Cranfïeld
College of Aeronautics Report No. 8312May 1983
A Note on the Estimation of Longitudinal and Lateral Aircraft Derivatives using
Semi-Empirical Methods
by R. THORNE
College of Aeronautics Cranfield Institute of Technology
Cranfield, Bedford, U.K.
TECHNISCHE HOGESCHOOL DELFT
LUCHTVAART- EN RUIMTEVAARHECHNIEK BIBLIOTHEEK
Kluyverweg 1 - DELFT
Cranfield
College of Aeronautics Report No. 8312May 1983
A Note on the Estimation of Longitudinal and Lateral Aircraft Derivatives using
Semi-Empirical Methods
by R. THORNE
College of Aeronautics Cranfield Institute of Technology
Cranfield, Bedford, U.K.
ISBN 0 902937 85 5 £7.50
"The views expressed herein are those of the authors alone and do not necessarily represent those of the Institute."
Summary
A number of semi-empirical and theoretical techniques are used to estimate the longitudinal and lateral aerodynamic derivatives of a twin turbo prop aircraft, the H.P. Jetstream.
Estimates of the aircraft derivatives are performed
for the aircraft in a typical cruise condition. Longitudinal parameters are calculated using classical theory and lateral derivatives are calculated using the semi-empirical methods of ESDU.
(Note prepared for C.I.T. Short Course -'Flight Mechanics and Aircraft Control' 23 May - 3 June 1983)
Contents Notation 1. Introduction 2. Data 3. Estimation of Derivatives 3.1 General 3.2 Longitudinal Derivatives 3.2.1 Preliminary calculations 3.2.2 Calculations 3.2.3 Summary of estimates 3.3 Lateral Derivatives 3.3.1 Preliminary calculations 3.3.2 Calculations 3.3.3 Summary of estimates References Table Figure
Appendix A Definition of Derivatives
Appendix B Linearised Equations of Motion
Notation
The notation used is based on that proposed in ARC R&M
3562 (Hopkin, 1970). In particular for aeronormalised derivatives, 'dip' notation will be used, e.g. L .
Only some of the more important parameters are listed here and for quantities not listed here or defined in the text, the reader should refer to the appropriate ESDU item or other reference given.
2 A Aspect ratio of wing = b /S Ap Effective aspect ratio of fin A-, Tailplane aspect ratio
aji^ Tail lift curve slope b Span of wing
b ^ Tailplane span
(Cj^^)p Lift-curve slope of fin (ESDU Item 70011) c Fin root chord
c. „ Fin tip chord
c Tailplane tip chord d Maximum body width h Maximum body height
h ' Wing vertical position relative to body (defined in ESDU Item 73006)
h, jh- Body section heights at 0.251, , 0.75^,
h„„ Body height at fin root quarter chord section tit
h Body height at wing root quarter chord section h_ Height of fin from fin root chord
1 Distance of yaw axis from nose ^, Body length
m Fin root quarter chord station aft of moment reference centre
S Maximum cross-sectional area of body S-, Fin area
F
S„ Tailplane area
S Wing reference area
S , Area of side elevation of body x„ Fin arm F
z Height of fin root chord from fuselage datum crp
Height of centre of pressure position of fin from fin root chord.
Height of intersection of fin mounted tailplane from fin root chord.
Height of wing root quarter chord above local body centre line (positive for low wings).
Dihedral of wing
1. Introduction
In this note, the aerodynamic derivatives for the H.P. Jestream aircraft are estimated at one flight condition using
semi-empirical and theoretical techniques. Sufficient data for the aircraft is given to enable estimates to be made for all the significant longitudinal and lateral derivatives at a typical cruise condition of 150 kts, 6500 ft altitude.
Longitudinal derivatives are estimated using simple theory based on consideration of the lift, drag and pitching moment acting on the aircraft. Ref. 1 contains details of the derivation of formulae used.
Lateral derivatives are estimated using semi-empirical methods based on experimental test data. Theoretical
corrections are used as guides to the effects of aircraft geometry, angle of attack, etc. Widely available sources of data are
those of ESDU^ and DATCOM "^ , and in this note, ESDU items will be used for the lateral derivatives. Wind axes (aerodynamic body axes) will be used except where stated but transformations to other axes systems can be carried out as detailed by Hopkin'^.
This note is only intended to give an introduction to the practical estimation of aerodynamic derivatives and so only major effects are considered. The aim has been to introduce
the reader to the techniques used. Once familiar with these techniques, reference works such as ESDU or DATCOM can be
consulted and worked from in order to make allowance for effects not covered in this introductory exercise.
2. Data
The subject aeroplane, shown in 3-view in Fig. 1, is the H.P. Jetstream, Series 100. Data and dimensions for the aircraft are given in Appendix C, along with datum flight conditions.
Additional information may be found in Ref. 5. The main parameters used from this reference, appropriate to the
estimation exercises are:
(^La^WB = 5.2/rad
9CLT
a, = = 4.3/rad 3a
de
Rate of change of downwash, — = 0.312 da
The reader is referred to Ref. 5 pp. 4-10 for geometrical data on the Jetstream collected from original authoritative drawings.
3. Estimation of Derivatives
3.1 General
The Jetstream is a turbo prop aircraft with wing mounted engines. In a detailed analysis of the aerodynamic derivatives the engine effects would be accounted for. However, these are beyond the scope of the present excersise and so are not
considered. The engine nacelles will have a small effect on the lateral derivatives and as a first approximation, the calculations described should provide a reasonable first estimate of the derivatives.
Propeller power effects on the longitudinal aerodynamic characteristics are mentioned in Ref. 5. Van Rooyen° describes the influence of propellers on the lateral-directional stability of multi-engined aircraft, including the Jetstream.
Control derivatives are not estimated in this exercise. For details of these the reader is referred to such works as Ref. 1.
3.2 Longitudinal Derivatives
3.2.1 Preliminary Calculations dC
Quantities such as C-., — - etc will first be ^^L
evaluated for the flight condition under consideration (details given in Appendix C ) .
3.2.1.1 Cj^
From Ref. 5, at a C, of 0.72, C„ = 0.079
Ref. 5, Table 1.6.1, gives C, = 5.85/rad for M = 0.2.
3.2.1.3 Vjj
Tail volume coefficient, V =
^ cS = 1.119
3.2.1.4 dC /dC^ m' L
dCj^
Ref. 5 gives = -O.236 including effects of fuselage ^^L
and nacelles.
3.2.2 Calculations
The following longitudinal derivatives are estimated;
X,,, Z „ , X, , Z , M , Z „ , M , M ' . D e r i v a t i o n s of t h e formula
« - l u w w w q q w
used in estimating these can be found in Ref. 1.
For practical purposes some of the longitudinals can be equated to zero and consequently are not estimated here, These are X , M and Z..
q u w 3.2.2.1 X u X^ = - 2C. - V
^S
D 9V 8Cp X = - 0.158 ignoring the contribution from Vu 3V 3.2.2.2 Z u 8C^ Z = - 2C^ - V ^ 3V
7 = - 1.4 4 iqnorinq the contribution from second u
3.2.2.3 X^
3a
X = 0.72 ignoring contribution from the second term. 3.2.2.4 Z w 3C 3a = -0.079 - 5.85 3 . 2 . 2 . 5 ^w
K
•" = s - 5 . 9 3 m 3a ^^m ^^L 3a = -0.236 X 5.85 M^ = -1.381 3.2.2.6 Z_2
Zg - . a^ S.c = -^H^l = -1.119 X 4.3 Zg = -4.83 . 2 . 2 . 7 M 2 M^ - ~ — • T • ^ 1 1 T = . Vy. — . a^ c M = - 1 7 . 3 4 q 3 . 2 . 2 . 8 M. w "w - ^H — • ^ 1 • — c da = M — "^ da M. = - 5 . 4 1 w 3 . 2 . 3 Summary of e s t i m a t e s X u Z u ^w Z w M w \ M q ^w — = = = = = =-= - 0 . 1 5 8 - 1 . 4 4 0 . 7 2 - 5 . 9 3 - 1 . 3 8 1 - 4 . 8 - 1 7 . 3 - 5 . 4 1
3.3 Lateral Derivatives
3.3.1 Preliminary calculations
Using the data given in Appendix C for the Jetstream, we first of all derive parameters needed in the ESDU methods of estimating lateral derivatives.
The span of the aircraft, b, is used as the characteristic 4 length in forming the aeronormalised derivatives (see Hopkin ) .
3.3.1.1 Body
The moment reference centre (jn.r.c.) will be taken at a distance i = 5.66 m from the nose.
Height of wing root quarter chord above local body centre -line, z = 0.8 m.
w
h„^, height of body at wing root quarter chord section = 1.981 m.
S , maximum cross-sectional area of body = 3.08 m^.
S , , area of side elevation of body = 19.60 m^.
h-,, h^, body section heights at 0.251,, 0.751, = 1.78 m, 1.0 m respectively.
3.3.1.2 Wing
A = 0.3333
A = 1 O
^c/4 ^
For M = 0.23, g = A - M = 0.973 which gives the equivalent E
^E = 1
^^^lO^M ^ „ ,
K = = 0.774 2ff — = 12.57 3.3.1.3 Tailplane ^E = 3.1° < = ^ ^ ^ = 0.496 2lT _§A = 10.99 3.3.1.4 Fin3.3.1.4.1 Lift curve slope
ESDU Item 70011 is used to estimate the fin lift curve slope using the following parameters.
^ c / 4 ^ F ^ F z ^^F ^BF ^ F
V
^ F = = = = = = = 4 2 . 7 ° 5 . 2 3 m^ 2 . 5 1 m 0 . 6 9 m 1 . 1 8 1 m 2 . 4 1 0 . 2 7 8 = Ap t a n Ajjp -U + A„Ap tan A,p = 1.659
6A„ =2.345 F
For A = 0.25, ESDU Item 70011, Fig. Ic gives
1 ^^L
A d ^ = 1.12/rad
fdC
^°ld^jp = (CLa>F = 2.70/rad
3.3.1.4.2 Centre of pressure position
Using ESDU Item 82010, the centre of pressure (c.p.)
position on the fin is estimated using the following parameters
z_ = 0 . 7 4 5 m ^T — = 0 . 2 9 7 hp ^T - i = 2 . 6 3 2
Fig. 4 of ESDU Item 82010 gives
— = 0.435 ^F
Thus 2 = 1.092 m F
Empirical correction factors put the c.p. position at 0.85 z above and 0.7z tan h^ behind the fin root quarter-chord station.
*Figs. referred to in section 3.3 will be those in the ESDU item under discussion unless otherwise stated.
Moment arms of the c.p. relative to the moment reference centre are given by
X = m + 0.7 Zp tan A,p = 5.98 m
^cr "•" °*^^ ^F " 1.618 m F
3.3.2 Calculations
The lateral derivatives estimated are as follows; Derivatives due to sideslip, y„, N and L .
^ V V V
Derivatives due to roll rate, L , and N . (Y is negligible) Derivatives due to yaw rate, L and N . (Y negligible). 3.3.2.1 y^
The main contribution to this derivative is from the fin.
3.3.2.1.1 Fin contribution
ESDU Item 82010 provides a method for estimating this and includes an allowance for interference between the body, wing, tailplane and fin.
<^V)F = - J B ^ T J W (^La^F T ^
Fig.l of ESDU 82010 gives J^, the effect of the body on the
D
basic fin lift curve slope as a function of h ^BF -^ ^F
J^ alows for the interference effect of the tailplane on the sideforce derivative and for ^ = 0.297,
hf ^T
-^ = 2.632, J = 0.98
^ F •
J allows for the interference between the wing and the body-fin-tailplane assembly and for z = +n 41^4
^BW it has the value 1.23.
So (Y„)^ = -1.10(0.98) (1.23) (2.70) (5.23)/25.08
V r
= -0.747
3.3.2.1.2 Body contribution
(Y )_, is calculated using ESDU Item 79006. The maximum V c
body width, d, is taken as 1.981 m. (Y^)g is given by:
(— . 2 h b F F I I -n ^K I (Y ) „ = - 0 . 0 7 1 4 + 0 . 6 7 4 ^^ + ^ ( 4 . 9 5 — - - 0.12V — + 0 . 0 0 6 r
- ^ L Sj^ - Sj, h -J S
which includes allowances for body, wing-body and dihedral effects. Graphs in the above ESDU Item give values for the effect of wing height and body width function, F f as 0.057 and the wing planform factor, F , as 0.88.
w
Using these and the rest of the Jetstream data gives
3.3.2.1.3 Total Y
V
^v = (^v^F + (^V^B
= -0.747 + (-0.236) Y^ = -0.983
Comments on the calculation of Y for a complete aircraft are given in ESDU Item 82011. Difficulties with setting the centre of pressure position for the load on the fin can lead to
inaccuracies in the estimate of Y .
3.3.2.2 N^
This derivative is difficult to estimate as it is a balance between a large destabilising component from the body and a large stabilising contribution from the fin.
Interference effects are accounted for implicitly in the standard methods of estimation.
3.3.2.2.1 Fin contribution
ESDU Item 82010 again provides the estimate for the fin contribution to this derivative and is based on the value of (Y„)„ calculated in section 3.3.2.1.1.
V r
For small a, we can use the approximations cos a = 1 and sin a = O. The formula for (N„)„ is
then:-V r
( V P = -(Y^)p (nip + 0.7 ^p tan A^p)/b giving
3.3.2.2.2 Body contribution
Preliminary quantities required are 2 \ -^ = 9.09 'b ^1 — = 1.78 ^2
Fig. 1 ESDU Item 79006 gives
-N„ - ^ = 0.16
V mid c , ^K 1 b lb
So that, about a yaw axis through the mid-point of the body, N , .^ = -0.105
V mid
Using the m.r.c. in section 3.3.1.1, at \ = 5.66 m, we have
^ ^ (1 - ^b/2) ^
(N )„ = N + -^ ^^-^ Y
v'B V mid u ^
(N^)g = -0.090
Effect of the engine nacelles is considered negligible.
3.3.2.2,3 Total N
ESDU Item 82011 provides some move information on the Lation of N for a complete aire]
aircraft under consideration, we have
calculation of N for a complete aircraft. For the Jestream
N = (N ) ^ + (N )„ V v'F v B = 0.282 - 0.090 N = 0 . 1 9 2
3.3.2.3 L
This is one of the most important derivatives with the main contribution coming from the wing. The wing geometrical features with the most effect are aspect ratio, dihedral, sweep and twist. Body effects and fin side force due to sideslip also make
contributions to L^. As with Y^, difficulties in setting the c.p. position for the load on the fin can lead to inaccuracies in estimating the fin contribution. ESDU Item 81032 provides further information on the calculation of L for a complete aircraft.
3.3.2.3.1 Fin contribution
This can only be regarded as an approximation because
the fin c.p. position is influenced considerably by the position of the fin/rudder with respect to the wing/body slipstream.
ESDU Item 82010 gives
(^v^F = ^^v^F [^"^crp "^ 0-85 Zp) cos a- (mp + O. 7 Zp tan A^p) sina]/b
Assuming, for low a, that cos a = 1 and sin a = 0 gives:
(L^)p = -0.076
3.3.2.3.2 Wing contribution
The contribution of full span dihedral to L^ is covered in ESDU Item A.06.01.03. For A = 0.25 Fig. lb of this item
(L)„e
gives - — ' = 0.0158 and for A = 0.5 Fig. Ic gives:
(L ),B
— = 0.0172
<T
Thus for r = 7° and A =0.333 we have (L^) J, = -0.091
Wing planform contribution, (L ) is covered in ESDU
hr
Item 80033. The method uses A for the half-chord line but in the case of slightly negative sweep for the half-chord line, as on the Jetstream, A„ = 0 is used. Figures lb and Ic of
ESDU 80033 give values of_ (L ) for A = 0.25, 0.50 X c.
^L (L^)
and interpolation gives - ^ = -0.023 (A = 0.3333) C.
Thus (L ) = - 0.017 V p
For the current exercise we will neglect the effect of twist. Effects of flaps on (L ) are dealt with in ESDU Iter
V w
80034, but for the current exercise we have flaps up.
( V w = ( V r
•"
( V p
= -0.091 - 0.017 = -0.108
3.3.2.3.3 Body contribution
This consists of two parts (described in ESDU Item 73006), tL ), , the isolated body contribution and (L )u the interference effect due to the vertical positioning of the wing on the body.
With the height of the equivalent body reference cross-sectional area, H, equal to h and h (ESDU Item 73006) equal to z then the equivalent wing vertical position relative to the body, h' (shown as h in Item 73006), is given by
h' h
— — K r H H
Using Fig. 2 of Item 73006 to give K = 0.0098 deg~ (N.B. not the K calculated in section 3.3.1.2), we obtain
h'
— = 0.331 H
(^)h
From Fig. la for A = 6, = 0.0165 (l + |)f(A)
where W is the width of the body reference cross-section, 1.98 m. From Fig. lb, the aspect ratio function, f(A), for A = 10, is 1.17 so
(L^)j^ = 0.0165 x 2 X 1.17
(L^)j^ = 0.039
The isolated body contribution, (^^)u,i is given by
^^V^b r. r^^A ^'b ^C , -1 = -0.014 — . — degree «b
= -0.00145.
Say a, = 2°, then {\)-^ = -0.0029 so the total body contribution = (L ), + (L ) L .
= 0.036
As before we will neglect the effects of turboprop nacelles and propeller arrangements for a conventional aircraft layout.
3.3.2.3.4 Total L
Summing the components estimated in the preceding sections we
have:-L = (have:-L )„ + (have:-L ) „ + (have:-L ) + (have:-L ), + (have:-L ), V v'F v'r V p v'b V h
= -0.076 + (-0.091) - 0.017 - 0.0029 + 0.039 L^ = -0.148
3.3.2.4 N^
Contributions to N are estimated from the wing and P
fin. The fin makes a significant contribution which is difficult to estimate due to inaccuracies in estimating fin
c.p. position.
3.3.2.4.1 Wing contribution
ESDU Item 81014 is used to estimate (N ) . We assume p w
the yawing axis is at the wing aerodynamic centre giving
X
- ^ = 0.
Wing taper has a minor effect and is accounted for in ESDU Item 81014 by using the sweepback related to the quarter chord line. The non-linear dependence of (N^),, on C- at high
p W li
C_ is ignored here and the variation of viscous drag coefficient L
with a, dc' , is taken to be zero. da
Fig. 1 of ESDU Item 81014 presents a carpet plot of the linear contribution to
CN^),,:-p w
'M
' "*'
C^ „ A + 4 c o s A / c o s A A / X - C 1 + 6 ( 1 + l A J ^ - Ë H t a n Aj^+ 2 \^'!— " i t a n A A | (N ) i k\\. p w' \ 12 y V ^ 12 /,, C^ 0,A = 0 1 1 1 ^ac ( t a n Aj. + - ) - ^ 8A * A 2A b F o r A = 1 0 , we h a v eI ^^pK ; = -0.038
^ ^L : o so t h a t f o r C_ = 0.72 (Np)^ = - 0 . 0 2 7 43.3.2.4.2 Fin contribution
This may be estimated using the relationship
(Np)F = - ^ . ! F . (Y^)p b b .^_._L:09 (_o.747) 15.85 15.85 = 0.0194 3.3.2.4.3 Total N
E
Summing the fin and wing components for N we have
N = (N ) + (N )-^ p p w ' p F -0.0274 + 0.0194 N = -0.008 P 3.3.2.5 L
The main contribution to this derivative comes from the wing but an estimate will be made of the contribution from the tailplane. For this exercise, the factored contribution to L from the fin will be ignored as would be usual for initial estimates. L is important in considering rolling performance as well as dynamic stability.
3.3.2.5.1 Wing contribution
6L
ESDU Item A.06.01.01 gives plots of - — ^ as a
K
BA
6L For A = 0.25, 2 = 0.28 6L and for A = 0.5, ^ = 0.285 K 6L giving ^ = 0.282 for A = 0.3333. K
So with B = 0.973 and K = 12.57 we have; (L ) = -0.224
p w
3.3.2.5.2 Tailplane contribution
This is estimated using ESDU Item A.06.01.01 as the L of the isolated tailplane based on its own area. The
result is then factored for summing with the wing contribution.
Using the appropriate data for the tailplane, we have
= 0.496 (using a-, = 3.2/rad obtained from flight test in ref. 5 ) . -^ = 10.99 and Ap = 3.1 For A = 0.25, _ ^ ^ = 0.26 6L For A = 0.5, - — 2 = 0.28 K 6L so for A = 0.41, E = 0.272 K giving (L ) ' = -0.139 P T
This result must be factored to take into account the relative sizes of the tailplane and wing before summing with (L ) .
The tailplane contribution to L is thus given by S b 2 .
(L )„ = K. — (-^) (L ) ' P ^ S b P T
where K is to account for induced velocity effects and will be taken as ^.
so (Lp)^ = 0.027(£p)'^ = -0.0037
3.3.2.5.3 Total L
Using the results of the preceding sections we have;
S = ^^P^ ""
%^T-= -0.224 - 0.004 Lp = -0.228
3.3.2.6 N r
The main contribution to N is from the fin but the body also makes a significant contribution.
3.3.2.6.1 Fin contribution
The dynamic derivative (Nj,)p can be expressed in terms of the static derivative (N ) by assuming that the yaw
rate produces a local sideslip velocity at the fin equal to the product of the yaw rate and the yawing moment arm of the fin sideforce. ESDU Item 82017 then gives (assuming for low
a, that cos a - 1, sin a - 0 ) .
^ V F =
-^\'>F
^"V "" °-^ ^F
^^"^
H F ^ / ^
= -(0.282) (5.977)/15.85 = -0.106
3.3.2.6.2 Body contribution
Information on the contribution of the body to N is scarce. Slender body theory can be adapted.
A reasonable estimate, bearing in mind the relatively large fuselage side area for the Jetstream, would be to put it equal to or slightly under the fin contribution, say
(N^)g = -0.09
3.3.2.6.3 Wing contribution
ESDU Item 71017 gives (N ) as the sum of two components, one due to the asymmetric distribution of the profile drag and the other due to the asymmetric distribution of the
lift-dependent drag due to the trailing vortex system.
(N ) = ^ . C -H ^ . C 2 ^""r'w _ "-DO ^ ^ 2 ^L ^Do ^L / N I ro Fig. la gives = -0.167 Cn./ Do/ A = 1 N For A = 0.3333, ' ^° ^Do/ A = 0.333 = 0.75 So /N ^° ' = -0.167 X 0.75 ^^Do / A = 0.333 -0.125
Which, for C^, = 0.032 means Do
N = -0.0040 ro
Fig. 2b, for A = 0.25 gives N ^L rv 2 = -0.005 and for so for A = 0.5 ^rv - ^ = -0.006 ^L A = 0.3333 N - ^ = -0.0053 giving N = -0.00275 for C^ = 0.72 Thus (N ) N -f- N r w = ro rv = -0.004 - 0.0028 (N^)„ = -0.0068 r w 3.3.2.6.4 Total N r
Summing the fin, body and wing contributions gives
\ = (N^^p -f (N^)^ -H (N^)^
= -0.106 - 0.09 - 0.007
N^ = -0.203
3.3.2.7 L^
The wing provides the main contribution to L for which ESDU Item 72021 provides a semi-empirical method of estimation. The fin also provides a significant contribution. The tailplane contribution to L could be estimated in a similar way to the wing contribution but the accuracy of the estimation is
questionable due to interference effects from the wing, fin and body.
3.3.2.7.1 Wing contribution
ESDU Item 72021 provides the means of estimating i
flaps,
contributions to (Lj.)^ from planform, dihedral, twist and
For Aj^ = 1 Fig. lb gives a value of sweep factor, 4
g(Ai^) = 1.01. From Fig. la,
- ^ ^ ^ = 0.107 g(A^) c (L ) So - ^ ^ P = 0.108 • ^L At C^ = 0.72,(L ) = 0.078 L ro p
The dihedral contribution is estimated from Fig. 3.
r = + 7°
o ^^ro^r -1
For A, = 1 , —i^:^-^ = 0.00005 deg ^ r
so ( L ^ Q ) p = 7 X 0.00005
= 0.00035
If the dihedral contribution had been significant the sweep factor from Fig. lb could also have been applied.
Fig. 4 ESDU Item 72021 shows the effects of twist on (L^) . Washout, e = +2° (convention as in ESDU Item 72021)
(L ) _, — — ^ = -0.0019 deg (L ) = -0.0038 ro E So (L ) = (L ) + (L ) ^ + (L ) r'w ro p ro r ro e = 0.078 + 0.00035 - 0.0038 (L„)^ = 0.075 r w 3.3.2.7.2 Fin contribution
This is expressed in terms of the static derivative (L„)p and the moment arm to an estimated fin c.p. position by assuming that yaw rate produces a local sideslip at the fin given by
yaw rate multiplied by the yawing moment arm of the fin sideforce,
For low a, with cos a = 1 and sin a == 0. ESDU Item 82017 gives;
(L^)p = -(L^)p(mp -H 0.7 Zp tan A^p)/b = -(-0.076) (5.98)/15.85
= 0.0288
3.3.2.7.3 Total L^
Using the results of the preceding sections, we have
= 0.0288 + 0.075
3.3.3 Summary of estimates
Table 1 shows a summary of the ESDU Items used to
calculate the contributions to the various lateral derivatives. The following is a summary of the lateral derivatives estimated
for the Jetstream aircraft at an altitude of 2000 m, speed 77 m/s (150 kts) and C^ = 0.72, clean configuration:
s/ ^v = \ = «p = \ ' N = r ^r = -0.983 0.192 -0.148 -0.008 -0.228 -0.203 0.104 - 0 . 3 4 6 / - 0 . 4 6 7 ] 0 . 1 8 2 / 0.190 ]-0.098/-0.124 [-0.124/-0.069 - 0 . 2 4 5 / - 0 . 3 1 7 _ 0.207/-0.135 [ o . l l 7 / 0.133^
Values in the square brackets are some typical values
obtained using parameter identification techniques by Van Rooyen for flaps up and flaps down cases respectively. Inaccuracies in the estimate of Y.^. will lead to inaccuracies in N (see section 3.3.2.4.2). Apart from these two derivatives, the
other predictions are quite close to the typical flight values shown.
1. Etkin, B. "Dynamics of Flight-Stability and Control' (2nd Edition, Wiley, 1982)
2. Engineering Sciences Data Unit.
Aerodynamics sub-series to the aeronautical series of data items.
3. Ellison, D.E. et al USAF Stability and Control Handbook
(DATCOM), Flight Control Laboratory, Wright-Patterson A.F.B., Dayton, Ohio, U.S.A.
4. Hopkin, H.R. A Scheme of Notation and Nomenclature
for Aircraft Dynamics and Associated Aerodynamics.
ARC R&M 3562 (5 parts), 1966.
5. Soronda, V A Preliminary Design Study for a
Variable Stability Jetstream. M.Sc. September 1979 College of Aeronautics, Cranfield.
6. Van Rooyen, R.S. Propeller Influence on the
Lateral-Directional Stability of Multi-Engined Aircraft.
PhD. November 1978. College of Aeronautics, Cranfield.
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PRINCIPAL DIMENSIO.'IS
4 (See Hopkin ) .1 Longitudina X = u z = u X = w
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-o \ hpVS o ipVS o \ èpVS o 1 - TTr* 1 X n o z n M n o ^ èpv^s o Z n ipV^S o M n — ipV'Sc ipVS o M M = - ^ w ipvsc o z ^ ipVSc o M = - ^ ^ ipvsc o M-M. = - ^ ^ ipSc^ 2o Y ^ v = JjpVS o N. N = ^ p V S b o N N = P Ï5pVSb2 O N. N = r I j p V S b ^ o L. L = V ^ p V S b o L L = P Ï5pVSb2 O L L = ^ p V S b 2
B.1 Longitudinal - = I P V S / j ^ Ü + X W + X q + n X ) - ^ c o s GL, m " ^ ^ Y ^ V ® ^ ^ ispVS ' " ^ -( z U - l - Z w + Z q + n Z ) + q c o s a *• u w q Y n ' ^ ^ -^ . ^ ^PV S c / J ^ u + M . — -I-WM + q M + n M n ) ^ T- ^ u W , , W ^ y q / I v V = q q c w h e r e q = — ^ V
and - indicates the quantity is divided by V.
B.2 Lateral
- ^SDVS / " - ^
V = -^^— Y . v + Y - . C + Y . c l + p sin a - r cos a + ^cos 0 cos 0 . ó m ^ ^ 5 r. '' ^ ^ ^ e e Y .V + Y^.C + Y .c + p sin a -V -V i, ^ / ^xz . + r , . V V V I x Ixz r = i P V ^ / N .V -.N H^ + N ^ + N,.C + N . 0 + z x J = p o
with Y_ = ^— (similarly Y.)
^ ^ p V 2 s ^ o
and L^ = ^ — ( s i m i l a r l y L , N^, N ) ^ Ï5pV2sb ^' ^ '^
C.1 Datum Flight Condition Altitude = 2000 m Mass, m = 5500 kg e.g. position = 24%c U = 7 7 m/s o
Flaps up, landing gear up,
C.2 Inertias ^x I y ^z = 22050 = 40080 = 69491 2 kg m 2 kg m 2 kg m 2 I = 9655 kg m xz ^ (See Ref. 6)
r
n A 1 A _ A^ U__[) __ J C I S T R L A M 100 SLRILS i M [ % i O r : :.> c - A > ; i ; i [•:::) f , - A x i . i ; ; TURHOMECA A S I A / O U X I V iriGiNi.s v/ING s p a n b Gross Area SMean Aerodynamic Chord c-Taper Ratio Tip/Centre Line Sweep of 30% Chord'Line Dihedral Angle
Root Datu;n Chord Section Tip Chord Section
Root Chord Setting to Longitudinal
Datu.Ti
Wing Twist about 30Ï Chord
(52 f t . ) (270 f t . 2 ) ( 5 . 6 3 f t . ) 1 5 . 8 5 0 m 2 5 . 0 8 4 m2 1.717 m 0 . 3 3 3
0°
70N.A.C.A. 63A418 (Modified) N.A.C.A. 63A';i2 (Modified)
+ 2°
- 2° (Washout)
FUS,eLAGr. Length Maximu.Ti Dia,T,:iter 13.35 n . (4 3 . 3 f t . ) 1.98 m. ( 6 . 5 f t . ) T A I LP L A : ; E Span b-j-Gross AreaS-j-Me an Ae rody n 2;r,i c Cli 0 rd T a i l Arn \j
Tailplane S e t t i n g to Fuselage Datu.ii
6 . 6 0 m ( 2 1 . 6 7 f t . ) 7.785 n 2 ( 8 3 . 8 f t . ^ ) • 1.25 n ( 4 . 1 0 5 f t . )
6.184 n ( 2 0 . 2 9 f t . ) "
+ 1.5°
CONTROL AREAS AND GEARINGS
Total Aileron Area Aft of Hinge Aileron Mean Chord Aft of Hinge Aileron Travel
Aileron Gearing
Total Elevator Area Aft of Hinge Elevator Mean Chord Aft of Hinge Elevator Tfavel
Elevator Gearing
Rudder Area Aft of Hinge
Rudder Mean Chord Aft of Hinge
Rudcfer Travel \ Rudder Gear-iruj 1 .598 r . 2 ( 1 7 . 2 0 ft.2) 0 . 2 7 p ( 0 . 8 8 5 f t . ) 25° up t o 150 Dov/n 2 . 2 5 R a d i a n s p e r m e t r e ( 0 . 6 8 6 r a d s / f t . ) 2.397 rn'^(25.80 f t . ^ ) 0.387 m 1.27 f t . ) - 2 8 ° t o + 220 3.71 R a d i a n s p e r Metre 1.942 in ( 2 0 . 9 ft^)^^-"'^^ r a d s / f t . ) 0.674 m ( 2 . 2 1 f t ) ± 250 • • •3.745 Radians per Metre (1.141
