Flight dynamis II Stability and ontrol hapter 2 Leture 8 Longitudinal stik fied stati stability and ontrol 5 Topis 2.6 ontributions of power plant to mg and mα 2.6.1 Diret ontributions of powerplant to mg and mα 2.6.2 Indiret ontributions of powerplant to mg and mα 2.7 General remarks slope of lift urve ( Lα ) and angle of zero lift (α 0L ) of airplane 2.7.1 Slope of lift urve ( Lα ) of the airplane 2.7.2 Angle of zero lift of the airplane 2.8 mg and mα of entire airplane 2.9 Stik-fied neutral point 2.9.1 Neutral point power-on and power-off 2.10 Stati margin 2.11 Neutral point as aerodynami entre of entire airplane 2.6 ontributions of power plant to mg and mα The ontributions of power plant to mg and mα have two aspets namely diret ontribution and indiret ontribution. 2.6.1 Diret ontribution of power plant to mg and mα The diret ontribution appears when the diretion of the thrust vetor does not oinide with the line passing through the.g.(fig.2.24). The diret ontribution is written as : M gp = T Z p (2.59) where, T is the thrust and Zp is the perpendiular distane of thrust line from FRL; positive when.g. is above thrust line. In non-dimensional form Eq.(2.59) is epressed as: Dept. of Aerospae Engg., IIT Madras 1
Flight dynamis II Stability and ontrol mgp = M gp /(½ ρv 2 S ) (2.60) Fig.2.24 ontribution of thrust to mg The thrust required varies with flight speed and altitude. Hene, mgp would vary with flight ondition. However, the thrust setting does not hange during the disturbane and hene, there is no ontribution to mα. This fat is also mentioned in Ref.1.9. p.506. The ontribution to mα omes from another ause. onsider a propeller at an angle of attak as shown in Fig.2.25. The free stream veloity (V) is at an angle (α)to the propeller ais. As the air stream passes through the propeller it leaves in a nearly aial diretion. This hange of diretion results in a normal fore (N p ) in addition to the thrust (T). Fig.2.25 Propeller at angle of attak Dept. of Aerospae Engg., IIT Madras 2
Flight dynamis II Stability and ontrol Fig.2.26 ontribution to mα from normal fore due to propeller N p ats at distane l p from the.g. (Fig.2.26) and hene, produes a moment N p l p. The value of N p depends on the angle of attak of the propeller and hene the term N p l p depends on α. This will ontribute to mα. mα due to normal fore depends on many fators like thrust setting, number of blades in the propeller and advane ratio. Remarks: i) It is evident from Fig.2.26 that when the propeller is ahead of.g., the ontribution to mα due to normal fore would be positive or destabilizing. In a pusher airplane, where the propeller is near the rear end of the airplane, the ontribution of normal fore to mα will be negative and hene stabilizing. ii) In the ase of a jet engine at an angle of attak, the air stream enters the intake at that angle and its diretion has to hange as the stream passes through the engine. This hange of diretion will also produe a normal fore N p and onsequently ontribute to mα. 2.6.2 Indiret ontributions of power plant to mg and mα The effet of propeller on the horizontal tail has been disussed in setion 2.4.3. In the ase of an airplane with a jet engine, the ehaust epands in size as it moves downwards and entrains the surrounding air. This would indue an angle to the flow; the indued angle would be positive in the region below the jet Dept. of Aerospae Engg., IIT Madras 3
Flight dynamis II Stability and ontrol and negative in the region above the jet. In military airplanes where the engine is loated in rear fuselage the engine ehaust would affet the horizontal tail, generally loated above the rear fuselage, by induing a downwash in addition to that due to wing. This effet will also ome into piture in ase of passenger airplanes with rear mounted engines. To alleviate this, the horizontal tail is mounted above the vertial tail (see onfigurations of Boeing MD-87 and Gulf stream V in Ref.2.3). Remarks: i) The ontribution of engine depends also on the engine power setting whih in turn depends on flight ondition or L. Hene, the level of stability ( mα ) will depend on L and also will be different when engine is off or on. ii) It is diffiult to aurately estimate the effets of power on mα. A rough estimate would be (Ref.1.7, hapter 5) : (d m / d L ) p = 0.04 or mαp = 0.04 Lα (2.60a) 2.7 General Remarks: 2.7.1 Slope of lift urve ( Lα ) and angle of zero lift (α 0L ) of the airplane: Let, L denote lift of airplane. Then, L = L wb +L t. For airplanes with large aspet ratio wings (A>5), the lift of the wing body ombination is approimately equal to lift produed by the gross wing i.e.. L wb L w Noting that L t = ½ρV 2 t S t (α ε + i t ) and L w = ½ ρv 2 S Lw ; the slope of the lift urve of the airplane ( Lα ) an be written as : Lα = Lαw + η (S t /S) {1-(dε/dα)} (2.60b) Referene 1.8 b gives epressions for orretions to obtain Lαwb from Lαw (see also Appendi setion 5 ). 2.7.2 Angle of zero lift (α 0L ) for airplane: Assuming that the wing is set suh that during ruise the angle of attak of the airplane (α r ) is zero, the lift oeffiient during ruise ( Lr ) an be written as : Lr = Lα (α r - α 0L ) = Lα (0 - α 0L ) Hene, α 0L = - Lr / Lα (2.60) Dept. of Aerospae Engg., IIT Madras 4
Flight dynamis II Stability and ontrol 2.8 mg and mα of entire airplane The important result of the last few setions an be reapitulated as follows. mg = ( mg) w +( mg) f +( mg) n +( mg) p +( mg) ht (2.12) mα = ( mα) w+( mα) f +( mα) n+( mα) p+( mα) ht (2.13) The wing ontribution is: g a mgw = maw + Lw( - ) = (α +i - α ) Lw Lαw w 0Lw = + α ; = (i - α ) L0w Lαw L0w Lαw w 0Lw g a g a mgw = maw + L0w( - )+Lαw α( - ) g a m0w = maw + L0w( - ) g a ( mα) w = Lαw( - ) The tail ontribution is: (2.17) (2.18) (2.19) (2.19a) (2.20) mgt = - V H η Lt (2.37) Lt = α t + Lδe δ e + Lδt δ t (2.38) α t = α - ε + i t = αw- iw - ε + i t (2.39) dε dε ε = ε 0+ α; ε 0 = (iw - α 0Lw ) dα dα dε = i - ε + α(1- ) + δ + δ dα Lt t 0 Lδe e Lδt t (2.41) (2.46) dε mgt = -VH η {it - ε 0+ α (1- )+ δ e+ tabδ t} (2.47) dα Lδe = ; tab = Lδ t Lα t dε ( mαt ) stik-fied = -VH η (1- ) (2.50) dα The ontributions of fuselage, naelle and power are epressed together as: Dept. of Aerospae Engg., IIT Madras 5
Flight dynamis II Stability and ontrol ( m) f,n,p = ( m0) f,n,p + ( mα) f,n,p α (2.61) Substituting various epressions in Eqs.(2.12) and (2.13) gives: mg = m0 + mα α + mδe δ e (2.62) g a m0 = maw + L0W ( - )+ ( m0) f,n,p-vh η {it - ε 0+ tab δ t} (2.63) mδe = -VH η (2.64) g a ( mα) stik fied = dε Lαw( - )+( mα) f,n,p - VH η (1- ) dα (2.65) Typial ontributions of the individual omponents and their sum, namely mg for a low subsoni airplane are shown in Fig.2.27. The details of the alulations are given in eample 2.4. Fig.2.27 mg vs α for a low subsoni airplane Following observations an be made in this ase. (a) mow has an appreiable negative value. Dept. of Aerospae Engg., IIT Madras 6
Flight dynamis II Stability and ontrol (b) mαw depends on the produt of Lαw and g - a. In the ase onsidered in eample 2.4, the.g. is at 0.295 and the a.. is at 0.25. Sine,.g. is aft of the aerodynami entre, the ontribution of wing is destabilizing (Fig.2.27). ( ) mof has small negative value and mαf has small positive value, indiating a slight destabilizing ontribution from fuselage (Fig.2.27). (d) mot is positive and mαt has a large negative value (Fig.2.27). (e) The line orresponding to the sum of all the ontribution ( wing+ fuselage+ power+tail) is the mg vs α urve for the whole airplane. The ontribution of naelle is ignored. It is seen that the large negative ontribution of tail renders mα negative and the airplane is stable. 2.9 Stik-fied neutral point It may be pointed out that the.g. of the airplane moves during flight due to onsumption of fuel. Further, the ontribution of wing to mα depends g a sensitively on the loation of the.g. as it is proportional to( - ). When the.g. moves aft, g inreases and the wing ontribution beomes more and more positive. There is a.g. loation at whih ( mα ) stik-fied beomes zero. This loation of.g. is alled the stik-fied neutral point. In this ase, the airplane is neutrally stable. Following Ref.1.1 this loation of the.g. is denoted as NP. If the.g. moves further aft, the airplane will beome unstable. The m vs. α urves for the statially stable, neutrally stable and unstable ases are shematially shown in Fig.2.28. Dept. of Aerospae Engg., IIT Madras 7
Flight dynamis II Stability and ontrol Fig.2.28 hanges in stati stability with movement of.g. (Shemati) An epression for NP an be obtained by putting mα = 0 and g = NP, in Eq.(2.65) i.e. NP a dε 0 = Lαw( - )+( mα) f,n,p - VH η (1- ) dα 1 dε NP a Hene, = - {( mα) f,n,p - VH η (1- )} Lαw dα Eample 2.4 illustrates the steps involved in arriving at the neutral point. 2.9.1 Neutral point power-on and power-off The ontribution of power is generally destabilizing and hene, the airplane will be more stable when engine is off. In other words, NP power off is behind NP power on. 2.10 Stati margin Noting the definition of NP g NP ( mα) stik-fied = Lαw( - ) (2.66) (2.67) from Eq.(2.67), the Eq.(2.65) an be rewritten as : (2.68) Dept. of Aerospae Engg., IIT Madras 8
Flight dynamis II Stability and ontrol g NP Thus, ( mα ) stik-fied is proportional to ( - ) and a term alled stati margin is defined as: Stati margin = NP g ( - ) (2.69) onsequently, ( mα ) stik-fied = - Lαw (stati margin) (2.70) and d d m L stik-fied = -(stati margin) 1 = ( mα) Lα stik-fied (2.71) It may be noted that stati margin, by definition, is positive for a stable airplane. 2.11 Neutral point as the aerodynami entre of entire airplane To eplain the above onept, the derivation of the epression for neutral point in the Ref.1.10. hapter 2 is briefly desribed. The wing ontribution ( mgw ) is epressed as: g a mgw = maw +αw Lαw ( - ) The ontributions of fuselage and naelle are aounted for by treating them as hanges in the following quantities: (a) pithing moment oeffiient is hanged from maw to mawb, (b) the angle of attak is hanged from α w to α wb, () the slope of the lift urve is hanged from Lαw to Lαwb and (d) aerodynami entre is hange from a to awb. The suffi wb indiates ombined effets of wing body and naelle. onsequently, g awb mgwb = mawb +αwb Lαwb ( - ) The ontribution of power is epressed as mgp. The ontribution of the horizontal tail is epressed as : mgt = - V H Lt ; note η = 1.0(assumed) Where, S t t V H = S l l t = distane between the aerodynami entre of the wing-body-naelle Dept. of Aerospae Engg., IIT Madras 9
Flight dynamis II Stability and ontrol ombination ( awb ) and the aerodynami entre of the horizontal tail. It is assumed that the L and Lα of the airplane are approimately equal to Lwb and Lαwb respetively. The epression for mg an now be written as: = + ( - ) - V + g awb mg mawb L H Lt mgp or = ( - )- V + g awb mα Lα H mαp The neutral point, NP, is given by: 1 NP awb = - (mαp - VH ) Lα (2.62a) (2.65a) (2.67a) g NP or mα = Lα ( - ) = - Lα (stati margin) (2.70a) It may be realled that the aerodynami entre of an aerofoil is the point about whih the pithing moment is onstant with angle of attak. Similarly, the aerodynami entre of the wing ( a ), by definition, is the point about whih maw is onstant with angle of attak. With this bakground, the quantity awb an be alled as the aerodynami entre of the wing - body - naelle ombination. Further, when the.g. is at neutral point, mα is zero or mg is onstant with α. This may be the reason Ref.1.10, hapter 2 refers the neutral point as the aerodynami entre of the entire airplane. Remark: There are some differenes in the epressions on the right hand sides of Eq.(2.67) and (2.67a) and Eq.(2.70) and (2.70a). These differenes are due to slight differene in treatment of the ontributions of individual omponents. The differenes in Eq.(2.70) and (2.70a) an be reoniled by noting that for airplanes with large aspet ratio wings, Lα Lαw. Referene 1.12, hapter 3 also mentions of this approimation to Lα. It may be realled that epression for slope of lift urve of the airplane is obtained in subsetion 2.7.1. Referene 1.8b also epresses d m mα = ( )Lα dl Dept. of Aerospae Engg., IIT Madras 10
Flight dynamis II Stability and ontrol where, Lα is the slope of the lift urve of the airplane and d m g = - L a d where, a is the loation of neutral point. Thereby treating neutral point as the aerodynami entre of the airplane (see Appendi setion 5.3). Dept. of Aerospae Engg., IIT Madras 11