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SOLO HERMELIN
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE
APPROACH
http://www.solohermelin.com
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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)
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SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED 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
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SOLO EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Assumptions1. 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
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 .
iopenSiflowR
Therefore
( ) ( )I
iflow
I
iopen
td
tRd
td
tRd
≠
and
( ) ( )iopeniflow
I
iopen
I
iflowSi VV
td
tRd
td
tRdV
−=−=:,
is the velocity of flow relative to the opening iopenS
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
6
SOLO
The Inertial Velocity and Acceleration of the mass dmi are given by
I
ii td
RdV
=I
i
I
ii td
Rd
td
Vda
2
2
==
Total Mass of the System
( )( )∫∫∑ ==→=
→
∞→
= tvmdmdm
NN
ii dvdmmmdtm
i
ρ1
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
ii mmdttm
1
( ) ( ) ( ) ( )∑∑ =∆
∆=
∆−∆+=
→∆→∆openings
iflowopenings
iflow
tttm
t
m
t
tmttmtm
00limlim
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
The mass rate (flow) , entering / leaving the system, is given by
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
Cdmmd
NN
iiiC dvRdmRtRmmdRtRm
i
ρ
::1
( ) ( )[ ] ( ) ( )∑∑ ∆+∆+∆+=∆+= openings
iflowiflowiflow
N
iiiiCC RRmmdRRtRmtRm
1
( ) ( )[ ] ( )∑+∑=
∆
∑ ∆+∆+∑∆=
∆∆=
=
=
→∆→∆ openingsiflowiflow
N
ii
I
iopeningsiflowiflowiflow
N
iii
t
C
tC Rmmd
td
Rd
t
RRmmdR
t
tRmRm
td
d
1
1
00limlim
Now let add the constraint that at time t the flow at the opening is such that
iopenS
( ) ( )tRtR iflowiopen
=
to obtain( ) ∑∑ −=
= openingsiopeniflow
I
C
N
ii
I
i RmRmtd
dmd
td
Rd
1
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
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
ii
I
i RmRmtd
dmd
td
Rd
1
( )
( )
∑
∑∑∑
∑∑
−=
−−=−+=
=−+=−
openingsCiopeniflow
I
C
openingsCiopenflowi
I
C
openingsiopeniflow
I
C
openingsCiflow
openingsiopeniflow
I
CC
openingsiopeniflow
I
C
rmtd
Rdm
RRmtd
RdmRm
td
RdmRm
Rmtd
RdmRmRmRm
td
d
,
Therefore
( ) ∑∑∑ −=−−== openings
Ciopeniflow
I
C
openingsCiopeniflow
I
CN
ii
I
i rmtd
RdmRRm
td
Rdmmd
td
Rd,
1
The First Moment of Inertia of the System, relative to the point O, is defined as:Oc,
( ) ( ) OCOC
N
iiO
N
iii
N
iiOi
N
iiOiO rmRRmmdRmdRmdRRmdrc ,
1111,, :
=−=∑−∑=∑ −=∑=====
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
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SOLO
( ) ∑∑∑ −=−−== openings
Ciopeniflow
I
C
openingsCiopeniflow
I
CN
ii
I
i rmtd
RdmRRm
td
Rdmmd
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
iii
N
ii
I
i mdVdmtd
RdtPmdVmd
td
RdtP
i
::
11
( ) ( )( ) ( ) ∑∑
∑∑∑
−=−−=
→−=−−==∞→
→=
openingsCiopeniflowC
openingsCiopeniflowC
N
mdmdopenings
Ciopeniflow
I
C
openingsCiopeniflow
I
CN
ii
I
i
rmVmRRmVmtP
rmtd
RdmRRm
td
Rdmmd
td
RdtP
i
,
,1
Differentiate ( )( ) OCOC
N
iiO
N
iii
N
iiOi
N
iiOiO
rmRRm
mdRmdRmdRRmdrc
,
1111,, :
=−=
−=−== ∑∑∑∑====
to obtain ( ) ( ) ( )
+−
+=
−+
−=−+
−=
∑∑
∑
openingsOiflowO
openingsCiflowC
OCopenings
iflow
I
O
I
COC
I
O
I
C
I
O
RmVmRmVm
RRmtd
Rd
td
RdmRRm
td
Rd
td
Rdm
td
tcd
,
to obtain
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
10
SOLO
Linear Momentum of the System (continue-1)
Substitute
( ) ( ) ∑−=∑ −−=openings
CiopeniflowCopenings
CiopeniflowC rmVmRRmVmtP ,
to obtain
( )
∑+−
∑+=
openingsOiflowO
openingsCiflowC
I
O RmVmRmVmtd
tcd
,
into
( ) ( ) ∑∑ −+=−−+=openings
OiopeniflowO
I
O
openingsOiopeniflowO
I
O rmVmtd
cdRRmVm
td
cdtP ,
,,
At the time t + Δ t the Linear Momentum of the System (including the mass entering/leaving through S) is:
( ) ( ) ( ) ( )∑∑ ∆+∆+∆+=∆+= openings
iflowiflowiflow
N
ii RRmmRRtPtP
1
:
By subtracting those two equations, dividing by Δt, and taking the limit, we get
( )
∑ ∑∑∑
∑∑∑
−−+=+=
∆
−
∆+∆+
∆+
=∆
∆=
=
==
→∆→∆
openingsI
openingsI
Ciopen
iflowopenings
Ciopeniflow
I
C
I
Ciflow
iflow
N
ii
I
i
N
ii
I
i
openingsI
iflow
I
iflow
iflow
N
ii
I
i
I
i
tt
td
rdmrm
td
Rdm
td
Rdm
td
Rdmdm
td
Rd
t
mdtd
Rd
td
Rd
td
Rdmmd
td
Rd
td
Rd
t
tP
td
Pd
,
,2
2
12
2
11
00limlim
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
11
SOLO
Linear Momentum of the System (continue-2)
We obtain
( ) ∑∑ −−+=i I
Ciflowiflow
openingsCiopeniflow
I
C
I
C
Itd
rdmrm
td
Rdm
td
Rdm
td
tPd ,,2
2
( )
( ) ( ) ( )∑∑
∑∑
∑∑
−−−−+=
→−−+=
++=
→∞
→
=
openingsCiopeniflow
openingsCiopeniflowC
I
C
I
N
mdmdopeningsI
Ciopen
iflowopenings
Ciopeniflow
I
C
I
C
openingsI
Ciflow
iflow
I
CN
ii
I
i
I
VVmRRmVmtd
Vdm
td
tPd
td
rdmrm
td
Rdm
td
Rdm
td
rdm
td
Rdmmd
td
Rd
td
tPd
i
,
,2
2
,
12
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EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
An equivalent result could be obtained by differentiating
( ) ∑−=openings
Ciopeniflow
I
C rmtd
RdmtP ,
We obtained
Table of Content
12
SOLO
Force Equation
Applying the 2nd Newton’s Law to the particle of mass mi, we obtain:
∑=
+==N
jijiexti
I
ii
I
i fdfdmdtd
Rdmd
td
Vd
1int2
2
where
iextfd
- External forces acting on the mass mi
ijfd 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
01 1
int
=∑∑
=≠=
N
i
N
ijj
ijfd
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
13
SOLO
( ) ∞→
→→∑−∑−+=N
mdmdopeningsI
Ciopeniflow
openingsCiopeniflow
I
C
I
C
Iitd
rdmrm
td
Rdm
td
Rdm
td
tPd ,,2
2
Force Equation (continue – 1)
We have ∑∑∑∑∑ =+==
≠===
ext
N
i
N
ijj
ij
N
iiext
N
ii
I
i Ffdfdmdtd
Vd
0
1 1int
11
∑∑=
=N
ii
I
iext md
td
RdF
12
2
Substitute this equation into
to obtain
∑∑∑∑ −−+=++=openings I
Ciopeniflow
openingsCiopeniflow
I
C
I
C
openings I
Ciflowiflow
I
Cext
Itd
rdmrm
td
Rdm
td
Rdm
td
rdm
td
RdmF
td
Pd ,,2
2,
Rearranging we obtain
∑∑∑∑ ++
−+=
openings I
Ciopeniflow
openingsCiopeniflow
openings I
Ciopen
I
Ciflowiflowext
I
C
td
rdmrm
td
rd
td
rdmF
td
Rdm ,
,,,
2
2
2
( ) ∑∑∑∑
−+−+
−+=
openings I
C
I
iopeniflow
openingsCiopeniflow
openingsI
iopen
I
iflowiflowext td
Rd
td
RdmRRm
td
Rd
td
RdmF
2or
( ) ( ) ( )∑∑∑∑ −+−+−+=openings
Ciopeniflowopenings
Ciopeniflowopenings
iopeniflowiflowext
I
C RRmVVmVVmFtd
Vdm
2
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
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
ii
I
iOi
N
iiiOi
N
iiOiO dm
td
RdrdmVRRPdRRH
1,
11, :
By taking a very large number N of particles, we go from discrete to continuous
∫⇒∑∞→
=
NN
i 1
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
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:
( ) ( )∑∑ ∆
∆+×∆++
∆+×∆+=∆+
= openingsiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOiOO m
td
Rd
td
Rdrrdm
td
Rd
td
RdrrHH
,,1
,,,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
15
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 1)
By subtracting
I
O
tI
O
t
H
td
Hd
∆∆
=→∆
,
0
, lim
( ) ( )
t
dmtd
Rdrm
td
Rd
td
Rdrrdm
td
Rd
td
Rdrr
openings
N
ii
I
iiOiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOi
t ∆
×−∆
∆+×∆++
∆+×∆+
=
∑ ∑∑==
→∆
1,,
1,,
0lim
∑∑∑ ×+×+×=== openings
iflow
I
iflowOiflow
N
ii
I
iOiN
ii
I
iOi m
td
Rdrdm
td
Rd
td
rddm
td
Rdr
,1
,
12
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 SYSTEMSIMPLIFIED PARTICLE APPROACH
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
OO dmVrdmVRRdvVRRH
,, ρ
dividing by Δt, and taking the limit, we get
from ( ) ( )∑∑ ∆
∆+×∆++
∆+×∆+=∆+
= openingsiflow
I
iflow
I
iflowOiflowOiflow
N
ii
I
i
I
iOiOiOO m
td
Rd
td
Rdrrdm
td
Rd
td
RdrrHH
,,1
,,,,
16
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 2)
∑∑∑ ×+×+×=== openings
iflow
I
iflowOiopen
N
ii
I
i
I
OiN
ii
I
iOi
I
O mtd
Rdrdm
td
Rd
td
rddm
td
Rdr
td
Hd
,1
,
12
2
,,
( )∑ ×−+∑ ×
−+∑ ×=
== openingsiflow
I
iflowOiopen
N
ii
I
i
I
O
I
iN
ii
I
iOi m
td
RdRRdm
td
Rd
td
Rd
td
Rddm
td
Rdr
112
2
,
( )∑ ×−+∑×−∑ ×=== openings
iflow
I
iflowOiopen
N
ii
I
i
I
ON
ii
I
iOi m
td
RdRRdm
td
Rd
td
Rddm
td
Rdr
112
2
,
By taking a very large number N of particles, we go from discrete to continuous
∫⇒∑∞→
=
NN
i 1
( )( )∑ ×−+×−∫ ×=
openingsiflowiflowOiopenO
tmI
O
I
O mVRRPVdmtd
Rdr
td
Hd
2
2
,,
( ) ( ) ∑∑ −+=−−+=openings
OiopeniflowO
I
O
openingsOiopeniflowO
I
O rmVmtd
cdRRmVm
td
cdtP ,
,,
Substitute to obtain
( )( ) ( )∑ ×−+×
∑ −−++∫ ×=
openingsiflowiflowOiopenO
openingsiflowOiopenO
I
O
tmI
O
I
O mVRRVmRRVmtd
cddm
td
Rdr
td
Hd
,
2
2
,,
or
( )( )∑ −×+×+∫ ×=
openingsiflowOiflowOiopenO
I
O
tmI
O
I
O mVVrVtd
cddm
td
Rdr
td
Hd
,,
2
2
,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
17
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 3)
We obtained
( )( )
( )( ) ( )
∫ ×=∫ ×−=∫ ×−=tm
Otm
Otv
OO dmVrdmVRRdvVRRH
,, ρ
Substitute in the previous equation
OIO
O
OO
I
O
I
O
I
OO rtd
rdV
td
rd
td
Rd
td
RdVrRR ,
,,, :&
×++=+==+= ←ω
( )( ) ( )
∫
×++×=∫ ×−= ←
tmOIO
O
OOO
tmOO dmr
td
rdVrdmVRRH ,
,,,
ω
( )( )
( ) ( )∫
×+∫ ××+×
∫= ←tm O
OO
tmOIOOO
tmO dm
td
rdrdmrrVdmr ,,,,,
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 SYSTEMSIMPLIFIED PARTICLE APPROACH
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 ×=×=×
∫ ,,,
( )( )
( )[ ]( )
IOOIOtm
OOOOtm
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
OO
tmOIOOO
tmO
tvOO dm
td
rdrdmrrVdmrvdVRRH ,,,,,, :
ωρ
( )∫
×+⋅+×= ←
tm O
OOIOOOO dm
td
rdrIVc ,,,,
ω
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
19
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 5)
(c)( )
( ) ( )( )
=∫
+×+=∫
×
tm O
OCCOCC
tm O
OO dm
td
rrdrrdm
td
rdr ,,
,,,
,
( ) ( ) ( ) O
OC
tmC
tm O
CC
tm O
COC
O
OCOC td
rddmrdm
td
rdrdm
td
rdrm
td
rdr ,
0
,,
,,
,,
,
×
∫+∫
×+∫
×+×=
(c1) (c2) (c3)
mtd
rdr
O
OCOC
,,
×(c1) - Change in the relative position of C (varies with time) and O.
(c2)( )
∑×−=∫
×
openingsiflowCiopenOC
tm O
COC mrrdm
td
rdr
,,
,,
(c3)( )∫
×
tm O
CC dm
td
rdr ,,
- 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 SYSTEMSIMPLIFIED PARTICLE APPROACH
20
SOLO
Absolute Angular Momentum Relative to a Reference Point O (continue – 6)
(c3)
( ) ( )∑+∫
×=∫
× ←
jOjrotorCjrotor
tmFrozenRotors
O
CC
tm O
CC Rj
Idmtd
rdrdm
td
rdr ω
,,
,,
,
where
Consider a system with a number of rigid rotorsI
R
CR
C
( )tS
OR
OOCr
Bx
Bz
shaftr
rotorr
By
Ix
Iy
Iz
Cshaftr
Crotorr
OyOx
Oz
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 SYSTEMSIMPLIFIED PARTICLE APPROACH
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
Bx
Bz
shaftr
rotorr
By
Ix
Iy
Iz
Cshaftr
Crotorr
OyOx
Oz
∫
×+∑ ⋅+⋅+×= ←←
mFrozemRotor
O
OO
jORjCjrotorIOOOOO md
td
rdrIIVcH
Rj
,,,,,
ωω
OIO
O
O
I
O Htd
Hd
td
Hd,
,,
×+= ←ω
( )
O
mFrozemRotor
O
OO
jORjCjrotor
jORjCjrotorIOOIOO
I
OO
mdtd
rdr
td
d
IIIIVctd
dRjRj
∫
×+
∑ ⋅+∑ ⋅+⋅+⋅+×= ←←←←
,,
0
,,,,,
ωωωω
∫
××+
∑ ⋅×+⋅×+ ←←←←←
mFrozemRotor
O
OOIO
jORjCjrotorIOIOOIO md
td
rdrII
Rj
,,,,
ωωωωω
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
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
iOi
N
jijiextOi dm
td
VdRRfdfdRR
×−=
∑+×−
=1int
from which
( ) ( ) ( )∑ ×−=∑ ∑ ×−+∑ ×−==
≠==
N
ii
I
iOi
N
i
N
ijj
ijtOi
N
iiextOi dm
td
VdRRfdRRfdRR
11 1int
1
We define the moment of external forces, relative to O, on the system, as:
( )∑∑=
×−=N
iiextOiOext fdRRM
1, :
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
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 SYSTEMSIMPLIFIED 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
SOLO
( )( )
( ) ( )∑ −×−+×+∫ ×−=openings
iflowOiflowOiopenO
I
O
tmI
O
I
O mVVRRVtd
cddm
td
RdRR
td
Hd
,
2
2
Moment Relative to a Reference Point O (continue – 2)
We have:
( ) ( )∑ ×−=∑ ×−=∑==
N
ii
I
iOi
N
ii
I
iOiOext dm
td
RdRRdm
td
VdRRM
12
2
1,
∞→↓ N
( )( )
( )( )∫ ×−=∫ ×−=∑tv
I
Otm
I
OOext dvtd
VdRRdm
td
VdRRM ρ
,
to obtain
( ) ( )∑ −×−+×+∑=openings
iflowOiflowOiopenO
I
OOext
I
O mVVRRVtd
cdM
td
Hd
,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Substitute the previous equation with inII
td
Rd
td
Vd2
2
=
25
SOLO
( ) ( ) ∑∑ −+=−−+=openings
OiopeniflowO
I
O
openingsOiopeniflowO
I
O rmVmtd
cdRRmVm
td
cdtP ,
,,
Moment Relative to a Reference Point O (continue – 3)
Let substitute in this equation the following
to obtain
( ) ( )∑ −×−+×+∑=openings
iflowOiflowOiopenO
I
OOext
I
O mVVRRVtd
cdM
td
Hd
,,
( ) ( ) O
I
O
openingsiflowOiflowOiopenOext
I
O Vtd
cdmVVRRMH
td
d
×+∑ −×−+∑= ,,,
( ) ( ) ( ) Oopenings
iflowOiopenOopenings
iflowOiflowOiopenOext VmRRmVPmVVRRM
×
∑ −+−+∑ −×−+∑= ,
( )∑ ×−+∑ ×+=openings
iflowiflowOiopenOOext mVRRVPM
,
or
∑ ×+∑ ×+=openings
iflowiflowOiopenOOext
I
O mVrVPMHtd
d
,,,
( )OCOCO RRmrmc −== ,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
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 SYSTEMSIMPLIFIED PARTICLE APPROACH
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
1openSg
σn1
t1
OR
O
Or,
OCr ,
jR
jF
kM
• Gravitation acceleration (E center of Earth).E
E
RR
MGg
3
=
• Force per unit surface applied by the surroundings on the surface of the system.( )2/mNσ
( ) dstfnpsdTsdnsd111 +−==⋅=⋅ σσ
where:
( ) ndsnnsdsd111 =⋅= - vector of surface differential
( )2/mN p - pressure on (normal to) the surface .
( ) ( )∑∫∫∑∑∑ +⋅+=→=
jj
tStv
exti
iextext FsddvgFfdF
σρ
( )( )
( )( )
( )( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑→
∑ ×−=∑
kk
jjOj
tSO
tvOOext
iiextOiOext
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 ∑
kkM
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
28
SOLO
( ) ( ) ( )∑∑∑∑ −+−+−+=openings
Ciopeniflowopenings
Ciopeniflowopenings
iopeniflowiflowext
I
C RRmVVmVVmFtd
Vdm
2
External Forces Equations (continue -2)
( ) ( ) ( )( )
( )∑∫∫∑∫∫∑ ++−+=+⋅+=
jj
tStvjj
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
( ) ( )( ) ( )[ ]
( )∑∫∑∫∫∑ ++−+=+⋅+= ∞
jj
tSjj
tStv
ext FdstfnpptmgFsddvgF
11σρ
( ) ( )[ ] ( )[ ] ∑∫∫ ∑ ∫∫ ++−++−+= ∞∞j
j
S openings S
FdstfnppdstfnpptmgW iopen
1111
Finally since on Sopen i (t) the openings (i=1,2,…) the friction force f = 0
( ) ( )[ ] ( ) ∑∫∫ ∑ ∫∫∑ +−++−+= ∞∞j
j
S openings S
ext FdsnppdstfnpptmgFW iopen
111
Substitute this equation in
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
29
SOLO
External Forces Equations (continue – 3)
or
( ) ( )[ ] ( ) ( )
( ) ( )∑ −+∑ −+
∑+∑
∫∫ −+−+∫∫ +−+= ∞∞
openingsiflowCiopen
openingsiflowCiopen
jj
openings Siflowopeniflow
SIC
mRRmVV
FdsnppmVVdstfnpptmgmVdt
d
iopenW
2
111 1
( ) ( ) ( )∑∑∑∑∑ −+−++++=openings
iflowCiopenopenings
iflowCiopenj
ji
TiAI
C mRRmVVFFFtmgmVdt
d
2
where
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
v(t)
I
ds
R
CR
dm
C
( )tS
2openS
1openSg
σn1
t1
OR
O
Or,
OCr ,
jR
jF
kM
( ) ( )∫∫ −+−= ∞
iopenS
iflowiopeniflowTi dsnppmVVF
1:Thrust Forces
( )[ ]∫∫∑ +−= ∞
WS
A dstfnppF11: Aerodynamic Forces
30
SOLO
External Forces Equations (continue – 4)
Let substitute
( ) ( ) ( )∑ −+∑ −+∑+∑+∑+=openings
iflowCiopenopenings
iflowCiopenj
ji
TiAI
C mRRmVVFFFtmgmVdt
d
2
in
CIO
O
C
I
C Vtd
Vd
td
Vda
×+== ←ω
to obtain
RIGID-BODY TERMSmVtd
VdCIO
O
C
×+ ←
ω
∑−∑
×+− ←
openingsiflowCiopen
openingsiflowCiopenIO
O
Ciopen mrmrtd
rd
,,,2 ω FLUID-FLOW TERMS
AERODYNAMIC & PROPULSIVE ∑+∑=
iTiA FF
v(t)
I
ds
R
CR
dm
C
( )tS
2openS
1openSg
σn1
t1
OR
O
Or,
OCr ,
jR
jF
kM
∑++j
jFmg
GRAVITATIONAL & DISCRETE TERMS
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
31
SOLO
External Moments Equations (continue – 5)
The moments of the external forces relative to the point O are given by
( )( )
( )( )
( ) ∑+∑ ×−+∫ ⋅×−+∫ ×−=∑k
kj
jOjtS
OStv
OOext MFRRsdRRdvgRRM
σρ ~,
( )( )
( ) ( )( )
( ) ∑+∑ ×−+∫ +−×−+×
∫ −=k
kj
jOjtS
OStv
O MFRRdstfnpRRgdvRR
11ρ
Let add to this equation the following
( )( )
( ) ( ) 010
5
=−×∇=×− ∫∫∫∫ ∞∞V
OS
GGauss
tS
OS dvRRpdsnpRR
to obtain
( )( )
( ) ( )[ ]( )
( ) ∑+∑ ×−+∫ +−×−+×
∫ −=∑ ∞k
kj
jOjtS
OStv
OOext MFRRdstfnppRRgdvRRM
11, ρ
( ) ( ) ( )[ ] ( ) ( )
( ) ∑+∑ ×−+
∫∫ ∑ ∫∫
+−×−++−×−+×−= ∞∞
kk
jjOj
S openings SSon
OOOC
MFRR
dstfnppRRdstfnppRRgmRRW iopen
W
1111
0
v(t)
I
ds
R
CR
dm
C
( )tS
2openS
1openSg
σn
1
t
1
OR
O
Or,
OCr ,
jR
jF
kM
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
32
SOLO
( ) ( ) ( )[ ] ( ) ( )[ ]
( ) ∑+∑ ×−+
∫∫ ∑ ∫∫ −×−++−×−+×−=∑ ∞∞
kk
jjOj
S openings SOOOCOext
MFRR
dsnppRRdstfnppRRgmRRMW iopen
111,
( ) ( )∑ −×−+×+∑=openings
iflowOiflowOiopenO
I
OOext
I
O mVVRRVtd
cdM
td
Hd
,,
External Moments Equations (continue -6)
Using
together with
we obtain
( ) ( ) ( ) ∑+∑ ×−+∑ −×−+
×+∑+∑+×=
kk
jjCj
openingsiflowOiopenOiopen
O
I
O
openingsOTiOAO
I
O
MFRRmVVRR
Vtd
cdMMgc
td
Hd
,,,,
,
( ) ( ) ( ) ( )∑ −×−+∑ −×−+
×+∑=
openingsiflowOiopenOiopen
openingsiflowiopeniflowOiopen
O
I
OOext
mVVRRmVVRR
Vtd
cdM
,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
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
1openSg
σn
1
t
1
OR
O
Or,
OCr ,
jR
jF
kM
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
34
SOLO
( )( )
Itm O
OO
Ij
ORjCjrotor
I
IOOIO
I
O
I
OOO
I
O
I
O dmtd
rdr
td
dI
td
d
td
dI
td
Id
td
VdcV
td
cd
td
HdRj
∫
×+∑+⋅+⋅+×+×= ←
←←
,,,,
,,
,,
ωωω
External Moments Equations (continue -8)
Using
together with
we obtain
( )( )∑+∫
×+⋅×+⋅+⋅ ←←←←
←
jI
ORjCjrotor
I
tmFrozenRotors
O
OOIOOIOIO
O
O
O
IOO Rj
Itd
ddm
td
rdr
td
dI
td
Id
td
dI ωωωωω
,
,,,
,,
( )( ) ( ) ( ) ∑+∑ ×−+∑ −×−+
×+∑+∑+×−=
kk
jjCj
openingsiflowOiopenOiopen
O
I
O
openingsOTiOAOC
I
O
MFRRmVVRR
Vtd
cdMMgmRR
td
Hd
,,,
( ) ( ) ( ) ∑+∑ ×−+∑ −×−+
∑+∑+
−×=
kk
jjCj
openingsiflowOiopenOiopen
openingsOTiOA
I
OO
MFRRmVVRR
MMtd
Vdgc
,,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED PARTICLE APPROACH
Table of Content
35
SOLOSUMMARY 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
SOLO
( )( )
∑∑ ∫∫∫
===openings iopenopenings S
i
tm td
mdmdmd
td
dtm
iopen
MASS EQUATION
FORCE EQUATION
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM
RIGID-BODY TERMSmVtd
VdCIO
O
C
×+ ←
ω
∑−∑
×+− ←
openings
iiflowiopen
openings
iiflowiopenIO
B
iopen mrmrtd
rd
ˆˆˆ
2 ωFLUID-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
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
OjrotorCrotorjIOj
OjrotorCrotorj RjRjII ωωω
,, ROTORS TERMS
( )
( )
∫
××+
∫
×+
←tm
FrozenRotorO
OOIO
O
tmFrozenRotor
O
OO
dmtd
rdr
dmtd
rdr
td
d
,,
,,
ω
BODY FLUIDS, MOVING PARTS, ELASTICITY,… TERMS
FLUID CROSSING OPENINGS TERMS∑
×+×− ←
openingsiflowOiopenIO
O
OiopenOiopen mr
td
rdr
,
,, ω
AERODYNAMIC & PROPULSIVE
∑+∑=i
OTiOA MM ,,
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEM
( ) ∑+∑ ×−+k
kj
jOj MFRR DISCRETE FORCES
& MOMENTS TERMS
−×+
I
OO td
Vdgc
, NON-CENTROIDAL MOMENTS TERMS
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
SOLO
References
1. Meriam, J.L., “Dynamics”, John Wiley & Sons, 1966
EQUATIONS OF MOTION OF A VARIABLE MASS SYSTEMSIMPLIFIED 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
January 5, 2015 40
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA
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