Chapter 2 Lecture 7 Longitudinal stick fixed static stability and control 4 Topics

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1 hapter 2 Lecture 7 Longitudinal stick ixed static stability and control 4 Topics Revised expression or mcgt mαt in stick-ixed case 2.5 ontributions o uselage to mcg and mα ontribution o body to mα based on slender body theory orrection to moment contribution o uselage or ineness ratio orrection to moment contribution o uselage or non-circular cross-section orrection to moment contribution o uselage or uselage camber ontribution o nacelle to mα Revised expression or mcgt Substituting or Lt in Eq.(2.37) gives: mcgt = -VHη Lαt[it - ε 0+α(1- )+ δ e+ tabδ tab] = mot - V H η Lαt [α(1- )+ δ e+ tab δ tab] (2.47) where, = / ; tab = / δ α δ α Lt Lt Lt Lt e t (2.48) and mot = - V H η Lαt (i t - ε 0 ) (2.49) mαt in stick-ixed case It may be pointed out that the pilot moves the elevator through the orward and backward movements o the control stick. Further, depending on the values o chosen light speed and altitude, the pilot adjusts the positions o the elevator to make mcg equal to zero. In small airplanes, like the general aviation airplanes, the pilot continues to hold the stick and maintain the elevator delection. In this background the analysis o the static stability o an airplane where the control Dept. o Aerospace Engg., IIT Madras 1

2 delection remains same even ater disturbance is called static stability stickixed. Hence, to obtain an expression or in stick-ixed case it is assumed that the mαt elevator delection (δ e ) and the tab delection (δ t ) remain unchanged ater the disturbance. Accordingly when Eq.(2.47) is dierentiated with respect to α,the derivatives o δ e and δ t are zero i.e., in this case, (dδ e /) = (dδ t /) = 0 and the ollowing result is obtained: ( mαt ) stick ixed = - VH η Lαt(1- ) Remarks: i) mαt is negative. To illustrate this, consider typical values as: η = 0.9, V H = 0.5, Lαt = 4.0 per radian and / = 0.4. Then, mαt = -0.5 x 0.9 x 4 x (1-0.4) = /radian. ii) V H depends on (S t /S) and (l t / c ). Hence, the contribution o tail to stability ( mαt ) can be increased in magnitude by increasing (S t /S) or (l t /c ) i.e. by increasing the area o the horizontal tail or by shiting the tail backwards. iii) m0 is the value o mcg when α is zero. It ( m0 ) is the sum o terms like (2.50) mow, mot etc. This value ( m0 ) can be adjusted by changing m0t. In this context we observe rom Eq.(2.49) that: m0t = - VH η Lαt(it - ε 0) This suggests that by choosing a suitable value o i t, the value o mo can be adjusted. This would permit trim, with zero elevator delection, at a chosen value o lit coeicient (see Fig.2.6 and example 2.5). The chosen value o L or this purpose is invariably the value o L during cruise. This serves as criterion or selecting tail setting. iv) In the beginning o this section a reason or examination o the stick-ixed stability was given by considering the case o general aviation airplane. However, the analysis o stick-ixed stability is carried out or all airplanes and the level o ( mαt ) stick-ixed decides the elevator delection required in steady light and in manoeuvres (see subsections and 4.2). Dept. o Aerospace Engg., IIT Madras 2

3 2.5 ontributions o uselage to mcg and mα Fuselage and nacelle are classiied as bodies. The steps or estimating the contributions o a body to mcg and mα are based on the descriptions in chapter 2, o Re.1.1, chapter 5 o Re.1.7 and chapter 3 o Re In this approach, the contribution o the body to mα is estimated based on the slender body theory and subsequently applying corrections or the eects o (a) inite ineness ratio, (b) non-circular cross section, (c) uselage camber and (d) downwash due to wing ontribution o body to mα based on slender body theory The potential low past and an axisymmetric slender body was studied by Munk in 1924 (see Re.1.1, chapter 2 or bibliographic details). He showed that a body at an angle o attack has a pressure distribution as shown in Fig.2.18 and produces no net orce, but a moment. He showed that the rate o change o moment with angle o attack α, in radians, is given by: dm 1 = 2q volume o body ;q = ρv 2 Alternatively when α is in degrees, 2 dm Volume o body q = 28.7 (2.51) Fig.2.18 Streamlines and potential low pressure distribution on an axisymmetric body; the negative and positive signs indicate respectively that the local pressure is lower or higher than the ree stream pressure Dept. o Aerospace Engg., IIT Madras 3

4 Remark: In a viscous low, the pressure distribution about the body changes and it experiences lit and drag orrection to moment contribution o uselage or ineness ratio Generally the uselage has a inite length. For such a uselage Multhopp in 1942 (see Re.1.1, chapter 2 or bibliographic details) suggested the ollowing correction to Eq.(2.51). dm (k2-k 1) = q x volume o body 28.7 (2.52) Where, (k 2 -k 1 ) is a actor which depends on the ineness ratio (l /d e ) o the body; l is the length o the body and d e is the equivalent diameter deined as: (π/4)d e 2 = max. cross sectional area o uselage. Figure 2.19 presents variation o (k 2 -k 1 ) with ineness ratio Fig.2.19 orrection to moment contribution o uselage or ineness ratio (Adapted rom Re.2.2, section ) orrection to moment contribution o uselage or non-circular cross section For a uselage o non-circular cross-section, Eq.(2.52) is modiied as: l l dm (k2-k 1)q π 2 (k2-k 1)q = w dx = w 2 dx (2.53) 0 0 Dept. o Aerospace Engg., IIT Madras 4

5 where, w is the local width o the uselage. Hence, the contribution o uselage to mα can be expressed as : dm mα = (2.54) 1ρV 2 Sc orrection to moment contribution due to uselage or uselage camber and downwash due to wing In an airplane, the low past a uselage is aected by the upwash-downwash ield o the wing (Fig.2.12). Further, the midpoints o the uselage cross sections may not lie in a straight line. In such a case the uselage is said to have a camber (Fig.2.20). A uselage with camber would produce a pitching moment coeicient ( mo ) even when FRL is at zero angle o attack. Fig.2.20 Fuselage with camber For a uselage with camber, mcg is expressed as: mcg = m0 + mαα (2.55) with k -k = w (α +i )Δx and (2.56) 36.5Sc l m0 0L x=0 Dept. o Aerospace Engg., IIT Madras 5

6 k -k mα = w Δx 36.5Sc l x=0 (2.57) where, (a) w is the average width over a length Δx o uselage (Fig.2.21) (b) i is the incidence angle o uselage camber line with respect to FRL. It is taken negative when there is nosedrop or at upsweep (Fig.2.20). (c ) α 0L is the zero lit angle o wing relative to FRL i.e. α 0L = α 0Lw + i w and (d) / is the derivative with α o the local value o upwash / downwash along the uselage. Fig.2.21 Division o uselage or calculation o m0 Though / along the uselage can be calculated rom an approach like the liting line theory, the ollowing emprical procedure is generally regarded adequate or evaluating mα. a) The uselage is divided into segments as shown in Fig.(2.22). b) The local value o, / ahead o the wing is denoted by u /. It is estimated rom Fig.(2.23). For the segment immediately ahead o the wing (section 5 in Fig.2.22) the value o u / varies rapidly and is estimated rom the curve b in Fig.2.23 (see example 2.4). For other segments ahead o wing, the curve a in the same igure is used to estimate u /. (c) For the portion o the uselage covered by the wing root (length c indicated in Fig.2.22) / is taken as zero. Actually, the contribution o this portion is taken to be zero as, this portion is accounted or under the wing area. Dept. o Aerospace Engg., IIT Madras 6

7 Fig.2.22 Division o uselage or calculation o mα Fig.2.23 Upwash ield ahead o wing (Adapted rom Re.2.2, section ) Dept. o Aerospace Engg., IIT Madras 7

8 (d) For the portion o the uselage behind the wing (segments 6 to 11 in Fig.2.22), / is assumed to vary linearly rom 0 to {1- (/) tail } where (/) tail is the value o (/) at a.c. o tail. Hence, x i = [1- ( ) tail] lt (2.58) The procedure is illustrated in example 2.4 or a low speed airplane and in Appendix or a jet airplane Remarks: i) In Re.2.2, the quantity / o Eq.(2.57) is written as (1+ / ) and values o / therein are accordingly lower by one as compared to those in Fig ii) The values in Fig.2.23 are or a LαWB o /deg. LαWB is the slope o lit curve o the wing-body combination which is roughly equal to LαW when the aspect ratio o the wing is greater than ive. For other values o LαW multiply the values o / by a actor o ( LαW /0.0785). Note that LαW is in deg -1. See also example ontribution o nacelle to mα The contribution o nacelle to mα can be calculated in a manner similar to that or the uselage. Generally it is neglected. Dept. o Aerospace Engg., IIT Madras 8