H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 52
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We mill considerhere the problem of rigid and flexible vehicles,where the propellant is treated as rigid mass,and determine some of the basic requirementsfor the location of the sensors and their vibrational characteristics. Some criteria can bedeveloped which yield enhanced stability of thevehicle without resorting to electrical filternet~vorks,as is performed in a complete rootlocus analysis of the dynamic system. Thesensors treated here comprise accelerometersand rate gyroscopes. With these assumptions,the equations of motion, considering only onebending mode, reduce to the following form:Translational motion: Equation (7.16)237Accelerometer: Equation (7.28)Rate gyro: Equation (7.29)The control equation is used in its simplifiedformFor a given codiguration, the control requirements are strongly dependent on the mutualsettings of the control system.I n the following discussion, numerical resultshave been presented for a Saturn I-type vehiclefor which half of the thrust is available forcontrol purposes (refs.
7.9 and 7.15).Criteria for Position Gyrosro~emy= F#+ F4-FTY'(xE)(7.36)where Y is the first bending mode.Pitching motion: Equa t,ion (7.17)I++F#EB=Ff x ~ ~ ~ Y ' ( x ~ ~ ) - T Y} ( z(7.37)E)Lateral bending uibration: Equation (7.22)The dynamic characteristics of the controlsensors are represented by the followingequations:must vanish. The stability polynomial is ofsixth degree. For p2 =0.0084 and pl~ 0 . 0 5 it,found that for a gyroscope location closeto the tintinode, the degree of structuralThis investigation presents the stabilityboundaries in the (Y;, gB) plane, indicating a twhat location a gyroscope should be mountedto maintain stability for a certain structuraldamping g, and for various gain values and lagcoefficients.The control equation, equation (7.27), wasemployed with an accelerometer gain value ofg,=O, that is, no additional accelerometercontrol.
Neglecting the effect of translationalmotion and combining the equations for pitching motion, equation (7.37), and bendingoscillations. equation (7.38), with the controlequation, equation (7.41)) leads to the conditionthat for this system of three homogeneousalgebraic equations the coefficient determinantdamping usually provided is sufficient tomaintain stability. For low bending frequencies the location behind the antinode seemsreasonable, but for larger bending frequencies238THE DYNAMIC BEHAVIOR OF LIQUIDSa location in front, of the antinode also yieldsstability; this is due to the lag terms in thecontrol equation.If only the bending equation, equation (7.38),is combined with the simple control equationb=ao+iwhereby setting the coefficient determinant equal tozero. Thus++<=--?Y'(X~)the stability polynomial is of second order andexhibits stability if the coefficients are positive,and is of the formwhich yields a fourth-order stability polynomial:with the coefficientsSince all of the values in these coefficients arepositive by their very nature except for Y'(xc),which can be either positive or negative, thesystem definitely exhibits stability for Y'(xc)>0, that, is, for a positire bending mode slopea t the location of the gyroscope.
This meansthat the gyroscope should be located behindthe antinode in the aft section of the vehicle.The stnbility bour~daryis at,that is, slightly forward of the antinode.Criteria for Rate GyroscopeWe shall consider the stability criteria for arate gyroscope by dividing our investigationinto two parts: one in which the vehicle isconsidered as a rigid body, and the other inwhich the flexibility of the vehicle is describedby the fundamental bending mode.withw;=-andgXae%k2-qtsa1c k,2cewcwhereis the longitudinal acceleration of t,he vehicleand k 2 = I / m is the square of the radius ofgyration.
This yieldsRigid Body StabilityThe stability boundary is obtained by settingBlB2B3=B&+ B,B: which leads t oTreating the equation for pitching motion,equation (7.37), together with the rate gyroscope equation, equation (7.40), together withthe simple control equation in its ideal formrepresenting a hyperbola in theand assuming again time dependency of theform e"~", the stnbility polynomial is obtainedplane. Only the branch in the first quadrant'of t,he coordinate system is of physical significance and is shown in figure 7.7. The crosshatched aren represents the stable area. From-])k(':[VEHICLE STABILITY AND CONTROLthis we can conclude that rigid body stabilitycan be accomplished by choosing the naturalfrequency of the rate gyroscope larger than thecontrol frequency and adjust its damping factorto an appropriate value.
For a ratio of thedamping factors of unity, a frequency ratiorange of OSL< (4'-1)wonldmaintninstnwG720I. 53:u1.0u lUnstablebility of the rigid vehicle.0.5Bending M o d e StabilityTaking, in this case, the bending equation,equation (7.38), and rate gyroscope equation,equation (7.40), together with the control equation, equation (7.41),while omitting all termsto rigidmotion, yields thecoefficient determinantThe stability polynomial, which is of sixth degree, is reduced to a quartic by setting thephase through making the coefficients pe=pl=O, which yields finally the stability boundarywhere??IandThe value (x,,,( represents the distance between the center of instantaneous rotation andthe mass center of the vehicle.I n the (t;, 6;)-plane, as shown in figureoa51.01.52owclwrFIGURE7.?.--Stabilityboundary for rigid vehicle withrate gyroscope.7.8, this represents a family of straight lines.The distance of the intersection point with theaxis from the origin is decreased by increasingthe frequency ratio ( w ~ / w ~that) , is, by decreasing the rate gyroscope frequency towardthe bending frequency.
The slope of the stability boundary line decreases with decreasingdamping ratioand increasing frequencyratio (w,/wo). Since the frequency ratio is 'limited to a small value because of the rigidbody stability, the slope is most effectivelychanged by the damping ratio. This indicatesthat a supercritical damping of the rate gyro;cope is deskzb!~. A f::rther e ~ n c l u s i ~ tr ?h ~ tcan be drawn from figure 7.8 with regard to theproper location of the rate gyroscope is thatpositive slope Y1(xR)minimizes stability.Therefore, it can be concluded that thenatural frequency for the rate gyroscope shouldbe quite large compared with the controlfrequency and the highest to-be-controlledbending frequency under consideration.
Thei240THE DYNAMIC BEHAVIOR OF LIQUIDSbe located behind the last antinode of theto-be-controlled bending modes.Criteria for AccelerometerI n many vehicles, an accelerometer is usedas an additional control element. By properchoice of its gain value in the control equationand its vibrational characteristics, such anelement can help to diminish loads on the spacevehicle and to reduce deflection requirementsof the swivel engines. In this section, therefore, we shall investigate the effect of thevibrational characteristics and the gain valueg, of an accelerometer and shall try to determinesimple criteria for the preliminary determination of those values (ref. 7.16).Rigid Body StabilityIFIGURE7.8.-Stabilitystability boundaryboundary for flexible vehicle withrate gyroscope.damping should preferably be supercritical,or large in comparison with the control damping.
Furthermore, the rate gyroscope shouldSince the translational motion of the vehicleis sensed by an accelerometer, this has to beconsidered in a simplified stability analysis.Therefore, for rigid body stability, we treatthe translational motion, equation (7.36),together with the pitching motion, equation(7.37), the accelerometer equation, equation(7.39), and the control equation, equation(7.41), together with the usual assumptione'*ctfor the solution.
The coeffic,ient determinant that has to vanish yields, with q=0,4,=4 and the phase coefficients p1=p2=0,t,he expression-3'20Il o-( a o + s u ~ , )The stability polynon~ial is of fourt.11 degreewith t'he coefficients-921IThe stability condition, \vit,h B , B Z B ~ ~ B O B : +B4Bq and ( x e ,I =k2/xE,~is then given by+(?)[l-tgg2(l+a)l20This is a polynomial of fourth degree inw,/w..VEHICLE241STABILITY AND CONTROLA case of special interest is the ideal accelerometer for which o,>>w,.With a =z./1~,,~1,the result of the above equation withegg,=X, yields the simple expression (IxelRIisthe distance of the center of instantaneousrotation from the mass center)which is shown in figure 7.9.
From this it canbe concluded that an accelerometer should belocated forward of the center of instantaneousrotation to allow all possible gain valuesX(>O).With decreasing gain values X, thelocation of the accelerometer becomes lessimportant for rigid body stability. Figures 7.10andsho'v theof fhequency, a , , of the accelerometer and of thedamping factor, fa. The influence of thechange of these parameters is not very pronounced and does not exhibit a large differencefrom the results for an ideal accelerometer. Ifthe natural frequency of the accelerometer.however, is small (which should be avoided inmany respects), the stability boundary curvesspread a little more. One can conclude fromthe above results that a location forward of themass center will insure stability of the rigidbody. With decreasing gain values, the location of the accelerometer becomes less impor--3.2-2 4-1.6I-0.8u IIoa8II ]FIGURE7.10.-Stability boundaries for rigid vehicle withaccelerometer control of various natural frequency.FIGURE7.11.-Stability boundaries for rigid vehicle withaccelerometer control of various damping factor, fa.tant, indicating that a location aft of the masscenter also becomes permissible.It shouid be mentioned that the expression-FIGURE1.9.-stabilityboundary for rigid vehicle withideal accelerometer control.represents the undamped natural frequency ofthe rigidbody(referred to as the controlfre...quency) without accelerometer control.
Theactual control frequency with accelerometercontrol depends upon the gain value and loca-242THE DYNAMIC BEHAVIOR OF LIQUIDStion of the accelerometer and is giren by theespressionegx~aow2 Ca-k2(l-~-ha)control accelerometer v-odd therefore not resultin an abrupt change of the control frequencyand would not drastically influence the stabilitybehavior of the vehicle.From this it can be seen that a location, xu, ofthe accelerometer near the center of instantaneous rotation, that is, a = -1, makes thecontrol frequency nearly independent of thegain value, X. For this value, ha, failure of theBending Modc StabilityConsidering only the vehicle, equation (7.38),and its interaction with the accelerometer,equation (7.39), and the control equation,equation (7.41), yields the stability determinantThe stability polynomial is of sixth degree.Assuming, for reasons of simplification as before,that the phase-lag coefficients vanish, that is,p,=p,=O, t.he stability polynomial reduces toone of fourth degree.
The coefficients aregiven byThe static s t a b i l i t , limit~is established from thecoefficient B,, which yieldsForever1 more simple criterion, let us assumethat the position gyroscope is mounted at thenntinode of the bending mode, that is, a t thelocntiorl where the slope of the bending modevanishes (EL=0). With this assumption, thecoefficients of the stability polynomial yieldwhere the equal sign expresses the boundary.I n the (E,, t'o)-plane, this represents a lineparallel to the c,-axis and ~ v i t ha distance fromthe e,-axis which increases with the increase ofthe frequency ratio osjwc, the bending frequencyover the control frequency (\vithout accelerometer).
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