Mobility Calculation
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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)
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$$ae^{j \theta_a} + be^{j \theta_b} + ce^{j \theta_c} = 0$$
$$a(\cos\theta_a + j\sin\theta_a) + b(\cos\theta_b + j\sin\theta_b) + c(\cos\theta_c + j\sin\theta_c) = 0$$
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 $c$:
$$c = -a\sin\theta_a -+ b\sin(\arccos(-\frac{a\cos\theta_a}{b}))$$ |
This relationship is plotted below:
Figure 3.3: Graph of distance of container from center of turntable
Figure 3.34: Crank slider and turntable (to scale)
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:
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Recall the position vector loop equation:
$$ae^{j \theta_a} + be^{j \theta_b} + ce^{j \theta_c} = 0$$
Differentiate this:
$$\frac{d}{dt}(ae^{j \theta_a} + be^{j \theta_b} + ce^{j \theta_c}) = aj\omega_ae^{j \theta_a} + bj\omega_be^{j \theta_b} + \dot ce^{j \theta_c} = 0$$
$$aj\omega_a(\cos\theta_a + j\sin\theta_a) + bj\omega_b(\cos\theta_b + j\sin\theta_b) + \dot cj = 0$$
$$a\omega_a(j\cos\theta_a - \sin\theta_a) + b\omega_b(j\cos\theta_b - \sin\theta_b) + \dot cj = 0$$
Separate into real and imaginary:
$$\text{Re:}~~-a\omega_a\sin\theta_a-b\omega_b\sin\theta_b = 0$$
$$\text{Im:}~~a\omega_a\cos\theta_a+b\omega_b\cos\theta_b+\dot c = 0$$
$a$ and $b$ are known lengths. $\theta_a$ and $\omega_a$ are inputs. $\theta_b$ can be determined from position analysis with the formula $\theta_b = \arccos(-\frac{a\cos\theta_a}{b})$. $\omega_b$ and $\dot c$ are unknown.
$$\omega_b=\cfrac{-a\omega_a\sin\theta_a}{b\sin\theta_b}$$
$$\omega_b=\cfrac{-a\omega_a\sin\theta_a}{b\sin(\arccos(-\frac{a\cos\theta_a}{b}))}$$
$$a\omega_a\cos\theta_a+\cfrac{-a\omega_a\sin\theta_a}{\sin(\arccos(-\frac{a\cos\theta_a}{b}))} \cos(\arccos(-\frac{a\cos\theta_a}{b}))+\dot c = 0$$
$$a\omega_a\cos\theta_a+\cfrac{a\omega_a\sin\theta_a}{\sin(\arccos(-\frac{a\cos\theta_a}{b}))} \cfrac{a\cos\theta_a}{b}+\dot c = 0$$
$$\dot c=-a\omega_a\cos\theta_a-\cfrac{a^2\omega_a\sin\theta_a\cos\theta_a}{b\sin(\arccos(-\frac{a\cos\theta_a}{b}))}$$
We now have expressions for $\omega_b$ (the angular velocity of the connecting link) and $\dot c$ (the linear velocity of the container) in terms of known/input quantities. These relationships are graphed below as a function of the rotation of the crank. |
Figure 3.5: Graph of angular velocity of connecting link
Figure 3.6: Graph of velocity of container