- Email: [email protected]
Preliminary aircraft design: lateral handling qualities
Aircraft Design 4 (2001) 63}73
Preliminary aircraft design: lateral handling qualities Paolo Teo"latto* Scuola di Ingegneria Aerospaziale, Universita degli Studi di Roma, La Sapienza, via Eudossiana 18, I-00184 Roma, Italy Received 15 August 2000; received in revised form 15 October 2000; accepted 20 October 2000
Abstract Handling qualities of airplanes can be improved by the use of an automatic control system (augmented stability control system), however, in the preliminary design phase it is sometimes better to implement suitable design changes. These changes can be performed according to iterative and trial}error procedures in order to get the required #ight qualities. In the present paper a di!erent approach to improve lateral handling qualities of aircraft is followed, based on necessary and su$cient conditions for the roots of the lateral characteristic equation to lie in prescribed region of the complex plane. As a result of this study, handling quality region of levels 1}3 can be visualized in the space of the physically relevant parameters. As an example, the vertical tail surface S and the dihedral angle C are chosen and the regions of levels 1}3 corresponding to some aircraft are shown in the (S , C) plane. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Preliminary aircraft design; Lateral stability; Flight-handling qualities
Flight-handling qualities are related to the pilot's opinion about the ease or di$culty of controlling the aircraft [1]. Based on several extensive #ight test campaigns, a list of speci"cations are available to the aircraft designer, so that, if the design satis"es certain requirements, the aircraft will have good #ying qualities. Of course #ight-handling qualities depend upon the type of aircraft and the #ight phase. Table 1 reports the well-known classi"cation of aircraft according to aircraft size and maneuverability, Table 2 classi"es #ight phases (see, for instance [2]).
* Tel.: #39-6-44585888; fax: #39-6-4881381. E-mail address: [email protected] (P. Teo"latto). 1369-8869/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 9 - 8 8 6 9 ( 0 0 ) 0 0 0 2 5 - 2
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P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Nomenclature AR b C * C I ,I V X k K l m p G o S S Z
aspect ratio wing span lift coe$cient in cruise #ight dihedral angle inertia moments along the longitudinal and normal axis "1.8, see Ref. [8] swept angle distance from the mass centre to the aerodynamic tail centre along the longitudinal axis aircraft mass coe$cients of the characteristic equation of lateral stability air density wing area vertical tail area distance from the mass centre to the aerodynamic tail centre along the normal axis
Table 1 Classi"cation of aircraft according to size and maneuverability Class Class Class Class
I II III IV
Small light airplanes Medium weight, low to medium maneuverability Large, heavy, low to medium maneuverability Height maneuverability airplanes
Table 2 Classi"cation of #ight phase Category A
Nonterminal #ight phase requiring rapid maneuvering (military type maneuvers)
Category B
Nonterminal #ight phase requiring gradual maneuvering (climb, cruise, descent)
Category C
Terminal #ight phase requiring gradual maneuvering (take-o!, approach, landing)
There are three levels of #ight qualities: E E
E
Level 1: #ying qualities adequate for the related #ight phase. Level 2: #ying qualities adequate but with some increase in pilot's work or degradation in mission e!ectiveness. Level 3: the airplane can be controlled safely, but with extra workload for the pilot or inadequate mission e!ectiveness.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
65
The #ight qualities are related to the aircraft stability and control characteristics, then to the response under longitudinal or lateral perturbations. Therefore, the handling qualities can be made equivalent to the requirement that the roots of the longitudinal and lateral characteristic equations j#p j#p j#p j#p "0 (1) (obtained by linearization of the equation of motion about some (equilibrium) reference #ight) are placed in prescribed regions of the complex plane. The regions of lateral-handling quality have boundaries de"ned in Table 3: the roots of Eq. (1) must lie to the left of the x line (x condition) and inside the cone of half-amplitude a (a condition), as shown in Fig. 1. Regions of the complex plane corresponding to level 1}3 handling qualities are hereafter called stability regions I}III. Table 3 If j are solutions of (1), then j"x $ ix tg(a) M M Level Category
x
a
1
A B C
!0.35 !0.15 !0.15
793 85.43 85.43
2
A B C
!0.05 !0.05 !0.05
88.83 88.83 88.83
3
A B C
0.0 0.0 0.0
Fig. 1. The j s stability regions in the complex plane. G
88.83 88.83 88.83
66
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Note that the characteristic roots j"x $ix tg(a) are generally expressed in terms of un damped frequency u and damping ratio f: j"!fu $iu (1!f. The two expressions are related by f"cos(a), u "!x /cos(a), then the above stability regions are de"ned as f'cos(a), u '!x /cos(a). The root's placement in the stability regions makes it possible for the designer to consider #ight-handling qualities in the preliminary design phase, namely, (i) the roots depend on the coe$cients p of the characteristic Eq. (1) G (ii) the coe$cients p are polynomial functions of the aircraft stability derivatives G (iii) the stability derivatives are related to the aircraft geometry, mass and aerodynamic properties. Then, the knowledge of the stability regions I}III can help the designer, since a relationship will be assessed between such regions and the physical parameters which can be varied in the preliminary design. A "rst step in de"ning such relationship is the de"nition of a set of conditions on the coe$cient p of the characteristic Eq. (1) in order to have the roots within the regions of the complex plane G de"ned in Table 3. These conditions on p are developed in Ref. [3]: for the present case, they are p '0 and G G I : a "6x #3p x #p '0, I : a "x #p x #p x #p x #p '0, I : b "4x #p '0, I : b "4x #3p x #2p x #p '0, I : b b a !b a !b '0. These are the necessary and su$cient conditions for the roots to be on the left of the x -straight line. Note that for x "0 one gets the usual Routh conditions. The limiting conditions for x "!0.35, x "!0.15, x "!0.05, or x "0.0 I "0, 2, I "0, (2) de"ne the boundaries of the levels I}III stability regions [4]. Conditions (2) on the coe$cients p determine the equations on the stability derivatives, hence on G geometrical, mass and aerodynamical aircraft parameters. Now, if an already existing aircraft design has to be improved, it is conceivable to reduce the number of changes as much as possible, and to restrict the number of variable physical parameters. Another approach is, of course, to implement an augmented stabilization system. However, it is preferable not to increase the complexity of the control system if small design changes can be e!ective. For lateral-handling qualities, two crucial parameters are the vertical tail surface S and the dihedral angle C. Changes of these parameters imply changes of the stability derivatives C , C , respectively, and L@ J@ historically the approach has been to choose these derivatives in order to satisfy both Spiral and From Table 3 one can notice that the a conditions are weak (a"903 is the whole stability half-plane) so that parameters satisfying the x conditions often satisfy the a condition. This occurs, for instance, in the examples shown in the following.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
67
Dutch-Roll convergence according to the Routh conditions [5] R"C C !C C '0 J@ LP L@ JP E"p p p !p p !p '0. The stability bounds E"0 in the (C , C ) plane are also of interest [6] and of particular elegance L@ J@ is the work [7], where the lateral stability region in the same plane is obtained as the region bounded by the E"0 hyperbola and the p "0 straight line. It should be remarked here that the condition p "0 is similar to the well-known condition for Routh's discriminant R"0. However, it is not possible to change C , C by variations of S and C while leaving the other derivatives L@ J@ una!ected and an iterative procedure can be followed. However, iterative methods or extensive numerical work can be avoided following the above method which can be summarized in the following steps: (i) Determine the stability derivatives as a function of the dihedral angle C and of the vertical tail surface S [8,9]: explicit formulas are recalled in Appendix A. (ii) De"ne the coe$cients p of the lateral characteristic Eq. (1) (see e.g. [5]): because of step (i) G these coe$cients are algebraic functions of C, S : p "p (C, S ). Appendix B reports the G G functions p (C, S ). G (iii) Impose the conditions I '0 on p so that the roots lie in the stability regions I}III I G (corresponding to the handling qualities of levels 1}3) [3]. (iv) Equating the above stability conditions to zero (see Eq. (2)) one gets the boundaries of the stability regions. (v) These boundaries are algebraic functions of C, S and determine the boundaries of the levels 1}3 handling qualities in the plane of the physical parameters C, S . Following steps (i)}(v), the handling qualities levels 1}3 regions are visualized for several aircraft whose stability derivatives can be found for instance, in [10]. For instance the level 3 stability region for the class of airplane similar to NT33A (data of NT33A are reported in Appendix C) but with variable (C, S ) parameters, is shown in Fig. 2. The stability region, called III, is bounded by the curves p "0, p "0, I "0, I "0, (the other inequalities being ine!ective) namely it is the region of the (C, S ) plane above the curves p "0, p "0, I "0 and inside the parabola-like curve I "0 (marked as stability region in Fig. 2). Analogous plots can be drawn for the stability regions I and II. Fig. 3 shows all the three stability regions in the (C, S ) plane. The outer line binds the stability region III, the middle and the inner bind the stability regions II and I respectively. The point in region II corresponds to the actual values (C, S ) of the NT33A. Fig. 4 shows similar (C, S ) plot for aircraft similar to CESSNA-Navion, corresponding to the point &o' in the "gure (CESSNA-Navion data are in Appendix C). Since the analysis is based on classical arguments of stability theory and computations are based on easy algebraic formulae, it turns out that the above plots are easily attainable and can be useful for the aircraft designer.
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P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Fig. 2. The x "0 stability region is the intersection of the area above the curves, p "0, p "0, I "0 and the region inside I "0.
Fig. 3. Stability regions I}III in the ( S , C) plane for NT33A-like aircraft.
Appendix A The dependence of the lateral stability derivatives on vertical tail surface S and dihedral angle C is recalled here.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
69
Fig. 4. Stability regions I}III in the ( S , C) plane for CESSNA-Navion-like aircraft.
The S "m dependent stability derivatives are C "k m#k m, 7@ W WW l C "C ( fus)! C , L@ L@ b W@
l C " C , W@ LP b C l Z C " * !2 C , JP 4 b b W@
(3)
where k C k "! *? (0.724#0.009AR), W S
k C k "! *? (3.06(1#cos(K) ) ), WW S
l C ( fus)"C #C . L@ L@ 7@ b The C-dependent derivative is 1 C "c C where c "!0.00045 . J@ D D deg The following derivatives are assumed constant C ,C ,C . JN LN WN
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P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Appendix B The functions p ( S , C) are explicitly reported here. Let the following dimensionless variables be G introduced m , k"2 oSb
I h" V , V m
I h" X , X m
b i "C k "b #b m#b m, L@ L@ h X b i "C "r m#r m, LP LP h X b "l #l m#l m, i "C JP JP h V b b b i "C , i "C , g"C k . JN JN h LN LN h J@ h V X V In the above formulas the dependence on the variable m"S is made explicit by the coe$cients b b l b l b "C ( fus)k , b "!k k , b "!k k , L@ W h b WW h b h X X X b l b l , r "k , r "k W h WW h b b X X C b Z Z l " * , l "!2l k , l "!2l k . h W h WW 4 h V V V The coe$cients p of the stability equation are related to the dimensionless parameters by G p "!(i #i #C ), LP JN W@ p "(i i !i i )#C (i #i )#i , JN LP LN JP W@ JN LP L@ p "!i i #(i !C )g!C (i i !i i ), JN L@ LN * W@ JN LP LN JP p "!C i i #C i g. (4) * JP L@ * LP Because of the polynomial dependence of i , C on m, g, the coe$cients p also are polynomial HI W@ G functions of m, g, namely
p " p mI, I I p " p mI, I I p "q g# p mI, I I
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
p "q gm#q gm# p mI, I I where the terms p , q , q , in Eq. (5) are HI I I p "!i , JN b l , p "!k !k W W hz b
b l , p "!k !k WW WW hz b p "b !i l , LN p "b #i k !i l #i r , JN W LN JN p "b #i k !i l #k r #i r , JN WW LN W JN p "k r #k r , WW W p "k r , WW q "(i !C ), LN * p "!b i , JN p "i k l !b i , LN W JN p "i k l !b i #i k l !i k r , LN WW JN LN W JN W p "i k l #i k l !i k r !i k r , LN WW LN W JN WW JN W p "i k l !i k r , LN WW JN WW q "C r , q "C r , * * p "!b C l , * p "!C (b l #b l ), * p "!C (b l #b l #b l ), * p "!C (b l #b l ), * p "!C (b l ). *
71
(5)
(6)
(7)
(8)
(9)
Appendix C In this appendix the parameters of the airplanes NT-33A and CESSNA-NAVION airplanes are reported. The lateral stability derivatives of the NT-33A are related to a #ight in a cruise condition with velocity < "69.3 m /s.
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P. Teoxlatto / Aircraft Design 4 (2001) 63}73
The stability derivatives values are C "!0.60, C "!0.077, C "0.049, 7@ J@ L@ C "!0.154, C "!0.012, C "0.21, C "!0.16. JN LN JP LP Mass parameters are m"5352.5 kg,
I "17219 kg m, V Geometric parameters are S"21.81 m,
I "43386 kg m. X
b"11.44 m,
S "1.29 m, C"33.
l "6 m, Z "0.2 m, The above parameters give the p coe$cients G p "9.53, p "52.06, p "310.70, p "43.60, hence the characteristic roots j
"!0.82$i6.11, j "!7.74, j "!0.148. Then x "!0.148 and a"82.353, so the aircraft is of Level 2 (close to the boundary of the stability region I). The lateral stability derivatives of the CESSNA-Navion are related to a #ight in a cruise condition with velocity < "53.6 m/s. The stability derivative values are C "!0.73, C "!0.075, C "0.016, 7@ J@ L@ C "!0.419, C "!0.058, C "0.132, C "!0.113. JN LN JP LP Mass parameters are m"1246.9 kg,
I "1420.6 kg m, V Geometric parameters are S"17.09 m,
I "4785 kg m. X
b"10.18 m,
S "1.29 m C"2.93
l "4 m, Z "0.5 m, It follows that x "!0.18, a"54.23 and that the aircraft is of level 1. References [1] Cooper GE, Harper RP. The use of pilot rating in the evaluation of aircraft handling qualities. NASA TN-D 5153, 1969. [2] Nelson RC. Flight stability and automatic control. New York: McGraw-Hill, 1989. [3] Teo"latto P. Modi"ed Routhian algorithms applied to aircraft stability. Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni, 1995;19:77}99. [4] Teo"latto P. A systematic approach to improve lateral handling qualities of aircraft. L'Aerotecnica Missili e Spazio 1999;7:55}62. [5] Etkin B. Dynamics of #ight. New York: Wiley, 1965.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
73
[6] Babister AW. Aircraft dynamical stability and responce. Oxford: Pergamon Press, 1980. [7] Crocco GA. Lateral stability hyperbola in #ight dynamics. Ponti"cia Academia Scientiarum Commentationes 1937;11:175}95 (in Italian). [8] Roskam J. Flight dynamics of rigid and elastic airplanes. University of Kansas Press, Lawrence, 1972. [9] Torenbeek E. Synthesis of subsonic aircraft design. Delft: Delft University Press, p. 198, 1982. [10] He%ey RK et al. Aircraft handling qualities data. NASA CR 2144, 1972.
Preliminary aircraft design: lateral handling qualities Paolo Teo"latto* Scuola di Ingegneria Aerospaziale, Universita degli Studi di Roma, La Sapienza, via Eudossiana 18, I-00184 Roma, Italy Received 15 August 2000; received in revised form 15 October 2000; accepted 20 October 2000
Abstract Handling qualities of airplanes can be improved by the use of an automatic control system (augmented stability control system), however, in the preliminary design phase it is sometimes better to implement suitable design changes. These changes can be performed according to iterative and trial}error procedures in order to get the required #ight qualities. In the present paper a di!erent approach to improve lateral handling qualities of aircraft is followed, based on necessary and su$cient conditions for the roots of the lateral characteristic equation to lie in prescribed region of the complex plane. As a result of this study, handling quality region of levels 1}3 can be visualized in the space of the physically relevant parameters. As an example, the vertical tail surface S and the dihedral angle C are chosen and the regions of levels 1}3 corresponding to some aircraft are shown in the (S , C) plane. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Preliminary aircraft design; Lateral stability; Flight-handling qualities
Flight-handling qualities are related to the pilot's opinion about the ease or di$culty of controlling the aircraft [1]. Based on several extensive #ight test campaigns, a list of speci"cations are available to the aircraft designer, so that, if the design satis"es certain requirements, the aircraft will have good #ying qualities. Of course #ight-handling qualities depend upon the type of aircraft and the #ight phase. Table 1 reports the well-known classi"cation of aircraft according to aircraft size and maneuverability, Table 2 classi"es #ight phases (see, for instance [2]).
* Tel.: #39-6-44585888; fax: #39-6-4881381. E-mail address: [email protected] (P. Teo"latto). 1369-8869/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 1 3 6 9 - 8 8 6 9 ( 0 0 ) 0 0 0 2 5 - 2
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P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Nomenclature AR b C * C I ,I V X k K l m p G o S S Z
aspect ratio wing span lift coe$cient in cruise #ight dihedral angle inertia moments along the longitudinal and normal axis "1.8, see Ref. [8] swept angle distance from the mass centre to the aerodynamic tail centre along the longitudinal axis aircraft mass coe$cients of the characteristic equation of lateral stability air density wing area vertical tail area distance from the mass centre to the aerodynamic tail centre along the normal axis
Table 1 Classi"cation of aircraft according to size and maneuverability Class Class Class Class
I II III IV
Small light airplanes Medium weight, low to medium maneuverability Large, heavy, low to medium maneuverability Height maneuverability airplanes
Table 2 Classi"cation of #ight phase Category A
Nonterminal #ight phase requiring rapid maneuvering (military type maneuvers)
Category B
Nonterminal #ight phase requiring gradual maneuvering (climb, cruise, descent)
Category C
Terminal #ight phase requiring gradual maneuvering (take-o!, approach, landing)
There are three levels of #ight qualities: E E
E
Level 1: #ying qualities adequate for the related #ight phase. Level 2: #ying qualities adequate but with some increase in pilot's work or degradation in mission e!ectiveness. Level 3: the airplane can be controlled safely, but with extra workload for the pilot or inadequate mission e!ectiveness.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
65
The #ight qualities are related to the aircraft stability and control characteristics, then to the response under longitudinal or lateral perturbations. Therefore, the handling qualities can be made equivalent to the requirement that the roots of the longitudinal and lateral characteristic equations j#p j#p j#p j#p "0 (1) (obtained by linearization of the equation of motion about some (equilibrium) reference #ight) are placed in prescribed regions of the complex plane. The regions of lateral-handling quality have boundaries de"ned in Table 3: the roots of Eq. (1) must lie to the left of the x line (x condition) and inside the cone of half-amplitude a (a condition), as shown in Fig. 1. Regions of the complex plane corresponding to level 1}3 handling qualities are hereafter called stability regions I}III. Table 3 If j are solutions of (1), then j"x $ ix tg(a) M M Level Category
x
a
1
A B C
!0.35 !0.15 !0.15
793 85.43 85.43
2
A B C
!0.05 !0.05 !0.05
88.83 88.83 88.83
3
A B C
0.0 0.0 0.0
Fig. 1. The j s stability regions in the complex plane. G
88.83 88.83 88.83
66
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Note that the characteristic roots j"x $ix tg(a) are generally expressed in terms of un damped frequency u and damping ratio f: j"!fu $iu (1!f. The two expressions are related by f"cos(a), u "!x /cos(a), then the above stability regions are de"ned as f'cos(a), u '!x /cos(a). The root's placement in the stability regions makes it possible for the designer to consider #ight-handling qualities in the preliminary design phase, namely, (i) the roots depend on the coe$cients p of the characteristic Eq. (1) G (ii) the coe$cients p are polynomial functions of the aircraft stability derivatives G (iii) the stability derivatives are related to the aircraft geometry, mass and aerodynamic properties. Then, the knowledge of the stability regions I}III can help the designer, since a relationship will be assessed between such regions and the physical parameters which can be varied in the preliminary design. A "rst step in de"ning such relationship is the de"nition of a set of conditions on the coe$cient p of the characteristic Eq. (1) in order to have the roots within the regions of the complex plane G de"ned in Table 3. These conditions on p are developed in Ref. [3]: for the present case, they are p '0 and G G I : a "6x #3p x #p '0, I : a "x #p x #p x #p x #p '0, I : b "4x #p '0, I : b "4x #3p x #2p x #p '0, I : b b a !b a !b '0. These are the necessary and su$cient conditions for the roots to be on the left of the x -straight line. Note that for x "0 one gets the usual Routh conditions. The limiting conditions for x "!0.35, x "!0.15, x "!0.05, or x "0.0 I "0, 2, I "0, (2) de"ne the boundaries of the levels I}III stability regions [4]. Conditions (2) on the coe$cients p determine the equations on the stability derivatives, hence on G geometrical, mass and aerodynamical aircraft parameters. Now, if an already existing aircraft design has to be improved, it is conceivable to reduce the number of changes as much as possible, and to restrict the number of variable physical parameters. Another approach is, of course, to implement an augmented stabilization system. However, it is preferable not to increase the complexity of the control system if small design changes can be e!ective. For lateral-handling qualities, two crucial parameters are the vertical tail surface S and the dihedral angle C. Changes of these parameters imply changes of the stability derivatives C , C , respectively, and L@ J@ historically the approach has been to choose these derivatives in order to satisfy both Spiral and From Table 3 one can notice that the a conditions are weak (a"903 is the whole stability half-plane) so that parameters satisfying the x conditions often satisfy the a condition. This occurs, for instance, in the examples shown in the following.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
67
Dutch-Roll convergence according to the Routh conditions [5] R"C C !C C '0 J@ LP L@ JP E"p p p !p p !p '0. The stability bounds E"0 in the (C , C ) plane are also of interest [6] and of particular elegance L@ J@ is the work [7], where the lateral stability region in the same plane is obtained as the region bounded by the E"0 hyperbola and the p "0 straight line. It should be remarked here that the condition p "0 is similar to the well-known condition for Routh's discriminant R"0. However, it is not possible to change C , C by variations of S and C while leaving the other derivatives L@ J@ una!ected and an iterative procedure can be followed. However, iterative methods or extensive numerical work can be avoided following the above method which can be summarized in the following steps: (i) Determine the stability derivatives as a function of the dihedral angle C and of the vertical tail surface S [8,9]: explicit formulas are recalled in Appendix A. (ii) De"ne the coe$cients p of the lateral characteristic Eq. (1) (see e.g. [5]): because of step (i) G these coe$cients are algebraic functions of C, S : p "p (C, S ). Appendix B reports the G G functions p (C, S ). G (iii) Impose the conditions I '0 on p so that the roots lie in the stability regions I}III I G (corresponding to the handling qualities of levels 1}3) [3]. (iv) Equating the above stability conditions to zero (see Eq. (2)) one gets the boundaries of the stability regions. (v) These boundaries are algebraic functions of C, S and determine the boundaries of the levels 1}3 handling qualities in the plane of the physical parameters C, S . Following steps (i)}(v), the handling qualities levels 1}3 regions are visualized for several aircraft whose stability derivatives can be found for instance, in [10]. For instance the level 3 stability region for the class of airplane similar to NT33A (data of NT33A are reported in Appendix C) but with variable (C, S ) parameters, is shown in Fig. 2. The stability region, called III, is bounded by the curves p "0, p "0, I "0, I "0, (the other inequalities being ine!ective) namely it is the region of the (C, S ) plane above the curves p "0, p "0, I "0 and inside the parabola-like curve I "0 (marked as stability region in Fig. 2). Analogous plots can be drawn for the stability regions I and II. Fig. 3 shows all the three stability regions in the (C, S ) plane. The outer line binds the stability region III, the middle and the inner bind the stability regions II and I respectively. The point in region II corresponds to the actual values (C, S ) of the NT33A. Fig. 4 shows similar (C, S ) plot for aircraft similar to CESSNA-Navion, corresponding to the point &o' in the "gure (CESSNA-Navion data are in Appendix C). Since the analysis is based on classical arguments of stability theory and computations are based on easy algebraic formulae, it turns out that the above plots are easily attainable and can be useful for the aircraft designer.
68
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Fig. 2. The x "0 stability region is the intersection of the area above the curves, p "0, p "0, I "0 and the region inside I "0.
Fig. 3. Stability regions I}III in the ( S , C) plane for NT33A-like aircraft.
Appendix A The dependence of the lateral stability derivatives on vertical tail surface S and dihedral angle C is recalled here.
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
69
Fig. 4. Stability regions I}III in the ( S , C) plane for CESSNA-Navion-like aircraft.
The S "m dependent stability derivatives are C "k m#k m, 7@ W WW l C "C ( fus)! C , L@ L@ b W@
l C " C , W@ LP b C l Z C " * !2 C , JP 4 b b W@
(3)
where k C k "! *? (0.724#0.009AR), W S
k C k "! *? (3.06(1#cos(K) ) ), WW S
l C ( fus)"C #C . L@ L@ 7@ b The C-dependent derivative is 1 C "c C where c "!0.00045 . J@ D D deg The following derivatives are assumed constant C ,C ,C . JN LN WN
70
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
Appendix B The functions p ( S , C) are explicitly reported here. Let the following dimensionless variables be G introduced m , k"2 oSb
I h" V , V m
I h" X , X m
b i "C k "b #b m#b m, L@ L@ h X b i "C "r m#r m, LP LP h X b "l #l m#l m, i "C JP JP h V b b b i "C , i "C , g"C k . JN JN h LN LN h J@ h V X V In the above formulas the dependence on the variable m"S is made explicit by the coe$cients b b l b l b "C ( fus)k , b "!k k , b "!k k , L@ W h b WW h b h X X X b l b l , r "k , r "k W h WW h b b X X C b Z Z l " * , l "!2l k , l "!2l k . h W h WW 4 h V V V The coe$cients p of the stability equation are related to the dimensionless parameters by G p "!(i #i #C ), LP JN W@ p "(i i !i i )#C (i #i )#i , JN LP LN JP W@ JN LP L@ p "!i i #(i !C )g!C (i i !i i ), JN L@ LN * W@ JN LP LN JP p "!C i i #C i g. (4) * JP L@ * LP Because of the polynomial dependence of i , C on m, g, the coe$cients p also are polynomial HI W@ G functions of m, g, namely
p " p mI, I I p " p mI, I I p "q g# p mI, I I
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
p "q gm#q gm# p mI, I I where the terms p , q , q , in Eq. (5) are HI I I p "!i , JN b l , p "!k !k W W hz b
b l , p "!k !k WW WW hz b p "b !i l , LN p "b #i k !i l #i r , JN W LN JN p "b #i k !i l #k r #i r , JN WW LN W JN p "k r #k r , WW W p "k r , WW q "(i !C ), LN * p "!b i , JN p "i k l !b i , LN W JN p "i k l !b i #i k l !i k r , LN WW JN LN W JN W p "i k l #i k l !i k r !i k r , LN WW LN W JN WW JN W p "i k l !i k r , LN WW JN WW q "C r , q "C r , * * p "!b C l , * p "!C (b l #b l ), * p "!C (b l #b l #b l ), * p "!C (b l #b l ), * p "!C (b l ). *
71
(5)
(6)
(7)
(8)
(9)
Appendix C In this appendix the parameters of the airplanes NT-33A and CESSNA-NAVION airplanes are reported. The lateral stability derivatives of the NT-33A are related to a #ight in a cruise condition with velocity < "69.3 m /s.
72
P. Teoxlatto / Aircraft Design 4 (2001) 63}73
The stability derivatives values are C "!0.60, C "!0.077, C "0.049, 7@ J@ L@ C "!0.154, C "!0.012, C "0.21, C "!0.16. JN LN JP LP Mass parameters are m"5352.5 kg,
I "17219 kg m, V Geometric parameters are S"21.81 m,
I "43386 kg m. X
b"11.44 m,
S "1.29 m, C"33.
l "6 m, Z "0.2 m, The above parameters give the p coe$cients G p "9.53, p "52.06, p "310.70, p "43.60, hence the characteristic roots j
"!0.82$i6.11, j "!7.74, j "!0.148. Then x "!0.148 and a"82.353, so the aircraft is of Level 2 (close to the boundary of the stability region I). The lateral stability derivatives of the CESSNA-Navion are related to a #ight in a cruise condition with velocity < "53.6 m/s. The stability derivative values are C "!0.73, C "!0.075, C "0.016, 7@ J@ L@ C "!0.419, C "!0.058, C "0.132, C "!0.113. JN LN JP LP Mass parameters are m"1246.9 kg,
I "1420.6 kg m, V Geometric parameters are S"17.09 m,
I "4785 kg m. X
b"10.18 m,
S "1.29 m C"2.93
l "4 m, Z "0.5 m, It follows that x "!0.18, a"54.23 and that the aircraft is of level 1. References [1] Cooper GE, Harper RP. The use of pilot rating in the evaluation of aircraft handling qualities. NASA TN-D 5153, 1969. [2] Nelson RC. Flight stability and automatic control. New York: McGraw-Hill, 1989. [3] Teo"latto P. Modi"ed Routhian algorithms applied to aircraft stability. Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e Applicazioni, 1995;19:77}99. [4] Teo"latto P. A systematic approach to improve lateral handling qualities of aircraft. L'Aerotecnica Missili e Spazio 1999;7:55}62. [5] Etkin B. Dynamics of #ight. New York: Wiley, 1965.
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[6] Babister AW. Aircraft dynamical stability and responce. Oxford: Pergamon Press, 1980. [7] Crocco GA. Lateral stability hyperbola in #ight dynamics. Ponti"cia Academia Scientiarum Commentationes 1937;11:175}95 (in Italian). [8] Roskam J. Flight dynamics of rigid and elastic airplanes. University of Kansas Press, Lawrence, 1972. [9] Torenbeek E. Synthesis of subsonic aircraft design. Delft: Delft University Press, p. 198, 1982. [10] He%ey RK et al. Aircraft handling qualities data. NASA CR 2144, 1972.