Equation of motion of a variable mass system1
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This is the first of three presentations (the easiest one) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. Each presentation uses a different method of derivation. The presentation is at undergraduate (physics, engineering) level. Please sent comments for improvements to solo.hermelin@gmail.com. Thanks! For more presentations on different subjects please visit my website at http://www.solohermelin.com.
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Equation of motion of a variable mass system1
- 1. 1 SOLO HERMELIN EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH http://www.solohermelin.com
- 2. 2 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM • Simplified Particle Approach (this 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 (see Power Point Presentation)
- 3. 3 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH TABLE OF CONTENT Sir Isaac Newton 1643-1727 • Assumptions • Inertial Velocity and Acceleration • Instantaneous Mass Center or Centroid C of the System • Linear Momentum of the System • Force Equation • Moment Relative to a Reference Point O • Absolute Angular Momentum Relative to a Reference Point O • External Forces and Moments Applied on the System • Summary of the Equations of Motion of a Variable Mass System • References
- 4. 4 SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Assumptions 1. The system at time t contains N particles. 2. The particle i, of mass dmi, is located at a point (relative to an inertial system – I ). iR 3. We define a reference point O by the vector (relative to I).OR 4. We obtain the equation of motion for the continuous by taking a very large number N of particles. ∫⇒∑ ∞→ = NN i 1 We have a system of particles enclosed at the time t by a surface S(t) that bounds the volume v(t). There are no sources or sinks in the volume v(t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2, …). In addition the particles are free to move relative to each other. OiOi RRr −=:, The particle relative position to O is given by:
- 5. 5 SOLO Assumptions (Continue - 1) We have 5. The position of the opening ,relative to I, is given by .iopenR iopenS ( ) ( ) ( ) ( )ttRttR tRtR iflowiopen iflowiopen ∆+≠∆+ = & The position of the mass particle flowing through the opening , relative to I, is given by . iopenS iflowR Therefore ( ) ( ) I iflow I iopen td tRd td tRd ≠ and ( ) ( ) iopeniflow I iopen I iflow Si VV td tRd td tRd V −=−=:, is the velocity of flow relative to the opening iopenS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 6. 6 SOLO The Inertial Velocity and Acceleration of the mass dmi are given by I i i td Rd V = I i I i i td Rd td Vd a 2 2 == Total Mass of the System ( ) ( ) ∫∫∑ ==→= → ∞→ = tvm dmdm NN i i dvdmmmdtm i ρ 1 At a given time t At the time t + Δ t the mass change is due to the flow through the openings ( ),2,1=iS iopen ( ) ∑∑ ∆+=∆+ = openings iflow N i i mmdttm 1 ( ) ( ) ( ) ( )∑∑ = ∆ ∆ = ∆ −∆+ = →∆→∆ openings iflow openings iflow tt tm t m t tmttm tm 00 limlim EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content The mass rate (flow) , entering / leaving the system, is given by
- 7. 7 SOLO Instantaneous Mass Center or Centroid (C) of the System At the time t + Δ t By subtracting those two equations, dividing by Δt, and taking the limit, we get The mass center (Centroid) , of the system, relative to I, at time t, is defined as ( )tRC ( ) ( ) ( ) ∫∫∑ ==→= → ∞→ = tvm C dmmd NN i iiC dvRdmRtRmmdRtRm i ρ :: 1 ( ) ( )[ ] ( ) ( )∑∑ ∆+∆+∆+=∆+ = openings iflowiflowiflow N i iiiCC RRmmdRRtRmtRm 1 ( ) ( )[ ] ( ) ∑+∑= ∆ ∑ ∆+∆+∑∆ = ∆ ∆ = = = →∆→∆ openings iflowiflow N i i I iopenings iflowiflowiflow N i ii t C t C Rmmd td Rd t RRmmdR t tRm Rm td d 1 1 00 limlim Now let add the constraint that at time t the flow at the opening is such that iopenS ( ) ( )tRtR iflowiopen = to obtain ( ) ∑∑ −= = openings iopeniflow I C N i i I i RmRm td d md td Rd 1 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 8. 8 SOLO Instantaneous Mass Center or Centroid (C) of the System (continue - 1) Let develop the right side of this equation ( ) ∑∑ −= = openings iopeniflow I C N i i I i RmRm td d md td Rd 1 ( ) ( ) ∑ ∑∑∑ ∑∑ −= −−=−+= =−+=− openings Ciopeniflow I C openings Ciopenflowi I C openings iopeniflow I C openings Ciflow openings iopeniflow I C C openings iopeniflow I C rm td Rd m RRm td Rd mRm td Rd mRm Rm td Rd mRmRmRm td d , Therefore ( ) ∑∑∑ −=−−= = openings Ciopeniflow I C openings Ciopeniflow I C N i i I i rm td Rd mRRm td Rd mmd td Rd , 1 The First Moment of Inertia of the System, relative to the point O, is defined as:Oc, ( ) ( ) OCOC N i iO N i ii N i iOi N i iOiO rmRRmmdRmdRmdRRmdrc , 1111 ,, : =−=∑−∑=∑ −=∑= ==== EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 9. 9 SOLO ( ) ∑∑∑ −=−−= = openings Ciopeniflow I C openings Ciopeniflow I C N i i I i rm td Rd mRRm td Rd mmd td Rd , 1 Linear Momentum of the System Substitute At a given time t the Linear Momentum of the system is defined as ( ) ( ) ( ) ( ) ∫∫∑∑ ==→== ∞→ → == tmtm I N mdmd N i ii N i i I i mdVdm td Rd tPmdVmd td Rd tP i :: 11 ( ) ( ) ( ) ( ) ∑∑ ∑∑∑ −=−−= →−=−−== ∞→ → = openings CiopeniflowC openings CiopeniflowC N mdmd openings Ciopeniflow I C openings Ciopeniflow I C N i i I i rmVmRRmVmtP rm td Rd mRRm td Rd mmd td Rd tP i , , 1 Differentiate ( ) ( ) OCOC N i iO N i ii N i iOi N i iOiO rmRRm mdRmdRmdRRmdrc , 1111 ,, : =−= −=−== ∑∑∑∑ ==== to obtain ( ) ( ) ( ) +− += −+ −=−+ −= ∑∑ ∑ openings OiflowO openings CiflowC OC openings iflow I O I C OC I O I C I O RmVmRmVm RRm td Rd td Rd mRRm td Rd td Rd m td tcd , to obtain EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 10. 10 SOLO Linear Momentum of the System (continue-1) Substitute ( ) ( ) ∑−=∑ −−= openings CiopeniflowC openings CiopeniflowC rmVmRRmVmtP , to obtain ( ) ∑+− ∑+= openings OiflowO openings CiflowC I O RmVmRmVm td tcd , into ( ) ( ) ∑∑ −+=−−+= openings OiopeniflowO I O openings OiopeniflowO I O rmVm td cd RRmVm td cd tP , ,, At the time t + Δ t the Linear Momentum of the System (including the mass entering/leaving through S) is: ( ) ( ) ( ) ( )∑∑ ∆+∆+∆+=∆+ = openings iflowiflowiflow N i i RRmmRRtPtP 1 : By subtracting those two equations, dividing by Δt, and taking the limit, we get ( ) ∑ ∑∑∑ ∑∑∑ −−+=+= ∆ − ∆+∆+ ∆+ = ∆ ∆ = = == →∆→∆ openings I openings I Ciopen iflow openings Ciopeniflow I C I Ciflow iflow N i i I i N i i I i openings I iflow I iflow iflow N i i I i I i tt td rd mrm td Rd m td Rd m td Rd mdm td Rd t md td Rd td Rd td Rd mmd td Rd td Rd t tP td Pd , ,2 2 1 2 2 11 00 limlim EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 11. 11 SOLO Linear Momentum of the System (continue-2) We obtain ( ) ∑∑ −−+= i I Ciflow iflow openings Ciopeniflow I C I C I td rd mrm td Rd m td Rd m td tPd , ,2 2 ( ) ( ) ( ) ( )∑∑ ∑∑ ∑∑ −−−−+= →−−+= ++= →∞ → = openings Ciopeniflow openings CiopeniflowC I C I N mdmdopenings I Ciopen iflow openings Ciopeniflow I C I C openings I Ciflow iflow I C N i i I i I VVmRRmVm td Vd m td tPd td rd mrm td Rd m td Rd m td rd m td Rd mmd td Rd td tPd i , ,2 2 , 1 2 2 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH An equivalent result could be obtained by differentiating ( ) ∑−= openings Ciopeniflow I C rm td Rd mtP , We obtained Table of Content
- 12. 12 SOLO Force Equation Applying the 2nd Newton’s Law to the particle of mass mi, we obtain: ∑= +== N j ijiexti I i i I i fdfdmd td Rd md td Vd 1 int2 2 where iextfd - External forces acting on the mass mi ij fd int - Internal forces that particle j exercise on the mass mi From the 3rd Newton’s Law the internal force that particle j applies on particle i is of equal magnitude but of opposite direction to the force that particle i applies on particle j : jiij fdfd intint −= Therefore 0 1 1 int =∑∑= ≠ = N i N ij j ij fd EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 13. 13 SOLO ( ) ∞→ → →∑−∑−+= N mdmdopenings I Ciopen iflow openings Ciopeniflow I C I C I itd rd mrm td Rd m td Rd m td tPd , ,2 2 Force Equation (continue – 1) We have ∑∑∑∑∑ =+= = ≠ === ext N i N ij j ij N i iext N i i I i Ffdfdmd td Vd 0 1 1 int 11 ∑∑ = = N i i I i ext md td Rd F 1 2 2 Substitute this equation into to obtain ∑∑∑∑ −−+=++= openings I Ciopen iflow openings Ciopeniflow I C I C openings I Ciflow iflow I C ext I td rd mrm td Rd m td Rd m td rd m td Rd mF td Pd , ,2 2 , Rearranging we obtain ∑∑∑∑ ++ −+= openings I Ciopen iflow openings Ciopeniflow openings I Ciopen I Ciflow iflowext I C td rd mrm td rd td rd mF td Rd m , , ,, 2 2 2 ( ) ∑∑∑∑ −+−+ −+= openings I C I iopen iflow openings Ciopeniflow openings I iopen I iflow iflowext td Rd td Rd mRRm td Rd td Rd mF 2 or ( ) ( ) ( )∑∑∑∑ −+−+−+= openings Ciopeniflow openings Ciopeniflow openings iopeniflowiflowext I C RRmVVmVVmF td Vd m 2 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 14. 14 SOLO Absolute Angular Momentum Relative to a Reference Point O The Absolute Momentum Relative to a Reference Point O, of the particle of mass dmi at time t is defined as: ( ) ( ) iiOiiiOiiOiO dmVrdmVRRPdRRHd ×=×−=×−= ,, : The Absolute Momentum Relative to a Reference Point O, of the mass m (t) is defined as: ( ) ( ) ∑∑∑ === ×=×−=×−= N i i I i Oi N i iiOi N i iOiO dm td Rd rdmVRRPdRRH 1 , 11 , : By taking a very large number N of particles, we go from discrete to continuous ∫⇒∑ ∞→ = NN i 1 ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH ,, ρ The Absolute Momentum Relative to a Reference Point O, of the system (including the mass entering (+)/leaving (-) through surface S), at time t + Δt is given by: ( ) ( )∑∑ ∆ ∆+×∆++ ∆+×∆+=∆+ = openings iflow I iflow I iflow OiflowOiflow N i i I i I i OiOiOO m td Rd td Rd rrdm td Rd td Rd rrHH ,, 1 ,,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 15. 15 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 1) By subtracting I O t I O t H td Hd ∆ ∆ = →∆ , 0 , lim ( ) ( ) t dm td Rd rm td Rd td Rd rrdm td Rd td Rd rr openings N i i I i iOiflow I iflow I iflow OiflowOiflow N i i I i I i OiOi t ∆ ×−∆ ∆+×∆++ ∆+×∆+ = ∑ ∑∑ == →∆ 1 ,, 1 ,, 0 lim ∑∑∑ ×+×+×= == openings iflow I iflow Oiflow N i i I iOi N i i I i Oi m td Rd rdm td Rd td rd dm td Rd r , 1 , 1 2 2 , Now let add the constraint that at time t the flow at the opening is such that iopenS ( ) ( ) ( ) ( )trtrtRtR OiflowOiopeniflowiopen ,, =→= to obtain (next page) EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH ,, ρ dividing by Δt, and taking the limit, we get from ( ) ( )∑∑ ∆ ∆+×∆++ ∆+×∆+=∆+ = openings iflow I iflow I iflow OiflowOiflow N i i I i I i OiOiOO m td Rd td Rd rrdm td Rd td Rd rrHH ,, 1 ,,,,
- 16. 16 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 2) ∑∑∑ ×+×+×= == openings iflow I iflow Oiopen N i i I i I Oi N i i I i Oi I O m td Rd rdm td Rd td rd dm td Rd r td Hd , 1 , 1 2 2 , , ( )∑ ×−+∑ × −+∑ ×= == openings iflow I iflow Oiopen N i i I i I O I i N i i I i Oi m td Rd RRdm td Rd td Rd td Rd dm td Rd r 11 2 2 , ( )∑ ×−+∑×−∑ ×= == openings iflow I iflow Oiopen N i i I i I O N i i I i Oi m td Rd RRdm td Rd td Rd dm td Rd r 11 2 2 , By taking a very large number N of particles, we go from discrete to continuous ∫⇒∑ ∞→ = NN i 1 ( ) ( )∑ ×−+×−∫ ×= openings iflowiflowOiopenO tm I O I O mVRRPVdm td Rd r td Hd 2 2 , , ( ) ( ) ∑∑ −+=−−+= openings OiopeniflowO I O openings OiopeniflowO I O rmVm td cd RRmVm td cd tP , ,, Substitute to obtain ( ) ( ) ( )∑ ×−+× ∑ −−++∫ ×= openings iflowiflowOiopenO openings iflowOiopenO I O tm I O I O mVRRVmRRVm td cd dm td Rd r td Hd , 2 2 , , or ( ) ( )∑ −×+×+∫ ×= openings iflowOiflowOiopenO I O tm I O I O mVVrV td cd dm td Rd r td Hd , , 2 2 , , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 17. 17 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 3) We obtained ( )( ) ( )( ) ( ) ∫ ×=∫ ×−=∫ ×−= tm O tm O tv OO dmVrdmVRRdvVRRH ,, ρ Substitute in the previous equation OIO O O O I O I O I OO r td rd V td rd td Rd td Rd VrRR , ,, , :& ×++=+==+= ←ω ( )( ) ( ) ∫ ×++×=∫ ×−= ← tm OIO O O OO tm OO dmr td rd VrdmVRRH , , ,, ω ( ) ( ) ( ) ( ) ∫ ×+∫ ××+× ∫= ← tm O O O tm OIOOO tm O dm td rd rdmrrVdmr , ,,,, ω We obtain (a) (b) (c) Let develop those three expressions (a), (b) and (c). where is the angular velocity vector from I to O.IO←ω EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 18. 18 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 4) (a) ( ) ( ) ( ) ( ) ( ) ( ) OOC tm OC tm OC tm OC tm C tm O cmRRdmrdmrdmrdmrdmr ,,,,,, =−===+= ∫∫∫∫∫ Where we used because C is the Center of Mass (Centroid) of the system. ( ) 0, =∫tm C dmr ( ) OOOOCO tm O VcVrmVdmr ×=×=× ∫ ,,, ( ) ( ) ( )[ ]( ) IOOIO tm OOOO tm OIBO Idmrrrrdmrr ←←← ⋅=⋅∫ −⋅=∫ ×× ωωω ,,,,,,, 1(b) where ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1: 2nd Moment of Inertia Dyadic of all the mass m(t) relative to O We obtain (a) + (b) + (c) ( )( ) ( ) ( ) ( ) ( ) ∫ ×+∫ ××+× ∫=∫ ×−= ← tm O O O tm OIOOO tm O tv OO dm td rd rdmrrVdmrvdVRRH , ,,,,, : ωρ ( ) ∫ ×+⋅+×= ← tm O O OIOOOO dm td rd rIVc , ,,, ω EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 19. 19 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 5) (c) ( ) ( ) ( ) ( ) =∫ + ×+=∫ × tm O OCC OCC tm O O O dm td rrd rrdm td rd r ,, ,, , , ( ) ( ) ( ) O OC tm C tm O C C tm O C OC O OC OC td rd dmrdm td rd rdm td rd rm td rd r , 0 , , , , , , , × ∫+∫ ×+∫ ×+×= (c1) (c2) (c3) m td rd r O OC OC , , ×(c1) - Change in the relative position of C (varies with time) and O. (c2) ( ) ∑×−=∫ × openings iflowCiopenOC tm O C OC mrrdm td rd r ,, , , (c3) ( ) ∫ × tm O C C dm td rd r , , - Change due to Elasticity, Sloshing, Moving Parts (Rotors, Pistons,..) If we choose O=C the first two terms (c1), (c2) will be zero, and the third (c3) describes the non-rigidity of the system. EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 20. 20 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 6) (c3) ( ) ( ) ∑+∫ ×=∫ × ← j OjrotorCjrotor tm FrozenRotors O C C tm O C C Rj Idm td rd rdm td rd r ω , , , , , where Consider a system with a number of rigid rotors I R CR C ( )tS OR OOCr Bxˆ Bzˆ shaftr rotorr Byˆ Ixˆ Iyˆ Izˆ Cshaftr Crotorr OyˆOxˆ Ozˆ System with Rotors RjCjrotorI , Ojrotor ←ω - Second Moment of Inertia Dyadic of the Rotor j, relative to it’s Centroid - Angular Velocity Vector of the Rotor j, relative to O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 21. 21 SOLO Absolute Angular Momentum Relative to a Reference Point O (continue – 7) We obtained Let differentiate this equation, relative to the inertial system I R CR C ( )tS OR OOCr Bxˆ Bzˆ shaftr rotorr Byˆ Ixˆ Iyˆ Izˆ Cshaftr Crotorr OyˆOxˆ Ozˆ ∫ ×+∑ ⋅+⋅+×= ←← m FrozemRotor O O O j ORjCjrotorIOOOOO md td rd rIIVcH Rj , ,,,, ωω OIO O O I O H td Hd td Hd , ,, ×+= ←ω ( ) O m FrozemRotor O O O j ORjCjrotor j ORjCjrotorIOOIOO I OO md td rd r td d IIIIVc td d RjRj ∫ ×+ ∑ ⋅+∑ ⋅+⋅+⋅+×= ←←←← , , 0 ,,,,, ωωωω ∫ ××+ ∑ ⋅×+⋅×+ ←←←←← m FrozemRotor O O OIO j ORjCjrotorIOIOOIO md td rd rII Rj , ,,, ωωωωω EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 22. 22 SOLO Moment Relative to a Reference Point O Multiplying (vector product) the 2nd Newton’s Law on the particle of mass dmi, by we obtain:OiOi RRr −=:, ( ) ( ) i I i Oi N j ijiextOi dm td Vd RRfdfdRR ×−= ∑+×− =1 int from which ( ) ( ) ( )∑ ×−=∑ ∑ ×−+∑ ×− == ≠ == N i i I i Oi N i N ij j ijtOi N i iextOi dm td Vd RRfdRRfdRR 11 1 int 1 We define the moment of external forces, relative to O, on the system, as: ( )∑∑ = ×−= N i iextOiOext fdRRM 1 , : EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 23. 23 SOLO Moment Relative to a Reference Point O (continue – 1) Since for any particles i and j the internal forces are of equal magnitude but of opposite directions we have jiij fdfd intint −= ( ) ( ) ( ) ( ) ( ) collinearfandrfdrfdRR fdRRfdRR fdRRfdRR jitijjitijjitij jitOjjitOi jitOjijtOi intintint intint intint 0 ←=×=×−= =×−+×−−= =×−+×− EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH We assumed that the equal but opposite forces between i and j act along the line joining them; i.e. Note collineararefandr jitij int This is not always true (see H. Goldstein “Classical Mechanics”, 2nd Edition, pg.8, R. Aris “Vectors, Tensors and the Basic Equations of Fluid Mechanics”, pp.102-104, Michalas & Michalas “Radiation Hydrodynamics”, pg.72, Jaunzemis “Continuous Mechanics” Sec. 11, pg.223) End Note
- 24. 24 SOLO ( )( ) ( ) ( )∑ −×−+×+∫ ×−= openings iflowOiflowOiopenO I O tm I O I O mVVRRV td cd dm td Rd RR td Hd , 2 2 Moment Relative to a Reference Point O (continue – 2) We have: ( ) ( )∑ ×−=∑ ×−=∑ == N i i I i Oi N i i I i OiOext dm td Rd RRdm td Vd RRM 1 2 2 1 , ∞→↓ N ( )( ) ( )( ) ∫ ×−=∫ ×−=∑ tv I O tm I OOext dv td Vd RRdm td Vd RRM ρ , to obtain ( ) ( )∑ −×−+×+∑= openings iflowOiflowOiopenO I O Oext I O mVVRRV td cd M td Hd , , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Substitute the previous equation with in II td Rd td Vd 2 2 =
- 25. 25 SOLO ( ) ( ) ∑∑ −+=−−+= openings OiopeniflowO I O openings OiopeniflowO I O rmVm td cd RRmVm td cd tP , ,, Moment Relative to a Reference Point O (continue – 3) Let substitute in this equation the following to obtain ( ) ( )∑ −×−+×+∑= openings iflowOiflowOiopenO I O Oext I O mVVRRV td cd M td Hd , , ( ) ( ) O I O openings iflowOiflowOiopenOext I O V td cd mVVRRMH td d ×+∑ −×−+∑= , ,, ( ) ( ) ( ) O openings iflowOiopenO openings iflowOiflowOiopenOext VmRRmVPmVVRRM × ∑ −+−+∑ −×−+∑= , ( )∑ ×−+∑ ×+= openings iflowiflowOiopenOOext mVRRVPM , or ∑ ×+∑ ×+= openings iflowiflowOiopenOOext I O mVrVPMH td d ,,, ( )OCOCO RRmrmc −== ,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 26. 26 SOLO External Forces and Moments Applied on the System We have a system of particles enclosed at the time t by a surface S(t) that bounds the volume v(t). There are no sources or sinks in the volume v(t). The change in the mass of the system is due only to the flow through the surface openings Sopen i (i=1,2, …). The surface S(t) can be divided in: • Sw(t) the impermeable wall through which the flow can not escape .( )0, =sV • Sopen i(t) the openings (i=1,2,…) through which the flow enters or exits .( )0>m ( )0<m EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 27. 27 SOLO External Forces and Moments on the System (continue -1 ) The external forces acting on the system are: v(t) I ds R CR dm C ( )tS 2openS 1openS g σ n 1 t 1 OR O Or, OCr , jR jF kM • Gravitation acceleration (E center of Earth).E E R R M Gg 3 = • Force per unit surface applied by the surroundings on the surface of the system.( )2 /mNσ ( )dstfnpsdTsdnsd 111 +−==⋅=⋅ σσ where: ( ) ndsnnsdsd 111 =⋅= - vector of surface differential ( )2 /mNp - pressure on (normal to) the surface . ( ) ( ) ∑∫∫∑∑∑ +⋅+=→= j j tStv ext i iextext FsddvgFfdF σρ ( ) ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑→ ∑ ×−=∑ k k j jOj tS O tv OOext i iextOiOext MFRRsdRRdvgRRM fdRRM σρ, , The moment of the external forces, relative to a point O, is: f - friction force per (parallel to) unit surface .( )2 / mN • Discrete force exerting by the surrounding on the point , and discrete moments .∑j jF jR ∑ k kM EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 28. 28 SOLO ( ) ( ) ( )∑∑∑∑ −+−+−+= openings Ciopeniflow openings Ciopeniflow openings iopeniflowiflowext I C RRmVVmVVmF td Vd m 2 External Forces Equations (continue -2) ( ) ( ) ( ) ( ) ( ) ∑∫∫∑∫∫∑ ++−+=+⋅+= j j tStvj j tStv ext FdstfnpdvgFsddvgF 11ρσρ ( ) ( ) ( ) 0111 0 =⋅∇== ∫∫∫ ∞∞∞ tv Gauss tStS dvnpdsnpdsnp Since the pressure far away from the body is constant∞p Let add this equation to the previous one ( ) ( ) ( ) ( )[ ] ( ) ∑∫∑∫∫∑ ++−+=+⋅+= ∞ j j tSj j tStv ext FdstfnpptmgFsddvgF 11σρ ( ) ( )[ ] ( )[ ] ∑∫∫ ∑ ∫∫ ++−++−+= ∞∞ j j S openings S Fdstfnppdstfnpptmg W iopen 1111 Finally since on Sopen i (t) the openings (i=1,2,…) the friction force f = 0 ( ) ( )[ ] ( ) ∑∫∫ ∑ ∫∫∑ +−++−+= ∞∞ j j S openings S ext FdsnppdstfnpptmgF W iopen 111 Substitute this equation in EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 29. 29 SOLO External Forces Equations (continue – 3) or ( ) ( )[ ] ( ) ( ) ( ) ( )∑ −+∑ −+ ∑+∑ ∫∫ −+−+∫∫ +−+= ∞∞ openings iflowCiopen openings iflowCiopen j j openings S iflowopeniflow SI C mRRmVV FdsnppmVVdstfnpptmgmV dt d iopenW 2 111 1 ( ) ( ) ( )∑∑∑∑∑ −+−++++= openings iflowCiopen openings iflowCiopen j j i TiA I C mRRmVVFFFtmgmV dt d 2 where EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH v(t) I ds R CR dm C ( )tS 2openS 1openS g σ n 1 t 1 OR O Or, OCr , jR jF kM ( ) ( )∫∫ −+−= ∞ iopenS iflowiopeniflowTi dsnppmVVF 1: Thrust Forces ( )[ ]∫∫∑ +−= ∞ WS A dstfnppF 11: Aerodynamic Forces
- 30. 30 SOLO External Forces Equations (continue – 4) Let substitute ( ) ( ) ( )∑ −+∑ −+∑+∑+∑+= openings iflowCiopen openings iflowCiopen j j i TiA I C mRRmVVFFFtmgmV dt d 2 in CIO O C I C V td Vd td Vd a ×+== ←ω to obtain RIGID-BODY TERMSmV td Vd CIO O C ×+ ← ω ∑−∑ ×+− ← openings iflowCiopen openings iflowCiopenIO O Ciopen mrmr td rd ,, , 2 ω FLUID-FLOW TERMS AERODYNAMIC & PROPULSIVE∑+∑= i TiA FF v(t) I ds R CR dm C ( )tS 2openS 1openS g σ n 1 t 1 OR O Or, OCr , jR jF kM ∑++ j jFmg GRAVITATIONAL & DISCRETE TERMS EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 31. 31 SOLO External Moments Equations (continue – 5) The moments of the external forces relative to the point O are given by ( )( ) ( )( ) ( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑ k k j jOj tS OS tv OOext MFRRsdRRdvgRRM σρ ~ , ( )( ) ( ) ( ) ( ) ( ) ∑+∑ ×−+∫ +−×−+× ∫ −= k k j jOj tS OS tv O MFRRdstfnpRRgdvRR 11ρ Let add to this equation the following ( ) ( ) ( ) ( ) 01 0 5 =−×∇=×− ∫∫∫∫ ∞∞ V OS GGauss tS OS dvRRpdsnpRR to obtain ( )( ) ( ) ( )[ ] ( ) ( ) ∑+∑ ×−+∫ +−×−+× ∫ −=∑ ∞ k k j jOj tS OS tv OOext MFRRdstfnppRRgdvRRM 11, ρ ( ) ( ) ( )[ ] ( ) ( ) ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫ +−×−++−×−+×−= ∞∞ k k j jOj S openings S Son OOOC MFRR dstfnppRRdstfnppRRgmRR W iopen W 1111 0 v(t) I ds R CR dm C ( )tS 2openS 1openS g σ n 1 t 1 OR O Or, OCr , jR jF kM EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 32. 32 SOLO ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ∑+∑ ×−+ ∫∫ ∑ ∫∫ −×−++−×−+×−=∑ ∞∞ k k j jOj S openings S OOOCOext MFRR dsnppRRdstfnppRRgmRRM W iopen 111, ( ) ( )∑ −×−+×+∑= openings iflowOiflowOiopenO I O Oext I O mVVRRV td cd M td Hd , , External Moments Equations (continue -6) Using together with we obtain ( ) ( ) ( ) ∑+∑ ×−+∑ −×−+ ×+∑+∑+×= k k j jCj openings iflowOiopenOiopen O I O openings OTiOAO I O MFRRmVVRR V td cd MMgc td Hd , ,,, , ( ) ( ) ( ) ( )∑ −×−+∑ −×−+ ×+∑= openings iflowOiopenOiopen openings iflowiopeniflowOiopen O I O Oext mVVRRmVVRR V td cd M , EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 33. 33 SOLO External Moments Equations (continue -7) where ( ) ( )[ ]∫∫∑ +−×−= ∞ WS OOAero dstfnppRRM 11:, Aerodynamic Moments ( ) ( ) ( ) ( )∫∫ −×−+−×−= ∞ iopenS OiflowiopeniflowOiopenOTi dsnppRRmVVRRM 1:, Thrust Moments on the opening i discrete forces exerting by the surrounding at point∑ j jF ∑ k kM jR discrete moments exerting by the surrounding on the system v(t) I ds R CR dm C ( )tS 2openS 1openS g σ n 1 t 1 OR O Or, OCr , jR jF kM EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH
- 34. 34 SOLO ( ) ( ) I tm O O O I j ORjCjrotor I IO OIO I O I O OO I O I O dm td rd r td d I td d td d I td Id td Vd cV td cd td Hd Rj ∫ ×+∑+⋅+⋅+×+×= ← ← ← , ,,, , , ,, ω ω ω External Moments Equations (continue -8) Using together with we obtain ( ) ( )∑+∫ ×+⋅×+⋅+⋅ ←←←← ← j I ORjCjrotor I tm FrozenRotors O O OIOOIOIO O O O IO O Rj I td d dm td rd r td d I td Id td d I ωωωω ω , , ,, , , ( ) ( ) ( ) ( ) ∑+∑ ×−+∑ −×−+ ×+∑+∑+×−= k k j jCj openings iflowOiopenOiopen O I O openings OTiOAOC I O MFRRmVVRR V td cd MMgmRR td Hd ,, , ( ) ( ) ( ) ∑+∑ ×−+∑ −×−+ ∑+∑+ −×= k k j jCj openings iflowOiopenOiopen openings OTiOA I O O MFRRmVVRR MM td Vd gc ,,, EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH Table of Content
- 35. 35 SOLO SUMMARY OF THE EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM FIRST MOMENT OF INERTIA SECOND MOMENT OF INERTIA DYADIC EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM ( )[ ] ( ) ∫ −⋅= tm OOOOO dmrrrrI ,,,,, 1: 2nd Moment of Inertia Dyadic of all the mass m(t) relative to O The First Moment of Inertia of the System, relative to the point O, is defined as:Oc, ( ) ( ) ( ) ( ) OCOC tm O tm OO rmRRmmdrmdRRc ,,, : =−==−= ∫∫
- 36. 36 SOLO ( ) ( ) ∑∑ ∫∫∫ === openings iopenopenings S i tm td md mdmd td d tm iopen MASS EQUATION FORCE EQUATION EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM RIGID-BODY TERMSmV td Vd CIO O C ×+ ← ω ∑−∑ ×+− ← openings i iflowiopen openings i iflowiopenIO B iopen mrmr td rd ˆˆ ˆ 2 ω FLUID-FLOW TERMS GRAVITATIONAL, AERODYNAMIC, PROPULSIVE & ∑+∑+= i TiA FFmg ∑+ j jF DISCRETE TERMS SUMMARY OF THE EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM (CONTINUE – 1)
- 37. 37 SOLO SUMMARY OF THE EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM (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 iflowOiopenIO O Oiopen Oiopen mr td rd r , , , 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
- 38. 38 SOLO SUMMARY OF THE EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM (CONTINUE – 2) ( )[ ]∫∫∑ +−= ∞ WS A dstfnppF 11: AERODYNAMIC FORCES ( ) ( )∫∫ −+−= ∞ iopenS iflowiopeniflowTi dsnppmVVF 1: THRUST FORCES ( ) ( )[ ]∫∫ +−×−=∑ ∞ WS OOA dstfnppRRM 11:, AERODYNAMIC MOMENTS RELATIVE TO O ( ) ( ) ( ) ( )[ ]∫∫ −×−+−×−= ∞ iopenS OiflowiopeniflowOiopenOTi dsnppRRmVVRRM 1:, THRUST MOMENTS RELATIVE TO O EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM Table of Content
- 39. 39 SOLO References 1. Meriam, J.L., “Dynamics”, John Wiley & Sons, 1966 EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM SIMPLIFIED PARTICLE APPROACH 2. Greensite, A.L., “Elements of Modern Control Theory”,Vol. 2, Spartan Books, 1970 3. Greenwood, D.T., “Principles of Dynamics”, Prentice-Hall Inc., 1965 Table of Content
- 40. January 5, 2015 40 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