Algorithme de Varignon |
➔ |
Composantes du vecteur position | Composantes du vecteur vitesse | Composantes du vecteur accélération | |
1 | \(\displaystyle\mathrm{ x_1(t) \ = \ a \ t^2+b \ t+c }\) | \(\displaystyle\mathrm{ {\dot x}_1(t) \ = 2 \ a \ t + b }\) | \(\displaystyle\mathrm{ {\ddot x}_1(t) \ = 2 \ a }\) |
\(\displaystyle\mathrm{ y_1(t)=\frac{d}{t}+e }\) | \(\displaystyle\mathrm{ {\dot y}_1(t)=-\frac{d}{t^2} }\) | \(\displaystyle\mathrm{ {\ddot y}_1(t)= 2 \frac{d}{t^3} }\) | |
\(\displaystyle\mathrm{z_1(t)=f}\) | \(\displaystyle\mathrm{{\dot z}_1(t)=0}\) | \(\displaystyle\mathrm{{\ddot z}_1(t)=0}\) | |
2 | \(\displaystyle\mathrm{ x_2(t) \ = \frac{a}{3} (8-t^3) - t }\) | \(\displaystyle\mathrm{ {\dot x}_2(t) \ = - at^2 -1 }\) et \(\displaystyle\mathrm{x_2(2) \ = \ -2 }\) |
\(\displaystyle\mathrm{ {\ddot x}_2(t) \ = - 2 \ a \ t }\) |
\(\displaystyle\mathrm{ y_2(t) \ = \ - \ 2 \ cos\left(3 \ t + \frac{π}{6}\right) }\) | \(\displaystyle\mathrm{ {\dot y}_2(t) \ = \ 6 \ sin\left(3 \ t + \frac{π}{6}\right) }\) et \(\displaystyle\mathrm{y_2(0) \ = - \sqrt{3} }\) |
\(\displaystyle\mathrm{ {\ddot y}_2(t) \ = \ 18 \ cos\left(3 \ t + \frac{π}{6}\right) }\) | |
\(\displaystyle\mathrm{ z_2(t) \ = 2 t - 2 }\) | \(\displaystyle\mathrm{ {\dot z}_2(t) \ = \ 2}\) et \(\displaystyle\mathrm{z_2(2) \ = \ 2 }\) |
\(\displaystyle\mathrm{ {\ddot z}_2(t) \ = 0 }\) | |
3 | \(\displaystyle\mathrm{ x_3(t) = - \frac{A}{ω^2} sin(ω t) + \frac{A}{ω} t }\) | \(\displaystyle\mathrm{ {\dot x}_3(t) = \frac{A}{ω} (1- cos(ω t)) }\) | \(\displaystyle\mathrm{ {\ddot x}_3(t) \ = \ A \ sin(ω \ t) }\) et \(\displaystyle\mathrm{{\dot x}_3(0) = 0}\) et \(\displaystyle\mathrm{x_3(0) = 0 }\) |
\(\displaystyle\mathrm{y_3(t) \ = \ B \ sin(ω' \ t)}\) | \(\displaystyle\mathrm{{\dot y}_3(t) = ω' \ B \ cos (ω' t) }\) | \(\displaystyle\mathrm{{\ddot y}_3(t) =- ω'^2 \ B \ sin (ω' t) }\) | |
\(\displaystyle\mathrm{z_3(t) \ = \frac{C}{2}\ t^2 + D \ t }\) | \(\displaystyle\mathrm{{\dot z}_3(t) \ = \ C \ t + D}\) et \(\displaystyle\mathrm{z_3(0) \ = \ 0 }\) |
\(\displaystyle\mathrm{{\ddot z}_3(t) \ = \ C }\) |
mètre | seconde | mètre | seconde | ||
A | 1 | -2 | a | 1 | -2 |
B | 1 | 0 | b | 1 | -1 |
C | 1 | -2 | c | 1 | 0 |
D | 1 | -1 | d | 1 | 1 |
f | 1 | 0 | e | 1 | 0 |
ω | 0 | -1 | ω' | 0 | -1 |