Summary

1.

${\displaystyle j={\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle a={\frac {3\pi v_{0}}{2\mathrm {T} }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle v={\frac {v_{0}}{2}}\left(1-\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle x={\frac {v_{0}}{2}}\left(t-{\frac {\mathrm {T} }{3\pi }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle a_{1}=0}$

${\displaystyle v_{1}=v_{0}}$

${\displaystyle x_{1}={\frac {v_{0}\mathrm {T} }{6}}}$

2.

${\displaystyle j=0}$

${\displaystyle a=0}$

${\displaystyle v=v_{0}}$

${\displaystyle x=v_{0}\left(t-{\frac {\mathrm {T} }{6}}\right)}$

${\displaystyle a_{2}=0}$

${\displaystyle v_{2}=v_{0}}$

${\displaystyle x_{2}={\frac {v_{0}\mathrm {T} }{2}}}$

3.

${\displaystyle j=-{\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle a=-{\frac {3\pi v_{0}}{2\mathrm {T} }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)}$

${\displaystyle v={\frac {v_{0}}{2}}\left(1+\cos \left({\frac {3\pi t}{\mathrm {T} }}\right)\right)}$

${\displaystyle x={\frac {v_{0}}{2}}\left(t+{\frac {\mathrm {T} }{3\pi }}\sin \left({\frac {3\pi t}{\mathrm {T} }}\right)+{\frac {\mathrm {T} }{3}}\right)}$

${\displaystyle a_{3}=0}$

${\displaystyle v_{1}=v_{0}}$

${\displaystyle x_{\mathrm {f} }={\frac {2v_{0}\mathrm {T} }{3}}}$

Max.

${\displaystyle v_{\max }=v_{0}={\frac {3x_{\mathrm {f} }}{2\mathrm {T} }}}$

${\displaystyle a_{\max }={\frac {3\pi v_{0}}{2\mathrm {T} }}={\frac {9\pi x_{\mathrm {f} }}{4\mathrm {T} ^{2}}}}$

${\displaystyle j_{\max }={\frac {9\pi ^{2}v_{0}}{2\mathrm {T} ^{2}}}={\frac {27\pi ^{2}x_{\mathrm {f} }}{4\mathrm {T} ^{3}}}}$

Scilab source

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 < 1/3);
    t2 = (t >= 1/3) & (t < 2/3);
    t3 = (t >= 2/3);
endfunction

function [j]=jerk(t)
    [t1, t2, t3] = indice(t);
    j = zeros(t);
    j(t1) = (9*%pi^2/2)*cos(3*%pi*t(t1));
    j(t2) = 0;
    j(t3) = -(9*%pi^2/2)*cos(3*%pi*t(t3));
endfunction

function [a]=acceleration(t)
    [t1, t2, t3] = indice(t);
    a = zeros(t);
    a(t1) = (3*%pi/2)*sin(3*%pi*t(t1));
    a(t2) = 0;
    a(t3) = -(3*%pi/2)*sin(3*%pi*t(t3));
    endfunction

function [v] = vitesse(t)
    [t1, t2, t3] = indice(t);
    v = zeros(t);
    v(t1) = (1/2)*(1 - cos(3*%pi*t(t1)));
    v(t2) = 1;
    v(t3) = (1/2)*(1 + cos(3*%pi*t(t3)));
endfunction

function [x] = position(t)
    [t1, t2, t3] = indice(t);
    x = zeros(t);
    x(t1) = (1/2)*(t(t1) - (1/3/%pi)*sin(3*%pi*t(t1)));
    x(t2) = t(t2) - 1/6;
    x(t3) = (1/2)*(t(t3) + (1/3/%pi)*sin(3*%pi*t(t3))) + 1/6;
endfunction

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", "$x/(v_0 \mathrm{T})$")

subplot(4, 1, 2)
plot(t, v)
xtitle("vitesse", "t/T", "$v/(v_0)$")

subplot(4, 1, 3)
plot(t, a)
xtitle("accélération", "t/T", "$a\times \mathrm{T}/v_0$")

subplot(4, 1, 4)
xtitle("à-coup", "t/T", "$j \times \mathrm{T}^2/v_0$")
plot(t, j)

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue

This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/3.0CC BY-SA 3.0 Creative Commons Attribution-Share Alike 3.0 truetrue

You may select the license of your choice.