Lay out the profile of a cam so that the follower: - is moved outwards through 30 mm during 180° of cam rotation with cycloidal motion - dwells for 20° of the cam rotation - return... Lay out the profile of a cam so that the follower: - is moved outwards through 30 mm during 180° of cam rotation with cycloidal motion - dwells for 20° of the cam rotation - returns with uniform velocity during the remaining 160° of the cam rotation The base circle diameter of the cam is 28 mm and the roller diameter 8 mm. The axis of the follower is offset by 6 mm to the left. What will be the maximum velocity and acceleration of the follower during the outstroke if the cam rotates at 1500 rpm counter-clockwise? Read more

Steps to Solve

1. Convert cam rotation speed to radians per second

The cam rotates at 1500 rpm. Convert this to radians per second: $$ \omega = 1500 \frac{\text{rev}}{\text{min}} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} = 50\pi \text{ rad/s} \approx 157.08 \text{ rad/s} $$

2. Calculate maximum velocity during outstroke (cycloidal motion)

For cycloidal motion, the maximum velocity $v_{max}$ is given by: $$ v_{max} = \frac{2h\omega}{\theta} $$ where $h$ is the total displacement (30 mm), $\omega$ is the angular velocity of the cam ($50\pi$ rad/s), and $\theta$ is the angle of cam rotation during the outstroke (180 degrees or $\pi$ radians). Plugging in the values: $$ v_{max} = \frac{2 \times 30 \times 10^{-3} \text{ m} \times 50\pi \text{ rad/s}}{\pi \text{ rad}} = 3 \text{ m/s} $$

3. Calculate maximum acceleration during outstroke (cycloidal motion)

For cycloidal motion, the maximum acceleration $a_{max}$ is given by: $$ a_{max} = \frac{2h\omega^2}{\theta} $$ Plugging in the values: $$ a_{max} = \frac{2 \times 30 \times 10^{-3} \text{ m} \times (50\pi \text{ rad/s})^2}{\pi \text{ rad}} = 150\pi^2 \text{ m/s}^2 \approx 1480.44 \text{ m/s}^2 $$

4. Consider the effect of the follower offset

The follower offset does influence the cam profile design. However, it does not affect the maximum velocity and maximum acceleration of the follower during the outstroke if the type of motion is already defined as cycloidal. The offset only changes the pressure angle. Therefore we do not need to consider the offset in the calculation of the max velocity and acceleration.

Is the answer in step 3 correct?

Check the book "Theory of Machines" by R.S. Khurmi and J.K. Gupta equation 7.28, it shows that for cycloidal motion, the max acceleration is actually:

$$ a_{max} = \frac{2h\omega^2}{\theta} $$

Therefore our calculation is incorrect, we need to divide by $ \pi $, so it will be:

$$ a_{max} = \frac{2S}{\theta} \cdot \omega^2 = \frac{2 \cdot 0.03 \cdot 157.08^2}{\pi} = 471.24 m/s^2 $$