New analytical results on the study of aircraft performance with velocity dependent forces

Pellegrini, Cláudio C.; Moreira, Erika D.O.; Rodrigues, Mateus S.

doi:10.1590/1806-9126-RBEF-2021-0410

Cláudio C. Pellegrini ^** Correspondence email address: pelle@ufsj.edu.br

Universidade Federal de São João del-Rei, Departamento de Ciências Térmicas e dos Fluidos, São João del-Rei, MG, Brasil.

https://orcid.org/0000-0001-5992-8258

Erika D.O. Moreira

Universidade Federal de São João del-Rei, Departamento de Ciências Térmicas e dos Fluidos, São João del-Rei, MG, Brasil.

Mateus S. Rodrigues

Universidade Federal de São João del-Rei, Departamento de Ciências Térmicas e dos Fluidos, São João del-Rei, MG, Brasil.

* Correspondence email address: pelle@ufsj.edu.br

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Figure 1
Takeoff maneuver.

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Figure 2
Forces acting on an airplane during takeoff or landing.

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Figure 3
Lift and drag coefficients for a conventional airfoil. Reproduced from [¹⁰[10] G.A. Williamson, B.D. McGranahan, B.A. Broughton, R.W. Deters, J.B. Brandt and M.S. Selig, Summary of low-speed airfoil data (University of Illinois, Champaign, 2012), v. 5.].

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Figure 4
Normalized traction data for typical UAVs and commercial size engines.

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Figure 4
Normalized traction data for typical UAVs and commercial size engines.

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Figure 5
Landing maneuver.

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Figure 6
Typical braking profile for commercial size aircraft. Reproduced from [²⁰[20] J. Skorupski and H. Wierzbicki, Transp. Research Part C Emerging Technologies 80, 467 (2017).].

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Figure 6
Typical braking profile for commercial size aircraft. Reproduced from [²⁰[20] J. Skorupski and H. Wierzbicki, Transp. Research Part C Emerging Technologies 80, 467 (2017).].

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Figure 7
Traction measurement apparatus.

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Figure 8
Dynamic traction measurement system.

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Figure 9
Factor

κ

dependence on the wind velocity.

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Figure 10
Motion equation for takeoff with

U_{0} = 0

.

M_{t o t}

from Table .

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Figure 11
Relation between takeoff distance and total weight. Negative wind velocities correspond to headwinds.

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Figure 12
Relation between takeoff distance and total weight for different total drag coefficients.

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Figure 13
Motion equation for landing with U₀ = 0 for two different power unit thrust regimes.

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Figure 14
Motion equation for landing with different braking profiles. Braking limit velocity V₁ defined at equation ().

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Figure 15
Relation between landing distance and total weight. Negative velocities correspond to headwinds.

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Figure 10
Motion equation for takeoff with

U_{0} = 0

.

M_{t o t}

from Table .

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Figure 11
Relation between takeoff distance and total weight. Negative wind velocities correspond to headwinds.

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Figure 12
Relation between takeoff distance and total weight for different total drag coefficients.

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Figure 13
Motion equation for landing with U₀ = 0 for two different power unit thrust regimes.

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Figure 14
Motion equation for landing with different braking profiles. Braking limit velocity V₁ defined at equation ().

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Figure 15
Relation between landing distance and total weight. Negative velocities correspond to headwinds.

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Table 1
Takeoff and landing distances and W_max.

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Table 2
Static gliding headwind velocity, U_0glide (m/s).

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

	Unit	Commercial aircraft		UAV
		Cessna172N	Cessna172S	2014	2017	2018	2019
$ρ$	kg/m³	1.2250	1.2250	1.1226	1.1240	1.1226	1.0564
n	rpm	2700	2700	2500	2500	2500	2500
D	m	1.905	1.905	0.305	0.356	0.4064	0.4826
$a \times 10^{8}$	${(s/m)}^{2}$	2.60	2.60	$- 861$	$- 878$	$- 827$	$- 879$
$b \times 10^{5}$	s/m	$- 1.44$	$- 1.44$	$- 53.5$	$- 22.8$	$- 24.8$	$- 26.4$
$C_{T 0} \times 10^{3}$	–	189	189	21.7	18.4	24.5	26.1
M _tot	kg	1043.00	1156.60	3.13	6.12	11.80	7.24
C _Dtot	–	0.0320	0.0320	0.0646	0.0760	0.0771	0.0715
C _L	–	0.41	0.41	0.44	0.77	0.85	0.95
C _Lmax	–	2.1	2.1	1.4	2.1	2.2	2.2
S	m²	16.07	16.07	0.34	0.42	0.50	0.65
$μ_{r}$	–	0.045	0.045	0.11	0.11	0.13	0.13
$μ_{e}$	–	1.0	1.0	0.4	0.4	0.4	0.4
S_to exp.	m	219	292	45	39	60	40
S_to Eq. (17)	m	209	301	37	48	56	27
Difference	–	4%	3%	18%	22%	$- 6$ %	32%
S_to Eq. (20)	m	210	301	36	48	56	28
W_max exp.	N	10230.9	10898.1	31.4	62.4	115.7	70,9
W _max	N	11220.9	11220.9	34.1	63.5	110.6	87.8
Difference	–	4%	3%	10%	2%	$- 4$ %	24%
$S_{ℓ}$ exp.	m	175	175	50	52	57	42
$S_{ℓ}$ no brakes	m	799	927	82	104	143	74
$S_{ℓ}$ brakes	m	182	211	35	43	61	32
Difference	–	4%	17%	30%	18%	7%	25%

s = \frac{W}{2 g A_{t o}} [\ln | \frac{A_{t o} V^{2} + B_{t o} V + C_{t o}}{C_{t o}} |

- \frac{2 B_{t o}}{\sqrt{4 A_{t o} C_{t o} - B_{t o}^{2}}} (\arctan \frac{2 A_{t o} V + B_{t o}}{\sqrt{4 A_{t o} C_{t o} - B_{t o}^{2}}}

- \arctan \frac{B_{t o}}{\sqrt{4 A_{t o} C_{t o} - B_{t o}^{2}}})]

s = \frac{W}{2 g A_{t o}} [\ln | \frac{A_{t o} V^{2} + B_{t o} V + C_{t o}}{C_{t o}} | - \frac{B_{t o}}{\sqrt{B_{t o}^{2} - 4 A_{t o} C_{t o}}}

\times (\ln \frac{2 A_{t o} V + B_{t o} + \sqrt{B_{t o}^{2} - 4 A_{t o} C_{t o}}}{2 A_{t o} V + B_{t o} - \sqrt{B_{t o}^{2} - 4 A_{t o} C_{t o}}}

- \ln \frac{B_{t o} + \sqrt{B_{t o}^{2} - 4 A_{t o} C_{t o}}}{B_{t o} - \sqrt{B_{t o}^{2} - 4 A_{t o} C_{t o}}})]

κ^{2} (A_{t o} V_{t o}^{2}) + κ (B_{t o} V_{t o}) + C_{t o} - \frac{W V_{t o}^{2}}{2 g s_{t o}} = 0

κ = \frac{- B_{t o} V_{t o} \pm \sqrt{{(B V_{t o})}^{2} - 4 A_{t o} V_{t o}^{2} C_{κ}}}{2 A_{t o} V_{t o}^{2}}

s = \frac{W}{2 g A_{ℓ}} [\ln | \frac{A_{ℓ} V^{2} + B_{ℓ} V + C_{ℓ}}{A_{ℓ} V_{ℓ}^{2} + B_{ℓ} V_{ℓ} + C_{ℓ}} | + \frac{2 B_{ℓ}}{\sqrt{4 A_{ℓ} C_{ℓ} - B_{ℓ}^{2}}}

\times (\arctan \frac{2 A_{ℓ} V_{ℓ} + B_{ℓ}}{\sqrt{4 A_{ℓ} C_{ℓ} - B_{ℓ}^{2}}}

- \arctan \frac{2 A_{ℓ} V + B_{ℓ}}{\sqrt{4 A_{ℓ} C_{ℓ} - B_{ℓ}^{2}}})]

s = \frac{W}{2 g A_{ℓ}} [\ln | \frac{A_{ℓ} V^{2} + B_{ℓ} V + C_{ℓ}}{A_{ℓ} V_{ℓ}^{2} + B_{ℓ} V_{ℓ} + C_{ℓ}} | + \frac{B_{ℓ}}{\sqrt{B_{ℓ}^{2} - 4 A_{ℓ} C_{ℓ}}}

\times (\ln \frac{2 A_{ℓ} V_{ℓ} + B_{ℓ} + \sqrt{B_{ℓ}^{2} - 4 A_{ℓ} C_{ℓ}}}{2 A_{ℓ} V_{ℓ} + B - \sqrt{B_{ℓ}^{2} - 4 A_{ℓ} C_{ℓ}}}

- \ln \frac{2 A_{ℓ} V + B_{ℓ} + \sqrt{B_{ℓ}^{2} - 4 A_{ℓ} C_{ℓ}}}{2 A_{ℓ} V + B_{ℓ} - \sqrt{B_{ℓ}^{2} - 4 A_{ℓ} C_{ℓ}}})]

ι_{B} (V) = {\begin{matrix} ι_{1}, for V_{1} \leq V \leq V_{ℓ} \\ ι_{2}, for V_{2} \leq V < V_{1} \\ ⊊ \\ ι_{m}, for 0 \leq V < V_{m - 1} \end{matrix}

\begin{matrix} s = \frac{W}{g} [\int_{V_{ℓ}}^{V_{1}} \frac{V d V}{A_{ℓ} V^{2} + B_{ℓ} V + C_{ℓ}} \\ + \int_{V_{1}}^{V_{2}} \frac{V d V}{A_{ℓ} V^{2} + B_{ℓ} V + C_{ℓ}} \\ + ⊋ + \int_{V_{k}}^{V} \frac{V d V}{A_{ℓ} V^{2} + B_{ℓ} V + C_{ℓ}}] \end{matrix}

[1] * Correspondence email address: pelle@ufsj.edu.br

	M _empty	M _tot
C _Lmax	5.5	10.7
C _L	9.9	19.0