Question

In a cam design, the rise motion is given by a simple harmonic motion (SHM) \(s = \frac{h}{2}\left( {1 - \cos \frac{{\pi \theta }}{\beta }} \right)\) where h is total rise, θ is camshaft angle, β is the total angle of the rise interval. The jerk is given by
1. \(\frac{h}{2}\left( {1 - \cos \frac{{\pi \theta }}{\beta }} \right)\)
2. \(\frac{\pi \omega}{\beta }\frac{h}{2}\sin \left( {\frac{{\pi \theta }}{\beta }} \right)\)
3. \(\frac{{{\pi ^2 \omega^2}}}{{{\beta ^2}}}\frac{h}{2}\cos \left( {\frac{{\pi \theta }}{\beta }} \right)\)
4. \(- \frac{{{\pi ^3\omega^3}}}{{{\beta ^3}}}\frac{h}{2}\sin \left( {\frac{{\pi \theta }}{\beta }} \right)\)

Answer 1

Correct Answer - Option 4 : \(- \frac{{{\pi ^3\omega^3}}}{{{\beta ^3}}}\frac{h}{2}\sin \left( {\frac{{\pi \theta }}{\beta }} \right)\)

Concept:

The rise motion is given by a simple harmonic motion (SHM)

\(s = \frac{h}{2}\left[ {1 - \cos \left( {\frac{{\pi \theta }}{\beta }} \right)} \right]\)

Velocity \(\dot s = \frac{{ds}}{{dt }} = \frac{{ds}}{{d\theta }}\frac{{d\theta }}{{dt }} = \frac{h}{2}\left[ { \sin \left( {\frac{{\pi \theta }}{\beta }} \right)} \right].\frac{\pi }{\beta }.\frac{d\theta}{dt}= \frac{h}{2}\left[ { \sin \left( {\frac{{\pi \theta }}{\beta }} \right)} \right].\frac{\pi }{\beta }\omega\)

And acceleration \(\ddot s = \frac{{d\dot s}}{{d\theta }} = \frac{h}{2}\left[ {\cos \left( {\frac{{\pi \theta }}{\beta }} \right)} \right]{\left( {\frac{\pi \omega }{\beta }} \right)^2}\)

And jerk \(\dddot{s}= \frac{{d\ddot s}}{{d\theta }} =- \frac{h}{2}\left[ {\sin \left( {\frac{{\pi \theta }}{\beta }} \right)} \right].\frac{{{\pi ^3 \omega^3}}}{{{\beta ^3}}}\)