5 FIIE ELEE AALYSIS OF HE PROPELLER I this chapte we descibe the solid odel ad the fiite eleet aalysis of the popelle I ode to educe the coputatioal cost, we have doe odal codesatio o the fiite eleet odel of the popelle We peset the theoy of odal codesatio ad the calculatio of the steady state espose Fially, the ethod developed fo the calculatio of the secod ode statistics of the espose of a liea syste subjected to CS excitatios is exteded to the case of odal codesatio 5 Popelle odel ad Fiite Eleet Aalysis Usig IDEAS Popelle geoety has udegoe cosideable chages duig the last two decades he use of wide blades with iceasig skewback ade the olde bea ad shell theoies iadequate fo static ad dyaic stegth calculatio O the othe had, fiite eleet ethod (FE) has becoe a poweful tool fo such static ad dyaic aalysis because of its successful applicatios (Politis, 984) We use IDEAS fo the odelig ad the fiite eleet aalysis (FEA) of the popelle Data of the popelle is povided by David aylo odel Basi ad is show i Appedix A Figue 5 shows the poits ad the lies joiig those poits to costuct the hydofoils hese hydofoils wee joied to ake a blade Costuctig thee blades of idetical shape ad size, we joied the with a cylidical hub to costuct the odel of the popelle Figue 5 shows the wiefae geoety of the popelle blade ad hub he solid odel was eshed usig E0 eleets (Beek, 978) It has the shape of tetahedo ad iplies 0 odes of which 4 ae located at the vetices ad 6 i the cete of the edges (Fig 53) Each ode has thee degees of feedo ad cosequetly the eleet stiffess atix 5 FIIE ELEE AALYSIS OF HE PROPELLER 47
5 FIIE ELEE AALYSIS OF HE PROPELLER 48 cotais 30 30 copoets We select E0 eleet because ) the sooth cuvatue of a popelle blade eables a fai appoxiatio by eas of flatsided tetahedos ) the oot sectio of the blade ad hub ae elatively thick ad 3D eleets ae suited fo that, ad 3) the E0 eleet solutio cotais stesses, that vay liealy i all diectios, so the pedoiat blade bedig is epeseted easily he ueical esults of FEA of popelle ae peseted i chapte 6 5 odal Codesatio he goveig syste of equatio fo the popelle espose ca be give by F C (5) whee,, C, ad ae ass, dapig, ad stiffess atices, espectively hese atices ae obtaied by the FEA of the popelle ad F ae the displaceet ad foce vectos F is obtaied usig the expessio fo lift ad dag developed i chapte 4 o calculate the ode shapes, we costuct the udaped fee vibatio poble as 0 (5) Puttig {} {P}e jωt i Eq 5, we get 0 P ω (53) Eq 53 ca be ewitte as
Iλ A P {} 0 (54) whee λ ω ad λ otivial solutio of Eq 54 iplies Iλ A 0 (55) Equatio 55 is the eigevalue poble esultig to eigevalues λ, λ, λ ad eigevectos ), ), ) We ae ot cosideig the case of the epeated eigevalues I geeal, fo a accuate estiate of the espose (t), we eed lage ube of eleets i the odel ad hece the lage ube of odes he disadvatage of such coplicated odel is that it akes the calculatio of secod ode statistics of the espose coputatioally vey expesive While a lage ube of odes ae coputatioally expesive, it ay ot be also eeded i soe cases Fequecy doai aalysis of focig fuctio soeties shows that the agitude of the foces coespodig to fequecy above a cetai level is ot sigificat ad hece odes of the stuctue, whose fequecies ae uch highe tha this, will ot be sigificat i the aalysis Fo these easos, it is beeficial to educe the diesios of the atices i Eq 5 A full odal aalysis would iclude all the eigevectos, but fo the aboveetioed easos, we will coside oly fo to (< ) eigevectos Costuctig the odal atix ) cosistig of eigevectos ), ), ), we get,, (56) Let ( t ) { ( t )} (57) whee (t) is espose vecto i picipal coodiate syste 5 FIIE ELEE AALYSIS OF HE PROPELLER 49
5 FIIE ELEE AALYSIS OF HE PROPELLER 50 Puttig (t) fo Eq 57 ito 5, we get F C (58) Peultiplyig Eq 58 by the ) ie, taspose of ), we obtai F C (59) We assue hee a special case of viscous dapig such that ) C) is diagoal, called odal dapig atix his assuptio is adequate i epesetig the dapig of the stuctue if the dapig is sall which is the case fo a popelle o obtai it we set th diagoal coefficiet C of the odal dapig atix equal to ξ ω, whee ξ is the dapig atio ad ω is atual fequecy coespodig to ode At this poit, we eplace ) ) by, called odal ass atix, ) ) by, called odal stiffess atix, ) C) by C, the odal dapig atix, ad ) F by F, called odal foce vecto hus all the odal atices ae diagoal atices, odal foce vecto has a diesio of, ad Eq 59 a syste of decoupled liea equatios give by F' ' C' ' (50) Equatio 50 ca be solved to obtai the displaceets, (t), i picipal coodiate syste ad the displaceet i the physical coodiate syste ca be obtaied usig Eq 57 5 Steady State Respose
As etioed ealie, Eq 50 is syste of decoupled liea equatio o obtai the steady state espose we set {F} {Fo} cos Ω t, whee eleets of {Fo} ae f o s, i Eq 50, we get fo η ( t ) ξ ω η ( t ) ω η ( t ) cos Ωt (5) ad the atual fequecies ω is give by ω (5) ad odal dapig facto ξ is C ξ (53) ω whee C,, ad is the eleet fo th ow ad th colu of the diagoal dapig, stiffess ad ass atices, espectively Eq 5 gives the solutio fo / η ( t) cos( Ωt α ) (54) ( ) (ξ ) whee taα ξ (55) 5 FIIE ELEE AALYSIS OF HE PROPELLER 5
ad Ω (56) ω 5 odal Codesatio ad IputOutput Poble As etioed ealie, to educe the diesio of the atices ivolved i Eq 5 ad hece to educe the coputatioal cost, the odal codesatio ethod ca be adopted ad the ube of odes cosideed i the fial calculatio will deped upo the fequecy cotet of the excitatios I this sectio we develop a ethod to calculate the espose of a liea syste subjected to CS excitatio if the diesio of the syste has bee educed usig odal codesatio Without loss of ay geeality, we assue that the eas of the excitatios ae zeo his iplies that the eas of the esposes ae also zeo Coelatio atix of the espose (t) i tes of (t) ca be witte as R ( t,t ) E ( t ) ( t ) E ( t ) ( t ) (57) akig the costats ) ad ) out of the expectatio sig, we get R t,t ) E ( t ) ( t ) 5 ( t,t ) ( (58) Whee R (t, t ) ca be obtaied usig Eq 50 ad the ethod to calculate the coelatio atix of the espose developed i chapte Howeve, we eed a elatio, which will elate the coelatio atix of the foces i the physical coodiate syste to the coelatio atix of the foces i the picipal coodiate syste akig the steps siila to the above, we wite the coelatio atix of F as R F' F' ( t,t ) E F' ( t ) F' ( t ) E F( t ) F( t ) (59) 5 FIIE ELEE AALYSIS OF HE PROPELLER 5
akig the costats ) ad ) out of the expectatio sig, we get R F' F' ( t, t FF t ) ) E F ( t ) F ( t ) R ( t, (50) Fo a give poble, we kow the coelatio atix of the foces R FF (t, t ) ad the usig Eq 50, we calculate the coelatio atix of the foces i picipal coodiate syste R F F (t, t ), which is the used i the calculatio of coelatio atix of the espose usig the atices ivolved i Eq 50 ad the ethod to calculate the secod ode statistics of the espose developed i chapte 4 5 FIIE ELEE AALYSIS OF HE PROPELLER 53