magnus force in ressistance enviroment

5
1 Dynamics and resistance of enviorment ]] [ [ ]]) [ [ ]] [ [ 2 ( ) 2 2 ) ( ) ( ) ( 2 1 ( ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 ) ( ) ( ) ( 2 1 ( ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 ) ( ) ( ) ( 2 1 ( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 t Cos t Cos r t Sin h r dt dx dt d h dt dy dt d h dt dz dt z d dt y d dt x d dt z d m direction z t Cos t Sin r t Cos h r dt dz dt d r dt dx dt d h dt dy dt z d dt y d dt x d dt y d m direction y t Cos t Sin r t Cos h r dt dy dt d r dt dz dt d h dt dx dt z d dt y d dt x d dt x d m direction x ] [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 ( ] [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 ( ) ( ] [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 2 ( ] [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 2 ( ) ( ], [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 2 ( ] [ ]] [ [ ]]) [ [ ]] [ [ 2 )( 2 ( ) ( 2 2 2 2 2 2 2 2 2 2 2 2 t x t Cos t Sin r t Cos h r dt dz dt d r dt dx dt d h t y t Cos t Sin r t Cos h r dt dy dt d r dt dz dt d h dt d dt d J J dt d J t z t Cos t Sin r t Cos h r dt dy dt d h dt dz dt d h t x t Cos t Cos r t Sin h r dt dx dt d h dt dy dt d h dt d dt d J J dt d J t y t Cos t Cos r t hSin r dt dx dt d h dt dy dt d h t z t Cos t Sin r t Cos h r dt dz dt d h dt dx dt d r dt d dt d J J dt d J y x z x z y z y x

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1

Dynamics and resistance of enviorment

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]][[2()22

)()()(2

1(

]][[]])[[

]][[2)(2

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1(

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2

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2

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2

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2

2

2

2

2

2

2

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2

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2

2

2

2

tCostCosr

tSinhrdt

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yd

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xd

dt

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directionz

tCostSinr

tCoshrdt

dz

dt

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dx

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dt

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dt

xd

dt

ydm

directiony

tCostSinr

tCoshrdt

dy

dt

dr

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dx

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yd

dt

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dt

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directionx

][]][[]])[[]][[2)(2

(

][]][[]])[[]][[2)(2

()(

][]][[]])[[]][[2)(22

(

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2

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d

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tztCostSinrtCoshrdt

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txtCostCosrtSinhrdt

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tytCostCosrthSinrdt

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tztCostSinrtCoshrdt

dz

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dt

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yxz

xzy

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2

Magnus forces

];[][][2

1tAtxtvF iii

Scalar projected;

xF ]][[]])[[]][[2)()()()(2

1 222

22

2

22

2

2

tCostSinrthCosrdt

dx

dt

zd

dt

yd

dt

xd

]][[]])[[]][[2)()()()(2

1 222

22

2

22

2

2

tCostSinrthCosrdt

dy

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zd

dt

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dt

xdFy

]][[]])[[]][[2)()()()(2

1 222

22

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22

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tCostCosrthSinrdt

dz

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3,1]],[][][[)]([ ittrtvSinrtvmF iiiiiii ;

3

kdt

dx

dt

dh

dt

dy

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dhj

dt

dz

dt

dr

dt

dx

dt

dhi

dt

dy

dt

dr

dt

dz

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dhmAF ii )

22()

2()

2(

;

Crossections of resistance

]][[]])[[]][[2(

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2

2

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tCostCosrtSinhr

tCosSinrtCoshr

tCostSinrtCoshr

Ai

Force-(Magnus)

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2/12

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tCosCosrhSinr

tCosSinrhCosr

tCosSinrhCosr

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Det

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Scalar projected Magnus momentus

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(

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(

]];[[]])[[]][[2)(2

(

2

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tCosCosrtSinhrdt

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tCosSinrtCoshrdt

dz

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tCostSinrtCoshrdt

dy

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Mz

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Momentum and Oiler equation as function of time

;2

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222

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tytxmCosCos

rmJ

tztxmCosCos

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tztymCosCos

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Final form of equations, projected in 6 directions.

k

dt

dx dt

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2

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txtCostSinrthCosrdt

tdz

dt

tdr

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tytCostSinrthCosrdt

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tdtztxmCosCos

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tztymCosCos

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tztCostSinrthCosrdt

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hrm

tytxmCosCos

rm

dt

tdtztxmCosCos

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tytCostCosrthSinrdt

tdx

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tztCostSinrthCosrdt

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tCostCosrthSinr

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dt

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txdm

osezMoment

oseyMoment

osexMoment

directionz

pravacy

directionx

5

Those equations are solved and animated in Mathematica 10 softwer, contribution of this eq is that now we can engage felsh trajectory even where is not translation at all. So you can have a curve path, even if you don’t have initial translation speed. Trkulja Goran-Fast and furios