Mathcad Professional 14.0 Orni 2, Rechenverfahren für Ornithopter Rechenprogramm zur Beschreibung der Aerodynamik und Dynamik von profilierten Schlagflügeln unter quasistationären Bedingungen. Horst Räbiger Ornithopter, Schlagflügel, Rechenprogramm, Rechenmodell, Orni, quasistationären Bedingungen, Kräftegleichgewicht Horst Räbiger 2 E08E1199-E3C3-43FC-9AD0-AC7A0997AA49 8D5B2E37-7280-4956-903E-A2B7BD8EFFB4 00000000-0000-0000-0000-000000000000 00000000-0000-0000-0000-000000000000
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Orni 2

Rechenverfahren für den Schlagflug

Das Programm ermöglicht zumindest näherungsweise eine quantitative Beschreibung der Dynamik und Aerodynamik von profilierten Schlagflügeln. Vor allem der zahlenmäßige Vergleich bei Veränderungen diverser Schlagflügel-Einflussgrößen ist damit möglich.

Für die Berechnung werden profilierte Schlagflügel und quasistationäre Strömungsbedingungen vorausgesetzt. Die Berechnungen führen also nur beim schnellen Vorwärtsflug mit relativ kleinen Schlagfrequenzen zu brauchbaren Ergebnissen (große Vögel, Fliegen mit dem Auftrieb).

Zum zum besseren Verständnis der nachstehenden Ausführungen sollten die Webseite <http//:www.ornithopter.de> und derInhalt des dort bereit gestellten Handbuches

" Wie Ornithopter fliegen " bekannt sein.

Das Rechenverfahren konnte bisher bei der Anwendung im Modellbau nur per Augenschein, aber noch nicht durch Messungen bestätigt werden. Daher ist Vorsicht geboten. Der Autor übernimmte keinerlei Gewähr für die Richtigkeit und Vollständigkeit der Berechnung und der gemachten Angaben.

- Das Programm ist mit der Software "Mathcad 13" geschrieben.

- Der Schutz des Arbeitsblattes kann mit Mathcad nach dem Speichern im xmcd- Format
entfernt werden (ohne Passwort).

- Im Normalfall sind nur die nachfolgend gelb markierten Felder zur Eingabe erforderlich.

.

Das hier aufgeführte Rechenprogramm ist als Open-Source-Software zu verstehen. Damit ist es jedem möglich, Einblick in das Rechenverfahren zu nehmen. Jeder hat die Erlaubnis diesen Quellcode beliebig weiter zu geben, zu verändern und vor allem zu korrigieren und zu verbessern.

Inhalt

1. Paramter für das Rechenmodell ..................................3

2. Gleitflug .........................................................................5

2.1 Modellabmessungen......................................................5

2.2 Gleitflug- Konstanten .......................................................6

2.3 Flügelortdaten...............................................................6

2.3.1 Integrationsverfahren ..................................................6

2.4 Flügekräfte....................................................................9

2.5 Örtliche Profilwerte.........................................................12

2.5.1 Drehmoment ................................................................13

2.6 Gleitflug- Ergebnis..........................................................14

3. Kraftflug .........................................................................15

3.1Kraftflug allgemein........................................................15

3.1.1 Quasistationäre Strömungsbedingungen .....................15

3.1.2 Rechenstützstellen ......................................................16

3.1.3 Integrationsverfahren.................................................17

3.1.4 Zirkulationsfaktor........................................................18

3.2Kraftflug-Konstanten.....................................................18

3.3 Phasenwerte zum Betrachtungszeitpunkt.......................18

3.3.1 Zirkulationskennzahlen allgemein...............................21

3.3.2 Zirkulationskennzahl in Aufschlagmitte........................21

3.3.3 Zirkulationskennzahl in Abschlagmitte.........................23

3.3.4 Zirkulationskennzahlen für den Horizontalflug ..............23

3.3.5 Zirkulation im Phasenverlauf.......................................24

3.3.6 Querkraftverteilung.....................................................25

3.3.7 Örtliche Profilwerte......................................................26

3.3.8 Drehmoment...............................................................29

3.3.9 Kräfte am Schlagflügel................................................31

3.3.10 Schlagmomente .........................................................36

3.3.11 Kurbelantrieb .............................................................38

3.3.12 Schlagleistungen......................................................38

3.4Flugdaten....................................................................40

3.5Kräftegleichgewicht .......................................................42

3.5.1 Kräftegleichgewicht in z-Richtung ................................42

3.5.2 Kräftegleichgewicht in x-Richtung ................................42

3.5.3 Kraftflug-Gleichgewichtsuche.....................................44

4. Gleitflugprogramm ........................................................47

5. Fehlerbetrachtung ........................................................50

Eingangsparamter für das Rechenmodell

Die Größe der Eingabewerte wird nicht kontrolliert. Sie stehen untereinander in Beziehung und sollten nach modellbautechnischen Gesetzmäßigkeiten plausibel sein. Bei nicht so gebräuchlichen Größen werden zur Information hinter den Wertzuweisungen Wertebereiche aufgeführt, die etwa einzuhalten sind.

Das Beispiel entspricht etwa den Werten eines EV -Ornithoptermodells.

Modell

Modellmasse

 m 4.0 kg

Modellrestwiderstand

 c 0.02

bezogen auf die

Flügelfläche

Schlagflügel

Spannweite

 b 2.8 m

Streckung

 Λ 10

relativer Abstand des Flügel-

umrissknicks von der Flügelwurzel,

bezogen auf die Halbspannweite

 y 0.8

Werte 0 bis 1

relative Flügeltiefe an der

Flügelspitze, bezogen auf die Flügelwurzeltiefe

 l 0.7

Werte 0 bis 1

Bei rechteckigem

Umriss (yk = 1)

ist ls wirkungslos

relative Flügelmasse

bezogen auf die Modellmasse

 m 0.2

etwa 1/5 der

Modellmasse

relativer Flügelschwerpunkt-

abstandvom Schlaglager,

bezogen auf die Halbspannweite

 r 0.44

Abstand des Flügelholms vor

dem 1/4-Punkt des Flügelholms

 H 22 mm

Gleitflug

mittlerer Auftriebsbeiwert

c 0.65

Zirkulationskennzahl

(G = griech. Gamma)

 c 8.00

Werte7 bis 8

8 für elliptische Auftriebsverteilung

Kraftflug

Fluggeschwindigkeitsfaktor *)

Kraftflug/Gleitflug

 k 1.119935

Werte0.9 bis 1.2

Modell-Steiggeschwindigkeit *)

 v 0.452812 m s

Werte0 bis +1

*)Diese Parameter werden nachstehend im Zuge einer Kraftflug-Gleichgewichtssuche auch automatisch bestimmt.

Die dabei ermittelten Werte sind dann im Bedarfsfalle hier einzusetzen.

minimal zulässige

Zirkulationskennzahl

beim Aufschlag

c 0

Werte 0 .. 5

-20 = ohne

Begrenzung

maximal zulässige

Zirkulationskennzahl

beim Abschlag

 c 10

Werte8 .. 10

10 = ohne

Begrenzung

Vorschlag für die

Flügelschlagfrequenz

(wie bei einem Vogel

mit diesem Gewicht)

 f e log 10 kg m 1 s f 1 s 1.4887547148558855

gewählte Flügelschlagfrequenz

(Sie wird insbesondere durch die

Flügelkonstruktion begrenzt.)

 f 1.50 1 s

Werte 1 bis 4

Schlag-Endlagenwinkel,

gemessen von Schlagmitte ±

φ 30 deg

Werte20 bis 45

Wirkungsgrad des Antriebs

(Leistungsverluste von Motor,

Getriebe, Mechanik und

Flügelverwindung)

 η 0.50

Akku-Energie

(Zellenzahl [8] x

Zellenspannung [1,0 V] x

Ladung [2Ah*3600 = 7200 As bzw. C])

 E 57600 C V

Physik

Viskosität der Luft

ν 0.00001464 m 2 s

Luftdichte

 ρ 1.225 kg m 3

Fallbeschleunigung

 g 9.81 m s 2

Rechnun g

Anzahl der Rechenstützpunkte längs der Halbspannweite

und während eines Taktes

 n 10

Werte etwa

10 bis 40,

nur geradzahlige,

ganze Werte

Profildaten

Es werden die Profildaten vom CLARK-Y (11.7) verwendet. Sie sind in einem eigenen Arbeitsblatt enthalten (siehe folgender Verweis).

Der Nullauftriebswinkel und der Auftriebsgradient der Profildaten sind auf den Rezahlbereich der EV-Modelle abgestimmt. Auch die Grenzwerte des ca-Arbeitsbereiches folgen aus der Flügelbauweise dieser Modelle.

Bei hohen Genauigkeitsanforderungen ist ihre Festlegung an die gewählten Abmessungen anzupassen.

 3d52573c2bce968b545b5da9a6790bee Profilprogramm für das Schlagflügel-Rechenverfahren "Orni"

Orni 2 und die Profildatei sind im gleichen Verzeichnis zu speichern. Andernfalls ist der Verweis zur Profildatei anzupassen.
Das Rechenprogramm funktioniert nur mit korrektem Dateiverweis.

Hinweis

Mathcad unterscheidet Namen nicht nur nach den verwendeten Zeichen, sondern auch nach der Schriftart. Die hier verwendete Schriftarten sind:

Schriftart Formatvorlage

______________________________________________________________

Variable Arial StandardVariable

Vektor Times New Romanfett Benutzer 1

Zahlenfeld Arial fett, kursivBenutzer 2

Variable "s" Roman kursiv Benutzer 3

Eine Matrix ist ein rechteckiges Zahlenfeld.

Ein Vektor ist eine Matrix mit nur einer Spalte.

Indizes für zeitliche Phasen oder Ort am Schlagflügel

Gim Gleitflug

Nin einem Zeitabschnitt (Takt oder Gleitflug)

Kim Kraftflug

0an der Flügelwurzel

2Gleitflug

2.1 Modellabmessungen

mittlere Flügeltiefe

l b Λ l m 0.27999999999999997

Flügelfläche

 A l 2 Λ A m 2 0.78399999999999981

spezifische Flächenbelastung

 p m A p kg m 2 5.1020408163265323

Modellgewichtskraft

 F m g F N 39.24

Flügelgewichtskraft

 F m m g F N 7.8480000000000008

Halbspannweite

 s b 2 s m 1.4

Um Kollisionen mit der Einheit s = sec zu vermeiden wird beim Schreiben der Halbspannweite s die Formatvorlage "Benutzer 3" verwendet.

2.2 Gleitflug-Konstanten

Fluggeschwindigkeit im Gleitflug

 v 2 F ρ c A v m s 11.212352507342905

Gesamtzirkulation im Gleitflug

 Γ F ρ v b Γ m 2 s 1.0203240781682044

Staudruck im Gleitflug

 q ρ v 2 2 q N m 2 77.001569858712728 F c q A F N 1.2073846153846153

Restwiderstand des Modells

mittlere Rezahl im Gleitflug

 Re v l ν Re 214443.90041366211

Auf "s" bezogener,

relativer Druckpunktabstand

von der Flügelwurzel

 y c 6 π y 0.42441318157838759 l l y 1 y l 1 2

Flügelwurzeltiefe

 l m 0.28865979381443296

Um die bei den Flügel-Eingangsparametern y k und l k beschriebene Bedingungen zu erzielen, gilt nun

l if y 1 l 1 l 0.7

2.3Flügelortdaten

Iterationswerte

Für die Zählung der Flügelorte längs der Halbspannweite wird die Variable "j" verwendet.

j 0 1 n

Alle mit diesem Index versehenen Werte sind Größen an einem Ort des Flügels längs der Halbspannweite.

Abstand von der Flügelwurzel

 y j j n s

2.3.1Integrationsverfahren

Zur Integration einer Größe längs des Flügels wird hier das numerische Rechenverfahren von Simpson angewendet. Die allgemeine Gleichung für die Integration, beispielsweise für eine Kraft an einer Flügelhälfte mit der Halbspannweite s, lautet dann

F s n 3 f 0 4 f 1 2 f 2 4 f 3 2 f 4 ... 4 f n 2 2 f n 1 f n

Bei diesem Rechenverfahren dürfen nur geradzahlige Werte von n zur Anwendung kommen.

Die einzelnen Funktionsgrößen f 1 , f 2 ... sind differentielle Größen, ohne Angabe der Flügelbreite für die sie gelten. Die Einheit dieser Größen ist also immer auch 1/m. Erst nach der Integration entfällt bei Mathcad diese Einheit. Dies ist bei der Betrachtung der Zwischenergebnisse an einem Flügelort zu beachten.

Die ganzzahligen Faktoren, mit denen beim Verfahren von Simpson die Größen am jeweiligen Flügelort multipliziert werden, sollen hier zentral beschrieben werden. Sie sind dann bei allen Integrationsmaßnahmen nach folgendem Schema anzuwenden.

F s n 3 c j F j

Bei Berechnungen längs der ganzen Spannweite ist anstelle der Halbspannweite s die Spannweite b einzusetzen.

Bei der Festlegung des Faktors c y wird der Umstand genutzt, dass Flügelorte mit geradzahligen Indizes den Faktor 4 und ungeradzahlige den Faktor 2 haben, ausgenommen erstes und letztes Glied).

Faktor für örtliche Integration

c j j 0 1 j n 1 floor j 2 j 2 2 4

oder in Form einer Funktion geschrieben

 f j1 n1 j1 0 1 j1 n1 1 floor j1 2 j1 2 2 4

deren Aufruf erfolgt mit

 c j f j n

Flügeltiefe

Bei dem früher von mir beim EV6 verwendeten Gleichungsystem von Albert Betz wird für jede Zirkulationsverteilungsform mit dem Verteilungsparameter a 1 auch die dazu gehörige Flügeltiefenverteilung mit dem Verteilungsparamter a 1 mit angegeben.

liter 0 l 4 π 1 a 4 a 2 3 1 Λ liter j l 1 j n 2 1 a j n 2 1 a 4 a 2 3 1 4 j n 2 1 Λ 4 π

Abb. 2.3-1Flügeltiefenverteilung nach A. Betz für etwa rechteckigen

Flügelumriß (Verteilungsparameter a 1 = 0.5)

Dagegen fehlen bei der Zirkulationsbeschreibung von R. T. Jones Angaben über die dazugehörige Flügeltiefenverteilung. Es wäre allerdings selbst bei vorhandenen Angaben schwierig (wie bei A. Betz), einen schlüssigen Kompromiss zwischen den unterschiedlichen theoretischen Flügeltiefenverteilungen vom Auf- und vom Abschlag zu finden. Beide Takte arbeiten ja mit unterschiedlichen Zirkulationsverteilungen und demzufolge mit entsprechend unterschiedlichen, theoretischen Flügeltiefenverteilungen.

Das Verfahren von A. Betz wurde früher insbesondere für manntragende Flugzeuge mit starren Tragflügeln verwendet. Die Fehlerabweichungen durch die dann aber praktisch angewendeten, eckigen Flügelumrisse hat man dabei inkauf genommen. Das soll hier auch für Flugmodelle und dem Verfahren von R. T. Jones so gehandhabt werden. Hier wird der eckige Flügelumriss sogar schon bei der theoretischen Berechnung eingesetzt.

 liter j y j y s l y j y s l y j y s l l l 1 y s

In Funktionsschreibweise

 f jx nx y jx nx y y liter jx l liter jx l 1 1 l y y 1 y liter jx

Ihr Aufruf erfolgt mit

 liter j f j n j 0 1 2 3 4 5 6 7 8 9 10 liter 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.28865979381443296 0.245360824742268 0.20206185567010307

Trägheitsmoment des Schlagflügels

Das Trägheitsmoment des Schlagflügels wird an Hand des Flügelflächenmomentes ermittelt. Es wird also so getan, als ob das spezifische Gewicht pro Flächeneinheit in allen Flügelbereichen gleich groß ist.

In der Praxis wird man natürlich versuchen den Flügel an der Flügelspitze deutlich leichter zu machen als an der Flügelwurzel. Auf diese Weise wird das praktisch erzielte Trägheitsmoment kleiner sein als hier angegeben. Die hier gewählte Berechnungsmethode führt also nur zu einem ersten Näherungswert.

Mit Hilfe der Flügeltiefenverteilung kann man für das Trägheitsmoment J F schreiben

 J j y j 2 liter j A m m

Durch Integration über die Spannweite erhält man

 J b n 3 c J J kg m 2 0.49647945017182138

Dieser Wert gilt für beide Flügelhälften zusammen.

2.4 Flügelkräfte

Zirkulationsverteilung

In der klassischen Tragflügeltheorie gibt es zwei Möglichkeiten sich der Aerodynamik eines Tragflügels zu nähern.

Allgemein üblich ist heute, die Auftriebsverteilung und den Widerstand bei gegebener Flügelform zu bestimmen, also aus Profildaten, Umriss und Flügelverwindung. Die andere Möglichkeit ist die Bestimmung der Flügelform bei gegebener Auftriebsverteilung. Letzteres Verfahren wird hier angewendet.

Die Beschreibung der Auftriebs- bzw Zirkulationsverteilung erfolgt nach einem Verfahren von R. T. Jones. Außerdem werden Profil und Flügelumriss vorgegeben. Damit lässt sich dann die dazu passende Flügelverwindung bestimmen.

Nach R. T. Jones gilt für die Beschreibung der Zirkulation längs der Halbspannweite

 Γ j Γ 12 π 6 y 1 j n 2 18 y 24 π j n 2 acosh n j

Für den Ausdruck in der eckigen Klammer ist zur leichteren Handhabung das Schreiben einer Funktion zweckmäßig. Um gleichzeitig eine Division durch Null zu vermeiden, wird an der Stelle j = 0 nur der relevante Teil der Gleichung verwendet.

 f y jx nx jx 0 f 12 π 6 y f 12 π 6 y 1 jx nx 2 18 y 24 π jx nx 2 acosh nx jx

Für den Zirkulationsverlauf längs der Halbspannweite gilt dann

Γ j Γ f y j n

Damit lassen sich nun alle anderen erforderlichen Größen am Schlagflügel berechnen.

Rezahl

 Re j v liter j ν

Auftriebsbeiwert

 c j 2 Γ j liter j v

Die Einhaltung der Grenzwerte des c a -Arbeitsbereiches lässt sich recht gut an Hand einer Grafik kontrollieren.

 c 0.80277753295552 0.798753560082188 0.78655813308453226 0.76580095870305465 0.73575776219571554 0.69522573712687974 0.64222202636441617 0.57329782956440456 0.48166651977331187 0.4116736635644711 0

Der nicht ganz harmonisch aussehende Verlauf im Bereich der Flügelspitze ergibt sich infolge der Zuspitzung des Flügels bzw. dem eckigem Flügelumriss.

Auftriebskraft

 F j ρ v Γ j F b n 3 c F F N 39.05783265927414

Abwindverteilung

v j Γ 18 b 1 π 2 3 y π 2 y 2 3 j n

Abwindwinkel

 α j atan v j v

induzierter Widerstand

 F j F j tan α j F b n 3 c F F N 0.80811212744334227

Zur Kontrolle kann man beim Gleitflug, bei dem der induzierte Widerstand genau entgegen der Flugrichtung arbeitet, den gesamten induzierten Widerstand noch auf eine andere Weise berechnen.

nach R. T. Jones gilt

F Γ 2 ρ 2 π 9 2 π 2 y 2 12 π y 9 F N 0.8118811957003782

bzw. wenn die Auftriebsverteilung elliptisch geformt ist.

F c 2 q A π Λ F N 0.81188119570037653

Schlagmoment

Das Drehmoment des Flügels um seine Schlagachse wird hier generell als Schlagmoment M Schl bezeichnet. Das soll hier auch bei der gestreckten Flügelstellung im Gleitflug gelten.

Das Schlagmoment wird, wie im Kraftflug, für beide Flügelhälften zusammen angegeben. Es folgt aus dem Abstand des betrachteten Flügelortes y und der dort wirkenden Auftriebskraft F A .

M j y j F j

über die Spannweite integriert ergibt das

 M b n 3 c M M N m 23.056076358775009

Für den Gleitflug, bei dem die Auftriebskraft senkrecht zur Flugrichtung steht, kann man zur Kontrolle das Schlagmoment auch einfacher ermitteln. Es ist das Produkt aus der Auftriebskraft F AG und dem Druckpunktabstand von der Flügelwurzel.

 M F y s M N m 23.207322634270309

Mit steigender Anzahl der Betrachtungsorte "n" gleichen sich beide Werte einander an.

Biegemoment des Flügelholms

Das Biegemoment des Flügelholms M BH an der Flügelwurzel einer Flügelhälfte ist nur halb so groß wie das Schlagmoment beider Flügelhälften.

M M 2 M N m 11.528038179387504

2.5 Örtliche Profilwerte

Mit Hilfe von c a und Re kann man nun die dazugehörigen Profildaten bestimmen (nähere Einzelheiten dazu siehe Profil-Arbeitsblatt, z. B. clark_y.xmcdz).

Anstellwinkel der Profilsehne

 α j f c j Re j

Profilwiderstandsbeiwert

 c j f c j Re j

Profilwiderstand

 F j c j q liter j F b n 3 c F F N 0.61579229446433936 c j f c j Re j

Momentenbeiwert

Einstellwinkel

 α j α j α j σ

Flügelverwindung

Für den Bau des Schlagflügels ist es vorteilhaft, auch die Flügelverwindung des Gleitfluges zu kennen. Dazu wird die Differenz des Einstellwinkels längs der Spannweite a EG(y) gegenüber dem Einstellwinkel an der Flügelwurzel a EG(0) gebildet.

Δα j α j α 0 Δα deg 0 -0.047340857333317207 -0.19081646907044422 -0.43501852061723961 -0.78846789129181671 -1.2653152450428278 -1.8861649820879305 -2.6902806114213984 -3.7414551013555286 -4.4114127568706945 -8.0276971402447774

2.5.1Drehmoment

Das Drehmoment des Flügels wird aus der Normalkraft F N und deren Abstand e vom Drehpunkt berechnet.

Abb. 2.5.1-1Druckpunktrücklage e und Normalkraft F N

c j c j cos α j c j sin α j

Beiwert der Normalkraft

 F j c j q liter j

Normalkraft

Sie steht immer senkrecht zur Profilsehne

Druckpunktrücklage

Um eine Division durch Null zu vermeiden, schreibt man für den Druckpunktabstand e

 e j c j 0 e j liter j e j liter j c j c j

Bei großen Anstellwinkeländerungen wächst die Druckpunktrücklage e ins Plus- oder Minus-Unendliche. Aus programmtechnischen Gründen wird sie hier auf 3/4 der Flügeltiefe begrenzt ( = Flügeltiefe hinter dem 1/4 -Punkt). Es kommen aber auch andere Werte von e max in Betracht.

e j liter j 0.75 e j if e j e j e j if e j e j e j e j

Druckpunktrücklage hinter dem Hauptholm

H j e j H H mm -49.821475954295913 -50.002456809991614 -50.562350955228261 -51.556635911223282 -53.09584503710829 -55.384809219342571 -59.005533509801289 -64.970156309858524 -75.144718710785583 -73.183443856395741 129.54639175257731

Drehmoment

 M j H j F j

Das Drehmoment MD des Schlagflügels um den Hauptholm ist insbesondere bei der konstruktiven Gestaltung der Flügelverwindung hilfreich. Da diese hier nicht festgelegt ist, werden die Auswirkungen des Drehmoments nicht weiter untersucht.

2.6 Gleitflug-Ergebnis

Folgende Werte wurden bisher ermittelt.

Auftrieb

 F N 39.05783265927414

induzierter Widerstand

 F N 0.80811212744334227

Profilwiderstand

 F N 0.61579229446433936

Restwiderstand

 F N 1.2073846153846153

Gesamtwiderstand

 F F F F F N 2.631289037292297

Daraus lassen sich folgende Beiwerte ableiten.

mittlerer Auftriebsbeiwert

 c 0.65

induzierter Widerstand

 c F q A c 0.013386159093735283

Profilwiderstand

 c F q A c 0.010200433012278814 c F q A c 0.02

Restwiderstand

 c F q A c 0.0435865921060141

Gesamtwiderstand

Fluggeschwindigkeit

 v m s 11.212352507342905

Gleitzahl des Modells

 ε c c ε 14.912842885698161

oder

 ε F F ε 14.843611669307995

Mit steigender Anzahl der Betrachtungsorte "n" gleichen sich beide Werte von e einander an.

Sinkgeschwindigkeit

 v v ε v m s 0.75536552404739787

Gleitwinkel

 γ atan 1 ε γ deg -3.8541384273433335

Schwebeverlustleistung

 P v F P W 29.502940234828184

3. Kraftflug

3.1 Kraftflug allgemein

3.1.1 Quasistationäre Strömungsbedingungen

Bei dem hier angewandten Rechenverfahren werden am Schlagflügel quasistationäre Bedingungen vorausgesetzt. Dabei wird angenommen, dass sich die Strömung innerhalb eines Zeitschrittes (einer Phase) der Berechnung nicht ändert. Auch instationäres Strömungsverhalten bleibt unberücksichtigt.

In jeder Phase des Flügelschlags werden insbesondere die jeweilige Anströmung und die Zirkulationsverteilung längs der Spannweite vorgegeben und dann die Flügelwerte wie bei einem stationären Zustand (z. B. Gleitflug) berechnet.

Auf diese Weise - also durch Aneinanderreihung unterschiedlicher, stationärer Zustände - ergibt sich der zeitliche Kraftverlauf unter "quasistationären Bedingungen".

Am Ende werden die einzelnen Phasenwerte einer berechneten Größe zum Ergebnis der ganzen Schlagperiode zusammengefasst.

Dieses Verfahren führt aber nur beim schnellen Vorwärtsflug mit relativ kleinen Schlagfrequenzen zu brauchbaren Ergebnissen (große Vögel, Fliegen mit dem Auftrieb). Bei hoher Schlagfrequenz gelten andere aerodynamische Gesetze.

Nach Walter Birnbaum kann man noch wie unter stationären Bedingungen rechnen, wenn die Wellenlänge des Flügelschlages mehr als etwa 30 Flügeltiefen beträgt. Erich v. Holst sieht stationäre Bedingungen noch erfüllt, wenn das Ergebnis aus Halbspannweite x Schlagfrequenz / Fluggeschwindigkeit kleiner 0,1 bis 0,2 ist. Dieser Wert wird als "Reduzierte Frequenz" bezeichnet. Bei Überschreitung der Grenze kommt es insbesondere zu einer Phasenverschiebung zwischenSchlagbewegung und Kraftentfaltung.

Die reduzierte Frequenz wird insbesondere im Fachgebiet der Aeroelastik bei der Betrachtung der Aerodynamik schwingender Bauteile angewendet. Sie gilt dort, wenn der Auftrieb in den Umkehrpunkten der Schwingung verschwindet und anschließend jeweils in entgegengesetzter Richtung mit gleicher Verteilungsform komplett neu aufgebaut wird. Wenn man in so einem Falle in einem Zeitpunkt der Schwingung die Luftkräfte so berechnen will wie unter stationären Bedingungen, so darf die reduzierte Frequenz nicht größer als 0,2 werden.

Bei einem Schlagflügel im Reiseflug wird der Auftrieb in den Umkehrpunkten aber nicht Null. Er wirddaher während eines Takte auch nicht komplett neu aufgebaut. Der Auftrieb wird während einer Schlagperiode in seiner Größe nur verändert. Andererseits wird er aber längs der Spannweite verlagert, was bei der Defenition der reduzierten Frequenz völlig unberücksichtig bleibt.

Die Veränderungen des Auftriebs sind am Schlagflügel insgesamt sicher kleiner als bei der Defenition der reduzierten Frequenz angegeben. Man darf daher den Geltungsbereich stationärer Bedingungen beim Schlagflug sicherlich etwas erweitern. Gesicherte Aussagen gibt es darüber aber nicht.

Zur Kontrolle der gewählten Eingangsparameter wird im späteren Rechnungsgang die "reduzierte Frequenz" m berechnet und die Grenze für quasistationäre Bedingungen weiterhin bei 0,2 gesehen.

Reduzierte Frequenz

 μ s f v

bzw. nach anderer Quelle

μ 2 π f l 2 v

3.1.2 Rechenstützstellen

Die Anzahl der Rechenstützstellen während einer Schlagperiode sollte etwa zur gleichen Rechengenauigkeit führen, wie die Anzahl der Rechenstützstellen längs des Flügels. Das wirdschätzungsweise erreicht, wenn die Anzahl der Rechenstützstellen während eines Taktes derjenigen längs der Halbspannweite gleicht.

Abb. 3.1.1.-1 Rechenstützstellen längs der Halbspannweite und während eines Taktes

Anzahl der Rechenstützstellen während einer Schlagperiode

 z 2 n

Iterationswerte

i 0 z

Alle Größen mit dem Index "i" sind Momentan- bzw. Phasenwerte während einer Schlagperiode

3.1.3 Integrationsverfahren

Auch zur Integration der Kräfte während eines Zeitabschnitts wird das numerische Integrationsverfahren von Simpson angewendet. Die allgemeine Gleichung für den zeitlichen Verlauf eine Kraftfunktion während einer Schlagperiode lautet in diesem Falle.

 Funktion i T 3 z f 0 4 f 1 2 f 2 4 f 3 2 f 4 ... 4 f z 2 2 f z 1 f z

Die ganzzahligen Faktoren, mit denen die Größen f 1 , f 2 ,... zum jeweiligen Zeitpunkt multipliziert werden, sollen hier zentral berechnet werden. Sie werden dann bei allen Kraftberechnungen nach dem gleichem Schema angewendet.

F T 3 z c i F i

In diesem Produkt aus Kraft mal Zeitwird korrekter Weise die Wirkungsdauer einer Kraft berücksichtigt. Das Ergebnis ist jeweils ein Kraftimpuls. Vor einer Gleichgewichtssuche wären in diesem Falle auch die Kraftimpulse der Gewichtskraft, des Restwiderstandes usw. zu bilden.

Kraftimpulswerte mit ihrer Einheit [mkg/s] sind aber wenig anschaulich. Da die übrigen Randbedingungen relativ übersichtlich sind, wird hier daher auf die Impulsbildung einfach verzichtet und statt dessen nur mit Kräften bzw. Kraftgleichgewichten gerechnet. Es gilt also

F 1 3 z c i Funktion i

Bei der Festlegung des Faktors c t wird der Umstand genutzt, dass bei dem Verfahren nach Simpson Flügelorte mit geradzahligen Indizes den Faktor 4 und ungeradzahlige den Faktor 2 haben, ausgenommen erstes und letztes Glied.

Faktor für zeitliche Integration

c i i 0 1 i z 1 floor i 2 i 2 2 4

oder in Form einer Funktion

 f ix zx ix 0 1 ix zx 1 floor ix 2 ix 2 2 4

Der Aufruf der Funktion erfolgt mit

 c i f i z

3.1.4 Zirkulationsfaktor

Der Zirkulationsfaktor k G beschreibt, wie sich die Größe der Gesamtzirkulation infolge der Variation des Druckpunktabstandes y G und der Fluggeschwindigkeit v ändert, wenn man den Einstellwinkel an der Flügelwurzel immer konstant hält.

Die Ableitung des Zirkulationsfaktors ist im Handbuch, Anhang D beschrieben. Hier wird zu seiner Beschreibung folgende Funktion verwendet.

 f y k 2 l c 12 π 6 y 18 b 1 π 2 3 y 2 l c 12 π 6 y 18 b 1 π 2 3 y k

3.2 Kraftflug-Konstanten

(Das sind Daten des Kraftfluges, die während der ganzen Schlagperiode konstant bleiben.)

Dauer einer Schlagperiode

 T 1 f T s 0.66666666666666663

Modellfluggeschwindigkeit

v v k v m s 12.557106005311075

Staudruck

q ρ 2 v 2 q N m 2 96.579558127529424

Restwiderstand

 F c q A F N 1.5143674714396611

3.3 Phasenwerte zum Betrachtungszeitpunkt

(Das sind Daten, welche die zeitlichen Veränderungen am Schlagflügel beschreiben, aber in dem betrachteten Zeitabschnitt der jeweiligen Phase konstant gehalten werden.)

Für Schlagflügelmodelle wird häufig ein Kurbelantrieb verwendet. Die zeitlich sinusförmige, schwingende Bewegung eines Kurbelzapfens wird hier der Schlagbewegung des Flügels zu Grunde gelegt. Sie lässt sich wie folgt beschreiben.

Zeitlicher Verlauf

 tonne i T i z

Phasenwinkel F (groß Phi)

Die Untersuchung der Schlagflügelbewegung startet hier in der unteren Flügelendlage mit Beginn des Aufschlags. Damit ist die Aufschlagbewegung um 90 Grad bzw. p/2 gegenüber einer bei Null beginnenden Sinuswelle verschoben. Der Phasenwinkel F der Schwingung ist entsprechend anzupassen.

Anstelle der allgemeinen Gleichung für den Phasenwinkel einer Sinusschwingung

 Φ i 2 π f tonne i Φ

gilt hier

 Φ i 2 π 1 T tonne i π 2

Schlagwinkel f (klein Phi)

An einem Schlagflügel kann man mehrere Drehbewegungen feststellen

1. Die Drehbewegung um seine Schlagachse an der Flügelwurzel.

2. Eine Drehung der Flügelwurzel in Richtung einer Anstellwinkeländerungen

3. Eine ansteigende Verdrehung um eine Achse längs der Halbspannweite,

zur Verwindung des Flügels.

Zur leichteren und eindeutigen Unterscheidung wird hier die erstgenannte Drehbewegung als "Schlagbewegung" bezeichnet.

In gleicher Weise werden auch andere Größen dieser Bewegungsrichtung umbenannt (z. B. in Schlagwinkel, Schlaggeschwindigkeit, Schlagmoment, Schlagleistung usw.).

 φ i φ sin Φ i φ i deg -29.999999999999996 -28.531695488854602 -24.270509831248422 -17.633557568774194 -9.27050983124842 0 9.2705098312484147 17.633557568774183 24.270509831248422 28.531695488854602 29.999999999999996 28.531695488854602 24.270509831248429 17.633557568774194 9.27050983124845 3.6738190614671318E-15 -9.2705098312484164 -17.63355756877419 -24.270509831248418 -28.531695488854602 -29.999999999999996

Aufschlag

Abschlag

Winkelgeschwindigkeit

Die Winkelgeschwindigkeit eines rotierenden Teiles mit umlaufender Bewegung ist allgemein

 ω 2 π n

wobei n die Drehzahl pro Zeiteinheit ist

Für die mittlere Winkelgeschwindigkeit des Schlagflügels, mit seiner begrenzten Winkelbewegung während einer ganzen Schlagperiode, gilt entsprechend

 ω 2 2 φ T

der Drehwinkel f (Phi) ist in [rad] einzusetzen

Hat der Schlagflügel einen zeitlich sinusförmigen Bewegungsablauf, so lässt sich analog zur allgemeinen Sinushalbwelle der Mittelwert der Winkelgeschwindigkeit auch aus ihrem Scheitelwert berechnen.

ω 2 π ω

mit dem Scheitel w max der Winkelgeschwindigkeit

Daraus ergibt sich der Scheitelwert der Winkelgeschwindigkeit zu

ω 2 π φ T

und daraus wiederum der zeitliche Verlauf der Winkelgeschwindigkeit

 ω i ω cos Φ i

Aufschlag

Abschlag

Umfangsgeschwindigkeit eines Flügelortes

Dies ist die Geschwindigkeit eines Flügelpunktes auf seiner Bahn um die Schlagachse.

 v j i y j ω i

effektive Anströmgeschwindigkeit

Steht die Schlagebene senkrecht zur x-Achse des Modells bzw. zur Flugrichtung, so kann man dafür schreiben (siehe Abb. 3.3.9 -1 im Abschnitt "Kräfte am Schlagflügel")

 v j i v j i 2 v 2

Staudruck

 q j i ρ 2 v j i 2

Rezahl

 Re j i v j i liter j ν

Neigung der Bahnlinie eines Flügelortes

Definiert man den Neigungswinkel der Bahnlinie beim Aufschlag als negativ und beim Abschlag als positiv, so gilt

δ j i atan v j i v δ n i deg -1.9301780390647006E-15 -9.6489536219674985 -17.920701130814145 -23.994251933064742 -27.621101424154702 -28.818902062394343 -27.621101424154702 -23.994251933064756 -17.920701130814145 -9.6489536219674985 -1.9301780390647006E-15 9.6489536219675074 17.920701130814134 23.994251933064742 27.621101424154691 28.818902062394343 27.621101424154702 23.994251933064749 17.920701130814148 9.6489536219675021 5.7905341171941029E-15

Aufschlag

Abschlag

3.3.1 Zirkulationskennzahlen allgemein

Der Schub des Schlagflügels wird um so größer, je größer der Unterschied zwischen den Zirkulationsverteilungen von Auf- und Abschlag ist. Da mit profilierten Flügeln beim Fliegen mit dem Auftrieb generell nur wenig Schub erzeugt werden kann, wird hier bei nachstehender, automatischer Zirkulationskennzahlbestimmung versucht, diesen Unterschied immer so groß wie irgend möglich zu machen.

Begrenzt wird diese Zielsetzung insbesondere durch die Grenzwerte des Profilauftriebsbeiwertes c a_min und c a_max . Indirekt wird sie dadurch auch vom Verlauf der Flügeltiefe l (y) längs der Spannweite und von der Schlaggeschwindigkeit der beiden Takte begrenzt (Je größer die Schlaggeschwindigkeit ist, umso größer wird zur Flügelspitze hin die Anströmgeschwindigkeit. Dadurch kann dort die Zirkulation größer werden.).

Will man dagegen im Horizontalflug fliegen, ist kein so großer Schub erforderlich. Die Unterschiede der Zirkulationsverteilungen von Auf- und Abschlag können dann kleiner sein. Die Zirkulationsverteilungen werden also gegenüber dem Steigflug näher an die des Gleitfluges heran gerückt. Der Auftrieb wird dadurch gleichmäßiger und der induzierte Widerstand beim Abschlag kleiner. Um das im Rechengang zu erreichen, wird hier die Möglichkeit eingeführt, die Zirkultaionskennzahl durch Eingangsparamter (c G 1_min und c G 2_max ) zu begrenzen.

Die für einen Horizontalflug optimalen Zirkulationskennzahlen können hier nicht automatisch bestimmt werden. Sie sind manuell zu ermitteln. Die vorher berechneten Zirkulationskennzahlen des Steigfluges dienen dabei als Orientierungshilfe und als Grenzwerte.

3.3.2 Zirkulationskennzahl in Aufschlagmitte

Nachstehendes Verfahren zur Bestimmung der Zirkulationskennzahl beim Aufschlag basiert auf der Nullstellensuche der von Mathcad bereit gestellten Wurzelfunktion. Dabei wird zunächst eine mittlere Zirkulationskennzahl c G 1a vorgegeben. Daraus lässt sich die dazugehörige Auftriebsbeiwertverteilung und deren minimale Differenz zum c a -Grenzwert ermitteln. Von der Wurzelfunktion wird dann die Nullstelle dieser Differenz gesucht und die dazugehörige Zirkulationskennzahl zurück gegeben. Auf diese Weise erhält man eine Zirkulationskennzahl, deren Auftriebsbeiwertverteilung an den jeweiligen c a -Grenzwert heranreicht.

Um bei diesem Verfahren die Variation der Kraftfluggeschwindigkeit v K und des Fluggeschwindigkeits- faktors k v im späteren Rechnungsgang mit berücksichtigen zu können, werden den Funktionen diese Paramter mit übergeben (beide Paramter können auch zusammengefasst werden).

 f k v c y c 6 π k f y k j 0 n v j n s ω v v 2 v 2 Γ Γ f y j n k c j 2 Γ liter j v Diff c min c

Bei der Nullstellensuche wird hier der Start- bzw. Anfangswert der Aufschlag-Zirkulationskennzahl mit Null vorgegeben.

c 0 c root f k v c c c -0.65124149510715035

Im Gleitflug beträgt der Druckpunktabstand von der Flügelwurzel etwa 0.42 x Halbspannweite. Beim Aufschlag wird dieser Druckpunktabstand kleiner. Bei einer Zirkulationskennzahl von c G 1 = 0 liegt der Druckpunkt direkt an der Flügelwurzel.

Wandert der Druckpunktabstand über die Flügelwurzel hinaus auf die andere Seite des Schlaglagers (ins Negative), so wechselt das Schlagmoment des Flügels die Richtung. Der Flügel wird dann nicht mehr vom Flügelauftrieb bewegt, sondern vom Antrieb. Dieser Last- bzw. Vorzeichenwechsel des Schlagmoments findet während der Aufschlagbewegung statt und soll hier erst einmal vermieden werden. Außerdem führt eine kleiner werdende Zirkulationskennzahl zu immer weniger Auftrieb. Diesen Schwund auszugleichen wird immer schwieriger.

Die Aufschlag-Zirkulationskennzahl c G 1 bzw. der Druckpunktabstand y G 1 soll hier also zu keinem Zeitpunkt kleiner als c G 1_min = 0 werden (siehe Eingangsparameter). Je nach verwendetem Profil ist aber selbst dieser Grenzwert nicht immer erreichbar.

c if c c c c c 0

Dies ist nun die minimal zulässige Zirkulationskennzahl in Aufschlagmitte.

3.3.3 Zirkulationskennzahl in Abschlagmitte

Im Prinzip erfolgt diese Berechnung nach der gleichen Methode wie beim Aufschlag. Lediglich die Differenzbildung ändert sich. Außerdem gibt es zunächst keine Begrenzung von c G 2 .

f k v c y c 6 π k f y k j 0 n v j n s ω v v 2 v 2 Γ Γ f y j n k c j 2 Γ liter j v Diff max c c

Der Start- bzw. Anfangswert der Abschlag-Zirkulationskennzahl wird hier mit 9 vorgegeben.

c 9 c root f k v c c c 8.9739882917190421

Dies ist nun die maximal zulässige Zirkulationskennzahl in Abschlagmitte, wenn Strömungsabriss vermieden werden soll.

Um aber nicht immer mit der maximalen Zirkulationskennzahl arbeiten zu müssen (z. B. beim Horizontalflug), wird wie beim Aufschlag ein Eingangsparameterc G 2_max zur Begrenzung eingeführt. Bei einer Wertzuweisung von 10 oder größer, ist praktisch keine Begrenzung mehr gegeben.

 c if c c c c c 8.9739882917190421

3.3.4 Zirkulationskennzahlen für den Horizontalflug

Folgende Zirkulationskennzahlen wurden zuletzt für den Kraftflug ermittelt.

Aufschlag

 c 0

Abschlag

 c 8.9739882917190421

Für den Horizontalflug werden die Zirkulationsverteilungen des Kraftfluges an die des Gleitfluges angenähert. Man kann dies hier einfach erreichen, indem man die Eingangsparameter der Zirkulationskennzahlbegrenzung beispielsweise wie folgt abändert.

Aufschlag

 c 5

Abschlag

 c 8.8

3.3.5 Zirkulation im Phasenverlauf

Von der Zirkulation ist zu erwarten, dass sie sich zusammen mit dem Phasenwinkel F der Winkelgeschwindigkeit ändert. Außerdem wird angenommen, dass die Form der Zirkulationsverteilung in den Flügelendlagen - wo der Flügel für einen kurzen Moment stillsteht - derjenigen des Gleitfluges entspricht.

Bei der Beschreibung der Zirkulation nach R. T. Jones werden Änderungen der Zirkulation durch die Zirkulationskennzahl beschrieben. Für die soll also hier gelten.

 c i i n c c c cos Φ i c c c cos Φ i

Für den zeitlichen Verlauf des relativen Druckpunktabstandes y G K gilt dann

 y i c i 6 π

Aufschlag

Abschlag

Zirkulationsfaktor

 k i f y i k

Aufschlag

Abschlag

Die Zirkulationsverteilungen längs der Spannweite und in Schlagmitte sehen dann wie folgt aus.

Γ j i Γ Γ f y i j n f y i k

Gleitflug

Aufschlag

Abschlag

Die Größe der Gesamtzirkulation zum Betrachtungszeitpunkt wird einfach mittels dem betreffenden Zirkulationsfaktor beschrieben.

 Γ i Γ k i Γ 0 m 2 s 1.1426966464833077

Die Zirkulation zum Betrachtungszeitpunkt i kann man auch mit Hilfe der Zirkulationsbeschreibung des Kraftfluges ermitteln. Da sie auf der Gesamtzirkulation des Gleitfluges längs der ganzen Spannweite basiert, entfällt in diesem Falle bei der Summierung der Faktor b der Spannweite.

Γ i 1 3 n c Γ i Γ 0 m 2 s 1.1373918042471656

Wegen dem etwas groben Integrationsverfahren erhält man jedoch geringfügig kleinere Werte. Die Unterschiede verlieren sich aber bei hoher Anzahl der Rechenstützpunkte n.

3.3.6Querkraftverteilung

Die Querkraft liegt quer zur auftriebsfreien Anströmrichtung und wird sonst üblicherweise als "Auftrieb" bezeichnet. Beim Schlagflug wird aber der Auftrieb erst aus der Querkraft berechnet. Um Verwechslungen zu vermeiden wurde in Anlehnung an entsprechende Literatur für diese Luftkraft die Bezeichnung "Querkraft" gewählt.

Auftriebsbeiwert

 c j i 2 Γ j i liter j v j i

Gleitflug

Aufschlag

Abschlag

F j i ρ v j i Γ j i

Querkraft

v j i Γ i 18 b 1 π 2 3 y i π 2 y i 2 3 j n

Abwind

Abwindwinkel

 α j i atan v j i v j i

3.3.7 Örtliche Profilwerte

Mit Hilfe von c a und Re kann man nun nach dem Gleitflug-Muster die dazugehörigen Profildaten bestimmen (nähere Einzelheiten siehe Arbeitsblatt Profil.mcd).

Auftriebsbeiwert

 c j i 2 Γ j i liter j v j i

Anstellwinkel

α j i f c j i Re j i

Profilwiderstandsbeiwert

 c j i f c j i Re j i

Momentenbeiwert

 c j i f c j i Re j i

Einstellwinkel

α j i α j i α j i δ j i σ

Gleitflug

Aufschlag

Abschlag

α deg 5.6620886230061176 5.6147477656727993 5.471272153935673 5.2270701023888781 4.8736207317143 4.39677337796329 3.7759236409181876 2.9718080115847192 1.9206335216505892 1.2506758661354227 -2.3656085172386589
α 0.25 z deg 3.7990527347895915 3.3008421855232912 2.2329515661530253 1.0661585299605576 -0.20906060134689017 -1.6787388583169538 -2.7511133417477294 -3.5763111513598536 -4.1410272108808792 -4.5272166180329707 -2.774126211698078
α 0.75 z deg 6.9881394336268343 7.2425995301456014 7.693413604038942 8.1552182664192863 8.5408612917470155 8.6714285836311618 8.4214364385075946 7.6926505621480663 6.3187789357076376 5.2721478847887084 -2.774126211698078

Gleitflug

Aufschlag

Abschlag

α deg 4.8473773573336141 4.8000365000002976 4.65656088826317 4.4123588367163746 4.0589094660417979 3.582062112290787 2.961212375245684 2.1570967459122161 1.1059222559780859 0.43596460046291935 -3.1803197829111625
α 0.25 z deg 4.65057753748916 6.7015904255427863 8.1603828181797073 9.4850864179584988 10.654576456614191 11.572550401156937 12.822029086921935 14.245772505374058 15.851615845173297 17.553716465471588 21.310404408444704
α 0.75 z deg 5.0343979646620767 2.5471333824503311 0.27128798733619552 -1.9637948102338148 -4.2315286331286392 -6.6984456912735011 -9.4792917223881457 -12.663381835995136 -16.409972211649176 -19.741550929935563 -29.981555005105104

Verwindungskennwert

Je größer die theoretische Flügelverwindung ist muss, um so größer sind die Schwierigkeiten bei der Konstruktion des Schlagflügels. Die erforderliche Verwindung ist daher ein wichtiges Kriterium bei Beurteilung des Rechenergebnisses.

Um die Flügelverwindung leicht abschätzen zu können, wurde der sogenannte Verwindungskennwert kreiert. Dazu wird zunächst zum Zeitpunkt der maximalen Verwindung, in Schlagmitte der Anstellwinkel von Auf- und Abschlag in Halbspannweitenmitte ermittelt. Dann bildet man die Differenz gegenüber dem Gleitfluganstellwinkel an dieser Stelle. Die Angabe erfolgt in Grad pro Meter.

Der Kennwert entspricht etwa der Flügelverwindung im Armflügelbereich. Er dient aber nur als Anhaltspunkt. Der Verwindungsverlauf ist ja längs der Spannweite nicht konstant. Im Bereich der Flügelspitze ist die Verwindung [Grad/m] etwa doppelt so groß.

für den Aufschlag

 V α z 0.25 n 2 α n 2 2 s V deg m 11.414983269808785

für den Abschlag

 V α z 0.75 n 2 α n 2 2 s V deg m -14.686439719377555

für den Kraftflug insgesamt gilt dann als Richtwert

 V V V V deg m 26.101422989186339

3.3.8Drehmoment

Das Drehmoment des Flügels um seine Längsachse wird aus der Normalkraft F N und deren Abstand e vom Drehpunkt berechnet.

Beiwert der Normalkraft

 c j i c j i cos α j i c j i sin α j i

Normalkraft

 F j i c j i q j i liter j

Druckpunktrücklage vom 1/4-Punkt des Profils

Um eine Division durch Null zu vermeiden, schreibt man dafür

 e j i c j i 0 e j i liter j e j i liter j c j i c j i

Bei großen Anstellwinkeländerungen wächst die Druckpunktrücklage e ins Plus- oder Minus- Unendliche. Aus programmtechnischen Gründen wird sie hier auf 3/4 der Flügeltiefe begrenzt

( = Flügeltiefe hinter dem 1/4 -Punkt). Es kommen aber auch andere Werte von e max in Betracht.

e j liter j 0.75 e j i if e j i e j e j if e j i e j e j e j i

Druckpunktrücklage gegenüber dem Hauptholm

 H j i e j i H

Gleitflug

Aufschlag

Abschlag

Drehmoment um den Hauptholm

 M j i H j i F j i

Gleitflug

Aufschlag

Abschlag

Drehmoment des ganzen Flügels um den Hauptholm im jeweiligen Betrachtungszeitpunkt

 M i b n 3 c M i

Die Kenntnis des Drehmoments M D des Schlagflügels um den Hauptholm ist insbesondere bei der konstruktiven Gestaltung der Flügelverwindung erforderlich. Da diese hier nicht festgelegt ist, werden die Auswirkungen des Drehmoments nicht weiter untersucht.

3.3.9 Kräfte am Schlagflügel

Die prinzipielle Anordnung der verschiedenen Schlagflügelkräfte kann man dem folgenden Bild entnehmen.

Abb. 3.3.9-1Kräfte am Schlagflügel

 F j i F j i tan α j i

Induzierter Widerstand

Da Querkraft und induzierter Anstellwinkel längs der Spannweite stellenweise unterschiedliche Vorzeichen haben, treten im Vorzeichen-Wechselbereich in einigen Phasen auch negative induzierte Widerstände auf.

Profilwiderstand

F j i c j i q j i liter j

Abb. 3.3.9-2Ausschnittvergrößerung von den Kräften am Schlagflügel

Flügelwiderstand in z-Richtung

F j i F j i sin δ j i F j i sin δ j i

Die Flügelwiderstände wirken beim Schlagflug nicht nur entgegen der Flugrichtung sondern auch entgegen der Schlagrichtung.

Widerstand in z-Richtung im Betrachtungszeitpunkt

F i b n 3 c F i

Widerstand in z-Richtung einer ganzen Schlagperiode

 F 1 z 3 c F F N 0.14815517561102595

Dieser Auftrieb durch Widerstand im Kraftflug ist - über eine ganze Schlagperiode gesehen - zwar relativ gering, aber insgesamt positiv. Das Prinzip, mit Widerstand Auftrieb zu erzeugen, ist bemerkenswert. Es spielt vielleicht bei schnell schlagenden Flügeln oder aufgefächerten Flügelspitzen eine größere Rolle.

x-Anteil des induzierten Widerstandes

 F j i F j i cos δ j i

induzierter Widerstand im jeweiligen Betrachtungszeitpunkt

 F i b n 3 c F i

Induzierter Widerstand einer ganzen Schlagperiode

F 1 z 3 c F F N 1.1825095900277276

Aufschlag

Abschlag

x-Anteil des Profilwiderstandes

 F j i F j i cos δ j i

Profilwiderstand im jeweiligen Betrachtungszeitpunkt

F i b n 3 c F i

Profilwiderstand einer ganzen Schlagperiode

F 1 z 3 c F F N 0.94341878365866561

Aufschlag

Abschlag

Flügelauftrieb

Der auf den Flügel bezogene Auftrieb F AF setzt sich aus einem Querkraftanteil und den z-Anteilen der Widerstände zusammen.

 F j i F j i cos δ j i F j i

Flügelauftrieb im jeweiligen Betrachtungszeitpunkt

 F i b n 3 c F i

Modellauftrieb

Um den auf das Modell bezogenen Auftrieb F AM zu bestimmen, ist der Schlagwinkel des Flügels zum jeweiligen Zeitpunkt zu berücksichtigen.

Abb. 3.3.9-3Bestimmung des Modellauftriebs F AM

Die in Bezug auf die Modell-Symetrieebene nach innen und außen gerichteten Kraftanteile FAy heben sich gegenseitig auf. Da sie aber erst einmal erzeugt werden müssen und zum induziertem Widerstand beitragen, ist dies ein Nachteil großer Schlagwinkel, zumindest unter quasistationären Bedingungen.

Durch die relativ längere Verweildauer des Schlagflügels im Bereich der Endlagen wird zusammen mit einem steigendem Schlagendlagenwinkel der Modellauftriebsimpuls sogar überproportional kleiner.

 F i F i cos φ i

Modellauftrieb einer ganzen Schlagperiode

 F 1 z 3 c F F N 39.21443736176095

Aufschlag

Abschlag

Schub

Ausgangspunkt für der Schubberechnung ist der Vortrieb F V , also die Kraft in x-Richtung des Modells, ohne Berücksichtigung der Widerstände.

F j i F j i sin δ j i

Gleitflug

Aufschlag

Abschlag

Vortrieb im jeweiligen Betrachtungszeitpunkt

 F i b n 3 c F i

Vortrieb einer ganzen Schlagperiode

 F 1 z 3 c F F N 5.0543788781872587

Der Schub des Schlagflügels ist die Differenz aus der Vortriebskraft und den x-Anteilen der Flügelwiderstände.

 F F F F F N 2.9284505045008657

Aufschlag

Abschlag

Bei der Modell-Gesamtbetrachtung wird nachfolgend auch noch der Rest- bzw. der schädliche Widerstand F WrK berücksichtigt (Abschnitt 3.4).

3.3.10Schlagmomente

Die Momente der Schlagbewegung M Schl sind hier so definiert, dass ein in Richtung Heben wirkendes Moment des Flügels positiv ist.

Schlagmoment aerodynamischer Kräfte

Von besonderem Interesse ist das Schlagmoment aerodynamischer Kräfte M SchlA , also infolge des Flügelauftriebs F A . Zur Berechnung wird dessen Produkt mit dem jeweiligen Abstand y des Betrachtungsortes von der Flügelwurzel gebildet .

 M j i F j i y j

Das Schlagmoment aerodynamischer Kräfte während einer Schlagperiode beschreibt man mit

 M i b n 3 c M i

Aufschlag

Abschlag

Flügelbeschleunigungsmoment

Ein ähnlich großes Schlagmoment wie das der aerodynamischen Kräfte ist beim Beispiel der EV -Modelle für die Beschleunigung der Flügelmassen erforderlich M SchlB . Es ist abhängig vom Massenträgheitsmoment der Flügel J F und von der Winkelbeschleunigung a B .

Winkelbeschleunigung

α i 2 π T 2 φ sin Φ i

Aufschlag

Abschlag

Beschleunigungsmoment

M i J α i

Das Beschleunigungmoment des Flügels kann auch von Endlagenfedern aufgebracht werden (siehe unten). Es belastet dann den Antrieb nicht weiter. Der Flügelholm muss dieses Moment aber immer aufnehmen.

Moment zum Heben des Flügelgewichts

Das erforderliche Schlagmoment zur Hebung des Flügelmasse ist dagegen relativ klein und bei mittleren Schlagwinkeln fast konstant.

M i F r s cos φ i

Gesamtes Schlagmoment

(nach oben gerichtete Flügelkräfte positiv)

 M i M i M i M i

Aufschlag

Abschlag

Scheitelschlagmoment

Dies ist das größte, aber nur kurzzeitig auftretende Schlagmoment während einer Schlagperiode.

 M max M M min M M if M M M M M N m 53.098520566499779

Biegemoment des Flügelholms

Das maximale Biegemoment des Flügelholms im Kraftflug M BHK an der Flügelwurzel einer Flügelhälfte ist nur halb so groß wie das Scheitelschlagmoment beider Flügelhälften.

M M 2 M N m 26.54926028324989

Das maximale Biegemoment des Holmes ist in diesem Beispiel im Kraftflug etwa doppelt so groß wie im Gleitflug.

3.3.10 Kurbelantrieb

Die Berechnungen der Kurbelantriebsdaten sind in einem eigenen Arbeitsblatt enthalten (siehe k_antrieb.xmcd).

Die dort aufgeführten Berechnungen ermöglichen die Beschreibung der Antriebskräfte eines Ornithopters mit einem bestimmten Kurbelantrieb. Ziel ist es dabei, den Verlauf des Kurbeldrehmoments zu beschreiben, das zur Überwindung des Flügelschlagmomentes erforderlich ist. Außerdem wird versucht, die Antriebsbelastung durch den Einbau von Federn zu vergleichmäßigen.

3.3.12Schlagleistungen

Es werden hier die verschiedenartigen Leistungen am Schlagflügel untersucht (ohne Anwendung von Federn). Die Vorzeichen werden aus der Sicht der vom Antrieb aufzubringenden Leistung gesetzt.

Leistung für aerodynamische Kräfte

aerodynamische Leistung im Betrachtungszeitpunkt

 P i M i ω i

Leistung einer ganzen Schlagperiode

 P 1 z 3 c P P W 65.0533983480738

Leisung für die Flügelbeschleunigung

Beschleunigungsleistung im Betrachtungszeitpunkt

 P i M i ω i

Leistung einer ganzen Schlagperiode

 P 1 z 3 c P P W 9.42110482990402E-15

Die Leistung zur Beschleunigung der Flügelmasse gleicht sich über eine ganze Schlagperiode gesehen im Mittel aus. Sie beeinflußt jedoch den Verlauf der Gesamtleistung.

Leistung zum Heben der Flügelmasse

Leistung im Betrachtungszeitpunkt

 P i M i ω i

Leistung einer ganzen Schlagperiode

 P 1 z 3 c P P W -1.5065422732250659E-15

Auch die Leistung zum Heben der Flügelmasse gleicht sich - über eine ganze Schlagperiode gesehen - im zeitlichen Mittel aus. Sie ist aber sowieso generell relativ klein.

Gesamtleistung im Kraftflug

Leistung im Betrachtungszeitpunkt

 P i P i P i P i P i M i ω i

oder auch

mittlere Kraftflugleistung einer ganzen Schlagperiode ist dann

P 1 z 3 c P P W 65.0533983480738

Dies ist die mittlere Antriebsleistung zur Bewegung des Schlagflügels.

Wie nachfolgende Abbildung zeigt, unterliegt die Gesamtleistung P K am Schlagflügel innerhalb einer Schlagperiode deutlichen Schwankungen.

Aufschlag

Abschlag

Antriebsmotor-Eingangsleistung

Auf dem Weg der Energie vom Akku zur Flügelbewegung summieren sich die Verluste von Akku, Kabel, Motorsteller, Motor, Getriebe, Mechanik und Flügelverwindung und bestimmen den Wirkungsgrad h Antr (Eingangsparameter).

P i P i η

Mittlere Antriebsmotor-Eingangsleistung

 P 1 z 3 c P P W 130.1067966961476

Aus obigem Diagramm lässt sich schon der ungefähre Verlauf der Motorbelastung ablesen. Für den genauen zeitlichen Verlauf ist aber noch die verwendete Schlagflügelmechanik zu berücksichtigen.

Für die Anwendung einer einfachen Kurbel mit Kreuzschleife wird die Motorbelastung in einem separaten Datenblatt berechnet (siehe k_antrieb.xmcd).

3.4 Flugdaten

Energie ist ein anderes Wort für "Arbeitsvermögen". Der Akku enthält also Energie, mit der manfolgende mechanische Arbeit vollbringen kann.

Arbeitsvermögen der Akkuenergie

(Umwandlung in Kraft x Weg)

 E N m 57600

Diese Umwandlung der Einheiten gegenüber dem Eingangsparameter dient aber nur zur Klarstellung. Mit dem gegebenen Arbeitsvermögen des Akkus kann man direkt die Flugdauer und die Flugstrecke berechnen.

Die vom Akku gelieferte Energie wird während der Betriebsdauer des Motors in die Schlagbewegung der Flügel umgewandelt. Die Betriebsdauer entspricht der Kraftflugzeit t K des Modells.

 F P v

Kraftflugdauer

 t E P t s 442.71322838359845

Das Modell fliegt während dieser Zeit mit der Kraftfluggeschwindigkeit v K und legt dabei ein Flugstrecke s x zurück.

Flugstrecke in x-Richtung

 s v t s 5559.1969387663376

Fliegt das Modell im flachen Steigflug, so fliegt es sinngemäß einen Hang hinauf und muss dabei den sogenannten Hangabtrieb überwinden.

 E F h

Abb. 3.4-1Kräfte an einem Schlagflügelmodell im Steigflug

Bildet man die Differenz aus dem Schub des Schlagflügels F S und dem Restwiderstand F WrK des Modells, so verbleibt eine Kraft, die zur Überwindung eines Hangabtriebes zur Verfügung steht.

Hangabtrieb

 F F F F N 1.4140830330612046

Flugbahn-Steigungswinkel g

γ atan F F

oder auch

γ atan v v γ deg 2.0638619869024404

Die damit erzielbare Steighöhe ergibt sich damit aus der zurück gelegten Flugstrecke s x .

Steighöhe

 h s sin γ h 200.20557050586058

Steiggeschwindigkeit

 v h t v m s 0.45222405311184455

Spezifische Transportlenergie

Die spezifische Transportenergie CT beschreibt die aufgewendete Energie pro Kilogramm Fluggerätemasse und pro Kilometer Flugstrecke. Sie ist ein guter Maßstab für die Leistungsfähigkeit eines Schlagflügelgerätes. Um die verschiedenen Fluggeräte miteinander vergleichen zu können, ist als Datenbasis der unbeschleunigte Horizontalflug zu wählen (hier nicht der Fall).

CT E m s CT W s kg km 2590.3021890776113

Spezifische Transportleisung

Anschaulicher ist vielleicht die spezifische Leistung.

 CL P m v CL W s kg km 2590.3021890776117

3.5 Kräftegleichgewicht

Da die Randbedingungen relativ leicht überschaubar sind, wird hier anstelle der Kraftimpulsgleichgewichte mit Kräftegleichgewichten gearbeitet.

3.5.1 Kraftgleichgewicht in z-Richtung

Dieses Gleichgewicht liegt vor, wenn alle Kräfte parallel zur z-Achse des Modells, über eine ganze Schlagperiode gesehen, sich im Gleichgewicht befinden. Die Summe aller Kräfte bzw. die folgende Kraft Fz muss also gleich Null sein.

 F F F cos γ F N -0.0001079323825905476

Als Variable für die Veränderung des Auftriebs kommen insbesondere folgende Größen in Betracht.

G 1 Zirkulationsgröße beim Aufschlag

G 2 Zirkulationsgröße beim Abschlag

k G Schlagzeitverhältnis Aufschlag/Abschlag

k v Fluggeschwindigkeitsfaktor

Die Zirkulationen von Auf- und Abschlag werden hier im Interesse der Schuberzeugung bereits anderweitig festgelegt und kommt daher als Variable für die Auftriebserzeugung nicht mehr in Frage.

Das Schlagzeitverhältnis k G ist hier nicht veränderbar. Außerdem sind seine Auswirkungen auf den Auftrieb relativ begrenzt.

Bleibt als einzige Möglichkeit zur Auftriebsvariation die Fluggeschwindigkeit in Form des Fluggeschwindigkeitsfaktors k v (Eingangsparameter).

3.5.2 Kraftgleichgewicht in x-Richtung

Dieses Gleichgewicht liegt vor, wenn alle Kräfte parallel zur x-Achse des Modells über eine ganze Schlagperiode gesehen, sich im Gleichgewicht befinden. Die Summe aller Kräfte bzw. die Kraft F x muss also gleich Null sein.

F F F F

F N 0

Als Variable zu Beeinflussung des Kräftegleichgewichts in x-Richtung kommen insbesondere folgende Größen in Betracht.

G 1 Zirkulation beim Aufschlag

G 2 Zirkulation beim Abschlag

fFlügelschlagfrequenz

g Flugbahn-Steigungswinkel

Um den Schub zu vergrößern, ist der Unterschied zwischen Auf- und Abschlagzirkulation möglichst groß zu machen. Die Zirkulationskennzahlen beider Takte sollen sich also möglichst stark unterscheiden. Dies wird vorstehend bei der Festlegung der Zirkulationskennzahlen bis an die Grenze der c a -Arbeitsbereiches bereits praktiziert. Als Variable für das Kräftegleichgewicht fallen die Zirkulationsbeschreibungen also aus.

Die Flügelschlagfrequenz f und der Flugbahn-Steigungswinkel g sind theoretisch beide sehr wirkungsvoll, um das Kräftegleichgewicht in x-Richtung zu beeinflussen. Zumindest im Steigflug gibt es bei der Schlagfrequenz jedoch gewisse Einschränkungen.

Bei einer Erhöhung der Schlagfrequenz steigt das Flügelbeschleunigungsmoment im Bereich der Flügelschlagendlagen deutlich an. Dies führt bei hohen Schlagfrequenzen zu Verformungen des Flügels oder sogar zum Bruch. Auch der Antrieb muss in die Lage sein, die großen Beschleunigungskräfte aufzubringen. Schon alleine durch diese Zusammenhänge sind einer Steigerung der Schlagfrequenz enge Grenzen gesetzt.

Außerdem darf bei hoher Schlagfrequenz die Luftströmung am Flügel nicht abreißen. Die Verwindung des Flügels muß sich also an die hohe Schlagfreqeunz anpassen können. Man wird daher schon bei der Antriebs- und der Schlagflügelkonstruktion eine Grenze der maximal zulässigen Schlagfrequenz festlegen müssen. Sie wird in der Praxis, wegen der generell kleinen Steigleistung profilierter Schlagflügel, meist voll auschöpft und lässt sich dann nur nach unten variieren.

Eine weitere Hürde für die Steigerung der Schlagfrequenz ist die Einhaltung quasistationärer Bedingungen. Relativ schnell erreicht man den Grenzwert der reduzierten Frequenz. Als variable Größe zur Erlangung eines Kräftegleichgewichtes ist die Schlagfrequenz also relativ ungeeignet..

Als Variable für das Kräftegleichgewicht in x-Richtung bleibt also beim hier interessierenden Steigflug nur noch der Flugbahnsteigungswinkel g übrig.

Anmerkung

Wäre hier bei diesem Rechenverfahren auch das Taktzeitverhältnis von Auf- und Abschlag veränderbar, so wäre es aus folgendem Grunde dennoch für die Schubvariation ungeeignet.

Man kann beispielsweise bei konstanter Schlagfrequenz den Abschlag schneller ausführen als den Aufschlag. Dadurch erhöht sich zwar die maximale Schubkraft des Abschlags, verkürzt aber dessen Wirkungsdauer. Auf diese Weise bleibt - bei sonst gleichbleibenden Schlagflügelparametern - der Schubimpuls einer Schlagperiode konstant. Veränderungen des Taktzeitverhältnisses können folglich nur bei der Variation des Auftriebs hilfreich sein. (Wegen veränderten Anströmgeschwindigkeiten ändern sich im Rechenmodell mit der Schlaggeschwindigkeit allerdings auch die Zirkulationskennzahlen von Auf- und Abschlag geringfügig. Und in der Praxis ändern sich zusätzlich auch noch andere Parameter, z. B. die Beschleunigungskräfte der Flügel- und Modellmassen.)

Bei Erhöhung der Schlagfrequenz und konstantem Taktzeitverhältnis ändert sich zwar der Schubimpuls einer Schlagperiode praktisch auch nicht. Pro Zeiteinheit (beispielsweise 1 Minute) werden aber mehr Schlagperioden ausgeführt. Insgesamt wird auf diese Weise der Schubimpuls pro Zeiteinheit in diesem Falls dann doch größer.

3.5.3 Kraftflug-Gleichgewichtssuche

Für einen stationären Flug muss das Kräftegleichgewicht in z- und x-Richtung gleichzeitig vorhanden sein. Die Variablen für Auftrieb und Schub beeinflussen aber auch das jeweils andere Kräftegleichgewicht ein wenig. Ändert man beispielsweise die Schubvariable, so ändert sich auch der Auftrieb und umgekehrt. Die manuelle Gleichgewichtssuche ist also etwas langwierig.

Um diese Aufgabe zu erleichtern, wurde zunächst mal die Funktion f Kraftflug geschrieben. Sie berechnet unter Verwendung der vorstehenden Eingangsparameter in Kurzform die Kräfte einer ganzen Schlagperiode, soweit sie für die Beschreibung des Kräftegleichgewichtes erforderlich sind. Als Variationsparameter kommen für den Auftrieb der Fluggeschwindigkeitsfaktor k v und für den Schub die Kraftflug-Steiggeschwindigkeit v sK zur Anwendung. Ihre Startwerte werden als Funktionsparamter beim Aufruf der Funktion mit übergeben.

f k v k k v v m s v v k c 0 c root f k v c c c if c c c c c 9 c root f k v c c z 2 n i 0 z Φ 2 π i z π 2 φ φ sin Φ ω ω cos Φ c i z 2 c c c cos Φ c c c cos Φ y c 6 π Γ Γ f y k j 0 n liter f j n v j n s ω v v 2 v 2 Re v liter ν Γ Γ f y j n c 2 Γ liter v F ρ v Γ δ atan v v F f c Re ρ 2 v 2 liter v Γ 18 b 1 π 2 3 y π 2 y 2 3 j n F F v v F F sin δ F sin δ F j F sin δ F cos δ F cos δ F j F cos δ F cos φ c j f j n F i b n 3 c F F i b n 3 c F c i f i z γ atan v v F 1 z 3 c F F cos γ F 1 z 3 c F c ρ 2 v 2 A F sin γ μ s f v Force F 1 N F 1 N μ

Das Ergebnis dieser Funktion sind die resultierenden Kräfte in z- und x-Richtung und die reduzierte Frequenz (gerundet auf die eingestellte Nullschwelle, hier 10 -4 ).

Die eigentliche Gleichgewichtssuche in beiden Kraftrichtungen ist nun relativ einfach. Es werden so lange abwechselnd die beiden Kraft-Nullstellen in x- und z-Richtung gesucht, bis gleichzeitig ein Kräftegleichgewicht in beiden Richtungen besteht. Das Fluggerät befindet sich dann in einem stationären Flugzustand.

f k v k k v v z3 0 10 GG_Variable k v F f k v F 0 10 -3 F 1 10 -4 k root f k v 0 k v root f k v 1 v k k v v Ergebnis GG_Variable

Achtung !

Die Berechnung des Kräftegleichgewichts kann einige Minuten dauern. Um unbeabsichtigte Rechnungsgänge zu vermeiden sollte man die automatische Berechnung dieses Arbeitsblattes abschalten.

Variable f 1 1 k Variable 0 k 1.1199353825297242 v Variable 1 v 0.45281159935627485

Als Ergebnis erhält man die Werte der beiden Variablen, die zum gleichzeitigem Kräftegleichgewicht in x- und z-Richtung führen.

Bei ungeeigneter Modellkonfiguration ist ein Kräftegleichgewicht nicht möglich. In diesem Fall entfällt die Ergebnisanzeige und Mathcad meldet einen Fehler. Ein oder mehrere relevante Eingangsparamter sind dann zu ändern.

Das Fenster geeigneter Modellkonfigurationen ist relativ klein.

Bei erfolgreicher Gleichgewichtsuche werden die Variablenwerte zunächst der Kraftflugfunktion f k v übergeben um das Ergebnis zu kontrollieren.

f k v -4.8581125334123954E-05 -9.3258734068513149E-15 0.1672359285513064

Nur wenn die Kräfte F z und F x gleichzeitig Null sind (Index 0 und 1) und die reduzierte Frequenz m (Index 2) kleiner 0,2,ist das Rechenergebnis auf die Praxis übertragbar.

Sind die Werte von k v und v sK in Ordnung, werden sie oben als Eingangsparamter in dieSchlagflügelberechnung eingesetzt. Das Rechenprogramm ermittelt dann alle bisher beschriebenen Größen des Schlagflügels bei vorhandenem Kräftegleichgewicht.

4.Gleitflugprogramm

Für die Untersuchung der Gleitflugeigenschaften des gerechneten Modells mit Bereichsvariablen, wurde folgendes Gleitflugprogramm geschrieben.

f m bx Λx y l c c m m kg b bx m Λ Λx y y l l c c c c s b 2 l b Λ A l 2 Λ v 2 m g ρ c l b Γ m g ρ v b q ρ v 2 2 F c q A y c 6 π l l y 1 y l 1 2 l if y 1 l 1 j 0 n y j n s liter j n y liter l liter l 1 1 l j n y 1 y Γ j Γ f y j n c 2 Γ j liter v Re v liter ν α f c Re v Γ 18 b 1 π 2 3 y π 2 y 2 3 j n α atan v v α j α α σ F j ρ v Γ j F j F j v v F j f 2 Γ j liter v v liter ν q liter J j y 2 liter A m m c j j 0 1 j n 1 floor j 2 j 2 2 4 F b n 3 c F F b n 3 c F F b n 3 c F F F F F ε F F v v ε γ atan 1 ε P v F J b n 3 c J Ergebnis ε

Um einen anderen Ausgabeparameter als e zu erhalten, ist dieser in die letzte Zeile der Gleiflugfunktion einzutragen.

Anzeige und Übertragung der Eingangsparameter

m kg 4 b m 2.8 Λ 10 y 0.8 l 0.7 m m 1 kg b1 b 1 m Λ1 Λ y y l l c 0.65 c 8 c c c c

Bereichsvariable:

Es sollte jeweils nur eine der Bereichsvariablen aktiviert werden !

m 1 5 b1 2 5 Λ1 5 15 y 0.5 0.6 1 l 0.5 0.6 1 c 0.4 0.5 1 c 7 7.2 8.5

Ergebnisanzeige

(hier Gleitzahl e bei Variation der Flügelstreckung)

f m b1 Λ1 y l c c 11.398096043811561 12.370486241032303 13.16931096827609 13.839334976580084 14.400918116414541 14.843611669307995 15.224355087038894 15.555041593627301 15.64360590685658 15.659452052761917 15.664896728176844

Fügt man vorstehende Kraftflug- und Gleitflugfunktion zusammen und übergibt an die gemeinsame Funktion gleichzeitig die zu variierenden Eingangsparamter, so gelangt man vielleicht etwas schneller zu einem optimierten Schlagflügel. Das ist im Rechenprogramm Orni 3 ausgeführt (siehe orni3.xmcdz).

5.Fehlerbetrachtung

Nur Fehler die bekannt sind, können vermieden werden!

Die zu erwartende Fehlerabweichung dieses Rechenverfahrens soll im Gesamtzusammenhang,

also auch zusammen mit seiner praktischen Anwendung betrachtet werden.

Folgende Fehlerquellen sind nach den Erfahrungen bei den EV -Modellen bekannt.

Fehler durch Idealisierung der physikalischen Zusammenhänge

Rumpfpendelbewegung

Die senkrechte Rumpfpendelbewegung des Ornithoptermodells wird vernachlässigt. Es wird im Rechenprogramm so getan, als ob der Rumpf auf einer geraden Bahn dahinfliegt. Die bei einer Pendelbewegung auftretenden Anstellwinkelschwankungen bleiben also unberücksichtigt.

Annahme quasistationärer Bedingungen

Die Zulässigkeit dieser Annahme ist bei Einhaltung des Grenzwertes der reduzierten Frequenz allgemein anerkannt. Erst bei Überschreitung des Grenzwertes wirkt sich instationäres Strömungsverhalten immer stärker aus. Es kommt dabei insbesondere zu einer kleinen Phasenverschiebung zwischen Schlagbewegung und Kraftentfaltung.

Periodisch schwankende Fluggeschwindigkeit

Die Fluggeschwindigkeit während des Kraftfluges wird im Rechenmodell als konstant angenommen. Die Schubkräfte des Schlagflügels sind aber bei Auf- und Abschlag recht unterschiedlich. Demzufolge ist auch die Fluggeschwindigkeit veränderlich (siehe Video vom EV6 ). Die Größe der Veränderungen hängen von der Masse des Ornithopters und vom Kraftverlauf ab.

Fehler durch Unzulänglichkeiten der Berechnungsmethode

Fehler durch Anwendung beliebiger Flügelumrisse

Ähnlich wie beim Gleichungssystem von A. Betz zur Beschreibung der Zirkulation, sollte vermutlich auch bei dem Gleichungssystem von R. T. Jones der Flügelumriss auf die jeweilige Zirkulationsverteilung abgestimmt sein. Dies ist hier aber mangels solcher Angaben nicht der Fall. Es wird daher, entsprechend der praktischen Anwendung, mit eckigen Flügelumrissen gerechnet.

Im Bedarfsfalle könnte man die mit einem bestimmten Flügelumriss verbundene Fehlerabweichung wahrscheinlich quantifizieren. Man muss "nur" aus der hier ermittelten Flügelform die daraus resultierende Zirkulationsverteilung berechnen - z. B. nach dem Traglinienverfahren von Multhopp - und sie dann mit der von R. T. Jones vergleichen.

Fehler durch Konstrunktions- und Herstellungsmängel

Zeitlicher Ablauf der Schlagbewegung

Zumindest im Prüfstand zeigten Messungen an einer EV -Schlagflügel-Antriebsmechanik mit zeitlich sinusförmigem Bewegungsablauf (Kardankurbel), dass der zeitliche Verlauf der Flügelschlagbewegung deutlich vom theoretisch vorgesehenen abweicht. Schon allein die starken Drehzahlschwankungen des Antriebs bleiben unberücksichtigt. Aber auch die Eigenschwingungen des ganzen Systems und seiner Teile sind deutliche Störgrößen. Das Spiel der Lager und Gelenke sowie die Elastizität der verwendeten Bauteile bleiben ebenfalls unberücksichtigt.

Konstruktion der Flügelverwindung

Der konstruierte Verlauf der Flügelverwindung längs der Halbspannweite entsprach beim EV6 und EV7 nur annähernd den theoretischen Vorgaben. Erst beim EV8 wurde weitgehende Übereinstimmung von Theorie und Praxis erzielt.

Zeitlicher Verlauf der Flügelverwindung

Infolge der Flügelmassenträgheit weicht der praktisch erzielte, zeitliche Verlauf der Flügelverwindung deutlich vom theoretisch geforderten ab (siehe Bilder und Videos in <http://www.ornithopter.de>).

Hauptsächlich im Bereich der oberen Schlagendlage kommt es zu einem Überschwingen des Anstellwinkels. Statt einer dem Gleitflug entsprechenden Flügelverwindung zeigen die Bilder vom Flug eines Ornithoptermodells in der oberen Flügelendlage und im Bereich der Flügelspitze stark negative Anstellwinkel.

Profilform

Durch Einfallen der Bespannung zwischen den Flügelrippen weicht die Profilform eines Folie-bespannten Schlagflügels deutlich von der theoretischen Profilform ab. Auch die aerodynamische Güte der Flügelhinterkante lässt zu wünschen übrig.

Die im Rechenmodell angewandte Methode zur Berücksichtigung der Profilformabweichung (ca-Begrenzung), ist nur ein sehr grobes, theoretisches Näherungsverfahren.

Fehlerbewertung

Erst ab dem Ornithoptermodell EV6 wurden die EV -Schlagflügelmodelle nach den Ergebnissen eines vergleichbaren Rechenmodells gebaut. Die damit erzielten praktischen Flugergebnisse waren deutlich besser als vorher. Sie wichen - abgesehen von der unzureichenden Flugstabilität - gar nicht so weit vom Rechenergebnis ab (satt einem theoretischen Flugbahnsteigungswinkel von +1,8 Grad wurde nur der Horizontalflug erreicht).

Die Rechenergebnisse erscheinen plausibel. An der Antriebsmechanik, der Schlagflügelform und im Flugbild der EV-Schlagflügelmodelle sieht man aber deutlich Abweichungen von den theoretischen Vorgaben. Dabei handelt es sich offensichtlich um Mängel der Konstruktion und der Herstellung. Die Auswirkungen dieser technischen Mängel werden wesentlich größer eingeschätzt, als die Fehlerabweichungen infolge der bekannten Mängel des angewandten Rechenverfahrens.

Nach Abwägung dieser Zusammenhänge kann man sagen, dass das hier beschriebene Rechnungsverfahren ein brauchbares Hilfsmittel bei der Entwicklung großer Ornithoptermodelle mitProfilschlagflügeln ist. Verbesserungen sind insbesondere bei der Konstruktion der Modelle erforderlich. Aber auch das Rechenverfahren sollte weiter entwickelt werden.

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