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qstruct:teoria:qsection:integrali_rotazione

Integrali notevoli con rotazione del sistema di riferimento

Momenti statici

Le formule per il calcolo dei momenti statici nel caso di rotazione del sistema di riferimento di un angolo $\theta$ sono

$$S_{z}^{\odot} = S_z \cdot \cos \theta + S_y \cdot \sin \theta$$

$$S_{y}^{\odot} = - S_z \cdot \sin \theta + S_y \cdot \cos \theta $$

Per il calcolo analitico vedi la pagina del wiki sulla geometria delle aree.

Momenti di inerzia

Le formule per il calcolo dei momenti di inerzia nel caso di rotazione del sistema di riferimento di un angolo $\theta$ sono

$$I_{yy}^\odot = I_{yy} \cos ^2 \theta - 2 I_{yz} \sin \theta \cos \theta + I_{zz} \sin ^2 \theta$$

$$I_{zz}^\odot = I_{yy} \sin ^2 \theta + 2 I_{yz} \sin \theta \cos \theta + I_{zz} \cos ^2 \theta$$

$$I_{yz}^\odot = \left( I_{yy} - I_{zz} \right) \sin \theta \cos \theta + I_{yz} \left( \cos^2 \theta - \sin^2 \theta \right) = \frac{I_{yy} - I_{zz}}{2} \sin 2 \theta + I_{yz} \cos 2 \theta$$

Per maggiori dettagli su come si determinano tali formule si rimanda alla sezione del wiki sulla geometria delle aree.

Momenti di terzo ordine

Le formule che ci permettono di calcolare i momenti di terzo ordine in caro di rotazione di angolo $\theta$ sono

$$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\ \cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \\ - \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \\ - \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ + \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \\ \sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ + \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

Si riporta di seguito il dettaglio dei calcoli

$$\iint \limits_{S} y_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( \cos^3 \theta \, y^3 + 3 \cos^2 \theta \, \sin \theta \, y^2 \, z + 3 \cos \theta \, \sin^2 \theta \, y \, z^2 + \sin^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ \cos^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \cos^2 \theta \, \sin \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 3 \cos \theta \, \sin^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} z_{\odot}^3 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( - \sin \theta \, y + \cos \theta \, z \right)^3 \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( - \sin^3 \theta \, y^3 + 3 \sin^2 \theta \, \cos \theta \, y^2 \, z - 3 \sin \theta \, \cos^2 \theta \, y \, z^2 + \cos^3 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ - \sin^3 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z + 3 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 3 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot}^2 \, z_{\odot} \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right)^2 \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( \cos^2 \theta \, y^2 + 2 \sin \theta \, \cos \theta \, y \, z + \sin^2 \theta \, z^2 \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right) \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( - \sin \theta \, \cos^2 \theta \, y^3 - 2 \sin^2 \theta \, \cos \theta \, y^2 \, z - \sin^3 \theta \, y \, z^2 + \cos^3 \theta \, y^2 \, z + 2 \sin \theta \, \cos^2 \theta \, y \, z^2 + \sin^2 \theta \, \cos \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ - \sin \theta \, \cos^2 \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - \sin^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ + \cos^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin^2 \theta \, \cos \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$

$$\iint \limits_{S} y_{\odot} \, z_{\odot}^2 \; \mathrm{d}y \mathrm{d}z = \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( - \sin \theta \, y + \cos \theta \, z \right)^2 \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( \cos \theta \, y + \sin \theta \, z \right) \, \left( \sin^2 \theta \, y^2 - 2 \sin \theta \, \cos \theta \, y \, z + \cos^2 \theta \, z^2 \right) \; \mathrm{d}y \mathrm{d}z = \\ \iint \limits_{S} \left( \sin^2 \theta \, \cos \theta \, y^3 - 2 \sin \theta \, \cos^2 \theta \, y^2 \, z + \cos^3 \theta \, y \, z^2 + \sin^3 \theta \, y^2 \, z - 2 \sin^2 \theta \, \cos \theta \, y \, z^2 + \sin \theta \, \cos^2 \theta \, z^3 \right) \; \mathrm{d}y \mathrm{d}z = \\ \sin^2 \theta \, \cos \theta \iint \limits_{S} y^3 \; \mathrm{d}y \mathrm{d}z - 2 \sin \theta \, \cos^2 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z + \cos^3 \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z \\ + \sin^3 \theta \iint \limits_{S} y^2 \, z \; \mathrm{d}y \mathrm{d}z - 2 \sin^2 \theta \, \cos \theta \iint \limits_{S} y \, z^2 \; \mathrm{d}y \mathrm{d}z + \sin \theta \, \cos^2 \theta \iint \limits_{S} z^3 \; \mathrm{d}y \mathrm{d}z $$


qstruct/teoria/qsection/integrali_rotazione.txt · Ultima modifica: 2014/11/26 13:51 da mickele

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