The Recovery of Rockets.

Transcription

1 . Iroducio The Recoery of Roces. Jørge Frac, Daish Aaeur Roce Club The priary objecie of ay recoery sequece, of course, is he reducio of he roces elociy o soe alue for which adequae ipac surial ay be obaied. The parachue is ejeced close o apogee due o he relaie low elociy, o aoid daage o he recoery syse ad he roce iself. Typically a cross-ype parachue cosisig of fie squares is used because i is easy o fabricae. The legh of all he riggig lies are equal o wo ies he side i oe square. Fig. Cross-ype parachue A ier is use as he sole ehod o iiiae he recoery. The ier is se o a predeeried ie based o he rajecory calculaios. A ae-off he ier is iiiaed by use of a acceleraio swich. The ier for he recoery syse is self-coaied wih is ow power supply. This desig assures a reliable ad safey lauch procedure. Whe he roce is oued o he laucher, he oly requiree regardig he recoery syse is o swich o he ier circui. Uforuaely he locaio of he roce afer ouchdow ca be difficul, because he roce ca drif subsaially whe he parachue is deployed a apogee. A sall pilo parachue or sreaer ca be used isead a apogee o guide he roce fas dow o aoid excessie drif. The laer he ai parachue is deployed o achiee he ecessary ipac elociy. Oe drawbac is he subsaially ore coplex recoery syse requirig wo iers. Jørge Frac The Recoery of Roces page of

2 The recoery syse diide he roce wih a ieral separaor i wo pars respeciely he egie ad he payload secio. The wo pars are cobied o he parachue wih a lie. The os ipora par is he payload secio which carries he isrues o coduc he experies. The lie bewee he parachue ad he egie is loges o achiee ouchdow of he lesser ipora egie firs. This arragee reduce he ipac elociy addiioal for he payload secio. Parachue Payload secio Egie Fig. Roce raellig erically dowward i a parachue Usig differe lie leghs reduce he probabiliy for ipac bewee he egie ad payload secio whe he roce is diided. The lies are aached a he eds of he wo secios. This assure aily logiudial sress a separaio ad ouchdow o he secios. Jørge Frac The Recoery of Roces page of

3 . Moio wih air resisace equal o ² The roce i parachue (cosidered as a paricle of ass ) haig weigh of agiude is raellig erically dowward wih he elociy. The air resisace acig o he syse has agiude proporioal o he square of he isaaeous elociy, ad opposed o i. -² Fig. 3 Forces acig o a roce raellig erically dowward i a parachue The by Newo s secod law a = d d = which ay be wrie d = d Jørge Frac The Recoery of Roces page 3 of

4 Assuig g ad cosa ad ha < we ge by iegraig d = d ah = ah g ah = The fucio ah (x) is he ierse hyperbolic age of x. The elociy a ay subseque isa is he g = ah + ah, < Siilar we ge g = coh + coh, > As icreases he hyperbolic age fucio approaches he lii resulig i he liiig alue for he elociy erial = This is called he erial elociy, he lii o which he elociy of he parachue eds. I ca also be obaied a oce fro he equaio of oio. A he oe he air resisace balace graiy he acceleraio a of he parachue is zero. Iserig i he equaio of oio he sae resul for he erial elociy is obaied. The disace a ay subseque isa is dy = d Jørge Frac The Recoery of Roces page 4 of

5 Jørge Frac The Recoery of Roces page 5 of dy d g = + ah ah dy g d y = + ah ah y g g = + l cosh ah y g = + < l cosh ah cosh ah, Siilar we ge y g = + > l sih coh sih coh, Fially he acceleraio a ay subseque isa is a d d = a g g = + ah ah a g g = + < ah ah, Siilar we ge a g g = + > coh coh,

6 3. Moio wih air resisace equal o A a relaiely low elociy, he air resisace ay be approxiaed by assuig ha i is proporioal o he isaaeous elociy, ad opposed o i. - Fig. 4 Forces acig o a roce raellig erically dowward i a parachue The equaio i his case is a = d d = which ay be wrie d = d Assuig g ad cosa, we ge by iegraig Jørge Frac The Recoery of Roces page 6 of

7 d = d l( ) = l = = ( ) exp The elociy a ay subseque isa is he = + exp The erial elociy is erial = because as icreases he expoeial fucio approaches he lii, as is liewise eide fro he equaio of oio. The disace a ay subseque isa is dy d dy = d = + exp y dy = + y = exp d exp y = + exp Fially he acceleraio a ay subseque isa is a d = d Jørge Frac The Recoery of Roces page 7 of

8 a = exp a = g exp 4. Deeriaio of he coefficies ad The alue is equal o where ρ A C D = ½ρAC D : Desiy of he air. : Cross secioal area 5 a² of he cross-ype parachue, where a is he legh of a square. : Drag coefficie of he parachue (diesio -less). The alue is equal o where η K = ηk : Coefficie of iscosiy of he air. : Coefficie relaed o he shape of he parachue. The coefficie η depeds o he fricioal force bewee differe layers of he air oig wih differe elociies. The coefficies C D ad K ca be deeried heoreical ad experieal. The heoreical approach is coplex because ay paraeers ifluece o he size of he coefficies as: Shape of he parachue. Porosiy of he caopy aerial. Legh of riggig lies. Ierferece fro he roce aached o he parachue. Ierferece fro he riggig lies. A pracical approach is o derie forulas base o experieal daa o obai he coefficies. The os coo experieal echique o deerie he coefficies is by he use of a wid uel. Pracical liiaios i he size of he wid uel ad o aoid ierferece fro he wid uel walls requires ypically a scale odel of he parachue o perfor he easurees. Aoher experieal approach is he erical drop es fro a buildig or ower where he acual parachue ca be used. To aoid daage o he roce iself a siilar weigh ca be used as duy. The coefficies ca be deeried by easurig he ie ae by he parachue o fall a specified disace whe released fro res. I he wo cases Jørge Frac The Recoery of Roces page 8 of

9 we hae he followig relaioship bewee disace ad ie whe he iiial elociy is zero: y = lcosh g y = g exp Uforuaely he coefficies ad ca o be deeried direcly i he aboe equaios by iserig he easured disace ad ie. I he laer equaio he expoeial fucio ca be wrie as a ifiie series as follows: x x x x exp( x) = + x ! 3! 4! 5! For sall alues of x his series coerge ery fas ad herefore oly he firs four ers i he series are isered i he equaio, we ge 3 y = g + 3 ( 3 ) 6 y = g g = y g Uforuaely he aboe equaios is ore suied for objecs wih a fixed geoery durig he whole fall. Whe he parachue is released fro res i aes soe ie before i ge fully iflaed causig iaccurae deeriaio of coefficies. A ore accurae ehod is o easure oer a disace where he parachue rael wih he erial elociy, we hae y = erial ad i he wo cases we ge = y = y Jørge Frac The Recoery of Roces page 9 of

10 5. Exaple I his exaple we will use he forulas where he air resisace is proporioal o he square of he isaaeous elociy. Firs we us deerie he cosa for he parachue. The firs proble you ypically ecouer usig he erical drop es is he dead for a exreely accurae iig deice o achiee a reasoable accurae deeriaio of. Usig a large drop disace y ad a saller ass ha he acual prolog he drop ie aoidig a exreely accurae iig deice. The roce hae a burou ass icludig he parachue equal o = 9.5g I he erical drop es he roce is replaced by a sadbag wih a reduced weigh o prolog he drop ie. The ass of he sadbag icludig he parachue is equal o =.g Fially you aoid o cushio he roce a he ed of is fall o aoid daage. A buildig is used wih a drop disace of y = 4.65 Acually we dropped he parachue fro a poi 3 higher o assure ha he parachue has achieed he erial elociy a he begiig of he disace of ieres. Seeral erical drop ess resuled i a aerage drop ie equal o = 3.6s The cosa ca ow be foud, we ge = = y 3.6s. g 9.8 / s =.65g / 4.65 The drag coefficie of he parachue is. 65g / CD = = ρa 5. g / 5. 3 = 85. where ρ is he desiy of he air ad A is he cross secioal area of he cross parachue. The erial elociy of he roce is erial = = 9. 5g 9. 8 / s. 65g / = / s Use a scaled parachue based o he oe fro he experie o adjus he erial elociy. This is possible because a scaled parachue do o ifluece o he size of he experieal foud drag coefficie aoidig furher experies. A parachue wice he size used i he experie reduce he erial elociy by Jørge Frac The Recoery of Roces page of

11 . 7 We assue ha he parachue oe wih he erial elociy hroughou he disace of ieres, which is erial = =. g 9. 8 / s. 65g / = 4.7 / s To chec his assupio we will fid he elociy of he parachue whe i eers he disace of ieres. We firs deerie he ie is aes o rael 3 = g cosh exp y =. g. 65g / 9. 8 / s cosh 3. 65g / exp. g =. s Reeber he iiial elociy is zero because we drop i ad furher we eglec he ie i aes he parachue o fully deploy. Fially he elociy ca be obaied afer raellig 3, we ge = ah g =. g 9. 8 / s. 65g / 9. 8 / s ah 65.s. g /. g = 4. / s Apparely he parachue hae reached a elociy ery close o he erial elociy, so 3 is adequae o esure his. Ayway we always force he parachue o be fully deployed before we release i, o creae a sceario as close o he assupios aboe. Las we will deerie he ouchdow elociy of he payload. Due o differe leghs of he lies respeciely o he egie ad payload, he payload secio is 4 aboe groud a egie ouchdow. This ie he iiial elociy is greaer ha he erial elociy ad he ass is equal o = roce egie = 95. g 6. 9g =.6g As before we calculae firs he ie i aes o rael he disace, i his case 4, we ge Jørge Frac The Recoery of Roces page of

12 = g y sih sih coh exp coh =. 45s Fially he elociy ca be obaied afer raellig 4, we ge g = coh + coh = 7. 3 / s This siple arragee reduce he ouchdow elociy for he payload secio fro /s o 7.3/s. The erial elociy of he payload is erial = =. 6g 9. 8 / s. 65g / = 6. 3 / s Fially I will o recoed o use he heory based o air resisace proporioal o he isaaeous elociy i he aboe experie. Especially he use of a reduced ass i he experie o prolog he drop ie resul i a ery iaccurae deeriaio of he roces erial elociy. Iserig he alues resul i a erial elociy of he roce equal o erial roce y = exp erial = 9. 5g = / s. g 3. 6s 6. Refereces. Theoreical Mechaics, Murray R. Spiegel. Schau s Oulie Series I Sciece. McGraw-Hill Boo Copay.. Fudaeal Uiersiy Physics, Volue I, Mechaics, Marcelo Aloso ad Edward J. Fi. Addiso-Wesley Publishig Copay. 3. A Iroducio To The Ifiiesial Calculus, G. W. Cau. Oxford Uiersiy Press. 4. Maheaical Hadboo, Murray R. Spiegel. Schau s Oulie Series I Maheaics. McGraw-Hill Boo Copay. 5. Maheaische Forelsalug, Hochschulaschebücher, Bad 3, Karl Roa. Bibliographisches Isiu AG, Mahei. Jørge Frac The Recoery of Roces page of