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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}}^2
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&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} = -\frac{\Delta_{\theta_{arm}}k - \tau_{external} + b_{gears}\omega_{arm} + b_{lid}\omega_{lid} + \frac{1}{2} m_{arm} g l_{arm}cos(180 - \theta_{arm}) - \frac{1}{2} m_{lid} g l_{lid} cos(180 - \theta_{lid})}{I_{lid}\frac{\alpha_4}{\alpha_2} + I_{gear_2} + m_{coin}r_{coin}^2 + I_{arm} + I_{gear_1}} \end{align} |
The above equation is valid during contact between the cam and follower. When there is no contact, the equation is split into two, with the relevant terms for the lid forming their own equation of motion.
Overall, the dynamic simulation was a success. The acceleration of the lid and coin due to the spring were synchronized such that the coin was able to enter the bank while the lid was open. This type of dynamic simulation can be a powerful tool when designing physical systems.