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공학수학 | 미분방정식의 해를 구하는 방법의 비교 |

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작성자 cemtool 작성일14-06-11 12:18 조회9,394회 댓글12건

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발사체의 운동방정식은 다음과 같다.
여기에서 r(t)는 위치벡터, v(t)는 속도벡터, m은 발사체의 질량,  는 공기저항으로 인한 힘, g 는 중력가속도, y n 는 y 축 방향의 단위벡터 (unit vector)이다. 2 차원 문제의 경우, xy 평면에서 y 축을 높이, x 축을 수평거리로 하여 발사체의 운동을 나타낼 수 있다. 그러면, 발사체의 운동방정식에서 속도벡터 v(t)와 위치벡터 r(t)를 다음과 같이 나타낼 수 있다.
일반적으로 공기저항은 속도의 제곱에 비례한다고 가정한다. 이 힘은 운동방향의 반대쪽으로 작용하기 때문에, 다음과 같은 방법으로 표현할 수 있다.
여기서, 계수 a 의 값은 발사체의 모양과 면적, 그리고 공기밀도에 따라 달라진다. 실제 발사 체에서는 계수의 값이 속도에 따라 변하지만, 이 예제에서는 상수라고 가정한다. 발사체 운동방정식의 해를 구하는 프로그램을 작성하기 전에, 몇 가지 사항에 대한 결정을 내 려야 한다. 첫째로, 발사체가 지면에 도달하면, 프로그램이 중단되어야 한다. 이것은 y(t) = 0 (단, t > 0 )인 경우에 해당된다. 발사체가 지면에 도달하는 시각은 미지수이므로, 해를 구하는 알고리즘에서 이 값을 결정해야 한다. 초기위치는 이고, 최종위치는  이다. 여기서  는 수평 이동거리이고,  는 발사체가 지면에 도달하는 시 간이다. 다음으로, 수치해(numerical solution)를 구하기 위한 간격크기를 결정해야 한다. 공기저항을 무 시하고 수치 알고리즘을 적용하면, 적당한 간격크기를 결정할 때 필요한 정보를 얻을 수 있다. 공기저항을 무시하면, 발사체 운동방정식은
가 되고, 이 된다. 문제를 간단히 하기 위해, 초기위치는 r(0)=[0,0]으로 한다. 그러면, 해를 나타내는 궤적 r(t)는 포물선이 된다. 이 방정식을 x(t)와 y(t)에 대하여 풀면,
가 된다. 위치에 대한 초기조건 x(0) = 0, y(0) = 0으로부터, 계수 이다. 발 사체가 초기속도 v(0) = ? 수평축 사이의 각도 로 발사되거나 던져진다고 가정하면, 속도에 대한 초기조건을 다음과 같이 정의할 수 있다.
\vec{v}(0)=v_x(0)\hat{i}+v_y(0)\hat{j}=v_0cos\theta\hat{i}+v_0sin\theta\hat{j}
X축 방향으로는 힘이 작용하지 않으므로, x(t)=v_xt 가 되고, 이로부터 c_3=v_0cos\theta 가 된 다. 관계식 로부터  를 구할 수 있다. 가속도 관계식으로부터  가 되므로,  이다. 따라서, 발사체의 궤적에 대한 최종 결과는
이고, 지면 도달시간  
이 된다. 공기저항의 영향을 고려하면 비행거리가 짧아지기 때문에, 이 식으로 구한 지면 도달 시간은 발사체의 최대 비행시간이 된다. 가능한 간격크기의 값으로는 다음 식으로부터 구할 수 있다.
여기서, M은 공기저항을 고려하지 않은 경우의 점의 개수이다. 이상의 내용을 바탕으로, 2차원 평면에서 발사체의 궤적을 구하고, 그래프를 그리는 CEMTool 프로그램을 작성하여라. 작성한 프로그램을 사용하여, 다음과 같은 상황에 대한 발사체의 궤적 을 구하여라.
v(0)=[40,40][m/s] a =0.0052[1/m] M = 50[개] h = 0.2[sec]
단, 이번 예제에서는 초기시간 t_0=0, 초기위치 r(0) = [0,0]으로 한다.
ex6_11.cem
function;
T,X,Last <> f,t0,m,h,x0,drag,g
 
/*
 [T,X,Last]=clrk4pj(f,t0,m,h,x0,drag,g) Solve Xdot=f(T,X) INPUTS: function f; m points; h step size; t0,x0 initial values
         drag drag coefficient,g gravitational acceleration
 Output: T abscissa, X solution of differential equation system
 Last is index of last nonzero value of X(:,2)*/
 
T=zeros(m,1);           // T array m rows (mx1)X=zeros(m,length(x0));  // X array m rows, 4 columns (mx4)T(1)=t0;
X(1;:)=x0;
//There are m-1 steps and m points maximum
for(I=1;I<=m-1;I++){    tI=T(I);
    xI=X(I;:);
    eval(["k1=h*" f "(tI,xI,drag,g)"]);
    eval(["k2=h*" f "(tI+h/2,xI+k1/2,drag,g)"]);
    eval(["k3=h*" f "(tI+h/2,xI+k2/2,drag,g)"]);
    eval(["k4=h*" f "(tI+h,xI+k3,drag,g)"]);
    X(I+1;:)=xI+(k1+2*k2+2*k3+k4)/6;
    T(I+1)=t0 + h*I;
    if(X(I+1;2) < 0){        break;  // Quit upon impact
    }}Last=(I+1);     // Index of last nonzero value
ex6_11.cem
function;
xdot <> t,x,drag,g
 
/*
 xdot = clprojf(t,x) Define equations of motion with drag
 x''=-drag*v*x'     x1=x, xdot1=x'=x3  xdot3=x''
 y''=-drag*v*y'-g   x2=y, xdot2=y'=x4  xdot4=y''
 OUTPUT: xdot is column matrix of solutions for time point t
*/
 
V=sqrt(x(3)^2 + x(4)^2);  
xdot(1)=x(3);                 // represents x'
xdot(2)=x(4);                 // represents y'
xdot(3)=-drag*V*x(3);         // represents x''
xdot(4)=-drag*V*x(4)-g;       // represents y''
ex6_11.cem
/*
 Example 6.11 Projectile motion
*/
 
// Solve the equations of flight for a projectile
//  with air resistance and plot trajectory
// INPUTS:  drag, initial velocity [Vox Voy] = [X0(1),X0(2)],
//          M number of points in interval, h step size.
// Fixed Parameter Values g, t(0), r(0) 
g=9.81;         // Gravity  9.81 m/sec-sect0=0.0;         // Initial time
r=[0 0];        // Initial [x y]// Input Parameters
drag=input("drag= ");
// drag=0.0 is ideal casev=input("Enter initial velocity v=[Vx(0) Vy(0)] = ");
M=input("Number of points= ");
h=input("Step in time= ");
 
Vx=v(1);        // Initial x velocity
Vy=v(2);        // Initial y velocity
X0=[r(1) r(2) Vx Vy];
// Let X(:,1)=x; X(:,2)=y; X(:,3)=x'; X(:,4)=y' state vector
// Create system matrix and solution
//  Last is the index of the last nonzero value of X(:,2), i.e. y[T,X,Last]=clrk4pj("clprojf",t0,M,h,X0,drag,g);
 
msgprint("Strike ESC key to plot results")pause// Plot range versus height
plot(X(1:Last;1),X(1:Last;2))title(["Projectile-Tfinal is ",num2str(T(Last)),"sec","; Drag= ",num2str(drag)])xtitle("Range in meters")ytitle("Height in meters")
 
 

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theresa님의 댓글

theresa 작성일

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ronghui13님의 댓글

ronghui13 작성일

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darryl님의 댓글

darryl 작성일

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ronald님의 댓글

ronald 작성일

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antonio님의 댓글

antonio 작성일

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Coach Outlet http://www.coachbags.us/
Off White Shoes http://www.offwhitecanada.ca/
Nike http://www.nikeairmaxoutlet.us/
Balenciaga http://www.balenciagashoes.ca/
Ralph Lauren http://www.polo-ralphlauren.ca/
Hydro Flask 32 OZ http://www.hydroflask.us.org/
Salomon Speedcross 4 http://www.salomons.us/
Suicoke Shoes http://www.suicoke.us/
Mont Blanc Refills http://www.mont-blancpens.us/
Longchamp Backpack http://www.longchampbackpack.us/
Fjallraven Bags http://www.fjallraven.in.net/
Christian Louboutin http://www.redbottomsshoes.us/
Golden Goose Sale http://www.goldengoosesneakers.ca/
Puma Suede http://www.pumacanada.ca/
Birkenstock Sandals http://www.birkenstock-canada.ca/
Pandora Bracelet http://www.pandoracanada.ca/
Nike Kyrie http://www.kyrie5-shoes.us/
Adidas Outlet http://www.adidas-outlet.ca/
Adidas NMD http://www.adidasboost.us/
Ralph Lauren Outlet http://www.ralph-laurencanada.ca/
Coach Purses http://www.coach-bags.us/
Pandora Canada http://www.pandoracanadas.ca/
Kyrie 5 http://www.kyrie5shoes.us/
Jordan 11 Space Jam http://www.jordan11.org.uk/
Off White Hoodie http://www.off-white.us.org/
Pandora Charms http://www.pandoracanadarings.ca/
Christian Louboutin http://www.louboutintrainers.co.uk/
Karen Millen Dresses http://www.karen-millen.us/
Yeezy Shoes http://www.yeezy-boost.co.uk/
Salomon Gtx http://www.salomon-boots.us/
Birkenstock http://www.birkenstocksandalssale.us/
hampion Clothing http://www.championhoodie.co.uk/
Bape Shirt http://www.bape.org.uk/
Christian Louboutin Shoes http://www.christianlouboutinredbottoms.us/
Balenciaga Shoes http://www.balenciagas.us/
Ferragamo Canada http://www.ferragamocanada.ca/
Nike Air Max http://www.airmax2019.us/
Supreme T Shirt http://www.supremeshirt.us/
Fjallraven Kanken Backpack http://www.fjallravenkanken.us.com/
Kate Spade Outlet http://www.katespadebags.ca/
Bape http://www.bapecanada.ca/
Converse All Star http://www.converses.us/
Adidas NMD http://www.adidasnmdwomens.us/
Nike Air Max 270 http://www.nikeairmax-uk.co.uk/
Fitflop Shoes http://www.fitflopshoes.us/
Golden Goose Sneakers Sale http://www.goldengooses.us.org/
Michael Kors Outlet http://www.michaelkors-bags.us/
Pandora Rings http://www.pandorajewelrystore.us/
Yeezy 350 http://www.yeezytrainers.co.uk/
Air Jordan http://www.jordanshoesoutlet.us/
Golden Goose http://www.goldengoose-shoes.us/
Swarovski Jewelry http://www.swarovskijewelry.us/
Birkenstock Sandals http://www.birkenstock-sale.us/
Adidas http://www.adidas-trainersuk.org.uk/
Five Finger Shoes http://www.vibram.us.org/
Vans Old Skool http://www.vansslipon.us/
Fitflop Shoes http://www.fitflop.org.uk/
Bape http://www.bapehoodie.co.uk/
Pandora Charms http://www.pandorarings.us/
Louboutin http://www.louboutinredbottoms.us/
Fjallraven Kanken Backpack http://www.fjallraven-kanken.us.org/
Balenciagas http://www.balenciaga-sneakers.us.org/
CK Underwear http://www.calvinklein.org.uk/
Converse UK http://www.converse-uk.co.uk/
KD Shoes http://www.kd11.ca/
Jordan Shoes http://www.jordanshoes.ca/
Off White Hoodie http://www.offwhites.us/
Karen Millen Sale http://www.karenmillendresses.co.uk/
Coach Canada http://www.coachbags.ca/
Giuseppe http://www.giuseppeshoes.us/
Bape Hoodie http://www.bape-hoodie.us/
Bape Shorts http://www.bapeshirt.us/
Ultra Boost Uncaged http://www.adidasultraboost.us.com/
Supreme Sweatshirt http://www.supremes.us.org/
Yeezy http://www.yeezyboost350v2.us.com/
Nike Shox http://www.nikeshoxwomen.us/
Supreme T Shirt http://www.supremeclothings.us.com/
Fila Trainers http://www.filashoes.co.uk/
MCM Handbags http://www.mcmbags.us/
Nike Shoes http://www.nikeuptempo.us/
Vans Shoes http://www.vanscanada.ca/
Louboutin Shoes http://www.redbottoms.us.com/
Fitflop Shoes http://www.fitflopsandals.us/
Adidas Yeezy http://www.adidasyeezy.ca/
Ferragamo http://www.ferragamo.in.net/
Yeezy Boost 350 http://www.yeezy-supply.us/
Nike Air Max 97 http://www.airmax-720.us/
Longchamp Bags http://www.longchampbags-outlet.us/
Air Max 97 http://www.airmax-2019.us/
Harden Shoes http://www.hardenvol3.us/
Karen Millen http://www.karen-millen.co.uk/
Pandora http://www.pandoras.us/
New Balance http://www.newbalanceshoes.us/
Supreme Clothing http://www.supreme-canada.ca/
Lauren Ralph Lauren http://www.ralphlauren-polo.us/
Asics Running Shoes http://www.asicscanada.ca/
Birkenstock Sandals http://www.birkenstock-sandals.co.uk/
Birkenstock Sale http://www.birkenstockcanadasandals.ca/
Adidas Canada http://www.adidascanadashoes.ca/
Aike Air Max 270 http://www.nikeairmax720.us/
Birkenstock UK http://www.birkenstockuk.org.uk/
Skechers Shoes http://www.skechersshoes-outlet.us/
Balenciaga http://www.balenciagasneakers.us/
Curry 5 Shoes http://www.curry5.ca/
Champion Clothing http://www.championhoodie.us.com/
Supreme Clothing http://www.supremehoodie.co.uk/
Lebron 16 Shoes http://www.lebron16.ca/
Kate Spade http://www.katespadecanada.ca/
Fjallraven Bags http://www.fjallraven.us.org/
ED Hardy Dress http://www.edhardyclothing.us/
Fjallraven Kanken Sale http://www.fjallraven.org.uk/
Goyard Wallet http://www.goyard-bags.us/

katherine님의 댓글

katherine 작성일

http://www.superdryhoodie.co.uk/ Superdry Hoodie
http://www.pandorajewelrycom.us/ Pandora Earrings
http://www.yezzys.org.uk/ Yezzys
http://www.pumaschuhedamen.de/ Puma Sneaker
http://www.vibramshoes.us.com/ Vibram
http://www.saucony-shoes.us/ Saucony Running Shoes
http://www.puma-shoes.co.uk/ Puma Shoes
http://www.riverislandshoes.co.uk/ River Island Shoes
http://www.balenciagaus.us/ Balenciaga Shoes
http://www.filaschuhe.de/ Fila Winterschuhe
http://www.pradapurse.us/ Prada Handbags
http://www.underarmourtracksuit.co.uk/ Under Armour UK
http://www.superdryjackets.co.uk/ Superdry
http://www.conversefemme.fr/ Converse
http://www.pandora-outlet.us/ Pandora Rings
http://www.skecherssandals.co.uk/ Skechers Sandals
http://www.adidas-yeezy.fr/ Yeezy
http://www.reeboks.us/ Reebok Outlet
http://www.ultra-boost.us/ Adidas Shoes
http://www.air-max.fr/ Shaussure Nike
http://www.tommyhilfiger-outlet.us/ Tommy Hilfiger
http://www.underarmour-outlet.ca/ Under Armour Shoes
http://www.yeezymafia.us/ Adidas Yeezy
http://www.prada-purse.us/ Prada Handbags
http://www.vansshoes-outlet.us/ Vans
http://www.yeezychaussureadidas.fr/ Yeezy Boost 350 V2
http://www.swarovskijewelryoutlet.us/ Swarovski Rings
http://www.pandorajewelryofficialsite.us/ Pandora Official Site
http://www.anellipandora.eu/ Collane Pandora
http://www.offwhiteshirt.us/ Nike Off White
http://www.timberlandbootsstore.us/ Timberland Shoes
http://www.offwhite-hoodie.us/ Nike Off White
http://www.yeezys.eu/ Yeezys
http://www.vans-shoes.ca/ Vans
http://www.yeezyboost350.fr/ Yeezy
http://www.valentinosandals.us/ Valentino Sneakers
http://www.pandoraringsuk.co.uk/ Pandora Charms
http://www.ralphlauren-poloshirts.us/ Ralph Lauren
http://www.thenorthface-jacke.de/ The North Face Rucksack
http://www.puma-canada.ca/ Puma Canada
http://www.uggslipperscanada.ca/ UGG Outlet
http://www.vans-canada.ca/ Vans Outlet
http://www.scarpe-hogan.eu/ Hogan
http://www.ralphlaurenshirt.org.uk/ Ralph Lauren UK
http://www.off-whiteshoes.us/ Off White Hoodie
http://www.pandora-charms.ca/ Pandora Charms
http://www.pandoracharms-outlet.us/ Pandora Bracelet
http://www.offwhiteshoes.org.uk/ Off White T Shirt
http://www.bottes-ugg.fr/ Boots UGG
http://www.adidas-yeezy.de/ Adidas Jacke
http://www.poloralph-lauren.us/ Polo Ralph Lauren
http://www.suicoke-sandals.us/ Suicoke Womens
http://www.tomsboots.us/ Toms Boots
http://www.scarpehogan.eu/ Hogan Donne
http://www.the-northface.org.uk/ The North Face
http://www.salvatoreferragamo.org.uk/ Ferragamo Shoes
http://www.ugg-uk.co.uk/ UGG
http://www.supremehoodie.org.uk/ Supreme
http://www.salomonshoes.org.uk/ Salomon Running Shoes
http://www.northface-jacke.de/ North Face Jacke
http://www.riverislandsale.co.uk/ River Island Clothes
http://www.superdryclothing.us/ Superdry Jacket
http://www.swarovskicrystalcanada.ca/ Swarovski Bracelet
http://www.pumarihannafenty.us/ Puma Fenty
http://www.adidasyeezyschuhe.de/ Adidas NMD
http://www.reebok.org.uk/ Reebok Sneakers
http://www.supremeclothing-uk.co.uk/ Supreme Jacket
http://www.skechers-boots.us/ Skechers Sneakers
http://www.pandorajewelryoutlet.us.org/ Pandora Bracelet
http://www.sperry.us.org/ Sperry Outlet
http://www.chaussurenikeofficiel.fr/ Nike Air Max
http://www.stevemaddensneakers.us/ Steve Madden
http://www.skechersoutletcanada.ca/ Skechers Canada
http://www.pradabags.org.uk/ Prada Purse
http://www.timberlandshoes.co.uk/ Timberland Sale
http://www.riverislandclothes.us/ River Island Dresses
http://www.prada-canada.ca/ Prada Purse
http://www.scarpeadidas.eu/ Scarpe Adidas
http://www.bijouxpandora2019.fr/ Pandora Bracelet
http://www.michaelkorsborse.eu/ Michael Kors Borse Outlet
http://www.salvatore-ferragamo.co.uk/ Ferragamo Belt
http://www.vansshoes.org.uk/ Vans Trainers
http://www.supremehoodiestore.us/ Supreme Sweatshirt
http://www.doudounethenorthface.fr/ Veste North Face
http://www.thenorth-face.fr/ Doudoune The North Face
http://www.toryburchhandbags.us/ Tory Burch Crossbody
http://www.redbottomschristianlouboutin.us/ Christian Louboutin Shoes
http://www.adidasschuheherren.de/ Adidas Yeezy
http://www.sperryoutlet.us/ Sperry
http://www.puma.org.uk/ Puma Shoes
http://www.riverislanddresses.co.uk/ River Island Coats
http://www.underarmourtrainers.co.uk/ Under Armour Outlet
http://www.scarpesaucony.eu/ Saucony Uomo
http://www.nikeschuheairmax.de/ Nike Air Max
http://www.superdryjacket.us/ Superdry Clothing
http://www.borsemichaelkors.eu/ Michael Kors Borse Outlet
http://www.toryburch-outlet.us/ Tory Burch Purse
http://www.uggslippersoutlet.us/ UGGS On Sale
http://www.supremeclothingcanada.ca/ Supreme Canada
http://www.scarpenikeairmax.eu/ Nike
http://www.pumafentyshoes.us/ Puma Shoes
http://www.salomonshoes.us/ Salomon Hiking Boots
http://www.converseallstar.fr/ Converse Noir
http://www.reebok-classic.us/ Reebok Shoes
http://www.scarpevans.eu/ Vans Scarpe
http://www.schuhe-nike.de/ Nike Air Max
http://www.supremeclothingshop.us/ Supreme Coats
http://www.pandora-canada.ca/ Pandora Rings
http://www.tommyhilfigerdresses.us/ Tommy Hilfiger Jacket
http://www.uggbootsstore.us/ UGGS For Women
http://www.vibramshoes.us/ Vibram 5 Fingers
http://www.bapehoodieoutlet.us/ Bape Shirt
http://www.the-northface.us/ North Face Jacket
http://www.underarmour.us.com/ Under Armour Sweatshirt
http://www.swarovskicrystaloutlet.us/ Swarovski Bracelet

jennifer님의 댓글

jennifer 작성일

http://www.ralphlaurendresses.ca/ Ralph Lauren Canada
http://www.uggbootssaleuk.co.uk/ Ugg Boots
http://www.calvin-klein.us.org/ Calvin Klien Store
http://www.boseheadphones.us.org/ Bose Headphones
http://www.columbiajackets.us.com/ Columbia Jacket
http://www.jewelry-pandora.us.com/ Pandora Rings
http://www.goldengoose-sneakers.co.uk/ Golden Goose Superstar
http://www.adidasclothing.us.org/ Adidas Tracksuit
http://www.jordan11.us.org/ Nike Jordan
http://www.riverislanddresses.org.uk/ River Island Shoes
http://www.beatssolo3.ca/ Beats Solo 3
http://www.katespadebagscanada.ca/ Kate Spade Purses
http://www.suicokesandals.ca/ Suicoke Moto Sandals
http://www.fjallravenbackpack.us.org/ Fjallraven
http://www.kedssneakers.us/ Keds Slip Ons
http://www.asicsshoes.us.com/ Asics Gel Nimbus
http://www.katespadepurse.us.com/ Kate Spade Purses
http://www.asicsrunningshoes.us.com/ Asics Gel Nimbus
http://www.goldengoose-sneakers.org.uk/ Golden Goose Sneakers
http://www.skechersshoesoutlet.us.com/ Skechers Shoes
http://www.pandoracanadacharmsjewelry.ca/ Pandora Jewelry Canada
http://www.coach-bags.co.uk/ Coach Bags
http://www.montblanc-pens.org.uk/ Mont Blanc Pens
http://www.coachhandbags.us.com/ Coach Outlet
http://www.ralphlaurencanadaoutlet.ca/ Polo Ralph Lauren Canada
http://www.canadagoose-canada.ca/ Canada Goose Outlet
http://www.airmax-720.us.com/ Air Max 720
http://www.longchampsbags.us/ Longchamp Bags
http://www.ecco-canada.ca/ Ecco Shoes
http://www.pandorajewelry-store.us/ Pandora Charms
http://www.conversetrainers.org.uk/ Converse Trainers
http://www.aldoshoes.us.com/ Aldo
http://www.christian-louboutin.ca/ Christian Louboutin
http://www.hogan.us.com/ Hogan Shoes
http://www.gstar-jeans.us/ G Star Hoodie
http://www.nikeoutletcanada.ca/ Air Max 270
http://www.fjallravenkankenbackpack.co.uk/ Fjallraven
http://www.pandoracanada-charms.ca/ Pandora Earrings
http://www.juicycouture.us.org/ Juicy Couture Bags
http://www.adidasnmd.us/ Adidas Nmds
http://www.burberryshirt.co.uk/ Burberry Shirt
http://www.eccocanadashoes.ca/ Ecco
http://www.coachbagsoutlet.ca/ Coach Outlet
http://www.keds-shoes.us/ Keds Sneakers
http://www.nikeoffwhite.co.uk/ Off White
http://www.christian-louboutin.us.org/ Christian Louboutin
http://www.beatssolo3wireless.us/ Beats Wireless Headphones
http://www.katespade-bags.us.com/ Kate Spade Purses
http://www.armaniclothing.us/ Armani Shoes
http://www.brooksshoes.ca/ Brooks Ghost
http://www.monclers.co.uk/ Moncler Coat
http://www.pandorajewelrycharms.ca/ Pandora Charms
http://www.ecco-shoes.us.org/ Ecco Shoes
http://www.monclerjackets.us.com/ Moncler
http://www.air-max.org.uk/ Air Max 270
http://www.ferragamo-shoes.org.uk/ Salvatore Ferragamo Belt
http://www.balenciaga-trainers.co.uk/ Balenciagas
http://www.hoganshoesoutlet.co.uk/ Hogan Shoes
http://www.jordan-shoes.org.uk/ Jordan Shoes
http://www.supreme-clothing.ca/ Supreme T Shirt
http://www.geox-shoes.us/ Geox Kids Shoes
http://www.louboutinshoes.ca/ Christian Louboutin Shoes
http://www.adidas.us.org/ Adidas Yeezy
http://www.pandorarings.us.org/ Pandora Rings
http://www.brooksrunningshoes.co.uk/ Brooks Running Shoes
http://www.birkenstocksandalssale.ca/ Birkenstock Men
http://www.niketrainersoutlet.org.uk/ Nike Trainers
http://www.golden-goosesneakers.us.org/ Golden Goose Outlet
http://www.uggsboots.us/ Ugg Slippers
http://www.fendishirt.us/ Fendi
http://www.hydro-flask.ca/ Hydro Flask Sale
http://www.armani.us.org/ Armani Shoes
http://www.air-max.us.org/ Air Max
http://www.adidas-yeezy.us.org/ Yeezy Boost 350
http://www.balenciagashoes.ca/ Balenciaga Sneakers
http://www.birkenstock.us.org/ Birkenstock Sandals
http://www.adidastrainersoutlet.org.uk/ Adidas Uk
http://www.northface-jacket.ca/ North Face Backpack
http://www.pandoracharmsonline.us/ Pandora Jewelry
http://www.pumashoescanada.ca/ Puma
http://www.ecco.us.org/ Ecco
http://www.hugo-boss.us.org/ Hugo Boss T Shirt
http://www.clarksshoescanada.ca/ Clarks Sandals
http://www.pandoracharmsofficial.us/ Pandora
http://www.thenorthfacejackets.ca/ North Face
http://www.ferragamoshoesoutlet.us.org/ Ferragamo Shoes
http://www.offwhiteshoes.us.org/ Off White Jacket
http://www.asics-trainers.co.uk/ Asics Running Shoes
http://www.christianlouboutinredbottomshoes.us/ Louboutin Shoes
http://www.hoganshoes.ca/ Hogan Shoes
http://www.karenmillen-dresses.org.uk/ Karen Millen Dresses
http://www.nikeshox.us.org/ Nike Outlet
http://www.merrellshoes.us.com/ Merrell
http://www.uggsbootsforwomen.us.com/ Ugg Slippers
http://www.monclercanada.ca/ Moncler
http://www.asicsrunningshoes.ca/ Asics Shoes
http://www.brooksshoes.org.uk/ Brooks Trainers
http://www.northfacecanada.ca/ North Face Jacket
http://www.belstaffjackets.org.uk/ Belstaff Jackets
http://www.newbalanceshoes.us.com/ New Balance Outlet
http://www.woolrichparka.us/ Woolrich Coat
http://www.curry7.us.com/ Curry 6
http://www.northfacecoat.co.uk/ North Face Coat
http://www.newbalance.us.org/ New Balance Trainers
http://www.parajumperjacket.us.com/ Parajumper Jacket
http://www.beatsheadphones.us.org/ Beats
http://www.curry7.ca/ Curry 7 Shoes
http://www.bape-hoodie.co.uk/ Bape
http://www.birkenstock-canada.ca/ Birkenstock Shoes
http://www.fjallravenkankenuk.org.uk/ Fjallraven Sale
http://www.mcmhandbags.us.com/ Mcm Purse
http://www.air-max270.us.org/ Air Max 270
http://www.pandoracanada-jewelry.ca/ Pandora Rings
http://www.lecoqsportifshoes.us/ Le Coq Sportif Tennis Shoes
http://www.uggsforwomen.us.org/ Ugg Slippers
http://www.montblancpens.ca/ Mont Blanc Canada
http://www.longchamp-bags.ca/ Longchamp Outlet
http://www.aldo.us.com/ Aldo Shoes Women
http://www.jimmychoo-shoes.us/ Jimmy Choo Flats
http://www.michaelkorshandbags.ca/ Michael Kors
http://www.adidassuperstar.us.org/ Adidas Shoes
http://www.brooksshoes.us.org/ Brooks Ghost
http://www.hydroflasksale.us.org/ Hydro Flask Stickers
http://www.adidasshoes.ca/ Adidas Stan Smith
http://www.belstaff.org.uk/ Belstaff Leather Jacket
http://www.adidas-nmd.ca/ Nmd R1
http://www.uggs-onsale.us/ Ugg Slippers
http://www.cartierlovering.ca/ Cartier Ring
http://www.calvinklein-canada.ca/ Calvin Klein
http://www.calvin-klein.ca/ Calvin Klein Underwear
http://www.bape-clothing.us.com/ Bape Clothing
http://www.bosesoundlink.ca/ Bose Speakers
http://www.brooksshoes.us.com/ Brooks Shoes
http://www.conversecanada.ca/ Converse Canada
http://www.offwhite-hoodie.ca/ Off White Belt
http://www.parajumper.us.com/ Parajumper Coat
http://www.ed-hardy.us/ Ed Hardy Shoes
http://www.champion-hoodie.us.com/ Champion Sweatshirt
http://www.hydroflasksale.ca/ Hydro Flask Sale
http://www.katespadehandbags.co.uk/ Kate Spade Purses
http://www.karenmillendressesoutlet.co.uk/ Karen Millen Sale
http://www.golden-goose.ca/ Golden Goose Shoes
http://www.lebron17.ca/ Lebrons 14
http://www.bose-headphones.ca/ Bose Bluetooth Headphones
http://www.vansoldskool.ca/ Vans Canada
http://www.birkenstock-sandals.us.org/ Birkenstock Sale
http://www.northfacejackets.us.com/ North Face Vest
http://www.dieselunderwear.us/ Diesel Sale
http://www.redbottomshoesforwomen.us/ Red Bottoms
http://www.bosesoundlink.org.uk/ Bose Headphones
http://www.fjallravenkankencanada.ca/ Fjallraven

linda님의 댓글

linda 작성일

http://www.asicsshoes.ca/ Asics Running Shoes
http://www.yeezyadidas.co.uk/ Yeezy
http://www.aldoshoesoutlet.us/ Aldo Sandals
http://www.asics-shoes.ca/ Asics Running Shoes
http://www.salomons.co.uk/ Salomon Speedcross
http://www.montblancs.ca/ Mont Blanc
http://www.pandorajewelryca.ca/ Pandora Rings
http://www.pandoracom.us.com/ Pandora Rings
http://www.toms.us.org/ Toms Womens Shoes
http://www.balenciagacanada.ca/ Balenciaga Canada
http://www.hydro-flask.us.org/ Hydro Flask Lids
http://www.merrell.us.org/ Merrell Hiking Boots
http://www.nike-outlet.ca/ Nike Air Max
http://www.vansshoes.ca/ Vans Shoes
http://www.ralphlaurenoutlet.ca/ Ralph Lauren
http://www.pumacanada.ca/ Puma Suede
http://www.champion-hoodie.co.uk/ Champion Sweatshirt
http://www.ralphlaurenhoodie.org.uk/ Ralph Lauren Polo Shirts
http://www.bosewirelessheadphones.us.com/ Bose Wireless Earbuds
http://www.nikeoffwhiteshoes.us.com/ Off White
http://www.goldengoosesale.us.com/ Golden Goose Superstar
http://www.monclerjacketsoutlet.co.uk/ Moncler Coat
http://www.jordanshoes.ca/ Jordan 11 Shoes
http://www.nikehuarache.ca/ Nike Shoes
http://www.ferragamoshoessale.us/ Ferragamo Outlet
http://www.michaelkorscanadaoutlet.ca/ Michael Kors Canada
http://www.wirelessbeats.us/ Wireless Beats
http://www.vansoldskool.co.uk/ Vans Shoes
http://www.beatsstudio3.ca/ Powerbeats 3
http://www.ralphlaurendresses.us.com/ Polo Ralph Lauren
http://www.adidas-shoes.us.org/ Adidas Shoes For Men
http://www.puma-shoes.us.org/ Puma Fenty
http://www.ultraboost.ca/ Yeezy Boost
http://www.birkenstock-shoes.ca/ Birkenstock Arizona
http://www.saucony.us.org/ Saucony Shoes
http://www.ralphlauren-dresses.us/ Ralph Lauren Polo Shirts
http://www.bosewirelessheadphones.co.uk/ Bose Wireless Headphones
http://www.karenmillen.us.org/ Karen Millen Outlet
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