H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 50
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1; n = l , 2...(7.20)The subscript n indicates the number of thepropellant mode under consideration, while Xindicates the container number. cnr is thedamping factor of the propellant and U,A isthe undamped circular natural frequency.Y,(X,,~~and i7i(xnA) &re Z i e p k c e ~ e f i t nndslope, respectively, of the vth lateral bendingmode a t the location of the nth sloshing massin the Xth container.A=ln = l A=In(7.18)and the equation for the conservation of theangular moment has been observed asLateral Bending Equation of MotionAs vehicles increase in size, the lateral fundamental bending frequency approaches moreand more closely to the control frequency and230THE DYNAMIC BEHAVIOR OF LIQUIDSthe lo\{-er natural frequencies of the propellant.This indicates that in many cases the bendingvibrations of the vehicle cannot be neglectedin a dynamic analysis of the vehicle that alsoincludes propellant sloshing and the controlsystem.
(See ch. 9.)The equation of motion of the vth bendingmode is obtained from equation (7.1) togetherwith equations (7.2) through (7.7) and bynoting the results of equations (7.12), (7.19),and the orthogonality relations between normalmodes, as expressed bySj " m : ~ ~ ( Y.(x)~zx+x)I:Y:(x)Y:(x)~IxI1+C~ O A Y V (~'IS(TOA)X O A +C) A - 1 IOAY;(~OA)Y: (XOA)A-1Control System Equation of MotionAct,ually, t,he control equation cannot beexpressed as a linear equation; however,translational and rotational motions of thespace vehicle usually occur a t sufficiently smallfrequencies so that the control elements canbe considered as essentially linear.
Nonlinearities usually occur a t higher frequenciesin the form of saturation of amplifiers, limitedoutput of velocities, and so forth. The control equation is written, therefore, in the formwhere the operators j,andfi are functions thatdepend on the character of the system. Ai isthe indicated acceleration, as measured b y anto the longitudinal axisof the vehicle. I n linear form one can expressHere, the p , are the so-called phase-lag coefficients, and d i is the indicated angulardeviation from the trajectory as indicated bythe gyroscope--I197 ?nnitiniYv(zni)(7.22)A ~ B Vn= l A-iHere, w , represents the natural circular frequency of the vtll lateral bending mode and y,is the corres~'nding structural damping.
Thegeneralized mass of the vtll lateral bendingmode, MB", is give11 by the expression+C mOrYZ (rod+ZI2X=1A-IIokZ712(X ~ A )I4-2C ? ~ ~ A I ' ; ( X ~(7.23)A)n-lA-1values I7v(~oh)and 1': (x,~)represent nothiIlgbut the displucerilent and slope of tile ,,tll1atel.al bending mode tit the location of ther~onsloshinpmass in the h t h propellant tank.where 1 ( ) is the derivative of thevth lateral bending- mode at the locationof the gyroscope.
If the fundamental lateralbending frequency is \\-ell above the controland propellant sloshing frequencies, of u-hichthe corresponding sloshing masses create prothe flexibility ofllounced dynamicmace vehicle can be neglectedand the controlequation can be written in the simplified form( v = O ; j = l , 2 , K = O , and p o = l )~=a~++a~i+gtAt(7.26)Here. time derivatives in B, which produceincreasing phase lags with increasing frequency,hare been neglected.
I n order to include theeffect of the flexible structure, the phase-lagcoefficients are of importance and have to beconsidered. A still simplified control equationfor these cases ~vouldbe of the formp 2 8 + P l P + ~ = a o ~ t + a l ~ , + ~ z(7-27)Al.VEHICLESTABILITY AND CONTROLwhich only approximately describes the complexity of an actual control system. By properchoice of the phase-lag coefficients, pl and p,,and by proper choices of the attitude value, a,,,and the rate value, a,, the control system can beapproximated.
A simple system is preferred forthis type of analysis in order to keep the analysisand the interpretation lucid, and still rendergood results. If no accelerometer is employedfor control purposes, the gain value, g,, vanishes.This term, g2, is a measure of the strength withwhich the control accelerometer influences thecontrol of the space vehicle.The attitude gyroscope is a free gyroscopethat measures the position of the vehicle, themain function of the gyroscope being to maintain a space-fixed angular reference. Since werestrict our motion to one plane only (pitching),we consider only the appropriate gyroscope ofthe stable platform. The essential property ofa gyroscope is used, which represents an angularvelocity about an orthogonal axis (output axis),if a torque is applied about an input axis.
Ifthe platform on which the gyroscope is mountedis perfectly balanced and the bearings arefrictionless, no torque will be experienced bythe platform, thus maintaining its orientationregardless of the motion of the space vehicle(ref. 7.9). The pickoffs of the ,gyroscope mustbe very light in weight and should not introduceany torques; this can be achieved by variousforms of capacitive, inductive, or optical pickoffs. Because of unbalances and friction, whichcannot be eliminated completely, a disturbancetorque is exerted on the stable platform; a servosystem counteracts this disturbing torque andproduces essentially a torque-free system.
Forthe purposes of this chapter, we consider aproperly designed gyroscope which exhibits atransfer function of unity for a very largebandpass. This means that the indicatedattitude angle mill be the same as the inputangle and is described by equation (7.25).the accelerometer is required to provide areference for translational motion.
An accelerometer and a gyroscope are therefore capable ofsupplying information about the motion of arigid space vehicle in the pitch plane (x, 7~plane). I n addition to the ,gyroscope, an accelerometer provided for control purposes can, byproper choice of the gain value and its vibrational characteristics, also diminish loads andreduce engine angle requirements of the swivelengines. Mounting an accelerometer to thestructure of the vehicle such that its sensitivedirection is perpendicular to the longitudinalaxis of the vehicle, this instrument is thencapable of sensing the accelerations due totranslation, pitching, and bending motions.The equation of motion of an accelerometer(shown schematically in fig. 7.3) can be obtainedfrom Lagrange's equation.
The kinetic energyis given bywhere ya is the relative deflection of the uccelerometer mass, ma, with respect to the spacevehicle, and (-xu) is the location of the accelerometer measured from the mass center of thevehicle. The dissipation function corresponding to linear damping can be represented bywhere ca is the damping coefficient. The potential energy isAccelerometer Equation of MotionA vehicle moving in the pitch plane possessestwo rigid body degrees of freedom, one translational degree of freedom in the y-direction andone rotational degree of freedom denoted by 6.While the gyroscope detects rotational motion,231FIGURE7.3.-Schematic of accelerometer..232THE DYNAMIC BEHAVIOR OF LIQUIDSThe first term is the energy stored in the spring,while the second term is the potential energydecrease resulting from the change of locationof the mass, ma, in t.he equivalent accelerationfield, g.
With these results, the equation ofmotion of the accelerometer is obtained byapplying Lagrange's equat,ion to yieldWith k,/m,= wz as the square of the naturalcircular frequency of the accelerometer, andco/rna=2~,,~,,,one obtainswhere la is the damping factor of theacceleronleter.B y scaling the output in such a mannerthat it is equal to the input, for frequenciessmall in comparisoli with the natural frequencyof the acceleron~eter,one has to transform withwhere A irepresents the indicated acceleration.The response characteristic of the accelerometeris therefore determined by the differentialequationRate Gyro Equation of MotionI n some space vehicles, the rate of theattitude angle, o,, in the control equation isnot obtained by n differentiating network operating on the angle, $,, but rather by the outputsignal of a rate gyroscope.The function of nrSntegyroscope is to give nn output signal whichis proportional to the angular ~ e l o c i t ~aboutyaperpendicular axis.
Figure 7.4 represents the/-+a/InputFIGURE7.4.-Schematic of rate gyroscope mounting.mounting of such a rate gyroscope. I n contrast to the free gyroscope, the outer frame ofthe rate ,Toscope is rigidly fastened to thestructure of the vehicle. The inner gimbalsupport is restrained by a spring and damperwhich permit a limited rotation about theouter frame.
The z-axis about which the rehicle turns is called the input axis, and theaxis of rotation of the inner gimbal frame iscalled the output axis. Any rotation of the~ as a forcedvehicle frame a t a rate ;PiR I V actprecession of the spin axis and will induce agyroscopic reaction torque. This torque isgiven bywhere IR is the moment of inertia of the rotor~~theund w, its angular velocity. t j representsangular velocit,y of the structure of the vehiclea t the location of the rate gyroscope. Thetorque, AdR, is balanced by the inertial, damping, and restoring moments of the gyroscopeabout its output axis. Damping has beenintroduced t o prevent the undesirable conditionof excessive o\~ershoot and oscillations aboutt.he steady-state angle.
The behavior of therate gyroscope can therefore be establishedfrom t,he differential equation-1,9+c&+K,e=~~w&~where I, represents the rllonlent of inertia of233VEHICLE STABILITY AND CONTROLthe rotor and the inner gimbal frame withrespect to the output axis. cc is the coefficientof viscous damping and KG is the restoringmoment per unit angle. By definingthe angle O is scaled in such a fashion that theindicated value, O,, for frequencies small incomparison with the natural frequency, o,,of the rate gyroscope (w:= KG/I,), representsthe angular velocity of the vehicle at the location of the gyroscope. The response characteristic of a rate gyroscope of natural circularfrequency, wc, a damping factor, lo, and anangular velocity, w,, of the rotor is thereforedescribed by the differential equationwhere Oi represents the indicated angularvelocity of the airframe of the vehicle at thelocation of the rate gyroscope.
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