As an illustration of the preceding theory, the angle-of-attack, velocity and thrust are plotted (figures 4.2 to 4.7) as a function of time both in dimensional and dimensionless form for a typical airliner and a Starfighter on approach to land. These three parameters are determined for the cases of numerical solution, second-order and third-order approximation of the angle-of-attack.
3.2.1 Angle-of-attack, velocity and thrust as a function of time
Time t is made dimensionless (3.45a,3.45b) multiplying by the modified short period frequency (3.26):
τ≡wt, (3.45a)
w2=ρScV02
2I |CM α|αM −αL α0−αL
. (3.45b)
The angle-of-attack is made dimensionless (3.46) using the difference from the angle of zero lift
(2.5a) as a relative difference from the initial value (3.6).
ψ(t) =ϕ(t)
ϕ0 −1 = α(t)−αL
α0−αL −1 = α(t)−α0
α0−αL . (3.46)
The dimensionless angle-of-attack specifies the dimensionless velocity by (3.48) and the dimension- less thrust by (3.54) involving the function (3.51b), the parameter (3.53a) and the initial velocity. The dimensionless angle-of-attack is given by: (I) second-order approximation (3.23) that is a sinusoidal de- pendence; (II) third-order approximation (3.37) involving Weierstrasian elliptic function with parameters (3.35a,3.35b); (III) unrestricted by numerical integration of (3.31, 3.32) starting with ψ(0) = 0at t = 0 implyingc2= 0:
˙ ϕ0
ϕ0
t= Z ψ
0
dξ
p1 + 2n(ϕ0/α1)ξ+ 2nlog(1 +ξ). (3.47) The velocity is made dimensionless dividing by the initial velocity (2.21b) and is specified (3.48) by the dimensionless angle-of-attack (3.46):
V˜(t) = V(t) V0
= r ϕ0
ϕ(t) = 1
p1 +ψ(t). (3.48)
The thrust needed to keep a constant glide slope (3.49a) is made dimensionless (2.10) dividing by the weight (3.49b):
γ(t) =const≡γ0 (3.49a)
T˜(t) =T(t)
mg −sinγ0= V˙
g +V2e(ϕ)
mg . (3.49b)
The first term on the r.h.s. of (3.49b) is evaluated from (3.48):
V˙ g =V0
g [(1 +ψ)−12]˙ =−V0
2g(1 +ψ)−32ψ.˙ (3.50)
Rewriting (2.9,2.8) in the form:
e(ϕ) = ρS
2mh(ψ), (3.51a)
h(ϕ)≡CD0+λCLαϕ+kCLα2ϕ, (3.51b)
the second term on the r.h.s. of (3.49b) becomes:
V2e(ϕ)
mg = ρSV2
mg h(ϕ) = ρSV02 mg
V2
V02h(ϕ). (3.52)
Introducing the dimensionless mass ratio (3.53a) and using (3.48) leads to (3.53b):
µ≡ ρSV02
mg : (3.53a)
V2e(ϕ)
mg = µ
p1 +ψ(t)h(ϕ0+ϕ0ψ(t)). (3.53b) Substituting (3.50) and (3.53b) on (3.49b), it specifies the dimensionless thrust needed to keep a con-
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stant glide slope:
T˜(t)≡ T(t) mg =−V0
2g[1 +ψ(t)]−32dψ
dt + µ
p1 +ψ(t)h(ϕ0+ϕ0ψ(t)), (3.54) as a function of time using the dimensionless angle-of-attack (3.46).
In each of the figures 4.2 to 4.7 there are 3 curves:
• (I) The second-order approximation (3.23), for which the dimensionless relative angle-of-attack (3.6) is a sinusoidal function of time, with frequency (3.24) related by (3.26) to the short-period frequency (3.25);
• (II) The third-order approximation (3.37), for which the dimensionless relative angle-of-attack is specified by a Weierstraussian elliptic function, with parameters (3.35a,3.35b) specifying the two periods;
• (III) The numerical solution with unrestricted order using the integral (3.31, 3.32) to obtaint(ψ) followed by inversion forψ(t).
The (I) second, (II) third and (III) unrestricted order approximation are compared for the relative dimensionless angle-of-attack, for the dimensionless velocity relative to the initial airspeed and for thrust made dimensionless dividing by weight. Each figure has both dimensional and dimensionless scales for angle-of-attack, velocity, thrust and time. The linear case of sinusoidal small amplitude oscillation (I) is compared with the cases of moderate (II) and strong (III) non-linear perturbations.
3.2.2 Non-linear stability data for a large airliner
The Boeing 747-100 is chosen as one of the two examples for application and the relevant data listed in table 4.1. This is a commercial jet airliner and cargo aircraft and the first wide-body airplane produced.
The balance of transverse forces (2.1b, 2.2a, 2.3a) for steady descent with glide slope angle¯γ:
mgcos ¯γ=1
2ρSV¯2CL( ¯α) = 1
2ρSV¯2CLα( ¯α−αL) (3.55a) specifies the angle-of-attack relative to the angle of zero lift:
¯
α−αL = 2mgcosγ
ρSV¯2CLα = 0.2884rad, (3.55b) using an approach speedV¯ = 70m/s, glide slope angle ¯γ = 3◦ and air densityρ = 1.218kg.m−3 for ISA at sea level. For an angle of zero liftαL = −0.162rad = −9.28◦ the stabilized angle-of-attack is
¯
α = 0.2884−0.162 = 0.1264rad = 7.24◦. The steady descent is perturbed with unchanged airspeed V¯ = V0 = 70m/sand initial angle-of-attack α0 = 6.00◦ < |αL| smaller in modulus than the zero lift angle, so that in the second-order approximation the relative angle-of-attack never vanishes ϕmin =
−α0−αL=|αL| −α0>0and there is no singularity of the velocity in (2.21b).
In the second-order approximation the angle-of-attack is a sinusoidal function of time (3.27), corre- sponding to an oscillation at a modified short-period frequency (3.25, 3.26). In the third-order approxima- tion the angle-of-attack is a Weierstrassian elliptic function of time (3.37). Although the Weierstrassian elliptic function always has two periods, if the periods are complex it may not exhibit oscillatory behaviour for real variable. It will be shown that the third-order approximation of the angle-of-attack is a monotonic function of time, as is the case for the numerical solution.
The modified short-period frequency is calculated for zero angle of zero pitching momentαM = 0 leading to (3.56):
w= V0
pϕ0/αL
rρSc
2I |CM α|12 = 0.4847s−1. (3.56) The mass factor (3.53a) equalsµ= 1.2151.
In the integral for the numerical solution (3.47) as in previous expressions must be chosen the initial rate-of-change of angle-of-attack (3.57a):
˙
ϕ0= 5◦s−1, (3.57a)
bV02
=ρScV02
2I CM α, (3.57b)
and the parameter (3.57b) follows from (2.12a).
Figure 3.1: Boeing 747-100 [31]
3.2.3 Non-linear stability data for an interceptor aircraft
The second aircraft chosen to implement the preceding theory is the Lockheed F-104, whose sig- nificant data is listed in table 4.1. This Starfighter is a single-engine supersonic interceptor aircraft developed to the United States Air Force. It is maintained an approach speed ofV¯ = 70m/s, glide slope angle of¯γ= 3◦and air density ofρ= 1.218kg.m−3forISAat sea level.
The initial rate-of-change of angle-of-attack is also kept atϕ˙0= 5◦s−1. The stabilized angle-of-attack isα¯= 10◦= 0.175rad, and beingCL= 0.735thenαL=−0.1611rad. Once againα0= 0.1047rad <|αL|, causing the absence of singularities on the velocity (2.21b) relative to the second-order approximation of the angle-of-attack.
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Figure 3.2: Lockheed F-104 [32]
The mass factor isµ= 1.73and for zero angle of zero pitching moment the modified short-period fre- quency isw= 0.9007s−1. The periods of the Weierstrassian elliptic function remain complex, therefore the third-order approximation is not oscillatory.
Chapter 4
Results
4.1 Numerical model
In order to study the behavior of the angle-of-attack under a constant glide slope angle, it is plot- ted the numerical solution in time, a sinusoidal oscillation at a modified short-period frequency and a Weierstrassian elliptic function. It is also computed the evolution of velocity and consequently thrust for each approximation. These plots are obtained with MATLABTM, a numerical analysis environment, for the Boeing 747-100 and Lockheed F-104 in approach to land.
4.1.1 Program overview
After some research, it was possible to collect the Boeing 747-100 and the Lockheed F-104 airplane data presented in table 4.1. These values were then acquired by the program, making it possible to determine the parameters from table 4.2 and evaluate the numerical response, the second-order and the third-order approximation of the angle-of-attack.
The first case is obtained from (3.47). The value ofψis determined making use of afor cycle that uses the functionvpasolveto solve (3.47) with the functionvpaintegral for a range of time, as illustrated in figure 4.1. Both functionsvpasolveandvpaintegralare numeric solvers that return the approximation of the numerical symbolic solution, since the usualsolveandintegral orint were not able to find any solutions for the referred integral.
Figure 4.1: For cycle in MATLABTM
The second-order approximation (3.23) is a sinusoidal oscillation at a modified short-period fre- quency.
The third-order approximation is specified by a Weierstrassian elliptic function. Since (3.44) is a fourth order equation in the formt(ψ), MATLABTMfunctionrootsfinds the real and positive solution for a
23
scope of time.
The angle-of-attack (3.46), velocity (3.48) and thrust (3.54) are then plotted in function of time re- garding the results previously obtained from the three approximations. For the calculation of thrust, the parameter dψdt is given by the functiondiff.
4.1.2 Inputs and outputs
The aircraft selected to validate the proposed solution are a large commercial airplane, a Boeing 747-100, and an interceptor aircraft Lockheed F-104. Their parameters in a situation of approach to land are listed in table 4.1. From the graphics [33] replicated in appendix A it is possible to identify the minimum drag coefficientCD min and from this value is determined the minimum lift coefficient for minimum dragCLminD. Both of these parameters regard the solid line correspondent to the sea level, since we are in a situation of approach to land. Being aware of the drag and lift coefficient for this case, the drag polar coefficientskandλare estimated from the equation (4.1):
CD=CD min+k(CL−CLminD)2, (4.1) where λ = −2kCLminD. The drag coefficient at zero angle-of-attack is a result of CD0 = CD min+ kCLminD2. The remaining parameters are acknowledged from textbooks [1, 25, 33].
Table 4.1: Boeing 747-100 [25, 33] and Lockheed F-104 [1, 33] data Parameter Boeing 747-100 Lockheed F-104
m(kg) 288756.9 6407.5
I(kg.m2) 4.4878×107 75654.64
S(m2) 510.97 18.22
c(m) 8.324 2.91
CLα(rad−1) 5.70 3.44 CM α(rad−1) −1.26 −0.64
CM0(rad) 0 0
CD0 0.021 0.01966
αL(rad) −0.162 −0.1611
λ −0.0367 −0.10563
k 0.0988 0.59414
ρ(kg.m−3) 1.218 1.218
The program also received some values established in subchapter 3.2.2, such as initial velocity, glide slope angle and initial rate-of-change of angle-of-attack (3.57a). With this data it is possible to determine other parameters introduced throughout this work, as presented in the table 4.2.
It was settled a time scope of 50 seconds since it is the sufficient time for the plotted curves to stabilize.
Table 4.2: Boeing 747-100 and Lockheed F-104 calculated parameters Parameter Boeing 747-100 Lockheed F-104
a(m−1) 0.0069 0.0060
b(m−2) 7.8925×10−5 27315×10−4 e0(m−1) 2.5544×10−5 3.4046×10−5 e1(m−1) −2.5445×10−4 −6.2925×10−4 e2(m−1) 0.0039 0.0122 h0 −0.5400 −1.0267 h1 −2.6889 −2.6995
n 2.2075 7.5734
wsp(s−1) 0.0055 0.0191
w(s−1) 0.4847 0.9007
α1(rad) −0.1620 −0.1611
α0(rad) 0.1047 0.1047