ENGINEERING RESEARCH INSTITUTE DEPARTMENT OF AERONAUTICAL ENGINEERING UNIVERSITY OF MICHIGAN DC ANALOGUE COMPUTER SOLUTION OF B-47 AND F-86D LONGITUDINAL FLIGHT EQUATIONS DeQpartment.:of the Air Force C.ontr.ac t.No.. A.F 3 3('616)-21B1 E. O -No.' R68-4'21 PO-3a Interi'm: Techitical Re)port September, 1954 Project 2164 Submitted for the project by: E. G. Gilbert

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ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN DC ANALOGUE COMPUTER SOLUTION OF B-47 AND F-86D LONGITUDINAL FLIGHT EQUATIONS 1. Introduction The equations for longitudinal motionfor the F-86D and B-47 aircraft were set up and solved on the dc electronic differential analyzer of the Department of Aeronautical Engineering,-University of Michigan. Results and conclusions of this investigation are presented. The simulation of the longitudinal motion was complete. All nonlinearities in the equations were included with the exception of the variations produced in aerodynamic force coefficients by air density and Mach number. The assumption of constant density and Mach number is valid for the changes in altitude and velocity encountered during a single computer run.- Scaling was adequate to allow maximum aircraft motion. Thus, results obtained for small motions are realistic from the standpoint of complete simulation. Computer solutions indicate that duplication of small motion dynamic characteristics by the general purpose dc differential analyzer is good./ Since accurate simulation of the longitudinal characteristics is most difficult it would appear that similar accuracy would be obtained in the complete simulation of all motions. 2. Computing Equipment Used The computer used was of the REAC* dc analogue type. Resolvers and multipliers were standard Reeves servo units, S-101 Mod 4, capable of good resolution and fair dynamic response (natural frequency of 3 cps). The maximum velocity and acceleration ratings of 50 volts/sec and 500 volts/sec2 respectively were somewhat inadequate for the largest scale motions. All runs were selected to avoid these limitations., The summing and integrating amplifiers were chopper stabilized and of good bandwidth with computing components of. 1% accuracy. Brush recorders were used to chart important variables. *Reeves Elc.tronic AnalogueComputer, Reeves Instrument Corp., N.Y. 28, N.Y. *..::'- -1

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 3. F-86D Equations and Computer Realization The longitudinal equations of motion are obtained from those of page I-il of the previous final report for this project* by setting V, P, and R and all their time derivatives equal to zero. Bank (0) and heading (p) angles are arbitrarily assumed zero. The equations of equilibrium then become.. X+P +X U = -gsine-WQ (1) m Z+P +Z W + g cos 0 +UQ (2) m. M+T +M Q y. c (3) I YY The angle of attitude is given by. = GQ. (4) and the rate of climb by - Sz = U sin O- W cos 8. (5) The angle of attack is given within a good approximation by a W (6) U The force and moment equations were obtained from a report by the Engineering and Research Corporation (ERCO Report No. 620-8-20) and are those used in the F-86D flight simulator. In the notation of the ERCO report Zt = - 288 q CL (7) 2 Xt = - 288 q (a +ba +ca ) (8) * Reference Project 2164, Final Report __ _ _ _..___ _ _ __ _ _ __ __._ _ _ _ 2

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Mt = 2329 q [.030 - f8 (Ma) CL' -f13 (Ma) 6SE - fll (q) T. 10- f6 (Ma) (9) U 0. 162 a] U X+Px +X = Xt cos a - Zt sin a +T Xt Zt a +T (10) Z +P + Z = Zt cos a + Xt sin a Zt +Xt a (11) M +Ty+M = Mt Zd (12) y C where 2.' 2 q = 1/2 p V - 1/2 p U dynamic pressure M Mach number 6SE elevator, stick deflection in degrees T thrust in pounds CL, C' lift coefficient (function of l L L a and Ma) a a, b, c, f6I ffl l, f13 functions of Ma d distance from C. G. to reference point The computer set up diagram is shown in Figure 1. All variables have been scaled to allow the maximum range of variation as given in the ERCO report. Computer symbols are conventional and indicated variables are in volts. Angles are in radians except at inputs of resolvers. Coefficient potentiometer settings are given in Table 1f. The airplane is in clean configuration with a gross weight of 15, 500 pounds. 4. Results of F-86D Simulation Computer runs were made at the following combinations of altitude and Mach number: 10, 000 ft., Mach. 5 10, 000 ft., Mach.7; 30, 000 ft., _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~~~3 _ _ _ _ _ _ _ _ _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN x+ Px + X 000IQ 2 VOLTS I DEGREE AO I I L, 0_ SINe II.0 10I~~~~ "+"''",oP-J~.,'N -ASI. z+Fz+ zc i - 000 4*~~~~ tO~~~ ~0 -'o10,ou H I2 I 000 N I 1000 2 2i4 Ho_ IOO O I!4 ~x'4 -"q + 10s2 L - "-' 4 Cgo41 |2q 2q ttl | qT^10M 2000r I - 200 0 2000 I 10 a20B 2 220 A0's - _BIN __ A I~ 2000 I 10' L OK -12 I' 2 0 0 0 —'-H0 0 Lo -200T-oq L __ A qTOIO 2 —~"~C —---- I 000 I 20000 I1 ~~~0 COS 2 I00 I 0 2000 1000 0 USE 21 I00a FOR I-. I S 2000 4 I - 4 D 0104 0 I00029 ~* ~ooa Figure 1_ 4

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Table 1 Pot. No. P ot. Setting 1 100/m =.208 2.200 4 100/m =.208 5.002 7 20, 000/I =.680 yy 8 g/I000 =.032 9.268 10 1/2 x 103p =.876(10,000 ft),.444 (30,000 ft) 230 (45, 000 ft) 11.400 12 T x 10 13.0239 f6 =.675 (M.5).735 (M =.7).756 (Ma =.8) a 14 d =.513 15.286 (Ma.5),.310 (Ma =.7),.351 (M =.8) 16 2.88 a =.043 (M =.5),.043 (Ma =.7),.045 (M =.8) a 17 82. 5b =.007 (M.5).010(M =.7).005 (Ma.8) a 18 592 c =.243 (M.5).272 (M =.7).367 (Ma =.8) a 19.700 20 4.02 f8 =.830 (Ma =.5,.7,.8) 21 23.29 f13 6SE 22.96 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _5 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Mach. 8; 45, 000 ft., Mach. 8. Altitude and speed variations were limited for each combination so that aerodynamic force coefficients, as functions of Mach number and air density, could be assumed constant. For each combination Q, 0, a, and sz were recorded, first, for an initial pitch rate, and second, for a step in elevator deflection. Sample recordings are shown in Figure 2 and Figure 3. In all cases results were excellent. The computer was easily adjusted to maintain level flight or a given rate of climb. Solution stability was good (e. g., a 1000 ft/min rate of climb would hold within + 50 ft/min over a three minute period with no computer adjustment). Figure 2 and Figure 3 indicate accurate dynamic simulation and fine resolution. Transient response also checks favorably with previous solutions of the linearized equations. Figure 4 shows that the long period longitudinal oscillation was faithfully reproduced. This is a direct result of excellent resolution, especially in integration. None of the present ac simulators tested by this group have approached such performance in the duplication of small motion, long period oscillations. The only trace in Figure 2 which shows the effect of finite re'solution is the rate of climb. This is due to granularity and sticky friction in the resolving servo which implements Equation (5). Other multiplications and resolutions are less critical as they are all followed by integrations which smooth fine irregularities. This advantage is only present when the integrations themselves are linear for small motions. It is believed that the superior simulation achieve in this study is due in large part to the excellence of dc electronic integration. Figure 5a illustrates more fully the errors due to resolution in rate of climb. Though jumps appear quite large none of them are much greater than 100 ft/min. It should be noted that the rate of climb does not appear directly in any simulator presentation. Figure 5b shows how rate of climb would be indicated on a panel instrument which has a linear system first order time delay of three seconds. The resulting smoothing makes the response completely acceptable. Similar action would take place upon integration of rate of climb to obtain altitude. It was proved that the irregularities of the trace in Figure 5a were due to poor trigonometric resolution of Equation (5) by using the approximation (for small 6) - s ~ Ue -W. (13) z _______ _______ _______ _______ _______ ___ 6 _ _______ _______ _______ _______ ______.

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN rad - h = 1,000 ft M =.5.2 sec/div Initial P it h R=ate Re sponse I 1111 11 11 11 11 11.F1igureI

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN.... __ -... -1-1 —-_- -1. -: = —-: —=~,- -t-~ — I-= —-— ~ -: — --— =- -....... ~~ —-:':! -_ —~ —: —,-__~ -: — E-:~-: t-' —-I~ ", —.-j t__ r~ a': _s e C _ —':~ — e - _~!,2-3-i _~ _~_ _:__ ____._ "Q __ _ ~.2. rad- set~V h - 30,O00 ft M~ =.8 5,SE from.4O to.09.2 sec/div, 1 ~l t - —.[I %ii'I. I fI I, I-! I_!' W m - —'[7.-7 —t, I -I I, _I —' I- t- __=m-H Z.WW-..-f =-t_ 1 —---- —' —_' ~.1 rad~l; ~I.:I I IS~ \'X0 tm \ l= ~. ~ -10,000 rmi. I~~~~~~ — t I t' frr t', / 1' 1-1X Elevator Response Figure 3 t f t f f f f f I f f f S f f f8

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ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN Scaling was such that the product U e was performed with high accuracy. Figure 5b shows the improvement. In practice small motion resolution could be improved by letting sin 8 = 8 +(sin - ). The term (sin 8 - O) would be generated by a function potentiometer and would be effective for large angle motions only. The linear term in 8 would provide high resolution for small angles. 5. B-47 Equations and Computer Realization To simulate B-47 longitudinal motion it was decided to use Link Aviations' simplified equations. In this system an additional attitude angle is defined. The angle, P, is measured in a vertical plane between the horizontal plane and flight path velocity Vp. Approximate equations are used to determine accelerations X+P +X U X c (14) G m. W z c (15). M+-TyM +M *' Q c Q 57.3 Y (16) YY Note that UG and WG do not include gravity terms. The angle of attitude is given as before e = Q. (17) The angle of attack for longitudinal motion is given simply by a =: - p. (18) Approximate equations yield p and the flight path velocity, Vp,. WG +g cos P P -57. 3 (19) V p Vp =UG- g sin p (20) * In this section all angles and their rates are measured in degrees 11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Rate of climb is simply s = V sinai (21) z p The force and moment equations- are taken directly from Link Aviation sources. X + Px + = q [-. 376 (a + 2. 7)2 - 26.85- 1428 fl (Ma)] +T (22) Y+ P + Y = - 109 q (a +2. 7) (23) y c M +T + M = q [-643 a - 466 6SE 61000 ( Q )+109 (a + 2. 7) d V p (24) + 10, 000 (Ma. 6) + 210] +1. 77 T The computer diagram is shown in Figure 6. The statements of Section 3 regarding symbols, scaling, etc. apply here. Coefficient potentiometer settings are given in Table 2. 6. Results of B-47 Simulation Computer runs were made under two sets of conditions: altitude - 20, 000 ft., Ma =. 7, gross weight - 192, 000 pounds; altitude - 40, 000 ft., Ma =.8, gross weight - 112, 000 pounds. The tests and conditions of Section 4 were duplicated. The pitch rate Q and rate of climb were recorded. No attempt was made to determine the errors introduced by the approximate Link equations. It seems likely that the simplification is justifiable when aircraft motion is restricted to near level flight. Figures 7 and 8 indicate that excellent simulation was again achieved. Characteristics were similar to those of the F-86D simulation and need no furthe elaboration. Figure 9 compares a response made- during this study with one obtained from the Link training simulator. Considering the somewhat different flight conditions agreement is good. There is a noteworthy(about one second) lag in Link response. This might result from the mechanical devices used in multiplication and integration. 7. Conclusions The longitudinal equations of motion, the most difficult to simulate accuirately, were solved with all important non-linearities on the dc electronic analogue computer. Results obtained were far superior to those of any present ac trainer type simulation. Outstanding characteristics of both the F-86D and 12

40 I VP VP 10 4a- 4 (a +2.7),~ -4(a+2.7) H I 0 4 n~~~~~~~~~~~~~~ IIN ~ ~~~~~ 10 __ ~~~z V x ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I O ~ I 16s ( Q + 2. 7 )2. i. " —— l-' ~~m I O ~~~~~~~~~~~~~~~IN L0~~~Lr wG + 9 Cos B. a -A Vp 3!'* EvV SIN B, RATE OF CLIMB I-L IN M+Ty4Mc~~~~~~~~~~~~~~~~~~~~~~~~~~~.A I I 61 6 oo FT/M 2, A I S7 4-. 10 B N SIN I r~~~WG+9 COS B ~Ig BI IN SIN B VP c La - 0 r H0 m H -.40 10 10 VP +100 2 LDG +g GOS B SP q Cos B 10 L I 0 I 0~~~~~~ to 4 -9.4Q x~~~~~~~~~~~~~~~~~~~~~xlo4 10 VP IIN -3M +Ty-4- Mc n T x 10o A 4 40 I0 +0z Figure 6 L 4 a+ -.7r 10 A 10 4 (a.+ 2.7).16 W +2.f7) x+pP + Xc + 100 A 4103 2 L L Z L~~~~~~~L-~ ~ ~ ~ ~ ~~4 i O 4= 0'd3 2 VOLTS I DEGREE TxII 100 Figure 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN TABLE 2 Pot. No. Pot. Setting 1.802 2 5 (M.6)2 a 233 6SE 4.105.760 6 -.088 7.730 8.586 9 Txl105 10 100/m =.017 (192, 000 lbs),.029 (112, 000 lbs) 11.119 (Ma =.7),.162 (Ma =.8) 12 1.07x103/4m=.445 (192, 000 lbs),.763 (112, 000 lbs) 13. 630 (20, 000 ft),.283 (40, 000 ft) 15. 287 16.032 17.573 18.108 19.500 14

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m Z rl,. =.I 40,000 FT -- ~'~..'~.~ ~~~GROSS WEIGHT - 112,0'00 POUNDS,'LINK SIMULATION';'/~~~~~~~~~~~~~~~~~~~~~~~~~~~r.MO,=-.8, h =3-5,:000 FT'_GROSS ~WEIGHT — 140,000 PONDS'! 0 24 6 O 10 1.2 14 16 18 20 S DSECONDS -4 00'0 8SE 0 2 4 6 8 t 0 12 14 16 A8 2 0 r SECONDS ~ =M Fig. 9. Pitch Rate Response to Impulse of Elevator

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN B-47 simulation were: 1. -accuracy of steady-state solution; 2. ease of trimming to level flight or constant rate of climb; 3. duplication of short term dynamic response; 4. duplication of long term phugoid oscillations of small magnitude (something not achieved in any present ac simulation); 5. good resolution even for smallest motions encountered in normal flight. The study also indicates that high resolution one-turn servomultiplying potentiometers are adequate for the bulk of operationrs. This would allow use of simply, easy to maintain, high speed servos. Several critical trigonometric resolutions might require careful treatment to obtain high accuracy in small-motion simulation. Finally, it should be pointed out that these results were obtained without component trimming, lead relocation or shielding, phase shifting, or other critical adjustments often required of large ac computers. UNIVERSITY OF MICHIGAN II/lllllll llslslBlRlllll l 8!l /IIl7l11 1 3 9015 02826 1082 * - 1R - - - -.~~~~~~~~-I