Mobility Calculation
...
The most complex part of the position analysis is determining the location of the slider with respect to the crank angle.
Figure 3.2: Crank slider mechanism (and vector representation)
...
| Latex formatting |
|---|
$$ae^{j \theta_a} + be^{j \theta_b} + ce^{j \theta_c} = 0$$
$$a(\cos\theta_a + j\sin\theta_a) + b(\cos\theta_b + j\sin\theta_b) + c(\cos\theta_c + j\sin\theta_c) = 0$$
Since $\theta_c = 90^270^\circ{}$, $\cos\theta_c = 0$ and $\sin\theta_c = -1$
$$a(\cos\theta_a + j\sin\theta_a) + b(\cos\theta_b + j\sin\theta_b) +- cj = 0$$
Separating the terms into real and imaginary:
$$\text{Re:}~~a\cos\theta_a+b\cos\theta_b = 0$$
$$\text{Im:}~~a\sin\theta_a+b\sin\theta_b+-c = 0$$
$a$ and $b$ are known lengths, while $\theta_b$ and $c$ are unknown. $\theta_a$ is our input (known). Thus, we have two equations and two unknowns and can solve:
$$\cos\theta_b = -\cfrac{a\cos\theta_a}{b}$$
$$\theta_b = \arccos(-\frac{a\cos\theta_a}{b})$$
We can now substitute for $\theta_b$ that into the second equation to solve for $c$:
$$c = -a\sin\theta_a -+ b\sin(\arccos(-\frac{a\cos\theta_a}{b}))$$ |
...
Figure 3.3 shows a to-scale representation of the crank slider mechanism in combination with the turntable. The end of the connecting link (position of the "slider") is offset from the actual topping container by a fixed amount. Adding this quantity onto c will provide the distance of the edge of the topping container from the crank axis. This distance should range between a maximum and a minimum value, determined by the lengths of the crank and connecting links. The minimum distance should correspond to the near (w.r.t. the crank axis) edge of the topping container reaching the center of the turntable. This position occurs when the crank is at 270° (when measured from the positive x-axis). The maximum distance of the container from the crank axis should correspond to the far (w.r.t. the crank axis) edge of the topping container reaching the edge of the turntable. This position occurs when the crank is at 90° (when measured from the positive x-axis). The lengths of the linkages were chosen such that the desired behavior was produced, as shown in the following animation:
| Multimedia | ||
|---|---|---|
|
...