Question

Consider a rotating disk cam and a translating roller follower with zero offset. Which one of the following pitch curves, parameterized by t, lying in the interval 0 to 2π, is associated with the maximum translation of the follower during one full rotation of the cam rotating about the center at (x, y) = (0, 0)?
1. x(t) = cos t, y(t) = sin t
2. x(t) = cos t, y(t) = 2 sin t
3. x(t) = ½ + cos t, y(t) = 2 sin t
4. x(t) = ½ + cost t, y(t) = sin t

Answer 1

Correct Answer - Option 3 : x(t) = ½ + cos t, y(t) = 2 sin t

Explanation:

Translation of the follower is, d = \(√{{x^2}+{y^2}}\)

Case 1: 

x(t) = cos t, y(t) = sin t

\(d=√{{cos^2t}+{sin^2t}} = 1\)

d_max = 1

Case 2:

x(t) = cos t, y(t) = 2 sin t

\(d=√{{cos^2t}+{\left(2\;sin\;t\right)^2}} = √{{cos^2t}+{4\;sin^2t}} = √{1+3\;sin^2t} \)

'd' will be maximum when sin²t will be maximum.

sin t will be maximum at t = π /2 , 3π /2 in the interval of (0, 2π)

\(d_{max}=√{1+3} = 2\)

Case 3:

x(t) = ½ + cos t, y(t) = 2 sin t

\(d=√{{(\frac{1}{2}+cos\;t)^2}+{\left(2\;sin\;t\right)^2}} = √{\frac{1}{4}+cos\;t+{cos^2t}+{4\;sin^2t}} = √{\frac{5}{4}+cos\;t+3\;sin^2t} \)

'd' will be maximum when leading term 'sin²t' will be maximum.

sin²t will be maximum at t = π /2 , 3π /2 in the interval of (0, 2π)

\(d_{max}=√{\frac{5}{4}+0+3} = 2.06\)

Case 4:

x(t) = ½ + cos t, y(t) = sin t

\(d=√{{(\frac{1}{2}+cos\;t)^2}+{\left(sin\;t\right)^2}} = √{\frac{1}{4}+cos\;t+{cos^2t}+{sin^2t}} = √{\frac{5}{4}+cos\;t} \)

'd' will be maximum when cos t will be maximum.

cos t will be maximum at t = 0 , 2π in the interval of (0, 2π)

\(d_{max}=√{\frac{5}{4}+1} = 1.5\)

Option ‘C’ has maximum net amplitude.