ime hisory [m 3 ] 3 volume displacemen 4 6 8 Fig. 8: CASE #- acual displaced volume ime hisory Figures 9- show he dynamic apparen wind angle and speed ime hisories, due o he wave induced pich oscillaion which have been used o evaluae he dynamic aerodynamic driving force coefficien (fig. ). In paricular a each ime sep he aerodynamic driving force coefficien has been obained by means of numerical inegraion of he rheological model equaions (); siffness and damping parameers appearing in eq.() have been seleced from red curves of figure 3 in correspondence of he reduced velociy value resuling from he considered mean apparen wind speed and encouner frequency.

Figure shows he comparison beween he driving force coefficien hysereic loop obained by he numerical model and he eperimenal one. 4 β dyn [deg] 8 4 6 8 Fig. 9: CASE #- dynamic apparen wind angle ime hisory Vris [knos] 4 4 6 8 Fig. : CASE #- dynamic apparen wind speed ime hisory Vris.5 C [-].45.4.35 4 6 8 Fig. : CASE #- dynamic aero-coefficien ime hisory.5 Eperimenal Numerical Saic coefficien.45 C [-].4.35.5.5.5 3 3.5 β din [deg] Fig. : CASE #- aerodynamic driving force coefficien Finally figure 3 shows he hydrodynamic oal resisance and aerodynamic force ime hisories.

[N] Toale Resisance Aerodynamic force 4 6 8 Fig. 3: CASE #- oal hydrodynamic resisance and aerodynamic hrus ime hisories In he following he same resuls obained for CASE# are repored. ξ [m] - 3 4 5 6 Fig. 4: CASE #- wave profile ime hisory 6 Tangenial boa speed [m/s] 5 4 3 4 5 6 Fig. 5: CASE #- angenial yach speed ime hisory In his case yach speed oscillaions becomes saionary in he neighbourhood of a mean value which is abou 3% less of he seady sae problem soluion. [m 3 ] 3 3 4 5 6 Fig. 6: CASE #- acual displaced volume ime hisory 8 β dyn [deg] 7 6 5 3 4 5 6 Fig. 7: CASE #- dynamic apparen wind angle ime hisory

Vris [knos] 3 8 Apparen wind speed 6 3 4 5 6 Fig. 8: CASE #- dynamic apparen wind speed ime hisory.8 C [-].6.4 3 4 5 6 Fig. 9: CASE #- dynamic aero-coefficien ime hisory Finally resuls of CASE#3 are repored in he following. ξ [m] - 4 6 8 Fig. 3: CASE #3- wave profile ime hisory 5 Tangenial boa speed [m/s] 4 3 4 6 8 Fig. 3: CASE #3- angenial yach speed ime hisory ory In his case yach speed oscillaions becomes saionary in he neighbourhood of a mean value which is abou 5-6% less of he seady sae problem soluion. [m 3 ] 3 4 6 8 Fig. 3: CASE #3- acual displaced volume ime hisory

4 β dyn [deg] 35 3 5 4 6 8 Fig. 33: CASE #3- dynamic apparen wind angle ime hisory Vris[knos] 6 5 4 Vris 3 4 6 8 Fig. 34: CASE #3- dynamic apparen wind speed ime hisory.8 C [-].6.4 4 6 8 Fig. 35: CASE #3- dynamic aero-coefficien ime hisory As a general commen even if he obained resuls could be cerainly, o a cerain een, considered overall unrealisic in erms of yach speed and displaced volume due o he model simpliciy, on he oher side i can be said ha he proposed model, which moves from basic principle of hydrodynamics and mechanics, is leading o a descripion of yach moions in waves ha agrees qualiaively wih many aspecs of he real life siuaion. 6. IRREGULAR WAVES The proposed model has been also used o invesigae he yach behavior in irregular sea. Realisic irregular waves have been obained using an idealised wave energy specrum: several formulas eis for epressing in analyical form he idealized wave energy specra as a funcion of characerisic parameers such as he significan wave heigh (H /3 ) and he average period (T ) of he specrum and for he presen sudy he Breschneider energy specrum, which is paricularly suied for represening condiions in mid ocean, was used. Waves ime hisories have been synhesized by adding a large number of componen sine waves according o he following epression: n n ( k + ω φ ) ξ ( ( ), )) ξ cos + (34) n en n where phase angles φ n are chosen from a random disribuion. Again wave sea specrum significan wave heigh and modal period are linked o he rue wind speed using relaionships repored in figure 5.

Analyses have been carried ou considering he same rue wind speed and rue wind angles used in he previously shown regular waves cases (ab. ) and for each case relevan wave heigh and period (repored in able 4) have been assumed as significan wave heigh and modal period values defining he Breschneider energy specrum. The maimum frequency considered in he eq. (34) has been limied in such a way ha he presence of irregular wave moion could no affec subsanially he dynamic response of he yach. A his aim he yach pich ampliude and frequency range caused by he waves passing beneah i were esimaed using he main yach dimensions (Table ) and he heave and pich Response Ampliude Operaor (RAO) for Model 455 from he DSYHS, obained from owing ank ess and kindly provided by Prof. Keuning of Delf Universiy of Technology. This model has similar characerisics o he yach here under analysis and figure 36 (lef) shows he abovemenioned pich/wave slope RAO as a funcion of he LWL/wavelengh raio. θ / waveslope [rad/rad].7.6.5.4.3. RAO pich/w aveslope (L/ (/3) 5, φ [deg]) θ / ζ [deg/m] 8 6 4 RAO in funzione della frequenza di inconro ω e TWS3.867 [m/s] TWS4.56 [m/s] TWS5.444 [m/s] TWS6.733 [m/s] TWS7. [m/s] TWS8.3 [m/s] TWS.889 [m/s]...4.6.8..4 L/λ Pich Response Ampliude Operaor vs. L WL /λ..5.5.5 3 3.5 4 4.5 ω e [rad/s] Pich Response Ampliude Operaor vs. encouner frequency Figure 36 In order o esimae he yach encouner frequency range, performance predicions a differen rue wind speeds and rue wind angles were obained using he longiudinal seady equaions of moion (eq.33). The yach VMG and he corresponding apparen wind angle were evaluaed for each TWS (wihin he 6- [knos] TWS range). Then he corresponding yach encouner frequency ranges were calculaed using he abovemenioned values and fig. 36 (righ) shows he obained pich RAOs as a funcion of he encouner frequency. The final sep is o evaluae he pich response in waves. A his aim he abovemenioned Breschneider energy specra have been considered for he same rue wind inensiies (fig.37 lef) and finally he pich response specra have been calculaed by he produc of he square of he moion RAOs and he wave energy specra a he various wind inensiies (fig.37 righ) showing ha he yach pich response specra is defined wihin he range (-4 [rad/s]).

SBenc [rad /(rad/s)].35.3.5..5. spero Breschneider in ω inconro TWS3.867 [m/s] TWS4.56 [m/s] TWS5.444 [m/s] TWS6.733 [m/s] TWS7. [m/s] TWS8.3 [m/s] TWS.889 [m/s] SBpichenc [rad /(rad/s)].5-3.5 spero pich Breschneider in ω inconro TWS3.867 [m/s] TWS4.56 [m/s] TWS5.444 [m/s] TWS6.733 [m/s] TWS7. [m/s] TWS8.3 [m/s] TWS.889 [m/s].5.5.5.5 3 3.5 4 4.5 ω e [rad/s] Yach course vs wind and waves direcion.5.5.5 3 3.5 4 4.5 ω e [rad/s] Yach environmen Figure 37 As an eample, resuls repored in he following refer o a seasae synheized wih he reference values repored in Tab.5 and fig. 38 shows he relevan energy specrum. TWA (deg) TWS (knos) Hs (m) T (s) 6.5 6 Table 5. wind characerisics and irregular sea significan wave heigh and modal period S Bξ [m /(rad/s)].4..8..4.6 ω e [rad/s] Fig. 38 Comparing fig. 37 righ wih figure 38 i can be seen ha frequency yach pich response specrum and wave energy specrum have a very weak overlapping according o he maimum frequency chosen o define irregular wave ime hisory in order o comply wih he model basic assumpions. Fig. 4 shows he irregular wave surface encounered by he yach and figures 4 and 4 show respecively he angenial componen of he yach speed and of he acual displaced volume ime hisories. ξ [m] - 4 6 8 Fig. 4: irregular wave profile ime hisory

5 Tangenial boa speed [m/s] 4 3 4 6 8 Fig. 4: irregular wave - angenial yach speed ime hisory [m 3 ] 3 4 6 8 Fig. 4: irregular wave - acual displaced volume ime hisory Figures 43-44 show he dynamic apparen wind angle and speed ime hisories, due o he wave induced pich oscillaion which have been used o evaluae he dynamic aerodynamic driving force coefficien (fig. 45). 4 β dyn [deg] 35 3 5 4 6 8 Fig. 43: irregular wave - dynamic apparen wind angle ime hisory Vris [knos] 8 6 4 Apparen wind speed 4 6 8 Fig. 44: irregular wave - dynamic apparen wind speed ime hisory.8 C [-].6.4 4 6 8 Fig. 45: irregular wave - dynamic aero-coefficien ime hisory In his case, in order o selec he rheological model parameers values, a reference reduced velociy value has been considered based on he seady sae apparen wind speed and on he modal encouner frequency appearing in he wave specrum (fig. 38).

7 CONCLUSIONS AND FUTURE DEVELOPMENTS This paper describes a very simple ime domain model which aims o simulae yach moion when under sail in a seaway. The proposed approach should be considered jus a firs sep of a more comprehensive general research wih he final goal o develop mehods and ools for he ime domain analysis of yach behaviour in a realisic environmen. A presen well-esablished resuls are available on dynamic effecs on hydrodynamic forces acing on he yach hull and appendages, while he physics of unseady sail aerodynamics have received far less aenion and are consequenly less well undersood. In he presen paper unseady sail aerodynamics is embedded in a very simple ime domain model which aims o simulae yach moion in a seaway: in paricular he yach is modelled as a single poin mass consrained o move on a surface governed by he equaions of wave moion and sail aerodynamics is inroduced moving from he idea o compue sail aeroelasic forces induced by a variaion in he insananeous angle of aack by means of rheological models able o represen hysereic effecs of unseady sail aerodynamics highlighed by eperimenal resuls obained from dynamic wind unnel ess. The proposed model is srongly relying on he assumpion ha cerain ime and lengh scales for he yach and is wave paern are shor compared wih he ime and lengh scales of he wave moion and represens jus a firs aemp o reproduce yach induced pich moion effecs on sail aerodynamics aking ino accoun hree dimensional effecs and unseady mainsail-jib ineracion. Auhor s view is ha he proposed model, which moves from basic principle of hydrodynamics and mechanics could be improved in order o include some imporan addiional feaures leading o a more realisic descripion of yach moions in waves; in paricular sail aerodynamics dependence on pich moion ampliude and frequency could be included in he proposed aero-model, as well as a more realisic descripion of he yach ime scales oscillaions including added mass and damping effecs due o pich, heave and surge moions. Finally an added resisance in waves formulaion could be included in order o allow he designer o make comparison beween various sailing yach designs in he acual environmenal condiions. REFERENCES [] Spens P.G., DeSai P., Brown P.W., Eperimenal sudies of he sailing yach, SNAME Transacion, 967. [] Lecher, J.S., Seering qualiies off he wind, AIAA Symposium on he Aero- Hydronauics of Sailing, AIAA, 97. [3] Gerrisma J.: Course keeping qualiies and moions in waves of a sailing yach - Technical Repor, Delf Universiy of Technology, May, 97. [4] Gerrisma J., Moyes G., The seakeeping performance and seering properies of Sailing Yachs, 3 rd HISWA Symposium on Yach Design and Consrucion, Amserdam 973. [5] Masuyama Y., Fukasawa T., Sasagawa H., Tacking simulaion of a sailing yach - Numerical inegraion of equaions of moion and applicaion of neural nework echnique, Proceedings of he h Chesapeake Sailing Yach Symposium, Annapolis, 995.

[6] Jacquin E., Rou Y., Guillerm P.E., Alessandrini B., Toward numerical VPP wih he full coupling of hydrodynamic and aerodynamic solvers for ACC yach, Proceedings of he 7h Chesapeake Sailing Yach Symposium, Annapolis, 5. [7] Keuning J.A., Vermeulen K.J., de Ridder E.J., A generic mahemaical model for he maneuvering and acking of a sailing yach, Proceedings of he 7h Chesapeake Sailing Yach Symposium, Annapolis, 5. [8] Baisin D., Ledri M., A Tool for Time Dependen Performance Predicion and Opimizaion of Sailing Yachs, Proceedings of he 8h Chesapeake Sailing Yach Symposium, Annapolis, 7. [9] Claughon, A., Developmens in he IMS VPP formulaions, Proceedings of he 4h Chesapeake Sailing Yach Symposium, Annapolis, 999. [] Fossai F., Claughon A., Muggiasca S., Baisin D.,: Changes and Developmen o Sail Aerodynamics in he ORC Inernaional Rule h HISWA Symposium on Yach Design and Consrucion, Amserdam, 8 [] Gerhard F., Flay R., Richards P.: Unseady Aerodynamic Phenomena Associaed wih Sailing Upwind in Waves, 3 rd High performance Yach Design Conference, Auckland 8. [] Gerhard, F. C.; Flay, R.; Richards, P. (): Unseady aerodynamics of wo ineracing yach sails in wo-dimensional poenial flow. J. Fluid Mech., 668. 55-58 [3] Fossai F., Muggiasca S., Sails Aerodynamic Behaviour in Dynamic Condiions, Proceedings of he 9h Chesapeake Sailing Yach Symposium, Annapolis, 9. [4] Fossai F., Muggiasca S., Eperimenal Invesigaion of Sail Aerodynamic Behavior in Dynamic Condiions, Journal of Sailboa Technology, - SNAME [5] Viola I., Flay R., Pressure disribuions on sails invesigaed using hree mehods: on waer measuremens, wind unnel measuremens and CFD, Proceedings of he h Chesapeake Sailing Yach Symposium, Annapolis,. [6] Keuning, J.A., Vermeulen K.J., H.P. en Have., An approimaion Mehod for he added resisance in waves of a sailing yach, nd Inernaional Symposium on Design and Producion of Moor and Sailing Yachs MDY 6, Madrid, Spain, 6. [7] Jackson, P.S.: An improved Upwind Sail Model for VPP s, Proceedings of he 5h Chesapeake Sailing Yach Symposium, Annapolis,. [8] Augier B., Bo P., Hauville F., Durand M.: Eperimenal validaion of unseady models for wind/sails/ rigging fluid srucure ineracion, Inernaional Conference on Innovaion in High Performance Sailing Yach, Lorien,. [9] Augier B., Bo P., Hauville F.: Eperimenal full scale sudy on yach sails and rig under unseady sailing condiions and comparison o fluid srucure ineracion unseady models, Proceedings of he h Chesapeake Sailing Yach Symposium, Annapolis,. [] Richard T., Harries S., Hochkirch K., Manoeuvring simulaions for ships and sailing yachs using FRIENDSHIP Equilibrium as an open modular workbench, COMPIT, Hamburg 5.

[] Harris D.H., Simulaion of upwind manoeuvring of a sailing yach, Disseraion for Docor of Philosophy degree, Univ. of Maryland-College Park,. [] Coneno G., Ledri M., Codiglia L., Eperimenal Analysis of he Verical Moions in Waves of an IACC Yach wih Calm Waer-Opimized Bulb Shapes, Proc. of nd High Performance Yach Design Conference, pp.8-88, Auckland 6 [3] Gerhard F., Le Pelley D., Flay R., Richards P.: Tacking in he wind unnel, 9h Chesapeake Sailing Yach Symposium, Annapolis, 9. [4] Fossai F., Muggiasca S., Numerical modeling of sail aerodynamic behaviour in Dynamic Condiions, Inernaional Conference on Innovaion in High Performance Sailing Yach, Lorien,. [5] Kerwin, JE A velociy Predicion Program for Ocean racing yachs, Rep 78- MIT, July 978 [6] Le Pelley D., Morris D., Richards P.: Aerodynamic force deducion on yach sails using pressure and shape measuremens in real ime, 4 h High performance Yach Design Conference, Auckland. [7] Fossai F., Muggiasca S.: An eperimenal invesigaion of unseady sail aerodynamics including sail fleibiliy, 4 h High performance Yach Design Conference, Auckland [8] Fossai F. Aero-hydrodynamics and he performance of sailing Yachs Adlard Coles Nauical, London 9 APPENDI In he following he yach s velociy and acceleraion normal and angenial componens given by eqs (5), (6), (7) and (8) will be derived. Le s consider he vessel on he sea surface a posiion: ( ) z f ( ( ), )) (A) moving wih velociy: V Z d( ) V & ( ) d df ( ( ), )) f & + d f (A) Using he free surface angle σ which is defined by: σ g ( f ) (A3)

he angenial and normal componens of he yach s velociy V n and V are given by: V V cosσ + V sinσ (A4) Z V n V cosσ V sinσ (A5) Z From eq. (A5): Vn V cosσ Bu: gσ f cosσ Z V gσ ( + gσ ) ( + ) f (A6) (A7) And subsiuing in eq. (A6) we obain: V n ( V V f ) ( + gσ ) [ f & + f f & ] ( + f ) ( + f ) f Z (A8) ha is eq.(5). Analogously from eq. (A4) we obain: V V cosσ + V gσ Z (A9) and subsiuing eq. (A7) in eq. (A5) we obain V [ & + ( f & + f ) f ] ( + f ) [ & ( + f ) + f f ] ( + f ) (A) & ( ) + f + f f ( + f ) ha is eq.(6). In he same way, considering he yach s acceleraion componens:

a Z dv d Z a dv d f && + f && ( ) & + f & + f (A) he angenial and normal componens of he yach s velociy a n and a are given by: a a cosσ + a sinσ (A) Z a n a cosσ a sinσ (A3) Z and aking ino accoun for eq,(a7) subsiuing eq.(a) in eq.(a3) a n a Z a gσ a ( + f ) ( f & + f & + f ) Z a f ( ) ( f && ) + f & + f & + f && f + f (A4) ha is eq.(7). Analogously subsiuing eq.(a) in eq.(a) and aking ino accoun for eq,(a7): a cosσ ( a + az f ) cosσ [&& + ( f && + f & + f & + f ) f ] ( + f ) [&& ( + f ) + f ( f & + f & + f )] && ( + f ) + f ( + f ) ( f & + f & + f ) (A5) Bu from eq.(a4) we recognize ha: ( + f ) ( f & + f & + f ) an (A6) So from eq.(a6) follows: ( + f ) f an a & + (A7) ha is eq.(8).