1、. - ye, .-,: “p-,:“”-j”rouy. . . . . -l)onefeim-Pmrwm- . 222222bl12humberofrkta pointspmfuncton.:241Contmcy*fermctmm,m-.-.- .L, -0.5Et:sTlmore-ru$ridntmrequencywsponse,hruResultswereobtainedinform continuouswith frequencybut muncrierdevaluationwasI1ldoat numberOffWpCIICICeS11OWILMctbodis printardyr.
2、fricwblto impulseorstepinputs;thcrcfocc,umdysiso!the inputfunctionis not required.FOUEIf2JfANALYSIS OF TRANSIENT RESPOIW4Ehdhw well-known method of determining tIw frequencywsponw is to determine the eoeffkients of the Fouriertransform of the. input and output fUIKtiOIISover a frequencyrange by zmdy
3、zing the response (m a function of time) ofthe uircmft to an tirbitrary input The process is indictit.eclby the expressionJlag (jw)= “mqot)ejtdts (4) ql(t)e-dtwhich represents the M t io of the Fourier int cgrtd of the outp-ut to the Fourier integral of the input. The dc.riwttion andProvided by IHSN
4、ot for ResaleNo reproduction or networking permitted without license from IHS-,-,-APPLICATION OF SEVERAL METHODS FOR DETERMINING TRANSFER FUNCTIONS AND FREQUENCY RESPONSE 5used as well M the value of the hnsient q(t) at this point,The coefficients thus determined give an equation that maybe used to
5、check the fit of the transient by the expressionbefore further work is initiated.This approach to the evaluation of the Fourier integralmay be expressed analytically as follows:n-wQ()= ,2, Q?2(4=K()-jK2() (6)whereJ(Jn()= t”+(a#3+bJ2+cJ+dJ e-” dtnandn “L.1lJ tf?e-i.tt= (6+6jtu3tu2jt3u3) e-” +71 nJb t
6、:+“ k+lt-e-iufdt=j (2+2jtwt2a2) e-iiu n nJtn+l L+lc, t“ t -80 . -. - -0 4 8 12 16 20Frequency, W, radians/seeFIf.iLiRltI tl.-Comparkon of frequency-response curves of the fighterat 2 .6- .Triangular pulse Time, t,sec Time, t, secgcre the pitching velocity appeared torwwh a stetidy state before the e
7、levator was again disturbed.Time inc.rwnents of 0.05 seconcl were wed to obtain thefrequency response of this portion of tk time histories bythe Prory, Donegmi-Pewson, and manual Fourier methock.(In this umdysis, the Prony method requirocl 16 man-hours;the Donegan-Pearson method required 9 man-hours
8、.) Inwldit,ion ? check points were oh tained by the manual Fouriermethod (b? using Simpsons hwe-point- rde of intgra.tion)with time mtwvals of 0.10, 0.025, and 0.0125 seconcl. Thefrqlelcy-respollse -results of this analysis are- shown infigure 14 and, although all the met.hocs closely agree, thefreq
9、ue.nc.y response a.ppems to be quite cliflerent from thatindicat ecl in figure 6 which WM ob tainecl for the swne airphme-at, the smnc flight conditions.The source of this discrepunry was determined when thetime histories shown in figure 12 were again wmlyzc.d by usingthe Donegun-Pea.rson method at
10、0.05 -scrond intervtds to atime of 1.40 seconcls where the response still had not. ap-proached a- stwdy state too C1OSCJY,Lrut the length of therecord used had been doubkl and k effective amplitudehacl been more than doubled. .4 Lhird analysis was madeby using the Donegnn-Pemson methocl at 0,10-seco
11、nd inter-vals to 2.IO seconds at which time its st,eudy-state. value wasclosely attuiuecl. The frequency- response obtained b-y usingetich of the three record lengths is shown in figure 15 togetherwith the frequency response oht,aimxl for the fighter (by the,j+J 0:, Method At sec Ptony- Doneqon-Peor
12、son-%4-1-!- -i-=-=- - -.- -“Frequency,w, rodionsjsecFIGURE 14.FrecIuency response of Ihc figLlw rchit ing pil rhilwvelocity to elcwt or defieut icm aiJ .31=0.0 imd 1,-=10,000 fwl :isdetermined from first one-third of rwl :Iugi W-pldsu-inpl Il. t illu,history.Donegan-Pwwm method) from iguw 6. It nm.v
13、 IN swnthat, when the first one-third of its response wls tmtljxl?ll,the record was short, and a steady stutc 1A not l.wnreached; these fuct.ors preclu(ltd tin adequately prwiw lh+ini-tion of the time response and an crroncws frcqurnly rwponwwas obttiined. When the hmgth of thrword was LNLIM,a more
14、correct trend bewne tippwnl, but, bmwuse IL St CWJ”stmte had still not be-en defined, sonw fuirly large disrr(pnwil+persisted, part,ic.ularly with regard to the static WLIW of thufrequency response (the frecllell?:y-respt311si!vurvw of figllw f;being used as a basis for comparison). Whm t hc iuhisis
15、inchxlecl the entire response, even t lougtl the t inw inl wvulused in the analysis was doubkd, a ek agrmmwnt withthe freqnem!y response ob ttiined from the sitUP inpu, wwobtained. Reference 17 rec.onmwnds that enough )f t IUIresponse time history should be t tdwn to rwver t hr ntlt umlperiocl of th
16、e system.,.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Application OF SEVERAL METHODS FOR DETERMINING TRANSFER FUNCTIONS AND FREQUENCY RESPONSE 15I Ii m.1 . I I I I I III I I Io-20Eo I I I N I Ill IllI l x I I I I I I-otil h% I I I I I I . -%. -8
17、0 - . -1000 4 8 12 1 20Frequency, U, radians/seeFIGURE 15.Frequency response of the fighter relating pitchingvelocity to elevator deflection at M= 0.6 and hn= 10,000 feet asdetermined from portions of rectangular-pulse- ancl step-inputtime histories by using the Donegan-Pearson method.OTHER CAUSES A
18、ND EFFECTS OF ERRORSIn Lhe dctmnination of transfer functions from inputs andou tiputs having regions of low harmonic. cent t!nt an a.cvan-tage has been indicated to the approach of fitting an analytic-al expression to experimental data. In the authorsopinion, this curve-fitting technique, as compar
19、ecl to theFourier analysis, is of partic.uhw merit if hhe.re is reasonableconfidence that the assumed analytical expression is of thecorrect form for the system being analyzed. In this mtmner,another condition (the form of the transfer function) isst,ip ulated which the antil,vsis must obey. In math
20、cmmticalprocesses, the more concliions correctly stipulated, the momprecise the results, On the other hand, errors in the transferfunction or frequency response as obtained from the curve-fitting methods, due to either the. wrong assumption of thoform of the transfer function or to the errors in the
21、 cwkula-tions, are not readily a.ppment since the assumption of agiven form will usually give variaions that appear logical.However, as has been pointed out, mrtain checks, such asthe use of the inverse Laplace transformation, are. mmilablefor comparing the time response predicted from the transferf
22、unctions with the time-response curves from which thetransfer functions were derived.In the Fourier methods, inaccuracies arc, in general, momreadily discernible than in curve-fit Ling mcthock. In tht!use of Fourier methods, there has been found evidence ofdiscrepancies attributable to three muses (
23、as pointed out inref. 10): the lack of harmonic content of the Fourier integrcd,the usc of too largi” tille intervals in the time clomain toafford accuracy in the frequency domain, and the incorrectsynchronization of input and response data in the timeclomain.The first of these errors hm already bee
24、n discussed in theconsiclemtion of the effect of input on Fourier methods ancl,as has been pointed out, is usually discernible by divergenceof the curves in some smzdl range of frequencies.The second of these errors, that of too large time intervals,is generally indicated by a scattering of the data
25、 points inthe frequency domain where the magnitude of scatter usuallydiverges rapidly with increasing frequency. Insight into thecause of this scatter may be seen in the characteristics of theFourier transform where, at each frequency, the transientq(t) is multiplied by a sine and cosine wave of uni
26、t amplitudeand whero the resulting area under the two procluct, curvesdetermine tho coefic.ients of the real ancl imaginary parts ofthe complex variable in the frequency clomain. As fre-quencies greater than the naturul freqmmc.y are investigated,the differences in the positie and negative areas of
27、theproduc L curves grow smaller (compared with the magnit ucleof the individual me.as) so that. the eficwt of small errors ismagnifiecl. Thus, small inaccuracies in the representationof the trausient curve become more prominent as higherfrequencies arc investigated ancl appear in the frequencydomain
28、 as scatter. Several estimates of the frequency atwhich scatter will become important, for the clifferent Fouriermethods, based on the time interval chosen, have bee-n givenin the section entitled “Descripticm ancl Discussion ofJlethocls.”A typical occurrence of scatter due to the choicl? of toolarg
29、e a time interval was ob ta.inwl when the response of afree-fall model, the c.haracterist ics of which are given intable I, was analyzed at O.10-second intervals by the manualFourier (numerical-integration) method. The ckator inputused and the response of the model in angle of attack areshown in fig
30、ure 16. The frecluency resptmse as determinedby the numerical manual Fourier, the Coradi harmonic.analyzer, and the exponential-approximation methods ofanalysis are shown in figure 17. The scatter of pointsobtained by the manual Fourier method of numericalintegration occurs at frequencies greater th
31、an about 8 radiansper seconcl. Further analysis with snmller time intervalsof, say, 0.05 second should provide better results in thisregion, ,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 REPORT 1204NA!tTOhAL ADVISORY COJMMITTE13 FOR AEIK)N.4WWS
32、I 1 I I I3 I I I 11 I Measuredg o Computed (Donegan-Peorson method)u Commzted (exponential-aaaromotion method)c1 LJiiiiiiiiiiiiilo .4 .8 1.2 1.6 2.0 2.4 2.8Time,t, secFIGLRE lraphicd find rchmlictd (tJmpIIl :It itJti, .l,IIIIIIYiky rwn lkmiw Flight J M w.NACA Rep. 1070, 1!).52, (Supmds S ACA TX 237O
33、.I18. Diederich, Franklin IV.: Cakulatim of 1INAcrudytmnic .uaditlgof Swept aml lTnswept I%!xiI)lv Vings of Arhit rury ,Stifflw.AACA Rep. 1000, 1!)50. (Supmcvbs NA( !A TX ltJ7li,!, “ -,.? ”. ,”.,. - . . . . . . , .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
