Results and Conclusion.

Here's the results of the kinematic analysis when the fan is on the high setting with RPM of 900.

Here's the result when the fan is on a low setting with a RPM of 400.

No matter what the RPM of the fan is, the range of oscillation for the fan head is 84 degrees (137-53); this is expected as this range is dependent only on the lengths of the 4-bar and is always constant. The gear ratio between the input crank and fan RPM was found to be 160:1 

These results show:

1. The angular position's critical point (maximum/minimum) is when the angular velocity is zero.  
2. The angular velocity's critical point (maximum/minimum) is at the angular position's inflection point, (head is midway in its oscillation) and the angular acceleration is zero.
3. The angular acceleration's critical point (maximum/minimum) is at the angular velocity's inflection point, (when the velocity is midway through its cycle).

These conclusions agree with basic calculus concepts. 

Calculating Mechanical Advantage can be obtained from the input RPM and the angular velocity.  

At a fan rotation of 400 RPM, the range of mechanical advantages for the fan head is as follows:

Interestingly, the range of mechanical advantage stays the same regardless of what the fan RPM is. This is because the mechanical advantage is a "system" property. One may notice that the magnitude of the fan angular velocity is almost double for the 900 RPM vs 400 RPM, but the input RPM being almost doubled must be taken into consideration as well. Thus, the effects cancel each other out. The very high values of MA can be attributed to when dividing by a low ( ideally zero) angular velocity.

The mechanical advantage at a particular RPM is not constant because the the angular velocity the fan head experiences is not constant. This is because of the different amounts of angular accelerations experienced at the upper and lower bounds of oscillation. To be more specific, the angular velocity changes as the position of the fan head. Here's an a visualization on Mechanical Advantages at its position for reference. 

Here we see the system has infinite MA at the end of its oscillation cycle (when its angular rotation is zero), and that MA is minimized when it is midway through its cycle (at its peak angular rotation).

Some recommendation for future work include a varying range of RPMs writing the MATLAB code. This would output a 3D graphs and would have to be color-visualized. Currently, the Matlab code accepts a singular input RPM but by varying it, the magnitude changes in angular velocity and acceleration may be observed.

Additional refinement include better resolution in the lengths of the 4 bar mechanism; as lengths were recorded to the nearest quarter inch.Better system representation may be achieved if better resolution measurements can be made, especially as the scale of the 4 bar is not relatively large, (.5-3 inches).

A third recommendation is to further define the ranges of fan RPM. The range of fan speeds, (from a low setting at 400 RPM and a high setting of 900 RPM), are estimates and efforts to precisely define all fan setting RPMs will produce even greater system representation. 

To conclude, this project was very enjoying and made the best of the situation regarding the COVID-19 outbreak of spring 2020. This was still enriching experience and a kinematic analysis of an oscillating fan was discussed. Lastly, some future recommendations were made to further enhance the kinematic analysis.