Laboratorium Mechaniki Lotu

Aproksymacja biegunowej oporowej samolotu.

Łukasz Krawczyk

III LiK

nr ind. 113442

1. Schemat aproksymacji metodą najmniejszych kwadratów:

APROKSYMACJA 1:

y=Ax²+Bx+C

S(A,B,C) → Σ(Ax_i²+Bx_i+C-y_i)² →min S

Σ(Ax_i²+Bx_i+C-y_i)²=A² Σx_i⁴+2AB Σx_i³+B² Σx_i²+2AC Σx_i²-2A Σx_i²y_i+2BC Σx_i-2B Σx_iy_i+nC²-2C Σy_i+ Σy_i²

$$\frac{\text{δS}}{\text{δA}} = 2A\sum_{}^{}x_{i}^{4} + 2B\sum_{}^{}x_{i}^{3} + 2C\sum_{}^{}x_{i}^{2} - 2\sum_{}^{}x_{i}^{2}y_{i} = 0$$

$$\frac{\text{δS}}{\text{δB}} = 2A\sum_{}^{}x_{i}^{3} + 2B\sum_{}^{}x_{i}^{2} + 2C\sum_{}^{}x_{i} - 2\sum_{}^{}x_{i}^{2}y_{i} = 0$$

$$\frac{\text{δS}}{\text{δC}} = 2A\sum_{}^{}x_{i}^{2} + 2B\sum_{}^{}x_{i} + 2Cn - 2\sum_{}^{}y_{i} = 0$$

$$A\sum_{}^{}x_{i}^{4} + B\sum_{}^{}x_{i}^{3} + C\sum_{}^{}x_{i}^{2} = \sum_{}^{}x_{i}^{2}y_{i}$$

$$A\sum_{}^{}x_{i}^{3} + B\sum_{}^{}x_{i}^{2} + C\sum_{}^{}x_{i} = \sum_{}^{}x_{i}^{2}y_{i}$$

$$A\sum_{}^{}x_{i}^{2} + B\sum_{}^{}x_{i} + Cn = \sum_{}^{}y_{i}$$

W =$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{4} & \sum_{}^{}x_{i}^{3} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}x_{i}^{3} & \sum_{}^{}x_{i}^{2} & \sum_{}^{}x_{i} \\ \sum_{}^{}x_{i}^{2} & \sum_{}^{}x_{i} & n \\ \end{matrix} \right| = \ $n$\sum_{}^{}x_{i}^{4}\sum_{}^{}x_{i}^{2} + 2\sum_{}^{}x_{i}^{2}\sum_{}^{}{x_{i}\sum_{}^{}x_{i}^{3}} - {(\sum_{}^{}{x_{i}^{2})}}^{3} - \sum_{}^{}x_{i}^{2}\sum_{}^{}x_{i}^{4} - n{(\sum_{}^{}{x_{i}^{3})}}^{2}$

W_A=$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{2}y_{i} & \sum_{}^{}x_{i}^{3} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}{x_{i}y_{i}} & \sum_{}^{}x_{i}^{2} & \sum_{}^{}x_{i} \\ \sum_{}^{}y_{i} & \sum_{}^{}x_{i} & n \\ \end{matrix} \right| = n\sum_{}^{}x_{i}^{2}\sum_{}^{}{x_{i}^{2}y_{i}} + \sum_{}^{}x_{i}\sum_{}^{}x_{i}^{2}\sum_{}^{}{x_{i}y}_{i} + \sum_{}^{}x_{i}\sum_{}^{}x_{i}^{3}\sum_{}^{}y_{i} - {(\sum_{}^{}{x_{i}^{2})}}^{2}\sum_{}^{}y_{i} - {\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - (\sum_{}^{}{x_{i}^{2})}}^{2}\sum_{}^{}{x_{i}^{2}y_{i}} - n\sum_{}^{}x_{i}^{3}\sum_{}^{}{x_{i}y}_{i}\ $

W_B=$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{4} & \sum_{}^{}{x_{i}^{2}y_{i}} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}x_{i}^{3} & \sum_{}^{}{x_{i}y_{i}} & \sum_{}^{}x_{i} \\ \sum_{}^{}x_{i}^{2} & \sum_{}^{}y_{i} & n \\ \end{matrix} \right| = \ n\sum_{}^{}x_{i}^{4}\sum_{}^{}{x_{i}y_{i}} + \sum_{}^{}x_{i}^{2}\sum_{}^{}x_{i}^{3}\sum_{}^{}y_{i} + \sum_{}^{}x_{i}^{2}\sum_{}^{}{x_{i}\sum_{}^{}x_{i}^{2}}y_{i} + {(\sum_{}^{}{x_{i}^{2})}}^{2}\sum_{}^{}{x_{i}y_{i}} + \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \sum_{}^{}x_{i}\sum_{}^{}x_{i}^{4}\sum_{}^{}{y_{i} - n}\sum_{}^{}x_{i}^{3}\sum_{}^{}{x_{i}^{2}y_{i}}$

W_C=$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{4} & \sum_{}^{}x_{i}^{3} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}x_{i}^{3} & \sum_{}^{}x_{i}^{2} & \sum_{}^{}x_{i} \\ \sum_{}^{}x_{i}^{2} & \sum_{}^{}x_{i} & n \\ \end{matrix} \right| = \sum_{}^{}x_{i}^{4}\sum_{}^{}x_{i}^{2}\sum_{}^{}y_{i} + \sum_{}^{}x_{i}^{3}\sum_{}^{}{x_{i}\sum_{}^{}x_{i}^{2}}y_{i}$+$\sum_{}^{}x_{i}^{2}\sum_{}^{}x_{i}^{3}\sum_{}^{}{x_{i}y_{i}} - {(\sum_{}^{}{x_{i}^{2})}}^{2}\sum_{}^{}{x_{i}^{2}y_{i} - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - \sum_{}^{}x_{i}\sum_{}^{}x_{i}^{4}\sum_{}^{}{x_{i}y_{i}} - {(\sum_{}^{}{x_{i}^{3})}}^{2}\sum_{}^{}y_{i}}$

$A = \frac{W_{A}}{W}$ ; $B = \frac{W_{B}}{W}$; $C = \frac{W_{C}}{W}$

APROKSYMAJCA 2:

y=Ax²+B

S(A,B) → Σ(Ax_i²+B -y_i)² →min S

Σ(Ax_i²+B-y_i)²=A² Σx_i⁴+2AB Σx_i²+nB²-2A Σx_i²y_i-2B Σy_i+ Σy_i²

$$\frac{\text{δS}}{\text{δA}} = 2A\sum_{}^{}x_{i}^{4} + 2B\sum_{}^{}x_{i}^{2} - 2\sum_{}^{}x_{i}^{2}y_{i} = 0$$

$$\frac{\text{δS}}{\text{δB}} = 2A\sum_{}^{}x_{i}^{2} + 2nB - 2\sum_{}^{}x_{i}^{2}y_{i} = 0$$

$$A\sum_{}^{}x_{i}^{4} + B\sum_{}^{}x_{i}^{2} = \sum_{}^{}x_{i}^{2}y_{i}$$

$$A\sum_{}^{}x_{i}^{2} + nB = \sum_{}^{}y_{i}$$

W =$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{4} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}x_{i}^{2} & n \\ \end{matrix} \right| = \ $n$\sum_{}^{}x_{i}^{4} - (\sum_{}^{}x_{i}^{2})$²

W_A =$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{2}y_{i} & \sum_{}^{}x_{i}^{2} \\ \sum_{}^{}y_{i} & n \\ \end{matrix} \right| = \ n\sum_{}^{}x_{i}^{2}y_{i} - \sum_{}^{}x_{i}^{2}y_{i}$

W_B =$\ \left| \begin{matrix} \sum_{}^{}x_{i}^{4} & \sum_{}^{}x_{i}^{2}y_{i} \\ \sum_{}^{}x_{i}^{2} & \sum_{}^{}y_{i} \\ \end{matrix} \right| = \sum_{}^{}x_{i}^{4}y_{i} - \sum_{}^{}x_{i}^{2}y_{i}\sum_{}^{}x_{i}^{2}\ $

$A = \frac{W_{A}}{W}$ ; $B = \frac{W_{B}}{W}$;

$$A = \frac{\overset{\overline{}}{x_{i}^{2}y_{i}} - \overset{\overline{}}{x_{i}^{2}}\overset{\overline{}}{\text{\ y}_{i}}}{\overset{\overline{}}{x_{i}^{4}} - \left( \overset{\overline{}}{x_{i}^{2}} \right)^{2}}$$

$$B = \frac{\overset{\overline{}}{x_{i}^{4}} \bullet \overset{\overline{}}{y_{i}} - \overset{\overline{}}{x_{i}^{2}} \bullet \overset{\overline{}}{\text{\ x}_{i}^{2}y_{i}}}{\overset{\overline{}}{x_{i}^{4}} - \left( \overset{\overline{}}{x_{i}^{2}} \right)^{2}}$$

1. Dane do obliczeń:

	Cz				Cx
	x	x^2	x^3	x^4	y	xy	x^2y
1	-1,19	1,4161	-1,685159	2,00533921	0,087	-0,10353	0,1232007
2	-1,02	1,0404	-1,061208	1,08243216	0,070	-0,0714	0,072828
3	-0,85	0,7225	-0,614125	0,52200625	0,057	-0,04845	0,0411825
4	-0,68	0,4624	-0,314432	0,21381376	0,045	-0,0306	0,020808
5	-0,51	0,2601	-0,132651	0,06765201	0,031	-0,01581	0,0080631
6	0,00	0	0	0	0,026	0	0
7	0,51	0,2601	0,132651	0,06765201	0,031	0,01581	0,0080631
8	0,68	0,4624	0,314432	0,21381376	0,045	0,0306	0,020808
9	0,85	0,7225	0,614125	0,52200625	0,057	0,04845	0,0411825
10	1,02	1,0404	1,061208	1,08243216	0,070	0,0714	0,072828
11	1,19	1,4161	1,685159	2,00533921	0,087	0,10353	0,1232007
12	1,36	1,8496	2,515456	3,42102016	0,108	0,14688	0,1997568
13	1,53	2,3409	3,581577	5,47981281	0,133	0,20349	0,3113397
14	1,7	2,89	4,913	8,3521	0,169	0,2873	0,48841
E	4,59	14,8835	11,010033	25,0354198	1,016	0,63767	1,5316711
średnia	0,3278571	1,06310714	0,78643093	1,78824427	0,0725714	0,04554786	0,10940508

Aproksymacja I	Aproksymacja II
model matematyczny	y=ax^2+bx+c
W	Wa
0,4370991	0,02129628
	A
	0,04872184

aproksymacja 1		aproksymacja 2
Cxa 1		Cxa 2
0,089102311		0,089873201
0,070872283		0,071458449
0,055458379		0,055876736
0,042860596		0,043128062
0,033078936		0,033212427
0,020630692		0,020463753
0,03352755		0,033212427
0,043458748		0,043128062
0,056206068		0,055876736
0,071769511		0,071458449
0,090149077		0,089873201
0,111344765		0,111120991
0,135356575		0,13520182
0,162184508		0,162115687

1. Wykres: