Nombres complexes

Posons la vibration résultante sous la forme :

\(s(t) = a \cos \left(\omega t + \phi\right) = \Re \left\{\underline{s}(t)\right\}\)

avec \(\underline{s}(t) = a e^{j\left(\omega t + \phi\right)} = \underline{A} e^{j \omega t}\)

où \(\underline{A} = a e^{j \phi}\)

d'où :

\(s(t) = s_{1}(t) + s_{2}(t) \Rightarrow \underline{s}(t) = \underline{s}_{1}(t) + \underline{s}_{2}(t) \Rightarrow \underline{A} = \underline{A}_{1} + \underline{A}_{2} \Rightarrow\)

\(a e^{j \phi} = a_{1} + a_{2} e^{j \varphi}\)

Relation entre les modules :

\(a = \left\arrowvert a_{1} + a_{2} e^{j \varphi}\right\arrowvert =\left\arrowvert a_{1} + a_{2} \cos \varphi + j a_{2} \sin \varphi \right\arrowvert\)

\(a= \left[\left(a_{1} + a_{2} \cos \varphi\right)^{2} + a_{2}^{2} \sin^{2} \varphi\right]^{\tfrac{1}{2}} = \left[a_{1}^{2} + a_{2}^{2} + 2 a_{1}a_{2} \cos \varphi\right]^{\tfrac{1}{2}}\) ( 4 points )

Relation entre les arguments :

\(\phi = \arg \left(a_{1} + a_{2}e^{j \varphi}\right) = \arg\left(a_{1} + a_{2} \cos \varphi + j a_{2} \sin \varphi\right)\)

\(\phi = \textrm{Arctan} \left(\frac{a_{2} \sin \varphi}{a_{1} + a_{2} \cos \varphi}\right)\) ( 4 points )