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internship formulas

$\frac{\partial ω_{x}}{\partial t} = [\frac{2 \cdot a^{2}}{a^{2} + c^{2}} - \frac{2 \cdot a^{2}}{a^{2} + b^{2}}] \cdot ω_{z} \cdot ω_{y} + P_{o} \sin (t) \cdot \frac{2 \cdot a^{2}}{a^{2} + b^{2}} \cdot ω_{Z} + \frac{2 \cdot a^{2}}{a^{2} + c^{2}} \cdot ω_{y} + P_{o} \sin (t) + L \cdot τ_{x}$ $(delomega_x)/(delt) = [(2*a^2)/(a^2+c^2)-(2*a^2)/(a^2+b^2)]*omega_z*omega_y+P_osin(t)*(2*a^2)/(a^2+b^2)*omega_Z+(2*a^2)/(a^2+c^2)*omega_y+P_osin(t)+ L*tau_x$ $\frac{\partial ω_{y}}{\partial t} = [\frac{2 \cdot b^{2}}{a^{2} + b^{2}} - \frac{2 \cdot b^{2}}{b^{2} + c^{2}}] \cdot ω_{x} \cdot ω_{z} + P_{o} \cos (t) \cdot \frac{2 \cdot b^{2}}{a^{2} + b^{2}} \cdot ω_{Z} - \frac{2 \cdot b^{2}}{b^{2} + c^{2}} \cdot ω_{x} + P_{o} \cos (t) + L \cdot τ_{y}$ $(delomega_y)/(delt) = [(2*b^2)/(a^2+b^2)-(2*b^2)/(b^2+c^2)]*omega_x*omega_z+P_ocos(t)*(2*b^2)/(a^2+b^2)*omega_Z-(2*b^2)/(b^2+c^2)*omega_x+P_ocos(t)+ L*tau_y$ …

Amith Ganta
updated on 30 Oct 2021

$\frac{\partial ω_{y}}{\partial t} = [\frac{2 \cdot b^{2}}{a^{2} + b^{2}} - \frac{2 \cdot b^{2}}{b^{2} + c^{2}}] \cdot ω_{x} \cdot ω_{z} + P_{o} \cos (t) \cdot \frac{2 \cdot b^{2}}{a^{2} + b^{2}} \cdot ω_{Z} - \frac{2 \cdot b^{2}}{b^{2} + c^{2}} \cdot ω_{x} + P_{o} \cos (t) + L \cdot τ_{y}$ $(delomega_y)/(delt) = [(2*b^2)/(a^2+b^2)-(2*b^2)/(b^2+c^2)]*omega_x*omega_z+P_ocos(t)*(2*b^2)/(a^2+b^2)*omega_Z-(2*b^2)/(b^2+c^2)*omega_x+P_ocos(t)+ L*tau_y$

$\frac{\partial ω_{z}}{\partial t} = [\frac{2 \cdot c^{2}}{b^{2} + c^{2}} - \frac{2 \cdot c^{2}}{a^{2} + c^{2}}] \cdot ω_{x} \cdot ω_{y} - P_{o} \cos (t) \cdot \frac{2 \cdot c^{2}}{a^{2} + c^{2}} \cdot ω_{y} - P_{o} \sin (t) \cdot \frac{2 \cdot c^{2}}{b^{2} + c^{2}} \cdot ω_{x} + L \cdot τ_{z}$ $(delomega_z)/(delt) = [(2*c^2)/(b^2+c^2)-(2*c^2)/(a^2+c^2)]*omega_x*omega_y-P_ocos(t)*(2*c^2)/(a^2+c^2)*omega_y-P_osin(t)*(2*c^2)/(b^2+c^2)*omega_x+ L*tau_z$

$L = \frac{15}{16 \cdot π} [\begin{matrix} (\frac{b^{2} + c^{2}}{b^{2} \cdot c^{2}}) & 0 & 0 \\ 0 & (\frac{a^{2} + c^{2}}{a^{2} \cdot c^{2}}) & 0 \\ 0 & 0 & (\frac{b^{2} + a^{2}}{b^{2} \cdot a^{2}}) \end{matrix}]$ $L = 15/(16*pi)[[((b^2+c^2)/(b^2*c^2)),0,0],[0,((a^2+c^2)/(a^2*c^2)),0],[0,0,((b^2+a^2)/(b^2*a^2))]]$