The Mathematics of Flat Parachutes

Transcription

1 he Mathematic of Flat aachute epaed By: J.R. Bohm NR #78048 CR #S680 Date: Septembe 4, 004

2 1.0 Intoduction all ocketee know, the pevailing device fo model ocket ecovey i by fa the ubiquitou paachute. aachute fo model ockety pupoe ae available in a boad ange of ize and a ainbow of colo, and ae made fom a vaiety of diffeent mateial. Howeve, the modele may chooe to make hi own paachute, ometime to ave the cot of commecial paachute, but moe often becaue hi poject equie a non-tandad ize. hi i often the cae fo competition o pecial payload model, whee a paticula non-tandad diamete i needed fo a duation event o a cetain ate of decent. ehap the mot complete hape fo a paachute i a hemiphee. Many ocketee will ecall that thi wa the hape of the paachute ued in the Space ogam, uccefully deliveing manned payload to et in the ocean fo ubequent ea ecovey opeation. While hemipheical paachute function vey well, they can be complex to make, a the hape i 3 dimenional. Making a hemipheical hape equie the modele to cut piece of mateial into pecial cuved egmental hape, called goe, which when fitted togethe will fom the hemiphee. Fotunately hemipheical paachute aen t eally needed in mot modele application. Mot commonly available paachute ae in fact ceated fom tandad two-dimenional (i.e.: flat) geometic figue uch a hexagon o octagon. When billowed fom the weight of a model, thee paachute do a vey good job appoximating the pefomance of a hemipheical paachute. Moe impotantly, becaue thee paachute ae deived fom flat figue, thei natue make them eaie fo the modele to wok with. hi pape exploe the polygonal geomety that chaacteize the conventional twodimenional (i.e.: flat) paachute figue that mot ocketee ue a thei ecovey method today. We will develop a geneal olution that will pemit u to calculate the ize (diamete) of the paachute needed to delive a equied minimum canopy aea. hi olution will wok pefectly well fo a hexagonal o octagonal paachute, o fo any othe egula polygonal haped paachute. Mathematic of aachute 1

3 .0 aachute Conideation Befoe examining the geomety aociated with flat paachute, the fit quetion that need to be anweed i - How big of a paachute do I need? he ate of decent will be dependent on the aea of the paachute; once we know the equied minimum aea, a little geomety will tell u the diamete (ize) we need to make the paachute. In hi book, Model Rocket Deign and Contuction, nd Edition, im Van Milligan (pogee Component) povide a ueful fomula fo calculating the minimum paachute aea needed fo a afe decent peed fo a given model ocket ma. he fomula i given a: gm ρc V d Whee: g the acceleation due to gavity, 9.81 m/ at ea level m the ma of the ocket (popellant conumed) ρ the denity of ai at ea level (15 g/m 3 ) Cd the coefficient of dag of the paachute etimated to be 0.75 fo a ound canopy V the decent velocity of the ocket, 11 to 14 ft/ (3.35 m/ to 4.6 m/) being conideed a afe decent peed. With thi decent ate equation, and a good calculato, one can eadily find the needed minimum paachute aea fo a paticula model o miion. o detemine it ize (diamete), we mut geneate an expeion that elate aea to ize, and we mut take into conideation the hape we chooe fo the paachute, a hape and diamete will dictate available uface aea. Mathematic of aachute

4 3.0 aachute Geomety Let incibe an n-ided polygon inide a cicle. an incibed polygon, it vetice will be tangent to the cicle, and the ditance fom it cente to any vetex will be, the adiu of the cicle. Figue 3-1 illutate what an incibed polygon look like; in thi cae we have choen to incibe a egula octagon inide the cicle. Figue 3-1: Incibed olygon In thi illutation two line ae hown, each oiginating fom the cente of the cicle and extending to a vetex; togethe they fom an iocele tiangle, the tiangle having two identical ide and a bae the length of the polygon ide. If imila line wee dawn to each emaining vetex, we would eadily ee that the octagon i made up of 8 identical iocele tiangle. We can extend thi pinciple geneally and ay that an n-ided polygon i made up of n identical tiangle, each tiangle coeponding to one of the polygon ide. We can alo ee that the aea of the polygon i jut the um of the aea of the tiangle that compie it; hee, in the cae of thi paticula polygon, it aea i equal to 8 time the aea of one tiangle, o: O 8, whee i the aea of the tiangle and O i the aea of the octagon. Genealizing fo any n-ided polygon, ou expeion fo aea i: n With thi concept now etablihed, a little geomety will pemit u to calculate the dimenion of ou paachute. o do thi, we imply need to etablih the aea of the elemental tiangle that make up the polygon, and then ue the elationhip above to calculate the total aea. Mathematic of aachute 3

5 Let extact thi tiangle fom the polygon and take a cloe look at it: θ/ h Figue 3-: he Elemental iangle We know that the aea of thi tiangle i: h h We will now manipulate thi elationhip o that it can be expeed entiely in tem of. o do thi, we will ue ome tigonomety. eaoned ealie, an n-ided polygon i made up of n identical, elemental tiangle. he angle ubtended at the tiangle apex i θ; ince thee ae n tiangle making up the polygon, the value of θ mut be 360 /n. Fo ou pupoe, we ae inteeted in the angle between and h; ince thi i half of θ (emembe, we ae dealing with an iocele tiangle), it value mut be 360 /n, o 180 /n once educed. We can deive the following expeion fom the chaacteitic of the tiangle: / and θ 180 ; h co ; h co Mathematic of aachute 4

6 Recalling that the aea of the tiangle i and h and get: h, we can then make the ubtitution fo co n co hee i a tigonometic identity that can be ued to futhe educe thi expeion, a follow: Fo an angle α, in α inα coα Hee, if we et α, then in α co n pplying thi identity to ou expeion fo, we get: co Subtitute thi eult into ou expeion fo become, and the aea of ou n-ided polygon n We now have a geneal expeion in tem of the adiu,, which we can ue to calculate the aea of any n-ided polygon. ypically, we meaue the diamete of a paachute a oppoed to it adiu, o with a little moe algeba, we can tanfom the eult into one expeed in tem of diamete. Recall that d, whee d i the diamete of the paachute/cicle: hen d n nd 8 Fo pactical pupoe, we would calculate the equied paachute aea fo a paticula model fom the decent ate equation. Once we know the aea, we can ue the expeion fom above to detemine the equied diamete, depending on the type of the paachute we intend to make (hexagonal, octagonal, o othe). Reaanging ou equation, we can olve fo d: Mathematic of aachute 5

7 d n We can complete the execie definitively by ubtituting the decent ate equation fo paachute aea in the place of ; then we get: d nρc V d 4gm Let wok out ome pactical example. Fo a hexagonal paachute, we know n 6. So plugging 6 in fo n, we get: d Fo an octagonal paachute, we know n8: d Why hould it make ene fo the coefficient (the multiplie) of the paachute aea to be malle fo an octagon? Well, if we ecall Figue 3-1, it can be eadily een that the aea of an octagon will be lage (cove moe of the cicle) than that of a hexagon fo the ame adiu. So to aive at the ame paachute aea, the diamete of a hexagonal paachute will need to be lage than that of an octagonal one. Mathematic of aachute 6

8 4.0 Othe Way of Looking at the Same hing In the peviou ection, we deived an expeion that elated paachute diamete to the paachute hape and aea. Finding the diamete i impotant, a thi paamete i the mot ueful one fo laying out the paachute. Howeve, with ome futhe algebaic manipulation we can e-wok the eult we found to expe the aea of the paachute in tem of the ize of it ide, and alo in tem of the ditance meaued fom one ide aco to an adjacent ide. hee e-woked expeion povide an altenate way of calculating the aea of a known paachute. Let go back and look at the Elemental iangle: θ/ h Figue 4-1: he Elemental iangle Recall that θ 180 hen Recall that 180 co Subtituting fo : Mathematic of aachute 7

9 co 4 in Reducing give: co 4 4 tan n n 4 tan hi eult give u the aea of the paachute expeed in tem of the length of a ide. Let e-expe the elation in tem of the ditance fom ide to adjacent ide let call thi ditance D. hen D h ; h 180 co h co D h D co Subtituting thi value fo into the equation fo, we get: D in D co 4 co 4 co D 4 tan n nd 4 tan hi eult give u the aea of the paachute expeed in tem of the ditance meaued fom ide to adjacent ide. Note that thi expeion i only valid fo polygon with an even numbe of ide. Mathematic of aachute 8

10 5.0 Concluion & Finding hi pape demontate eveal way to detemine the aea of a paachute depending on the paamete available. It povide a fomula fo detemining the minimum diamete needed to povide a paachute canopy of pecibed aea, a calculation that i impotant if the modele intend to make the paachute himelf. he following ummaize the finding of thi analyi: 5.1 Minimum aachute ea fo a Given Model gm ρc V d (Refeence Model Rocket Deign and Contuction, nd Edition, im Van Milligan (pogee Component)). 5. Diamete of a aachute, given it ea d n Fo a Hexagonal paachute, d Fo an Octagonal paachute, d ea of a aachute, given it Diamete nd 8 ; n the numbe of ide of the paachute. Fo a Hexagonal paachute, ( ) d Fo an Octagonal paachute, ( ) d H O 5.4 ea of a aachute, given the length of a Side n ; the length of a ide. 4 tan Mathematic of aachute 9

11 Fo a Hexagonal paachute, (.5981) Fo an Octagonal paachute, ( 4.884) H O 5.5 ea of a aachute, given the ditance between oppoite ide nd 4 tan ; D the ditance between adjacent ide, and n mut be even. Fo a Hexagonal paachute, ( ) D Fo an Octagonal paachute, ( 0.884) D H O Mathematic of aachute 10

12 ppendix : n Explicit Deivation fo the Hexagonal aachute We can co check the coectne of the geneal expeion by looking at the chaacteitic of a hexagonal paachute. Fo thi paachute, thee will be 6 ide and it will be compied of 6 elemental tiangle. Unique in thi cae i the fact that all of the inteio angle of each tiangle ae of the ame value. Since two of the tiangle ide ae known to be equal to, and with all angle equal, we can afely eaon that the thid ide, the bae, mut alo be equal to. hi lead to the concluion that the elemental tiangle in thi cae i an equilateal tiangle. Figue -1 illutate thi elemental tiangle: 30 h / 60 Figue -1: Elemental iangle fom a Hexagon Uing the ythagoa heoem, we can daw the following elationhip: + h ; 4 h h 3 h H 6 ; but 4 d 3 3 d H d Mathematic of aachute 11

13 nd d 8 H 3 3 H H hi i the ame eult we obtained ealie when we et n6 in the geneal fomula. Mathematic of aachute 1