H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 49
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Since the largest portion of thetotal weight of the space vehicle is in form ofliquid propellant, the problem of interactionof the sloshing propellant with the motion ofthe space vehicle remains an important consideration throughout the entire powered flight.The general problem we are concerned withhere is the motion of the center of mass, thevehicle attitude, the motion of the propellantsin the tanks, and the lateral bending of thevehicle structure under the action of a controlsystem.
For the purpose of the followinginvestigations, the rate of mass, the momentof inertia, and acceleration variations wereconsidered small enough to be negligible.'The questions to be answered therefore arehow to decrease the influence of propellantsloshing upon the stability of the vehicle byproper container geometry and location, properThe equations of motion derived in this chapterare quite simple and are only adequate to illustrategross effects of the interaction of propellant sloshing,structure, and control.choice of the control system and its gain values,and how to obtain the requirements for additional baffles to provide damping of the liquidin the containers.
To simplify the analysis,aerodynamic effects and the inertia of the swivelengines, as well as their compliances, areneglected. The control moments w i l l be produced by the swivel engines. The main energyis fed into the system by the feedback loopsbetween the structure of the space vehicleand its control system.The coordinate system has its origin a t thecenter of mass of the undisturbed vehicle.The accelerated coordinate system is substituted by an inertial system such that thespace vehicle is subjected to an equivalentfield of acceleration .(fig. 7.1). Centrifugaland Coriolis forces, which result from a rota-/bBending mcdeSpringFIGURE7.1.-Coordinate system of space vehicle.225Preceding page blankzzf226THE DYNAMIC BEHAVIOR OF LIQUIDStion, are considered negligible and the acceleration is in the direction of the t,rajectory.*7.2 SIMPLIFIED EQUATIONS OF MOTIONI n order to outline the problem a t hand, we~ r - i l lconsider the motion of a flexible body inthe plane perpendicular to the trajectory.
Wedefine a set of body fixed coordinates, x, y, z,with the origin coinciding with the center ofmass. The translatory motion, y, of thevehicle, the rotational motion, 4, about thecenter of mass, as well as the propellant motion,y,, and the bending vibrations, vn, are restrictedto the x-, y-plane. We follow the conventionalmethod of deriving the equations of motionfrom Lagrange's equation. I t is assumed thatthe motion of the space vehicle can be describedby a superposition of a finite number of preassigned bending mode shapes with a translatory and rotational motion of the vehicle. Theelastic mode shapes are introduced as normalmodes of vibration of the vehicle structure.Lagrange's EquationFor the derivation of the equations of motion,we employ Lagrange's equntion in the formwhere T is the kinetic energy of the syst,em, Dthc dissipation function, and 1' the potentialenergy. The generalized forces. Q,, correspondto those external forces I\-hich cannot be derived from n potential.
The generalized coordinates, p,, lire independent of each otherand specify the configuration of the system a tany time, their number representing the numberof degrees of freedom of the syst,em. Thesecoordinates are:?/=the lateral translation of the center ofmass, or centerline of the undeformedspace vehicle# = t h e rotational motion of the centerline ofthe l~ndefornledvehicle about the centerof nlass, relative to the x-coordinateSce refs.
7.1 slid 7.2. Other papers 011 the subjectof dynnmics of missilcs nre given in refs. 7.3 through7.6.y,=thedisplacement of the sloshing massesrelative to the container wallrl,=the elastic deflection of the normalizedbending mode shapeKinetic EnergyThe kinetic energy is composed of partsarising from motions of the empty structure ofthe space vehicle and the liquid propellant.The influence of the sivivel engines can beneglected.
The kinetic energy, T,, of the emptystructure is obtained by summation of thetranslational and rotational kinetic energy ofeach segment. The translational velocity, v,resulting from translation, rotation and bendingdisplacements and the angular velocity, w , arewhere x is the distance of the considered element from the center of mass of the spacevehicle, and 17,is the normalized bending deflection of the vth lateral bending mode.
(Thedot indicates differentiation with respect totime, while the prime stands for differentiationwith respect to x.) The kinetic energy, T,,ist,henHere, m: is the mass of the struclure per unitlength and I: is the mass moment of inertiaper unit length about the center of mass of theelemental segment. The integration is performed over the total vehicle length and thebending mode deflection curves, 17,,are normalized (to unity) at the swivel point of the engines.The kinetic energy, T,, of the liquid propellant can be obtained from an equivalentmechanical model.
(See ch. 6.) Such a modeldescribes the linearized liquid motion and mayconsist of a fixed mass, m,, and a n infinite setof oscillating masses, m,, which are attachedt o the -propellantcontainer wall by means ofsprings of stiffness k,. With this mechanicalanalogy, the kinetic energy of the propellantcall be written asVEHICLE STABILITY AND CONTROLwhere I indicates the number of propellantcontainers, m0x the fixed (nonsloshing) mass inthe hth propellant tank, m , ~the mass of thenth sloshing mode in the Xth container, x0xthe distance of the nonsloshing mass in theXth container to the center of mass of the vehicle, and xx, the distance of the nth sloshingmass in the Xth container to the center of messof the vehicle. The value I o X is the moment ofinertia of the nonsloshing mass in the Xthcontainer about its own center of mass.
Thedisplacement of the mass of the nth sloshingmode in the Xth container relative to the tank~vallis denoted by ynx.Potential EnergyThe potential energy is also composed of twomain parts; namely, that of the structure andthat of the propellant. The potential energyof the structure is again made up of two parts,one of which represents the elastic energy ofdeformation+J%dx)1=I2 ntv.u=l(7.4)and the other represents the work performedin raising the center of mass of the emptyvehicle in the gravitational field as the resultnf mtat.inn. This part becomes, in linearizedformEquation (7.4) can also be expressed as227Here, i i represents the bending moment andQ, the shear force acting on an elemental crosssection. The flexural stiffness is E I , G is theshear modulus, and A, represents the sheararea of the cross section.
The bending frequency of the vth lateral bending mode isw,, and MBvrepresents the generalized mass ofthe space vehicle. I t may be mentioned herethat the values MBu,Yv, w: are obtained froma lateral bending analysis.The potential energy of the propellant isobtained also by using the mechanical analogyand is composed of the energy stored in springsand the raising of the model masses in thegravitational field. The complete expressionis:D i d p a t i o n FunctionThe dissipation function of the struct.urearises from its structural damping, which isconsidered proportional to the amplitude of theelastic system and in phase with its velocity.This, unfortunately, would lead to complexelements which would complicate the analysisconsiderably.
To avoid this computationalcomplication, a dissipation function is employedwhich is based on an equivalent linear viscousdamping. This is justified as long as thedamping forces are small and only of importancein the neighborhood of the bending frequencies.The dissipation function of the structure istherefore given bywhere g, is the dimensionless structural dampingcoefficient of the vth lateral bending vibrationmode, and ranges in the neighborhood of0.001 I g v I0.05.The dissipation function of the liquid propellant arises from the equivalent linear viscousdamping as it was introduced by linear dashpots228THE DYNAMIC BEHAVIOR OF LIQUIDSin the mechanical model. (See ch.
6.)takes the form, with c h n = 2 { n x w n x m n xThisand propellant flow forces in the pipelines, theonly force 1s-e shall be concerned with wvill bethat of the vehicle thrust, F. If only a part,F2, of the thrust F=Fl+F2is employed forcontrol purposes, that is, if only this part ofthe total thrust can be girnbaled, while theremainder, Fl, of the thrust is stationary, thenthe derivation of the generalized forces withrespect to the thrust is as follows.The generalized force of the lateral translation resulting from the thrust vector of thevehicle is given by the thrust component in they-direction. The virtual work is (see fig.
7.2)rnhwhereis the damping factor and wnh is t.hecircular natural frequency of the nth sloshingmode in the Xth container.Generalized MassThe equations of motion are to be obtainedby performing differentiations for each generalized coordinate in the Lagrange equation,where the follo~vingrelations hare to be observed.The total mass, m, of the vehicle is given by(engine mass neglected)from which the generalized force, Q,, is obtainedasThe origin of the coordinate system is a t theequilibrium position of the center of mass ofthe total spnce vehicle, 11-hich is expressed bythe equationwhere 0 is the engine deflection as measuredfrom the centerline of the vehicle.
The generalized force of rotation is presented b y themoment of t,he thrust force about the center ofmass of the vehicle, and withFurthermore, the linear momentum for thenormal modes of vibmt,ion is conserved andespressed by,gives the expressionwhere the integrations nre t o be performednlong the total vehicle length.Generalized Thrust ForcesBefore proceeding to the derivation of theequations of motion, the generalized forces thatcannot be represented from a potential arederived. These are obtained by calculating thevirtual work done by the esternal forces throughvirtt~nl increments, 6q,.
of the generalizedcoordinates, q , . Thus,where 1.1' is the work and Q,are the generalizedforces. Since \re neglect nerodynamic forcesFICURE7.2.-Generalized forces resulting from thrust.229VEHICLE STABILITY AND CONTROLFinally, the generalized force of the thrust withrespect to the generalized coordinates arisingfrom the lateral bending of the space vehicle areobtained in a similar way by observing that thethrust force is always perpendicular to thelateral bending motion and that the virtualwork through a virtual displacement of thegeneralized coordinate, v,, isThe generalized force is thereforeThe generalized forces of the thrust withrespect to the sloshing displacement are zero;i.e., Q,,=O.Translational Equation of MotionThe equation of motion for translation, y, ofthe vehicle is obtained from equation (7.1),with equations (7.2) and (7.3), by employingthe results equations (7.10) through (7.13), thus~ Y +n-IC Z~ ~ A Y ~ ~ = F ~ + R2P - rlrYI(~s)F~ 3 1~ 3 1The value XE is the distance of the swivelpoint of the engines from the origin and k isthe radius of gyration of the vehicle.
Theright-hand side of equation (7.17) representsthe generalized force of the thrust with respecttoo lateral bending. The values Yv(xE) andY:(xE) are the lateral displacement and slopeof the vth bending mode at the location of theswivel point,, respectively.Propellant Equation of MotionThe equation of motion of the moving propellant in the containers of the vehicle is basedon the mechanical model and is obtained byapplying the Lagrange equation to the generalized coordinate ynx observing that thegeneralized force, Q,,,, vanishes.
This yieldsthe equation of the modal sloshing mass(refs. 7.7 and 7.8)(7.16)Pitching Equation of MotionThe equation of motion of the vehicle inpitching, 4, is obtained from equation (7.1)with p,=$, and employing equations (7.2)through (7.14), thus-II1 4 - n5= l AIL mnxxn~?Jn~-gn==lC AC~.AY~A+F~xEP=I=F{zEf:? . Y : ( x B ) - ~ ?~Yv(xE),=Iv-1(7.17)-1. .- \rllart:the effs~tiVcm c z e f i t nf in~rkifl.of thespace vehicle is given by--I=mP=sm:z2dx+oSx=11Iidx+C moA$rX=l, 2, . . .
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