Question 15.9: The Geneva mechanism shown is used in many counting instrume......
The Geneva mechanism shown is used in many counting instruments and in other applications where an intermittent rotary motion is required. Disk D rotates with a constant counterclockwise angular velocity V_D of 10 rad/s. A pin P is attached to disk D and slides along one of several slots cut in disk S. It is desirable that the angular velocity of disk S be zero as the pin enters and leaves each slot; in the case of four slots, this will occur if the distance between the centers of the disks is l=1 \overline{2} R.
At the instant when f = 150°, determine (a) the angular velocity of disk S, (b) the velocity of pin P relative to disk S.
We solve triangle OPB, which corresponds to the position f = 150°. Using the law of cosines, we write
r^{2}=R^{2}+l^{2}-2 R l~ \cos 30^{\circ}=0.551 R^{2} \quad\quad\quad r=0.742 R=37.1 \mathrm{~mm}From the law of sines,
\frac{\sin \mathrm{b}}{R}=\frac{\sin 30^{\circ}}{r} \quad\quad \sin \mathrm{b}=\frac{\sin 30^{\circ}}{0.742} \quad\quad \mathrm{~b}=42.4^{\circ}Since pin P is attached to disk D, and since disk D rotates about point B, the magnitude of the absolute velocity of P is
\begin{aligned}v_{P}=R \mathrm{v}_{D}=(50 \mathrm{~mm})(10 ~\mathrm{rad} / \mathrm{s}) & =500 \mathrm{~mm} / \mathrm{s} \\\mathrm{v}_{P} & =500 \mathrm{~mm} / \mathrm{s} ~\mathrm{d} ~60^{\circ}\end{aligned}We consider now the motion of pin P along the slot in disk S. Denoting by P^{\prime} the point of disk S which coincides with P at the instant considered and selecting a rotating frame S attached to disk S, we write
\text {v}_{P}=\text{v}_{P^{\prime}}+\text{v}_{P / S}Noting that \text{v}_{P^{\prime}} is perpendicular to the radius OP and that \text{v}_{P / S} is directed along the slot, we draw the velocity triangle corresponding to the equation above. From the triangle, we compute
g = 90° – 42.4° – 30° = 17.6°
\begin{aligned}v_{P^{\prime}} & =v_{P} \sin \mathrm{g}=(500 \mathrm{~mm} / \mathrm{s}) \sin 17.6^{\circ} \\\mathrm{v}_{P^{\prime}} & =151.2 \mathrm{~mm} / \mathrm{s} ~\mathrm{f}~ 42.4^{\circ} \\v_{P / S} & =v_{P} \cos \mathrm{g}=(500 \mathrm{~mm} / \mathrm{s}) \cos 17.6^{\circ} \\& \quad \mathrm{v}_{P / S}=\mathrm{v}_{P / S}=477 \mathrm{~mm} / \mathrm{s} \text { d } 42.4^{\circ}\end{aligned}Since \text{v}_{P^{\prime}} is perpendicular to the radius OP, we write
\begin{aligned}v_{P^{\prime}}=r \mathrm{v}_{S} \quad 151.2 \mathrm{~mm} / \mathrm{s}= & (37.1 \mathrm{~mm}) \mathrm{v}_{S} \\& \mathrm V_{S}=\mathrm V_{S}=4.08 ~\mathrm{rad} / \mathrm{s}~ \mathrm i\end{aligned}
