ARTUR BIEŃ 187210

Projekt nr. 4

Lot szybowy, biegunowa prędkości.

1. Lot szybowy

• Gęstość na danych wysokościach

$$\rho_{1} = \rho_{o}*\left( 1 - \frac{H_{1}}{44331} \right)^{4,256}$$

$\rho_{o} = 1,225\ \frac{\text{kg}}{m^{3}}$

$$\rho_{1} = \rho_{o}*\left( 1 - \frac{H_{1}}{44331} \right)^{4,256} = 1,225*\left( 1 - \frac{2000}{44331} \right)^{4,256} = 1,006\ \frac{\text{kg}}{m^{3}}$$

Tab. 4.1

	h[m]	ρ[kg/m^3]
h 1	0	1,225
h 2	2000	1,006
h 3	4000	0,819

1. Biegunowa prędkości

m = 758 kg

S = 11, 59 m²

( dla C_z = 1, 777 = 0 .138 ,C_x = 0, 1803 i H₁ = 0 m):

Prędkość lotu

$$V_{1} = \sqrt{\frac{2 \bullet m \bullet g}{\rho \bullet S} \bullet \frac{1}{\sqrt{C_{z}^{2} + C_{x}^{2}}}} = \sqrt{\frac{2 \bullet 758 \bullet 9,81}{1,225 \bullet 11,59} \bullet \frac{1}{\sqrt{{1,777}^{2} + {0,1803}^{2}}}} = 24,21\ \frac{m}{s}$$

Prędkość opadania

$$w_{1} = \sqrt{\frac{2 \bullet m \bullet g}{\rho \bullet S} \bullet \frac{C_{x}^{2}}{\sqrt{\left( C_{z}^{2} + C_{x}^{2} \right)^{3}}}} = \sqrt{\frac{2 \bullet 758 \bullet 9,81}{1,225 \bullet 11,59} \bullet \frac{{0,1803}^{2}}{\sqrt{\left( {1,777}^{2} + {0,1803}^{2} \right)^{3}}}} = 2,44\ \frac{m}{s}$$

Kąt toru lotu

$$\gamma = \arctan\left( \frac{\text{Cx}}{\text{Cz}} \right) = \arctan\left( \frac{0,1803}{1,777} \right) \cong 5,79$$

Cz	Cx	$$\sqrt{\frac{1}{\sqrt{C_{z}^{2} + C_{x}^{2}}}}$$	$$\sqrt{\frac{C_{x}^{2}}{\sqrt{{(C_{z}^{2} + C_{x}^{2})}^{3}}}}$$	γ	V1	w1	V2	w2	V3	w3
1.777	0.1803	0.560	0.006	5.792	24.21	2.44	26.71	2.70	29.61	2.99
1.445	0.1197	0.690	0.005	4.736	26.88	2.22	29.66	2.45	32.87	2.71
1.294	0.1018	0.770	0.005	4.495	28.40	2.23	31.34	2.46	34.74	2.72
1.048	0.0747	0.952	0.005	4.077	31.58	2.25	34.84	2.48	38.61	2.75
0.960	0.0661	1.039	0.005	3.938	32.99	2.27	36.40	2.50	40.35	2.77
0.864	0.0585	1.155	0.005	3.877	34.79	2.35	38.38	2.60	42.54	2.88
0.813	0.0548	1.227	0.006	3.855	35.86	2.41	39.56	2.66	43.85	2.95
0.754	0.0508	1.323	0.006	3.852	37.22	2.50	41.06	2.76	45.52	3.06
0.695	0.0470	1.437	0.007	3.871	38.79	2.62	42.80	2.89	47.44	3.20
0.634	0.0434	1.574	0.007	3.919	40.60	2.77	44.79	3.06	49.65	3.39
0.510	0.0371	1.955	0.010	4.156	45.26	3.28	49.93	3.62	55.34	4.01
0.386	0.0319	2.580	0.018	4.726	51.99	4.28	57.35	4.73	63.58	5.24
0.326	0.0300	3.052	0.026	5.252	56.54	5.18	62.37	5.71	69.14	6.33
0.263	0.0282	3.775	0.043	6.104	62.89	6.69	69.38	7.38	76.90	8.18
0.200	0.0268	4.950	0.087	7.609	72.01	9.53	79.44	10.52	88.06	11.66
0.138	0.0256	7.126	0.236	10.494	86.40	15.74	95.32	17.36	105.66	19.24

1. Przybliżone wartości ekonomicznych i optymalnych warunków lotu szybowego.

Przykłady obliczeń dla wysokości lotu H₁ = 0 m, Λ_e = 6, 63 , C_x0 = 0, 0234:

Obliczenie prędkości optymalnej lotu

$$V_{\text{opt}} = \sqrt{\frac{2 \bullet m \bullet g}{\rho \bullet S \bullet \sqrt{\pi \bullet \Lambda_{e} \bullet C_{x_{0}}}}} = \sqrt{\frac{2 \bullet 758 \bullet 9,81}{1,225 \bullet 11,59 \bullet \sqrt{\pi \bullet 6,63 \bullet 0,0234}}} = 38,74\frac{m}{s}$$

Obliczenie prędkości optymalnej opadania

$$w_{\text{opt}} = \sqrt{\frac{8 \bullet m \bullet g \bullet \sqrt{C_{x_{0}}}}{\rho \bullet S \bullet \left( \sqrt{\pi \bullet \Lambda_{e}} \right)^{3}}} = \sqrt{\frac{8 \bullet 758 \bullet 9,81 \bullet \sqrt{0,0234}}{1,225 \bullet 11,59 \bullet \left( \sqrt{\pi \bullet 6,63} \right)^{3}}} = 2,55\ \frac{m}{s}$$

Obliczenie prędkości ekonomicznej opadania

$$w_{\text{ek}} = \sqrt{\frac{32 \bullet m \bullet g \bullet \sqrt{C_{x_{0}}}}{\rho \bullet S \bullet \left( \sqrt{3 \bullet \pi \bullet \Lambda_{e}} \right)^{3}}} = \sqrt{\frac{32 \bullet 758 \bullet 9,81 \bullet \sqrt{0,0234}}{1,225 \bullet 11,59 \bullet \left( \sqrt{3 \bullet \pi \bullet 6,63} \right)^{3}}} = 2,28\ \frac{m}{s}$$

Obliczenie prędkości ekonomicznej lotu

$$V_{\text{ek}} = \sqrt{\frac{2 \bullet m \bullet g}{\rho \bullet S \bullet \sqrt{3 \bullet \pi \bullet \Lambda_{e} \bullet C_{x_{0}}}}} = \sqrt{\frac{2 \bullet 758 \bullet 9,81}{1,225 \bullet 11,59 \bullet \sqrt{3 \bullet \pi \bullet 6,63 \bullet 0,0234}}} = 29,44\frac{m}{s}$$

	w opt	V opt	w ek	V ek
OBLICZONE	2.55	38.74	2.28	29.44	h 1
	2.81	42.74	2.51	32.47	h 2
	3.12	47.38	2.79	36.00	h 3
ODCZYTANE			2.22	26.88	h 1
			2.45	29.66	h 2
			2.71	32.87	h 3