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Equation of motion of a variable mass system3

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Solo Hermelin

The document discusses the equations of motion for a variable mass system using a Lagrangian approach, highlighting key concepts such as generalized forces, principal coordinate frames, and energy forms. It outlines the derivation of Lagrange's equations, the role of kinetic and potential energy, and the methods for analyzing the system's dynamics. Additionally, the text covers aspects of structural modeling, modal analysis, and the orientation of the body frame relative to the inertial frame.

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1 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN 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 (see Power Point Presentation) • Lagrangian Approach (this Power Point Presentation)
3 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH TABLE OF CONTENT • Generalized Forces Joseph-Louis Lagrange 1736-1813 • Lagrange’s Equations of Motion • Principal Coordinate Frames • Inertial Coordinate Frame • Body Coordinate Frame • Body Mean System Axes • Orientation of Body Frame • Kinetic Energy of the System • Potential Energy of the System • Elastic Potential Energy • Gravitational Potential Energy • Computation of Lagrange’s Equations in Body Coordinates • Derivation of Equations of Motion • Summary of the Equations of Motion of a Variable Mass System • References • Appendix A: Lagrange Equations
4 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Lagrange’s Equations of Motion The Lagrange’s Equations of Motion for a dynamic system are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ - system kinetic energy.T - system potential energy.U - generalized coordinates (i=1,2,…, number of degrees of freedom of the system).iξ - generalized force along the generalized coordinate given by.iξiQ ( ) ( )i i W Q ξδ δ ∂ ∂ = -virtual work done on the system by all external forces/moments (excluding those accounted for in the potential energy term) during virtual displacement along all the generalized coordinates. Wδ Table of Contents
5 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames 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. Inertial Coordinate Frame (vector form) orIzIyIx zRyRxRR ˆˆˆ ++=  IzIyIx I zRyRxR td Rd V ˆˆˆ    ++== (vector form) or (vector form) orIzIyIx II zRyRxR td Rd td Vd a ˆˆˆ2 2    ++=== { }T zyx RRRR ,,=  (matrix form) ( ) { }T zyx I RRRV   ,,= (matrix form) ( ) { }T zyx I RRRa  ,,= (matrix form) Table of Contents
6 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 1) Cr,  - Position of the mass element dm relative to C. Body Coordinate Frame The origin of the Body Frame (B) is located at the instantaneous Centroid (C) of the system. 0,  =∫ m C mdr R  - Position of the mass element dm relative to I. CR  - Position of the centroid C relative to I. CC rRR ,  += 0,Cr  - Position of the same mass element dm in the un-deformed system, relative to C. e  - Change in position of the mass element dm due to elastic deformation of the system. err CC  += 0,, Table of Contents
7 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 2) Body Mean System Axes Mean Body System Axes are defined such that the relative linear and angular momentum, due to elastic deformation, are zero at every instant. The Body Mean Axes must satisfy the following: 0  =∫ md td ed Bm 0,   =∫ × md td ed r B m C 0, =∫ ⋅ md td ed td rd B m B C  Table of Contents
8 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Structural Model of the System Assume that the elastic deformations are small, and can be represented in terms the normal un-damped modes of vibration. ( ) ( )∑= ∞ =1i ii tRe ηφ  - are mode shape functions that depend on the position of the mass element of the system.( )Ri  φ - are generalized coordinates giving the magnitude of the modal displacements and are functions of time. ( )tiη Structural Dynamic Analysis (e.g. final element method) provides the mode shape functions component of each element of the system, as well as the vacuo modal frequencies ( ) , for a selected number of modes. ( )Ri  φ iω ii i td d ηω η 2 2 2 −= The mode shape functions are orthogonal. 0 ji m ji md ≠ =⋅∫ φφ  i m ii Mmd =∫ φφ  MV Bx By Bz Wz Wy Wx α β α β Bp Wp Bq WqBr Wr FIRST ELASTIC MODE SECOND ELASTIC MODE
9 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 3) Orientation of the Body Frame The orientation of the Body Frame relative to the Inertial Frame has three degrees of freedom. We will use 3 Euler Angles that define the orientation by three consecutive rotations around the consecutive frame axes. [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc The three basic Euler rotations around axes are described by the rotation matrices: ,3ˆ,2ˆ,1ˆ
10 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 4) Orientation of the Body Frame (continue – 1) Using the basic Euler Angles we can define the following 12 different rotations: (a) six rotations around three different axes: 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ (b) six rotations such that the first and third are around the sam axes, but the second is different: 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→ Suppose that the Transfer Matrix from Inertia to Body is defined by three consecutive Euler Angles: around (unit vector in Inertial Frame), around (unit vector in intermediate frame), around (unit vector in Body Frame). B IC iθ Iiˆ jθ Interjˆ kθ Bkˆ [ ] [ ] [ ] [ ] [ ]TB I B I I B I B B Ikkjjii B I CCCICCC ==→== −1 &θθθ
11 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 5) Orientation of the Body Frame (continue – 2) The angular velocity vector of rotation of the Body frame relative to Inertia frame is: kIjIntriBIB kji θθθω  ˆˆˆ ++=← In Body frame this is: ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] [ ] ( ) k I Ijjiij Intr Intriii B Bk B Ij B Intri B B B IB kjikji θθθθθθθθθω  ˆˆˆˆˆˆ ++=++=← [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ]           =           =           =← k j i k j i I Ijjii Intr Intrii B B B IB DDDkji r q p θ θ θ θ θ θ θθθω          321 ˆˆˆ: { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 321 : ˆˆ:,&ˆˆ:&ˆ: DDDD kkDjjDconstiD I Ijjii B Iji Intr Intrii B Intri B B = ====== → θθθθθθ ( ) [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω             321 333231 232221 131211 : where: or
12 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 6) Orientation of the Body Frame (continue – 3) The velocity vector of the system centroid C is given by: ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 : The following relations will be useful: and ( )           = k j i E θ θ θ θ : andwhere [ ] ( )           − − − =×← 0 0 0 : pq pr qr B IBω  Table of Contents ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω  Appendix [ ] [ ] [ ] ( )B IB TB I TB I CC td d ×= ←ω  Appendix
13 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 Kinetic Energy of the System We have: We can write ( ) ∫         +⋅         += tm I C C I C C md td rd V td rd VT ,, 2 1     ( ) ( ) ( ) ( ) ∫ ⋅+∫+∫⋅= tm I C I C tm I C C tm CC 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 erRrRR CCCC  ++=+= 0,, I C C I C I C I td rd V td rd td Rd td Rd ,,    +=+=
14 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 1) (a) ( ) ( ) ( ) mVVmdVV CC tm CC  ⋅=∫⋅ 2 1 2 1 (b) Use Reynolds’ Transport Theorem when we differentiate ( ) 0,  =∫ tm C mdr Therefore ( ) ∑∫ −= openings i ifluidCiopen tm B C mrmd td rd    , , ˆ and ( ) ( ) ∫         ×+⋅=∫⋅ ← tm CIB B C C tm I C C mdr td rd Vmd td rd V , ,,      ω ( ) ( ) ( ) ∑∫∑ ∫∫∫∫ +=+=      = openings ifluidCiopen tm B C i S C tm B C REYNOLDS B tm C mrmd td rd mdrmd td rd mdr td d iopen        , , , , , ˆ0 ( ) ( ) ∑⋅−=∫⋅=∫⋅ openings ifluidCiopenC tm B C C tm I C C mrVmd td rd Vmd td rd V       , ,, ˆ ( ) ( ) ( ) ∫∫∫ ⋅=             ×⋅+⋅= ← tm B C C tm CIBC tm B C C md td rd VmdrVmd td rd V , 0 , ,        ω
15 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 2) (c) ( ) ( ) ∫         ×+⋅         ×+=∫ ⋅ ←← tm CIB B C CIB B C tm I C I C mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ( ) ∫ ⋅= tm B C B C md td rd td rd ,, 2 1  (c1) ( ) ( ) ∫ ×⋅         + ← tm CIB B C mdr td rd , ,   ω (c2) ( ) ( ) ( ) ∫ ×⋅×+ ←← tm CIBCIB mdrr ,, 2 1  ωω (c3) (c1) ( ) ( ) ∫         +⋅         +∫ =⋅ tm BB C BB C tm B C B C md td ed td rd td ed td rd md td rd td rd  0,0,,, 2 1 2 1 ( ) ( ) ( ) ∫         ⋅         +∫         ⋅         ∫ +⋅= tm BB tm BB C tm B C B C md td ed td ed md td ed td rd md td rd td rd      2 1 2 1 0 0,0,0, ( ) ( ) ( )∑ ∫ ×⋅×+∫ ⋅= ←← rotors m CjrotorBjrotorCjrotorBjrotor tm FrozenRotors B C FrozenRotors B C jrotor mdrrmd td rd td rd ,, ,, 2 1 2 1   ωω ( ) ( )∑ ∫         ⋅×+∫         ⋅+ ← rotors m B CjrotorBjrotor tm BFrozenRotors B C jrotor md td ed rmd td ed td rd          0 , 0 , ω ( ) ∫         ⋅         + tm BB md td ed td ed  2 1 ( ) ∫ ⋅= tm FrozenRotors B C FrozenRotors B C md td rd td rd ,, 2 1  ( )∑       ∫ ××−⋅+ ←⋅ ←← rotors I m BjrotorCjrotorCjrotorBjrotor BjrotorCjrotor jrotor mdrr      ω ωω , ,, 2 1 ( ) ∫ ∑∑ ∞ = ∞ = + tm j i j i ji md td d td d ηη φφ 1 12 1 
16 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( )( ) ∑ ⋅⋅+∫ ∫ ⋅=⋅ ←← rotors BjrotorCjrotorBjrotor tm tm FrozenRotors B C FrozenRotors B C B C B C Imd td rd td rd md td rd td rd ωω   , ,,,, 2 1 2 1 2 1 Kinetic Energy of the System (continue – 3) (c3) (c2) ( ) ∫ ∑ ∞ =       + tm i i i md td d 1 2 2 2 1 η φ ( )[ ]∫ −⋅= jrotorm CjrotorCjrotorCjrotorCjrotorCjrotor mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the Rotor j Relative to C ( ) ( ) ( ) ( ) ( ) ∫         ×⋅=∫         ⋅×=∫ ×⋅         ←←← tm B C CIB tm B C CIB tm CIB B C md td rd rmd td rd rmdr td rd , , , ,, ,      ωωω ( ) ( )        0 , 0, , ∫         ×⋅+∫         ×⋅= ←← tm B CIB tm B C CIB md td ed rmd td rd r ωω ( ) ( )( )∑ ∫ ××⋅+∫         ×⋅= ←←← rotors m CjrotorBjrotorCjrotorIB tm FrozenRotors B C CIB mdrrmd td rd r ,, , ,    ωωω ( ) [ ] [ ] Bjrotor rotors m CjrotorCjrotorIB tm FrozenRotors B C CIB mdrrmd td rd r ←←← ∑       ∫ ××−⋅+∫         ×⋅= ωωω    ,, , , ( ) [ ]∑⋅+∫         ×⋅= ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , ,
17 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 4) (c3) ( )[ ] ( ) ∫ −⋅= tm OOOOO mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the System Relative to O ( ) ( ) ( ) [ ] [ ] ( ) IB tm CCIB tm CIBCIB mdrrmdrr ←←←←       ∫ ××−⋅=∫ ×⋅× ωωωω  ,,,, 2 1 2 1 IBCIB I ←← ⋅⋅= ωω  , 2 1
18 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , Table of Contents
19 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System We consider only (Electromagnetic, Chemical Potentials are not considered) ge UUU += Elastic Deformation Potential eU Gravitational Field Potential gU Table of Contents
20 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System Elastic Deformation Potential eU ( ) ( )∑ ∞ = = 1i iii tle ηφ  ii i td d ηω η 2 2 2 −= i m ii Mmd =∫ φφ  0 ji m ji md ≠ =∫ φφ  ∫ ∑∑∫               −⋅        −=         ⋅−= ∞ = ∞ =m i iii j jj m B e mdmd td ed eU 1 2 1 2 2 2 1 2 1 ηωφηφ   ( ) ( ) ∑∑ ∫∑∑ ∫ ∞ = ∞ = ∞ = ∞ = =        ⋅=        ⋅= 1 22 1 22 1 1 2 2 1 2 1 2 1 i iii i ii m ii j i jii m ij Mmdmd ηωηωφφηηωφφ  iii i e M U ηω η 2 = ∂ ∂ From We obtain From this we obtain Table of Contents
21 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System (continue – 1) Gravitational Field Potential gU ( ) 2 2 01 E Earth EE R R gRRg  −= 22 0 sec/17.32sec/78.9 ftmg == where mREarth 135.378.6= ( ) ( ) ( ) 2/1 2 0 2 02 2 0 2 0 2 2 0 1 1 CECE Earth CE Earth m CE CE Earth CE Earth m E Earth ECCE m CCEg RR R gm R R gmmdrR R R g R R gm md R R gRrRmdgrRU    ⋅ ==⋅+= ⋅+−=⋅+= ∫ ∫∫ ( ) CE CE CE Earth CE CECE Earth C g R R R R gmR RR R gm R U    2 2 02/3 2 0 2 2 1 −= ⋅ −= ∂ ∂ ( )CECE CE Earth C g RgmR R R gm R U   =−= ∂ ∂ 12 2 0 From this we obtain or Table of Contents
22 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces - generalized force along the generalized coordinate given by.iξiQ ( ) ( )i i W Q ξδ δ ∂ ∂ = -virtual work done on the system by all external forces/moments (excluding those accounted for in the potential energy term) during virtual displacement along all the generalized coordinates. Wδ The generalized forces are: PQ - due to position change, relative to inertial system ( ) [ ]T zyx I P RRRR ==  ξ - due to rotation of the system, around its centroid, relative to inertial systemRQ ( ) [ ]T kji E R θθθθξ ==  - due to elastic modal displacementsEQ [ ] T E   4321 ηηηηηξ ==
23 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( ) ( ) ( )∑+= openings i iopenW tStStS Generalized Forces (continue – 1) Virtual Work due to Position Change, Relative to Inertial Frame The virtual work done by change in position is due to pressure distribution and fluid flow through the openings, and to discrete forces applied on the system ( ) RF td Rd mdstfnpW openings j j I ifluid ifluid S P vehicle     δδ ⋅           +           ++−= ∑ ∑∫ ˆ 11 where • 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 ( )2 / mNp - pressure on (normal to) the surface . f - friction force per (parallel to) unit surface .( )2 / mN n  1 - outward unit vector normal to the surface element ds t  1 - local unit vector of tangential stress due to flow on the surface element ds ∑ j jF  - discrete forces applied to the system at the position j R  ( )N is the mean position vector of the flow and of the opening on iopenSifluidR ˆ fluidm - fluid rate flowing through the opening iopenS
24 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 2) Si I CiopenI C I iopenfluid I Ciopen I C I ifluid V td rd V td rd td rd td Rd td Rd , ,,, ˆ ˆˆˆˆ     ++=++= - velocity of the centroid C of the system relative to inertiaCV  I Ciopen td rd , ˆ - mean velocity of the opening i relative to the centroid C SiV, ˆ - mean velocity of the fluid relative the opening i Virtual Work due to Position Change, Relative to Inertial Frame (continue – 1)
25 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 5) A virtual rotation ( ) kIjIntriB E kji θδθδθδθδ ˆˆˆ ++=  will produce a virtual displacement: ( ) C E C rr ,,  ×= θδδ ( ) ( ) ∑∑∑∫ ⋅+⋅+           ⋅++−⋅= k E k j jCj openings I ifluid ifluidCiopen S CR MFr td Rd mrdstfnprW vehicle θδδδδδ     ,,, ˆ ˆ11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∫ ⋅+⋅×+           ⋅×++−⋅×= k E k j jCj E openings I ifluid ifluidCiopen E S C E MFr td Rd mrdstfnpr vehicle θδθδ θδθδ     , ,, ˆ ˆ11 ( ) ( )           +×+           ×++−×⋅= ∑ ∑∑∫ openings k k j jCj I ifluid ifluidCiopen S C E MFr td Rd mrdstfnpr vehicle     ,,, ˆ ˆ11θδ The virtual work done along the generalized coordinates (rotation around C relative to inertial frame around Euler axes) is done by the pressure distribution, the flow through the openings and the discrete forces and moments applied on the system: ( )E θ  Virtual Work due to Rotation, Relative to Inertial Frame
26 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 6) - due to elastic modal displacementsEQ [ ] T E   4321 ηηηηηξ == The virtual work done during the elastic deformations along thr generalized coordinates is done by the pressure distribution on the wetted area of the system and the discrete forces and moments applied on the system: e  ( ) ( ) ( ) ( ) ( )∑ ∑∑ ∑∫ ∑ ∑∑∫     ×∇⋅+    ⋅+    ⋅+−= ×∇⋅+⋅+⋅+−= ∞ = ∞ = ∞ = k i iik j i iij S i ii k k j j S E MFdstfnp eMeFdsetfnpW W vehicle 111 11 11 δηφδηφδηφ δδδδ   ( ) ( ) ( )[ ]   ,2,111 =∑ ×∇⋅+∑ ⋅+∫ ⋅+−= ∂ ∂ = ∞ iMFdstfnpp W Q k ik l il S i i P Ei W φφφ ηδ δ Generalized Elastic Forces
27 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 7) Let add to this equation the following ( )( ) ( ) 01 0 , 5 =∫∫∫ ×∇=∫ ×− ∞∞ V C GGauss tS CS dvrpdsnpRR   ( ) ( ) ( ) 0111 0 =⋅∇== ∫∫∫ ∞∞∞ tv Gauss tStS dvnpdsnpdsnp   ( )[ ] RF td Rd mdstfnppW openings j j I ifluid ifluid S P vehicle     δδ ⋅           +           ++−= ∑ ∑∫ ∞ ˆ 11 ( )[ ] ( )E openings k k j jCjSi I Ciopen CifluidCiopen S CR MFrV td rd VmrdstfnpprW vehicle θδδ      ⋅           +×+                 ++×++−×= ∑ ∑∑∫ ∞ ,, , ,, ˆ ˆ11 ( ) ( ) ( ) ( )∑ ∑∑ ∑∫ ∑     ×∇⋅+    ⋅+    ⋅+−= ∞ = ∞ = ∞ = k i iik j i iij S i iiE MFdstfnpW W 111 11 δηφδηφδηφδ  where is the pressure far away from the system, to obtain∞p
28 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 8) From those equations we obtain ( ) ( ) ( ) ( ) ( )I openings j j I iopenI Cifluid i TiAI C PI P F td rd VmFF R W Q           +                 +++= ∂ ∂ = ∑ ∑∑∑        ˆ δ δ Generalized Position Forces in Inertial Frame ( ) ( )[ ]( ) ∫∑ +−= ∞ WsS II A dstfnppF  11 Aerodynamic Forces in Inertial Frame ( ) ( )[ ]( ) ∑∑ −+= ∞ openings I iopenSiifluid i I Ti nppSVmF    1, Thrust Forces in Inertial Frame
29 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 9) Generalized Moments around Euler Axes (E), relative to C Aerodynamic Moments relative to C Thrust Moments relative to C ( ) ( ) ( ) ( ) ( )[ ]∫ +−×= ∂ ∂ = ∞ vehicleS CE RE R dstfnppr W Q    11, θδ δ ∑ ∑∑ +×+                 ++×+ openings k k j jCjSi I CiopenI CifluidCiopen MFrV td rd Vmr      ,, , , ˆ ˆ ( )[ ]∫ +−×= ∞ WS C dstfnppr  11 ( )[ ]∑ +−×+ ∞ openings SiifluidopeniCiopen VmnppSr ,, 1    ∑∑∑ +×+         +×+ k k j jCj openings I CiopenI CCiopenifluid MFr td rd Vrm     , , , ˆ ˆ ( ) ( ) ( ) ( ) ( )E openings k k j jCj I CiopenI CCiopenifluid i CTiCAE RE R MFr td rd VrmMM W Q           +×+         +×++= ∂ ∂ = ∑ ∑∑∑∑        , , ,,, ˆ ˆ θδ δ ( )[ ]∫∑ +−×= ∞ WS CCA dstfnpprM  11:, ( )[ ]∑∑ −+×= ∞ openings iopenSiifluidCiopen i CTi nppSVmrM    1ˆ: ,,,
30 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 10) We want to find the Generalized Moments around Body Axes. We must find the transformation from the non-orthogonal Euler Axes (E) to Body Axes (B). ( ) ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ] ( )E k j i k j i I Ijjii Intr Intrii B B B DDDDkji θδ θδ θδ θδ θδ θδ θδ θθθθδ    =           =           = 321 ˆˆˆ ( ) { } ( )[ ] ( ) ( ) { } ( )[ ] ( )B openings k k j jCj I ifluid ifluidCiopen S C TTE B openings k k j jCj I ifluid ifluidCiopen S C TE R MFr td Rd mrdstfnpprD MFr td Rd mrdstfnpprDW vehicle vehicle           +×+           ×++−×=           +×+           ×++−×= ∑ ∑∑∫ ∑ ∑∑∫ ∞ ∞         ,,, ,,, ˆ ˆ11 ˆ ˆ11 θδ θδδ ( ) ( ) ( ) ( ) ( )B openings k k j jCj I CiopenI CCiopenifluid i CTiCA T E RB R MFr td rd VrmMMD W Q           +×+         +×++= ∂ ∂ = ∑ ∑∑∑∑        , , ,,, ˆ ˆ θδ δ Table of Contents
31 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame The generalized coordinates are [ ]T E T R T P ξξξξ  = where ( ) [ ]T CzCyCx I CP RRRR ==  ξ ( ) [ ]T kji E R θθθθξ ==  [ ] T E   4321 ηηηηηξ == The velocity vector of the system centroid C is given by: ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 : The angular velocity of rotation of the Body relative to inertia is: [ ] [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω            321 333231 232221 131211 :
32 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 1) We want to perform a change of coordinates from to( )ξξ , ( )w,ξ { } { }ηθηηθθθξ ,,,,,,,,,,: 21321 TT C T CzCyCx RRRR   == { } { }ηωηη  ,,,,,,,,,,: 21 T B T C T Vrqpwvuw == - system potential energy.( )ξU - generalized force along the generalized coordinate given by.iξiQ ( ) ( )wTT ,, ξξξ = - system kinetic energy. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηωηθ ηωθηθηθηθ    ,,,,, ,,,,,,,,,, 1 B B B C EI C B B B C ETEI C EI C EI C VRT DVCRTRRT = =     − The coordinates are called quasi-coordinates (see Meirovitch [4], pg. 157), to differentiate from the Lagrangian’s coordinates that describe the degrees of freedom. ( )ξξ , ( )w,ξ
33 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 2) The Lagrange’s Equations of Motion are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ Let outline (full derivation of the equation on this page is done in Appendix ) the derivation of the Lagrange’s Equations in Body Coordinates ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ B C ET B C I C B C I C V T C V T R V R T       θ Appendix ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +         ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C E TB C B IB E TB IB EE V TVTTT      θωθ ω θθ Appendix ( ) ( )         ∂ ∂ =         ∂ ∂ I C I C R T R T  Appendix ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } ( ) ( ) [ ]( ) ( )       ∂ ∂ ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← ← ← B C B C T B IB E TB IB E B C E TB C B IB E TB IB EE V T VD TT V TVTTT           ωθ ω θ θωθ ω θθ Appendix
34 ( ) ( ) ( ) [ ] [ ] ( )I P CC B C TB I B C TB II C I C I C Q R U R T V T td Cd V T td d C R U R T R T td d    =        ∂ ∂ +       ∂ ∂ −       ∂ ∂ +       ∂ ∂ =        ∂ ∂ +       ∂ ∂ −         ∂ ∂ SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 3) From ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ B C ET B C I C B C I C V T C V T R V R T       θ ( ) ( )         ∂ ∂ =         ∂ ∂ I C I C R T R T  ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ +         ∂ ∂ = ←← EEB IB T B IB T UTT td DdT td d D θθωω  ( ) { } ( ) ( ) [ ]( ) ( ) ( )E RB C B C T B IB E TB IB Q V T VD T      =         ∂ ∂ ×+         ∂ ∂ ∂ ∂ − ← → ωθ ω From we obtain ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +         ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C E TB C B IB E TB IB EE V TVTTT      θωθ ω θθ ( ) ( ) ( ) { } ( ) ( ) [ ]( ) ( )       ∂ ∂ ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C B C T B IB E TB IB EE V T VD TTT      ωθ ω θθ we obtain
35 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 4) Using [ ] [ ] [ ] ( )B IB TB I TB I C td Cd ×= ←ω  and ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω   we can compute ( ) ( ) ( )         ∂ ∂ +         ∂ ∂ −         ∂ ∂ I C I C I C R U R T R T td d   [ ] ( ) [ ] ( ) ( ) ( )         ∂ ∂ +         ∂ ∂ −         ∂ ∂ +         ∂ ∂ = I C I C B C TB I B C TB I R U R T V T td Cd V T td d C  [ ] ( ) [ ] [ ] ( ) ( ) ( ) ( ) ( )I PI C I C B C B IB TB IB C TB I Q R U R T V T C V T td d C     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ = ←ω
36 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 5) Finally ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ +         ∂ ∂ = ←← EEB IB T B IB T UTT td DdT td d D θθωω  [ ]( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ −         ∂ ∂ ×+ ← → B IB E TB IB B C B C T T V T VD ωθ ω     ( ) [ ]( ) ( ) { } ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂         ∂ ∂ +×+         ∂ ∂ = ← → ← ← EEB IB E TB IBB IB T B IB T UTT D T td d D θθωθ ω ω ω     [ ]( ) ( ) ( ) { } ( ) ( ) ( )E R B IB E TB IB B C B C T Q T V T VD      =       ∂ ∂ ∂ ∂ −       ∂ ∂ ×+ ← → ωθ ω [ ]( ) ( ) ( ) ( )I P B II C B II C B IB C B IBB C QC R U C R T C V T V T td d     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ←ω ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( ) ( )E R T E T E T B C B CB IB B IBB IB QD U D T D V T V TT td d       −−− ← ← ← =       ∂ ∂ +       ∂ ∂ −       ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ θθω ω ω
37 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 6) Summarize ( ) ( ) [ ]( ) [ ]( ) [ ]( ) ( ) ( ) ( )               ∂ ∂ ∂ ∂           −               ∂ ∂       ∂ ∂             ×× × +               ∂ ∂ ∂ ∂ − ← ← ← ← E I T B I B IB B C B IB B C B IB B IB B C T R T D C T V T V T V T td d θω ω ω ω         0 00 ( ) ( ) ( )                     =               ∂ ∂ ∂ ∂           + −− E R I P T B I E I T B I Q Q D C U R U D C     0 0 0 0 θ See Meirovitch and Kwak [6] and Meirovitch [7]. In the same way, for the elastic modes, we have:   ,2,1== ∂ ∂ + ∂ ∂ −      ∂ ∂ = ∂ ∂ + ∂ ∂ −      ∂ ∂ iQ UTT td dUTT td d Ei iiiiii ηηηηηη Table of Contents
38 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion The Translational Lagrange Equations in Body Coordinates are given by: ( ) [ ]( ) ( ) ( ) ( ) ( )I P B II C B II C B IB C B IBB C QC R U C R T C V T V T td d     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ←ω Pre-multiplying by will give the Translational Lagrange Equation in Inertial Frame. ( ) I B TB I CC = ( ) ( ) ( ) ( )I PI C I C I C Q R U R T V T td d   =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ where we used ( ) ( ) [ ]( ) ( )               ∂ ∂ ×+       ∂ ∂ =       ∂ ∂ ← B C B IBB C I BI C V T V T td d C V T td d    ω
39 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 1) Since the kinetic energy of the system is given by: ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , we have: PmrVm V T openings ifluidCiopenC C     =−=       ∂ ∂ ∑ :ˆ , This equation gives the Linear Momentum of the system. The same expression was obtained using Simplified Particles and Reynolds’ Theorem Approaches.
40 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 2) we have: PmrVm V T openings ifluidCiopenC C     =−=       ∂ ∂ ∑ :ˆ , This equation gives the derivative of the Linear Momentum of the system. The same expression was obtained using Simplified Particles and Reynolds’ Theorem Approaches if we identify: ∑∑ −−+=       ∂ ∂ = openings ifluidCiopen openings ifluid I Ciopen C I C ICI mrm td rd Vm td Vd m V T td d td Pd          , , ˆ ˆ 0=       ∂ ∂ CR T  ( )CECE CE Earth C RgmR R R gm R U   −=−=       ∂ ∂ → 12 2 0 → +=−−+= ∑∑ E E Earth P openings ifluidCiopen openings ifluid I Ciopen C I C I R R R gmQmrm td rd Vm td Vd m td Pd 1ˆ ˆ 2 2 0, ,         Substitute those equation in the Lagrange’s Equation: ( ) ( ) ( ) ( )I PI C I C I C Q R U R T V T td d   =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ gmQR R R gmQF PE E Earth Pext  +=+= → ∑ 12 2 0
41 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 3) → +=−−+= ∑∑ E E Earth P openings ifluidCiopen openings ifluid I Ciopen C I C I R R R gmQmrm td rd Vm td Vd m td Pd 1ˆ ˆ 2 2 0, ,         Substitute: ( ) ( )I openings j jifluid I Ciopen C i TiA I P Fm td rd VmFFQ         ++++= ∑ ∑∑∑       , ˆ in to obtain ∑∑∑∑∑ +++++= j j openings ifluid I Ciopen openings ifluidCiopen i TiA I C Fm td rd mrFFgm td Vd m       , , ˆ ˆ2
42 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 4) Rotation Equations The Rotational Lagrange’s equations in Body Coordinates are given by: ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( ) ( )E R T E T E T B C B CB IB B IBB IB QD U D T D V T V TT td d       −−− ← ← ← =       ∂ ∂ +       ∂ ∂ −       ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ θθω ω ω ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , ( ) C rotors BjrotorCrotorj tm RotorsFrozen B C CIBC IB HImd td rd rI T ,, , ,, :     =⋅+         ×+⋅=      ∂ ∂ ∑∫ ←← ← ωω ω Since the kinetic energy of the system is given by: we obtain
43 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 5) Rotation Equations (continue – 1) ( ) C rotors BjrotorCrotorj tm RotorsFrozen B C CIBC IB HImd td rd rI T ,, , ,, :     =⋅+         ×+⋅=      ∂ ∂ ∑∫ ←← ← ωω ω we obtain ( ) ( )       ⋅×+⋅+         ××+                 ×+ ⋅×+⋅+⋅= ×+==      ∂ ∂ ∑∑ ∫∫ ←←← ← ←←←← ← ← rotors BjrotorCrotorIB rotors BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C IBCIBIBCIBC CIB I C I C IIB II md td rd rmd td rd r dt d III H td Hd td HdT td d ωωω ω ωωωω ω ω           ,, , , , , ,,, , ,,
44 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 6) Rotation Equations (continue – 2) we obtain ( ) ( ) { }0   =       ∂ ∂ =       ∂ ∂ EE UT θθ ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ×+         ∂ ∂ −− ← ← ← E T E T B C B CB IB B IBB IB U D T D V T V TT td d θθω ω ω      ( ) ( ) [ ]( ) ( ) ( ) ( ) ( ) [ ]( ) ( ) [ ]( ) ( )B openings ifluidCiopenC B C B rotors BjrotorCrotor B IB rotors B BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C B IBC B IB B IBC B IBC mrVmVII md td rd rmd td rd r dt d III       ∑−×+∑ ⋅×+∑ ⋅+ ∫         ××+         ∫         ×+ ⋅×+⋅+⋅= ←←← ← ←←←←          ,,, , , , , ,,, ˆωωω ω ωωωω ( )E openings k k j jCj I Ciopen CCiopenifluid i CTiCA T MFr td rd VrmMMD           +×+         +×++= ∑ ∑∑∑∑−      , , ,,, ˆ ˆ ( )B openings k k j jCj I Ciopen CCiopenifluid i CTiCA MFr td rd VrmMM           +×+         +×++= ∑ ∑∑∑∑      , , ,,, ˆ ˆ
45 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 7) Rotation Equations (continue – 3) Finally ( ) ( ) [ ]( ) ( ) ( ) ( ) ( ) [ ]( ) ( ) ( ) ( ) ∑∑∑∑ ∑∑ ∫∫ +×++= ⋅×+⋅+         ××+                 ×+ ⋅×+⋅+⋅ ←←← ← ←←←← k k j jCj i B CTi B CA B rotors BjrotorCrotor B IB rotors B BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C B IBC B IB B IBC B IBC MFrMM II md td rd rmd td rd r dt d III         ,,, ,, , , , , ,,, ωωω ω ωωωω
46 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 8) Elastic Equations Ei iii Q UTT td d = ∂ ∂ + ∂ ∂ −      ∂ ∂ ηηη ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , Since the kinetic energy of the system is given by: we have td d M T i i i η η = ∂ ∂  0= ∂ ∂ i T η 2 2 td d M TT td d i i ii η ηη = ∂ ∂ −      ∂ ∂  Therefore
47 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 9) Elastic Equations (continue – 1) Ei iii Q UTT td d = ∂ ∂ + ∂ ∂ −      ∂ ∂ ηηηFrom ∑= ∞ =1 22 2 1 i iiie MU ηω we obtain iii i e M U ηω η 2 = ∂ ∂ From ( ) ( ) ( )[ ] ∑∑∫ ×∇⋅+⋅+⋅+−= ∂ ∂ = ∞ k ik j ij S i i P Ei MFdstfnpp W Q W φφφ ηδ δ  11 Therefore ( )[ ] ∑∑∫ ×∇⋅+⋅+⋅+−=      + ∞ k ik j ij S iii i i MFdstfnpp td d M W φφφηω η  11 2 2 2 Table of Contents
48 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ∑       =∑ ∫∫=∫= openings i iopen openings i S i tm td md mdmd td d tm iopen  MASS EQUATION FORCE EQUATION 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 TiA FFmg  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ∑+ j jF  DISCRETE TERMS
49 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 1) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O RIGID-BODY TERMSIOOIOOIOIOO 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
50 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 2) ( )[ ]∫∫∑ +−= ∞ WS A dstfnppF  11: AERODYNAMIC FORCES ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowTi dsnppmVVF    1 ˆˆ : THRUST FORCES ( ) ( )[ ]∫∫ +−×−=∑ ∞ WS OOA dstfnppRRM  11:, AERODYNAMIC MOMENTS RELATIVE TO O ( ) ( )[ ]∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOTi dsnppRRmVVRRM    1 ˆˆˆ :, THRUST MOMENTS RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM Table of Contents
51 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH References 2. Meirovitch, L., “Method of Applied Dynamics”, John Wiley & Sons, 1986 3. Goldstein, H., “Classical Mechanics”, 1st , 2nd and 3rd Editions 4. Lanczos, C., “The Variational Principles of Mechanics”, 4th Edition, Dover Publications, 1970 5. Meirovitch, L., “General Motion of a Variable-Mass Flexible Rocket with Internal Flow”, J. Spacecraft, Vol. 7, No. 2, Feb. 1970, pp. 186-195 1. Bilmoria, K.D., Schmidt, D.K., “An Integrated Development of the of Motion for Elastic Hypersonic Flight Vehicles”, AIAA-92-4605-CP, and Journal of Guidance, Control and Dynamics, Vol.18, No.1, Jan.-Feb., 1995, pp. 73-81 6. Meirovitch, L., Kwak, M.K., “Dynamics and Control of Spacecraft with Retargeting Flexible Antennas”, Journal of Guidance, Control and Dynamics, Vol.13, No.2, March-April, 1990, pp. 241-248
52 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH References (continue – 1) 7. Meirovitch, L., “State Equation of Motion for Flexible Bodies in Terms of Quasi-Coordinates”, Proceedings of the IUTAM/IFAC Symposium on Dynamics of Controlled Mechanical Systems, Switzerland, May-June 1998 8. Weng, S-L., Greenwood, D.T., “General Dynamical Equations of Motion for Elastic Body Systems”, Journal of Guidance, Control and Dynamics, Vol.15, No.6, Nov.-Dec., 1992, pp. 1434-1442 Table of Contents
53 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations The generalized coordinates are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ ( ) [ ]T zyx I CP RRRR ==  ξ Position components relative to Inertial System in Inertial Coordinates [ ]T kjiR θθθξ =Γ=  Euler Angles around Euler Axes [ ] T E   4321 ηηηηηξ == Elastic Modes We want to obtain the Lagrange Equations in Body Coordinates. ( ) ( )wTT ,, ξξξ = Kinetic Energy of the System ( )ξU Potential Energy of the System kQ Generalized Forces along the Degrees of Freedom Axes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηωηθηωθηθηθηθ    ,,,,,,,,,,,,,,, 1 B B B B EIB B B B ETEIEIEI VRTDVCRTRRT ==      − { } { }ηθηηθθθξ ,,,,,,,,,,: 21321 TTT zyx RRRR   == { } { }ηωηη  ,,,,,,,,,,: 21 T B T B T Vrqpwvuw ==
54 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH The angular velocity vector of rotation of the Body frame relative to Inertia frame is: kIjIntriBIB kji θθθω  ˆˆˆ ++=← In Body frame this is: ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] [ ] ( ) k I Ijjiij Intr Intriii B Bk B Ij B Intri B B B IB kjikji θθθθθθθθθω  ˆˆˆˆˆˆ ++=++=← [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ]           =           =           =← k j i k j i I Ijjii Intr Intrii B B B IB DDDkji r q p θ θ θ θ θ θ θθθω          321 ˆˆˆ: { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 321 : ˆˆ:,&ˆˆ:&ˆ: DDDD kkDjjDconstiD I Ijjii B Iji Intr Intrii B Intri B B = ====== → θθθθθθ ( ) [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω             321 333231 232221 131211 : where: or Appendix A: Lagrange Equations (continue – 1)
55 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − z B B B B EI y B B B B EI x B B B B EI z B B EB B ETEI y B B EB B ETEI x B B EB B ETEI z y x R VRT R VRT R VRT R DVCRT R DVCRT R DVCRT R T R T R T                ηωηθ ηωηθ ηωηθ ηωθθηθ ηωθθηθ ηωθθηθ ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = w T v T u T R w R v R u R w R v R u R w R v R u w T R w v T R v u T R u w T R w v T R v u T R u w T R w v T R v u T R u zzz yyy xxx zzz yyy xxx       We have: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∂ ∂ =       ∂ ∂         ∂ ∂ =         ∂ ∂ B B ET B B I B B I V T C V T R V R T       θ ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 :
56 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 3) also: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − k B B B B EI j B B B B EI i B B B B EI k B B EB B ETEI j B B EB B ETEI i B B EB B ETEI k j i VRT VRT VRT DVCRT DVCRT DVCRT T T T θ ηωηθ θ ηωηθ θ ηωηθ θ ηωθηθθ θ ηωθηθθ θ ηωθηθθ θ θ θ                ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = r T q T p T rqp rqp rqp r Tr q Tq p Tp r Tr q Tq p Tp r Tr q Tq p Tp kkk jjj iii kkk jjj iii θθθ θθθ θθθ θθθ θθθ θθθ       ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ ←← ← B IB ET B IB E B IB E T D TT ω θ ωθ ω θ     
57 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 4) In the same way: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                     ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − z y x z B B EB B ETEI y B B EB B ETEI x B B EB B ETEI z y x R T R T R T R DVCRT R DVCRT R DVCRT R T R T R T ηωθθηθ ηωθθηθ ηωθθηθ    ,,,,, ,,,,, ,,,,, 1 1 1 ( ) ( )       ∂ ∂ =       ∂ ∂ II R T R T 
58 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 5) and: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − k B B B B EI j B B B B EI i B B B B EI j B B EB B ETEI j B B EB B ETEI i B B EB B ETEI k j i VRT VRT VRT DVCRT DVCRT DVCRT T T T θ ηωηθ θ ηωηθ θ ηωηθ θ ηωθθηθ θ ηωθθηθ θ ηωθθηθ θ θ θ       ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ = r Tr q Tq p Tp r Tr q Tq p Tp r Tr q Tq p Tp w Tw v Tv u Tu w Tw v Tv u Tu w Tw v Tv u Tu T T T kkk jjj iii kkk jjj iii k j i θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ ( ) { } ( ) ( ) { } ( )                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ → r T q T p T rqp rqp rqp w T v T u T wvu wvu wvu T T T E TB IB E TB BV          θ ω θ θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ 333 222 111 333 222 111 3 2 1
59 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 6)           ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ z k y k x k z k y k x k z j y j x j z j y j x j z i y i x i z i y i x i kkk jjj iii R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C wvu wvu wvu    θθθθθθ θθθθθθ θθθθθθ θθθ θθθ θθθ 232221131211 232221131211 232221131211           ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ z k y k x k z j y j x j z i y i x i R C R C R C R C R C R C R C R C R C    θθθ θθθ θθθ 333231 333231 333231 ( ) { } ( ) ( ) { } ( ) { } ( ) { } { } { } { } { } [ ][ ] { } [ ][ ] { } [ ][ ] { } [ ] { } [ ] { } [ ] ( ) { } [ ] ( ) { } [ ] ( ) { } [ ] [ ] [ ] [ ][ ] ( ) ( ) [ ] ( ) ( )TB B TB B T TB B T TB B T TB B TTT I TTT I TTT I TT I TT I TT I T k T I T j T I T i T I k TB B j TB B i TB B E TB B VDVDDD DV DV DV DCR DCR DCR CDR CDR CDR C R C R C R V V V V                    ×−=×××−=             ×− ×− ×− =                     ×− ×− ×− =                     ×− ×− ×− =                             ∂ ∂         ∂ ∂       ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ 321 3 2 1 3 2 1 3 2 1 θ θ θ θ θ θ θ In a shorthand notation form: [ ]( ) ( ) [ ]( ) ( ) [ ]( )B B TTB B TTB B VDVDDV ×−=×=×=  and ( ) { } ( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ → → → → k TB IB j TB IB i TB IB kkk jjj iii E TB IB rqp rqp rqp θ ω θ ω θ ω θθθ θθθ θθθ θ ω     
60 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( ) { } ( ) ( ) { } ( )                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ → r T q T p T rqp rqp rqp w T v T u T wvu wvu wvu T T T T T T E TB IB E TB BV k j i          θ ω θ θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ θ θ θ 333 222 111 333 222 111 3 2 1 Appendix A: Lagrange Equations (continue – 7) ( ) { } ( ) [ ]( ) ( ) [ ]( ) ( ) [ ]( )B B TTB B TTB BE TB B VDVDDV V ×−=×=×= ∂ ∂    θ We found ( ) { } ( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ → → → → k TB IB j TB IB i TB IB kkk jjj iii E TB IB rqp rqp rqp θ ω θ ω θ ω θθθ θθθ θθθ θ ω      [ ]( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ ×−                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ → → → r T q T p T w T v T u T VD T T T T T T k TB IB j TB IB i TB IB B B T k j i k j i θ ω θ ω θ ω θ θ θ θ θ θ     In a shorthand notation form: ( ) ( ) [ ] ( ) ( ) ( ) { } ( ) ( )       ∂ ∂ ∂ ∂ +         ∂ ∂ ×−         ∂ ∂ =         ∂ ∂ ← → B IB E TB IB B B B B T EE T V T VD TT ωθ ω θθ     
61 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 8) ( ) ( )       ∂ ∂ =       ∂ ∂ II R T R T ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∂ ∂ =       ∂ ∂         ∂ ∂ =         ∂ ∂ B B ET B B I B B I V T C V T R V R T       θUsing and [ ] [ ] ( )I P II B B TB I B B TB I III Q R U R T V T td Cd V T td d C R U R T R T td d    =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ +         ∂ ∂ =        ∂ ∂ +         ∂ ∂ −         ∂ ∂we obtain Using ( ) ( ) [ ] ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +       ∂ ∂ ×−       ∂ ∂ =       ∂ ∂ ← → B IB E TB IB B B B B T EE T V T VD TT ωθ ω θθ     ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ ←← ← B IB ET B IB E B IB E T D TT ω θ ωθ ω θ      and we obtain ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( ) [ ]( ) ( ) ( ) { } ( ) ( ) ( )E RB IB E TB IB B B B B T EEB IB T B IB T Q T V T VD UTT td DdT td d D       =       ∂ ∂ ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ +         ∂ ∂ −       ∂ ∂ +       ∂ ∂ = ← → ←← ωθ ω θθωω
62 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 9) Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → Basic Euler Rotations The three basic Euler rotations, around the axes are described by the Rotation Matrices: ,3ˆ,2ˆ,1ˆ [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc Let differentiate with respect to Euler Angles. [ ] [ ][ ] 11 11 11 11 11 11 11 11 11 1ˆ 0 0 001 010 100 000 0 0 000 0 0 001 θ θθ θθ θθ θθ θθ θθ θθ θ ×−=           −          −=           −− −=           − = cs sc sc cs cs sc d d d d [ ] [ ][ ] 22 22 22 22 22 22 22 22 22 2ˆ 0 010 0 001 000 100 0 000 0 0 010 0 θ θθ θθ θθ θθ θθ θθ θθ θ ×−=           −           − =           − −− =           − = cs sc sc cs cs sc d d d d [ ] [ ][ ] 3333 33 33 33 33 33 33 33 3ˆ 100 0 0 000 001 010 000 0 0 100 0 0 θθθ θθ θθ θθ θθ θθ θθ θ ×−=           −           −=           −− − =           −= cs sc sc cs cs sc d d d d
63 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 10) Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → Basic Euler Rotations (continue – 1) is the matrix representation of the cross product of the vector ; i.e. in Cartesian coordinates : [ ]×A  A  ( )zyx 1ˆ,1ˆ,1ˆ ( ) ( )zzyyxxzzyyxx BBBAAABA 1ˆ1ˆ1ˆ1ˆ1ˆ1ˆ ++×++=×  ( ) ( ) ( ) zxyyxyxzzxxyzzy zyx zyx zyx BABABABABABA BBB AAA 1ˆ1ˆ1ˆ 1ˆ1ˆ1ˆ −+−+−=             = [ ]( ) { }( )zyxzyx z y x xy xz yz BA B B B AA AA AA ,,,, : 0 0 0  ×=                     − − − = The matrix is skew-symmetric; i.e.:[ ]×A  [ ] [ ]×−=× AA T  [ ] [ ] [ ][ ] 1111 1 1111 1ˆ θθθ θ θθ  ×−== d d td d [ ] [ ] [ ][ ] 2222 2 2222 2ˆ θθθ θ θθ  ×−== d d td d [ ] [ ] [ ][ ] 3333 3 3333 3ˆ θθθ θ θθ  ×−== d d td d
64 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 11) Computations of td Dd T       = td Dd td Dd td Dd td Dd 321          ++++++= k k j j i i k k j j i i k k j j i i d Dd d Dd d Dd d Dd d Dd d Dd d Dd d Dd d Dd θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ  333222111 { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 3 2 1 : ˆˆ:, ˆˆ: ˆ: DDDD kkD jjD constiD I Ijjii B Iji Intr Intrii B Intri B B = == == == → θθθθ θθ From 0111 = ∂ ∂ = ∂ ∂ = ∂ ∂ kji DDD θθθ Using those equations, and [ ] [ ] ( ) [ ] 0ˆ 22 211 2 = ∂ ∂ = ∂ ∂ ×−=×−= ∂ ∂ kj Intr Intrii i DD DDjD D θθ θ θ    [ ] [ ] [ ] ( ) [ ] [ ] [ ]( ) [ ] ( ) [ ] [ ]( ) [ ] [ ] [ ] ( ) [ ]( ) [ ] [ ] ( ) [ ]             = ∂ ∂ ×−=×−= −×−=×−= ∂ ∂ ×−=×−= ∂ ∂ 0 ˆˆ ˆˆˆˆ ˆ 3 32 3 311 3 k I Ijjii B Intr I Ijjiiii Intr Intrii I Ijj Intr Intrii j I Ijjii i D DDkj kjkj D DDkD D θ θθ θθθθθθ θ θθ θ      01111 = ∂ ∂ + ∂ ∂ + ∂ ∂ = k k j j i i DDD td Dd θ θ θ θ θ θ       [ ] i k k j j i i DD DDD td Dd θ θ θ θ θ θ θ         21 2222 ×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] [ ] ji k k j j i i DDDD DDD td Dd θθ θ θ θ θ θ θ           3231 3333 ×−×−= ∂ ∂ + ∂ ∂ + ∂ ∂ =
65 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 12) Computations of td Dd T [ ] [ ] jik k j j i i DDDD DDD td Dd θθθ θ θ θ θ θ           3231 3333 ×−×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] ik k j j i i DD DDD td Dd θθ θ θ θ θ θ         21 2222 ×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = 01111 = ∂ ∂ + ∂ ∂ + ∂ ∂ = k k j j i i DDD td Dd θ θ θ θ θ θ       We found: Therefore: 01 = td Dd T  [ ] i T T DD td Dd θ   ×= 12 2 [ ] [ ] j T i T T DDDD td Dd θθ      ×+×= 2313 3
66 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 13) Computations of ( ) { } ( )E TB IB θ ω   ∂ ∂ → By differentiating [ ] { } { } { }[ ]           =← k j i B IB DDD θ θ θ ω      321 we obtain: ( ) [ ] [ ][ ] [ ] [ ] [ ] [ ] kjkj k j i k j i iiii B IB DDDDDDDD DDDD DDD θθθθ θ θ θ θ θ θ θθθθ ω                        13123121 3121 321 0 ×+×=×−×−=           ×−×−=                 ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← ( ) [ ][ ] [ ] [ ] kk k j i k j i jjjj B IB DDDD DD DDD θθ θ θ θ θ θ θ θθθθ ω                    2332 32 321 00 ×=×−=             ×−=                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← ( ) [ ] 0000321                 =           =                 ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← k j i k j i kkkk B IB DDD θ θ θ θ θ θ θθθθ ω ( ) { } [ ] [ ]{ } [ ] [ ] k T j T T kj i TB IB DDDD DDDD θθ θθ θ ω         ×−×−= ×+×= ∂ ∂ ← 3121 1312 ( ) { } [ ]{ } [ ] k TT k j TB IB DDDD θθ θ ω     ×−=×= ∂ ∂ ← 3223 ( ) { } 0  = ∂ ∂ ← k TB IB θ ω
67 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 14) Computations of [ ]( )B IB T D ×→ω  Since we have[ ] [ ]{ }( ) 0  ≡×=× T lll T l DDDD [ ][ ] { } { } { } [ ] [ ] [ ]( )kji T T T B IB T DDD D D D D θθθω  ×+×+×           =×← 321 3 2 1 [ ] [ ] [ ] [ ] [ ] [ ]                   ×+× ×+× ×+× = j T i T k T i T k T j T DDDD DDDD DDDD θθ θθ θθ      2313 3212 3121 Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ →
68 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 15) Computations of ( ) { } ( ) [ ]( )B IB T E TB IB D ×+ ∂ ∂ → → ω θ ω    Let add and and compare with ( ) { } ( )E TB IB θ ω ∂ ∂ ←  [ ] ( )B IB T D ×←ω  td Dd T Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → ( ) { } [ ] ( ) td Dd D T B IB T i TB IB 1 1 0 ==×+ ∂ ∂ ← ←   ω θ ω ( ) { } [ ] ( ) [ ] td Dd DDD T i TB IB T j TB IB 2 122 =×=×+ ∂ ∂ ← ← θω θ ω   ( ) { } [ ] ( ) [ ] [ ] td Dd DDDDD T j T i TB IB T k TB IB 3 23133 =×+×=×+ ∂ ∂ ← ← θθω θ ω   or ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω  
69 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH [ ] [ ] [ ] kkjjii B IC θθθ= Appendix A: Lagrange Equations (continue – 16) Computations of td Cd B I [ ] [ ][ ] iiiB ii i td d θθ θ ×−= ˆ [ ] [ ][ ] jjjr jj j td d θθ θ ×−= int ˆ [ ] [ ][ ] kkkI kk k td d θθ θ ×−= ˆ we can compute [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] B I B I B Bkkjjii B Bkkjjii ii B I CDCii d dC ×−=×−=×−=      = ∂ ∂ 1 ˆˆ θθθθθθ θθ [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ]kkjj Intr Intriikkjj j ii j B I j d dC θθθθθ θ θ θ ×−=         = ∂ ∂ ˆ [ ] [ ]( ) [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] B I B I B Intrkkjjii B Intrkkjjiiii Intr Intrii CDCjjj ×−=×−=×−=−×−= 2 ˆˆˆ θθθθθθθθ [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ]kk I Ijjiikk k jjii k B I k d dC θθθθ θ θθ θ ×−=      = ∂ ∂ ˆ [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] B I B I B Ikkjjiiiijj I Ijjii CDCkk ×−=×−=−−×−= 3 ˆˆ θθθθθθθ where are the unit vectors along the three consecutive Euler axes of rotations (in Body, Intermediate and Inertial coordinates). IIntrB kji ˆ,ˆ,ˆ we found:
70 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 17) Computations of td Cd B I we can compute [ ] [ ] B I i B I CD C ×−= ∂ ∂ 1 θ [ ] [ ] B I j B I CD C ×−= ∂ ∂ 2 θ [ ] [ ] B I k B I CD C ×−= ∂ ∂ 3 θ [ ] [ ] [ ] [ ] k k B I j j B I i i B I B I CCC td Cd θ θ θ θ θ θ  ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] [ ] [ ] k B Ij B Ii B I CDCDCD θθθ  ×−×−×−= 321 [ ] [ ] [ ]( ) [ ]( ) B I B IB B Ikji CCDDD ×−=×+×+×−= ← ωθθθ  321 therefore [ ] [ ]( ) B I B IB B I C td Cd ×−= ←ω  [ ]( ) [ ]×−= ∂ ∂ 1DC C TB I i B I θ [ ] ( ) [ ]×−= ∂ ∂ 2DC C TB I j B I θ [ ] ( ) [ ]×−= ∂ ∂ k TB I k B I DC C θ
January 5, 2015 71 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 system3

  • 1. 1 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN 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 (see Power Point Presentation) • Lagrangian Approach (this Power Point Presentation)
  • 3. 3 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH TABLE OF CONTENT • Generalized Forces Joseph-Louis Lagrange 1736-1813 • Lagrange’s Equations of Motion • Principal Coordinate Frames • Inertial Coordinate Frame • Body Coordinate Frame • Body Mean System Axes • Orientation of Body Frame • Kinetic Energy of the System • Potential Energy of the System • Elastic Potential Energy • Gravitational Potential Energy • Computation of Lagrange’s Equations in Body Coordinates • Derivation of Equations of Motion • Summary of the Equations of Motion of a Variable Mass System • References • Appendix A: Lagrange Equations
  • 4. 4 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Lagrange’s Equations of Motion The Lagrange’s Equations of Motion for a dynamic system are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ - system kinetic energy.T - system potential energy.U - generalized coordinates (i=1,2,…, number of degrees of freedom of the system).iξ - generalized force along the generalized coordinate given by.iξiQ ( ) ( )i i W Q ξδ δ ∂ ∂ = -virtual work done on the system by all external forces/moments (excluding those accounted for in the potential energy term) during virtual displacement along all the generalized coordinates. Wδ Table of Contents
  • 5. 5 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames 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. Inertial Coordinate Frame (vector form) orIzIyIx zRyRxRR ˆˆˆ ++=  IzIyIx I zRyRxR td Rd V ˆˆˆ    ++== (vector form) or (vector form) orIzIyIx II zRyRxR td Rd td Vd a ˆˆˆ2 2    ++=== { }T zyx RRRR ,,=  (matrix form) ( ) { }T zyx I RRRV   ,,= (matrix form) ( ) { }T zyx I RRRa  ,,= (matrix form) Table of Contents
  • 6. 6 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 1) Cr,  - Position of the mass element dm relative to C. Body Coordinate Frame The origin of the Body Frame (B) is located at the instantaneous Centroid (C) of the system. 0,  =∫ m C mdr R  - Position of the mass element dm relative to I. CR  - Position of the centroid C relative to I. CC rRR ,  += 0,Cr  - Position of the same mass element dm in the un-deformed system, relative to C. e  - Change in position of the mass element dm due to elastic deformation of the system. err CC  += 0,, Table of Contents
  • 7. 7 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 2) Body Mean System Axes Mean Body System Axes are defined such that the relative linear and angular momentum, due to elastic deformation, are zero at every instant. The Body Mean Axes must satisfy the following: 0  =∫ md td ed Bm 0,   =∫ × md td ed r B m C 0, =∫ ⋅ md td ed td rd B m B C  Table of Contents
  • 8. 8 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Structural Model of the System Assume that the elastic deformations are small, and can be represented in terms the normal un-damped modes of vibration. ( ) ( )∑= ∞ =1i ii tRe ηφ  - are mode shape functions that depend on the position of the mass element of the system.( )Ri  φ - are generalized coordinates giving the magnitude of the modal displacements and are functions of time. ( )tiη Structural Dynamic Analysis (e.g. final element method) provides the mode shape functions component of each element of the system, as well as the vacuo modal frequencies ( ) , for a selected number of modes. ( )Ri  φ iω ii i td d ηω η 2 2 2 −= The mode shape functions are orthogonal. 0 ji m ji md ≠ =⋅∫ φφ  i m ii Mmd =∫ φφ  MV Bx By Bz Wz Wy Wx α β α β Bp Wp Bq WqBr Wr FIRST ELASTIC MODE SECOND ELASTIC MODE
  • 9. 9 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 3) Orientation of the Body Frame The orientation of the Body Frame relative to the Inertial Frame has three degrees of freedom. We will use 3 Euler Angles that define the orientation by three consecutive rotations around the consecutive frame axes. [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc The three basic Euler rotations around axes are described by the rotation matrices: ,3ˆ,2ˆ,1ˆ
  • 10. 10 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 4) Orientation of the Body Frame (continue – 1) Using the basic Euler Angles we can define the following 12 different rotations: (a) six rotations around three different axes: 321 →→ 231 →→ 312 →→ 132 →→ 213 →→ 123 →→ (b) six rotations such that the first and third are around the sam axes, but the second is different: 121 →→ 131 →→ 212 →→ 232 →→ 313 →→ 323 →→ Suppose that the Transfer Matrix from Inertia to Body is defined by three consecutive Euler Angles: around (unit vector in Inertial Frame), around (unit vector in intermediate frame), around (unit vector in Body Frame). B IC iθ Iiˆ jθ Interjˆ kθ Bkˆ [ ] [ ] [ ] [ ] [ ]TB I B I I B I B B Ikkjjii B I CCCICCC ==→== −1 &θθθ
  • 11. 11 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 5) Orientation of the Body Frame (continue – 2) The angular velocity vector of rotation of the Body frame relative to Inertia frame is: kIjIntriBIB kji θθθω  ˆˆˆ ++=← In Body frame this is: ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] [ ] ( ) k I Ijjiij Intr Intriii B Bk B Ij B Intri B B B IB kjikji θθθθθθθθθω  ˆˆˆˆˆˆ ++=++=← [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ]           =           =           =← k j i k j i I Ijjii Intr Intrii B B B IB DDDkji r q p θ θ θ θ θ θ θθθω          321 ˆˆˆ: { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 321 : ˆˆ:,&ˆˆ:&ˆ: DDDD kkDjjDconstiD I Ijjii B Iji Intr Intrii B Intri B B = ====== → θθθθθθ ( ) [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω             321 333231 232221 131211 : where: or
  • 12. 12 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Principal Coordinate Frames (continue - 6) Orientation of the Body Frame (continue – 3) The velocity vector of the system centroid C is given by: ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 : The following relations will be useful: and ( )           = k j i E θ θ θ θ : andwhere [ ] ( )           − − − =×← 0 0 0 : pq pr qr B IBω  Table of Contents ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω  Appendix [ ] [ ] [ ] ( )B IB TB I TB I CC td d ×= ←ω  Appendix
  • 13. 13 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System ( ) ∫ ⋅= tm II md td Rd td Rd T  2 1 Kinetic Energy of the System We have: We can write ( ) ∫         +⋅         += tm I C C I C C md td rd V td rd VT ,, 2 1     ( ) ( ) ( ) ( ) ∫ ⋅+∫+∫⋅= tm I C I C tm I C C tm CC 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 erRrRR CCCC  ++=+= 0,, I C C I C I C I td rd V td rd td Rd td Rd ,,    +=+=
  • 14. 14 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 1) (a) ( ) ( ) ( ) mVVmdVV CC tm CC  ⋅=∫⋅ 2 1 2 1 (b) Use Reynolds’ Transport Theorem when we differentiate ( ) 0,  =∫ tm C mdr Therefore ( ) ∑∫ −= openings i ifluidCiopen tm B C mrmd td rd    , , ˆ and ( ) ( ) ∫         ×+⋅=∫⋅ ← tm CIB B C C tm I C C mdr td rd Vmd td rd V , ,,      ω ( ) ( ) ( ) ∑∫∑ ∫∫∫∫ +=+=      = openings ifluidCiopen tm B C i S C tm B C REYNOLDS B tm C mrmd td rd mdrmd td rd mdr td d iopen        , , , , , ˆ0 ( ) ( ) ∑⋅−=∫⋅=∫⋅ openings ifluidCiopenC tm B C C tm I C C mrVmd td rd Vmd td rd V       , ,, ˆ ( ) ( ) ( ) ∫∫∫ ⋅=             ×⋅+⋅= ← tm B C C tm CIBC tm B C C md td rd VmdrVmd td rd V , 0 , ,        ω
  • 15. 15 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 2) (c) ( ) ( ) ∫         ×+⋅         ×+=∫ ⋅ ←← tm CIB B C CIB B C tm I C I C mdr td rd r td rd md td rd td rd , , , ,,, 2 1 2 1     ωω ( ) ∫ ⋅= tm B C B C md td rd td rd ,, 2 1  (c1) ( ) ( ) ∫ ×⋅         + ← tm CIB B C mdr td rd , ,   ω (c2) ( ) ( ) ( ) ∫ ×⋅×+ ←← tm CIBCIB mdrr ,, 2 1  ωω (c3) (c1) ( ) ( ) ∫         +⋅         +∫ =⋅ tm BB C BB C tm B C B C md td ed td rd td ed td rd md td rd td rd  0,0,,, 2 1 2 1 ( ) ( ) ( ) ∫         ⋅         +∫         ⋅         ∫ +⋅= tm BB tm BB C tm B C B C md td ed td ed md td ed td rd md td rd td rd      2 1 2 1 0 0,0,0, ( ) ( ) ( )∑ ∫ ×⋅×+∫ ⋅= ←← rotors m CjrotorBjrotorCjrotorBjrotor tm FrozenRotors B C FrozenRotors B C jrotor mdrrmd td rd td rd ,, ,, 2 1 2 1   ωω ( ) ( )∑ ∫         ⋅×+∫         ⋅+ ← rotors m B CjrotorBjrotor tm BFrozenRotors B C jrotor md td ed rmd td ed td rd          0 , 0 , ω ( ) ∫         ⋅         + tm BB md td ed td ed  2 1 ( ) ∫ ⋅= tm FrozenRotors B C FrozenRotors B C md td rd td rd ,, 2 1  ( )∑       ∫ ××−⋅+ ←⋅ ←← rotors I m BjrotorCjrotorCjrotorBjrotor BjrotorCjrotor jrotor mdrr      ω ωω , ,, 2 1 ( ) ∫ ∑∑ ∞ = ∞ = + tm j i j i ji md td d td d ηη φφ 1 12 1 
  • 16. 16 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( )( ) ∑ ⋅⋅+∫ ∫ ⋅=⋅ ←← rotors BjrotorCjrotorBjrotor tm tm FrozenRotors B C FrozenRotors B C B C B C Imd td rd td rd md td rd td rd ωω   , ,,,, 2 1 2 1 2 1 Kinetic Energy of the System (continue – 3) (c3) (c2) ( ) ∫ ∑ ∞ =       + tm i i i md td d 1 2 2 2 1 η φ ( )[ ]∫ −⋅= jrotorm CjrotorCjrotorCjrotorCjrotorCjrotor mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the Rotor j Relative to C ( ) ( ) ( ) ( ) ( ) ∫         ×⋅=∫         ⋅×=∫ ×⋅         ←←← tm B C CIB tm B C CIB tm CIB B C md td rd rmd td rd rmdr td rd , , , ,, ,      ωωω ( ) ( )        0 , 0, , ∫         ×⋅+∫         ×⋅= ←← tm B CIB tm B C CIB md td ed rmd td rd r ωω ( ) ( )( )∑ ∫ ××⋅+∫         ×⋅= ←←← rotors m CjrotorBjrotorCjrotorIB tm FrozenRotors B C CIB mdrrmd td rd r ,, , ,    ωωω ( ) [ ] [ ] Bjrotor rotors m CjrotorCjrotorIB tm FrozenRotors B C CIB mdrrmd td rd r ←←← ∑       ∫ ××−⋅+∫         ×⋅= ωωω    ,, , , ( ) [ ]∑⋅+∫         ×⋅= ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , ,
  • 17. 17 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 4) (c3) ( )[ ] ( ) ∫ −⋅= tm OOOOO mdrrrrI ,,,,, 1:  where Second Moment of Inertia Dyadic of the System Relative to O ( ) ( ) ( ) [ ] [ ] ( ) IB tm CCIB tm CIBCIB mdrrmdrr ←←←←       ∫ ××−⋅=∫ ×⋅× ωωωω  ,,,, 2 1 2 1 IBCIB I ←← ⋅⋅= ωω  , 2 1
  • 18. 18 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Kinetic Energy of the System (continue – 5) To summarize, the Kinetic Energy of the system is given by ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , Table of Contents
  • 19. 19 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System We consider only (Electromagnetic, Chemical Potentials are not considered) ge UUU += Elastic Deformation Potential eU Gravitational Field Potential gU Table of Contents
  • 20. 20 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System Elastic Deformation Potential eU ( ) ( )∑ ∞ = = 1i iii tle ηφ  ii i td d ηω η 2 2 2 −= i m ii Mmd =∫ φφ  0 ji m ji md ≠ =∫ φφ  ∫ ∑∑∫               −⋅        −=         ⋅−= ∞ = ∞ =m i iii j jj m B e mdmd td ed eU 1 2 1 2 2 2 1 2 1 ηωφηφ   ( ) ( ) ∑∑ ∫∑∑ ∫ ∞ = ∞ = ∞ = ∞ = =        ⋅=        ⋅= 1 22 1 22 1 1 2 2 1 2 1 2 1 i iii i ii m ii j i jii m ij Mmdmd ηωηωφφηηωφφ  iii i e M U ηω η 2 = ∂ ∂ From We obtain From this we obtain Table of Contents
  • 21. 21 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Potential Energy of the System (continue – 1) Gravitational Field Potential gU ( ) 2 2 01 E Earth EE R R gRRg  −= 22 0 sec/17.32sec/78.9 ftmg == where mREarth 135.378.6= ( ) ( ) ( ) 2/1 2 0 2 02 2 0 2 0 2 2 0 1 1 CECE Earth CE Earth m CE CE Earth CE Earth m E Earth ECCE m CCEg RR R gm R R gmmdrR R R g R R gm md R R gRrRmdgrRU    ⋅ ==⋅+= ⋅+−=⋅+= ∫ ∫∫ ( ) CE CE CE Earth CE CECE Earth C g R R R R gmR RR R gm R U    2 2 02/3 2 0 2 2 1 −= ⋅ −= ∂ ∂ ( )CECE CE Earth C g RgmR R R gm R U   =−= ∂ ∂ 12 2 0 From this we obtain or Table of Contents
  • 22. 22 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces - generalized force along the generalized coordinate given by.iξiQ ( ) ( )i i W Q ξδ δ ∂ ∂ = -virtual work done on the system by all external forces/moments (excluding those accounted for in the potential energy term) during virtual displacement along all the generalized coordinates. Wδ The generalized forces are: PQ - due to position change, relative to inertial system ( ) [ ]T zyx I P RRRR ==  ξ - due to rotation of the system, around its centroid, relative to inertial systemRQ ( ) [ ]T kji E R θθθθξ ==  - due to elastic modal displacementsEQ [ ] T E   4321 ηηηηηξ ==
  • 23. 23 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( ) ( ) ( )∑+= openings i iopenW tStStS Generalized Forces (continue – 1) Virtual Work due to Position Change, Relative to Inertial Frame The virtual work done by change in position is due to pressure distribution and fluid flow through the openings, and to discrete forces applied on the system ( ) RF td Rd mdstfnpW openings j j I ifluid ifluid S P vehicle     δδ ⋅           +           ++−= ∑ ∑∫ ˆ 11 where • 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 ( )2 / mNp - pressure on (normal to) the surface . f - friction force per (parallel to) unit surface .( )2 / mN n  1 - outward unit vector normal to the surface element ds t  1 - local unit vector of tangential stress due to flow on the surface element ds ∑ j jF  - discrete forces applied to the system at the position j R  ( )N is the mean position vector of the flow and of the opening on iopenSifluidR ˆ fluidm - fluid rate flowing through the opening iopenS
  • 24. 24 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 2) Si I CiopenI C I iopenfluid I Ciopen I C I ifluid V td rd V td rd td rd td Rd td Rd , ,,, ˆ ˆˆˆˆ     ++=++= - velocity of the centroid C of the system relative to inertiaCV  I Ciopen td rd , ˆ - mean velocity of the opening i relative to the centroid C SiV, ˆ - mean velocity of the fluid relative the opening i Virtual Work due to Position Change, Relative to Inertial Frame (continue – 1)
  • 25. 25 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 5) A virtual rotation ( ) kIjIntriB E kji θδθδθδθδ ˆˆˆ ++=  will produce a virtual displacement: ( ) C E C rr ,,  ×= θδδ ( ) ( ) ∑∑∑∫ ⋅+⋅+           ⋅++−⋅= k E k j jCj openings I ifluid ifluidCiopen S CR MFr td Rd mrdstfnprW vehicle θδδδδδ     ,,, ˆ ˆ11 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ∑∑ ∑∫ ⋅+⋅×+           ⋅×++−⋅×= k E k j jCj E openings I ifluid ifluidCiopen E S C E MFr td Rd mrdstfnpr vehicle θδθδ θδθδ     , ,, ˆ ˆ11 ( ) ( )           +×+           ×++−×⋅= ∑ ∑∑∫ openings k k j jCj I ifluid ifluidCiopen S C E MFr td Rd mrdstfnpr vehicle     ,,, ˆ ˆ11θδ The virtual work done along the generalized coordinates (rotation around C relative to inertial frame around Euler axes) is done by the pressure distribution, the flow through the openings and the discrete forces and moments applied on the system: ( )E θ  Virtual Work due to Rotation, Relative to Inertial Frame
  • 26. 26 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 6) - due to elastic modal displacementsEQ [ ] T E   4321 ηηηηηξ == The virtual work done during the elastic deformations along thr generalized coordinates is done by the pressure distribution on the wetted area of the system and the discrete forces and moments applied on the system: e  ( ) ( ) ( ) ( ) ( )∑ ∑∑ ∑∫ ∑ ∑∑∫     ×∇⋅+    ⋅+    ⋅+−= ×∇⋅+⋅+⋅+−= ∞ = ∞ = ∞ = k i iik j i iij S i ii k k j j S E MFdstfnp eMeFdsetfnpW W vehicle 111 11 11 δηφδηφδηφ δδδδ   ( ) ( ) ( )[ ]   ,2,111 =∑ ×∇⋅+∑ ⋅+∫ ⋅+−= ∂ ∂ = ∞ iMFdstfnpp W Q k ik l il S i i P Ei W φφφ ηδ δ Generalized Elastic Forces
  • 27. 27 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 7) Let add to this equation the following ( )( ) ( ) 01 0 , 5 =∫∫∫ ×∇=∫ ×− ∞∞ V C GGauss tS CS dvrpdsnpRR   ( ) ( ) ( ) 0111 0 =⋅∇== ∫∫∫ ∞∞∞ tv Gauss tStS dvnpdsnpdsnp   ( )[ ] RF td Rd mdstfnppW openings j j I ifluid ifluid S P vehicle     δδ ⋅           +           ++−= ∑ ∑∫ ∞ ˆ 11 ( )[ ] ( )E openings k k j jCjSi I Ciopen CifluidCiopen S CR MFrV td rd VmrdstfnpprW vehicle θδδ      ⋅           +×+                 ++×++−×= ∑ ∑∑∫ ∞ ,, , ,, ˆ ˆ11 ( ) ( ) ( ) ( )∑ ∑∑ ∑∫ ∑     ×∇⋅+    ⋅+    ⋅+−= ∞ = ∞ = ∞ = k i iik j i iij S i iiE MFdstfnpW W 111 11 δηφδηφδηφδ  where is the pressure far away from the system, to obtain∞p
  • 28. 28 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 8) From those equations we obtain ( ) ( ) ( ) ( ) ( )I openings j j I iopenI Cifluid i TiAI C PI P F td rd VmFF R W Q           +                 +++= ∂ ∂ = ∑ ∑∑∑        ˆ δ δ Generalized Position Forces in Inertial Frame ( ) ( )[ ]( ) ∫∑ +−= ∞ WsS II A dstfnppF  11 Aerodynamic Forces in Inertial Frame ( ) ( )[ ]( ) ∑∑ −+= ∞ openings I iopenSiifluid i I Ti nppSVmF    1, Thrust Forces in Inertial Frame
  • 29. 29 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 9) Generalized Moments around Euler Axes (E), relative to C Aerodynamic Moments relative to C Thrust Moments relative to C ( ) ( ) ( ) ( ) ( )[ ]∫ +−×= ∂ ∂ = ∞ vehicleS CE RE R dstfnppr W Q    11, θδ δ ∑ ∑∑ +×+                 ++×+ openings k k j jCjSi I CiopenI CifluidCiopen MFrV td rd Vmr      ,, , , ˆ ˆ ( )[ ]∫ +−×= ∞ WS C dstfnppr  11 ( )[ ]∑ +−×+ ∞ openings SiifluidopeniCiopen VmnppSr ,, 1    ∑∑∑ +×+         +×+ k k j jCj openings I CiopenI CCiopenifluid MFr td rd Vrm     , , , ˆ ˆ ( ) ( ) ( ) ( ) ( )E openings k k j jCj I CiopenI CCiopenifluid i CTiCAE RE R MFr td rd VrmMM W Q           +×+         +×++= ∂ ∂ = ∑ ∑∑∑∑        , , ,,, ˆ ˆ θδ δ ( )[ ]∫∑ +−×= ∞ WS CCA dstfnpprM  11:, ( )[ ]∑∑ −+×= ∞ openings iopenSiifluidCiopen i CTi nppSVmrM    1ˆ: ,,,
  • 30. 30 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Generalized Forces (continue – 10) We want to find the Generalized Moments around Body Axes. We must find the transformation from the non-orthogonal Euler Axes (E) to Body Axes (B). ( ) ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ] ( )E k j i k j i I Ijjii Intr Intrii B B B DDDDkji θδ θδ θδ θδ θδ θδ θδ θθθθδ    =           =           = 321 ˆˆˆ ( ) { } ( )[ ] ( ) ( ) { } ( )[ ] ( )B openings k k j jCj I ifluid ifluidCiopen S C TTE B openings k k j jCj I ifluid ifluidCiopen S C TE R MFr td Rd mrdstfnpprD MFr td Rd mrdstfnpprDW vehicle vehicle           +×+           ×++−×=           +×+           ×++−×= ∑ ∑∑∫ ∑ ∑∑∫ ∞ ∞         ,,, ,,, ˆ ˆ11 ˆ ˆ11 θδ θδδ ( ) ( ) ( ) ( ) ( )B openings k k j jCj I CiopenI CCiopenifluid i CTiCA T E RB R MFr td rd VrmMMD W Q           +×+         +×++= ∂ ∂ = ∑ ∑∑∑∑        , , ,,, ˆ ˆ θδ δ Table of Contents
  • 31. 31 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame The generalized coordinates are [ ]T E T R T P ξξξξ  = where ( ) [ ]T CzCyCx I CP RRRR ==  ξ ( ) [ ]T kji E R θθθθξ ==  [ ] T E   4321 ηηηηηξ == The velocity vector of the system centroid C is given by: ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 : The angular velocity of rotation of the Body relative to inertia is: [ ] [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω            321 333231 232221 131211 :
  • 32. 32 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 1) We want to perform a change of coordinates from to( )ξξ , ( )w,ξ { } { }ηθηηθθθξ ,,,,,,,,,,: 21321 TT C T CzCyCx RRRR   == { } { }ηωηη  ,,,,,,,,,,: 21 T B T C T Vrqpwvuw == - system potential energy.( )ξU - generalized force along the generalized coordinate given by.iξiQ ( ) ( )wTT ,, ξξξ = - system kinetic energy. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηωηθ ηωθηθηθηθ    ,,,,, ,,,,,,,,,, 1 B B B C EI C B B B C ETEI C EI C EI C VRT DVCRTRRT = =     − The coordinates are called quasi-coordinates (see Meirovitch [4], pg. 157), to differentiate from the Lagrangian’s coordinates that describe the degrees of freedom. ( )ξξ , ( )w,ξ
  • 33. 33 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 2) The Lagrange’s Equations of Motion are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ Let outline (full derivation of the equation on this page is done in Appendix ) the derivation of the Lagrange’s Equations in Body Coordinates ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ B C ET B C I C B C I C V T C V T R V R T       θ Appendix ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +         ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C E TB C B IB E TB IB EE V TVTTT      θωθ ω θθ Appendix ( ) ( )         ∂ ∂ =         ∂ ∂ I C I C R T R T  Appendix ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) { } ( ) ( ) [ ]( ) ( )       ∂ ∂ ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ∂ ∂ +       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← ← ← B C B C T B IB E TB IB E B C E TB C B IB E TB IB EE V T VD TT V TVTTT           ωθ ω θ θωθ ω θθ Appendix
  • 34. 34 ( ) ( ) ( ) [ ] [ ] ( )I P CC B C TB I B C TB II C I C I C Q R U R T V T td Cd V T td d C R U R T R T td d    =        ∂ ∂ +       ∂ ∂ −       ∂ ∂ +       ∂ ∂ =        ∂ ∂ +       ∂ ∂ −         ∂ ∂ SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 3) From ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ B C ET B C I C B C I C V T C V T R V R T       θ ( ) ( )         ∂ ∂ =         ∂ ∂ I C I C R T R T  ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ +         ∂ ∂ = ←← EEB IB T B IB T UTT td DdT td d D θθωω  ( ) { } ( ) ( ) [ ]( ) ( ) ( )E RB C B C T B IB E TB IB Q V T VD T      =         ∂ ∂ ×+         ∂ ∂ ∂ ∂ − ← → ωθ ω From we obtain ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +         ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C E TB C B IB E TB IB EE V TVTTT      θωθ ω θθ ( ) ( ) ( ) { } ( ) ( ) [ ]( ) ( )       ∂ ∂ ×−       ∂ ∂ ∂ ∂ +       ∂ ∂ =       ∂ ∂ ← ← B C B C T B IB E TB IB EE V T VD TTT      ωθ ω θθ we obtain
  • 35. 35 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 4) Using [ ] [ ] [ ] ( )B IB TB I TB I C td Cd ×= ←ω  and ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω   we can compute ( ) ( ) ( )         ∂ ∂ +         ∂ ∂ −         ∂ ∂ I C I C I C R U R T R T td d   [ ] ( ) [ ] ( ) ( ) ( )         ∂ ∂ +         ∂ ∂ −         ∂ ∂ +         ∂ ∂ = I C I C B C TB I B C TB I R U R T V T td Cd V T td d C  [ ] ( ) [ ] [ ] ( ) ( ) ( ) ( ) ( )I PI C I C B C B IB TB IB C TB I Q R U R T V T C V T td d C     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ = ←ω
  • 36. 36 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 5) Finally ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ +         ∂ ∂ = ←← EEB IB T B IB T UTT td DdT td d D θθωω  [ ]( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ −         ∂ ∂ ×+ ← → B IB E TB IB B C B C T T V T VD ωθ ω     ( ) [ ]( ) ( ) { } ( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂         ∂ ∂ +×+         ∂ ∂ = ← → ← ← EEB IB E TB IBB IB T B IB T UTT D T td d D θθωθ ω ω ω     [ ]( ) ( ) ( ) { } ( ) ( ) ( )E R B IB E TB IB B C B C T Q T V T VD      =       ∂ ∂ ∂ ∂ −       ∂ ∂ ×+ ← → ωθ ω [ ]( ) ( ) ( ) ( )I P B II C B II C B IB C B IBB C QC R U C R T C V T V T td d     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ←ω ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( ) ( )E R T E T E T B C B CB IB B IBB IB QD U D T D V T V TT td d       −−− ← ← ← =       ∂ ∂ +       ∂ ∂ −       ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ θθω ω ω
  • 37. 37 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Computation of Lagrange Equations in Body Frame (continue – 6) Summarize ( ) ( ) [ ]( ) [ ]( ) [ ]( ) ( ) ( ) ( )               ∂ ∂ ∂ ∂           −               ∂ ∂       ∂ ∂             ×× × +               ∂ ∂ ∂ ∂ − ← ← ← ← E I T B I B IB B C B IB B C B IB B IB B C T R T D C T V T V T V T td d θω ω ω ω         0 00 ( ) ( ) ( )                     =               ∂ ∂ ∂ ∂           + −− E R I P T B I E I T B I Q Q D C U R U D C     0 0 0 0 θ See Meirovitch and Kwak [6] and Meirovitch [7]. In the same way, for the elastic modes, we have:   ,2,1== ∂ ∂ + ∂ ∂ −      ∂ ∂ = ∂ ∂ + ∂ ∂ −      ∂ ∂ iQ UTT td dUTT td d Ei iiiiii ηηηηηη Table of Contents
  • 38. 38 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion The Translational Lagrange Equations in Body Coordinates are given by: ( ) [ ]( ) ( ) ( ) ( ) ( )I P B II C B II C B IB C B IBB C QC R U C R T C V T V T td d     =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ←ω Pre-multiplying by will give the Translational Lagrange Equation in Inertial Frame. ( ) I B TB I CC = ( ) ( ) ( ) ( )I PI C I C I C Q R U R T V T td d   =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ where we used ( ) ( ) [ ]( ) ( )               ∂ ∂ ×+       ∂ ∂ =       ∂ ∂ ← B C B IBB C I BI C V T V T td d C V T td d    ω
  • 39. 39 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 1) Since the kinetic energy of the system is given by: ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , we have: PmrVm V T openings ifluidCiopenC C     =−=       ∂ ∂ ∑ :ˆ , This equation gives the Linear Momentum of the system. The same expression was obtained using Simplified Particles and Reynolds’ Theorem Approaches.
  • 40. 40 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 2) we have: PmrVm V T openings ifluidCiopenC C     =−=       ∂ ∂ ∑ :ˆ , This equation gives the derivative of the Linear Momentum of the system. The same expression was obtained using Simplified Particles and Reynolds’ Theorem Approaches if we identify: ∑∑ −−+=       ∂ ∂ = openings ifluidCiopen openings ifluid I Ciopen C I C ICI mrm td rd Vm td Vd m V T td d td Pd          , , ˆ ˆ 0=       ∂ ∂ CR T  ( )CECE CE Earth C RgmR R R gm R U   −=−=       ∂ ∂ → 12 2 0 → +=−−+= ∑∑ E E Earth P openings ifluidCiopen openings ifluid I Ciopen C I C I R R R gmQmrm td rd Vm td Vd m td Pd 1ˆ ˆ 2 2 0, ,         Substitute those equation in the Lagrange’s Equation: ( ) ( ) ( ) ( )I PI C I C I C Q R U R T V T td d   =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ gmQR R R gmQF PE E Earth Pext  +=+= → ∑ 12 2 0
  • 41. 41 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 3) → +=−−+= ∑∑ E E Earth P openings ifluidCiopen openings ifluid I Ciopen C I C I R R R gmQmrm td rd Vm td Vd m td Pd 1ˆ ˆ 2 2 0, ,         Substitute: ( ) ( )I openings j jifluid I Ciopen C i TiA I P Fm td rd VmFFQ         ++++= ∑ ∑∑∑       , ˆ in to obtain ∑∑∑∑∑ +++++= j j openings ifluid I Ciopen openings ifluidCiopen i TiA I C Fm td rd mrFFgm td Vd m       , , ˆ ˆ2
  • 42. 42 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 4) Rotation Equations The Rotational Lagrange’s equations in Body Coordinates are given by: ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( ) ( )E R T E T E T B C B CB IB B IBB IB QD U D T D V T V TT td d       −−− ← ← ← =       ∂ ∂ +       ∂ ∂ −       ∂ ∂ ×+       ∂ ∂ ×+       ∂ ∂ θθω ω ω ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , ( ) C rotors BjrotorCrotorj tm RotorsFrozen B C CIBC IB HImd td rd rI T ,, , ,, :     =⋅+         ×+⋅=      ∂ ∂ ∑∫ ←← ← ωω ω Since the kinetic energy of the system is given by: we obtain
  • 43. 43 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 5) Rotation Equations (continue – 1) ( ) C rotors BjrotorCrotorj tm RotorsFrozen B C CIBC IB HImd td rd rI T ,, , ,, :     =⋅+         ×+⋅=      ∂ ∂ ∑∫ ←← ← ωω ω we obtain ( ) ( )       ⋅×+⋅+         ××+                 ×+ ⋅×+⋅+⋅= ×+==      ∂ ∂ ∑∑ ∫∫ ←←← ← ←←←← ← ← rotors BjrotorCrotorIB rotors BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C IBCIBIBCIBC CIB I C I C IIB II md td rd rmd td rd r dt d III H td Hd td HdT td d ωωω ω ωωωω ω ω           ,, , , , , ,,, , ,,
  • 44. 44 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 6) Rotation Equations (continue – 2) we obtain ( ) ( ) { }0   =       ∂ ∂ =       ∂ ∂ EE UT θθ ( ) [ ]( ) ( ) [ ]( ) ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ ×+         ∂ ∂ −− ← ← ← E T E T B C B CB IB B IBB IB U D T D V T V TT td d θθω ω ω      ( ) ( ) [ ]( ) ( ) ( ) ( ) ( ) [ ]( ) ( ) [ ]( ) ( )B openings ifluidCiopenC B C B rotors BjrotorCrotor B IB rotors B BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C B IBC B IB B IBC B IBC mrVmVII md td rd rmd td rd r dt d III       ∑−×+∑ ⋅×+∑ ⋅+ ∫         ××+         ∫         ×+ ⋅×+⋅+⋅= ←←← ← ←←←←          ,,, , , , , ,,, ˆωωω ω ωωωω ( )E openings k k j jCj I Ciopen CCiopenifluid i CTiCA T MFr td rd VrmMMD           +×+         +×++= ∑ ∑∑∑∑−      , , ,,, ˆ ˆ ( )B openings k k j jCj I Ciopen CCiopenifluid i CTiCA MFr td rd VrmMM           +×+         +×++= ∑ ∑∑∑∑      , , ,,, ˆ ˆ
  • 45. 45 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 7) Rotation Equations (continue – 3) Finally ( ) ( ) [ ]( ) ( ) ( ) ( ) ( ) [ ]( ) ( ) ( ) ( ) ∑∑∑∑ ∑∑ ∫∫ +×++= ⋅×+⋅+         ××+                 ×+ ⋅×+⋅+⋅ ←←← ← ←←←← k k j jCj i B CTi B CA B rotors BjrotorCrotor B IB rotors B BjrotorCrotor tm RotorsFrozen B C CIB B tm RotorsFrozen B C C B IBC B IB B IBC B IBC MFrMM II md td rd rmd td rd r dt d III         ,,, ,, , , , , ,,, ωωω ω ωωωω
  • 46. 46 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 8) Elastic Equations Ei iii Q UTT td d = ∂ ∂ + ∂ ∂ −      ∂ ∂ ηηη ( ) ( ) ( ) ( ) ∫ ⋅+∫⋅+∫⋅= tm I C I C tm I C C tm CC md td rd td rd md td rd VmdVVT ,,, 2 1 2 1   ( ) ∑⋅−⋅= openings i ifluidCiopenCCC mrVmVV   , ˆ 2 1 IBCIB I ←← ⋅⋅+ ωω  , 2 1 ( ) ∑ ⋅⋅+∫ ⋅+ ←← rotors BjrotorCrotorjBjrotor tm FrozenRotors B C FrozenRotors B C Imd td rd td rd ωω   , ,, 2 1 2 1 ∑ ∞ =       + 1 2 2 1 i i i M td dη ( ) ∑ ⋅⋅+∫         ×⋅+ ←←← rotors BjrotorCrotorjIB tm FrozenRotors B C CIB Imd td rd r ωωω    , , , Since the kinetic energy of the system is given by: we have td d M T i i i η η = ∂ ∂  0= ∂ ∂ i T η 2 2 td d M TT td d i i ii η ηη = ∂ ∂ −      ∂ ∂  Therefore
  • 47. 47 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Derivation of the Equations of Motion (continue – 9) Elastic Equations (continue – 1) Ei iii Q UTT td d = ∂ ∂ + ∂ ∂ −      ∂ ∂ ηηηFrom ∑= ∞ =1 22 2 1 i iiie MU ηω we obtain iii i e M U ηω η 2 = ∂ ∂ From ( ) ( ) ( )[ ] ∑∑∫ ×∇⋅+⋅+⋅+−= ∂ ∂ = ∞ k ik j ij S i i P Ei MFdstfnpp W Q W φφφ ηδ δ  11 Therefore ( )[ ] ∑∑∫ ×∇⋅+⋅+⋅+−=      + ∞ k ik j ij S iii i i MFdstfnpp td d M W φφφηω η  11 2 2 2 Table of Contents
  • 48. 48 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( ) ( ) ∑       =∑ ∫∫=∫= openings i iopen openings i S i tm td md mdmd td d tm iopen  MASS EQUATION FORCE EQUATION 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 TiA FFmg  EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ∑+ j jF  DISCRETE TERMS
  • 49. 49 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 1) MOMENT EQUATIONS RELATIVE TO A REFERENCE POINT O RIGID-BODY TERMSIOOIOOIOIOO 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
  • 50. 50 SOLO SUMMARY OF EQUATIONS OF MOTION OF A VARIABLE MASS (CONTINUE – 2) ( )[ ]∫∫∑ +−= ∞ WS A dstfnppF  11: AERODYNAMIC FORCES ( )∫∫ −+     −= ∞ iopenS iflowiopeniflowTi dsnppmVVF    1 ˆˆ : THRUST FORCES ( ) ( )[ ]∫∫ +−×−=∑ ∞ WS OOA dstfnppRRM  11:, AERODYNAMIC MOMENTS RELATIVE TO O ( ) ( )[ ]∫∫ −×−+     −×     −= ∞ iopenS OiflowiopeniflowOiopenOTi dsnppRRmVVRRM    1 ˆˆˆ :, THRUST MOMENTS RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM Table of Contents
  • 51. 51 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH References 2. Meirovitch, L., “Method of Applied Dynamics”, John Wiley & Sons, 1986 3. Goldstein, H., “Classical Mechanics”, 1st , 2nd and 3rd Editions 4. Lanczos, C., “The Variational Principles of Mechanics”, 4th Edition, Dover Publications, 1970 5. Meirovitch, L., “General Motion of a Variable-Mass Flexible Rocket with Internal Flow”, J. Spacecraft, Vol. 7, No. 2, Feb. 1970, pp. 186-195 1. Bilmoria, K.D., Schmidt, D.K., “An Integrated Development of the of Motion for Elastic Hypersonic Flight Vehicles”, AIAA-92-4605-CP, and Journal of Guidance, Control and Dynamics, Vol.18, No.1, Jan.-Feb., 1995, pp. 73-81 6. Meirovitch, L., Kwak, M.K., “Dynamics and Control of Spacecraft with Retargeting Flexible Antennas”, Journal of Guidance, Control and Dynamics, Vol.13, No.2, March-April, 1990, pp. 241-248
  • 52. 52 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH References (continue – 1) 7. Meirovitch, L., “State Equation of Motion for Flexible Bodies in Terms of Quasi-Coordinates”, Proceedings of the IUTAM/IFAC Symposium on Dynamics of Controlled Mechanical Systems, Switzerland, May-June 1998 8. Weng, S-L., Greenwood, D.T., “General Dynamical Equations of Motion for Elastic Body Systems”, Journal of Guidance, Control and Dynamics, Vol.15, No.6, Nov.-Dec., 1992, pp. 1434-1442 Table of Contents
  • 53. 53 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations The generalized coordinates are: i iii Q UTT td d =      ∂ ∂ +      ∂ ∂ −      ∂ ∂ ξξξ ( ) [ ]T zyx I CP RRRR ==  ξ Position components relative to Inertial System in Inertial Coordinates [ ]T kjiR θθθξ =Γ=  Euler Angles around Euler Axes [ ] T E   4321 ηηηηηξ == Elastic Modes We want to obtain the Lagrange Equations in Body Coordinates. ( ) ( )wTT ,, ξξξ = Kinetic Energy of the System ( )ξU Potential Energy of the System kQ Generalized Forces along the Degrees of Freedom Axes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )ηωηθηωθηθηθηθ    ,,,,,,,,,,,,,,, 1 B B B B EIB B B B ETEIEIEI VRTDVCRTRRT ==      − { } { }ηθηηθθθξ ,,,,,,,,,,: 21321 TTT zyx RRRR   == { } { }ηωηη  ,,,,,,,,,,: 21 T B T B T Vrqpwvuw ==
  • 54. 54 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH The angular velocity vector of rotation of the Body frame relative to Inertia frame is: kIjIntriBIB kji θθθω  ˆˆˆ ++=← In Body frame this is: ( ) ( ) ( ) ( ) ( ) [ ] ( ) [ ] [ ] ( ) k I Ijjiij Intr Intriii B Bk B Ij B Intri B B B IB kjikji θθθθθθθθθω  ˆˆˆˆˆˆ ++=++=← [ ] ( ) [ ] ( ) [ ] [ ] ( ) [ ] { } { } { }[ ]           =           =           =← k j i k j i I Ijjii Intr Intrii B B B IB DDDkji r q p θ θ θ θ θ θ θθθω          321 ˆˆˆ: { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 321 : ˆˆ:,&ˆˆ:&ˆ: DDDD kkDjjDconstiD I Ijjii B Iji Intr Intrii B Intri B B = ====== → θθθθθθ ( ) [ ] [ ] ( ) { } { } { }[ ]           =                     ==           =           =← k j i k j i E k j i B IB DDD DDD DDD DDD DD r q p θ θ θ θ θ θ θ θ θ θ ω             321 333231 232221 131211 : where: or Appendix A: Lagrange Equations (continue – 1)
  • 55. 55 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 2) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − z B B B B EI y B B B B EI x B B B B EI z B B EB B ETEI y B B EB B ETEI x B B EB B ETEI z y x R VRT R VRT R VRT R DVCRT R DVCRT R DVCRT R T R T R T                ηωηθ ηωηθ ηωηθ ηωθθηθ ηωθθηθ ηωθθηθ ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = w T v T u T R w R v R u R w R v R u R w R v R u w T R w v T R v u T R u w T R w v T R v u T R u w T R w v T R v u T R u zzz yyy xxx zzz yyy xxx       We have: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∂ ∂ =       ∂ ∂         ∂ ∂ =         ∂ ∂ B B ET B B I B B I V T C V T R V R T       θ ( ) [ ] ( )                     ==           =           = Cz Cy Cx I C Cz Cy Cx B I B C R R R CCC CCC CCC RC R R R C w v u V         333231 232221 131211 :
  • 56. 56 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 3) also: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − k B B B B EI j B B B B EI i B B B B EI k B B EB B ETEI j B B EB B ETEI i B B EB B ETEI k j i VRT VRT VRT DVCRT DVCRT DVCRT T T T θ ηωηθ θ ηωηθ θ ηωηθ θ ηωθηθθ θ ηωθηθθ θ ηωθηθθ θ θ θ                ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = r T q T p T rqp rqp rqp r Tr q Tq p Tp r Tr q Tq p Tp r Tr q Tq p Tp kkk jjj iii kkk jjj iii θθθ θθθ θθθ θθθ θθθ θθθ       ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ ←← ← B IB ET B IB E B IB E T D TT ω θ ωθ ω θ     
  • 57. 57 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 4) In the same way: In a shorthand notation form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                     ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − z y x z B B EB B ETEI y B B EB B ETEI x B B EB B ETEI z y x R T R T R T R DVCRT R DVCRT R DVCRT R T R T R T ηωθθηθ ηωθθηθ ηωθθηθ    ,,,,, ,,,,, ,,,,, 1 1 1 ( ) ( )       ∂ ∂ =       ∂ ∂ II R T R T 
  • 58. 58 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 5) and: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )                       ∂ ∂ ∂ ∂ ∂ ∂ =                       ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ − − − k B B B B EI j B B B B EI i B B B B EI j B B EB B ETEI j B B EB B ETEI i B B EB B ETEI k j i VRT VRT VRT DVCRT DVCRT DVCRT T T T θ ηωηθ θ ηωηθ θ ηωηθ θ ηωθθηθ θ ηωθθηθ θ ηωθθηθ θ θ θ       ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, ,,,,, 1 1 1                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ = r Tr q Tq p Tp r Tr q Tq p Tp r Tr q Tq p Tp w Tw v Tv u Tu w Tw v Tv u Tu w Tw v Tv u Tu T T T kkk jjj iii kkk jjj iii k j i θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ ( ) { } ( ) ( ) { } ( )                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ → r T q T p T rqp rqp rqp w T v T u T wvu wvu wvu T T T E TB IB E TB BV          θ ω θ θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ 333 222 111 333 222 111 3 2 1
  • 59. 59 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 6)           ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ z k y k x k z k y k x k z j y j x j z j y j x j z i y i x i z i y i x i kkk jjj iii R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C R C wvu wvu wvu    θθθθθθ θθθθθθ θθθθθθ θθθ θθθ θθθ 232221131211 232221131211 232221131211           ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ z k y k x k z j y j x j z i y i x i R C R C R C R C R C R C R C R C R C    θθθ θθθ θθθ 333231 333231 333231 ( ) { } ( ) ( ) { } ( ) { } ( ) { } { } { } { } { } [ ][ ] { } [ ][ ] { } [ ][ ] { } [ ] { } [ ] { } [ ] ( ) { } [ ] ( ) { } [ ] ( ) { } [ ] [ ] [ ] [ ][ ] ( ) ( ) [ ] ( ) ( )TB B TB B T TB B T TB B T TB B TTT I TTT I TTT I TT I TT I TT I T k T I T j T I T i T I k TB B j TB B i TB B E TB B VDVDDD DV DV DV DCR DCR DCR CDR CDR CDR C R C R C R V V V V                    ×−=×××−=             ×− ×− ×− =                     ×− ×− ×− =                     ×− ×− ×− =                             ∂ ∂         ∂ ∂       ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ 321 3 2 1 3 2 1 3 2 1 θ θ θ θ θ θ θ In a shorthand notation form: [ ]( ) ( ) [ ]( ) ( ) [ ]( )B B TTB B TTB B VDVDDV ×−=×=×=  and ( ) { } ( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ → → → → k TB IB j TB IB i TB IB kkk jjj iii E TB IB rqp rqp rqp θ ω θ ω θ ω θθθ θθθ θθθ θ ω     
  • 60. 60 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH ( ) { } ( ) ( ) { } ( )                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ → r T q T p T rqp rqp rqp w T v T u T wvu wvu wvu T T T T T T E TB IB E TB BV k j i          θ ω θ θθθ θθθ θθθ θθθ θθθ θθθ θ θ θ θ θ θ 333 222 111 333 222 111 3 2 1 Appendix A: Lagrange Equations (continue – 7) ( ) { } ( ) [ ]( ) ( ) [ ]( ) ( ) [ ]( )B B TTB B TTB BE TB B VDVDDV V ×−=×=×= ∂ ∂    θ We found ( ) { } ( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ → → → → k TB IB j TB IB i TB IB kkk jjj iii E TB IB rqp rqp rqp θ ω θ ω θ ω θθθ θθθ θθθ θ ω      [ ]( ) ( ) { } ( ) { } ( ) { }                     ∂ ∂ ∂ ∂ ∂ ∂                     ∂ ∂ ∂ ∂ ∂ ∂ +                     ∂ ∂ ∂ ∂ ∂ ∂ ×−                     ∂ ∂ ∂ ∂ ∂ ∂ =                     ∂ ∂ ∂ ∂ ∂ ∂ → → → r T q T p T w T v T u T VD T T T T T T k TB IB j TB IB i TB IB B B T k j i k j i θ ω θ ω θ ω θ θ θ θ θ θ     In a shorthand notation form: ( ) ( ) [ ] ( ) ( ) ( ) { } ( ) ( )       ∂ ∂ ∂ ∂ +         ∂ ∂ ×−         ∂ ∂ =         ∂ ∂ ← → B IB E TB IB B B B B T EE T V T VD TT ωθ ω θθ     
  • 61. 61 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 8) ( ) ( )       ∂ ∂ =       ∂ ∂ II R T R T ( ) ( ) ( ) ( ) ( ) ( ) ( )       ∂ ∂ =       ∂ ∂         ∂ ∂ =         ∂ ∂ B B ET B B I B B I V T C V T R V R T       θUsing and [ ] [ ] ( )I P II B B TB I B B TB I III Q R U R T V T td Cd V T td d C R U R T R T td d    =        ∂ ∂ +         ∂ ∂ −         ∂ ∂ +         ∂ ∂ =        ∂ ∂ +         ∂ ∂ −         ∂ ∂we obtain Using ( ) ( ) [ ] ( ) ( ) ( ) { } ( ) ( )         ∂ ∂ ∂ ∂ +       ∂ ∂ ×−       ∂ ∂ =       ∂ ∂ ← → B IB E TB IB B B B B T EE T V T VD TT ωθ ω θθ     ( ) ( ) ( ) ( ) ( ) ( ) ( )         ∂ ∂ =         ∂ ∂         ∂ ∂ =         ∂ ∂ ←← ← B IB ET B IB E B IB E T D TT ω θ ωθ ω θ      and we obtain ( ) ( ) ( )       ∂ ∂ +       ∂ ∂ −         ∂ ∂ EEE UTT td d θθθ   ( ) ( ) ( ) ( ) [ ]( ) ( ) ( ) { } ( ) ( ) ( )E RB IB E TB IB B B B B T EEB IB T B IB T Q T V T VD UTT td DdT td d D       =       ∂ ∂ ∂ ∂ −         ∂ ∂ ×+         ∂ ∂ +         ∂ ∂ −       ∂ ∂ +       ∂ ∂ = ← → ←← ωθ ω θθωω
  • 62. 62 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 9) Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → Basic Euler Rotations The three basic Euler rotations, around the axes are described by the Rotation Matrices: ,3ˆ,2ˆ,1ˆ [ ]           − = 11 1111 0 0 001 : θθ θθθ cs sc [ ]           − = 22 22 22 0 010 0 : θθ θθ θ cs sc [ ]           −= 100 0 0 : 33 33 33 θθ θθ θ cs sc Let differentiate with respect to Euler Angles. [ ] [ ][ ] 11 11 11 11 11 11 11 11 11 1ˆ 0 0 001 010 100 000 0 0 000 0 0 001 θ θθ θθ θθ θθ θθ θθ θθ θ ×−=           −          −=           −− −=           − = cs sc sc cs cs sc d d d d [ ] [ ][ ] 22 22 22 22 22 22 22 22 22 2ˆ 0 010 0 001 000 100 0 000 0 0 010 0 θ θθ θθ θθ θθ θθ θθ θθ θ ×−=           −           − =           − −− =           − = cs sc sc cs cs sc d d d d [ ] [ ][ ] 3333 33 33 33 33 33 33 33 3ˆ 100 0 0 000 001 010 000 0 0 100 0 0 θθθ θθ θθ θθ θθ θθ θθ θ ×−=           −           −=           −− − =           −= cs sc sc cs cs sc d d d d
  • 63. 63 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 10) Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → Basic Euler Rotations (continue – 1) is the matrix representation of the cross product of the vector ; i.e. in Cartesian coordinates : [ ]×A  A  ( )zyx 1ˆ,1ˆ,1ˆ ( ) ( )zzyyxxzzyyxx BBBAAABA 1ˆ1ˆ1ˆ1ˆ1ˆ1ˆ ++×++=×  ( ) ( ) ( ) zxyyxyxzzxxyzzy zyx zyx zyx BABABABABABA BBB AAA 1ˆ1ˆ1ˆ 1ˆ1ˆ1ˆ −+−+−=             = [ ]( ) { }( )zyxzyx z y x xy xz yz BA B B B AA AA AA ,,,, : 0 0 0  ×=                     − − − = The matrix is skew-symmetric; i.e.:[ ]×A  [ ] [ ]×−=× AA T  [ ] [ ] [ ][ ] 1111 1 1111 1ˆ θθθ θ θθ  ×−== d d td d [ ] [ ] [ ][ ] 2222 2 2222 2ˆ θθθ θ θθ  ×−== d d td d [ ] [ ] [ ][ ] 3333 3 3333 3ˆ θθθ θ θθ  ×−== d d td d
  • 64. 64 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 11) Computations of td Dd T       = td Dd td Dd td Dd td Dd 321          ++++++= k k j j i i k k j j i i k k j j i i d Dd d Dd d Dd d Dd d Dd d Dd d Dd d Dd d Dd θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ  333222111 { } ( ) ( ){ } ( ) [ ] ( ) ( ){ } ( ) [ ] [ ] ( ) { } { } { }[ ]321 3 2 1 : ˆˆ:, ˆˆ: ˆ: DDDD kkD jjD constiD I Ijjii B Iji Intr Intrii B Intri B B = == == == → θθθθ θθ From 0111 = ∂ ∂ = ∂ ∂ = ∂ ∂ kji DDD θθθ Using those equations, and [ ] [ ] ( ) [ ] 0ˆ 22 211 2 = ∂ ∂ = ∂ ∂ ×−=×−= ∂ ∂ kj Intr Intrii i DD DDjD D θθ θ θ    [ ] [ ] [ ] ( ) [ ] [ ] [ ]( ) [ ] ( ) [ ] [ ]( ) [ ] [ ] [ ] ( ) [ ]( ) [ ] [ ] ( ) [ ]             = ∂ ∂ ×−=×−= −×−=×−= ∂ ∂ ×−=×−= ∂ ∂ 0 ˆˆ ˆˆˆˆ ˆ 3 32 3 311 3 k I Ijjii B Intr I Ijjiiii Intr Intrii I Ijj Intr Intrii j I Ijjii i D DDkj kjkj D DDkD D θ θθ θθθθθθ θ θθ θ      01111 = ∂ ∂ + ∂ ∂ + ∂ ∂ = k k j j i i DDD td Dd θ θ θ θ θ θ       [ ] i k k j j i i DD DDD td Dd θ θ θ θ θ θ θ         21 2222 ×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] [ ] ji k k j j i i DDDD DDD td Dd θθ θ θ θ θ θ θ           3231 3333 ×−×−= ∂ ∂ + ∂ ∂ + ∂ ∂ =
  • 65. 65 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 12) Computations of td Dd T [ ] [ ] jik k j j i i DDDD DDD td Dd θθθ θ θ θ θ θ           3231 3333 ×−×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] ik k j j i i DD DDD td Dd θθ θ θ θ θ θ         21 2222 ×−= ∂ ∂ + ∂ ∂ + ∂ ∂ = 01111 = ∂ ∂ + ∂ ∂ + ∂ ∂ = k k j j i i DDD td Dd θ θ θ θ θ θ       We found: Therefore: 01 = td Dd T  [ ] i T T DD td Dd θ   ×= 12 2 [ ] [ ] j T i T T DDDD td Dd θθ      ×+×= 2313 3
  • 66. 66 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 13) Computations of ( ) { } ( )E TB IB θ ω   ∂ ∂ → By differentiating [ ] { } { } { }[ ]           =← k j i B IB DDD θ θ θ ω      321 we obtain: ( ) [ ] [ ][ ] [ ] [ ] [ ] [ ] kjkj k j i k j i iiii B IB DDDDDDDD DDDD DDD θθθθ θ θ θ θ θ θ θθθθ ω                        13123121 3121 321 0 ×+×=×−×−=           ×−×−=                 ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← ( ) [ ][ ] [ ] [ ] kk k j i k j i jjjj B IB DDDD DD DDD θθ θ θ θ θ θ θ θθθθ ω                    2332 32 321 00 ×=×−=             ×−=                     ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← ( ) [ ] 0000321                 =           =                 ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ← k j i k j i kkkk B IB DDD θ θ θ θ θ θ θθθθ ω ( ) { } [ ] [ ]{ } [ ] [ ] k T j T T kj i TB IB DDDD DDDD θθ θθ θ ω         ×−×−= ×+×= ∂ ∂ ← 3121 1312 ( ) { } [ ]{ } [ ] k TT k j TB IB DDDD θθ θ ω     ×−=×= ∂ ∂ ← 3223 ( ) { } 0  = ∂ ∂ ← k TB IB θ ω
  • 67. 67 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 14) Computations of [ ]( )B IB T D ×→ω  Since we have[ ] [ ]{ }( ) 0  ≡×=× T lll T l DDDD [ ][ ] { } { } { } [ ] [ ] [ ]( )kji T T T B IB T DDD D D D D θθθω  ×+×+×           =×← 321 3 2 1 [ ] [ ] [ ] [ ] [ ] [ ]                   ×+× ×+× ×+× = j T i T k T i T k T j T DDDD DDDD DDDD θθ θθ θθ      2313 3212 3121 Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ →
  • 68. 68 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Appendix A: Lagrange Equations (continue – 15) Computations of ( ) { } ( ) [ ]( )B IB T E TB IB D ×+ ∂ ∂ → → ω θ ω    Let add and and compare with ( ) { } ( )E TB IB θ ω ∂ ∂ ←  [ ] ( )B IB T D ×←ω  td Dd T Computations of and td Dd T ( ) { } ( )E TB IB θ ω   ∂ ∂ → ( ) { } [ ] ( ) td Dd D T B IB T i TB IB 1 1 0 ==×+ ∂ ∂ ← ←   ω θ ω ( ) { } [ ] ( ) [ ] td Dd DDD T i TB IB T j TB IB 2 122 =×=×+ ∂ ∂ ← ← θω θ ω   ( ) { } [ ] ( ) [ ] [ ] td Dd DDDDD T j T i TB IB T k TB IB 3 23133 =×+×=×+ ∂ ∂ ← ← θθω θ ω   or ( ) { } ( ) [ ] ( )B IB T E TB IB T D td Dd ×+ ∂ ∂ = ← ← ω θ ω  
  • 69. 69 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH [ ] [ ] [ ] kkjjii B IC θθθ= Appendix A: Lagrange Equations (continue – 16) Computations of td Cd B I [ ] [ ][ ] iiiB ii i td d θθ θ ×−= ˆ [ ] [ ][ ] jjjr jj j td d θθ θ ×−= int ˆ [ ] [ ][ ] kkkI kk k td d θθ θ ×−= ˆ we can compute [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] B I B I B Bkkjjii B Bkkjjii ii B I CDCii d dC ×−=×−=×−=      = ∂ ∂ 1 ˆˆ θθθθθθ θθ [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ]kkjj Intr Intriikkjj j ii j B I j d dC θθθθθ θ θ θ ×−=         = ∂ ∂ ˆ [ ] [ ]( ) [ ] [ ] [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ]( ) [ ] B I B I B Intrkkjjii B Intrkkjjiiii Intr Intrii CDCjjj ×−=×−=×−=−×−= 2 ˆˆˆ θθθθθθθθ [ ] [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ]kk I Ijjiikk k jjii k B I k d dC θθθθ θ θθ θ ×−=      = ∂ ∂ ˆ [ ] [ ] [ ]( ) [ ] [ ] [ ] [ ] [ ] [ ]( ) [ ] B I B I B Ikkjjiiiijj I Ijjii CDCkk ×−=×−=−−×−= 3 ˆˆ θθθθθθθ where are the unit vectors along the three consecutive Euler axes of rotations (in Body, Intermediate and Inertial coordinates). IIntrB kji ˆ,ˆ,ˆ we found:
  • 70. 70 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM LAGRANGIAN APPROACH Table of Contents Appendix A: Lagrange Equations (continue – 17) Computations of td Cd B I we can compute [ ] [ ] B I i B I CD C ×−= ∂ ∂ 1 θ [ ] [ ] B I j B I CD C ×−= ∂ ∂ 2 θ [ ] [ ] B I k B I CD C ×−= ∂ ∂ 3 θ [ ] [ ] [ ] [ ] k k B I j j B I i i B I B I CCC td Cd θ θ θ θ θ θ  ∂ ∂ + ∂ ∂ + ∂ ∂ = [ ] [ ] [ ] k B Ij B Ii B I CDCDCD θθθ  ×−×−×−= 321 [ ] [ ] [ ]( ) [ ]( ) B I B IB B Ikji CCDDD ×−=×+×+×−= ← ωθθθ  321 therefore [ ] [ ]( ) B I B IB B I C td Cd ×−= ←ω  [ ]( ) [ ]×−= ∂ ∂ 1DC C TB I i B I θ [ ] ( ) [ ]×−= ∂ ∂ 2DC C TB I j B I θ [ ] ( ) [ ]×−= ∂ ∂ k TB I k B I DC C θ
  • 71. January 5, 2015 71 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