File:Loi de mouvement sinmod en acceleration.svg

DescriptionLoi de mouvement sinmod en acceleration.svg	Français : Loi de mouvement sinusoïdal modifié en accélération (sin-mod). English: Modified sinusoidal (mod-sin) acceleration motion law.
Date	31 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:

1. t ∈ [0 ; T/8] ${\displaystyle {\begin{cases}x={\frac {\pi }{\pi +4}}x_{\mathrm {f} }\left[{\frac {t}{\mathrm {T} }}-{\frac {1}{4\pi }}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)\right]\\v={\frac {\pi }{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left[1-\cos \left(4\pi {\frac {t}{\mathrm {T} }}\right)\right]\\a={\frac {4\pi ^{2}}{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)\\j={\frac {16\pi ^{3}}{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\cos \left(4\pi {\frac {t}{\mathrm {T} }}\right)\end{cases}}}$

2. t ∈ [T/8 ; 7T/8] ${\displaystyle {\begin{cases}x={\frac {\pi }{\pi +4}}x_{\mathrm {f} }\left[{\frac {t}{\mathrm {T} }}-{\frac {9}{4\pi }}\cos \left({\frac {4\pi }{3}}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{8}}\right)\right)+{\frac {2}{\pi }}\right]\\v={\frac {\pi }{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left[1+3\sin \left({\frac {4\pi }{3}}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{8}}\right)\right)\right]\\a={\frac {4\pi ^{2}}{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\cos \left({\frac {4\pi }{3}}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{8}}\right)\right)\\j={\frac {16\pi ^{3}}{3(\pi +4)}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\sin \left({\frac {4\pi }{3}}\left({\frac {t}{\mathrm {T} }}-{\frac {1}{8}}\right)\right)\end{cases}}}$

3. t ∈ [7T/8 ; T] ${\displaystyle {\begin{cases}x={\frac {\pi }{\pi +4}}x_{\mathrm {f} }\left[{\frac {t}{\mathrm {T} }}-{\frac {1}{4\pi }}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)+{\frac {4}{\pi }}\right]\\v={\frac {\pi }{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left[1-\cos \left(4\pi {\frac {t}{\mathrm {T} }}\right)\right]\\a={\frac {4\pi ^{2}}{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)\\j={\frac {16\pi ^{3}}{\pi +4}}{\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\cos \left(4\pi {\frac {t}{\mathrm {T} }}\right)\end{cases}}}$

This media was created with Scilab, a free open-source software. Here is a listing of the Scilab source used to create this file.

function [t1, t2, t3] = indice(t)
    t1 = (t <= 1/8);
    t2 = (t > 1/8) & (t <= 7/8);
    t3 = (t>7/8);
endfunction

function [j]=jerk(t)
    [t1, t2, t3] = indice(t)
    j = zeros(t);
    coef = 4*%pi*%pi/(4 + %pi)
    j(t1) = coef*4*%pi*cos(4*%pi*t(t1));
    j(t2) = -coef*4/3*%pi*sin(4*%pi/3*(t(t2) - 1/8));
    j(t3) = coef*4*%pi*cos(4*%pi*t(t3));
    endfunction

function [a]=acceleration(t)
    [t1, t2, t3] = indice(t)
    a = zeros(t);
    coef = 4*%pi*%pi/(4 + %pi);
    a(t1) = coef*sin(4*%pi*t(t1));
    a(t2) = coef*cos(4*%pi/3*(t(t2) - 1/8));
    a(t3) = coef*sin(4*%pi*t(t3));
    endfunction

function [v] = vitesse(t)
    [t1, t2, t3] = indice(t)
    coef = %pi/(%pi + 4);
    v = zeros(t);
    v(t1) = coef*(1 - cos(4*%pi*t(t1)));
    v(t2) = coef*(1 + 3*sin(4*%pi/3*(t(t2) - 1/8)));
    v(t3) = coef*(1 - cos(4*%pi*t(t3)));
endfunction

function [x] = position(t)
    [t1, t2, t3] = indice(t)
    x = zeros(t);
    coef = %pi/(%pi + 4);
    x(t1) = coef*(t(t1) - 1/(4*%pi)*sin(4*%pi*t(t1)));
    x(t2) = coef*(t(t2) - 9/(4*%pi)*cos(4*%pi/3*(t(t2) - 1/8)) + 2/%pi);
    x(t3) = coef*(t(t3) - 1/(4*%pi)*sin(4*%pi*t(t3)) + 4/%pi);
endfunction

// tracé

t = 0:0.005:1;

j = jerk(t);
a = acceleration(t);
v = vitesse(t);
x = position(t);

scf(0);
clf;

subplot(4, 1, 1)
plot(t, x)
xtitle("position", "t/T", "$x/x_\mathrm{f}$")

subplot(4, 1, 2)
plot(t, v)
xtitle("vitesse", "t/T", "$v/(x_\mathrm{f}/\mathrm{T})$")

subplot(4, 1, 3)
plot(t, a)
xtitle("accélération", "t/T", "$a/(x_\mathrm{f}/\mathrm{T}^2)$")

subplot(4, 1, 4)
plot(t, j)
xtitle("à-coup", "t/T", "$j/(x_\mathrm{f}/\mathrm{T}^3)$")

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