Angular momentum & RTT r X m M net , X = d t CV d V d r X - - PowerPoint PPT Presentation

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Angular momentum & RTT r X m M net , X = d t CV d V d r X - - PowerPoint PPT Presentation

Mar 08, 2024 416 likes •682 views

Angular momentum & RTT r X m M net , X = d t CV d V d r X m ( n ) + CS V rel V d A The vector sum of moments on the fluid = the rate of change of angular momentum in the cv + the net

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Angular momentum & RTT

⃗ Mnet,X = d dt ∭CV ⃗ rXm∧휌⃗ V d + ∬CS ⃗ rXm∧휌(⃗ Vrel ⋅ ⃗ n)⃗ V dA

The vector sum of moments on the fluid = the rate of change of angular momentum in the cv + the net flow of angular momentum through the cs

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Figure CC-0 Olivier Cleynen 2|25

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Figure CC-0 Olivier Cleynen 3|25

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If ⃗ rXm ∧ 휌(⃗ Vrel ⋅ ⃗ n)⃗ V is uniform across each inlet/outlet: ⃗ Mnet = d dt ∭CV ⃗ rXm ∧ 휌⃗ V d + ∑

  • ut

{ ⃗ rXm ∧ | ̇ m|⃗ V } − ∑

in

{ ⃗ rXm ∧ | ̇ m|⃗ V }

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Photo CC-by-sa by Commons User:Timhall 5|25

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Photo CC-by by JJ Harrison 6|25

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Drawing CC-by-sa U:Tosaka 7|25

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Photo by Andy Wolfe, US Army (public domain) 8|25

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Figure CC-by by Commons User:Tosaka 9|25

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Photo by Alex R. Lloyd, U.S. Air Force (public domain) 10|25

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∼ A worked-out example for the angular momentum equation ∼

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Falcon Heavy take-off on 2019-04-10

Photo by nasa (public domain) 12|25

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Falcon Heavy take-off on 2019-04-10

Photo CC-0 by SpaceX 13|25

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crs-8 landing on 2016-04-09

Photo CC-0 by SpaceX 14|25

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crs-8 landing on 2016-04-09

Photo CC-0 by SpaceX 15|25

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Figure CC-0 Olivier Cleynen 16|25

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Figure CC-0 Olivier Cleynen 17|25

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Figure CC-0 Olivier Cleynen 18|25

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Remove unsteady terms, split inlet/outlet ⃗ Mnet,X = d dt ∭CV ⃗ rXm ∧ 휌⃗ V d + ∬CS ⃗ rXm ∧ 휌(⃗ Vrel ⋅ ⃗ n)⃗ V dA = + ∬in ⃗ rXm ∧ 휌(⃗ Vrel ⋅ ⃗ n)⃗ V dA + ∬out ⃗ rXm ∧ 휌(⃗ Vrel ⋅ ⃗ n)⃗ V dA = + ∬in ⃗ rXm ∧ 휌V⟂ in⃗ V dA + ∬out ⃗ rXm ∧ 휌V⟂ out⃗ V dA

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Hoop #1: sign of V⟂ ⃗ Mnet,X = + ∬in ⃗ rXm ∧ 휌V⟂ in⃗ V dA + ∬out ⃗ rXm ∧ 휌V⟂ out⃗ V dA = − ∬in ⃗ rXm ∧ 휌|V⟂ in|⃗ V dA + ∬out ⃗ rXm ∧ 휌|V⟂ out|⃗ V dA

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Uniform velocity at inlets & outlets ⃗ Mnet,X = − ∬in ⃗ rXm ∧ 휌|V⟂ in|⃗ V dA + ∬out ⃗ rXm ∧ 휌|V⟂ out|⃗ V dA = − ⃗ rX-inlet ∧ 휌|V⟂ in|⃗ Vin Ain + ⃗ rX-outlet ∧ 휌|V⟂ out|⃗ Vout Aout = − 휌|V⟂ in| Ain ⃗ rX-inlet ∧ ⃗ Vin + 휌|V⟂ out| Aout ⃗ rX-outlet ∧ ⃗ Vout = − ̇ min ⃗ rX-inlet ∧ ⃗ Vin + ̇ mout ⃗ rX-outlet ∧ ⃗ Vout

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Vector cross-product: don’t panic ⃗ Mnet,X = − ̇ min ⃗ rX-inlet ∧ ⃗ Vin + ̇ mout ⃗ rX-outlet ∧ ⃗ Vout = − ̇ min ⃗ rX-inlet ∧ ⃗ Vin⟂r + ̇ mout ⃗ rX-outlet ∧ ⃗ Vout⟂r

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All right-hand-side vectors in same plane: we drop vectors Hoop #2: sign of V⟂r ⃗ Mnet,X = − ̇ min ⃗ rX-inlet ∧ ⃗ Vin⟂r + ̇ mout ⃗ rX-outlet ∧ ⃗ Vout⟂r Mnet,X = − ̇ min rX-inlet × (Vin⟂r) + ̇ mout rX-outlet × (Vout⟂r)

(V⟂r positive clockwise, negative anticlockwise)

Mnet,X = 0 + ̇ mout rX-outlet × (+ |Vout⟂r|)

(length in z-direction)

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Numbers! Mnet,X = 0 + ̇ mout rX-outlet × (+ |Vout⟂r|) = 0 + 2,5 × 40 × (+ |155 cos 20|) = +14,7 kN m ⃗ Mnet,X = ⎛ ⎜ ⎜ ⎝ +14,7 ⋅ 103 ⎞ ⎟ ⎟ ⎠ Moment exerted on fluid leaving the rocket (moment on rocket by fluid is opposite)

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Angular momentum & RTT

⃗ Mnet,X = d dt ∭CV ⃗ rXm∧휌⃗ V d + ∬CS ⃗ rXm∧휌(⃗ Vrel ⋅ ⃗ n)⃗ V dA

The vector sum of moments on the fluid = the rate of change of angular momentum in the cv + the net flow of angular momentum through the cs

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