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["Aerodynamic Model of Rocket Pitching Moment"]
["Copyright E.V.LaBudde 1998-1999 All rights reserved"]
[]
["The assumptions used in this model make it valid up to about +/- 45 degree angle of attack."]
["The design is limited to a simple rocket with one conical transition and trapezoidal fins."]
["The input definitions conform to the usual Barrowman method."]
["The model extends the Barrowman method to large angles of attack and unifies the Barrowman and Cutout methods of Cp prediction"]
["The main objective of the model is to estimate the critical stability point of positive pitching moment, if it exists, up to 45 degree Alpha."]
["Any length unit may be used, but, all dimensions must be of the same unit."]
["Blue cells are user input cells, Black cells contain formulas."]
[]
["INPUT PARAMETERS","VALUE","LABLE","UNIT","COMMENTS"]
["Name of Rocket","Long Arcas","Name","NA"]
["Length Of Nose",65,"Ln","Any Length"]
["Diameter Of Nose",13.818,"D_","Any Length"]
["Total Length Of Rocket",328,"Lr","Any Length","Includes nose, transition, body and engine overhang"]
["Distance To Front Of Transition",317,"Xp","Any Length","If no transition is used, set to nose length"]
["Diameter At Rear Of Transition",10.367999999999999,"Dr","Any Length","Only one transition is allowed, the front diameter is equal to the nose"]
["Length Of Transition",11,"Lt","Any Length","If no transition is used, set to zero"]
["Distance To Leading Edge Of Root Cord",308,"Xb","Any Length"]
["Root Cord",20,"Cr","Any Length"]
["Tip Cord",12.8,"Ct","Any Length"]
["Fin Span",13,"S","Any Length"]
["Distance To Fin Tip",7.2,"Xs","Any Length"]
["Number Of Fins",4,"N","NA","Can be 3, 4 or 6"]
["Center Of Gravity",230,"Cg","Any Length","Of complete rocket with engine"]
["Nose Shape","Ogive","Nose","Na","Limited to conical, ogive, elliptical, or parabolic"]
["Barrowman Or Extended","B","B_factor","Na","B for standard Barrowman, or E for \"Extended\" Barrowman with the body-wing interference factor"]
["Stall Angle Of Attack",11,"Peak_Alpha","Degree","11 is a conservative value"]
["Drag Coefficient For The Body ",0.4245,"Cd_body","NA","Cd of .4245 used to fit Arcas data at Mach .6 Should be based on Reynolds number"]
[]
["Output"]
["Stability at zero alpha","Stable","Stability_zero","NA","Slope must be negative at zero alpha"]
["Stability at non zero alpha","Stable","Stability_non_zero","NA","Cm must be negative somewhere"]
["Alpha for Zero Pitching Moment",0,"Alpha_zero","Degree","If one exists"]
["Point of Positive Pitching moment",17.72972181581791,"Alpha_positive","Degree","If one exists"]
[]
["Intermediate Output"]
["Center Of Pressure Of Rocket At Zero Alpha",254.44440347542752,"Cpo","Length","For the Complete Rocket"]
["Normal Force Coefficient",9.438763148954207,"Cnao","Force/rad","For the Complete Rocket"]
["Cp Location Of The Body Alone At Zero Alpha",-196.33060393380092,"Cpob","Length","Body only"]
["Cutout Cp Location Of The Body Alone",175.07015103917786,"CLAb","Length","Body only"]
["Body Alone Normal Force Coefficient at Zero Alpha",1.1259771986510072,"Cnaob","Force/rad","Body only"]
["Cutout Normal Force Coefficient Of Body Alone",9.85696389191629,"Cnacb","Force/rad","Body only"]
[]
["Barrowman Calculations"]
["Diameter At Front Of Transition",13.818,"Df","Length","Limited to the nose diameter"]
["Tail Radius",6.909,"R","Length","At fin location"]
["Eff Nose Length",30.290000000000003,"Xn","Length"]
["Nose Force Coeff",2,"Cn_n","Force/rad"]
["Nose Moment",60.580000000000005,"M_n","Length"]
["Transition Moment",-0.8740228013489928,"Cn_t","Force/rad"]
["Transition Length",322.23848507400976,"X_t","Length"]
["Transition Moment",-281.6437834268416,"M_t","Length"]
["Fin Mid Cord Height",3.6000000000000014,"Cm_c","Length"]
["Fin Mid Cord",13.4892549831338,"Lf","Length"]
["Fin Aspect Ratio",0.34702898186749714,"AR_f","NA"]
["Body-Wing Interference Factor",0,"BW_I","NA","This factor was ignored in the original Barrowman assumptions"]
["Wing-Body Interference Factor",1.3470289818674972,"WB_I","NA"]
["Fin Aspect Factor",1.3470289818674972,"AF_f","NA"]
["Fin Geometry Factor",6.171200517733849,"GF_f","NA"]
["Fin Force Coeff",8.3127859503032,"Cn_f","Force/rad"]
["Eff Fin Length",315.50243902439024,"Xf","Length"]
["Fin Moment",2622.704242408343,"M_f","Length"]
["Total Moment",2401.640458981501,"M_r","Length"]
["Static Margin",-24.44440347542752,"Zs","Length"]
["Reference Area",149.96166651399082,"Sref","Length^2","Maximum of D or Dr"]
[]
["Cutout Method Calculations"]
["Nose Area",556.8654,"A_nose","Length^2","Approximate Area"]
["Transition Area",133.023,"A_transistion","Length^2","Approximate Area"]
["Body Area Forward Of Transition",3482.136,"A_body_fore","Length^2"]
["Body Area Aft Of Transition",0,"A_body_aft","Length^2"]
["Body Area",3482.136,"A_body","Length^2"]
["Total Cutout Area",4172.0244,"A_cutout","Length^2"]
["Nose Distance",40.3,"X_nose","Length","Approximate Length"]
["Transition Distance",322.2547245749635,"X_transistion","Length","Approximate Length"]
["Body Distance Forward Of Transition",191,"X_body_fore","Length"]
["Body Distance Aft Of Transition",328,"X_body_aft","Length"]
["Total Moment",730396.9418471354,"Cutout_moment","Length^3"]
[]
["Cm Model",null,null,null,"The Cm model was derived to contain all the sign information.  This was done to be expedient at making a single function for 3DOF simulations without having to keep track of Cp variations with alpha"]
["Input Alpha For Making Graph",11,"Input_Alpha","Degree"]
["Conversion Factor",0.017453292519943295,"d2r","rad/degree"]
["Angle Of Attack In Radians",0.19198621771937624,"Alpha","rad"]
["Moment Normalization Factor ",24.44440347542752,"MNF","Length","To normalize Cm to initial Cp location (eliminate the need for separate Cp calculations)"]
["Cm Model of the Finned Rocket"]
["Barrowman Force Coefficient Of Rocket",9.438763148954207,"Cna","Force/rad","Corrected Cna of rocket"]
["Corrected Cp Location Of Rocket",325.31100841391185,"Cp","Length","Corrected Cp of rocket"]
["Corrected Static Margin Of Rocket",-95.31100841391185,"Margin","Length","Static Margin of rocket"]
["Pitching Moment Of Rocket",-899.6180339068956,"Moment","Force-Length/rad","Pitching Moment of rocket"]
["Non-Linear Barrowman Pitching Moment Approximation",-7.022270494142013,"Cm_Est","Force","Corrected  Pitching Moment Coefficient Corrected for Sin Alpha"]
["Cm Model of the Body Alone"]
["Corrected Barrowman Force Coefficient Of Body",2.791927998238937,"Cnaob_","Force/rad","Corrected Cna of body alone"]
["Corrected Cp Location Of Body",-125.46399899531656,"Cpob_","Length","Corrected Cp of body alone"]
["Corrected Static Margin Of Body",355.46399899531656,"Margin_","Length","Static Margin of body alone"]
["Pitching Moment Of Body",992.4298911610017,"Moment_","Force-Length/rad","Pitching Moment of body alone"]
["Corrected Force Coefficient Of The Body",7.746744595524336,"Cmb","Force","Corrected  Pitching Moment Coefficient of body alone"]
["Angle of the body for Stability",{"$error":"#NUM!"},"Alpha_stable","Degree","If CLAb is behind Cg"]
["Body Displacement terms"]
["Maximum Value Of Cm Of The Rocket At Stall Alpha",-7.022270494142013,"Cm Peak","Force","Corrected for sin(alpha)"]
["Maximum Value Of Cm Of The Body At Stall Alpha",7.746744595524335,"Cmb Peak","Force","Corrected for sin(alpha)"]
["Final Cm Estimates"]
["Linear Barrowman Pitching Moment Approximation",-1.8010009140491048,"Cm_Barrow","Force","Linear approximation corrected for sin(alpha)"]
["Total Corrected Barrowman  Force Coefficient",-7.022270494142013,"Cm_Correct","Force","Of Total Rocket"]
["Location of the Corrected Cp",325.31100841391185,"Cp_Correct","Length","Of Total Rocket"]
[]
["Solution Of The Cubic Equations For The Point Of + Cm"]
["Term a",-132.65592890403084,"a","NA"]
["Term b",135.16787396910024,"b","NA"]
["Term c",19.637973149932947,"c_","NA"]
["Term d",-14.769015089666349,"d","NA"]
["Exact Zero Crossing of Cm",17.72972181581791,"Zero","NA","The Point of Positive region, If one exists"]
[]
["Graphs and Tables"]
[null,null,"Xn","Cg"]
["Nose length table","Conical",0.6666666666666666,0.6666666666666666]
[null,"Elliptical",0.3333333333333333,0.57]
[null,"Ogive",0.466,0.62]
[null,"Parabolic",0.5,0.6]
["Cm Graph",null,null,null,"Validation Data for Long Arcas (requires MNF to be set to 22 to match Arcas data)"]
["Angle","Corrected Cm","Barrowman Cm","Corrected Cp","Note: To achieve these fits a \"fudge factor\" of 1.152  was used to multiply Dr on the boat tail (assumed cause is bad air flow separation off the boat tail)"]
[1,-7.022270494142013,-1.8010009140491048,325.31100841391185,"Long Arcas Finned Data",{"$nulls":6},"Long Arcas Body Data"]
[1e-10,-1.647374942695274e-11,-1.647374942651589e-11,254.44440347607573,"Angle","Corrected","Barrowman","Data","R^2 ",null,null,"Angle","Corrected","Barrowman","Data","R^2 "]
[1.0000000001,-0.2084097778984135,-0.16472913075727405,260.926240402978,0,0,0,0,0.9913853111639532,null,null,0,0,0,0,0.9992099420771333]
[2.0000000001,-0.5040774538161122,-0.32940808340907873,267.4061028995612,2,-0.5040741573490963,-0.3294086217963213,-0.5,null,null,null,2,0.8503505354020999,0.6902278805443115,1.25]
[3.0000000001,-0.886793275383278,-0.4939866951676293,273.88201713693695,4,-1.3562270992076995,-0.6584159099284922,-1.5,null,null,null,4,1.9966652284783666,1.3796148247376339,2.5]
[4.0000000001,-1.3562413437717709,-0.658414833793475,280.3520104888951,6,-2.5535092678363682,-0.9866210196956112,-2.6,null,null,null,6,3.4021085301064935,2.0673209207803867,4]
[5.0000000001,-1.9119999527780163,-0.8226424128827399,286.81411213278426,8,-4.091286739793518,-1.3136240837275486,-4.3,null,null,null,8,5.029070321238636,2.752508304727544,5.5]
[6.0000000001,-2.5535420568412905,-0.9866194071239245,293.26635364984395,10,-5.963262341007796,-1.639026699161742,-5.6,null,null,null,10,6.839586700271621,3.434342181296791,7.25]
[7.0000000001,-3.280235867457618,-1.1502958675361024,299.7067696248039,12,-6.087000575052764,-1.9624324130357156,-6.2,null,null,null,12,8.795754910394162,4.111991840937988,9]
[8.0000000001,-4.09134557727654,-1.3136219366838624,306.13339824456835,14,-4.037228947773551,-2.2834472053039447,-4.4,null,null,null,14,10.860137986585904,4.784631671924789,11.1]
[9.0000000001,-4.986032211012476,-1.4765478638643672,312.5442818958036,16,-1.9163308862606705,-2.601679968890589,-1.5,null,null,null,16,12.996154860806872,5.451442166235356,13.2]
[10.0000000001,-5.963354602147869,-1.6390240202618984,318.9374677612458,18,0.24059038255967558,-2.9167429861932166,0.5,null,null,null,18,15.16845186848477,6.11161091799666,15.35]
[11.0000000001,-7.022270494044819,-1.8010009140652756,325.3110084117369,20,2.3999969643868457,-3.228252401456983,2.2,null,null,null,20,17.34325184722124,6.764333613275877,17.5]
[12.0000000001,-6.035351741860445,-1.9624292055435415,305.1775262401371]
[13.0000000001,-5.021583567635329,-2.1232597220753195,287.8118698982697]
[14.0000000001,-3.9855416923679243,-2.2834434731272752,272.66547008626105]
[15.0000000001,-2.9317368744192702,-2.442931665177102,259.33547428429523]
[16.000000000100002,-1.864603526794106,-2.6016757165765063,247.5191399297976]
[17.000000000100002,-0.7884886829120052,-2.7596272723496424,236.98432563485244]
[18.000000000100002,0.2923586611672615,-2.916738218922511,227.54982019752586]
[19.000000000100002,1.3737978012941365,-3.072960698778818,219.07188505149634]
[20.000000000100002,2.4518061510501834,-3.2282471250378384,211.43483553815503]
[21.000000000100002,3.5224886146652157,-3.3825501959498405,204.54431480792928]
[22.000000000100002,4.582086484249809,-3.535822909304657,198.32240367990255]
[23.000000000100002,5.626985840014898,-3.688018576749013,192.70400775879756]
[24.000000000100002,6.653725433656616,-3.839090838008253,187.6341493395352]
[25.000000000100002,7.659004036641621,-3.9889936750081283,183.06591081741578]
[26.000000000100002,8.63968723673056,-4.137681425892346,178.9588542523916]
[27.000000000100002,9.592813667721208,-4.285108798931617,175.27779368943044]
[28.000000000100002,10.515600659073145,-4.431230886319953,171.99183209107653]
[29.000000000100002,11.405449293787532,-4.576003177854018,169.07359905142826]
[30.000000000100002,12.25994886465494,-4.719381574491369,166.4986424798606]
[31.000000000100002,13.076880720744656,-4.8613224017834575,164.24493952084921]
[32.0000000001,13.854221497787035,-5.001782423179273,162.2925006572782]
[33.0000000001,14.590145727890553,-5.140718853195635,160.62304725804282]
[34.0000000001,15.283027825831988,-5.27808937045005,159.21974747294428]
[35.0000000001,15.931443450957294,-5.413852130552226,158.0669988263174]
[36.0000000001,16.534170245526532,-5.5479657788502745,157.1502484471315]
[37.0000000001,17.090187952124175,-5.6803894630277325,156.4558438304789]
[38.0000000001,17.59867791453061,-5.811082845547586,155.97090851897082]
[39.0000000001,18.05902196820754,-5.940006115939459,155.68323824147163]
[40.0000000001,18.470800728283734,-6.067120002926287,155.58121393698866]
[41.0000000001,18.833791284633463,-6.192385786386713,155.653728786461]
[42.0000000001,19.14796431531355,-6.315765309149617,155.89012692119263]
[43.0000000001,19.413480631261514,-6.4372209886171605,156.28015190839417]
[44.0000000001,19.630687166751635,-6.5567158282128135,156.8139034577054]
[45.0000000001,19.800112431654917,-6.67421342865086,157.48180106729995]