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Equation of motion of a variable mass system2

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Solo Hermelin

The document discusses equations of motion for a variable mass system using Reynolds' transport theorem, exploring different approaches such as Lagrangian methods. It outlines key concepts including mass flow rates, inertial properties, and the mathematical foundation involving calculus and vector fields. The content is structured with a comprehensive table of contents that guides through various equations and their applications in fluid dynamics.

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1 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH SOLO HERMELIN http://www.solohermelin.com
2 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM • Simplified Particle Approach (see Power Point Presentation) The equations of motion can be developed using At a given time t the system has V(t) – system volume. m (t) – system mass. S (t) – system boundary surface. • Reynolds’ Transport Theorem Approach (this Power Point Presentation) • Lagrangian Approach (see Power Point Presentation)
3 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH TABLE OF CONTENT OSBORNE REYNOLDS 1842-1912 • Reynolds’ Transport Theorem • Inertial Velocity and Acceleration • Mass Equation • First Moment of Inertia Relative to a Reference Point O • Linear Momentum Equation • Force Equation • Absolute Angular Momentum Relative to a Reference Point O • Moment Relative to a Reference Point O • External Forces and Moments of the System • Summary of Equation of Motion for a Variable Mass System
4 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH TABLE OF CONTENT (Continue) • Kinetic Energy of the System • Quasi-Lagrangian Equations • Energy Flow • References • Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors) • Summary of the Equations of Motion of a Variable Mass System MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O
5 REYNOLDS’ TRANSPORT THEOREM -any system of coordinatesOxyz - any continuous and differentiable functions in ( ) ( )trtr OO ,,, ,,  ηχ ( )tandrO,  ( )trO ,,  ρ - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS   =⋅− ,ρ sd  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t( )trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O,∇ - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,  −= - differential of any vector , in O frame. O td d ζ  ζ 
6 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM Start with LEIBNIZ THEOREM from CALCULUS: ( ) ( )    ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ       −+= ∫∫ )),(()),(( ),( ),(:: )( )( )( )( ∂ ∂ and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4): ( ) ( ) ( ) ( ) ∫∫∫∫∫       ⋅∇+∇⋅+=⋅+ → = tv OSOOOSGAUSS Opotolative dsofMovement thetodueChage tS OS tv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d ,,,,)4( intRe )( ,      χχ ∂ χ∂ χ ∂ χ∂ χ This is REYNOLDS’ TRANSPORT THEOREM OsborneReynolds 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM Johann Carl Friederich Gauss 1777-1855
7 0,,,,,  =−=⇒= OSOSOOS VVVVV ( ) ∫∫∫∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+=⋅+= )( ,,,, )4( , )()()( tv OOOO O GAUSS O tStv OO tv FFFF vdVV t GAUSS sdVvd t vd td d     χχ ∂ χ∂ χ ∂ χ∂ χ SOLO REYNOLDS’ TRANSPORT THEOREM (CONTINUE -1) ( ) ∫∫∫∫∫∫∫∫ ⋅∇=⋅== )( ,,)4(, )()( )( tv OOGAUSSO tStv F FFF vdV GAUSS sdVvd td d td tvd  χ               =⋅∇ → td tvd tv V tv OO )( )( 1 lim0)( ,,  EULER 1755 ρχ == &,, OOS VV  ( )∫∫∫∫∫∫∫∫ ∫∫∫       ⋅∇+=⋅+=== )( ,, )( , )( )( )( 0 tV OO tS O tV tV FFF F vdV t sdVvd t dv td d td tmd  ρ ∂ ρ∂ ρ ∂ ρ∂ ρ or, since this is true for any attached volume vF(t) ( ) 0,,,,,, =⋅∇+∇⋅+=⋅∇+ OOOOOO VV t V t  ρρ ∂ ρ∂ ρ ∂ ρ∂ CONSERVATION OF MASS EQUATION EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM CASE 1 (Control Volume vF attached to the Fluid )OOS VV ,,  = CASE 2 (Control Volume vF attached to the Fluid and )1=χOOS VV ,,  = CASE 3 (Control Volume vF attached to the Fluid and )ρχ =OOS VV ,,  =
8 Define ( ) ( ) ( )trtrtr OOO ,,:, ,,,  ηρχ = ( )∫∫∫∫∫∫∫∫ ⋅+         += )( , )()( tS OS tv OO tv sdVvd tt vd td d    ηρ ∂ ρ∂ η ∂ η∂ ρηρ We have (for any volume v(t) bounded by the surface S(t)) But, from CONSERVATION OF MASS ( ) ( )OOOO V t V t ,,,, 0  ρη ∂ ρ∂ ηρ ∂ ρ∂ ⋅∇−=⇒=⋅∇+ CASE 4: Flow Equations SOLO We have ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫∫∫∫∫= ∫∫∫∫∫ ∫∫∫∫∫∫∫∫ ⋅−− ⋅+         ⋅∇+∇⋅−         ∇⋅+= ⋅+         ⋅∇−= + + )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OSO tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sdVVvd tD D sdVvdVVV t sdVvdV t vd td d       ρηρ η ρηρηηρη ∂ η∂ ρ ρηρηρ ∂ η∂ ρη where 0: ,, =⋅∇+= η ∂ η∂η  OO OO V ttD D is the MATERIAL DERIVATIVE, RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -2)
9 CASE 4: Flow Equations (continue – 1) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -3) ( )[ ]∫∫∫∫∫∫∫∫ ⋅−−= )( ,, )()( tS OSO tv OO tv sdVVvd tD D vd td d    ρηρ η ρη v(t) ds m> 0 . m< 0 . 0, <⋅ sdV S  dm ∑= + N j ext iji fdfd 1 int  ( )tSW 2openS openiS O 0, >⋅ sdV S  OCr , Oflowir ,  C Copenir ,  Cr,  SV,  Oiopenr ,  openiV  OV,  Or,  Cflowir ,  flowiV  Ozˆ Oyˆ Oxˆ openflowS VVV  −=, - mass flow rate through the element . mdsdVS   =⋅− ,ρ sd  SOSO VVV ,,, :  =− - flow velocity relative to .sd  There are no sources or sinks in the volume v (t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2,…). The surface S(t) can be divided in: • Sw(t) the impermeable wall through which the fluid can not escape .( )0,  =sV • Sopen i(t) the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m ( ) ( ) ( )∑+= openings i iopenW tStStS ( ) ( ) ( )( ) ∑=∑ ∫∫ ⋅−+∫∫         ⋅−=∫∫ ⋅− openings i ii openings i tS S tS S tS S msdVsdVsdV iopenW   ηρηρηρη ˆ , 0 , )( ,
10 CASE 4: Flow Equations (continue – 2) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -3) ( )∫∫∫∫∫∫∫∫ ⋅+= .. , .... SC O O VCVC O sdVvd td d vd tD D   ρηρηρ η - mass flow rate through the element . mdsdVS   =⋅− ,ρ sd  SO VV ,,  = - flow velocity relative to O and .sd  CONTROL VOLUME WITH FIXED SHAPE C.V.( ) 0,  =OS V
11 CASE 4: Flow Equations (continue – 2) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -4) ( )∫∫∫∫∫∫∫∫ ⋅−+= )( , )()( tS S tv OO tv sdVvd tD D vd td d    ρηρ η ρη v(t) ds m> 0 . m< 0 . 0, <⋅ sdV S  dm ∑= + N j ext iji fdfd 1 int  ( )tSW 2openS openiS O 0, >⋅ sdV S  OCr , Oflowir ,  C Copenir ,  Cr,  SV,  Oiopenr ,  openiV  OV,  Or,  Cflowir ,  flowiV  Ozˆ Oyˆ Oxˆ openflowS VVV  −=, ( )( ) ∫∫ ⋅−= tS Si iopen sdVm   , ρ ( )( )       = ≠ ∫∫ ⋅− = 00 0:ˆ , i i i tS S i m m m sdV iopen      ρη η where i openings i i tv OO tv mvd tD D vd td d     ∑∫∫∫∫∫∫ += ηρ η ρη ˆ )()( ( )∑ ∫∫∫∫∫∫∫∫ ⋅−+= openings i S S tv OO tv iopen sdVvd tD D vd td d    , )()( ρηρ η ρη We can write or or REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 Mean Vector of on the opening iη  iopenS Mass Rate entering through opening iopenS or ( )∫∫∫∫∫∫∫∫ ⋅+= .. , .... SC O O VCVC O sdVvd td d vd tD D   ρηρηρ η Table of Content
12 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Inertial Velocity and Acceleration v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or,  R  - Position of the mass element dm relative to I. I td Rd V   = - Velocity of the mass element dm relative to I. II td Rd td Vd a 2 2   == - Acceleration of the mass element dm relative to I. Mass Equation Use the REYNOLDS’ TRANSPORT THEOREM with 01 =→= O tD Dη η   ( ) ( ) ( ) ( ) ∑∑ ∫∫∫∫ =⋅−=== openings i iflow openings i S S REYNOLDS tvtm msdVvd td d md td d tm iopen    , ρρ The change of the mass of the system is due to the flow through the openings in the surface S (t). ( )∫∫ ⋅−= iopenS Siflow sdVm   ,: ρ Table of Content Table of Content
13 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH First Moment of Inertia Relative to a Reference Point O v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or, Define the First Moment of Inertia Relative to O ( ) ( ) ( ) mRmdRmdRRc O tmtm OO  −=−= ∫∫:, -Position vector of the instantaneous Mass Center (Centroid) of the system, relative to I ( ) ( ) ( ) ( ) ( ) ( ) 0: =−→== ∫ ∫ ∫ ∫ tm C tm tv tv C mdRR m mdR vd vdR tR    ρ ρ Therefore ( ) ( )( ) ( ) mrmRRmRmdRmdRRc OCOC tm O tm OO ,, :  =−=−=−= ∫ ∫ For O = C we have: ( ) ( )( ) ( ) 0:,  =−=−=−= ∫ ∫ mRRmRmdRmdRRc CC tm C tm CC Table of Content
14 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Linear Momentum Equation v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or,  The Linear Moment of the mass enclosed by v (t) is defined as ( ) ( ) ∫∫∫∫∫∫ == tv I tv I vd tD RD vdVP ρρ   , : vdVmdVPd II ρ,, :  == The Linear Momentum, of the differential mass dm = ρdv, with initial velocity ,is defined as I I tD RD VV   == , : Use the REYNOLDS’ TRANSPORT THEOREM with R  =η and O = I ( ) ( ) ( ) ( ) ( ) ∑ ∫∫∫∫∫∫∫∫∫∫∫ −+= ⋅−−=== openings i iflowiopenC V C tS S mR tv REYNOLDS tv I tv I mRmR td Rd m sdVRvdR td d vd tD RD vdVP C C              ˆ : ,, ρρρρ ( )       = ≠ ⋅− = ∫∫ 00 0: ˆ , iflow iflow iflow S S iopen mif mif m sdVR R iopen      ρ where: is the mean position vector of the flow and of the opening on iopenSiopenR ˆ
15 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Linear Momentum Equation (continue – 1) v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or, We can write the Linear Momentum ( ) ( )( ) ∑      −−= ∑ ∫∫ ⋅−−−=∫= openings i iflowCiopenC openings i S SCC tm mRRmV sdVRRmVmdVP iopen    ˆ : ,ρ We can write ( ) ∫= tm mdVP  : ( ) ( )( )∑ ∫∫ ⋅−−+−−+−= openings i S SCOOOOC iopen sdVRRRRmVmVmV  ,ρ ( ) ( ) ( ) ( )( )∑ ∫∫ ⋅−−−+∑ ∫∫ ⋅−−+−= openings i S SOO openings i S SOCOC iopeniopen sdVRRmVsdVRRmVV  ,, ρρ ( ) ( ) ∑∑       −−+−+−= openings i iflowOiopenO td cd m openings i iflowOCOC mRRmVmRRmVV I O           ˆ , or ( )( ) ∑      −−+= ∑ ∫∫ ⋅−−−+= openings i iflowOiopenO I O openings i S SOO I O mRRmV td cd sdVRRmV td cd P iopen       ˆ, , , ρ Table of Content
16 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation Applying the 2nd Newton’s Law to the differential mass dm = ρdv, we obtain: int2 2 fdfdmd td Rd md td Vd ext II   +== where ext fd  - External forces acting on the differential mass dm intfd  - Internal forces that surroundings exercises on the differential mass dm From the 3rd Newton’s Law the internal force that particle j applies on particle i is of equal magnitude but of opposite direction to the force that particle i applies on particle j : jiij fdfd intint  −= Therefore ( ) 00 int 1 1 int  =→= ∫∑∑ →∞ = ≠ = tv NN i N ij j ij fdfd
17 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 1) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫ =++== ext l l tvtv ext tv I tm I FFfdfddv tD VD dm tD VD   intρ discrete forces exerting by the surrounding at point jR  Use the REYNOLDS’ TRANSPORT THEOREM with V  =η and O = I (Inertial System) ( ) ( ) ( )∑ ∫∫∫∫ ⋅−+= openings i S S tv I REYNOLDS I tv iopen sdVVvd Dt VD vdV td d    ,ρρρ Since the Linear Momentum is ( ) ( ) ∫∫ == tvtm dvVmdVP ρ  : ( ) ( ) ( ) ( ) ∑+∑= ∑+∫= ∑ ∫∫ ⋅−+∫=∫= openings i iflowiflowext openings i iflowiflow tm I openings i S S tm II tm I mVF mVmd Dt VD sdVVmd Dt VD mdV td d P td d iopen         , , , ˆ ˆ ρ Integrate the force equation on ρ dv over the volume v (t)
18 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 2) where ( ) ( )        = ≠ ∫∫ ⋅− =→∫∫ ⋅−= 00 0: ˆ : ˆ , , iflow iflow iflow S S iflow S Siflowiflow mif mif m sdVV VsdVVmV iopen iopen        ρ ρ - is the mean flow velocity, relative to an inertial frame, at the opening iflowV  iopenS Let differentiate the Linear Momentum I openings i iflowCiopenC I mRRmV td d td Pd     ∑      −−=    ˆ ∑           −−∑       −−+= openings i iflow I C I iopen openings i iflowCiopenC I C m td Rd td Rd mRRmVm td Vd        ˆ ˆ ∑∑      −−     −−+= openings i iflowCiopen openings i iflowCiopenC I C mVVmRRmVm td Vd        ˆˆ where is the mean velocity of the opening , relative to I iopenS ( ) iflow iflow iflow I iflow S S I iopeniopen V mif mif m sdVR td d R td d V iopen ˆ 00 0ˆ : ˆ ,       ≠         = ≠             ⋅− == ∫∫ ρ
19 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 3) Using ∑      −−∑      −−+= openings i iflowCiopen openings i iflowCiopenC I C I mVVmRRmVm td Vd td Pd        ˆˆ and ( ) ∑+∑=∫= openings i iflowiflowext I tm I mVFmdV td d P td d   , ˆ we obtain ∑      −+∑      −+−∑+∑= openings i iflowCiopen openings i iflowCiopenC openings i iflowiflowext I C mVVmRRmVmVFm td Vd          ˆˆˆ ∑      −+∑      −+ ∑−∑       +−+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowC openings iflowiopeniopeniflowext mVVmRR mVmVVVF         ˆˆ ˆˆˆ ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ Therefore
20 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 4) Note We could obtain this result by performing ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ ( )( ) ( )( )∑ ∫∫∑ ∫∫ ⋅−−−+=         ⋅−−−= openings i I S SCC I C I openings S SCC I iopeniopen sdVRR td d mVm td Vd sdVRRmV td d td Pd       ,, ρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫∫ ∑ ∫∫∑ ∫∫∑ ∫∫ ⋅−−+ ⋅−−+⋅−−=⋅−− openings i SOFPOSITIONINCHANGE S S I C openings i STHROUGH FLOWINCHANGE S I SC openings i SOFSHAPEAND MAGNITUDEINCHANGE S td d SC openings i I S SC iopen iopen iopen iopen iopen I iopen iopen sdVRR td d sdV td d RRsdVRRsdVRR td d             , ,,, ρ ρρρ ( ) ( ) ( ) ( ) ∑∑ ∫∫∑ ∫∫      −=⋅−−+⋅−− openings i iflowCiopen openings i S I SC openings i S td d SC mRRsdV td d RRsdVRR iopen I iopen   ˆ ,, ρρ Therefore ( ) ( ) ( ) ( ) ( )∑∑ ∫∫∑ ∫∫ −=⋅−−=⋅−− openings i iflowCiopen openings i S SC openings i S S I C mVVsdVVVsdVRR td d iopeniopen   ,, ρρ ( ) ( ) ∑ ∑∑ ∫∫      −−      −−+=         ⋅−−−= openings i iflowCiopen openings i iflowCiopenC I C I openings i S SCC I mVV mRRmVm td Vd sdVRRmV td d td Pd iopen          ˆ ˆ ,ρ or End Note Table of Content
21 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH  ,, ρ Absolute Angular Momentum Relative to a Reference Point O Define the Absolute Angular Momentum Relative to O of the mass enclosed by the volume v (t), as Substitute in the previous equation OIO O O O I O I O I OO r td rd V td rd td Rd td Rd VrRR , ,, , :&      ×++=+==+= ←ω ( )( ) ( ) ∫         ×++×=∫ ×−= ← tm OIO O O OO tm OO dmr td rd VrdmVRRH , , ,,    ω ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫= ← tm O O O tm OIOOO tm O dm td rd rdmrrVdmr , ,,,,   ω We obtain (a) (b) (c) Let develop those three expressions (a), (b) and (c). where IO←ω  is the angular velocity vector of the O frame relative to I. ( ) ( ) PdRRvdVRRHd OOO  ×−=×−= ρ, The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as vdmd ρ= V 
22 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -1) (a) ( ) ( ) ( ) ( ) ( ) ( ) OOC tm OC tm OC tm OC tm C tm O cmRRdmrdmrdmrdmrdmr ,,,,,,  =−===+= ∫∫∫∫∫ Where we used because C is the Center of Mass (Centroid) of the system. ( ) 0, =∫tm C dmr  ( ) OOOOCO tm O VcVrmVdmr  ×=×=×         ∫ ,,, ( ) ( ) ( )[ ]( ) IOOIO tm OOOO tm OIOO Idmrrrrdmrr ←←← ⋅=⋅−⋅=×× ∫∫ ωωω  ,,,,,,, 1(b) where ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1:  2nd Moment of Inertia Dyadic of all the mass m(t) relative to O We obtain (a) + (b) + (c) ( )( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫∫∫∫         ×+⋅+×=         ×+××+×      =×−= ← ← tm O O OIOOOO tm O O O tm OIOOO tm O tv OO dm td rd rIVc dm td rd rdmrrVdmrvdVRRH , ,,, , ,,,,, :     ω ωρ
23 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -2) ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , Difference between O and System centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors • Moving Parts (Pistons, …) • Fluids • Elasticity
24 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -3) Let differentiate the Absolute Angular Momentum, relative to an Inertial System and apply REYNOLDS’ Transport Theorem with and O = I (Inertial System)( ) VRR O  ×−=η ( )( ) ( ) ( ) ( ) ( )∑ ∫∫∫∫ ⋅−×−+ ×− =×−= openings i S SO tv I O REYNOLDS I tv O I O iopen sdVVRRvd Dt VRRD vdVRR td d td Hd     , , ρρρ ( )( ) ( )( ) ( ) ( ) ( )∑ ∫∫∫∫∫ ⋅−×−+×−×−=×−= openings i S md SO P tv O tv I O REYNOLDS I tv O I O iopen sdVVRRvdVVvd Dt VD RRvdVRR td d td Hd          , , ρρρρ to obtain Use ( )( ) ( )( ) ( ) ( ) ( )( )∑ ∫∫∫∫ ⋅−−×−+×−+×+×−=×−= openings i S SOOOOCOC tv I O I tv O I O iopen sdVVVRRmVRRmVVvd Dt VD RRvdVRR td d td Hd       , , ρρρ And use ( ) ( ) ( )( )∑ ∫∫ ⋅−−−=∫=∫= openings i S SCC tmtv iopen sdVVRRmVmdVvdVP  ,ρρ ( )[ ] ( ) ( ) ( ) ( ) VV tD VD RR tD VD RRVVV tD VD RRV tD RD tD RD tD VRRD O I O I OO I O I O II O         ×−×−=×−+×−=×−+×         −= ×− to obtain ( )( ) ( )( ) O I O openings i iflowOiflowOiopen tv I O I tv O I O V td cd mVVRRvd Dt VD RRvdVRR td d td Hd        ×+      −×      −+×−=×−= ∑∫∫ ,, ˆˆ ρρ or Table of Content
25 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O Multiplying (vector product) the 2nd Newton’s Law on the particle of mass dm=ρ dv, by we obtain the Moment of forces applied, relative to O: OO RRr  −=:, ( ) ( ) ( ) dm td Vd RRfdfdRR I OextO   ×−=+×− int from which by integration over the volume v (t) ( )( ) ( )( ) ( )( ) ∫∫∫ ×−=×−+×− tv I O tv O tv extO dv td Vd RRfdRRfdRR ρ   int We define the moment of external forces, relative to O, on the mass of the volume v (t), as: ( )( ) ∫∑ ×−= tv extOOext fdRRM  :,
26 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 1) We assumed that the equal but opposite forces between i and j act along the line joining them; i.e. Note collineararefandr jitij int  This is not always true (see H. Goldstein “Classical Mechanics”, 2nd Edition, pg.8, R. Aris “Vectors, Tenors and the Basic Equations of Fluid Mechanics”, pp.102-104, Michalas & Michalas “Radiation Hydrodynamics”, pg.72, Jaunzemis “Continuous Mechanics” Sec. 11, pg.223) End Note Since for any particles i and j the internal forces are of equal magnitude but of opposite directions we have jiij fdfd intint  −= ( ) ( ) ( ) ( ) ( ) ( )( ) 0 0 int intintint intint intint     =×−⇒ ←=×=×−= =×−+×−−= =×−+×− ∫ tv tOj jitijjitijjitij jitOjjitOi jitOjijtOi fdRR collinearfandrfdrfdRR fdRRfdRR fdRRfdRR
27 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 2) Therefore ( )( ) ( )( ) ( )( ) ∫ ∫∫∑ ×−= ×−=×−= tv extO tv I O tm I OOext fdRR dv tD VD RRdm tD VD RRM      ρ, ( )( ) ( )( ) O I O openings i iflowOiflowOiopen tv I O I tv O I O V td cd mVVRRvd Dt VD RRvdVRR td d td Hd        ×+      −×      −+×−=×−= ∑∫∫ ,, ˆˆ ρρ Use to obtain O I O openings i iflowOiflowOiopenOext I O V td cd mVVRRMH td d     ×+      −×      −+= ∑∑ , ,, ˆˆ
28 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 3) Use ( ) ( )( )∑ ∫∫ ⋅−−−+=∫= openings i S SCO I O tm iopen sdVVRRmV td cd mdVP  , , ρ ( ) ( ) ( ) O I O openings i S SOOOext I O V td cd sdVVVRRMH td d iopen    ×+∑ ∫∫ ⋅−−×−+∑= , ,,, ρ ( ) ( ) ( ) ( ) ( ) O openings i S SOO openings i S SOOOext VsdVRRmVPsdVVVRRM iopeniopen  ×        ∑ ∫∫ ⋅−−+−+∑ ∫∫ ⋅−−×−+∑= ,,, ρρ ( ) ( )∑ ∫∫ ⋅−×−+∑ ×+= openings i S SOOOext I O iopen sdVVRRVPMH td d  ,,, ρ in to obtain
29 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 4) Use the centroid C instead of O Absolute Angular Moment Relative to Center of Mass C ( ) ( ) ( ) ( ) ∫∫ ×−=×−= tm C tv CC mdVRRvdVRRH  ρ:, and ( ) ( ) ( ) ∑∑ ∑ ∫∫∑       −×      −+= =⋅−−×−+= openings i iflowCiflowCiopenCext openings i S SCCCext I C mVVRRM sdVVVRRMH td d iopen    ˆˆ , ,,, ρ ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫ ×−+×−=×−= tm OC tm C tm OO mdVRRmdVRRmdVRRH  , ( ) ( ) ( ) PrHPRRHmdVRRH OCCOCC tm OCC  ×+=×−+=×−+= ∫ ,,,, The relation between and is obtained usingOH,  CH, 
30 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let compute OIO O O I O H td Hd td Hd , ,,   ×+= ←ω                 ××+                 ×+ ∫∫ ← m O O OIO Om O O O md td rd rmd td rd r td d , , , ,     ω Absolute Angular Momentum Relative to a Reference Point O (Continue - 5) Using ( ) ∫         ×+⋅+×= ← tm O O OIOOOOO dm td rd rIVcH , ,,,,   ω IOOI ←⋅+ ω  , IOOIOIOO II ←←← ⋅×+⋅+ ωωω  ,, Rotation from I to O coordinates I O OO I O td Vd cV td cd    ×+×= , , Difference between O and System centroid C Non-rigidity of System • Rotors • Moving Parts (Pistons,…) • Fluids • Elasticity Table of Content
31 SOLO External Forces and Moments of the System We have a system of particles enclosed at the time t by a surface S(t) that bounds the volume v(t). There are no sources or sinks in the volume v(t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2, …). The surface S(t) can be divided in: • Sw(t) the impermeable wall through which the flow can not escape .( )0,  =sV • Sopen i(t) the openings (i=1,2,…) through which the flow enters or exits .( )0>m ( )0<m EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
32 SOLO External Forces and Moments of the System (continue -1 ) The external forces acting on the system are: • Gravitation acceleration (E center of Earth).E E R R M Gg  3 = • Force per unit surface applied by the surroundings on the surface of the system.( )2 /mNσ  ( )dstfnpsdTsdnsd  111 +−==⋅=⋅ σσ where: ( ) ndsnnsdsd  111 =⋅= - vector of surface differential ( )2 /mNp - pressure on (normal to) the surface . ( ) ( ) ∑∫∫∑∑∑ +⋅+=→= j j tStv ext i iextext FsddvgFfdF  σρ ( ) ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑→ ∑ ×−=∑ k k j jOj tS O tv OOext i iextOiOext MFRRsdRRdvgRRM fdRRM   σρ, , The moment of the external forces, relative to a point O, is: f - friction force per (parallel to) unit surface .( )2 / mN • Discrete force exerting by the surrounding on the point , and discrete moments .∑j jF  jR  ∑ k kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH nT  1⋅= σ - force per unit surface ( )2 /mN
33 SOLO ∑      −+∑      −+∑      −+∑= openings i Ciopeniflow openings i Ciopeniflow openings i iopeniflowiflowext I C RRmVVmVVmF td Vd m         ˆˆ 2 ˆˆ External Forces Equations (continue -2) ( ) ( ) ( ) ( ) ( ) ∑∫∫∑∫∫∑ ++−+=+⋅+= j j tStvj j tStv ext FdstfnpdvgFsddvgF  11ρσρ ( ) ( ) ( ) 0111 0 =⋅∇== ∫∫∫ ∞∞∞ tv Gauss tStS dvnpdsnpdsnp   Since the pressure far away from the body is constant∞p Let add this equation to the previous one ( ) ( ) ( ) ( )[ ] ( ) ∑∫∑∫∫∑ ++−+=+⋅+= ∞ j j tSj j tStv ext FdstfnpptmgFsddvgF  11σρ ( ) ( )[ ] ( )[ ] ∑+∫∫ ∑ ∫∫ +−++−+= ∞∞ j j S openings i S Fdstfnppdstfnpptmg W iopen  1111 Finally since on Sopen i (t) the openings (i=1,2,…) the friction force f = 0 ( ) ( )[ ] ( ) ∑+∫∫ ∑ ∫∫ −++−+=∑ ∞∞ j j S openings i S ext FdsnppdstfnpptmgF W iopen  111 Substitute this equation in EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
34 SOLO External Forces Equations (continue – 3) or ( ) ( )[ ] ( ) ∑      −+∑      −+ ∑+∑       ∫∫ −+     −+∫∫ +−+= ∞∞ openings i iflowCiopen openings i iflowCiopen j j openings i S iflowopeniflow SI C mRRmVV FdsnppmVVdstfnpptmgmV dt d iopenW        ˆˆ 2 1 ˆˆ 11 1 ( ) ( ) ( )∑∑∑∑∑ −+−++++= openings i iflowCiopen openings i iflowCiopen j j i iThrustAero I C mRRmVVFFFtmgmV dt d     2 where ( )[ ]∫∫∑ +−= ∞ WS Aero dstfnppF  11: Aerodynamic Forces ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowiThrust dsnppmVVF    1 ˆˆ : Thrust Forces EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
35 SOLO External Forces Equations (continue – 4) Let substitute ( ) ∑∑∑∑∑      −+     −++++= openings i iflowCiopen openings i iflowCiopen j j i iThrustAero I C mRRmVVFFFtmgmV dt d     ˆˆ 2 in CIO O C I C V td Vd td Vd a    ×+== ←ω to obtain RIGID-BODY TERMSmV td Vd CIO O C         ×+ ←   ω ∑∑ −         ×+− ← openings i iflowCiopen openings i iflowCiopenIO O Ciopen mrmr td rd      ,, , ˆˆ ˆ 2 ω FLUID-FLOW TERMS AERODYNAMIC & PROPULSIVE∑∑ += i iThrustAero FF  v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  ∑+= j jFmg  GRAVITATIONAL & DISCRETE TERMS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
36 SOLO External Moments Equations (continue – 5) The moments of the external forces relative to the point O are given by ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑ k k j jOj tS OS tv OOext MFRRsdRRdvgRRM  σρ ~ , ( )( ) ( ) ( ) ( ) ( ) ∑+∑ ×−+∫ +−×−+×      ∫ −= k k j jOj tS OS tv O MFRRdstfnpRRgdvRR  11ρ Let add to this equation the following ( ) ( ) ( ) ( ) 01 0 5 =−×∇=×− ∫∫∫∫ ∞∞ V OS GGauss tS OS dvRRpdsnpRR     to obtain ( )( ) ( ) ( )[ ] ( ) ( ) ∑+∑ ×−+∫ +−×−+×      ∫ −=∑ ∞ k k j jOj tS OS tv OOext MFRRdstfnppRRgdvRRM  11, ρ ( ) ( ) ( )[ ] ( ) ( )  ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫         +−×−++−×−+×−= ∞∞ k k j jOj S openings i S Son OOOC MFRR dstfnppRRdstfnppRRgmRR W iopen W   1111 0 v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
37 SOLO ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫ −×−++−×−+×−=∑ ∞∞ k k j jOj S openings i S OOOCOext MFRR dsnppRRdstfnppRRgmRRM W iopen   111, ∑       −×      −+×+∑= openings i iflowOiflowOiopenO I O Oext I O mVVRRV td cd M td Hd    ˆˆ , , External Moments Equations (continue -6) Using together with we obtain ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ×+++×= k k j jOj openings i iflowOiopenOiopen O I O openings i OiThrustOAeroO I O MFRRmVVRR V td cd MMgc td Hd        ˆˆ , ,,, , ∑      −×     −+∑      −×     −+ ×+∑= openings i iflowOiopenOiopen openings i iflowiopeniflowOiopen O I O Oext mVVRRmVVRR V td cd M      ˆˆˆˆˆ , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
38 SOLO External Moments Equations (continue -7) where ( ) ( )[ ]∫∫∑ +−×−= ∞ WS OOAero dstfnppRRM  11:, Aerodynamic Moments ( ) ( )∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOiThrust dsnppRRmVVRRM    1 ˆˆˆ :, Thrust Moments on the opening i discrete forces exerting by the surrounding at point∑ j jF  jR  ∑ k kM  discrete moments exerting by the surrounding v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
39 SOLO ( ) Itm O O O I IO OIO I O I O OO I O I O dm td rd r td d td d I td Id td Vd cV td cd td Hd ∫         ×+⋅+⋅+×+×= ← ← , ,, , , ,,       ω ω External Moments Equations (continue -8) Using together with we obtain ( ) ( ) ∫∫         ××+         ×+⋅×+⋅+⋅ ←←←← ← tm O O OIO Otm O O OIOOIOIO O O O IO O dm td rd rdm td rd r td d I td Id td d I , , , ,, , ,      ωωωω ω ( ) ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ×+++×−= k k j jCj openings i iflowOiopenOiopen O I O openings i OiThrustOAeroOC I O MFRRmVVRR V td cd MMgmRR td Hd      ˆˆ ,, , ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ++         −×= k k j jCj openings i iflowOiopenOiopen openings i OiThrustOAero I O O MFRRmVVRR MM td Vd gc       ˆˆ ,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Table of Content
40 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ∑∑ ∫∫∫       === openings i iopen openings i S iflow tm td md mdmd td d tm iopen  MASS EQUATION LINEAR MOMENTUM EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( )( ) ( ) mrmRRmRmdRmdRRc OCOC tm O tm OO ,, :  =−=−=−= ∫ ∫ ( ) ( )( ) ∑ ∑ ∫∫∫       −−= ⋅−−−== openings i iflowCiopenC openings i S SCC tm mRRmV sdVRRmVmdVP iopen    ˆ : , ρ ( ) ( )( ) ∑ ∑ ∫∫∫       −−+= ⋅−−−+== openings i iflowOiopenO I O openings i S SOO I O tm mRRmV td cd sdVRRmV td cd mdVP iopen       ˆ : , , , ρ First Moment of Inertia Relative to O 0,  =Cc ( ) ( ) ( ) ( ) ( ) ( ) 0: =−→== ∫ ∫ ∫ ∫ tm C tm tv tv C mdRR m mdR vd vdR tR    ρ ρ Mass Centroid ( )       = ≠ ⋅− = ∫∫ 00 0: ˆ , iflow iflow iflow S S iopen mif mif m sdVR R iopen      ρ
41 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM (CONTINUE – 1) FORCE EQUATIONS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ( ) ( ) ∑∑∫∫∫∫ =++== ext j j tvtv ext tv I tm I FFfdfddv tD VD dm tD VD      0 int ρ ( ) ( ) ( ) ∑∑∑ ∫∫∫∫ +=⋅−+== openings i iflowiflowext openings i S S tm II tm I mVFsdVVmd Dt VD mdV td d P td d iopen     ,, ˆ ρ ( ) ( ) ∑∫∫∑ +⋅+= j j tStv ext FsddvgF  σρ ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ ∑      −−∑      −−+= openings i iflowCiopen openings i iflowCiopenC I C I mVVmRRmVm td Vd td Pd        ˆˆ ( )dstfnpsdTsdnsd  111 +−==⋅=⋅ σσ  int fdfddv tD VD ext mdI   +=ρ
42 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 2) ( )[ ]∫∫∑ +−= ∞ WS Aero dstfnppF  11: AERODYNAMIC FORCES ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowiThrust dsnppmVVF    1 ˆˆ : THRUST FORCES EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM RIGID-BODY TERMSmV td Vd CIO O C         ×+ ←   ω ∑−∑         ×+− ← openings i iflowiopen openings i iflowiopenIO B iopen mrmr td rd      ˆˆ ˆ 2 ω FLUID-FLOW TERMS GRAVITATIONAL, AERODYNAMIC, PROPULSIVE & ∑∑ ++= i iThrustAero FFmg  ∑+ j jF  DISCRETE TERMS FORCE EQUATIONS (CONTINUE – 1)
43 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 3) ABSOLUTE ANGULAR MOMENT RELATIVE TO A REFERENCE POINT O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , O different from body centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors, Shafts • Fluids • Elasticity ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1:  2nd Moment of Inertia Dyadic of all the mass m(t) relative to O PrHH OCCO  ×+= ,,,
44 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 4) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( )( ) ( )( ) ( ) ∑∑∫∫∑∑ +×−+⋅×−+×−=×−= k k j jOj tS O tv O i iextOiOext MFRRsdRRdvgRRfdRRM  σρ, ( )( ) ( )( ) ( ) ( ) ( )∑ ∫∫∫∫∫ ⋅−×−+×−×−=×−= openings i S SO P tv O tv I O REYNOLDS I tv O I O iopen sdVVRRvdVVvd Dt VD RRvdVRR td d H td d       ,, ρρρρ O I O openings i iflowOiflowOiopenOext I O V td cd mVVRRMH td d     ×+      −×      −+= ∑∑ , ,, ˆˆ ( )( ) ( )( ) ∫∫∑ ×−=×−= tv I O tm I OOext dv tD VD RRdm tD VD RRM ρ     , ∑∑       −×      −+= openings i iflowCiflowCiopenCext I C mVVRRMH td d   ˆˆ ,, ( ) ( ) ( ) int, fdRRfdRRvd tD VD RRMd OextO I OO    ×−+×−=×−= ρ
45 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 5) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O BODY TERMSIOOIOOIOIOO III ←←←← ⋅+⋅×+⋅ ωωωω  ,,, ( ) ( )                 ××+                 ×+ ∫ ∫ ← tm O O OIO O tm O O O dm td rd r dm td rd r td d , , , ,     ω ROTORS, FLUIDS, SHAFTS, ELASTICITY,… TERMS FLUID CROSSING OPENINGS TERMS ∑         ×+×− ← openings i iflowOiopenIO O Oiopen Oiopen mr td rd r     , , , ˆ ˆ ˆ ω AERODYNAMIC & PROPULSIVE ∑∑ += i OiThrustOAero MM ,,  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ∑+∑ ×−+ k k j jOj MFRR  DISCRETE FORCES & MOMENTS TERMS         −×+ I O O td Vd gc   , NON-CENTROIDAL MOMENTS TERMS
46 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 6) ( ) ( )[ ]∫∫∑ +−×−= ∞ WS OOAero dstfnppRRM  11:, AERODYNAMIC MOMENTS RELATIVE TO O ( ) ( )[ ]∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOiThrust dsnppRRmVVRRM    1 ˆˆˆ :, THRUST MOMENTS RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM discrete forces exerting by the surrounding at point∑ j jF  jR  ∑ k kM  discrete moments exerting by the surrounding Table of Content
47 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System Kinetic Energy as a Function of Parameters Defined at a Reference Point O ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 Kinetic Energy of the System Let choose a reference point O and use: OO rRR ,  += I O O I O I O I td rd V td rd td Rd td Rd ,,    +=+= We can write ( ) ∫         +⋅         += tm I O O I O O md td rd V td rd VT ,, 2 1     ( ) ( ) ( ) ( ) ∫∫∫ ⋅++⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVV ,,, 2 1 2 1   (a) (b) (c) Let develop each of the three parts of this expression
48 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 1) (a) ( ) ( ) ( ) mVVmdVV OO tm OO  ⋅=∫⋅ 2 1 2 1 ( ) ∫⋅ tm I O O md td rd V ,   (b) Use Reynolds’ Transport Theorem when we differentiate ( ) OOC tm O cmrmdr ,,,  ==∫ ( ) ( ) I O openings i ifluidOiopen tm I O REYNOLDS I tm O td cd mrmd td rd mdr td d , , , , ˆ      =+=      ∑∫∫ Therefore ( ) ∑−=∫ openings i ifluidOiopen I O tm I O mr td cd md td rd    , ,, ˆ and ( ) ∑⋅−⋅=∫⋅ openings i ifluidOiopenO I O O tm I O O mrV td cd Vmd td rd V     , , ˆ ∑⋅−         ×+⋅= ← openings i ifluidOiopenOOIO O O O mrVc td cd V    , ˆω
49 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 2) (c) ( ) ( ) ∫∫         ×+⋅         ×+=⋅ ←← tm OBIO O OB OBIO O OB tm I O I O B mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ∑ ∫         ×+×+⋅                       ×++×++ ←←←← rotors m CirotorORiOCIO O OC td rd CirotorORi Ri Cirotor td rd OCIO O OC Ri RiRi Ri Ii RiCirotor Ri Ri I ORiC Ri Ri dmrr td rd r td rd r td rd ,, , , 0 , , , , , 2 1                ωωωω ( ) ∫         ×+⋅         ×+= ←← tm OIO FrozenRotor O O OIO FrozenRotor O O mdr td rd r td rd , , , , 2 1     ωω ( ) ( )∑ ∫∑ ∫ ×⋅×+⋅         ×+ ←←← rotors m CirotorORiCirotorORi rotors I OC m CirotorORi Ri RiRi Ri Ri Ri dmrr td rd dmr ,, , 0 , 2 1       ωωω ( ) ∫ ⋅= tm FrozenRotor O O FrozenRotor O O md td rd td rd ,, 2 1  (c1) ( ) ( ) ∫ ×⋅         + ← tm OIO FrozenRotor O O mdr td rd , ,   ω (c2) ( ) ( ) ( ) ∫ ×⋅×+ ←← tm OIOOIO mdrr ,, 2 1  ωω (c3) ( ) ( )∑ ∫ ×⋅×+ ←← rotors m CirotorIRiCirotorIRi Ri RiRi dmrr ,, 2 1  ωω (c4)
50 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 3) Let develop those equations ( ) ∫ ⋅ tm O O O O md td rd td rd ,, 2 1  (c1) (c2) ( ) ( ) ( ) ( ) ∫∫         ⋅×=×⋅         ←← tm O O OIO tm OIO O md td rd rmdr td rd O , ,, ,    ωω ( ) ( ) ∫∫         ×⋅=         ×⋅= ←← tm O O OIO tm O O OIO md td rd rmd td rd r , , , ,     ωω (c3) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ×⋅×−=∫ ×⋅× ←←←← tm IOOOIO tm OIOOIO mdrrmdrr ωωωω  ,,,, 2 1 2 1 ( )[ ] ( ) ( )[ ]( ) IO tm OOOOIO tm IOOOIO mdrrrrmdrr ←←←← ⋅∫ −⋅⋅=∫ ××⋅−= ωωωω  ,,,,,, 1 2 1 2 1 IOOIO I ←← ⋅⋅= ωω  , 2 1 ( )[ ] ( ) ∫ −⋅= tm OOOOO mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the System, Relative to O
51 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫∫∫ ⋅+⋅+⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVV     ,, , ˆ 2 1 ω Since the kinetic energy is independent of the chosen reference point O, the previous relation is invariant to O. ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 ( ) ( ) ∫∫         ×⋅+⋅+ ← tm O O OIO tm O O O O md td rd rmd td rd td rd , , ,, 2 1    ω IOOIO I ←← ⋅⋅+ ωω  , 2 1 Table of Content
52 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Quasi-Lagrangian Equations Let perform the following calculations ∑−+=∑−         ×++= ∂ ∂ ← openings i ifluidOiopen I O O openings i ifluidOiopenOIO O O O O mr td cd mVmrc td cd mV V T          , , ,, , ˆˆω ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVVT    ,, ˆ 2 1 ω ( ) ( ) IOOIO tm O O OIO tm O O O O Imd td rd rmd td rd td rd ←←← ⋅⋅+∫         ×⋅+∫ ⋅+ ωωω     , , , ,, 2 1 2 1 Since P V T O   = ∂ ∂ Also ( ) OOIOO tm O O O IO VcImd td rd r T     ×+⋅+∫         ×= ∂ ∂ ← ← ,, , , ω ω Since O IO H T ,   = ∂ ∂ ←ω ( ) OOIOO tm O O OO VcImd td rd rH    ×+⋅+∫         ×= ← ,, , ,, ω We found ∑−+= openings i ifluidOiopen I O O mr td cd mVP     , , ˆ
53 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Quasi-Lagrangian Equations (continue – 1) Let develop those equations in frame O P V T O   = ∂ ∂ O IO H T ,   = ∂ ∂ ←ω We found ( ) ∑−=∫=∑ openings i iflowiflow I tm I ext mV td Pd md tD VD F     ˆ ∑+∑=       ∂ ∂ openings i iflowiflowext IO mVF V T td d    ˆ ∑ ×+∑ ×+= openings i iflowiflowOiopenOOext I O mVrVPMH td d   ˆˆ ,,, ∑ ×∑ +=       ∂ ∂ ×+       ∂ ∂ ← openings i iflowiflowOiopenOext O O IIO mVrM V T V T td d    ˆˆ ,, ω [ ] ( ) ( ) ( ) ( ) ∑+∑=       ∂ ∂ ×+       ∂ ∂ ← openings i iflow O iflow O ext O O O IO OO mVF V T V T td d      ˆ ω [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ) ∑∑ ×+=         ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ ← ← ← openings i iflow O iflow O Oiopen O Oext O O O O O IO O IO OIO mVrM V T V TT td d        ˆˆ ,, ω ω ω This equation is obtained, also, using the LAGRANGE’s EQUATIONS OF MOTION. Table of Content
54 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow Let compute the time derivative of the kinetic energy, using the REYNOLD’s Transport Theorem.      openingsthethroughaddedenergykinetic openings i ifluid iopenII m II REYNOLDS m II m td Rd td Rd md td Rd td Rd md td Rd td Rd td d td Td ∑         ⋅+∫         ⋅=∫         ⋅= 2 1 2 1 2 2 where ( )        = ≠ ∫∫ ⋅−         ⋅ =         ⋅ 00 0 2 1 , iflow iflow iflow S S II iopenII m m m sdV td Rd td Rd td Rd td Rd iopen       ρ Is the mean value of the kinetic energy flow that crosses through the openings .iopenS
55 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 1) Let express the kinetic energy flow as function of the parameters of the reference point O. OIO O O O I O I O I r td rd V td rd td Rd td Rd , ,,     ×++=+= ←ω int2 2 fdfdmd td Rd ext I   += ( )         openingsthethroughaddedenergykinetic openings i ifluid iopenII m extOIO O O O m td Rd td Rd fdfdr td rd V td Td ∑         ⋅+∫ +⋅         ×++= ← 2 1 int, , ω ( ) IOOIOextO O O ext O O extO fdrfdrfd td rd fd td rd fdfdV ←← ⋅         ∫ ×+⋅∫ ×+∫ ⋅+∫ ⋅+         ∫ ∫+⋅= ωω          0 int,, 0 int ,, 0 int      openingsthethroughaddedenergykinetic openings i ifluid iopenII m td Rd td Rd ∑         ⋅+ 2 1 ( ) ( )        openingsthethroughaddedenergykinetic openings i ifluid iopenII IOOextext O O extO m td Rd td Rd Mfd td rd FV ∑                 ⋅+⋅∑+∫ ⋅+∑⋅= ← 2 1 , , ω
56 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 2) where  ForcesDiscrete l l ForcesSurface S ForcesBody V extext FsdTvdGfdF ∑+∫+∫=∫=∑      ρ:           MomentsDiscrete k k Moments ForcesDiscrete l lOl MomentsSurface S O MomentsBody V OextOOext MFrsdTrvdGrfdrM ∑+∑ ×+∫ ×+∫ ×=∫ ×=∑ ,,,,, : ρ G  - body force per unit mass ( )3 /mN nT  1⋅= σ - force per unit surface ( )2 /mN σ  - stress tensor (dyadic) ( )2 /mN ∑ l lF  - discrete forces applied to the system at the position lR  ( )N ∑ k kM  - discrete moments applied to the system ( )mN ⋅
57 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 3) Since the kinetic energy is invariant to the reference point O, let choose O = C (system centroid) ( ) ( )                openingsthethroughaddedenergyKinetic openings i ifluid iopenII momentsexternal bydoneWork ICCext bodytheofrigiditynonof becausedoneWork ext C C forcesexternal bydoneWork extC m td Rd td Rd Mfd td rd FV td Td ∑∑∫∑                 ⋅+⋅+⋅+⋅= ← − 2 1, ω       Openings trough Added Massof Energy Internal flow Openings trough Added Energy Kinetic flow System to Added Flow Heat Change Work m II Change Energy Kinetic Change Energy Internal td Ud td Td td Qd td Wd md td Rd td Rd e td d td Td td Ud +++=         ⋅+=+ ∫ 2 1 Change in Internal Energy + Change in Kinetic Energy = Change in Total Energy due to Surroundings From thermodynamics we have:
58 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 4) -internal energy of the molecules (vibration, rotation, translation). ∫= m mde td Ud ∫         ⋅= m II md td Rd td Rd td d td Td  2 1 ( ) ( )                   ForcesSurface S I ForcesBody v I momentsexternal bydoneWork ICCext bodyrigidnon anondoneWork ext C C forcesexternal bydoneWork extC sdT td Rd vdG td Rd Mfd td rd FV td Wd ∫∫∑∫∑ ⋅+⋅=⋅+⋅+⋅= ← − ρω,  ∫∫ ⋅− ∂ ∂ = S Surface throughRate Radiation ConductionV Rate Transfer Heat sdqvd t Q td Qd  ρ ∑                 ⋅= openings i ifluid iopenII flow m td Rd td Rd td Td   2 1 - kinetic energy change. - rate of work done by the surroundings on the system -rate of heat transfer and conduction/radiation added to the system. - kinetic energy added to the system through the openings. td Ud flow -internal energy added by the flow entering the system through the openings. Table of Content
59 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ OzˆLet assume that the variable mass has internal rigid rotors and other rigid moving parts (shafts...) Bm - mass of the body (excluding rotors) irotorm - mass of the rotor i m - mass of the system ∑+= rotors irotorB mmm BC - centroid of the body located at , relative to the reference point O.OCB r ,  RiC - centroid of the rotor i located at , relative to the reference point O.OCRi r ,  C - centroid of the system located at , relative to the reference point O.OCr ,  Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors)
60 SOLO I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ OBc ,  - first moment of inertia of the body, relative to O. ORi c ,  - first moment of inertia of the rotor i, relative to O. Oc,  - first moment of inertia of the system, relative to O. ∑+ ∑+ =→ =∑+=∑ ∫+∫=∫= rotors irotorB rotors irotorOCBOC OC OC rotors irotorOCBOC rotors m O m O m OO mm mrmr r mrmrmrmdrmdrmdrc RiB RiB RB ,, , ,,,,,,, :    Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 1) EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
61 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ BP  - body linear momentum ∑      −−+= openings i iflowOiopenBO I OB B mRRmV td cd P     ˆ, irotorP  - rotor i linear momentum irotorO I C irotorO I ORi irotor mV td rd mV td cd P Ri         +=+=      , P  - system linear momentum ∑      −−+=∑+= openings i iflowOiopenO I O rotors i irotorB mRRmV td cd PPP     ˆ, Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 2)
62 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -2) ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , Difference between O and System centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors, • Moving parts (Pistons,…) • Fluids • Elasticity
63 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let go in more details by taking the rotors in consideration For a point on the rotor i, we have I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ Ri Ri Ri ORi ORi RiORi RiORi CirotorIRi C Cirotor CIO O C O I Cirotor I C I O I irotor irotor CirotorCOirotor r td rd r td rd V td rd td rd td Rd td Rd V rrRR , 0 , , , , , , , :          ×++×++= ++== ++= ←← ωω Substitute those in ( )( ) ∫ ×−= tm OO dmVRRH  , , to obtain ( )( ) ( ) ∑+=∑ ∫ ×+∫ ×=∫ ×−= rotors ORiOB rotors m I irotor Oirotor tm I B OB tm OO HHdm td Rd rdm td Rd rdmVRRH RiB ,,,,,      ( ) ∑ ∫         ×+×++×+ ∫         ×++×= ←← ← rotors m CirotorIRiOCIO O OC OOirotor tm OBIO O OB OOB irotor RiRi Ri B dmrr td rd Vr dmr td rd Vr ,, , , , , ,       ωω ω where OBH ,  - body absolute angular momentum relative to the reference point O. ORiH ,  - rotor i absolute angular momentum relative to the reference point O. Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 3)
64 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH OBH ,  - body absolute angular momentum relative to the reference point O. ( ) ( ) ( ) ( ) ( ) ∫ ×+∫ ××+×∫= ∫         +×+×= ← ← tm O OB OB tm OBIOOBO tm OB tm O OB OBIOOOBOB BBB B dm td rd rdmrrVdmr dm td rd rVrH , ,,,, , ,,,     ω ω (a1) (b1) (c1) ( ) OB tm OB cdmr ,,  =∫(a1) (b1) ( ) OOBOOBBO tm OB VcVrmVdmr B  ×=×=×         ∫ ,,, ( ) ( ) ( )[ ]( ) IOOBIO tm OBOBOBOB tm OBIOOB Idmrrrrdmrr BB ←←← ⋅=⋅∫ −⋅=∫ ×× ωωω  ,,,,,,, 1 ( )[ ]∫ −⋅= Bm OOOOOB mdrrrrI ,,,,, 1:  where Second Moment of Inertia of the body relative to O ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫= ← tm O OB OB tm OBIOOBO tm OBOB BBB dm td rd rdmrrVdmrH , ,,,,,   ω ( ) ∫         ×+⋅+×= ← tm O OB OBIOOBOOB dm td rd rIVc ,, ,,,   ω Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 4)
65 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH ORiH ,  - rotor i absolute angular momentum relative to the reference point O. ∫         ×+×++×=∫         ×= ←← Ri RiRi Ri Ri m CirotorIRiOCIO O OC OOirotor m I irotor OirotorORi mdrr td rd Vrmd td Rd rH ,, , ,,,      ωω ∫         ×+×++×= ←← R RiRi Ri Ri m CirotorIRiOCIO O OC OCirotor mdrr td rd Vr ,, , ,    ωω ∫         ×+×++×+ ←← R RiRi Ri Ri m CirotorIRiOCIO O OC OOC mdrr td rd Vr ,, , ,    ωω irotorOCCRi PrH RiRi  ×+= ,, ∫         ×+×++×= ←← Ri RiRi Ri RiRi m CrotorIRiOCIO O OC OCirotorCRi mdrr td rd VrH ,, , ,,    ωω ( ) IRi m CirotorCirotorOCIO O OC O m Cirotor Ri RiRiRi Ri Ri Ri mdrrr td rd Vmdr ←←       ∫ ××−+         ×++×             ∫= ωω        ,,, , 0 , ( )[ ] IRiCirotorIR m CirotorCirotorCirotorCirotor Ri Ri RiRiRiRi Imdrrrr ←← ⋅=⋅∫ −⋅= ωω  ,,,,, 1 ( )[ ]∫ −⋅= Ri RiRiRiRiRi m CirotorCirotorCirotorCirotorCirotor mdrrrrI ,,,,, 1:  Second Moment of Inertia of the rotor i relative to it’s centroid RiC Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 5)
66 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Therefore ( )[ ]( ) Ri O OC OOCIOOirotorORiCirotor Ri B OC OOCIORiOCOCOCOCCirotorORiCirotor ORiIORi O OC RiOOCIRiCirotorORi m td rd VrII m td rd VrmrrrrII cm td rd mVrIH Ri RiRi Ri RiRiRiRiRiRiRi Ri RiRi         +×++⋅=         +×+−⋅++⋅=         ×++×+⋅= ←← ←← ←← , ,,, , ,,,,,,, , , ,,, 1        ωω ωω ωω Second Moment of Inertia of the rotor i relative to O ( )[ ] RiOCOCOCOCCirotorOirotor mrrrrII RiRiRiRiRi ,,,,,, 1:  −⋅+= Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 6)
67 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH The system Absolute Angular Momentum relative to O, is IOOB m O OB OBOOB rotors ORiOBO Imd td rd rVcHHH B ←⋅+         ×+×=+= ∫∑ ω    , , ,,,,, ∑         +×+⋅∑+∑ ⋅+ ←← rotors Ri O OC OOCIO rotors Oirotor rotors BRiCirotor m td rd VrII Ri RiRi , ,,,   ωω or ∑∫ ∑         ×+         ×+ ⋅+⋅+×= ←← rotors Ri O OC OC m O OB OB rotors ORiCirotorIOOOOO m td rd rmd td rd r IIVcH Ri Ri B Ri , , , , ,,,      ωω Second Moment of Inertia of the system relative to O∑+= rotors OirotorOBO III ,,, :  Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 7)
68 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH If we compare with ∑∫ ∑         ×+         ×+ ⋅+⋅+×= ←← rotors Ri O OC OC m O OB OB rotors ORiCirotorIOOOOO m td rd rmd td rd r IIVcH Ri Ri B Ri , , , , ,,,      ωω ( ) ∫         ×+⋅+×= ← tm O O OIOOOOO dm td rd rIVcH , ,,,,   ω we can see that ∑∫ ←⋅=         × rotors ORiCirotor m O O O Ri Imd td rd r ω    , , , ∫         ×+ Bm O OB OB md td rd r , ,   ∑         ×+ rotors Ri O OC OC m td rd r Ri Ri , ,   - Rotors Absolute Angular Moment Relative to O - Body Non-rigidity Elasticity of the System Sloshing of Liquids Moving Parts (Pistons,…) - Movement of Relative to O RiC Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 8)
69 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let define ∑         ×+∫         ×=∫         × rotors Ri O OC OC m O Ob Ob m FrozenRotors O O O m td rd rmd td rd rmd td rd r Ri Ri B , , , , , , :       We obtain ∫∑         ×+⋅+⋅+×= ←← m FrozemRotor O O O rotors ORiCirotorIOOOOO md td rd rIIVcH Ri , ,,,,,   ωω Let compute OIO O O I O H td Hd td Hd , ,,   ×+= ←ω ( ) O m FrozemRotor O O O rotors ORiCirotor rotors ORiCirotorIOOIOO I OO md td rd r td d IIIIVc td d RiRi                 ×+ ⋅+⋅+⋅+⋅+×= ∫ ∑∑ ←←←← , , 0 ,,,,,        ωωωω                 ××+      ⋅×+⋅×+ ∫∑ ←←←←← m FrozemRotor O O OIO rotors ORiCirotorIOIOOIO md td rd rII Ri , ,,,   ωωωωω Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 9)
70 SOLO ( ) ( ) I tm O O O I j ORjCjrotor I IO OIO I O I O OO I O I O dm td rd r td d I td d td d I td Id td Vd cV td cd td Hd Rj ∫         ×+∑+⋅+⋅+×+×= ← ← ← , ,,, , , ,,      ω ω ω External Moments Equations (continue -8) Using together with we obtain ( ) ( )∑+∫         ×+⋅×+⋅+⋅ ←←←← ← j I ORjCjrotor I tm FrozenRotors O O OIOOIOIO O O O IO O Rj I td d dm td rd r td d I td Id td d I ωωωω ω     , , ,, , , ( ) ( ) ∑+∑ ×−+∑      −×     −+ ×+∑+∑+×−= k k j jCj openings i iflowOiopenOiopen O I O openings i OTiOAOC I O MFRRmVVRR V td cd MMgmRR td Hd      ˆˆ ,, , ( ) ∑+∑ ×−+∑      −×     −+ ∑+∑+         −×= k k j jCj openings i iflowOiopenOiopen openings i OTiOA I O O MFRRmVVRR MM td Vd gc       ˆˆ ,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Table of Content
71 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 1) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O RIGID-BODY TERMS IOOIOOIOIOO III ←←←← ⋅+⋅×+⋅ ωωωω  ,,,     ∑ ⋅×+∑ ⋅+ ←←← j OjrotorCrotorjIO j OjrotorCrotorj RjRj II ωωω  ,, ROTORS TERMS ( ) ( )         ∫         ××+         ∫         ×+ ← tm FrozenRotor O O OIO O tm FrozenRotor O O O dm td rd r dm td rd r td d , , , ,     ω BODY FLUIDS, MOVING PARTS, ELASTICITY,… TERMS FLUID CROSSING OPENINGS TERMS ∑         ×+×− ← openings i iflowOiopenIO O Oiopen Oiopen mr td rd r     , , , ˆ ˆ ˆ ω AERODYNAMIC & PROPULSIVE ∑+∑= i OTiOA MM ,,  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ∑+∑ ×−+ k k j jOj MFRR  DISCRETE FORCES MOMENTS TERMS         −×+ I O O td Vd gc   , NON-CENTROIDAL MOMENTS TERMS
72 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 4) In the same way (c4) ( ) ( ) ( )[ ] ∑ ∑∑ ∫ ←← ←←←← ⋅⋅= ⋅−⋅⋅=×⋅× rotors ORiCrotorORi rotors ORiCirotorCirotorCirotorCirotorORi rotors m CirotorORiCirotorORi Ri RiRiRiRi Ri RiRi I rrrrdmrr ωω ωωωω   , ,,,,,, 2 1 1 2 1 2 1 Therefore (c) ( ) ( ) ∫∫         ×+⋅         ×+=⋅ ←← tm OIO O O OIO O O tm I O I O mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ( ) ∫ ⋅= tm FrozenRotor O O FrozenRotor O O md td rd td rd ,, 2 1  ( ) ∫         ×⋅+ ← tm FrozenRotor O O OIO md td rd r , ,   ω IOOIO I ←← ⋅⋅+ ωω  , 2 1 ORiCirotorORi Ri I ←← ⋅⋅+ ωω  , (c1) (c2) (c4)(c3)
73 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫∫∫ ⋅+⋅+⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVV     ,, , ˆ 2 1 ω Since the kinetic energy is independent of the chosen reference point O, the previous relation is invariant to O. ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 ( ) ( ) ∫∫         ×⋅+⋅+ ← tm FrozenRotor O O OIO tm FrozenRotor O O FrozenRotor O O md td rd rmd td rd td rd , , ,, 2 1    ω ∑ ⋅⋅+⋅⋅+ ←←←← rotors ORiCirotorORiIOOIO Ri II ωωωω  ,, 2 1 2 1 Table of Content
74 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLD’s TRANSPORT THEOREM APPROACH References 1. Shames, I.H., “Mechanics of Fluids”, 2nd Ed., McGraw-Hill, 1982 Table of Content
January 5, 2015 75 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA

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Equation of motion of a variable mass system2

  • 1. 1 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH SOLO HERMELIN http://www.solohermelin.com
  • 2. 2 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM • Simplified Particle Approach (see Power Point Presentation) The equations of motion can be developed using At a given time t the system has V(t) – system volume. m (t) – system mass. S (t) – system boundary surface. • Reynolds’ Transport Theorem Approach (this Power Point Presentation) • Lagrangian Approach (see Power Point Presentation)
  • 3. 3 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH TABLE OF CONTENT OSBORNE REYNOLDS 1842-1912 • Reynolds’ Transport Theorem • Inertial Velocity and Acceleration • Mass Equation • First Moment of Inertia Relative to a Reference Point O • Linear Momentum Equation • Force Equation • Absolute Angular Momentum Relative to a Reference Point O • Moment Relative to a Reference Point O • External Forces and Moments of the System • Summary of Equation of Motion for a Variable Mass System
  • 4. 4 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH TABLE OF CONTENT (Continue) • Kinetic Energy of the System • Quasi-Lagrangian Equations • Energy Flow • References • Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors) • Summary of the Equations of Motion of a Variable Mass System MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O
  • 5. 5 REYNOLDS’ TRANSPORT THEOREM -any system of coordinatesOxyz - any continuous and differentiable functions in ( ) ( )trtr OO ,,, ,,  ηχ ( )tandrO,  ( )trO ,,  ρ - flow density at point and time t Or,  SOLO - mass flow through the element .mdsdVS   =⋅− ,ρ sd  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM - any control volume, changing shape, bounded by a closed surface S(t)v (t) - flow velocity, relative to O, at point and time t( )trV OOflow ,,,  Or,  - position and velocity, relative to O, of an element of surface, part of the control surface S(t). OSOS Vr ,, ,  - area of the opening i, in the control surface S(t).iopenS - gradient operator in O frame.O,∇ - flow relative to the opening i, in the control surface S(t).OSiOflowSi VVV ,,,  −= - differential of any vector , in O frame. O td d ζ  ζ 
  • 6. 6 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM Start with LEIBNIZ THEOREM from CALCULUS: ( ) ( )    ChangeBoundariesthetodueChange tb ta tb ta td tad ttaf td tbd ttbfdx t txf dxtxf td d LEIBNITZ       −+= ∫∫ )),(()),(( ),( ),(:: )( )( )( )( ∂ ∂ and generalized it for a 3 dimensional vector space on a volume v(t) bounded by the surface S(t). Using LEIBNIZ THEOREM followed by GAUSS THEOREM (GAUSS 4): ( ) ( ) ( ) ( ) ∫∫∫∫∫       ⋅∇+∇⋅+=⋅+ → = tv OSOOOSGAUSS Opotolative dsofMovement thetodueChage tS OS tv O LEIBNITZ O tv vdVV t GAUSS sdVvd t vd td d ,,,,)4( intRe )( ,      χχ ∂ χ∂ χ ∂ χ∂ χ This is REYNOLDS’ TRANSPORT THEOREM OsborneReynolds 1842-1912 SOLO GOTTFRIED WILHELM von LEIBNIZ 1646-1716 REYNOLDS’ TRANSPORT THEOREM Johann Carl Friederich Gauss 1777-1855
  • 7. 7 0,,,,,  =−=⇒= OSOSOOS VVVVV ( ) ∫∫∫∫∫∫∫∫∫∫∫         ⋅∇+∇⋅+=⋅+= )( ,,,, )4( , )()()( tv OOOO O GAUSS O tStv OO tv FFFF vdVV t GAUSS sdVvd t vd td d     χχ ∂ χ∂ χ ∂ χ∂ χ SOLO REYNOLDS’ TRANSPORT THEOREM (CONTINUE -1) ( ) ∫∫∫∫∫∫∫∫ ⋅∇=⋅== )( ,,)4(, )()( )( tv OOGAUSSO tStv F FFF vdV GAUSS sdVvd td d td tvd  χ               =⋅∇ → td tvd tv V tv OO )( )( 1 lim0)( ,,  EULER 1755 ρχ == &,, OOS VV  ( )∫∫∫∫∫∫∫∫ ∫∫∫       ⋅∇+=⋅+=== )( ,, )( , )( )( )( 0 tV OO tS O tV tV FFF F vdV t sdVvd t dv td d td tmd  ρ ∂ ρ∂ ρ ∂ ρ∂ ρ or, since this is true for any attached volume vF(t) ( ) 0,,,,,, =⋅∇+∇⋅+=⋅∇+ OOOOOO VV t V t  ρρ ∂ ρ∂ ρ ∂ ρ∂ CONSERVATION OF MASS EQUATION EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM CASE 1 (Control Volume vF attached to the Fluid )OOS VV ,,  = CASE 2 (Control Volume vF attached to the Fluid and )1=χOOS VV ,,  = CASE 3 (Control Volume vF attached to the Fluid and )ρχ =OOS VV ,,  =
  • 8. 8 Define ( ) ( ) ( )trtrtr OOO ,,:, ,,,  ηρχ = ( )∫∫∫∫∫∫∫∫ ⋅+         += )( , )()( tS OS tv OO tv sdVvd tt vd td d    ηρ ∂ ρ∂ η ∂ η∂ ρηρ We have (for any volume v(t) bounded by the surface S(t)) But, from CONSERVATION OF MASS ( ) ( )OOOO V t V t ,,,, 0  ρη ∂ ρ∂ ηρ ∂ ρ∂ ⋅∇−=⇒=⋅∇+ CASE 4: Flow Equations SOLO We have ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫∫∫∫∫= ∫∫∫∫∫ ∫∫∫∫∫∫∫∫ ⋅−− ⋅+         ⋅∇+∇⋅−         ∇⋅+= ⋅+         ⋅∇−= + + )( ,, )( 4 . )( , )( ,,,,,, )( , )( ,, )( tS OSO tv O MDG DerMat GAUSS tS OS tv OOOOOO O tS OS tv OO OO tv sdVVvd tD D sdVvdVVV t sdVvdV t vd td d       ρηρ η ρηρηηρη ∂ η∂ ρ ρηρηρ ∂ η∂ ρη where 0: ,, =⋅∇+= η ∂ η∂η  OO OO V ttD D is the MATERIAL DERIVATIVE, RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -2)
  • 9. 9 CASE 4: Flow Equations (continue – 1) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -3) ( )[ ]∫∫∫∫∫∫∫∫ ⋅−−= )( ,, )()( tS OSO tv OO tv sdVVvd tD D vd td d    ρηρ η ρη v(t) ds m> 0 . m< 0 . 0, <⋅ sdV S  dm ∑= + N j ext iji fdfd 1 int  ( )tSW 2openS openiS O 0, >⋅ sdV S  OCr , Oflowir ,  C Copenir ,  Cr,  SV,  Oiopenr ,  openiV  OV,  Or,  Cflowir ,  flowiV  Ozˆ Oyˆ Oxˆ openflowS VVV  −=, - mass flow rate through the element . mdsdVS   =⋅− ,ρ sd  SOSO VVV ,,, :  =− - flow velocity relative to .sd  There are no sources or sinks in the volume v (t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2,…). The surface S(t) can be divided in: • Sw(t) the impermeable wall through which the fluid can not escape .( )0,  =sV • Sopen i(t) the openings (i=1,2,…) through which the fluid enters or exits .( )0>m ( )0<m ( ) ( ) ( )∑+= openings i iopenW tStStS ( ) ( ) ( )( ) ∑=∑ ∫∫ ⋅−+∫∫         ⋅−=∫∫ ⋅− openings i ii openings i tS S tS S tS S msdVsdVsdV iopenW   ηρηρηρη ˆ , 0 , )( ,
  • 10. 10 CASE 4: Flow Equations (continue – 2) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -3) ( )∫∫∫∫∫∫∫∫ ⋅+= .. , .... SC O O VCVC O sdVvd td d vd tD D   ρηρηρ η - mass flow rate through the element . mdsdVS   =⋅− ,ρ sd  SO VV ,,  = - flow velocity relative to O and .sd  CONTROL VOLUME WITH FIXED SHAPE C.V.( ) 0,  =OS V
  • 11. 11 CASE 4: Flow Equations (continue – 2) SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM (CONTINUE -4) ( )∫∫∫∫∫∫∫∫ ⋅−+= )( , )()( tS S tv OO tv sdVvd tD D vd td d    ρηρ η ρη v(t) ds m> 0 . m< 0 . 0, <⋅ sdV S  dm ∑= + N j ext iji fdfd 1 int  ( )tSW 2openS openiS O 0, >⋅ sdV S  OCr , Oflowir ,  C Copenir ,  Cr,  SV,  Oiopenr ,  openiV  OV,  Or,  Cflowir ,  flowiV  Ozˆ Oyˆ Oxˆ openflowS VVV  −=, ( )( ) ∫∫ ⋅−= tS Si iopen sdVm   , ρ ( )( )       = ≠ ∫∫ ⋅− = 00 0:ˆ , i i i tS S i m m m sdV iopen      ρη η where i openings i i tv OO tv mvd tD D vd td d     ∑∫∫∫∫∫∫ += ηρ η ρη ˆ )()( ( )∑ ∫∫∫∫∫∫∫∫ ⋅−+= openings i S S tv OO tv iopen sdVvd tD D vd td d    , )()( ρηρ η ρη We can write or or REYNOLDS’ TRANSPORT THEOREM OSBORNE REYNOLDS 1842-1912 Mean Vector of on the opening iη  iopenS Mass Rate entering through opening iopenS or ( )∫∫∫∫∫∫∫∫ ⋅+= .. , .... SC O O VCVC O sdVvd td d vd tD D   ρηρηρ η Table of Content
  • 12. 12 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Inertial Velocity and Acceleration v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or,  R  - Position of the mass element dm relative to I. I td Rd V   = - Velocity of the mass element dm relative to I. II td Rd td Vd a 2 2   == - Acceleration of the mass element dm relative to I. Mass Equation Use the REYNOLDS’ TRANSPORT THEOREM with 01 =→= O tD Dη η   ( ) ( ) ( ) ( ) ∑∑ ∫∫∫∫ =⋅−=== openings i iflow openings i S S REYNOLDS tvtm msdVvd td d md td d tm iopen    , ρρ The change of the mass of the system is due to the flow through the openings in the surface S (t). ( )∫∫ ⋅−= iopenS Siflow sdVm   ,: ρ Table of Content Table of Content
  • 13. 13 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH First Moment of Inertia Relative to a Reference Point O v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or, Define the First Moment of Inertia Relative to O ( ) ( ) ( ) mRmdRmdRRc O tmtm OO  −=−= ∫∫:, -Position vector of the instantaneous Mass Center (Centroid) of the system, relative to I ( ) ( ) ( ) ( ) ( ) ( ) 0: =−→== ∫ ∫ ∫ ∫ tm C tm tv tv C mdRR m mdR vd vdR tR    ρ ρ Therefore ( ) ( )( ) ( ) mrmRRmRmdRmdRRc OCOC tm O tm OO ,, :  =−=−=−= ∫ ∫ For O = C we have: ( ) ( )( ) ( ) 0:,  =−=−=−= ∫ ∫ mRRmRmdRmdRRc CC tm C tm CC Table of Content
  • 14. 14 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Linear Momentum Equation v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or,  The Linear Moment of the mass enclosed by v (t) is defined as ( ) ( ) ∫∫∫∫∫∫ == tv I tv I vd tD RD vdVP ρρ   , : vdVmdVPd II ρ,, :  == The Linear Momentum, of the differential mass dm = ρdv, with initial velocity ,is defined as I I tD RD VV   == , : Use the REYNOLDS’ TRANSPORT THEOREM with R  =η and O = I ( ) ( ) ( ) ( ) ( ) ∑ ∫∫∫∫∫∫∫∫∫∫∫ −+= ⋅−−=== openings i iflowiopenC V C tS S mR tv REYNOLDS tv I tv I mRmR td Rd m sdVRvdR td d vd tD RD vdVP C C              ˆ : ,, ρρρρ ( )       = ≠ ⋅− = ∫∫ 00 0: ˆ , iflow iflow iflow S S iopen mif mif m sdVR R iopen      ρ where: is the mean position vector of the flow and of the opening on iopenSiopenR ˆ
  • 15. 15 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Linear Momentum Equation (continue – 1) v(t) I R  dm ( )tS 2openS 1openS I td Rd V   = I td Vd a   = Ix Oy Iz Ox Oz Iy OR  O Or, We can write the Linear Momentum ( ) ( )( ) ∑      −−= ∑ ∫∫ ⋅−−−=∫= openings i iflowCiopenC openings i S SCC tm mRRmV sdVRRmVmdVP iopen    ˆ : ,ρ We can write ( ) ∫= tm mdVP  : ( ) ( )( )∑ ∫∫ ⋅−−+−−+−= openings i S SCOOOOC iopen sdVRRRRmVmVmV  ,ρ ( ) ( ) ( ) ( )( )∑ ∫∫ ⋅−−−+∑ ∫∫ ⋅−−+−= openings i S SOO openings i S SOCOC iopeniopen sdVRRmVsdVRRmVV  ,, ρρ ( ) ( ) ∑∑       −−+−+−= openings i iflowOiopenO td cd m openings i iflowOCOC mRRmVmRRmVV I O           ˆ , or ( )( ) ∑      −−+= ∑ ∫∫ ⋅−−−+= openings i iflowOiopenO I O openings i S SOO I O mRRmV td cd sdVRRmV td cd P iopen       ˆ, , , ρ Table of Content
  • 16. 16 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation Applying the 2nd Newton’s Law to the differential mass dm = ρdv, we obtain: int2 2 fdfdmd td Rd md td Vd ext II   +== where ext fd  - External forces acting on the differential mass dm intfd  - Internal forces that surroundings exercises on the differential mass dm From the 3rd Newton’s Law the internal force that particle j applies on particle i is of equal magnitude but of opposite direction to the force that particle i applies on particle j : jiij fdfd intint  −= Therefore ( ) 00 int 1 1 int  =→= ∫∑∑ →∞ = ≠ = tv NN i N ij j ij fdfd
  • 17. 17 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 1) ( ) ( ) ( ) ( ) ∑∑∫∫∫∫ =++== ext l l tvtv ext tv I tm I FFfdfddv tD VD dm tD VD   intρ discrete forces exerting by the surrounding at point jR  Use the REYNOLDS’ TRANSPORT THEOREM with V  =η and O = I (Inertial System) ( ) ( ) ( )∑ ∫∫∫∫ ⋅−+= openings i S S tv I REYNOLDS I tv iopen sdVVvd Dt VD vdV td d    ,ρρρ Since the Linear Momentum is ( ) ( ) ∫∫ == tvtm dvVmdVP ρ  : ( ) ( ) ( ) ( ) ∑+∑= ∑+∫= ∑ ∫∫ ⋅−+∫=∫= openings i iflowiflowext openings i iflowiflow tm I openings i S S tm II tm I mVF mVmd Dt VD sdVVmd Dt VD mdV td d P td d iopen         , , , ˆ ˆ ρ Integrate the force equation on ρ dv over the volume v (t)
  • 18. 18 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 2) where ( ) ( )        = ≠ ∫∫ ⋅− =→∫∫ ⋅−= 00 0: ˆ : ˆ , , iflow iflow iflow S S iflow S Siflowiflow mif mif m sdVV VsdVVmV iopen iopen        ρ ρ - is the mean flow velocity, relative to an inertial frame, at the opening iflowV  iopenS Let differentiate the Linear Momentum I openings i iflowCiopenC I mRRmV td d td Pd     ∑      −−=    ˆ ∑           −−∑       −−+= openings i iflow I C I iopen openings i iflowCiopenC I C m td Rd td Rd mRRmVm td Vd        ˆ ˆ ∑∑      −−     −−+= openings i iflowCiopen openings i iflowCiopenC I C mVVmRRmVm td Vd        ˆˆ where is the mean velocity of the opening , relative to I iopenS ( ) iflow iflow iflow I iflow S S I iopeniopen V mif mif m sdVR td d R td d V iopen ˆ 00 0ˆ : ˆ ,       ≠         = ≠             ⋅− == ∫∫ ρ
  • 19. 19 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 3) Using ∑      −−∑      −−+= openings i iflowCiopen openings i iflowCiopenC I C I mVVmRRmVm td Vd td Pd        ˆˆ and ( ) ∑+∑=∫= openings i iflowiflowext I tm I mVFmdV td d P td d   , ˆ we obtain ∑      −+∑      −+−∑+∑= openings i iflowCiopen openings i iflowCiopenC openings i iflowiflowext I C mVVmRRmVmVFm td Vd          ˆˆˆ ∑      −+∑      −+ ∑−∑       +−+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowC openings iflowiopeniopeniflowext mVVmRR mVmVVVF         ˆˆ ˆˆˆ ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ Therefore
  • 20. 20 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Force Equation (continue – 4) Note We could obtain this result by performing ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ ( )( ) ( )( )∑ ∫∫∑ ∫∫ ⋅−−−+=         ⋅−−−= openings i I S SCC I C I openings S SCC I iopeniopen sdVRR td d mVm td Vd sdVRRmV td d td Pd       ,, ρρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∑ ∫∫ ∑ ∫∫∑ ∫∫∑ ∫∫ ⋅−−+ ⋅−−+⋅−−=⋅−− openings i SOFPOSITIONINCHANGE S S I C openings i STHROUGH FLOWINCHANGE S I SC openings i SOFSHAPEAND MAGNITUDEINCHANGE S td d SC openings i I S SC iopen iopen iopen iopen iopen I iopen iopen sdVRR td d sdV td d RRsdVRRsdVRR td d             , ,,, ρ ρρρ ( ) ( ) ( ) ( ) ∑∑ ∫∫∑ ∫∫      −=⋅−−+⋅−− openings i iflowCiopen openings i S I SC openings i S td d SC mRRsdV td d RRsdVRR iopen I iopen   ˆ ,, ρρ Therefore ( ) ( ) ( ) ( ) ( )∑∑ ∫∫∑ ∫∫ −=⋅−−=⋅−− openings i iflowCiopen openings i S SC openings i S S I C mVVsdVVVsdVRR td d iopeniopen   ,, ρρ ( ) ( ) ∑ ∑∑ ∫∫      −−      −−+=         ⋅−−−= openings i iflowCiopen openings i iflowCiopenC I C I openings i S SCC I mVV mRRmVm td Vd sdVRRmV td d td Pd iopen          ˆ ˆ ,ρ or End Note Table of Content
  • 21. 21 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH  ,, ρ Absolute Angular Momentum Relative to a Reference Point O Define the Absolute Angular Momentum Relative to O of the mass enclosed by the volume v (t), as Substitute in the previous equation OIO O O O I O I O I OO r td rd V td rd td Rd td Rd VrRR , ,, , :&      ×++=+==+= ←ω ( )( ) ( ) ∫         ×++×=∫ ×−= ← tm OIO O O OO tm OO dmr td rd VrdmVRRH , , ,,    ω ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫= ← tm O O O tm OIOOO tm O dm td rd rdmrrVdmr , ,,,,   ω We obtain (a) (b) (c) Let develop those three expressions (a), (b) and (c). where IO←ω  is the angular velocity vector of the O frame relative to I. ( ) ( ) PdRRvdVRRHd OOO  ×−=×−= ρ, The Absolute Angular Momentum, of the differential mass and Inertial Velocity ,relative to a reference point O is defined as vdmd ρ= V 
  • 22. 22 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -1) (a) ( ) ( ) ( ) ( ) ( ) ( ) OOC tm OC tm OC tm OC tm C tm O cmRRdmrdmrdmrdmrdmr ,,,,,,  =−===+= ∫∫∫∫∫ Where we used because C is the Center of Mass (Centroid) of the system. ( ) 0, =∫tm C dmr  ( ) OOOOCO tm O VcVrmVdmr  ×=×=×         ∫ ,,, ( ) ( ) ( )[ ]( ) IOOIO tm OOOO tm OIOO Idmrrrrdmrr ←←← ⋅=⋅−⋅=×× ∫∫ ωωω  ,,,,,,, 1(b) where ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1:  2nd Moment of Inertia Dyadic of all the mass m(t) relative to O We obtain (a) + (b) + (c) ( )( ) ( ) ( ) ( ) ( ) ( ) ∫ ∫∫∫∫         ×+⋅+×=         ×+××+×      =×−= ← ← tm O O OIOOOO tm O O O tm OIOOO tm O tv OO dm td rd rIVc dm td rd rdmrrVdmrvdVRRH , ,,, , ,,,,, :     ω ωρ
  • 23. 23 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -2) ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , Difference between O and System centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors • Moving Parts (Pistons, …) • Fluids • Elasticity
  • 24. 24 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -3) Let differentiate the Absolute Angular Momentum, relative to an Inertial System and apply REYNOLDS’ Transport Theorem with and O = I (Inertial System)( ) VRR O  ×−=η ( )( ) ( ) ( ) ( ) ( )∑ ∫∫∫∫ ⋅−×−+ ×− =×−= openings i S SO tv I O REYNOLDS I tv O I O iopen sdVVRRvd Dt VRRD vdVRR td d td Hd     , , ρρρ ( )( ) ( )( ) ( ) ( ) ( )∑ ∫∫∫∫∫ ⋅−×−+×−×−=×−= openings i S md SO P tv O tv I O REYNOLDS I tv O I O iopen sdVVRRvdVVvd Dt VD RRvdVRR td d td Hd          , , ρρρρ to obtain Use ( )( ) ( )( ) ( ) ( ) ( )( )∑ ∫∫∫∫ ⋅−−×−+×−+×+×−=×−= openings i S SOOOOCOC tv I O I tv O I O iopen sdVVVRRmVRRmVVvd Dt VD RRvdVRR td d td Hd       , , ρρρ And use ( ) ( ) ( )( )∑ ∫∫ ⋅−−−=∫=∫= openings i S SCC tmtv iopen sdVVRRmVmdVvdVP  ,ρρ ( )[ ] ( ) ( ) ( ) ( ) VV tD VD RR tD VD RRVVV tD VD RRV tD RD tD RD tD VRRD O I O I OO I O I O II O         ×−×−=×−+×−=×−+×         −= ×− to obtain ( )( ) ( )( ) O I O openings i iflowOiflowOiopen tv I O I tv O I O V td cd mVVRRvd Dt VD RRvdVRR td d td Hd        ×+      −×      −+×−=×−= ∑∫∫ ,, ˆˆ ρρ or Table of Content
  • 25. 25 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O Multiplying (vector product) the 2nd Newton’s Law on the particle of mass dm=ρ dv, by we obtain the Moment of forces applied, relative to O: OO RRr  −=:, ( ) ( ) ( ) dm td Vd RRfdfdRR I OextO   ×−=+×− int from which by integration over the volume v (t) ( )( ) ( )( ) ( )( ) ∫∫∫ ×−=×−+×− tv I O tv O tv extO dv td Vd RRfdRRfdRR ρ   int We define the moment of external forces, relative to O, on the mass of the volume v (t), as: ( )( ) ∫∑ ×−= tv extOOext fdRRM  :,
  • 26. 26 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 1) We assumed that the equal but opposite forces between i and j act along the line joining them; i.e. Note collineararefandr jitij int  This is not always true (see H. Goldstein “Classical Mechanics”, 2nd Edition, pg.8, R. Aris “Vectors, Tenors and the Basic Equations of Fluid Mechanics”, pp.102-104, Michalas & Michalas “Radiation Hydrodynamics”, pg.72, Jaunzemis “Continuous Mechanics” Sec. 11, pg.223) End Note Since for any particles i and j the internal forces are of equal magnitude but of opposite directions we have jiij fdfd intint  −= ( ) ( ) ( ) ( ) ( ) ( )( ) 0 0 int intintint intint intint     =×−⇒ ←=×=×−= =×−+×−−= =×−+×− ∫ tv tOj jitijjitijjitij jitOjjitOi jitOjijtOi fdRR collinearfandrfdrfdRR fdRRfdRR fdRRfdRR
  • 27. 27 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 2) Therefore ( )( ) ( )( ) ( )( ) ∫ ∫∫∑ ×−= ×−=×−= tv extO tv I O tm I OOext fdRR dv tD VD RRdm tD VD RRM      ρ, ( )( ) ( )( ) O I O openings i iflowOiflowOiopen tv I O I tv O I O V td cd mVVRRvd Dt VD RRvdVRR td d td Hd        ×+      −×      −+×−=×−= ∑∫∫ ,, ˆˆ ρρ Use to obtain O I O openings i iflowOiflowOiopenOext I O V td cd mVVRRMH td d     ×+      −×      −+= ∑∑ , ,, ˆˆ
  • 28. 28 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 3) Use ( ) ( )( )∑ ∫∫ ⋅−−−+=∫= openings i S SCO I O tm iopen sdVVRRmV td cd mdVP  , , ρ ( ) ( ) ( ) O I O openings i S SOOOext I O V td cd sdVVVRRMH td d iopen    ×+∑ ∫∫ ⋅−−×−+∑= , ,,, ρ ( ) ( ) ( ) ( ) ( ) O openings i S SOO openings i S SOOOext VsdVRRmVPsdVVVRRM iopeniopen  ×        ∑ ∫∫ ⋅−−+−+∑ ∫∫ ⋅−−×−+∑= ,,, ρρ ( ) ( )∑ ∫∫ ⋅−×−+∑ ×+= openings i S SOOOext I O iopen sdVVRRVPMH td d  ,,, ρ in to obtain
  • 29. 29 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Moment Relative to a Reference Point O (continue – 4) Use the centroid C instead of O Absolute Angular Moment Relative to Center of Mass C ( ) ( ) ( ) ( ) ∫∫ ×−=×−= tm C tv CC mdVRRvdVRRH  ρ:, and ( ) ( ) ( ) ∑∑ ∑ ∫∫∑       −×      −+= =⋅−−×−+= openings i iflowCiflowCiopenCext openings i S SCCCext I C mVVRRM sdVVVRRMH td d iopen    ˆˆ , ,,, ρ ( ) ( ) ( ) ( ) ( ) ( ) ∫∫∫ ×−+×−=×−= tm OC tm C tm OO mdVRRmdVRRmdVRRH  , ( ) ( ) ( ) PrHPRRHmdVRRH OCCOCC tm OCC  ×+=×−+=×−+= ∫ ,,,, The relation between and is obtained usingOH,  CH, 
  • 30. 30 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let compute OIO O O I O H td Hd td Hd , ,,   ×+= ←ω                 ××+                 ×+ ∫∫ ← m O O OIO Om O O O md td rd rmd td rd r td d , , , ,     ω Absolute Angular Momentum Relative to a Reference Point O (Continue - 5) Using ( ) ∫         ×+⋅+×= ← tm O O OIOOOOO dm td rd rIVcH , ,,,,   ω IOOI ←⋅+ ω  , IOOIOIOO II ←←← ⋅×+⋅+ ωωω  ,, Rotation from I to O coordinates I O OO I O td Vd cV td cd    ×+×= , , Difference between O and System centroid C Non-rigidity of System • Rotors • Moving Parts (Pistons,…) • Fluids • Elasticity Table of Content
  • 31. 31 SOLO External Forces and Moments of the System We have a system of particles enclosed at the time t by a surface S(t) that bounds the volume v(t). There are no sources or sinks in the volume v(t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2, …). The surface S(t) can be divided in: • Sw(t) the impermeable wall through which the flow can not escape .( )0,  =sV • Sopen i(t) the openings (i=1,2,…) through which the flow enters or exits .( )0>m ( )0<m EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 32. 32 SOLO External Forces and Moments of the System (continue -1 ) The external forces acting on the system are: • Gravitation acceleration (E center of Earth).E E R R M Gg  3 = • Force per unit surface applied by the surroundings on the surface of the system.( )2 /mNσ  ( )dstfnpsdTsdnsd  111 +−==⋅=⋅ σσ where: ( ) ndsnnsdsd  111 =⋅= - vector of surface differential ( )2 /mNp - pressure on (normal to) the surface . ( ) ( ) ∑∫∫∑∑∑ +⋅+=→= j j tStv ext i iextext FsddvgFfdF  σρ ( ) ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑→ ∑ ×−=∑ k k j jOj tS O tv OOext i iextOiOext MFRRsdRRdvgRRM fdRRM   σρ, , The moment of the external forces, relative to a point O, is: f - friction force per (parallel to) unit surface .( )2 / mN • Discrete force exerting by the surrounding on the point , and discrete moments .∑j jF  jR  ∑ k kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH nT  1⋅= σ - force per unit surface ( )2 /mN
  • 33. 33 SOLO ∑      −+∑      −+∑      −+∑= openings i Ciopeniflow openings i Ciopeniflow openings i iopeniflowiflowext I C RRmVVmVVmF td Vd m         ˆˆ 2 ˆˆ External Forces Equations (continue -2) ( ) ( ) ( ) ( ) ( ) ∑∫∫∑∫∫∑ ++−+=+⋅+= j j tStvj j tStv ext FdstfnpdvgFsddvgF  11ρσρ ( ) ( ) ( ) 0111 0 =⋅∇== ∫∫∫ ∞∞∞ tv Gauss tStS dvnpdsnpdsnp   Since the pressure far away from the body is constant∞p Let add this equation to the previous one ( ) ( ) ( ) ( )[ ] ( ) ∑∫∑∫∫∑ ++−+=+⋅+= ∞ j j tSj j tStv ext FdstfnpptmgFsddvgF  11σρ ( ) ( )[ ] ( )[ ] ∑+∫∫ ∑ ∫∫ +−++−+= ∞∞ j j S openings i S Fdstfnppdstfnpptmg W iopen  1111 Finally since on Sopen i (t) the openings (i=1,2,…) the friction force f = 0 ( ) ( )[ ] ( ) ∑+∫∫ ∑ ∫∫ −++−+=∑ ∞∞ j j S openings i S ext FdsnppdstfnpptmgF W iopen  111 Substitute this equation in EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 34. 34 SOLO External Forces Equations (continue – 3) or ( ) ( )[ ] ( ) ∑      −+∑      −+ ∑+∑       ∫∫ −+     −+∫∫ +−+= ∞∞ openings i iflowCiopen openings i iflowCiopen j j openings i S iflowopeniflow SI C mRRmVV FdsnppmVVdstfnpptmgmV dt d iopenW        ˆˆ 2 1 ˆˆ 11 1 ( ) ( ) ( )∑∑∑∑∑ −+−++++= openings i iflowCiopen openings i iflowCiopen j j i iThrustAero I C mRRmVVFFFtmgmV dt d     2 where ( )[ ]∫∫∑ +−= ∞ WS Aero dstfnppF  11: Aerodynamic Forces ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowiThrust dsnppmVVF    1 ˆˆ : Thrust Forces EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 35. 35 SOLO External Forces Equations (continue – 4) Let substitute ( ) ∑∑∑∑∑      −+     −++++= openings i iflowCiopen openings i iflowCiopen j j i iThrustAero I C mRRmVVFFFtmgmV dt d     ˆˆ 2 in CIO O C I C V td Vd td Vd a    ×+== ←ω to obtain RIGID-BODY TERMSmV td Vd CIO O C         ×+ ←   ω ∑∑ −         ×+− ← openings i iflowCiopen openings i iflowCiopenIO O Ciopen mrmr td rd      ,, , ˆˆ ˆ 2 ω FLUID-FLOW TERMS AERODYNAMIC & PROPULSIVE∑∑ += i iThrustAero FF  v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  ∑+= j jFmg  GRAVITATIONAL & DISCRETE TERMS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 36. 36 SOLO External Moments Equations (continue – 5) The moments of the external forces relative to the point O are given by ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑ k k j jOj tS OS tv OOext MFRRsdRRdvgRRM  σρ ~ , ( )( ) ( ) ( ) ( ) ( ) ∑+∑ ×−+∫ +−×−+×      ∫ −= k k j jOj tS OS tv O MFRRdstfnpRRgdvRR  11ρ Let add to this equation the following ( ) ( ) ( ) ( ) 01 0 5 =−×∇=×− ∫∫∫∫ ∞∞ V OS GGauss tS OS dvRRpdsnpRR     to obtain ( )( ) ( ) ( )[ ] ( ) ( ) ∑+∑ ×−+∫ +−×−+×      ∫ −=∑ ∞ k k j jOj tS OS tv OOext MFRRdstfnppRRgdvRRM  11, ρ ( ) ( ) ( )[ ] ( ) ( )  ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫         +−×−++−×−+×−= ∞∞ k k j jOj S openings i S Son OOOC MFRR dstfnppRRdstfnppRRgmRR W iopen W   1111 0 v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 37. 37 SOLO ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫ −×−++−×−+×−=∑ ∞∞ k k j jOj S openings i S OOOCOext MFRR dsnppRRdstfnppRRgmRRM W iopen   111, ∑       −×      −+×+∑= openings i iflowOiflowOiopenO I O Oext I O mVVRRV td cd M td Hd    ˆˆ , , External Moments Equations (continue -6) Using together with we obtain ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ×+++×= k k j jOj openings i iflowOiopenOiopen O I O openings i OiThrustOAeroO I O MFRRmVVRR V td cd MMgc td Hd        ˆˆ , ,,, , ∑      −×     −+∑      −×     −+ ×+∑= openings i iflowOiopenOiopen openings i iflowiopeniflowOiopen O I O Oext mVVRRmVVRR V td cd M      ˆˆˆˆˆ , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 38. 38 SOLO External Moments Equations (continue -7) where ( ) ( )[ ]∫∫∑ +−×−= ∞ WS OOAero dstfnppRRM  11:, Aerodynamic Moments ( ) ( )∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOiThrust dsnppRRmVVRRM    1 ˆˆˆ :, Thrust Moments on the opening i discrete forces exerting by the surrounding at point∑ j jF  jR  ∑ k kM  discrete moments exerting by the surrounding v(t) I ds R  CR  dm C ( )tS 2openS 1openS g  σ  n  1 t  1 OR  O Or,  OCr ,  jR  jF  kM  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 39. 39 SOLO ( ) Itm O O O I IO OIO I O I O OO I O I O dm td rd r td d td d I td Id td Vd cV td cd td Hd ∫         ×+⋅+⋅+×+×= ← ← , ,, , , ,,       ω ω External Moments Equations (continue -8) Using together with we obtain ( ) ( ) ∫∫         ××+         ×+⋅×+⋅+⋅ ←←←← ← tm O O OIO Otm O O OIOOIOIO O O O IO O dm td rd rdm td rd r td d I td Id td d I , , , ,, , ,      ωωωω ω ( ) ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ×+++×−= k k j jCj openings i iflowOiopenOiopen O I O openings i OiThrustOAeroOC I O MFRRmVVRR V td cd MMgmRR td Hd      ˆˆ ,, , ( ) ∑∑∑ ∑∑ +×−+     −×     −+ ++         −×= k k j jCj openings i iflowOiopenOiopen openings i OiThrustOAero I O O MFRRmVVRR MM td Vd gc       ˆˆ ,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Table of Content
  • 40. 40 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ∑∑ ∫∫∫       === openings i iopen openings i S iflow tm td md mdmd td d tm iopen  MASS EQUATION LINEAR MOMENTUM EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( )( ) ( ) mrmRRmRmdRmdRRc OCOC tm O tm OO ,, :  =−=−=−= ∫ ∫ ( ) ( )( ) ∑ ∑ ∫∫∫       −−= ⋅−−−== openings i iflowCiopenC openings i S SCC tm mRRmV sdVRRmVmdVP iopen    ˆ : , ρ ( ) ( )( ) ∑ ∑ ∫∫∫       −−+= ⋅−−−+== openings i iflowOiopenO I O openings i S SOO I O tm mRRmV td cd sdVRRmV td cd mdVP iopen       ˆ : , , , ρ First Moment of Inertia Relative to O 0,  =Cc ( ) ( ) ( ) ( ) ( ) ( ) 0: =−→== ∫ ∫ ∫ ∫ tm C tm tv tv C mdRR m mdR vd vdR tR    ρ ρ Mass Centroid ( )       = ≠ ⋅− = ∫∫ 00 0: ˆ , iflow iflow iflow S S iopen mif mif m sdVR R iopen      ρ
  • 41. 41 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM (CONTINUE – 1) FORCE EQUATIONS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ( ) ( ) ∑∑∫∫∫∫ =++== ext j j tvtv ext tv I tm I FFfdfddv tD VD dm tD VD      0 int ρ ( ) ( ) ( ) ∑∑∑ ∫∫∫∫ +=⋅−+== openings i iflowiflowext openings i S S tm II tm I mVFsdVVmd Dt VD mdV td d P td d iopen     ,, ˆ ρ ( ) ( ) ∑∫∫∑ +⋅+= j j tStv ext FsddvgF  σρ ∑      −+∑      −+∑      −+∑= openings i iflowCiopen openings i iflowCiopen openings i iflowiopeniflowext I C mVVmRRmVVFm td Vd        ˆ 2 ˆˆˆ ∑      −−∑      −−+= openings i iflowCiopen openings i iflowCiopenC I C I mVVmRRmVm td Vd td Pd        ˆˆ ( )dstfnpsdTsdnsd  111 +−==⋅=⋅ σσ  int fdfddv tD VD ext mdI   +=ρ
  • 42. 42 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 2) ( )[ ]∫∫∑ +−= ∞ WS Aero dstfnppF  11: AERODYNAMIC FORCES ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowiThrust dsnppmVVF    1 ˆˆ : THRUST FORCES EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM RIGID-BODY TERMSmV td Vd CIO O C         ×+ ←   ω ∑−∑         ×+− ← openings i iflowiopen openings i iflowiopenIO B iopen mrmr td rd      ˆˆ ˆ 2 ω FLUID-FLOW TERMS GRAVITATIONAL, AERODYNAMIC, PROPULSIVE & ∑∑ ++= i iThrustAero FFmg  ∑+ j jF  DISCRETE TERMS FORCE EQUATIONS (CONTINUE – 1)
  • 43. 43 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 3) ABSOLUTE ANGULAR MOMENT RELATIVE TO A REFERENCE POINT O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , O different from body centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors, Shafts • Fluids • Elasticity ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1:  2nd Moment of Inertia Dyadic of all the mass m(t) relative to O PrHH OCCO  ×+= ,,,
  • 44. 44 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 4) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( )( ) ( )( ) ( ) ∑∑∫∫∑∑ +×−+⋅×−+×−=×−= k k j jOj tS O tv O i iextOiOext MFRRsdRRdvgRRfdRRM  σρ, ( )( ) ( )( ) ( ) ( ) ( )∑ ∫∫∫∫∫ ⋅−×−+×−×−=×−= openings i S SO P tv O tv I O REYNOLDS I tv O I O iopen sdVVRRvdVVvd Dt VD RRvdVRR td d H td d       ,, ρρρρ O I O openings i iflowOiflowOiopenOext I O V td cd mVVRRMH td d     ×+      −×      −+= ∑∑ , ,, ˆˆ ( )( ) ( )( ) ∫∫∑ ×−=×−= tv I O tm I OOext dv tD VD RRdm tD VD RRM ρ     , ∑∑       −×      −+= openings i iflowCiflowCiopenCext I C mVVRRMH td d   ˆˆ ,, ( ) ( ) ( ) int, fdRRfdRRvd tD VD RRMd OextO I OO    ×−+×−=×−= ρ
  • 45. 45 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 5) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O BODY TERMSIOOIOOIOIOO III ←←←← ⋅+⋅×+⋅ ωωωω  ,,, ( ) ( )                 ××+                 ×+ ∫ ∫ ← tm O O OIO O tm O O O dm td rd r dm td rd r td d , , , ,     ω ROTORS, FLUIDS, SHAFTS, ELASTICITY,… TERMS FLUID CROSSING OPENINGS TERMS ∑         ×+×− ← openings i iflowOiopenIO O Oiopen Oiopen mr td rd r     , , , ˆ ˆ ˆ ω AERODYNAMIC & PROPULSIVE ∑∑ += i OiThrustOAero MM ,,  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ∑+∑ ×−+ k k j jOj MFRR  DISCRETE FORCES & MOMENTS TERMS         −×+ I O O td Vd gc   , NON-CENTROIDAL MOMENTS TERMS
  • 46. 46 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 6) ( ) ( )[ ]∫∫∑ +−×−= ∞ WS OOAero dstfnppRRM  11:, AERODYNAMIC MOMENTS RELATIVE TO O ( ) ( )[ ]∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOiThrust dsnppRRmVVRRM    1 ˆˆˆ :, THRUST MOMENTS RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM discrete forces exerting by the surrounding at point∑ j jF  jR  ∑ k kM  discrete moments exerting by the surrounding Table of Content
  • 47. 47 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System Kinetic Energy as a Function of Parameters Defined at a Reference Point O ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 Kinetic Energy of the System Let choose a reference point O and use: OO rRR ,  += I O O I O I O I td rd V td rd td Rd td Rd ,,    +=+= We can write ( ) ∫         +⋅         += tm I O O I O O md td rd V td rd VT ,, 2 1     ( ) ( ) ( ) ( ) ∫∫∫ ⋅++⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVV ,,, 2 1 2 1   (a) (b) (c) Let develop each of the three parts of this expression
  • 48. 48 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 1) (a) ( ) ( ) ( ) mVVmdVV OO tm OO  ⋅=∫⋅ 2 1 2 1 ( ) ∫⋅ tm I O O md td rd V ,   (b) Use Reynolds’ Transport Theorem when we differentiate ( ) OOC tm O cmrmdr ,,,  ==∫ ( ) ( ) I O openings i ifluidOiopen tm I O REYNOLDS I tm O td cd mrmd td rd mdr td d , , , , ˆ      =+=      ∑∫∫ Therefore ( ) ∑−=∫ openings i ifluidOiopen I O tm I O mr td cd md td rd    , ,, ˆ and ( ) ∑⋅−⋅=∫⋅ openings i ifluidOiopenO I O O tm I O O mrV td cd Vmd td rd V     , , ˆ ∑⋅−         ×+⋅= ← openings i ifluidOiopenOOIO O O O mrVc td cd V    , ˆω
  • 49. 49 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 2) (c) ( ) ( ) ∫∫         ×+⋅         ×+=⋅ ←← tm OBIO O OB OBIO O OB tm I O I O B mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ∑ ∫         ×+×+⋅                       ×++×++ ←←←← rotors m CirotorORiOCIO O OC td rd CirotorORi Ri Cirotor td rd OCIO O OC Ri RiRi Ri Ii RiCirotor Ri Ri I ORiC Ri Ri dmrr td rd r td rd r td rd ,, , , 0 , , , , , 2 1                ωωωω ( ) ∫         ×+⋅         ×+= ←← tm OIO FrozenRotor O O OIO FrozenRotor O O mdr td rd r td rd , , , , 2 1     ωω ( ) ( )∑ ∫∑ ∫ ×⋅×+⋅         ×+ ←←← rotors m CirotorORiCirotorORi rotors I OC m CirotorORi Ri RiRi Ri Ri Ri dmrr td rd dmr ,, , 0 , 2 1       ωωω ( ) ∫ ⋅= tm FrozenRotor O O FrozenRotor O O md td rd td rd ,, 2 1  (c1) ( ) ( ) ∫ ×⋅         + ← tm OIO FrozenRotor O O mdr td rd , ,   ω (c2) ( ) ( ) ( ) ∫ ×⋅×+ ←← tm OIOOIO mdrr ,, 2 1  ωω (c3) ( ) ( )∑ ∫ ×⋅×+ ←← rotors m CirotorIRiCirotorIRi Ri RiRi dmrr ,, 2 1  ωω (c4)
  • 50. 50 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 3) Let develop those equations ( ) ∫ ⋅ tm O O O O md td rd td rd ,, 2 1  (c1) (c2) ( ) ( ) ( ) ( ) ∫∫         ⋅×=×⋅         ←← tm O O OIO tm OIO O md td rd rmdr td rd O , ,, ,    ωω ( ) ( ) ∫∫         ×⋅=         ×⋅= ←← tm O O OIO tm O O OIO md td rd rmd td rd r , , , ,     ωω (c3) ( ) ( ) ( ) ( ) ( ) ( ) ∫ ×⋅×−=∫ ×⋅× ←←←← tm IOOOIO tm OIOOIO mdrrmdrr ωωωω  ,,,, 2 1 2 1 ( )[ ] ( ) ( )[ ]( ) IO tm OOOOIO tm IOOOIO mdrrrrmdrr ←←←← ⋅∫ −⋅⋅=∫ ××⋅−= ωωωω  ,,,,,, 1 2 1 2 1 IOOIO I ←← ⋅⋅= ωω  , 2 1 ( )[ ] ( ) ∫ −⋅= tm OOOOO mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the System, Relative to O
  • 51. 51 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫∫∫ ⋅+⋅+⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVV     ,, , ˆ 2 1 ω Since the kinetic energy is independent of the chosen reference point O, the previous relation is invariant to O. ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 ( ) ( ) ∫∫         ×⋅+⋅+ ← tm O O OIO tm O O O O md td rd rmd td rd td rd , , ,, 2 1    ω IOOIO I ←← ⋅⋅+ ωω  , 2 1 Table of Content
  • 52. 52 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Quasi-Lagrangian Equations Let perform the following calculations ∑−+=∑−         ×++= ∂ ∂ ← openings i ifluidOiopen I O O openings i ifluidOiopenOIO O O O O mr td cd mVmrc td cd mV V T          , , ,, , ˆˆω ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVVT    ,, ˆ 2 1 ω ( ) ( ) IOOIO tm O O OIO tm O O O O Imd td rd rmd td rd td rd ←←← ⋅⋅+∫         ×⋅+∫ ⋅+ ωωω     , , , ,, 2 1 2 1 Since P V T O   = ∂ ∂ Also ( ) OOIOO tm O O O IO VcImd td rd r T     ×+⋅+∫         ×= ∂ ∂ ← ← ,, , , ω ω Since O IO H T ,   = ∂ ∂ ←ω ( ) OOIOO tm O O OO VcImd td rd rH    ×+⋅+∫         ×= ← ,, , ,, ω We found ∑−+= openings i ifluidOiopen I O O mr td cd mVP     , , ˆ
  • 53. 53 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Quasi-Lagrangian Equations (continue – 1) Let develop those equations in frame O P V T O   = ∂ ∂ O IO H T ,   = ∂ ∂ ←ω We found ( ) ∑−=∫=∑ openings i iflowiflow I tm I ext mV td Pd md tD VD F     ˆ ∑+∑=       ∂ ∂ openings i iflowiflowext IO mVF V T td d    ˆ ∑ ×+∑ ×+= openings i iflowiflowOiopenOOext I O mVrVPMH td d   ˆˆ ,,, ∑ ×∑ +=       ∂ ∂ ×+       ∂ ∂ ← openings i iflowiflowOiopenOext O O IIO mVrM V T V T td d    ˆˆ ,, ω [ ] ( ) ( ) ( ) ( ) ∑+∑=       ∂ ∂ ×+       ∂ ∂ ← openings i iflow O iflow O ext O O O IO OO mVF V T V T td d      ˆ ω [ ]( ) ( ) [ ]( ) ( ) ( ) [ ]( ) ( ) ∑∑ ×+=         ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ ← ← ← openings i iflow O iflow O Oiopen O Oext O O O O O IO O IO OIO mVrM V T V TT td d        ˆˆ ,, ω ω ω This equation is obtained, also, using the LAGRANGE’s EQUATIONS OF MOTION. Table of Content
  • 54. 54 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow Let compute the time derivative of the kinetic energy, using the REYNOLD’s Transport Theorem.      openingsthethroughaddedenergykinetic openings i ifluid iopenII m II REYNOLDS m II m td Rd td Rd md td Rd td Rd md td Rd td Rd td d td Td ∑         ⋅+∫         ⋅=∫         ⋅= 2 1 2 1 2 2 where ( )        = ≠ ∫∫ ⋅−         ⋅ =         ⋅ 00 0 2 1 , iflow iflow iflow S S II iopenII m m m sdV td Rd td Rd td Rd td Rd iopen       ρ Is the mean value of the kinetic energy flow that crosses through the openings .iopenS
  • 55. 55 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 1) Let express the kinetic energy flow as function of the parameters of the reference point O. OIO O O O I O I O I r td rd V td rd td Rd td Rd , ,,     ×++=+= ←ω int2 2 fdfdmd td Rd ext I   += ( )         openingsthethroughaddedenergykinetic openings i ifluid iopenII m extOIO O O O m td Rd td Rd fdfdr td rd V td Td ∑         ⋅+∫ +⋅         ×++= ← 2 1 int, , ω ( ) IOOIOextO O O ext O O extO fdrfdrfd td rd fd td rd fdfdV ←← ⋅         ∫ ×+⋅∫ ×+∫ ⋅+∫ ⋅+         ∫ ∫+⋅= ωω          0 int,, 0 int ,, 0 int      openingsthethroughaddedenergykinetic openings i ifluid iopenII m td Rd td Rd ∑         ⋅+ 2 1 ( ) ( )        openingsthethroughaddedenergykinetic openings i ifluid iopenII IOOextext O O extO m td Rd td Rd Mfd td rd FV ∑                 ⋅+⋅∑+∫ ⋅+∑⋅= ← 2 1 , , ω
  • 56. 56 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 2) where  ForcesDiscrete l l ForcesSurface S ForcesBody V extext FsdTvdGfdF ∑+∫+∫=∫=∑      ρ:           MomentsDiscrete k k Moments ForcesDiscrete l lOl MomentsSurface S O MomentsBody V OextOOext MFrsdTrvdGrfdrM ∑+∑ ×+∫ ×+∫ ×=∫ ×=∑ ,,,,, : ρ G  - body force per unit mass ( )3 /mN nT  1⋅= σ - force per unit surface ( )2 /mN σ  - stress tensor (dyadic) ( )2 /mN ∑ l lF  - discrete forces applied to the system at the position lR  ( )N ∑ k kM  - discrete moments applied to the system ( )mN ⋅
  • 57. 57 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 3) Since the kinetic energy is invariant to the reference point O, let choose O = C (system centroid) ( ) ( )                openingsthethroughaddedenergyKinetic openings i ifluid iopenII momentsexternal bydoneWork ICCext bodytheofrigiditynonof becausedoneWork ext C C forcesexternal bydoneWork extC m td Rd td Rd Mfd td rd FV td Td ∑∑∫∑                 ⋅+⋅+⋅+⋅= ← − 2 1, ω       Openings trough Added Massof Energy Internal flow Openings trough Added Energy Kinetic flow System to Added Flow Heat Change Work m II Change Energy Kinetic Change Energy Internal td Ud td Td td Qd td Wd md td Rd td Rd e td d td Td td Ud +++=         ⋅+=+ ∫ 2 1 Change in Internal Energy + Change in Kinetic Energy = Change in Total Energy due to Surroundings From thermodynamics we have:
  • 58. 58 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Energy Flow (continue – 4) -internal energy of the molecules (vibration, rotation, translation). ∫= m mde td Ud ∫         ⋅= m II md td Rd td Rd td d td Td  2 1 ( ) ( )                   ForcesSurface S I ForcesBody v I momentsexternal bydoneWork ICCext bodyrigidnon anondoneWork ext C C forcesexternal bydoneWork extC sdT td Rd vdG td Rd Mfd td rd FV td Wd ∫∫∑∫∑ ⋅+⋅=⋅+⋅+⋅= ← − ρω,  ∫∫ ⋅− ∂ ∂ = S Surface throughRate Radiation ConductionV Rate Transfer Heat sdqvd t Q td Qd  ρ ∑                 ⋅= openings i ifluid iopenII flow m td Rd td Rd td Td   2 1 - kinetic energy change. - rate of work done by the surroundings on the system -rate of heat transfer and conduction/radiation added to the system. - kinetic energy added to the system through the openings. td Ud flow -internal energy added by the flow entering the system through the openings. Table of Content
  • 59. 59 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ OzˆLet assume that the variable mass has internal rigid rotors and other rigid moving parts (shafts...) Bm - mass of the body (excluding rotors) irotorm - mass of the rotor i m - mass of the system ∑+= rotors irotorB mmm BC - centroid of the body located at , relative to the reference point O.OCB r ,  RiC - centroid of the rotor i located at , relative to the reference point O.OCRi r ,  C - centroid of the system located at , relative to the reference point O.OCr ,  Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors)
  • 60. 60 SOLO I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ OBc ,  - first moment of inertia of the body, relative to O. ORi c ,  - first moment of inertia of the rotor i, relative to O. Oc,  - first moment of inertia of the system, relative to O. ∑+ ∑+ =→ =∑+=∑ ∫+∫=∫= rotors irotorB rotors irotorOCBOC OC OC rotors irotorOCBOC rotors m O m O m OO mm mrmr r mrmrmrmdrmdrmdrc RiB RiB RB ,, , ,,,,,,, :    Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 1) EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH
  • 61. 61 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ BP  - body linear momentum ∑      −−+= openings i iflowOiopenBO I OB B mRRmV td cd P     ˆ, irotorP  - rotor i linear momentum irotorO I C irotorO I ORi irotor mV td rd mV td cd P Ri         +=+=      , P  - system linear momentum ∑      −−+=∑+= openings i iflowOiopenO I O rotors i irotorB mRRmV td cd PPP     ˆ, Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 2)
  • 62. 62 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Absolute Angular Momentum Relative to a Reference Point O (continue -2) ( )( ) ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, :   ωρ OO Vc  ×= , Difference between O and System centroid C IOOI ←⋅+ ω  , Rotation from I to O coordinates ( ) ∫         ×+ tm O O O dm td rd r , ,   Non-rigidity of System • Rotors, • Moving parts (Pistons,…) • Fluids • Elasticity
  • 63. 63 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let go in more details by taking the rotors in consideration For a point on the rotor i, we have I R  CR  C ( )tS OR  OOCr  Bxˆ Bzˆ shaftr  rotorr  Byˆ Ixˆ Iyˆ Izˆ Cshaftr  Crotorr  OyˆOxˆ Ozˆ Ri Ri Ri ORi ORi RiORi RiORi CirotorIRi C Cirotor CIO O C O I Cirotor I C I O I irotor irotor CirotorCOirotor r td rd r td rd V td rd td rd td Rd td Rd V rrRR , 0 , , , , , , , :          ×++×++= ++== ++= ←← ωω Substitute those in ( )( ) ∫ ×−= tm OO dmVRRH  , , to obtain ( )( ) ( ) ∑+=∑ ∫ ×+∫ ×=∫ ×−= rotors ORiOB rotors m I irotor Oirotor tm I B OB tm OO HHdm td Rd rdm td Rd rdmVRRH RiB ,,,,,      ( ) ∑ ∫         ×+×++×+ ∫         ×++×= ←← ← rotors m CirotorIRiOCIO O OC OOirotor tm OBIO O OB OOB irotor RiRi Ri B dmrr td rd Vr dmr td rd Vr ,, , , , , ,       ωω ω where OBH ,  - body absolute angular momentum relative to the reference point O. ORiH ,  - rotor i absolute angular momentum relative to the reference point O. Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 3)
  • 64. 64 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH OBH ,  - body absolute angular momentum relative to the reference point O. ( ) ( ) ( ) ( ) ( ) ∫ ×+∫ ××+×∫= ∫         +×+×= ← ← tm O OB OB tm OBIOOBO tm OB tm O OB OBIOOOBOB BBB B dm td rd rdmrrVdmr dm td rd rVrH , ,,,, , ,,,     ω ω (a1) (b1) (c1) ( ) OB tm OB cdmr ,,  =∫(a1) (b1) ( ) OOBOOBBO tm OB VcVrmVdmr B  ×=×=×         ∫ ,,, ( ) ( ) ( )[ ]( ) IOOBIO tm OBOBOBOB tm OBIOOB Idmrrrrdmrr BB ←←← ⋅=⋅∫ −⋅=∫ ×× ωωω  ,,,,,,, 1 ( )[ ]∫ −⋅= Bm OOOOOB mdrrrrI ,,,,, 1:  where Second Moment of Inertia of the body relative to O ( ) ( ) ( ) ( ) ∫         ×+∫ ××+×      ∫= ← tm O OB OB tm OBIOOBO tm OBOB BBB dm td rd rdmrrVdmrH , ,,,,,   ω ( ) ∫         ×+⋅+×= ← tm O OB OBIOOBOOB dm td rd rIVc ,, ,,,   ω Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 4)
  • 65. 65 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH ORiH ,  - rotor i absolute angular momentum relative to the reference point O. ∫         ×+×++×=∫         ×= ←← Ri RiRi Ri Ri m CirotorIRiOCIO O OC OOirotor m I irotor OirotorORi mdrr td rd Vrmd td Rd rH ,, , ,,,      ωω ∫         ×+×++×= ←← R RiRi Ri Ri m CirotorIRiOCIO O OC OCirotor mdrr td rd Vr ,, , ,    ωω ∫         ×+×++×+ ←← R RiRi Ri Ri m CirotorIRiOCIO O OC OOC mdrr td rd Vr ,, , ,    ωω irotorOCCRi PrH RiRi  ×+= ,, ∫         ×+×++×= ←← Ri RiRi Ri RiRi m CrotorIRiOCIO O OC OCirotorCRi mdrr td rd VrH ,, , ,,    ωω ( ) IRi m CirotorCirotorOCIO O OC O m Cirotor Ri RiRiRi Ri Ri Ri mdrrr td rd Vmdr ←←       ∫ ××−+         ×++×             ∫= ωω        ,,, , 0 , ( )[ ] IRiCirotorIR m CirotorCirotorCirotorCirotor Ri Ri RiRiRiRi Imdrrrr ←← ⋅=⋅∫ −⋅= ωω  ,,,,, 1 ( )[ ]∫ −⋅= Ri RiRiRiRiRi m CirotorCirotorCirotorCirotorCirotor mdrrrrI ,,,,, 1:  Second Moment of Inertia of the rotor i relative to it’s centroid RiC Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 5)
  • 66. 66 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Therefore ( )[ ]( ) Ri O OC OOCIOOirotorORiCirotor Ri B OC OOCIORiOCOCOCOCCirotorORiCirotor ORiIORi O OC RiOOCIRiCirotorORi m td rd VrII m td rd VrmrrrrII cm td rd mVrIH Ri RiRi Ri RiRiRiRiRiRiRi Ri RiRi         +×++⋅=         +×+−⋅++⋅=         ×++×+⋅= ←← ←← ←← , ,,, , ,,,,,,, , , ,,, 1        ωω ωω ωω Second Moment of Inertia of the rotor i relative to O ( )[ ] RiOCOCOCOCCirotorOirotor mrrrrII RiRiRiRiRi ,,,,,, 1:  −⋅+= Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 6)
  • 67. 67 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH The system Absolute Angular Momentum relative to O, is IOOB m O OB OBOOB rotors ORiOBO Imd td rd rVcHHH B ←⋅+         ×+×=+= ∫∑ ω    , , ,,,,, ∑         +×+⋅∑+∑ ⋅+ ←← rotors Ri O OC OOCIO rotors Oirotor rotors BRiCirotor m td rd VrII Ri RiRi , ,,,   ωω or ∑∫ ∑         ×+         ×+ ⋅+⋅+×= ←← rotors Ri O OC OC m O OB OB rotors ORiCirotorIOOOOO m td rd rmd td rd r IIVcH Ri Ri B Ri , , , , ,,,      ωω Second Moment of Inertia of the system relative to O∑+= rotors OirotorOBO III ,,, :  Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 7)
  • 68. 68 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH If we compare with ∑∫ ∑         ×+         ×+ ⋅+⋅+×= ←← rotors Ri O OC OC m O OB OB rotors ORiCirotorIOOOOO m td rd rmd td rd r IIVcH Ri Ri B Ri , , , , ,,,      ωω ( ) ∫         ×+⋅+×= ← tm O O OIOOOOO dm td rd rIVcH , ,,,,   ω we can see that ∑∫ ←⋅=         × rotors ORiCirotor m O O O Ri Imd td rd r ω    , , , ∫         ×+ Bm O OB OB md td rd r , ,   ∑         ×+ rotors Ri O OC OC m td rd r Ri Ri , ,   - Rotors Absolute Angular Moment Relative to O - Body Non-rigidity Elasticity of the System Sloshing of Liquids Moving Parts (Pistons,…) - Movement of Relative to O RiC Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 8)
  • 69. 69 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Let define ∑         ×+∫         ×=∫         × rotors Ri O OC OC m O Ob Ob m FrozenRotors O O O m td rd rmd td rd rmd td rd r Ri Ri B , , , , , , :       We obtain ∫∑         ×+⋅+⋅+×= ←← m FrozemRotor O O O rotors ORiCirotorIOOOOO md td rd rIIVcH Ri , ,,,,,   ωω Let compute OIO O O I O H td Hd td Hd , ,,   ×+= ←ω ( ) O m FrozemRotor O O O rotors ORiCirotor rotors ORiCirotorIOOIOO I OO md td rd r td d IIIIVc td d RiRi                 ×+ ⋅+⋅+⋅+⋅+×= ∫ ∑∑ ←←←← , , 0 ,,,,,        ωωωω                 ××+      ⋅×+⋅×+ ∫∑ ←←←←← m FrozemRotor O O OIO rotors ORiCirotorIOIOOIO md td rd rII Ri , ,,,   ωωωωω Absolute Angular Momentum Relative to a Reference Point O (Body Containing Rotors - 9)
  • 70. 70 SOLO ( ) ( ) I tm O O O I j ORjCjrotor I IO OIO I O I O OO I O I O dm td rd r td d I td d td d I td Id td Vd cV td cd td Hd Rj ∫         ×+∑+⋅+⋅+×+×= ← ← ← , ,,, , , ,,      ω ω ω External Moments Equations (continue -8) Using together with we obtain ( ) ( )∑+∫         ×+⋅×+⋅+⋅ ←←←← ← j I ORjCjrotor I tm FrozenRotors O O OIOOIOIO O O O IO O Rj I td d dm td rd r td d I td Id td d I ωωωω ω     , , ,, , , ( ) ( ) ∑+∑ ×−+∑      −×     −+ ×+∑+∑+×−= k k j jCj openings i iflowOiopenOiopen O I O openings i OTiOAOC I O MFRRmVVRR V td cd MMgmRR td Hd      ˆˆ ,, , ( ) ∑+∑ ×−+∑      −×     −+ ∑+∑+         −×= k k j jCj openings i iflowOiopenOiopen openings i OTiOA I O O MFRRmVVRR MM td Vd gc       ˆˆ ,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Table of Content
  • 71. 71 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 1) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O RIGID-BODY TERMS IOOIOOIOIOO III ←←←← ⋅+⋅×+⋅ ωωωω  ,,,     ∑ ⋅×+∑ ⋅+ ←←← j OjrotorCrotorjIO j OjrotorCrotorj RjRj II ωωω  ,, ROTORS TERMS ( ) ( )         ∫         ××+         ∫         ×+ ← tm FrozenRotor O O OIO O tm FrozenRotor O O O dm td rd r dm td rd r td d , , , ,     ω BODY FLUIDS, MOVING PARTS, ELASTICITY,… TERMS FLUID CROSSING OPENINGS TERMS ∑         ×+×− ← openings i iflowOiopenIO O Oiopen Oiopen mr td rd r     , , , ˆ ˆ ˆ ω AERODYNAMIC & PROPULSIVE ∑+∑= i OTiOA MM ,,  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ∑+∑ ×−+ k k j jOj MFRR  DISCRETE FORCES MOMENTS TERMS         −×+ I O O td Vd gc   , NON-CENTROIDAL MOMENTS TERMS
  • 72. 72 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 4) In the same way (c4) ( ) ( ) ( )[ ] ∑ ∑∑ ∫ ←← ←←←← ⋅⋅= ⋅−⋅⋅=×⋅× rotors ORiCrotorORi rotors ORiCirotorCirotorCirotorCirotorORi rotors m CirotorORiCirotorORi Ri RiRiRiRi Ri RiRi I rrrrdmrr ωω ωωωω   , ,,,,,, 2 1 1 2 1 2 1 Therefore (c) ( ) ( ) ∫∫         ×+⋅         ×+=⋅ ←← tm OIO O O OIO O O tm I O I O mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ( ) ∫ ⋅= tm FrozenRotor O O FrozenRotor O O md td rd td rd ,, 2 1  ( ) ∫         ×⋅+ ← tm FrozenRotor O O OIO md td rd r , ,   ω IOOIO I ←← ⋅⋅+ ωω  , 2 1 ORiCirotorORi Ri I ←← ⋅⋅+ ωω  , (c1) (c2) (c4)(c3)
  • 73. 73 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLDS’ TRANSPORT THEOREM APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫∫∫ ⋅+⋅+⋅= tm I O I O tm I O O tm OO md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−         ×+⋅+⋅= ← openings i ifluidOiopenOOIO O O OOO mrVc td cd VmVV     ,, , ˆ 2 1 ω Since the kinetic energy is independent of the chosen reference point O, the previous relation is invariant to O. ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 ( ) ( ) ∫∫         ×⋅+⋅+ ← tm FrozenRotor O O OIO tm FrozenRotor O O FrozenRotor O O md td rd rmd td rd td rd , , ,, 2 1    ω ∑ ⋅⋅+⋅⋅+ ←←←← rotors ORiCirotorORiIOOIO Ri II ωωωω  ,, 2 1 2 1 Table of Content
  • 74. 74 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM REYNOLD’s TRANSPORT THEOREM APPROACH References 1. Shames, I.H., “Mechanics of Fluids”, 2nd Ed., McGraw-Hill, 1982 Table of Content
  • 75. January 5, 2015 75 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 –2013 Stanford University 1983 – 1986 PhD AA