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The dynamics of the system were simulated in MATLAB using kinematic relationships, Lagrangian Dynamics, and Euler's method for numerical integration. Results can be seen below:

Slowed-down videos of the forward and backward directions are shown below:

Masses of important components (left gear, right gear, arm, lid, coin) were measured using a scale. The spring constant was calculated by measuring the force of the spring at a known displacement using the scale Each gear's inertia were modeled using the thin disc formula. The arm and lid were treated as slender rods, and the coin as a point mass. Damping components for the gears and the lid were arbitrarily assigned. After some research, the suction cup was modeled simply as a time-delay, as the force imparted by it would not impact the dynamics of the system.

The Potential and Kinetic Energy of the system are shown by the following equations:

\begin{align}
&PE = m_{arm} g l_{arm} sin(180-\theta_{arm}) + m_{lid} g l_{lid} sin(180-\theta_{lid}) + \frac{1}{2} k \Delta_{\theta_{arm}}^2
\\
&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:

\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.