Kinematic Analysis Results

Calling the function fan(1450) gives the following results:

i) Position analysis

Figure 1a. Fan total range as a function of input angle

It can be seen from the plot above that for one full revolution of input theta, the fan oscillates about an angle of between approximately 40 degrees to 140 degrees This is consistent with the actual movement of the fan, and makes intuitive sense for symmetry as seen in the figure on the right.

ii) Velocity Analysis

Figure 2a. Fan output angular velocity as a function of input angle

It can be seen from the plot above that for one full revolution of input theta, the fan has constant velocity during the range of full oscillation. This is consistent with physical movement of the fan, since the change in speed with respect to time is constant as the head moves from side to side. Also, the constant values coincide with the position of the trough and peak of the position analysis figure above.

iii)Acceleration Analysis

Figure 3a. Fan output angular acceleration as a function of input angle

This angular acceleration figure corresponds to the acceleration of the fan's head. It can be seen that the points at which theta_4 (fans head position) hits a minimum, and where omega_4 (fan's head velocity hits a maximum, the acceleration is at it's highest point, just before the fan's head turns). Thereafter, there is constant acceleration as the fan traverses one point to the other, which corresponds to the constant velocity also seen in figure 2a. then the acceleration drops as it nears the end of a full rotation cycle, in anticipation of a change in direction to restart all over again.

The acceleration value is really high, as this is in RPM. When this is converted to rev/sec^2, values scale down to 13.5 rev/sec^2 at it's maximum value.

iv) Mechanical Advantage

Figure 4. Fan Mechanical Advantage as a function of input angle

It is an established fact that mechanical advantage is directly proportional to speed and inversely proportional to torque. Thus, the plot above indicates that at the start of input motion of theta2, just when the fan head switches position (around theta_2 ~50 degrees), there is a spike in MA. This corresponds to low torque and is the onset of high, constant omega_out (omega_4). Thereafter, the fan continues in it's path through the revolution. Again, just when the direction is about to switch (around theta_2 ~320 degrees), there is another spike in MA, where the toque decreases again, and allows for change in fan head position. This MA is determined as a the relationship between omega_out (omega4) and omega_in (omega_2).

The MATLAB function was called with the low end of the rpm values from the fan speeds above, and similar trend graphs were obtained, so they weren't included in this report.

Figure 1b. Approximate angle subtended by fan head as a function of one full revolution of input theta

This again, is consistent with what is seen when the fan is turned on.

Figure 2b. Fan output angular velocity as a function of input velocity

I was interested in seeing how the input velocity affects the output velocity. As can be seen, due to the constant nature of the velocities involved, the constant input of omega_2 produces a range of omega_4 depending on the theta_2 input on the left hand side. This gives a range of output velocity corresponding to 33.75 rpm and a minimum of -235.24 rpm at 1450 rpm fan speed. Omega experiences a speed reduction due to the gear box between the fan motor and the four bar mechanism; this speed reduction ensures that the same motor that spins the fan's blade is also the motor that allows for the fan's oscillation. This makes for an efficient design as it eliminates the existence of a second motor).

Figure 3a. Fan output angular acceleration as a function of input angle

I was interested in seeing how the intermediate link () behaved as the theta input changed. The resulting figure is indicative of the double rocker system, and intuitively makes sense, as it follows the resulting acceleration of the fan's head.