Düzgün doğrusal yatay uçuş yapan bir uçağın kararlılık türevlerinin bulunuşu ve kararlılığın analizi
Calculation of stability derivatives for an aircraft which has a longitudinal steady state flight
- Tez No: 66665
- Danışmanlar: DOÇ. DR. RAMAZAN TAŞALTIN
- Tez Türü: Yüksek Lisans
- Konular: Uçak Mühendisliği, Aeronautical Engineering
- Anahtar Kelimeler: Kararlılık, Uçaklar, Uçuş, Stability, Airplanes, Flight
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Uçak Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bir uçağın güvenilir bir uçuş yapabilmesi için statik ve dinamik kararlı olması gerekir. Statik kararlılık uçağın belli bir referans konumda kuvvet ve moment dengesine sahip olmasıdır. Dinamik kararlılık ise uçağın belli bir referans denge konumundan bozuntularla saptırıldığında tekrar denge haline gelmesidir. Uçağa etkiyen aerodinamik, ağırlık ve itki kuvvetleri ile bunların etkidiği yerler belli olduğunda statik kararlılık kolaylıkla incelenebilir. Dinamik kararlılığı incelemek için referans denge konumundan bozuntularla saptırılan uçakta meydana gelen ilave kuvvet ve momentlerin zamanla değişimlerinin bilinmesi gerekir. Bu çalışmada kararlılık türevleri adı verilen bu değişimlerin, bir uçak için analitik olarak nasıl hesaplanabileceği açıklanmıştır. Fakat kararlılık türevlerinin gerçek değerlerinin deneylerle hesaplanacağı belirtilmiştir. Bundan dolayı deney amaçlı bir uçak önerilmiş ve bu uçağın birim impuls cevaplarına bakılarak kararlı olduğu belirtilmiştir. Birinci bölümde kararlılık türevleri ile ilgili genel bilgiler verilip bu konuda yapılan çalışmalar belirtildi. ikinci bölümde uçak aerodinamiğinin temel karakteristikleri anlatılıp uçak rijit kabul edilerek genel uçuş hareket denklemleri oluşturuldu. Genel hareketin uzunlamasına ve yanlamasına hareket olarak iki bileşenden oluştuğu belirtildi ve uzunlamasına hareket ele alındı. Bu hareketin denklemleri lineerleştirildi ve kararlılık türevleri denilen katsayılar cinsinden lineer bir denklem sistemi olarak ifade edildi. Üçüncü bölümde uçağın uzunlamasına hareketteki kararlılık türevlerinin analitik ve deneysel olarak nasıl hesaplanabileceği açıklandı. Bu hesap ile ilgili bir metod önerildi. Dördüncü bölümde ise ikinci bölümde önerilen metod ile kararlılık türevlerini hesaplayan bir bilgisayar programı (KARTUR) hazırlandı. Uygulama olarak iki uçak seçildi. Bunlardan biri deney amaçlı kullanılabilecek bir uçak diğeri ise Convair 880 uçağıdır. Bu bölümde deney amaçlı kullanılacak uçağın geometrisi belirtildi. Kararlılık türevlerinin hesabında uçağın geometrisinin önemi vurgulandı. Yatay dümen birim impuls cevaplarına bakılarak önerilen uçağın dinamik kararlı olduğu belirtildi. Karşılaştırma yapılabilmesi açısından Convair 880 uçağının da birim impuls cevaplan verildi.
Özet (Çeviri)
SUMMARY An aircraft must have static and dynamic stabilty to fly in safety. Static and dynamic stability of an aircraft are defined as follows. When a disturbance is applied to an aircraft; if forces and moments are constituted so as to bring the aircraft to its equilibrium then the aircraft is statically stable. If forces and moments which is constituted after a disturbance, brings the aircraft to its equilibrium position then the aircraft is dynamically stable. The static stability prior condition for dynamic stability i.e. a dynamically stable aircraft is always staticaly stable. The reverse is not true. The analysis of static stabilty can be performed easily when aerodynamic forces, thrust forces and their acting points are known. The dynamic stability analysis on the other hand, needs the time history of the forces and moments after a disturbance is applied to the aircraft. In this study, the static and dynamic stability properties of a small model aircraft is analysed. The flight conditions and the geometrical properties of this model aircraft ( KARTUR ) are tabulated in Table 1. Time history of forces and moments after a disturbance is applied to this model aircraft is studied in detail. This thesis is organised as follows. First chapter contains a literature survey related to stability derivatives. Second chapter examines basics of aerodynamics, center of pressure, aerodynamic center, lift, drag and aircraft axes. Aircraft equations of motion are derived in this chapter as below. Aircraft Equations of Motion The model aircraft is assumed to be rigid. The equations of motion related to the aircraft are as follows [5]. X = m(Ü+QW-RV) Y = m(V + RU-PW) Z = m(W + PV-Q-U)XIIL=PIXX-RIXZ + QR(IZZ-IYY)-PQIXZ M=QIYY-PR(IXX-IZZ)-R2IXZ + P2IXZ N=RIZZ-PIXZ + PQ(IYY-IXX) + QRIXZ P = 6-*Sin© Q = © Cos O + ¥ Cos 0 Sin O R = - © Sin O + W Cos © Cos O where X, Y, Z are forces, L, M, N are moments that act on the aircraft, U, V, W are aircraft velocities, P, Q, R are angular velocities in X, Y, Z direction respectively. O, 0, i[/ are euler angles. 1^, Iyy, 1^ and IM are inertial moments about X, Y, Z directions respectively. The above equations are in general nonlinear. They can be approximated to linear differential equations for certain flight conditions. For linearisation purpose, the motion variables are assumed to consist of two elements: i.e. a constant value and a small perturbation value P = P0 + p, Q = Qo + q, etc. With these assumtion, aircraft equations of motion can be approximated to linear differantial equations around those constant flight values. Of course approximotion is only valid if perturbation values are small compared to the variable itself. The linearised equations of motion contains long notations. A shorthand notation is used to clarify the equations. X =-- Y =-- Z =-- ymöy y m 3y y m dy 1 dL 1 ÖM xr 1 3N Lv = Mv = Nv = y I*x dy y Iw öy y I“ dy These terms are called stability derivatives. The value of stability derivatives depend on the aircraft geometry and flight conditions. The linearised differential equations which represents the aircraft equations of motion can be brought to standart form as follows [ 5 ]. X=AX+BU where X=[uvwpqr0iji]' and U = [ 8e 8a 5r ]'. A and B are matrices that contains stability derivatives coefficients. The elements of B are also called control xniderivatives. For most flight conditions the above equation can be sipiitted into two matrix differantial equations as follows X^AjXj+BjU, X2=A2X2+B2U2 where X,=[uwq0], X2=[vpr<|)]5 U,=[5E], U2 =[ 8a 8r ]. The first equation represents longitudinal motion and the second one represents lateral motion. As seen above, if the matrices A, B (or Aj, A2, B,, B2 ) are known, then dynamics of aircraft can be analysed easily. So the main problem becomes the calculation of those matrices i.e calculation of stability derivatives. The third chapter includes the calculation of stability derivatives of the model experimental aircraft. As mentioned above stability derivatives depend on lift coefficient, drag coefficient and pitching moment coefficient. The following stability derivatives (CLa, CDa, CMa, CL&, CDi, CMs, CLq, CDq, CMq, C^, C^, CM ) are analysed in this work. The rest of the stability derivatives are ignored due to their small values. The stability derivatives related to angle of attact (CL, CD, CM ) depend on aircraft geometry. The stability derivative CL is the change in lift coefficient with angle of attack and is commonly known as the lift curve slope. This derivative is always positive for angles of attack below the stall. Ordinarily, the wing accounts for 85% to 90% of the total Cha. This derivative is very important in equilibrium flight and in dynamic conditions. The derivative also makes an important contribution to the damping of the longitudinal short period mode. The stability derivative CD is positive in sign, since the drag coefficient increases as the angle of attack increases. CD usually has little effect on short period mode and has only a small effect on the phugoid mode in that a decrease in CD usually increases stability. CM is perhaps the most important derivative related to longitudinal stability and control, since it primarily establishes the natural frequency of the short period mode and is a major factor in determining the response of the airframe to elevetor motions and gusts. The CL, CD and CM values, related to the model experimental aircraft are tabulated in Table 2. Table 2: KARTUR's the stability derivatives related to angle of attack XIVThe stability derivatives CLft, C”4 and C^ are the changes in lift, drag and pitching moment coefficient with the rate of change of angle of attack. C, arises from a type of“plunging”motion along the z-axis, during which the angle of pitch, 0, remains zero. For low speed flight, the derivative results primarily from the aerodynamic time lag effect at the horizontal tail, and its sign is positive. CL can also arise from aeroelastic effects at high speed, but these aeroelastic contributions are negligible for light aircraft. Like C^, CD& arises from the aerodynamiclag effect and various“dead-weight”aeroelastic effects. Consequently, CDt is taken to be zero. CM& is quite important in longitudinal dynamics, since it is involved in the damping of the short period mode. A negative value of C^ increases short period damping; thus, high negative values are desirable. This derivative is actually caused by a lag effect of the downwash at the horizontal tail of the aircraft. The CL, CD. and Cj^ values, related to the model experimental aircraft are tabulated in Table 3. Table 3: KARTUR's the stability derivatives related to rate of change of angle of attack The stability derivatives CL, CD and CM represent the changes in airplane lift, drag and pitching moment with varying pitching velocity while the angle of attack of the airplane as a whole remains constant. The general concensus is that CL plays only a minor part in estimating the longitudinal response of the aircraft. CD has contributions from both the wing and the fuselage but both the contributions are very small. In all of the literature for subsonic flight, CD is ignored because it is really unimportant in analyzing flight dynamics and very small in magnitude. If an airplane has a positive pitch rate with a constant angle of attack (flying a curved flight path), the angle of attack at the tail is increased, thereby adding more positive lift to the tail and creating a moment to oppose the pitching motion. For this reason, the derivative is sometimes referred to as the“pitch damping”derivative and is usually negative. The wing contribution to CM either opposes or increases the pitching motion, depending on the e.g. location; however, this is relatively insignificant compared to the tail contribution. The fuselage contribution is always neglected for light airplanes. This particular derivative is very important in longitudinal dynamics because it plays a major role in the damping of the short xvperiod mode and a minor role in phugoid damping. The C,, Cn and CM values, 1 1 i related to the model experimental aircraft are tabulated in Table 4. The changes in lift coefficient, drag coefficient and pitching moment coefficient due to elevetor deflection are the stability derivatives CL, CD and CMg respectively. Since downward deflection of the elevetor is defined as positive, Table 4: KARTUR's the stability derivatives related to pitching velocity producing a positive lift, CL is normally positive in sign. For an elevetor of reasonable size, the total airplane drag does not change appreciably with elevetor deflection. For this reason, CDg is often neglected. Cu& usually referred to as“elevator power”or“elevator effectiveness”. If CM and the maximum deflection of the elevator are known, the maximum rotation moment which the tail can exert can be estimated. The CLs, CDg and CMj. values, related to the model experimental aircraft are tabulated in Table 5. Table 5: KARTUR's the stability derivatives related to elevator angle Chapter four contains A MATLAB program which is written in conjuction with this work calculates stability derivatives for a given. The matlab program, KARTUR, makes the calculation using the above techniques. A simulation program is added to KARTUR. A model experimental aircraft's stability derivatives are calculated and simulated for impulse and step inputs. Results show that the stability derivative values are in reasonable range. The overall results are discussed in chapter five. XVI
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