Orni 3, Rechenmodell für Ornithopter

Mathcad Professional 14.0 Orni 3, Rechenmodell für Ornithopter Rechenprogramm für Daten-Reihenuntersuchungen von Antriebs- und Flugleistungen verschiedener Modellkonfigurationen. Horst Räbiger Ornithopter, Schlagflügel, Rechenprogramm, Rechenmodell, quasistationären Bedingungen, Kräftegleichgewicht Horst Räbiger 2 8AD6F86D-3095-43A7-A008-AADB5FACD345 D104467B-FADE-40A7-9EBD-166B98FC84D0 00000000-0000-0000-0000-000000000000 00000000-0000-0000-0000-000000000000

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Orni 3

Rechenmodell für Ornithopter

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 durchführbar.

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>, der Inhalt des dort bereit gestellten Handbuches " Wie Ornithopter fliegen " und das Rechenprogramm Orni 2 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. Datenvorschlag für das Rechenmodell................ 2

2. Berechnung der Zirkulationskennzahlen.............. 3

3. Schlagflug-Rechenprogramm.............................. 5

4. Gleichgewichtsuche in x- und z-Richtung.............12

5. Eingabe der Modelldaten ......................................13

6. Ergebnis der Modelldaten .....................................15

7. Horizontalflug.......................................................16

8. Bereichsberechnung...........................................16

8.1 Festlegung der Bereichsvariablen .................17

8.2 Ergebnis der Bereichsberechnung ................18

1. Datenvorschlag für das Rechenmodell

Die vorgegebenen Beispielwerte entsprechen etwa denen eines EV-Ornithoptermodells und sind änderbar. Viele von ihnen lassen sich am Ende dieses Arbeitsblattes auch in Form einer Bereichsvariablen schrittweise variieren.

Modell

Modellmasse

m 4 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

Flügel- spitze, bezogen auf die Flügelwurzeltiefe

l 0.7

Werte 0 bis 1

Bei rechteckigem

Umriss (y_k = 1)

ist l_s wirkungslos

relative Flügelmasse

bezogen auf die Modellmasse

m 0.2

m _F ist etwa 1/5

der Modellmasse

relativer Flügelschwerpunkt-

abstandvom Schlaglager,

bezogen auf die Halbspannweite

r 0.44

Gleitflug

mittlerer Auftriebsbeiwert

c 0.65

Zirkulationskennzahl

(8 für elliptische

Auftriebsverteilung)

c 8.00

Werte7 bis 8

minimal zulässige

Zirkulationskennzahl

beim Aufschlag

c 0

Werte 0 .. 5

-20 = ohne

Begrenzung

Kraftflug

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 1.500 1 s

Werte 1 bis 4

(große Vögel)

Schlag-Endlagenwinkel,

gemessen von der Schlagmitte $±$

φ 30 deg

Werte20 bis 45

Antriebswirkungsgrad

(für 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

Index der Variablen im

Dateneingabevektor data

für eine Bereichsberechnung

VAR 0

Werte 0 bis 12,

siehe unten

Kraftflugart-Schalter

zum Umschalten der Berechnung

zwischen Horizontal- und Steigflug

flag 1

Steigflug = 1

Horizontalflug = 0

Profil

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 c _a -Arbeitsbereiches folgen aus der Flügelbauweise dieser Modelle.

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

de764c7e9a6d14917151d19a2b6680e8 Profilprogramm für das Schlagflügel-Rechenverfahren "Orni"

Orni 3 und die Profildatei sind im gleichen Verzeichnis zu speichern.

Andernfalls ist der Verweis zur Profildatei anzupassen.
Das Rechenprogramm funktioniert nur mit korrektem Dateiverweis.

2. Berechnung der Zirkulationskennzahlen

Nachstehende Bestimmung der Zirkulationskennzahlen basiert auf der Nullstellensuche der von Mathcad bereit gestellten Wurzelfunktion. Dabei wird zunächst mit einer geschätzten, vorgegebenen Zirkulationskennzahl die Verteilung des Auftriebsbeiwertes beschrieben und die kleinste Differenz zum c_a-Grenzwert berechnet. Von der Wurzelfunktion wird dann die Nullstelle dieser Differenz gesucht und die dazugehörige Zirkulationskennzahl zurückgegeben.

Um bei diesem Rechenverfahren die Variation der verschiedenen Eingangsparameter im späteren Rechnungsgang berücksichtigen zu können, werden diese Eingangsparamter in Form des Vektors Para der Wurzelfunktion mit übergeben.

Zirkulationsverteilung

Die Berechnung der Zirkulation längs der Spannweite ist im Rechnungsgang an mehreren Stellen erforderlich. Zur einfacheren Handhabung soll gelten:

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

Beschreibung der c _a -Verteilung

Der Einfacheit halber wird in dieser Funktion ohne Einheiten gerechnet.

f Para c b Para 0 y Para 1 l Para 2 l Para 3 y Para 4 Γ Para 5 ω Para 6 k Para 7 v Para 8 y c 6 π s b 2 Γ Γ 2 l c 12 π 6 y 18 b 1 π 23 y 2 l c 12 π 6 y 18 b 1 π 23 y k j 0 n y j j n s y j y s liter j l liter j l 1 1 l j n y 1 y v y j ω v v 2 v 2 Γ Γ f y j n c j 2 Γ liter j v c

Zirkulationskennzahl für den Aufschlag

f Parameter c c f Parameter c Diff c min c

Zirkulationskennzahl für den Abschlag

f Parameter c c f Parameter c Diff max c c

3. Schlagflug-Rechenprogramm

Gleit- und Kraftflug des Rechenmodells sind in der folgenden Funktion zusammengefasst.

f datax k v fx VAR datax 0 flag datax 1 m datax 2 kg b datax 3 m Λ datax 4 y datax 5 l datax 6 c datax 7 c datax 8 c datax 9 c datax 10 φ datax 12 E datax 13 C V k k v v m s f fx 1 s s b 2 l b Λ A l 2 Λ m m m 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 j n s y j y s liter j l liter j l 1 1 l y j s y 1 y j 0 Γ j Γ 12 π 6 y Γ j Γ f y j n c 2 Γ j liter j v Re v liter j ν α f c Re v Γ 18 b 1 π 23 y π 2 y 23 j n α atan v v α j α α F j ρ v Γ j F j F j v v F j f 2 Γ j liter j v v liter j ν q liter j J j y j 2 liter j A 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 P v F ε F F J b n 3 c J v v k T 1 f ω 2 π φ T Param b 1 m y l l 1 m y Γ s m 2 ω s k v s m c 0 c root f Param c c c if c c c c c 9 c root f Param c c c if c c 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 2 l c 12 π 6 y 18 b 1 π 23 y 2 l c 12 π 6 y 18 b 1 π 23 y k j 0 n v y j ω v v 2 v 2 Re v liter j ν j 0 Γ Γ 12 π 6 y Γ Γ f y j n c 2 Γ liter j v α f c Re F ρ v Γ δ atan v v F f c Re ρ 2 v 2 liter j v Γ 18 b 1 π 23 y π 2 y 23 j n F F v v α j α atan v v δ σ F j F cos δ F sin δ F sin δ F j F sin δ F cos δ F cos δ M j F j y j c j j 0 1 j n 1 floor j 2 j 2 2 4 F i b n 3 c F cos φ F i b n 3 c F M i b n 3 c M M i m g r s cos φ M i J 2 π T 2 φ sin Φ M i M i M i M i P i M i ω η i z 0.25 V α n 2 α n 2 2 s i z 0.75 V α n 2 α n 2 2 s c i i 0 1 i z 1 floor i 2 i 2 2 4 P 1 z 3 c P F 1 z 3 c F F c ρ 2 v 2 A t E P s v t γ atan v v h s sin γ CL P m v M min M M max M M if M M M M F 1 z 3 c F m g cos γ F F F m g sin γ μ s f v V V V Ergebnis VAR flag F 1 N F 1 N μ l 1 m round v s m 2 round ε 1 round P 1 W 1 c c round k 3 round f s 3 round v s m 3 round P 1 W 0 round h 1 m 0 round s 1 m 0 round t 1 s 0 round CL kg km W s 0 round M 1 N m 0 round V m deg 1

4. Gleichgewichtsuche in x- und z-Richtung

Die Gleichgewichtsuche erfolgt durch abwechselnde Nullstellensuche der Kraftsummen in x- und in z-Richtung und zwar solange, bis gleichzeitig ein Kräftegleichgewicht in beiden Richtungen besteht. Dabei wird bei jedem Schritt mit dem Wert des Variationsparamter weiter gerechnet, der vorher in der anderen Kraftrichtung zur Nullstelle geführt hat.

- Bei der Suche nach dem Kräftegleichgewicht in z-Richtung wird immer der
Fluggeschwindikgeitsfaktor k _v variiert.

- Für das Kräftegleichgewichtsuche in x-Richtung wird beim Steigflug die Steiggeschwindigkeit
v _sK und beim Horizontalflug die Flügelschlagfrequenz f als Variationsparameter verwendet.

Die Startwerte der Variablen für die Nullstellensuche werden innerhalb der Funktion vorgegeben.

f data1 k 1 flag data1 1 flag 1 v 1 f data1 11 v 0 f 1 z3 010 GG_Variable k v f F f data1 k v f F 2 10 -3 F 3 10 -4 k root f data1 k v f 2 k flag 1 v root f data1 k v f 3 v v v f root f data1 k v f 3 f f f k k GG_Variable

5. Eingabe der Modelldaten

Nur als Anhaltspunkt werden die als Bereichsvariable in Betracht kommenden Parameter von obigem Modellvorschlag an einen neuen Datenvektor data übergeben und zunächst nur angezeigt.

data VAR flag m 1 kg b 1 m Λ y l c c c c f s φ E 1 C V

data 014 2.8 10 0.8 0.7 0.65 8010 1.5 0.52359877559829882 57600

Im nächsten Schritt werden diese Werte erneut an den data -Vektor übergeben. Es besteht nun im Nahbereich der Ergebnissanzeige die Möglichkeit, die Eingangsparamter zu verändern.

Die Dateneingabe erfolgt in nachstehendem, farblich hervorgehobenen Feld. Zur leichteren Orientierung sind daneben in Textform die Bezeichnungen der Parameter mit ihren Indizes im data -Vektor aufgelistet.

Die Größe der Eingabewerte wird nicht kontrolliert. Sie stehen untereinander in Beziehung und sollten nach modellbautechnischen Gesetzmäßigkeiten plausibel sein. Andernfalls ist wahrscheinlich kein Kräftegleichgewicht erreichbar.

0. Index der Variablen im data -Vektor VAR

1. Schalter Steigflug/Horizontalflug (1/0)flag

2. Modellmasse m _M [kg]

3. Spannweite b [m]

4. Flügelstreckung L

5. relativer Flügelumrissknick-Abstand y _kr

6. relative Flügeltiefe an der Flügelspitze l _sr

7. mittlerer Auftriebsbeiwert im Gleitflug c _amG

8. Zirkulationskennzahl für den Gleitflug c _G _G

9. Minimale Aufschlag-Zirkulationskennzahlc _G _{1_min}

10. Maximale Abschlag-Zirkulationskennzahl c _G _{2_max}

11. Flügelschlagfrequenz f [Hz]

12. Schlag-Endlagenwinkel f _E [rad]

13. Akku-Energie E _Akku [CV]

data 014 2.8 10 0.8 0.7 0.65 8010 1.5 30 deg 57600

Diese neuen Daten kommen nun beim Aufruf der Funktion für die Gleichgewichtsuche zum Einsatz.

Variable f data

Schließlich werden die zum Gleichgewicht führenden Werte von k _v und v _sK angewendet und das Modell im Gleichgewichtszustand berechnet. Für die Gleichgewichtsuche im Horizontalflug wird auch noch die Flügelschlagfrequenz f an die Funktion mit übergeben.

Flug f data Variable 0 Variable 1 Variable 2

6. Ergebnis der Modelldaten

Für die Auswertung wurden 21 Ausgabeparameter ausgewählt.

Beispielwerte

0. Index der Variablen im data -Vektor VAR

1. Schalter Steigflug/Horizontalflug (1/0)flag

2. Kraftsumme in z-Richtung F _z [N]

3. Kraftsumme in x-Richtung F _x [N]

4. Reduzierte Frequenz, Soll < 0.2 m

5. Flügelwurzeltiefe l₀ [m]

6. Gleitfluggeschwindigkeit v _G [m/s]

7. Gleitzahl e

8. Schwebeverlustleistung im Gleitflug P _VG [W]

9. Zirkulationskennzahl des Aufschlags c _G ₁

10. Zirkulationskennzahl des Abschlags c _G ₂

11. Fluggeschwindigkeitsfaktor Kraft/Gleit k _v

12. Schlagfrequenz f [1/s]

13. Modell-Steiggeschwindigkeit v_sK [m/s]

14. Motoreingangsleistung P _Mot [Watt]

15. Steighöhe h _s [m]

16. Flugstrecke s _x [m]

17. Flugdauer t _K [sec]

18. Speziefische Transportleistung CL[Ws/kg/km]

19. max. Schlagmoment M _{Schl_max} [Nm]

20. Verwindungskennwert V_Da _K [Grad/m]

Flug 01 -4.8581125341229381E-05 -9.3258734068513149E-15 0.1672359285513064 0.28865979381443296 11.21 14.8 29.5 0 8.9739882478279149 1.12 1.5 0.453 1302005559443259053 26.1

0100 0.167 0.289 11.21 14.8 29.5 0 8.974 1.12 1.5 0.453 1302005559443259053 26.1

Nur wenn die Kräfte F _z und F _x gleichzeitig gleich Null sind und die reduzierte Frequenz m kleiner 0,2,ist das Rechenergebnis auf die Praxis übertragbar.

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 im data -Vektor zu ändern.

Mit diesem Rechenmodell lässt sich nun sehr schön ein in Planung befindliches Ornithoptermodell in verschiedenen Zielrichtungen optimieren. Dazu wählt man beispielsweise ein bestimmtes Modellgewicht, eine bestimmte Spannweite oder einen anderen Paramter als Fixpunkt aus und versucht dann das bestmögliche Ergebnis zu erzielen. Als Zielrichtung interessieren vielleicht die

- größte Steighöhe

- höchste Steiggeschwindigkeit

- größte Flugstrecke im Horizontalflug

- geringste Antriebsleistung im Horizontalflug

- kleinste Flügelverwindung (womöglich sogar ohne Armflügelverwindung), usw.

Natürlich kommen auch Kombinatinen mehrerer Ziele in Betracht, beispielsweise die vorstehend genannten zusammen mit Gleitflugdaten.

Es werden zwar eine ganze Reihe möglicher Eingangsparamter automatisch festgelegt (c _G ₁ , c _G ₂ , v _sK ,

k _v ). Eine Vielzahl von Kombinationsmöglichkeiten bleibt aber bestehen. Die hinter den Entwicklungsrichtungen liegenden Gesetzmäßigkeiten sind daher gar nicht so leicht zu erkennen.Gerade das macht aber die Suche nach den Optima recht spannend (beispielsweise folgende Fragestellung: Große Flügelstreckungen sind gut für lange Flugstrecken und kleine Flügelstreckungen sind gut für große Steighöhen. Wie verhält sich dabei aber die Motorleistung?).

7.Horizontalflug

Das Flugverhalten eines Ornithopter mit Profilschlagflügeln ist ein wenig mit dem eines Motorseglers vergleichbar. Im Modellmotorflug strebt man damit hauptsächlich einen kräftigen Steigflug an und im Gleitflug eine große Gleitzahl. Man will schnell Höhe gewinnen um dann in der Termik zu Kurbeln. Aber auch die Erzielung großer Flugstrecken kann dabei eine Optimierungsrichtung sein.

Die Stärke des Schlagfluges dürfte im Streckenflug liegen. Vögel bringen es dabei zu phantastischen Leistungen. Im Flugmodellbau ist der Streckenflug aber eine eher seltene Disziplin. Hier liegen jedenfalls keine Daten vom Streckenflug motorisierter Flugmodelle vor. Um Propellermodelle mit Ornithoptern vergleichen zu können, sollten die Daten etwas über die zurückgelegte Flugstrecke (ohne Windeinfluss) pro Wattstunde und pro Kilogramm Modellmasse aussagen (speziefische Transportleistung).

Um bei Schlagflügelmodellen wenigstens theoretische Betrachtungen über den Horizontalflug anstellen zu können, bietet das Rechenmodell die Möglichkeit diesen gezielt zu untersuchen. Man kann dazu bei der automatischen Kraftgleichgewichtsuche zwischen Steigflug und Horizontalflugberechnung wechseln.

Die Umschaltung erfolgt mit dem Eingangsparamter "flag". Statt im Steigflug (flag = 1) mit der Variablen v _sK , wird im Horizontalflug (flag = 0) durch Variation der Flügelschlagfrequenz f ein Kräftegleichgewicht in x-Richtung gesucht. Die Flügelschlagfrequenz ist dabei in der Regel kleiner als beim Steigflug. Man sollte aber neben anderen Veränderungen zusätzlich auch die Zirkulationskennzahlen von Auf- und Abschlag weiter eingrenzen, also näher an die Zirkulationskennzahl des Gleitfluges heranrücken.

Durch geeignete Parameterwahl ist ein Modell mit dem Gewicht und der Energie wie im Rechenbeispiel durchaus so hinzutrimmen, dass es im Horizontalflug Flugstrecken über 10 km schafft. Dies wird insbesondere mit großen Flügelstreckungen erreicht. Allerdings ist dabei zu beachten, dass der hier praktizierte Bezug des Modellrestwiderstandbeiwertes c _wr auf die Flügelfläche A - statt auf die Rumpfquerschnitt oder Leitwerksfläche - ein Vereinfachung darstellt. Bei großen Veränderungen der Flügelstreckungen verzerrt das dieErgebnisse etwas.

8. Bereichsberechnung

Um bei der Auswertung die Veränderungen bei Variation eines Paramters leichter beurteilen zu können, soll nun die Berechnung in einem ganzen Datenbereich ermöglicht werden. Dazu dient die folgende Funktion f_Reihe.

f data2 VAR_Nr Anfang Anzahl Schrittweite data3 data2 w 0 Anzahl data3 VAR_Nr Anfang w Schrittweite Var f data3 data3 0 Anfang w Schrittweite Ergebnis w f data3 Var 0 Var 1 Var 2 Ergebnis

Neben den Variationswerten des Bereichsparameters wird dieser Funktion auch der data -Vektor mit den Eingabedaten übergeben.

8.1 Festlegung der Bereichsvariablen

Die Festlegung der Bereichsvariablen erfolgt in den nachstehenden, farblich hervorgehobenen Feldern. Mit dem "Index der Variablen" bestimmt man, welche der Eingangsparamter als Bereichsvariable verwendet werden soll. Zur Erinnerung sind daneben in Textform die Namen der Variablen und ihre Indizes im data -Vektor noch einmal aufgelistet.

Die Werte der übrigen drei farblich hervorgehobenen Felder beschreiben den gewünschten Untersuchungsbereich.

Beim Horizontalflug ist die Variation der Flügelschlagfrequenz wirkungslos !

Index_der_Variablen 11

0. Index der Variablen im data -Vektor VAR

1. Schalter Steigflug/Horizontalflug (1/0)flag

2. Modellmasse m _M [kg]

3. Spannweite b [m]

4. Flügelstreckung L

5. relativer Flügelumrissknick-Abstand y _kr

6. relative Flügeltiefe an der Flügelspitze l _sr

7. mittlerer Auftriebsbeiwert im Gleitflug c _amG

8. Zirkulationskennzahl für den Gleitflug c _G _G

9. Minimale Aufschlag-Zirkulationskennzahlc _G _{1_min}

10. Maximale Abschlag-Zirkulationskennzahl c _G _{2_max}

11. Flügelschlagfrequenz f [1/s]

12. Schlag-Endlagenwinkel f _E [rad]

13. Akku-Energie E _Akku [CV]

Anfangswert 1.2

Schrittweite 0.1

Endwert 1.5

Diese Werte werden nun an die Funktion f _Reihe übergeben.

Anzahl Endwert Anfangswert Schrittweite

Anzahl der Rechendurchgänge

Begrenzung auf 10 Rechendurchgänge

Anzahl if Anzahl 10 10 Anzahl

Achtung !

Die Berechnung im Datenbereich kann einige Minuten dauern. Um unbeabsichtigte Rechengänge zu vermeiden, sollte man vor der Aktivierung der nachstehenden Gleichung die automatische Berechnung des Arbeitsblattes abschalten.

8.2 Ergebnis der Bereichsberechnung

Matrix f data Index_der_Variablen Anfangswert Anzahl Schrittweite

8.2 Ergebnis der Bereichsberechnung

0. Wert der Bereichsvariablen VAR

1. Schalter Steigflug/Horizontalflug (1/0)flag

2. Kraftsumme in x-Richtung F _z [N]

3. Kraftsumme in x-Richtung F _x [N]

4. Reduzierte Frequenz, Soll < 0.2 m

5. Flügelwurzeltiefe l ₀ [m]

6. Gleitfluggeschwindigkeit v _G [m/s]

7. Gleitzahl e

8. Schwebeverlustleistung im Gleitflug P _VG [W]

9. Zirkulationskennzahl des Aufschlags c _G ₁

10. Zirkulationskennzahl des Abschlags c _G ₂

11. Fluggeschwindigkeitsfaktor Kraft/Gleit k _v

12. Schlagfrequenz f [1/s]

13. Modell-Steiggeschwindigkeit v _sK [m/s]

14. Motoreingangsleistung P _Mot [Watt]

15. Steighöhe h _s [m]

16. Flugstrecke s _x [m]

17. Flugdauer t _K [sec]

18. Speziefische Transportleistung CL [Ws/kg/km]

19. Maximales Schlagmoment M _{Schl_max} [Nm]

20. Verwindungskennwert V _Da _K [Grad/m]

Matrix 1.2 1 -4.3606254962469393E-05 -1.4210854715202004E-14 0.1333077632597402 0.28865979381443296 11.21 14.8 29.5 0 8.9501219211829675 1.124 1.2 0.114 103647044559204448 17.5 1.3 1 -6.4936991400088573E-05 -2.6423307986078726E-14 0.14457409203728178 0.28865979381443296 11.21 14.8 29.5 0 8.9575447160224559 1.123 1.3 0.226 1121166474514222449 20.4 1.4 1 -6.5689512162236952E-05 -2.19824158875781E-14 0.15588216202076755 0.28865979381443296 11.21 14.8 29.5 0 8.9655040654485134 1.121 1.4 0.339 1211615984476240651 23.3 1.5 1 -4.8581125341229381E-05 -9.3258734068513149E-15 0.1672359285513064 0.28865979381443296 11.21 14.8 29.5 0 8.9739882478279149 1.12 1.5 0.453 1302005559443259053 26.1

Bei ungeeigneter Modellkonfiguration ist ein Kräftegleichgewicht nicht möglich. Wenn dies auch nur bei einem der Bereichsschritte der Fall ist, entfällt die ganze Ergebnisanzeige und Mathcad meldet einen Fehler. Die Daten der Bereichsvariablen sind dann in geeigneter Weise abzuändern.

Das Fenster geeigneter Modellkonfigurationen ist relativ klein.

Nur wenn die Kräfte F _z und F _x eines Rechenschrittes gleichzeitig Null sind und seine reduzierte Frequenz m kleiner 0,2 ist, ist das Rechenergebnis auf die Praxis übertragbar.

Zur besseren Darstellung sind in nachfolgender Grafik einige Werte durch 10 dividiert.

In der Praxis werden bei einem Modell meist mehrere Ziele gleichzeitig verfolgt. Beispielsweise soll ein Ornithopter eine möglichst große Steighöhe, bei kleiner Motorleistung und mit kleiner Flügelverwindung erzielen und womöglich auch noch gut im Gleitflug sein. Die diesbezüglich mit dem Rechenmodell erzielbare Datenflut führt leicht zur Verwirrung. Besser ist es, eine zusammenfassende Benotungen jeder Modellkonfiguration zu haben.

Dazu wäre zunächst eine zahlenmäßige Bewertung (z. B. mit Werten von 1 bis 10) der Ergebnisse der einzelnen Ausgabeparameter erforderlich.

Außerdem ist jeder der Ausgabeparameter entsprechend der Aufgabenstellung und der persönlichen Beurteilung zu gewichten (z. B. ebenfalls mit Werten von 1 bis 10). Dies dient insbesondere der Betonung der einzelnen Ausgabeparamter unereinander.

Die Bewertung und Gewichtung kann dann durch Produktbildung in einer Gesamtnote jeder Modellkonfiguration zusammengefasst werden.

Mit so einer Gesamtnote sind zahlenmäßige Vergleiche des Modells als Ganzes möglich. Dieses beim EV7 und EV8 praktizierte Konzept ist aber hier mit Mathcad und für womöglich sehr unterschiedliche Modellgrößen erst noch zu verwirklichen.

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