H.N. Abramson - The dynamic behavior of liquids in moving containers. With applications to space vehicle technology (798543), страница 53
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With increasing attitude gain value, a,,this boundary shifts farther aIrag from thec,-asis. Increasing the accelerometer gain, thatis, providing more influence of the accelerometeron the control of the vehicle, decreases thedistance of the static stability boundary frornthe ca-axis. The dynamic stability is expressedbyVEHICLE STABILITY AND CONTROLwhere the equal sign represents the stabilityboundary (fig. 7.12).
I n the (t,-€:)-plane thestability boundary represents a straight line ofslope a,€(27 through the origin.Stability istherefore provided below the parallel line to the€,-axis (static stability boundary) and to theleft of the straight line (dynamic stabilityboundary). With increasing attitude controlgain, ao,the dynamic stability boundary rotatestoward the E:-axis, providing a hrge part ofthe left half plane for stability; in addition, thestatic stability boundary moves upward. Thisexpresses that a limited range for the locationalong the vehicle can be provided for themounting of a control accelerometer.
If e, isnegative, that is. the accelerometer is mounteda t a location where the bending mode exhibitsnegative values (vehicle normalized to unity a tthe gimbal station xE), then positive and negative values for the slope of the bending modeare available for the location of the accelerometer. With increasing frequency ratio ofbending to control frequency, the same trendoccurs.
For an increase of the availability ofthrust for control purposes, the static stabilityboundary remains a t its position, while thedynamic stability boundary rotates againtoward the vertical. For increasing accelerometer gain, A, the static stability boundarymoves toward the €,-axis while the dynamicstability boundary remains fixed, which indi-staticboundary1FIGURE7.12.-Stability/boundaries for flexible vehiclewith accelerometer control.243cates that a more restricted range of locationsfor the mounting of a control accelerometer isavailable.Propellant Sloshing in O n e Container of a VehicleWith Simple Control SystemI n order to obtain some of the basic effectsof the influence of propellant sloshing uponvehicle stability without expending any greatnumerical effort, the equations of motion of thevehicle will be treated with the propellant asfree to oscillate in one container only. Thevehicle will be considered as rigid (rlY=O)andonly the most pronounced modal sloshing mass(see ch.
6) will be retained in the analysis;that is, A = 1 and n= 1. Introducing ~ , = 0andn=A=l into equations (7.16), (7.17), (7.20),(7.26), and (7.28), one obtains the equations ofmotion for the rigid vehicle with propellantsloshing in one container and additionalaccelerometer control (refs. 7.17 and 7.18).The first equation represents translational motions and the second one represents pitchingmotion of the vehicle.
Equation (7.44) is thesloshing equation describing the motion of thefirst modal sloshing mass. The fourth equationrepresents the control, which indicates that aposition gyroscope and additional accelerometerare employed for the control of the vehicle.Finally, the last equation represents the dynamics of the couiio! acce!ernmeter.
If anideal accelerometer were employed, that is,w,>>w,,the first and second terms of equation(7.46) would be neglected. A control systememploying no accelerometer control \\-ould leadto the omission of the last equation and avanishing gain value g, in equation (7.45). Thestability of the solutions of such a system isagain obtained by assuming a solution of the244T H J ~ DYNAMIC BEHAVIOR OF LIQUIDSform ebc', where s is the complex frequency,s = r+ iw. This assump tion for the solutiontransforms the differential equations into ahomogeneous algebraic system, which exhibitsnontrivial solutions only if the determinants of(Y(@)Here, X=gg, and pll=mll/m is the ratio of thesloshing mass in the container to the total massof the vehicle.
The coefficient determinantyields then t,he characteristic polynomial in swhere the coefficients, B,, are represented aspolynomials of f , and 7,coefficients vanish. With only the part F2=cFavailable for control purposes, Fl= (1 -e)F andt,he longitudinal acceleration of the vehicle isg=F/m. The coefficient determinant is therefore(A,)(~11)on the mass ratio, pll=p, and the vibrationalcharacteristics of the accelerometer. With thenotations f , as the damping factor of the controlsystem, wc as the circular frequency of the control system, w: =wZo/(l - X E - A X E X ~ ~ /and~ ~ ) waas the circular frequency of the control system~vithoutaccelerometer (w:, = g x ~ e a ~ / k ~and) , {E=xE/k, {,=x,/k as the distance with respect tothe radius of gyration, and v,=w,/w,,va=wa/wcas the ratios of the eigenfr'equencies of sloshingand accelerometer (wll= w,) as well as the valueA = 1-tX(l + [ B E .
) , and withthe coefficients k, are given by the expressions:The location of the modal mass, E l l = x l l l k ~ f s(with respect to the mass center), nnd thedamping factor, ill E-y,, are extracted, sincethey are the ones by which the stability of thevehicle can be influenced most readily; therepresentation of the stability boundaries \\-illbe in these coordinates. This therefore yieldsnot only the magnitude of the damping required in the tanks but also its location.
Theabbreviations k,(j=O, 1, 2 . . . 18) depend onthe frequency and damping factors, and thegain values of the control system, as well as1 1k3=~{~(;+4tcta)~(a%+l)+l-p+vi-k4=2v,Ak6=2t d+a{%k+2(lvaaoEiv'.e-,)&!va245VEHICLE SFABILITY AND CONTROLys= damping factor of propellantva=wa/w,=frequency .ratio of undamped accelerometer frequency to undamped control frequencyla=damping factor of the accelerometerX =ggZ=produc t of longitudinal accelerationof the vehicle and gain value of theaccelerometerfa=za/k=ratio of the coordinate of the accelerometer location to radius of gyrationof the space vehiclef,=x,/k=ratio of the coordinate of the locationof the modal mass of the propellant toradius of gyration of the space vehicleao=gain value of the attitude control systemk7=A { ~ , + 4 l ~ l ~va" +va% )ks=-{leva+2PAV~EE:"1< a - ~k9--5- cp~+~(l-')va[1+4~ctva+v~lu',2vrk10=7 [icA+lavaIvakll='"[ao.+4lJaaoua~--lI -PA€EAV~S( h tG)vr~IFP(XE- 1); k 1 2(1-'3 =) 7 [lava+ {CAI; k14=?2 r l a i k17---1- p;k 18-- - Lk 1 52pAcC;= 7 k1s---tavav','a2vaThe various parameters are as follo~vs:~=ms/m=ratio of modal mass of liquid tototal mass of space vehicles(,=control damping factorv,= w,/wc=frequency ratio of undamped propellant frequency to undamped controlfrequencyThe stability boundaries are characterizedby the roots, s, one of which at least 1vil1 havea zero real part, while the others are stableroots.
The theorem of Hurwitz for a stabilitypolynomialof thenthdegree(ref.7.13)Bn=O,H,-l=Owhere Hn-1represents the Hurmitz determinantof the formB,B3B6....BoB2B4....0B1B3. . . . (n-1)0BoB,....lines and columns......................Representing the stability boundaries in the(f,, y,)-plane, the Hurwitz determinant H5=0results in a polynomialwhere the functions C,(f,) are polynomials inf*. The stability boundary for the undampedliquid is Co([,)=O, and represents the intersection points with the [,-axis. For all points(f,, y,) above the stability boundary, oneobtains stability.
Because of B,=O, thestability is interrupted a t the left and a t theright so that only within these boundaries isstability guaranteed. The stability boundariesof Bn=O to the right and left are given in theform of straight lines perpendicular to the[,-axis asFor most vehicles, however, these boundariesplay no practical role. Substitution of f,=y,=O into the Hunvitz determinants determines whether the origin is in the stable orunstable region.
A necessary and s&cientcondition for stability is (ref. 7.19)246THE DYNAMIC BEHAVIOR OF LIQUIDS(1) The coefficients:B,, Bn-], Bn-3- - - - - - - - - - - - - - - - >OB,, Bo>O for even nBo>O for odd n(2) The Hurrvitz determinants:Hn-l, Hn-3- - - - - - - - - - - - - - - - - - - - >oH3>0 if n is evenH2>0 if n is oddand give a first estimate for the critical area.The value l / a o e has been considered to be ofsmall magnitude. This assumption is satisfiedin some cases if the control frequency is farenough away from the first natural frequencyof the liquid. Therefore, the result expressesthat the stability boundary for small valuesof l/aoe intersects the (,-axis in the vicinity ofthe center of mass (origin) and the instantaneRigid Space Vehicle Without Accelerometer Controlous center of rotation t d = x d / k .One can see that the second point of interUsing a simple control system without ansection,&, becomes sensitive to changes ofaccelerometer (X=O) results in a stability polyl/a,e,whichindicates that for decreasing gainThecoefficientsB,nomial of fourt'h degree.values, ao, the intersection point shifts towardIF-ould be obtained from equations (7.48) andthe tail of the vehicle.
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