1、NASA Technical Memorandum 78678Airplane Stability CalculationsWith a Card ProgrammablePocket CalculatorWindsor L. ShermanAUGUST 1978NASAProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ERRATANASA Technical Memorandum 78678AIRPLANE STABILITY CALCULATIO
2、NS WITH A CARDPROGRAMMABLE POCKET CALCULATORWindsor L. ShermanAugust 1978JPlease make the following corrections:Page 15: Sentence after equation (11) should read as follows:Equations (9) and (10) were programmed for the calculator and theprogram is given in appendix B.Page 16: Equation (16) should r
3、ead as follows:b2Re(y) = S + TPage 24, Last sentence: Change step 49 to step 45.Page 25: Step 100 should read as follows:STOx9 (gT/2Uss) sin 2yssPage 26, Step 105: Change - to +Step 141: Change RCL8 to RCLBPage 29: Delete the last sentence.Page 49: In column headed “Output,“ change the values of 33,
4、 82, a-|,a0, and a2 toa3 = 1.3980958a2 = 1.1093007a-i = -0.0098076a0 = -0.0211448a12 = 0.0373094ISSUED NOVEMBER 1978Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ERRATANASA Technical Memorandum 78737DEVELOPMENT OF A NONLINEAR SWITCHINGFUNCTION AND
5、ITS APPLICATION TOSTATIC LIFT CHARACTERISTICSOF STRAIGHT WINGSDonald E. HewesSeptember 1978Page 5: Equation (3) should read/in e 1/2 / 1 V/2ISSUED NOVEMBER 1978Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA Technical Memorandum 78678Airplane St
6、ability CalculationsWith a Card ProgrammablePocket CalculatorWindsor L. ShermanLangley Research CenterHampton, VirginiaNASANational Aeronauticsand Space AdministrationScientific and TechnicalInformation Office1978Provided by IHSNot for ResaleNo reproduction or networking permitted without license fr
7、om IHS-,-,-SUMMARYPrograms are presented for calculating airplane stability characteristicswith a card programmable pocket calculator. These calculations include eigen-values of the characteristic equations of lateral and longitudinal motion aswell as stability parameters such as the time to damp to
8、 one-half amplitude orthe damping ratio. The effects of wind shear are included. Background infor-mation and the equations programmed are given. The programs are written for theInternational System of Units, the dimensional form for the stability deriva-tives, and stability axes. In addition to the
9、programs for stability calcula-tions, an unusual and short program is included for the Euler transformation ofcoordinates used in airplane motions. The programs have been written for aHewlett Packard HP-67 calculator. However, the use of this calculator does notconstitute an endorsement of the produ
10、ct by the National Aeronautics and SpaceAdministration.INTRODUCTIONOver the past several years, the programmable pocket calculator has devel-oped into a highly sophisticated device that has almost computer characteristics.Because of its sophistication, the newer models are capable of being programme
11、dto make very complicated calculations. Since different logics are used in pro-grammable calculators and since the available keyboard instructions vary withmodels of different manufacturers, it is necessary to identify the make andmodel of the calculator for which a program is written. The airplane
12、stabilityprograms presented in this paper were written for a Hewlett Packard HP-67 cardprogrammable calculator; however, its use and identification in this report doesnot constitute an endorsement of the product by the National Aeronautics andSpace Administration.Programs are given for the calculati
13、on of the coefficients of the airplanelateral and longitudinal characteristic equations, the eigenvalues, and thestability parameters such as the time to damp to one-half amplitude or thedamping ratio. In addition, a unique coordinate transformation program isgiven for transformations between inerti
14、al axes and airplane body axes. Thisprogram requires very few program steps and may be useful as part of a largerprogram. The equations on which the programs are based are given so that theprograms can be readily adapted to other calculators that have sufficient pro-gram capacity.The programs presen
15、ted herein evolved during the study of wind shear andits effect on airplane stability and control. These programs proved usefulin making stability calculations in this study and should be of use in otherinvestigations.Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
16、se from IHS-,-,-SYMBOLSA aspect ratioag,a-|,. . .,35 coefficients of characteristic equationsal2al3fa!4f elements of longitudinal stability determinantb wing spanb2,b,b0 coefficients of resolvent cubicc?drag coefficient,PSu|s/2CD,O drag coefficient for CL = 0Llift coefficient,CL o lift coefficient a
17、t zero angle of attackrolling-moment coefficient,MXPSbu|s/29CZ9PProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-clocnr3B34)Mypitch ing -moment coefficient,po total pitching-moment coefficient at zero angle of attackMZCn yawing-moment coefficient,8r83
18、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-thrust coefficient-YCYP 33CYside-force coefficient,3CxPY3C3Cclltc21/C30 I terms in lateral stability determinantb11 bi2,b13,. . .jc mean aerodynamic chordD dragFIJ thrustFT tr trim thrust3FT3uFX,FY,FZ f
19、orces along X, Y, and Z stability axis3Fx36eProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-3FY3FyX = 37;3FZFZ(5e = 36g acceleration of gravityIx,Iy,Iz moments of inertia, stability axesIxz product of inertia, stability axesIm( ) imaginary part of co
20、mplex rootNkX/kY I radii of gyration, stability axesL liftMX,MY,MZ moments about X, Y, and z stability axes3MXMx* =a 3 airplane yaw (heading), pitch, and roll angles, respectivelywn undamped circular frequencyDot over a symbol indicates differentiation with respect to time.EQUATIONS PROGRAMMED AND P
21、ROGRAM DESCRIPTIONSSix programs are presented in this paper. The first three calculate theelements of the lateral and longitudinal stability determinants and the coeffi-cients of the characteristic equations. In addition, program 3 extracts a realroot of a fifth-order polynomial when required. Progr
22、ams 4 and 5 complete theroot extraction process and calculate the stability parameters. Program 6implements the Euler angle transformation by using the polar-rectangular keysfound on calculators.Programs 1, 2, and 3 are written for the International System of Units,stability axes (fig. 1), and the d
23、imensional form of the stability derivatives.The equations programmed are the linearized form of the equations of motionderived in appendix A of reference 1; thus, the effects of wind shear areincluded.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
24、In deriving these equations, head winds and updrafts were taken as negative.Thus, a positive uw will change a head wind into a tail wind, and a positivew,w will change an updraft into a downdraft. The signs of and w,w setthe signs of a. and is a gradient with altitude and ww is agradient along the f
25、light path.In writing the programs, the following conventions were used for the labels:(1) Capital letters (A to E) are program labels(2) Lower-case letters (a to e) are subroutine labels(3) Numbers (0 to 9) are used for all other labelsTable I summarizes the programs presented in this paper. The ke
26、y entriesgiven in appendixes A to F are the standard HP-67 key entries given in theowners manual. Check cases for all programs are given in appendix G.TABLE I.- SUMMARY OF PROGRAMSProgram Description Key entriesgiven inCalculates the elements of longitudinal stability determi-nant and normalized coe
27、fficients for characteristicequationAppendix ACalculates the elements of lateral stability determinantand starts calculating coefficients of the characteristicequationAppendix BLabel A completes calculating coefficients of characteris-tic equations of lateral motion; label B calculates areal root of
28、 a fifth-order polynomial and reduces thefifth-order polynomial to a fourth-order one; t-| /2 ortD for the real root determined; label B can be used asa stand-alone programAppendix CUses Ferraris method to calculate the roots of a fourth-order polynomial and can be used as a stand-alone pro-gram; wi
29、ll also determine roots of cubic, quadratic, andfirst-order equationsAppendix DCalculates stability parameters such as t-|and NTAppendix EUses the polar-rectangular transformations of the calcula-tor to implement the Euler transformation between spaceand body axes or body and space axes; this method
30、 savesabout 57 program steps when compared with the more usualmethods of programmingAppendix FProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Programs 1 and 2 give solutions from an equilibrium flight condition.There are six parameters, Uss, YSs atr
31、FT tr aT and w that must beadjusted correctly to obtain the equilibrium flight condition. There are twoequations to accomplish this adjustment. Programs 1 and 2 were set up in thefollowing manner. The parameters Uss, YSSf T and w are specified by theuser. The program calculates ot, assuming that FT
32、r is 0. For the flightcondition Uss = 77.12 m/sec, Y ss = -0.05236 rad, OT = 2.0, and aw = o.O, theerror introduced in atr by this method is 0.00081 rad, which is consideredacceptable. If it is desired to monitor the calculated value of ar, insert apause after step 45 of program 1.Program 1sThe line
33、arized equation of longitudinal motion is in symbolic formddt314a21a22 + a23atd-a31a32 + a33dtThe characteristic equation for longitudinal stability is obtained fromthe determinant of the 3x3 matrix and has the forma4 + a3 I + a2 + al + a0 = 0 (2)where to 34 are given bya4 = a22 - a32a3 = a33 a25a32
34、 + a22a35)(3a)(3b)a2 = (a23a35 a25a33 a26a32)+ ai3(a31 - a21) a33 a25a32 + a22a35(3c)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-al = a12 1* =6 6r(6)and are based on equations (A5) , (A8) , and (A9) of reference 1 . In lineariz-ing these equation
35、s, it was assumed that no wind gradient existed in theYe derivative in Earth axes. If the wind gradients are zero (i.e., no windshear) , these equations reduce to the standard form of the linearized equa-tions of lateral motion that are given in many standard works, such as refer-ence 2. The equatio
36、ns are valid in the interval -0.17453 S yss S 0.17453.The characteristic equation is obtained from theleft-hand side of equation (6) and has the form33 matrix on thea5 + a4 + a3 + a2 + al + an = 04 J U (7)for 0.When a15b34 - b16b33 - bi7b32) + C3o(b15b24 - b17b23) - b22(b35 + b14b34 b43b33 - bi6b32)
37、 + b31 (b15 + b-|4b24 - b16b23 - b17b42) (9d)- b24b33) - b22(b14b35 + bj 5b34 - b16b33 - b17b32)- C21(bi5b35 - b17b33) + b31(b15b24 - b17b23) (9e)a0 = -b22(b15b35 - bi7b33) (9f)The C and b terms that are used to generate the coefficients of the charac-teristic equation are given byb = 0.0 (10a)9w gb
38、13 = - - co yss (TOc)“ Usssb!4 = -CZpk6OOe)= cnpk7= uw-Cnrk7J = Cn()k7 (lOg)= -CY k5 (lOh)13Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-b3i =(101)(10m)(10n)(10o)b42 =b43 = (10s)+ b12 Hot)+ b2iC30 = 1 + b30 (10v)PSUSSwhere kc = - , kc =2m b 2IX, a
39、nd 2I . The trim angle of attackProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-was calculated fromatr = / =-(aT sin2 yss - aw + cos ss) CL,O/ T,1 (11)PSUgSEquations (9), (10), and (11) were programmed for the calculator and the programis given in ap
40、pendix B. The stability derivatives Cg, cn an 0) (16)orwhereRe(y) = 2(R2 + f)V3 cos b2 (f S 0) (17)Q = (3b, - b22)9R = (9b2b-| - 27b0 - 2b23)/ 54f = R2 + Q3(18a)(18b)(18c)16Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-s = (R + /f)V3 (18d)T = (R -
41、/f)V3 (18e)The root Re(y) is any root of the resolvent cubic, equation (14); this pro-gram is written to calculate the largest real root of equation (14). OnceRe(y) is known, the roots of the quartic are obtained by solving the followingtwo quadratic equations:z2 + (A + C)z + (B + D) = 0z2 + (A - C)
42、 z + (3-D) =0(19)wherea3A =B = Re(y)D = JB2 - a0C = Av2 - a2 + Re(y)(D jt 0)(D = 0)(20)Equations (15) to (20) and a quadratic solution routine were programmed toobtain the roots of a quartic equation. The key codes for program 4 are givenin appendix D.Because f and D are tested to determine program
43、direction, special pro-gramming is required both to insure that nonsignificant digits do not influencethe test and to protect against the small difference of large numbers. Theexpressions for f and D were written asf -R2 1 +17Provided by IHSNot for ResaleNo reproduction or networking permitted witho
44、ut license from IHS-,-,-D =for programming. In each case, the quantity in the parenthesis was rounded tothe calculator display and then tested. Special routines were added to protectagainst R and B being equal to 0. The introduction of rounding will intro-duce some error if a significant number is t
45、runcated. As the rounding is con-trolled by the number of decimal digits in the calculator display, there isflexibility in the amount of rounding introduced. Experience with a set of20 test equations indicates that a display of 7 digits is satisfactory for mostcases.The roots of the quartic are stor
46、ed in registers R-j , R2, S-| , and 82- Theroot indicator (-1.0 for complex roots and 0.0 for real roots) is stored inregisters RQ and SQ. If the roots are complex, the real part is stored inregister 1 and the imaginary part in register 2.This program is a general program for the roots of a quartic
47、equation andmay be used as a stand-alone program if the coefficients of the quartic arestored in the following locations:33 in register RQ32 in register R-a-j in register R230 in register R3In addition, this program may be used to solve for the roots of lower orderequations. For the cubic where the
48、equation has the form33x3 + 32X + a-X + a0 = 0, 33 = 1.0the equation is multiplied by x so that it is converted to a quartic with azero root and the coefficients are stored as follows:32 in register RQ3-) in register R-3g in ,register R20.0 in register R3Quadratic and first-order equations may be solved in a similar manner bymultiplying through by x2 or x, respect
