2164-4- T ENGINEERING RESEARCH INSTITUTE DEPARTMENT OF AERONAUTICAL ENGINEERING UNIVERSITY OF MICHIGAN DIFFERENTIAL ANALYZER SOLUTION OF THE LINEARIZED F-86D EQUATIONS By John W. Peterson Assistant in Research Wright-Air Development Center Department of the Air Force Contract No. AF 33(616)-2131 E. O. No. R668-421 PO-3a Interim Technical Report July, 1954

TABLE OF CONTENTS Page LIST OF FIGURES iii TABLE OF SYMBOLS iv to vi SUMMARY 1 I. EQUATIONS OF MOTION 1 II. LINEARIZING THE EQUATIONS OF MOTION 3 A. Linearizing the Gravity Forces 4 B. Linearizing the Accelerations 5 C. Linearizing the Engine Forces 6 D. Linearizing the Air Loads 6 E. The Steady State Equations of Motion 8 F. Final Form of the Linearized Equations 8 III. DISCUSSION OF THE LINEARIZED EQUATIONS OF MOTION 9 IV. CORRELATION OF SIMULATOR RESPONSE WITH APPROXIMATE FORMULAS 14 V. CORRELATION WITH FLIGHT TEST DATA 31 VI. MECHANIZATION OF THE EQUATIONS OF MOTION 37 ii

LIST OF FIGURES Figure Page 1 Coordinate Axis System 1 2 Simulated Pitching Motion of F-86 D Showing Short Period and Phugoid Response 11 3 Simulated Lateral Motion of F-86 D Showing Short Period Response and Decaying Spiral 12 4 Pitching Motion Response with Simplified Equations 15 5 Lateral Motion Response with Simplified Equations 16 6 Pitching Motion Response with Simplified Equations 17 7 Lateral Motion Response with Simplified Equations 18 8 Pitching Motion Response 19 9 Lateral Motion Response 20 10 Lateral Motion Response Showing Damping of Rolling Motion 21 11 Undamped Natural Period for Pitching Motion (Short Period) 22 12 Damping in Pitch (Short Period) 23 13 Undamped Natural Period for Pitching Motion (Long Period) 26 14 Damping in Pitch (Long Period) 27 15 Undamped Natural Period for Yawing Motion 29 16 Damping of Lateral Motion 30 17 Time Constant of Decaying Spiral 32 18 Time Constant of Uncoupled Rolling Motion 33 19 Flight Test - Simulator Correlation; Short Period 34 20 Pitching Motion 20 Flight Test - Simulator Correlation; Dynamic Directional Stability Test 36 21 Flight Test - Simulator Correlation; Dynamic Directional Stability Test 38 22 Computer Circuit for Pitching Motion 41 23 Computer Circuit for Lateral Motion 42 iii

TABLE OF SYMBOLS ( ) arrow written over a variable indicates it is a vector. (,, ) triad notation indicating the three components of a vector in the body axes coordinate system. ~- ~bar written over a variable indicates the perturbation ( ) increment of the variable. ( )O subscript zero on a variable means the steady state value of the variable for zero perturbation. ( ) dot over a variable means differentiation with respect to time. b airplane wing span, feet. C nondimensional airplane coefficient, there are six of these with a single subscript. C{Li CD. CY airplane lift, drag, and side-force coefficients. C1, Cm, Cn airplane roll, pitch, and yaw moment coefficients. C with two subscripts indicates an airplane stability derivative of which there are fourteen in this analysis. CDn drag curve slope. CLa lift curve slope. CLt lift due to rate of change of angle of attack. CLq lift due to pitching rate. Cyp side force due to yaw. Cl3 rolling moment due to yaw. Clp damping in roll. Clr rolling moment due to yaw rate. Cnp yawing moment due to sideslip (directional stability). Cnp yawing moment due to roll. Cnr damping in yaw. Cma moment curve slope. Cmna moment due to rate of change of angle of attack. iv

Cmq damping in pitch. C as a subscript to a variable indicates the effect of control surface deflection. c as a variable is airplane mean aerodynamic chord, feet. D drag force, pounds. e as a subscript indicates effect of engine. F = (X, Y, Z) air load force vector, pounds. g acceleration of gravity, 32. 2 feet per second per second. H airplane angular momentum vector. h altitude, feet. 1XX Iyy I airplane polar moment of inertia relative to the x, y, z xx' yy' zz body axes, slug-feet squared. ~I ~ airplane product of inertia relative to the plane y = 0, xz slug-feet squared. I as a subscript means inertial frame of reference. i as a subscript means initial value. M = (L, M, N) air load moment vector, pound-feet. M mach number. m airplane mass slugs. P = (Px Py' Pz) components of engine thrust, pounds. (P, Q, R) or airplane angular rotation rate vector, radians per second. (p, q, r) o ~ dynamic pressure 1/2 p IV|, pounds per square foot. = (x, y, z) position vector in body axes coordinate system. S airplane wing area, square feet. s as a subscript indicates stability axes. t time, seconds. -4 V = (U, V, W) airplane velocity vector, feet per second. (u,v,w) perturbation of airplane velocity vector, feet per second. F = (X, Y, Z)airplane air load force vector, pounds. v

(x, y, z) position vector in body axes coordinate system. a airplane angle of attack. p sideslip angle. Y = xY Yy' Yz) unit gravity vector. C~~ ~ damping factor. 9 pitch angle. x root of characteristic equation of linear system. p mass density of air. ( Tx, Ty, Tz) torque reaction of engine on the airplane. roll angle. vi

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN SUMMARY This report describes an investigation of the dynamic behavior of the F-86D aircraft by means of electronic differential analyzer solutions of the linearized equations of motion. This work is part of an overall study program of the computer section of flight simulators. It is hoped that as a result of this report certain possible simplifications in the configuration of flight computers for fighter aircraft may result. The equations of motion used in this study are referred to body axes coordinates. These equations are linearized by assuming small perturbations from level flight. In this case linearization results in a system of equations which are uncoupled in such a way that longitudinal pitching motion is independent of lateral motion. The aerodynamic coefficients of the F-86D were substituted into these equations and airplane motion was then simulated on the electronic differential analyzer facility of the Department of Aeronautical Engineering, University of Michigan. The response to a step function input was obtained for four speed-altitude cases. The period and damping of this motion was measured from the traces and compared with approximate formulas. Correlation with flight test data is also shown. In all cases the agreement was considered to be satisfactory. I. EQUATIONS OF MOTION The body axes system of coordinates, shown in Figure 1, are used in this analysis. y V Figure 1 Coordinate Axis System 1_

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN These axes are orthogonal and fixed with respect to the airplane with the origin at the airplane center of gravity. The airplane velocity with respect to the air is resolved into components along these axes. V = (U, V, W) The rotation rate of the airplane which is the same as the rotation of the coordinate frame of reference is = (P, Q, R) The equations of motion are obtained from Newton' s laws; force equals rate of change of momentum and moment equals rate of change of angular momentum. The forces acting are gravity, mgy = mg(Y, Yyi Y.) engine thrust and torque, P = (Px PY P Z) = (T Ty TZ) and aerodynamic force and moment due to air loads F =(X, Y, Z) M = (L, M, N) Subscript c indicates loads due to control surface deflection. Therefore d (d mV) = mgy+P+F+c (1) dH T= T+M+M (2) (dt I 7 +M Since the time derivatives in Equations (1) and (2) are relative to inertial coordinates, the component due to rotation of the body axes must be taken into account according to the formula [dt] = [(*) +W x()] (3) where the dot means the rate of change relative to body axes coordinates. The angular momentum is ~H = f 7x( xh) dm. (4) -----------------------— 2 —------------

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN where r = (x, y, z) The integration is over the whole mass of the airplane. When expanded Equation (4) becomes H =(PIxx -QIxy RIx QIy - - PIxy, RI - PI -QIy (5) XX XZ' YY Y-'-' z zz x z yz r 2 2 where I =(y + z ) dm, Ix = xzdm, etc. The airplane is assumed to be symmetrical with respect to the plane y = 0 which leaves Iz the only nonzero product of inertia and (5) becomes H = (PIxx- RIxz, QI RIz -PIx) (6) Hence, by equation (3), dH' ( = (PIxx - RIx QIyy, RIz - PIx (7) dt I xx xz yy'' z z xy + [QR (I - Iyy)- PQI, PR (I - I) + (P - R2)I, PQ (Iyy - I ) + QRIxz * yy xx xz When equations (1) and (2) are finally expanded according to these formulas the result is m (U +QW - RV) = mgyx+P x + X +X m (V + RU- PW) = mgy + P + Y + Y (8) * y c m (W + PV- QU) = mgy + Z + Z PI' -RIx +QR(I - Iyy) - PQI = T +L +L QIyy + PR (Ix - I + (p2 R2) = Ty + M + Mc 0 ~ RIz - Pz + PQ (Iyy- I)+ QRIxz = +N +Nc II. LINEARIZING THE EQUATIONS OF MOTION The equations are linearized by considering small perturbations from the steady state of level flight. The steady state angle of attack is ___________________________ 3 ___________________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN determined by the lift required to balance the weight of the airplane. The angle of attack therefore depends on the steady state speed and altitude and is not assumed to be small. Perturbations from other steady states such as turning motion are possible and would perhaps be considered in more complet studies. It is also possible to compute from linearized equations the motion from an unstable initial state for a short space of time. The technique of linearizing employed here may be described as follows. The eight quantities U, V, W, P, Q, R, e, and q are regarded as independent variables and written as the sum of their steady state value plus a perturbation which is denoted by a bar. The absolute value of the steady state velocity vector is denoted by V. U = V cos a + u V = o +v W = V sin aCL + w o 0 P= 0 + (10) Q = 0+ q R o+ r 0 = a + 0 Each of the terms that appear in the equations of motion are written as explicit functions of these variables, developed in power series of the perturbation variables, and all quadratic and higher powers dropped. A. Linearizing the Gravity Forces The orientation of the airplane with respect to vertical is defined by the components of the unit gravity vector y. Equations which can be used to generate this variable may be derived by applying equation (3) to t, which is itself a constant vector with respect to inertial axes. Thus [dyi = 0 = x (11) In terms of the body axes components this becomes ='Yy- Q;y = PUz - RyX (12) ------------------------ 4 ___ -_________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Yz = QYX PYy In this application we want to introduce the pitch and bank angles and linearize. The gravity vector is related to the pitch and bank angles by = (- sin 0, cos e sin, cos e cos,). (13) In the steady state (13) reduces to Yo = (- sin ao, 0, cos a) (14) The perturbation approximation for the gravity vector is therefore y (- sin a0s - cos a os,, cos a - 0 sin a). (15) From Equations (12) and (13) we have e = - R sin 0 + Q cos 0 (16) = P + tan O (Q sin + R cos ). The linearized form is O = q (17) = p.+ rtan a 0 Equations (17) together with the six equations of motion provide a sufficient number of relations among the eight independent variables of the problem. B. Linearizing the Accelerations The Coriolis term in the acceleration does not vanish in the linearized case. Substituting (10) in (8) and dropping quadratic perturbation terms, U +WQ- VR = u + Vo sin a q V+UR - WP = + Vo cos ao - V sin aop (18) W +VP - UQ = w- Vo cos a q In the case of angular acceleration the Coriolis term vanishes. According to Equation (9) these terms are quadratic in the perturbation variables when the steady state is P = Q = R = 0. The linearized equations therefore do not take into account precession torques due to rotational airplane motion. Thus,.____.____________________________ 5 ______,,___

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN PIxx -RI + QR(Izz -I )- PI ) -z rIxz XX xz zz yy xx xz QIyy + PR (Ixx ) + (P - Rz) Iy (19) RIzz -PIxz + PQ (Iyy -I +QRIxz = rIzz - pIxz C. Linearizing the Engine Forces The engine forces are assumed to be constant in this analysis and therefore the thrust (p) and moments (T) do not appear in the linearized equations. The torques due to precession of the rotating engine rotor could be taken into account, if necessary, by the equation T = X (I ). (20) e' Since the axis of rotation of the engine is the x axis, approximately, (20) becomes T = 0 x T + Ie e r r (21) e e T| = - Ie q z e e These terms have the effect of coupling the longitudinal and lateral motions. D. Linearizing the Air Loads The six components of the air load are X = [- CD (a) cos a + CL (a, a, q) sin a] qS Y = +Cy(p)S Z = - [CL (a, a, q) cos a + CD (a) sin a] qS (22) L = [Cl(p, P, rs) cos a - Cn (p, ps, rs) sin a] qSb M = a Cm (a, a, q) qSc N =+ [Cn (p P, r ) cos a + C (P, Ps r) sin a] qSb The angle of attack factors appear because aerodynamic data is conventionally measured and computed relative to the stability axes rather than the body axes; the transformation involving a must therefore be applied. The parameters on __________________ 6 ________________

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN which these coefficients are assumed to depend are indicated. In some cases it might be desirable to take into account more factors. In Equation (22) the dependences indicated were those for which aerodynamic data was available. The following stability derivatives are therefore involved CD L,CL CL C Cy C I CCI a a a q P P p (23) C1r, CnC Cn, Cn, C, Cm I n n n m' m' m r. p r a a q It is assumed that the p and r stability derivative data used is relative to the stability axis, thus, for example, CI C1 p + Cl Ps + Cr rs (24) = C -v- + C ('P cos a +r sin a) V P + Cl (r cos a - p sin a) r The perturbation in a is derived from tan a - U ior V sin a + w ~tan (ao + a) = ~ (25) V0 cos a + u Linearizing each side of equation (25) gives the result - u a = - cos a - -- sin ao. (26) V~ V 0 0 In the case of the forces X and Z which have a nonzero steady state value it is necessary to consider the perturbation in dynamic pressure q, t== 1 + 2 -cos a +- sinao)]. (27) V V 0 0 Other useful relations are cos a = cos a -sinao a (28) sin|a = sin a + cos ao a. ------- 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN E. The Steady State Equations of Motion The steady state equations are found by equating all perturbation to zero. Equations (8) and (9), together with relations given in (A), (B), (C) and'(D) reduce to 0 = -mg sin ao + P + (- CD cos a +CL sin a ) oS + X x ji Du 0 L 0 0 0 C 0 =P +Y y c 0= mg os P - (CL cos + C sin a) q S + Z o o (29) 0 = T +L X C 0 = T + M y c 0= T +N J z C The first and third equations were solved for ao assuming that Xc, Z and Pz were all zero and making use of the airplane lift-drag polar. The remaining equations indicate that any asymmetrical engine thrust is compensated by trim. F. Final Form of the Linearized Equations When equations (8) and (9) have been subjected to all the linearizing operations outlined above, the following equations result 1*0S 2 2 -- + cosa ) CD - sin a cos ao (CD + CL )+sin a C ] u mV o a o a - [ (1 + sin2 ) CL + sin a cos a (CD CL )+ cos a CD o o a a * ~ qSc w u - a C q+ cos sin [C qC (cos 2mV~ q a V0 V~ + V0 sin a q + g cos a X (30) m'o0 0o a a + [C + cos2 a CL + sin a cos a C ] w 0 a a,,Sc s u + 0- cos aO [CL q +CL, (coso - a0 ZmV0 q a V' V

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - V0 cos a q + g sin ao q = {SCc sin au- C cos a w L m 0 m VI a a o yy - 2 [Cm q + Cm (cos aO - sin a q a Vo0 V - = -q - v = - v - V sin a P + V cos a r- g cos ao ~ G oSb rz _ p+ xz C = -p x r(- Cosa +C sina ) 2V Ixxb 2 2 + [- C cos a + (C1 + C ) sin a cos a C sin ] p r p r + [ -C cos aO + (C - Cl )sin a cos a + C sin ao r r r p p I. toSbZ 2 - r +~ xzp = - - ( - Cn[3 cos a - C1 sin a) V' (31) I 2V I }b b ~ zz 0{ozz C i + [- Cn cos2 a +(Cn - C1 ) sin ao cos ao C sin ] p P r p r 2 2 + [- cos ao (Cn +C1 ) sin ao cos aO C1 sin2 a r p r P - = - P -tan a r III. DISCUSSION OF THE LINEARIZED EQUATIONS OF MOTION The first four equations (30) contain only the variables u, w, q, and e, the second group of four equations (31) involve only the variables v, p, r, and 4. Since these equations are linear each system can be solved separately. Motions for which all eight variables are excited may be derived from these solutions by invoking the principle of superposition. Each syste ______L________________________ 9 _________.____

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN consists of four linear first-order equations in the four variables and is therefore a fourth-order system with four characteristic roots. The first system, which applies to longitudinal motion,turns out to have a characteristic equation with two pairs of complex conjugate roots for all the cases studied. The response involves a short period oscillatory component and a long period or phugoid component. When the airplane response to a step function input is simulated the short period motion is quickly damped out leaving the phugoid motion which is less rapidly dissipated. The amplitude of oscillation of w and q is very small for phugoid motion as can be seen in Figure 2. The period and damping of the short period motion can therefore be accurately measured from the w or q response traces when an initial value has been applied to w or q. The period and damping of the phugoid motion are obtained from the response to an initial value on u or e, measuring the damped oscillations which remain when the short period transient has damped out. The second system of equations for lateral motion also has a short period oscillatory component. However instead of a long period oscillation one finds in this case a pair of exponential functions, one with a short time constant and the other with a long time constant compared with the period of oscillatory motion. The long time constant motion is illustrated in Figure 3 - the short time constant motion in Figure 10. The period and damping of the short period motion can be best measured from the response to an initial value on v since this does not excite either exponential component to a great extent. The long time constant can be measured by observing the decay constant of the spiral motion resulting from applying an initial value to 0. The short time constant can be measured from the response to an initial value on p. The response will decay rapidly toward zero with very little residual oscillation. Equations (30) and (31) can be greatly simplified by setting angle of attack to zero. This may often be a reasonable approximation since the stability derivative data is not very accurately known for a given airplane and the angle of attack is usually not large. If the angle of attack is set to zero only in those terms which involve stability derivatives,equations (30) and (31) reduce to aSS - u =-.[L2CD u - (CL CD ) ~w] + V~ sin ao q + g cos a~ 1 mV o o a o~~~~1

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- ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIO(AN qeS _c w -[ 2CL u + (CD +CL )w +- (CL q + CL. -)] mV o o a 2 q a V - V cos a q + g sin a 0 (32) - V = -q _ CS v-c _ _ _ - q = - C - (C aO r - g cos a' +^ I r= t —[ — C1 v -C, p-C1 i] Ix 2VI b p p r 0 o yy o (33) - s = - P - tan a C r. A further simplification would be to set the angle of attack to zero in the remaining terms which are Coriolis acceleration terms and gravity forces. The resulting systems of equations are o w - w = ~ [2CL u + (CD + CL ) w +-(CL q + CL. -)] - V0 q ImV 2V q a Vr ~-q[ = q0Sc c m ~~~~~i*~C~(34) Ixz. 0Scb 2 r +- [ —.Cn. v - C p- Cn r] 2V0Ixx b p p r m —--— [oU (CL -CD W]+g1 ~ %sbz

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN - 1xz:- = Sb2 _ xz * ^z 2 - r +^"-p = ---— [ — C ^ C P-Ca F] r35) Izz ^2V I b P pr - = - i. The order of the systems of equations have not been reduced by these simplifications. The change in the response due to these approximations is shown for a moderate and a high angle of attack in Figures 4, 5, 6, and 7. It is seen that in the case of lateral motion the Coriolis terms cannot be approximated successfully, especially at high angles of attack. The other approximations all agree quite well with the exact linearized equations. Such appro) mations do not gain much in the present, application since the coefficients are evaluated before bringing the problem to the computer. How" ever a simulator which automatically generates these coefficients as functions of speed and altitude would be simpler in design if these approximations were made. IV. CORRELATION OF SIMULATOR RESPONSE WITH APPROXIMATE FORMULAS Measurements of the simulator response were made for four speed altitude cases. Eight parameters are measured, corresponding to the four characteristic roots of the pitching motion and four also for lateral motion. This data was taken from Brush recorder tapes in the case of the short period oscillating motions and damping in roll. Figures 8, 9, and 10 are examples of this data. This recording equipment has adequate frequency response for rapidly changing variables. Phugoid motion and spiral decay data were read directly from servo dials since these motions change very slowly. Approximate formulas are derived in this section for all cases. Figures 11 through 18 show these formulas plotted as curves and also points obtained from the more exact simulator response traces. The approximate formula for short period pitching motion may be derived from Equations (34) by assuming the change in the forward component of velocity, u, is small. The second and third equation then uncouple from the first and fourth, and equations (33) degenerate to the secondorder system. -^ -~ [(CD +CL )) Voq 0~~~~~~~ ---------— ) —-— C L 4 CV

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RT NO. BL-924 THE BRUSH DEVELOPMENT CO. PRINTED IN U.S.A. Figure 8. Pitching Motion Response Mach Number 0. 7 Altitude 10, 000 feet Damping factor 0. 32 Undamped natural period 1. 28 seconds 19

='-_. - 1 1 k7 \- \ \\\ CO. PRINTED IN U.S.A. CHAF Figure 9. Lateral Motion Response Mach Number 0. 7 Altitude 10, 000 feet Damping factor Undamped natural period 20

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN _ %qSc c w -q = [- C w —(C q +Cm )] (36) Vyy a 2 q a V Since CD, CL, and CL are small and have a negligible effect they are o a q also omitted. The characteristic equation of the system (36) is mV- VL maV CL + Xx| The roots are complex. The undamped natural period and the damping corresponding to these roots are | -~~T = 2V~ I oyy (38) Vo0 pSc (-Cm pScCLc ) C.X C 0 y -C - C -C - ~a a q o SCa l6irI mc a q a The rootsdamping factor can bmplex. The measurndamped from the amplitude of the decaying cori oscillations. In this case the damping factor cannot be found by comparing the amplitude of the overshoot with the initial displacement since the initial value does not correspond to a maximum excursion. Figures 11 and 12 show this data. The approximate formula for phugoid motion can be derived from (34) by assuming that the angle of attack, which is proportional to (38) does not change appreciably. Since mg = CL0 fo S these equations reduce to g _ 0 YY - V (39) -e =-q 2 L m 16wrr I mc a q a oscillations. In this case the damping factor cannot be found by comparing the amplitude of the overshoot with the initial displacement since the initial value does not correspond to a maximum excursion. Figures 1 1 and 12 show this data. The approximate formula for phugoid motion can be derived from;34) by assuming that the angle'of attack, which is proportional to w does not change appreciably. Since mg: CLo o S these equations reduce to g CD u = - U + go Vo CLo o 2- u - Vo q (39) V Z4...

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The second-order system (39) has period and damping given by the formulas To = orr Vo/g (40) = CDo/,/2 CLo The period is not a function of any airplane parameter. Figures (13) and (14) exhibit this data. The satisfactory simulator performance is significant since computer circuits can often be designed to function better over a restricted range or to work best in one range at the expense of another range of variables. In this case, however, the same computer circuit was used for these two greatly different types of motion. An approximate formula for the oscillatory lateral motion which is very similar to the formula for pitching motion can be obtained in certain cases. In equations (35) with I = 0, if 0 is dropped from the first equaxz tion and p from the third equation, a second-order system results. One would expect this approximation would be most satisfactory at high speeds and low altitudes since then the inequalities g < < Vo and cnp < < cn are strongest. This assumption does not mean the airplane does not roll much (as a matter of fact it rolls about twice as much as it yaws) but rather that the rolling does not affect the rest of the system very much due to the small coupling term. The second order system is *v = - y- Cv +V)r mV 0 (41) oSbz 2 -r (= -C (-C r). 2V I b nr 0 zz The period and damping for this system is _Zr 11 Zzz - 1\ pSb (Cn + p4 Cy Cn 4m Y nr (42) T pV oSb 2Izz C = _ ~- - (_ z Cy - C 16 I mb S r __________________________ 25 ______________,

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN These formulas are compared with the simulator responses in Figures 15 and 16. The periods check very well and also the damping except at the higher altitudes where the assumptions made are most questionable. When operating the simulator it was found that spiral motion was a very slowly changing phenomenon. Therefore it seemed reasonable to approximate this by setting all the accelerations to zero. Equations (35) then become %qSc + - C v - Vr+g = 0 mVo J3 z C + C1 + C1 r = 0 b p p r (43) 2 C v + C P +C r b n n nr p = p. The Cyp term proved to have a negligible effect so it was also dropped as a matter of convenience. The characteristic root therefore satisfies the equation o 0 -V0 g 2 c c C C C0 b p p r = 0 (44) C C C 0 b n np nr 0 1 0 -X Solving this equation for X one gets for the time constant of decaying motion 1 V Cn C Cn t P P -P (45) 0 X g C CC - C C nr r "p If the sign of to should turn out to be negative the spiral would become tighter rather than decay. A criterion for spiral stability is therefore 28

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN C, Cn - Cl Cn -P p P P_ > 0. (46) Cl C - CI Cn rP r r r Formula (45) is compared with the simulator response in Figure 17. This motion is much slower than phugoid motion, yet the correlation is quite good. When the simulator response to an initial value on P with vi, ri, and.i all zero is studied, it is found that the rolling velocity is rapidly damped and that very little oscillatory motion is excited, particularly at the lower altitudes. This can be seen in Figure 10. The effect of Ixz is not noticeable. This suggests that an approximate formula may be found by dropping all terms but P terms from the second of equations (35). The resulting equation is 2 * qt Sbz P = C (47) oIxx p The time constant of this motion is 41 1 t = xx (48) p VSbZ CI This is plotted in Figure 18. The time constant was calculated by measuring the time to damp to half-amplitude instead of measuring the decay over a longer period; this was done with the hope of minimizing the effect of the oscillating motion on the readings. V. CORRELATION WITH FLIGHT TEST DATA Figure 19 compares a simulator response trace with flight test data points for pitching motion. The initial conditions on the simulator variables were assumed to be zero for O and u. Also ai = ao, since for 2G acceleration the lift must be double the airplane weight, thus doubling the angle of attack. It was assumed that the pitching rate.i corresponds to motion on a circle with radius such that the centripetal acceleration is one gravity; thus qi = g/Vo. The normal acceleration, according to equation (18), is w - VO cos ao q. The simulator trace and the flight test points agree well with regard to shape although the simulator has a slightly longer period. The flight test points do not approach unity, which is abnormal. 31

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The angle.of pitch necessary to have this effect in steady flight is 45 degrees which is much too high. There were no other measurements in this set of data which correspond to other variables computed by the simulator. The lateral motion data shown in Figure 20 is more complete; it contains all four of the variables involved. No attempt was made to match the initial conditions of the flight test data but rather to obtain the best overall agreement through the first cycle taking all four traces into account. This seems to be the most reasonable approach since the test pilot is unable to center his controls instantly; this results in a distortion in the data during the first half second or so of motion. Many combinations of initial values were tried in an effort to get the best fit. Damping factor and period were measured from the simulator response and from each of the p, Pt and r recordings of the flight test data with the results shown in the table. Lateral motion M =.6 h - 40, 000 Trace Period Damping Ratio Simulator 2.75.087 Flight-Test.17 3. 3..*17 Flight-Test 3.1.13 r Flight-Test 3.0 14 The simulator, which is an almost ideal linear system, shows the same damping and period on all traces. The different values measured from the flight test data are an indication of nonlinearity or scatter in the data. The flight test points seem more reasonable if one follows the peak to peak values rather than zero to peak values. A variable zero error would explain many irregularities in the data. When the transient is finished the airplane appears to be left in a tighter spiral with more slip than the simulator. This is probably an indication of erroneous data since the airplane directional stability should reduce the sideslip nearly to zero in this case. It appears that the simulator roll rate is twice as great as the airplane roll rate. The apparent large discrepancy in the angle of bank probably is not important since a small amount of aileron control during this maneuver could make a big difference. ___~__________________________ ~35 ________________________eIt

................................................................................ - - - - - - -....................................................................................... - - -........... _L.LL.................................................................................. - ------ ---.........................:4F:FF -IE4:F

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This is because there are no large forces in the system which tend to reduce bank angle to zero. More lateral motion data is shown in Figure 21. At this speed several trends seem to be reversed. The simulator damping is now too much rather than too little, and the simulator roll rate is half rather than double the airplane roll rate. The yaw-rate flight-test data is doubtless plotted 180 degrees out of phase. Since the slope of the sideslip angle is principally the negative of the yaw rate, the given flight-test data appears to be physically impossible. The simulator yaw rate is twice too large in this case. There was no angle of bank data available. Measurements from these traces are shown in the table below. Lateral motion M = 0. 8 h = 40, 000 ft. Trace Period Damping Simulator 2.22.093 Flight-Test 2.53.061 Flight-Test 263.93 Flight-Test 2.50.066 p VI. MECHANIZATION OF THE EQUATIONS OF MOTION Four speed-altitude cases were computed for both systems M =.5 h = 10,000 M =.7 h = 10,000 (49) M =.8 h = 30,000 M =.8 h = 45,000 The coefficients of equations (25) and (26) were evaluated using data from North American report NA-50-107A. The stability derivatives Cmq and CLq vary with the position of the airplane center of gravity, which depends on the rockets and fuel carried. The cases considered were for no rockets and thirty percent of the fuel expended, which places the c. g. at the 24% | point of the mean aerodynamic chord. The following physical constants of the F-86 were used: S = 288 square feet b = 37.1 feet ------------------ 37

Figure 21 38

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN c = 8. 09 feet mg = 15,500 pounds Ixx = 8, 625 slug feet squared (50) Iyy = 29,500 " yy I = 35,300 " zz I = 0 and 2700 xz The two values of Ix were estimated to be the upper and lower limits. The equations for pitching motion derived from (30) are -u = +.008 u -.081)' +.296 100 q +.322.012.052.194 (4).006').049.325 003 L.049J L.648J (1) (2) (3) - w = + [.059 u + 4.2691 w - 10.533 100 q +.0 181 0.042.411 1.746.008.040.245.790 o.013 0411.124.772.027 (5) (6) (7) (8) -100 = -.1241 u + 4. 539 + 4.265 100 q.0841.768 i4.399.067. 398.228 (51).068j.2021.114 (9) (10) (11) - 100 O = - 100 q Each set of four numbers corresponds to each of the four speed-altitude cases (49). The equations for lateral motion derived from (31) are = 1.22v- r301 100 P + 10.540 100 r -.322 (100 ).325.197.757 (4).175.327.798.086.652.778 (1) (2) (3) ------------ - -39

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN -100 p +31.3 r =10{.540 V + 10.400 100 p - 4. 297} 100 r,.681.611 i.385.441.,360.268 267 K.184.157 (5) (6) (7) (52) - 100 r +7.65) p = -4.357 v + I.067 100 P + (.476 100 r.527.018.778 ).302.039.405 1.145 1. 054..186 (8) (9) (10) - 100 ~ = - 100 p -.056) r 1026.041 084, (11) The computer circuit diagrams for solving these equations are shown in Figures 22 and 23. The conventional symbols used there are as follows: A triangle indicates a high gain inverting DC amplifier with a one megohm feedback resistor; in this case the amplifier serves as a signchanging summer. A triangle with a double vertical bar indicates the feedback element is a one microfarad capacitor; thus the amplifier acts as a sign-changing integrator-summer. The input resistors are one megohm unless the numbers 4 or 10 are shown, in which case the resistor is 1/4 or 1/10 megohm. A circle with a number indicates a potentiometer which multiplies the voltage according to the coefficients of equations (51) and (52), identified by the circled numbers. All the circuit connections are not indicated in order that the diagram may be simplified. When two or more points are labeled by the same variable a common connection is implied. In the case of lateral motion two values of I were used. When xz I =0 the problem is straightforward because the equations (31) are xz solved explicitly for p and r. When I = 2700 equations (31) are solved for p' and r'where P' = p-.313r (53) F' = r -.0765 p. 40

- - ~1006'Q~~-0 o - tooq h -a w-_ oo. I -, -los- ~ 100 q C> -1~9 Figure 22. Computer Circuit for Pitching Motion. 41

loo r 3- O-v0- 1 loo r 4- W. — v toF -,o -/oop /1004 -/100 r -P o —IoO1 ooSwitch up for = o. Figure 23. Computer Circuit for Lateral Motion. Switch up for I = 0. Switch down for I = 2700. 42

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Equations (53) are solved for p' and f with the result P = 4 x.256 p'. +.320' (12) (13) (54 (54) r =.078 P' +4x.256 F' (14) (15) J The circuit for computing lateral motion (Figure 23) was designed so that either value of I could be chosen by throwing one switch. _xz43 _____________________43 ____________

UNIVERSITY OF MICHIGAN 3 9015 03483 3502