Chaptr 3 Lctur 14 Longitudinal stick fr static stability and control 3 Topics 3.4.4 Rquirmnt for propr stick forc variation 3.4.5 Fl of th stability lvl by th pilot Exampl 3.3 3.5 Dtrmination of stick-fr nutral point from flight tsts 3.4.4 Rquirmnt for propr stick forc variation Th stick forc variation is calld propr, whn (i) a pull forc is ndd to rduc th flight spd blow th trim spd and (ii) a push forc is ndd to incras th spd abov th trim spd. This rquirmnt is du to th following rasons. Whn th pilot wishs to rduc th spd blow th trim spd, h knows that th lift cofficint and hnc th angl of attack should incras or th nos should go up. For propr fl h should pull th stick or apply a pull forc. Whn h wishs to incras th flight spd abov th trim spd, a lowr angl of attack is ndd.thn, h should push th stick forward. Thus, for propr variation of stick forc, th gradint (df/dv) should b ngativ. In Eq.(3.31) it is obsrvd that C mδ is ngativ and (dc m /dc L ) stick-fr is also ngativ for a stabl airplan. Hnc, for (df/dv) to hav a small but ngativ valu, C hδ should hav a small ngativ valu. 3.4.5 Fl of th stability lvl by th pilot Th pilot fls th stability of th airplan through (df/dv). If (df/dv) is high, h fls that th airplan is stiff and hnc vry stabl. Howvr, if (df/dv) is vry low thn artificial mans ar mployd for propr fl. Friction in th control dflction linkag masks th fl and hnc, it nds to b kpt low. Dpt. of Arospac Engg., IIT Madras 1
Exampl 3.3 From th following additional data for th airplan in xampl 3., calculat and plot th stick-forc rquird vrsus quivalnt airspd for tab sttings of, 5, 1 and 15, Assum (dc m /dc L ) stick-fr = -.15. Cross plot ths curvs to giv tab stting for zro stick forc vrsus quivalnt airspd (1 to 3 kmph). S = 1.8 m, c =.6m,G = 1.6 pr mtr, α Lw = -, δ CL = + 4. Solution: Th data givn ar: C Lαw =.85 dg -1, C Lαt =.58 dg -1, dc Lt / dδ =.3 dg -1, C hαt = -.3 dg -1, C hδ = -.55 dg -1, C hδt = -.3 dg -1, i w =, i t = -1, α Lw = -, S t =.5 S, l t / c =3, a.c. at.5 c, η =1., (C mα ) f,n,p =.37/ rad. S = 1.8 m, c =.6m, G = 1.6/ m. δ CL = 4, (dc m /dc L ) stick-fr = -.15, W/S = 15 N/m From Eq.(3.4) : 1 W C F=K ρ V {A + C δ } - K (dc /dc ) S C K=G S c η; A=C {α - i + i )+ C δ K =1.6 1.8.6 1 = 1.78 hδ hδt t m L stick-fr mδ hαt Lw w t hδ CL A = -.3{- - -1}+(-.55)(+ 4 ) =.9-. = -.13 C = - V η C = -.5 3 1..3 57.3 = -1.375rad ; not C W Chδ dcm -.55 57.3 K ( ) stick-fr = 1.78 15 (-.15) = - 89.1 S Cmδ dcl -1.375 1 1 3 ρv = ρv; ρ = 1.5 kg/m 1 F = 1.78 ρv {-.13+(-.3)δ t} + 89.1 dc dδ -1 Lt mδ H Lαt Lαt = 1.59 V {-.13 -.3δ }+89.1; For δ t = : F = 89.1 -.1377 V t (E 3.3.1) Dpt. of Arospac Engg., IIT Madras
Th valus of F for diffrnt valus of V with δ t qual to,.5, 5, 7.5 and 1 ar tabulatd in Tabl E3.3a.Th valus ar plottd in Fig. E3.3a. V (kmph) V (m/s) F(N) From Eq. (E 3.3.1) δ t =.5 5 7.5 1 89.1 89.1 89.1 89.1 89.1 1 7.78 78.47 7.35 66. 6.1 54. 15 41.7 65.19 51.4 37.5 3.81 1. 55.56 46.59.8 -.43-6.97-51.5 5 69.44.7-15.58-53.87-9. 3 83.33-6.5-61.65 Tabl E3.3a F vs V with δ t as paramtr 1 F (N) 8 6 4 - -4-6 -8-1 δ t = V (kmph) 1 3 4 δ t =.5 δ t =1 δ t = 5 δ t =7.5 Fig. E3.3a Variations of stick forc vs quivalnt airspd with tab dflction as paramtr Dpt. of Arospac Engg., IIT Madras 3
Cross plotting th data in Fig.E3.3a givs th valus of (δ t ) trim for diffrnt valus of quivalnt airspd. Altrnativly, from Eq.(E3.3.1), (δ t ) trim can b valuatd as: = 1.59 V {-.13 -.3(δ t) trim} + 89.1 845 Or (δ ) = - 4.33 + with V in m/s t trim V 363463 =- 4.33 + withv in kmph V Th valus of (δ t ) trim at diffrnt quivalnt airspds ar tabulatd in Tabl E3.3b and plottd in Fig.E3.3b. Sinc, th wing loading is givn as 15 N/m w can calculat th lift cofficint in lvl flight (C L ) can b calculatd as: W 449 W C = = for = 15N / m and ρ = 1.5kg / m 3 L ρsv V S Th valus of C L ar also shown in Tabl E3.3b. Th plot of (δ t ) trim vs C L is shown in Fig. E 3.3 c. V (kmph) V (m/s) C L (δ t ) trim (dgrs) 15 41.67 1.41 11.8 55.56.793 4.76 5 69.44.58 1.49 89.7 8.47.378 3 83.33.35 -.9 Tabl E3.3b Tab dflction for trim at diffrnt quivalnt airspds and lift cofficints Dpt. of Arospac Engg., IIT Madras 4
(δ t ) trim (dgrs) 14 1 1 8 6 4 V (kmph) - 1 3 4-4 Fig. E3.3b Tab dflction for trim at diffrnt quivalnt airspds (δ t ) trim in dgrs 14 1 1 8 6 4 -.5 1 1.5 C L -4 Fig. E3.3c Variation of (δ t ) trim with C L Dpt. of Arospac Engg., IIT Madras 5
Rmark: Th (δ t ) trim vs C L curv in Fig. E3.3c is linar as th non-linar trms in th xprssion for C mα hav bn ignord. For actual airplans th curvs from flight tsts will b slightly non-linar. S Rf..5 chaptr 3 and Rf.1.7, chaptr 6. 3.5 Dtrmination of stick-fr nutral point from flight tsts In sction.13, a procdur was xplaind to obtain th stick-fixd nutral point (x NP ) from flight tsts. It was basd on th fact that whn c.g. is at x NP thn dδ /dc L is zro. Similarly to obtain stick-fr nutral point from flight tsts, two ways ar suggstd by Eqs. (3.7) and (3.9). Equation (3.7) suggsts that d(f/q) / dc L is zro whn (dc m /dc L ) stick-fr is zro and Eq.(3.9) suggsts that (dδ t /dc L ) is zro whn (dc m /dc L ) stick-fr is zro. Thus, by masuring ithr F or δ t during th flight tsts at diffrnt flight vlocitis and at diffrnt c.g. locations, d(f/q) /dc L or (dδ t /dc L ) can b obtaind for ths c.g. locations. Extrapolating ths curvs givs th nutral point stick-fr. For dtails s Rf.1.7 chaptr 6. Rmark: It may b rcalld that (a) th xact contribution of wing is slightly non-linar and (b) th contribution of powr changs with C L. Hnc, th δ t vs C L and (F/q) vs C L curvs from flight tsts show slight non- linarity. Consquntly, th stick fr nutral point location also shows slight dpndnc on th lift cofficint similar to that shown in Fig..37. Dpt. of Arospac Engg., IIT Madras 6