Summary edit

Equations:

mod-sin: see File:Loi_de_mouvement_sinmod_en_acceleration.svg#Description

3rd order Fourier series develomment:

${\displaystyle {\begin{cases}x=x_{\mathrm {f} }\left[{\frac {t}{\mathrm {T} }}-{\frac {1}{4\pi ^{2}}}\left(b_{1}\sin \left(2\pi {\frac {t}{\mathrm {T} }}\right)+{\frac {b_{2}}{4}}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)+{\frac {b_{3}}{9}}\sin \left(6\pi {\frac {t}{\mathrm {T} }}\right)\right)\right]\\v={\frac {x_{\mathrm {f} }}{\mathrm {T} }}\left[1-{\frac {1}{2\pi }}\left(b_{1}\cos \left(2\pi {\frac {t}{\mathrm {T} }}\right)+{\frac {b_{2}}{2}}\cos \left(4\pi {\frac {t}{\mathrm {T} }}\right)+{\frac {b_{3}}{3}}\cos \left(6\pi {\frac {t}{\mathrm {T} }}\right)\right)\right]\\a={\frac {x_{\mathrm {f} }}{\mathrm {T} ^{2}}}\left[b_{1}\sin \left(2\pi {\frac {t}{\mathrm {T} }}\right)+b_{2}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)+b_{3}\sin \left(6\pi {\frac {t}{\mathrm {T} }}\right)\right]\\j=2\pi {\frac {x_{\mathrm {f} }}{\mathrm {T} ^{3}}}\left[b_{1}\sin \left(2\pi {\frac {t}{\mathrm {T} }}\right)+2b_{2}\sin \left(4\pi {\frac {t}{\mathrm {T} }}\right)+3b_{3}\sin \left(6\pi {\frac {t}{\mathrm {T} }}\right)\right]\end{cases}}}$

with:

• b₁ = 5.3087039
• b₂ = 1.3819893
• b₃ = 0.6204979

Note that theoretically 1/(2π) × (b₁ + b₂/2 + b₃/3) = 1.

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