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| Latex formatting |
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\begin{align}
&PE = m_{arm} g l_{arm} sin(\pi180-\theta_{arm}) + m_{lid} g l_{lid} sin(\pi180-\theta_{lid}) + \frac{1}{2} k \Delta_{\theta_{arm}}
\\
&KE = \frac{1}{2}I_{arm}\omega_{arm}^2 + \frac{1}{2}m_{coin}v_{coin}^2 + \frac{1}{2}I_{gear_1}\omega_{gear_1}^2 + \frac{1}{2}I_{gear_2}\omega_{gear_2}^2 + \frac{1}{2}*I_{lid}\omega_{arm}^2;
\end{align} |
After solving the Euler Lagrange equation and substituting the appropriate kinematic relationships between accelerations of the components, the acceleration of the arm is found to be:
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\begin{align}
&\alpha_{arm} = -(\Delta_{\theta_{arm}}k - \Tau_{external} + b_{gears}\omega_{arm} + b_{lid}\omega_{lid} + \frac{1}{2}g l_{arm} m_{arm}cos(180 - \theta_{arm}) - \frac{1}{2} g l_{lid} m_{lid} cos(180 - \theta_{lid}/(ILid*Warm2Lid^2 + IGear2*Warm2Gear2^2 + mCoin*rToCoin^2 + IArm + IGear1);
\end{align} |