File:Loi de mouvement trapezoidale en acceleration.svg

DescriptionLoi de mouvement trapezoidale en acceleration.svg	Français : Lois de mouvement trapézoïdales en accélération : vert : deux phases (démarrage et freinage) 1/8-1/4-1/4-1/4-1/8 ; bleu : trois phases (démarrage, vitesse constante, freinage) 1/9-1/9-1/9-1/3-1/9-1/9-1/9. English: Movement which are traezoidal for the acceleration: green: two phases (starting and braking) 1/8-1/4-1/4-1/4-1/8; blue: three phases (starting, constant speed and braking) 1/9-1/9-1/9-1/3-1/9-1/9-1/9.
Date	30 August 2018
Source	Own work
Author	Cdang
SVG development InfoField	The SVG code is valid.   This structural formula was created with Batik.    This file uses embedded text that can be easily translated using a text editor.

Equations:

Two phases (green):

1. ${\displaystyle {\begin{cases}j={\frac {128}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {128}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}{\frac {t}{\mathrm {T} }}\\v={\frac {64}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}{\frac {t^{2}}{\mathrm {T} ^{2}}}\\x={\frac {64}{9}}x_{\mathrm {f} }{\frac {t^{3}}{\mathrm {T} ^{3}}}\\\end{cases}}}$

2. ${\displaystyle {\begin{cases}j=0\\a={\frac {16}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\\v={\frac {16}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{16}}\right)\\x={\frac {8}{3}}x_{\mathrm {f} }\left({\frac {t^{2}}{\mathrm {T} ^{2}}}-{\frac {1}{8}}{\frac {t}{\mathrm {T} }}+{\frac {1}{192}}\right)\\\end{cases}}}$

3. ${\displaystyle {\begin{cases}j=-{\frac {128}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {64}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left(-2{\frac {t}{\mathrm {T} }}+1\right)\\v={\frac {16}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left(4\left(-{\frac {t^{2}}{\mathrm {T} ^{2}}}+{\frac {t}{\mathrm {T} }}\right)-{\frac {5}{8}}\right)\\x={\frac {16}{3}}x_{\mathrm {f} }\left(-{\frac {4}{3}}{\frac {t^{3}}{\mathrm {T} ^{3}}}+2{\frac {t^{2}}{\mathrm {T} ^{2}}}-{\frac {5}{8}}{\frac {t}{\mathrm {T} }}+{\frac {7}{96}}\right)\\\end{cases}}}$

4. ${\displaystyle {\begin{cases}j=0\\a=-{\frac {16}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\\v={\frac {16}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left(-{\frac {t}{\mathrm {T} }}+{\frac {15}{16}}\right)\\x={\frac {16}{3}}x_{\mathrm {f} }\left(-{\frac {1}{2}}{\frac {t^{2}}{\mathrm {T} ^{2}}}+{\frac {15}{16}}{\frac {t}{\mathrm {T} }}-{\frac {97}{384}}\right)\\\end{cases}}}$

5. ${\displaystyle {\begin{cases}j={\frac {128}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {128}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left({\frac {t}{\mathrm {T} }}-1\right)\\v={\frac {64}{3}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left({\frac {t^{2}}{\mathrm {T} ^{2}}}-2{\frac {t}{\mathrm {T} }}+1\right)\\x={\frac {16}{3}}x_{\mathrm {f} }\left(4\left({\frac {1}{3}}{\frac {t^{3}}{\mathrm {T} ^{3}}}-{\frac {t^{2}}{\mathrm {T} ^{2}}}+{\frac {t}{\mathrm {T} }}\right)-{\frac {55}{48}}\right)\\\end{cases}}}$

Trois phases (blue):

1. ${\displaystyle {\begin{cases}j={\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}{\frac {t}{\mathrm {T} }}\\v={\frac {243}{8}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}{\frac {t^{2}}{\mathrm {T} ^{2}}}\\x={\frac {81}{8}}x_{\mathrm {f} }{\frac {t^{3}}{\mathrm {T} ^{3}}}\\\end{cases}}}$

2. ${\displaystyle {\begin{cases}j=0\\a={\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\\v={\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{18}}\right)\\x={\frac {27}{8}}x_{\mathrm {f} }\left({\frac {t^{2}}{\mathrm {T} ^{2}}}-{\frac {1}{9}}{\frac {t}{\mathrm {T} }}+{\frac {1}{243}}\right)\\\end{cases}}}$

3. ${\displaystyle {\begin{cases}j=-{\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {81}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left(-3{\frac {t}{\mathrm {T} }}+1\right)\\v={\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left(-{\frac {9}{2}}{\frac {t^{2}}{\mathrm {T} ^{2}}}+3{\frac {t}{\mathrm {T} }}-{\frac {5}{18}}\right)\\x={\frac {27}{4}}x_{\mathrm {f} }\left(-3{\frac {t^{3}}{\mathrm {T} ^{3}}}+3{\frac {t^{2}}{\mathrm {T} ^{2}}}-{\frac {5}{9}}{\frac {t}{\mathrm {T} }}+{\frac {1}{27}}\right)\\\end{cases}}}$

4. ${\displaystyle {\begin{cases}j=0\\a=0\\v={\frac {3}{2}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\\x={\frac {3}{4}}x_{\mathrm {f} }\left(2{\frac {t}{\mathrm {T} }}-{\frac {1}{3}}\right)\\\end{cases}}}$

5. ${\displaystyle {\begin{cases}j=-{\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {81}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left(-3{\frac {t}{\mathrm {T} }}+2\right)\\v={\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left(-{\frac {9}{2}}{\frac {t^{2}}{\mathrm {T} ^{2}}}-6{\frac {t}{\mathrm {T} }}-{\frac {16}{9}}\right)\\x={\frac {27}{4}}x_{\mathrm {f} }\left(-{\frac {3}{2}}{\frac {1}{3}}{\frac {t^{3}}{\mathrm {T} ^{3}}}+3{\frac {t^{2}}{\mathrm {T} ^{2}}}-{\frac {16}{9}}{\frac {t}{\mathrm {T} }}+{\frac {11}{27}}\right)\\\end{cases}}}$

6. ${\displaystyle {\begin{cases}j=0\\a=-{\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\\v={\frac {27}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left(-{\frac {t}{\mathrm {T} }}-{\frac {17}{18}}\right)\\x={\frac {27}{8}}x_{\mathrm {f} }\left(-{\frac {t^{2}}{\mathrm {T} ^{2}}}+{\frac {17}{9}}{\frac {t}{\mathrm {T} }}-{\frac {145}{243}}\right)\\\end{cases}}}$

7. ${\displaystyle {\begin{cases}j={\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\\a={\frac {243}{4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left({\frac {t}{\mathrm {T} }}-1\right)\\v={\frac {243}{8}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left({\frac {t^{2}}{\mathrm {T} ^{2}}}-2{\frac {t}{\mathrm {T} }}+1\right)\\x={\frac {27}{8}}x_{\mathrm {f} }\left(3{\frac {t^{3}}{\mathrm {T} ^{3}}}-9{\frac {t^{2}}{\mathrm {T} ^{2}}}+9{\frac {t}{\mathrm {T} }}-{\frac {73}{27}}\right)\\\end{cases}}}$

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