766 AIAA JOURNAL VOL. 4, NO. 4 Equations of Motion for the Variable Mass System WILLIAM T. THOMSON* University of California, Los Angeles, Calif. I. Introduction D URING the past years several papers have appeared on the equations of motion of the variable mass system. The equations presented appear to be considerably more complicated than those of classical development, and the ap- parent differences in the equations, because of the varied approaches of development, have not contributed to the clarification of the subject. This paper is an attempt to re- late these unfamiliar results to the classical equations and to interpret the results of the more recent presentations. Listing these papers in chronological order, Rankin 1 pre- sented an extensive document of 128 pages in 1949. The development of the basic force and moment equations, however, occupies only four pages (462-466). In 1957 Leitmann 2 presented a development for the momentum and force equations by considering the variation of the center of mass of the system caused by the ejected mass. Ap- proaching the problem from the point of view of a con- tinuum, Thorpe, 3 in 1962, presented the development for the force equation in integral form. Of the three papers men- tioned, the latter two dealt only with the force equation, whereas Rankin gave a formidable looking equation for the moment of momentum and the moment equation as well. Two texts, Halfman 4 and Thomson, 5 develop the moment equation by examining the acceleration of the mass particle, using the equations of relative motion. The equations pre- sented in (3-5) appear in forms quite unrelated to those of (1) and (2) that have raised questions as to which equations of motions are correct. It can be shown, however, that all these equations are valid and represent different forms of the equations of motion. II. Momentum and the Force Equation We will define the variable mass system, shown in Fig. 1, by a closed boundary B within which the mass at time t is m = ^, rrii (I) B The center of mass c of the system at time t, referred to inertial coordinates xyz, is then defined by the equation mr c = ]C mXi (2) B We also will define the momentum of the system at time t by the equation p = SniiTi (3) B To express p in terms of r c , we differentiate Eq. (2) mfc + m r c = Z mdi + ]C mXi (4) B 8 where the second summation associated with rhi represents summation for particles crossing over the boundary surface B (i.e., rhi = 0 for all particles within B which do not cross B). Thus, in view of Eq. (3), we obtain for the momentum the equation p = mf c + mr c — 2 w&»r» (5) s It is evident here that Eq. (5) contains two unclassical terms. To interpret them, consider a rocket with mass ejected at only one point e on the boundary B, as shown in Fig. 2. Then y"^ rhi = rh and ^P rhiTi = rht e Received December 8, 1965; revision received January 10, 1966. * Professor of Engineering; also Consultant TRW Systems. Associate Fellow Member AIAA. so that Eq. (5) becomes p = mr c + m(r c - r.) (6) We note here that m, for the case of the rocket, is negative, and that the second term is zero when the dimension $ ce = (r« — r c ) goes to zero. Thus, it is evident that the classical treatment, where the momentum is given only by the first term, is restricted to the system considered as a particle. To obtain the force equation, one must determine the change in the momentum of the same mass considered at two different times. The momentum of m at time t + dt is the sum of the momentum of the system within B at time t + dt and the momentum of the ejected masses during the time dt, (7) p(t + dt) - ^ nndt where V;* is the absolute velocity of the ejected mass — rhidt. Thus, from the principle "impulse equals change in momen- tum 77 , we obtain the force equation in terms of the momen- tum p F = p — (8) If Eq. (3) is differentiated and substituted into Eq. (8), we obtain F = E rniii + Z m t (i t - v t -*) B S (9) B S where U; is the relative velocity of the ejected mass (i.e. v* = To obtain the force equation in terms of the center of mass, we differentiate Eq. (5) and substitute into Eq. (8) F = = mr c + [ mr c — E wi»r,- ) V s ) (10) It is evident that several nonclassical terms result in this development. Again, it is instructive to consider the rocket ejecting mass at one point e. The previous equation then becomes F = mf c + m(r c - r.) + m(2f c - f e - v*) (11) Since absolute velocities are not convenient quantities, we replace them in terms of quantities relative to a fixed point 0 on the body as follows : f c = To + G> X 0oc + 00crel) r e = f 0 + G> X 0 0 e > (12) v* = r 0 + co X j>oe + u ) where u is the velocity of the ejected mass relative to a point e (the nozzle of the rocket engine). Then, Eq. (11) becomes F = mf c — m(p oe — 0 OC ) — m[2o> X (9oe - Poc) - 2{i ocrel + u] (13) Fig. 1 Variable mass system defined by closed boundary B. Downloaded by CARLETON UNIVERSITY LIBRARY on June 1, 2014 | http://arc.aiaa.org | DOI: 10.2514/3.3544
Transcript
766 AIAA JOURNAL VOL. 4, NO. 4
Equations of Motion for the VariableMass System
WILLIAM T. THOMSON*University of California, Los Angeles, Calif.
I. Introduction
DURING the past years several papers have appeared onthe equations of motion of the variable mass system.
The equations presented appear to be considerably morecomplicated than those of classical development, and the ap-parent differences in the equations, because of the variedapproaches of development, have not contributed to theclarification of the subject. This paper is an attempt to re-late these unfamiliar results to the classical equations and tointerpret the results of the more recent presentations.
Listing these papers in chronological order, Rankin1 pre-sented an extensive document of 128 pages in 1949. Thedevelopment of the basic force and moment equations,however, occupies only four pages (462-466). In 1957Leitmann2 presented a development for the momentum andforce equations by considering the variation of the centerof mass of the system caused by the ejected mass. Ap-proaching the problem from the point of view of a con-tinuum, Thorpe,3 in 1962, presented the development for theforce equation in integral form. Of the three papers men-tioned, the latter two dealt only with the force equation,whereas Rankin gave a formidable looking equation for themoment of momentum and the moment equation as well.Two texts, Halfman4 and Thomson,5 develop the momentequation by examining the acceleration of the mass particle,using the equations of relative motion. The equations pre-sented in (3-5) appear in forms quite unrelated to those of(1) and (2) that have raised questions as to which equationsof motions are correct. It can be shown, however, that allthese equations are valid and represent different forms ofthe equations of motion.
II. Momentum and the Force Equation
We will define the variable mass system, shown in Fig. 1,by a closed boundary B within which the mass at time t is
m = ^, rrii (I)B
The center of mass c of the system at time t, referred to inertialcoordinates xyz, is then defined by the equation
mrc = ]C mXi (2)B
We also will define the momentum of the system at time tby the equation
p = S niiTi (3)B
To express p in terms of rc, we differentiate Eq. (2)
mfc + m rc = Z mdi + ]C mXi (4)B 8
where the second summation associated with rhi representssummation for particles crossing over the boundary surfaceB (i.e., rhi = 0 for all particles within B which do not crossB). Thus, in view of Eq. (3), we obtain for the momentumthe equation
p = mfc + mrc — 2 w&»r» (5)sIt is evident here that Eq. (5) contains two unclassical
terms. To interpret them, consider a rocket with massejected at only one point e on the boundary B, as shown inFig. 2. Then
y"^ rhi = rh and ^P rhiTi = rhte
Received December 8, 1965; revision received January 10,1966.
* Professor of Engineering; also Consultant TRW Systems.Associate Fellow Member AIAA.
so that Eq. (5) becomesp = mrc + m(rc - r.) (6)
We note here that m, for the case of the rocket, is negative,and that the second term is zero when the dimension $ce =(r« — rc) goes to zero. Thus, it is evident that the classicaltreatment, where the momentum is given only by the firstterm, is restricted to the system considered as a particle.
To obtain the force equation, one must determine thechange in the momentum of the same mass considered at twodifferent times. The momentum of m at time t + dt is thesum of the momentum of the system within B at time t + dtand the momentum of the ejected masses during the time dt,
(7)p(t + dt) - ^ nndt
where V;* is the absolute velocity of the ejected mass — rhidt.Thus, from the principle "impulse equals change in momen-tum77, we obtain the force equation in terms of the momen-tum p
F = p — (8)
If Eq. (3) is differentiated and substituted into Eq. (8),we obtain
F = E rniii + Z mt(it - vt-*)B S
(9)
B S
where U; is the relative velocity of the ejected mass (i.e. v* =
To obtain the force equation in terms of the center of mass,we differentiate Eq. (5) and substitute into Eq. (8)
F == mrc + [ mrc — E wi»r,- )V s )
(10)
It is evident that several nonclassical terms result in thisdevelopment.
Again, it is instructive to consider the rocket ejecting massat one point e. The previous equation then becomes
F = mfc + m(rc - r.) + m(2fc - fe - v*) (11)
Since absolute velocities are not convenient quantities, wereplace them in terms of quantities relative to a fixed point0 on the body as follows :
fc = To + G> X 0oc + 00crel)
re = f0 + G> X 00e > (12)v* = r0 + co X j>oe + u )
where u is the velocity of the ejected mass relative to a pointe (the nozzle of the rocket engine). Then, Eq. (11) becomes
Fig. 1 Variable mass system defined by closed boundary B.
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APRIL 1966 TECHNICAL COMMENTS 767
Again this equation reduces to the classical resultF = mrc — mu (14)
when the system is considered to be a particle. Even for thefinite body, u for the rocket is very large compared to theother terms in the square bracket. Also, most rockets burnfuel at nearly a constant rate which makes the second termnegligible. Thus, the classical equation for F appears tobe quite justified for rockets.III. Moment of Momentum and the Moment Equation
The moment of momentum of the system about the fixedorigin in inertial coordinates at time t is
H = ZB
X (15)
The moment of the external forces about the inertially fixedorigin is simply found by noting the moment of momentumof the same mass at time t + dt, from which its time rate ofchange can be determined. If we let H(t + dt) be the mo-ment of momentum of the mass within the boundary B attime t + dt, we must add to this the moment of momentumof the mass which has crossed the boundary during the timedt. Thus, the new moment of momentum of the originalmass at time t + dt becomes
+ dt) - Z r» X rhi dt (16)
Subtracting Eq. (15) from Eq. (16) and dividing by dt, themoment equation becomes
M = H - hi v,* (17)
Differentiating Eq. (15) and recognizing that the masschanges only by crossing the boundary B, Eq. (17) can bewritten as
M = Z r»- X ZsX (18)
where U; is the velocity of the ejected mass relative to thepoint on the boundary from which it was ejected. Thus,the equation for the moment is basically quite simple; onejust determines the acceleration of each mass particle withinB and calculates its moment, which is summed over the body,and adds to this the second term, which is the moment of thethrust force required in changing the momentum of the massparticle passing across the boundary from — m» dt f; to-rhidtVi*.
Of course the moment about an inertially fixed point fora body moving in space is not a convenient quantity, and amore desirable quantity is a moment about some fixed pointin the body itself. Even the moment about the center ofmass of the body is not too convenient a quantity because forthe variable mass system the center of mass is not a fixedpoint in the body. One sees immediately from Eq. (18)that, to determine the moment about a fixed point 0 in thebody, one needs only to change the moment arm from r< top0i, so that we can easily write the equation for the momentabout the fixed point on the body as
Mo = Z ii — Z 0o; X (19)
This is essentially the equation given by Halfman andThomson, only that r; is expressed in rotating coordinates[i.e., r = r0 + w X 0 + <*> X (<o X p) + 2<a X 0rei + prei].
Now, the complicated equation given by Rankin comesabout because this simple equation has been expressed interms of the momentum p and the moment of momentumof the system about the center of mass. Although the centerof mass is not a desirable reference point for the moment equa-tion, it will be of interest to examine the moment equationabout this point.
We first recognize that the momentum of any mass par-ticle can be transferred to the center of mass provided we
Fig. 2 Rocket with mass exiting at one point e.
introduce its moment of momentum also at c. Thus, bysumming over the body, we obtain the resultant Hc and pacting at c. The relationship between H and Hc is then
whereH = Hc + rc X p
Hc = Z Pa X raf»5
(20)
(21)
Differentiating H and substituting into Eq. (17), we obtainthe relationship
M = He + fe X p + rc X p - Z r» X w» V,*s
= Ho + fc X p + re X f p - Z mm*} -\ S /
T^ f\ . V w xr * ^99^/ ^ Qci A niivi \££)sSince M = Mc + rc X F and
F = [ p — Z wv
the previous equation leads to the moment equation aboutthe center of mass which is
Me = He + fc X p - Z &a X mi v<* (23)sIf Eq. (6) is substituted for p, we obtain another form
Mc = Ho + fc X m(rc - r«) - Z Qot X mt v** (24)s
which is equal to Eq. (2.3.9), given by Rankin, when onenotes that Hc in this equation is the total momentum in-cluding the part contributed by the relative velocity of the par-ticles within the system, whereas Rankin defines Hc excludingthis component.
IV. ConclusionsThe classical equation for the momentum, given by Eq.
(3), can be expressed in terms of the kinematics of the centerof mass which results in Eq. (5) or (6) .
The force equation (8), in the classical sense, is given byEq. (9). When expressed in terms of the kinematics of thecenter of mass, Eq. (8) becomes Eq. (10, 11, or 13). Al-though these forms appear different, they are all equal.
The moment equation expressed classically by Eq. (17,18, or 19) take on a more complex form when expressed interms of the momentum and the moment of momentum.There is no advantage of going beyond Eq. (19), which canbe expressed in body coordinates asMo = — ?o X pocm + (Euler's Equation) + 2 Z Qoi X
B
mi(o> X potrei) +B
X ZS
(25)
which is Eq. (7.11-11) of Ref . (5).References
Rankin, R. A., "The mathematical theory of the motion of
2 Leitman, G., "On the equation of rocket motion," J. Brit.Interplanet. Soc. 16,141-147 (1957).
3 Thorpe, J. F., "On the momentum theorem for a continuoussystem of variable mass," Am. J. Phys. 30, 637-640 (1962).
4 Halfman, R. L., Dynamics (Addison Wesley Publishing Com-pany, Inc., Reading, Mass., 1962), Vol. I, pp. 153-167.
5 Thomson, W. T., Introduction to Space Dynamics (JohnWiley & Sons, Inc., New York, 1963), pp. 231-235.
Addendum: "Coupled Panel/CavityVibrations"
D. J. KETTER*The Boeing Company, Seattle, Wash.
IN a recent article/ the author presented an assumed modesolution for the effect of an underlying cavity on the
vibration characteristics of a thin, uniform, unstressed iso-tropic panel. For the sake of brevity, the structural stiffnessmatrix Kmn
2 in the final characteristic equation was writtenas a diagonal matrix of uncoupled nondimensional fre-quencies squared. However, for some of the boundaryconditions considered in Ref. 1, use of a diagonal stiffnessmatrix is approximate and possibly inadequate. For ex-ample, if a panel has two parallel edges clamped and theother two edges simply supported or if it has all four edgesclamped, the characteristic beam functions used for the as-sumed panel modes actually lead to an invacuo stiffnessmatrix Kmn,rs containing static coupling terms.
Practically speaking, this static coupling has a small effecton the natural (coupled) frequency spectrum. However,for those problems where natural panel mode shapes aredesired, the static coupling terms may be quite influentialand should be included in the characteristic equation. Equa-tion (24) in Ref. 1 is then written more completely as follows:
a. « rE E \
r =l s = i L
KflK2 XI GpqPprPpmPqsPt tnWr?\ = 0 (24)
where drs is the Kroneker-delta and m, n = 1, . . . , ». Theeigenvalues K2 may be obtained by established iterativeprocedures, whereas elements of the stiffness matrix are com-puted from the formula
Kmn,rs = Km + 2(L/w)*HmrHns + (L/wYKns (24a)This expression is readily obtained via a Galerkin solution
of Eq. (22),* for free vibrations. The terms in Eq. (24a) are:
K K"4Kmr = JQ
H» = ]j"2
K., = |G"4"" {0
TT __
8Gm2<
0(Gn?<.
rr
r
r =r -7
V(Gma(G/-
r +*m
2 - 2
= m^ m
T± m
sii
= m^ mm - GrOlr)
Gm*)m oddGmoim)
simply supported edgesat £ = 0 and £ = 1
r + m even r
(24b)
m
r ^- mr = m
clamped edges at £ = 0 and S = 1 (24c)
Received December 16,1965.* Research Engineer, Structures Technology Department,
Aerospace Group. Member AIAA.
Expressions similar to Eqs. (24b) and (24c) are obtainedfor terms having the subscripts ns by replacing m with nand r with s.
It also is to be noted that Eqs. (25) and (26) in Ref. 1 con-tain printing errors and should read as follows :
Gpq =
- = f' *«2(£)d£- f'K. »/ U »/ U
(25)
(26)
Here, K. is independent of m and n and is constant for givenboundary conditions. In the case of a panel simply sup-ported on all four edges, K. = J; for a panel simply supportedon two parallel edges and clamped on two parallel edges,K = \\ and when the panel is clamped on all four edgesK = 1.
Of additional interest to the subject matter is a recentBritish publication by Pretlove.2 He has independentlyobtained Eq. (24) for a panel simply supported on all fouredges and has included the results of calculations for twonumerical examples. The convergence trends, which arepresented for the fundamental coupled panel mode shape,suggest that a more detailed investigation of this acousticcoupling phenomenon may be warranted in connection withthe panel flutter problem.
References1 Ketter, D. J., "Coupled panel/cavity vibrations," AIAA J.
3,2164r-2166(1965).2 Pretlove, A. J., "Free vibrations of a rectangular panel
backed by a closed rectangular cavity," J. Sound Vibration 2,197-209(1965).
Errata: 6<;Aerodynamic BlastSimulation in Hypersonic Tunnels"
H. MlRELS* AND J. F. MuLLENfAerospace Corporation, El Segundo, Calif.
[AIAA J. 3, 2103-2108 (1965)]
^HE following equations contained typographical errorsand are given here in their correct form:
= [(7 (17)
(18)
Ms = R/ai ~ $(™-i)
Between Eqs. (15) and (16) in text
k = [2T/(7 - l)]1/2[2/(7
Between Eqs. (22) and (23) in textMs ~ ^(7-D-13/3
(21)
Received January 16,1966.* Head, Aerodynamics and Heat Transfer Department,
Laboratories Division. Associate Fellow Member AIAA.t Member of Technical Staff, Aerodynamics and Heat Transfer
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