Pitch Curve 2) (x(t) = cos t, y(t) = 2 sin t) is associated with the maximum translation of the follower during one full rotation of the cam. Therefore ,2) (x(t) = cos t, y(t) = 2 sin t) is correct .
To determine the pitch curve that yields the maximum translation of the follower during one full rotation of the cam, we can analyze the displacement of the follower for each given pitch curve.
Pitch Curve 1: x(t) = cos t, y(t) = sin t
In this case, the follower's displacement is represented by the parametric equation r(t) = (cos t, sin t).
The magnitude of the displacement, which represents the distance between the follower and the cam center, can be calculated using the distance formula:
|r(t)| = √(cos^2 t + sin^2 t) = √1 = 1
This indicates that the follower's displacement remains constant throughout the cam's rotation, and the maximum translation is equal to the radius of the cam.
Pitch Curve 2: x(t) = cos t, y(t) = 2 sin t
Similarly, the follower's displacement for this pitch curve is r(t) = (cos t, 2 sin t). The magnitude of the displacement is:
|r(t)| = √(cos^2 t + 4 sin^2 t) = √(1 + 4) = √5
This pitch curve results in a larger maximum translation compared to the first pitch curve, as the amplitude of the follower's vertical motion is doubled.
Pitch Curve 3: x(t) = ½ + cos t, y(t) = 2 sin t
For this pitch curve, the follower's displacement is r(t) = (½ + cos t, 2 sin t). The magnitude of the displacement is:
|r(t)| = √((½ + cos t)^2 + 4 sin^2 t) = √(0.25 + 2cos t + cos^2 t + 4 sin^2 t)
Using trigonometric identities, we can simplify the expression:
|r(t)| = √(0.25 + 2cos t + 1 - sin^2 t + 4 sin^2 t) = √(1.25 + 3cos t)
To find the maximum translation, we need to maximize the expression √(1.25 + 3cos t).
This occurs when cos t = 1, which corresponds to t = 0. Therefore, the maximum translation is:
|r(max)| = √(1.25 + 3) = √4.25 = 2.06
Pitch Curve 4: x(t) = ½ + cost t, y(t) = sin t
Similar to the previous pitch curve, the follower's displacement is r(t) = (½ + cost t, sin t).
The magnitude of the displacement is:
|r(t)| = √((½ + cost t)^2 + sin^2 t) = √(0.25 + cos t + cos^2 t + sin^2 t)
Using trigonometric identities, we can simplify the expression:
|r(t)| = √(0.25 + 2cos t + 1 - sin^2 t)
The maximum translation occurs when cos t = 1, which corresponds to t = 0. Therefore, the maximum translation is:
|r(max)| = √(1.25 + 2) = √3.25 = 1.81
Comparing Maximum Translations:
Analyzing the maximum translations for each pitch curve:
Pitch Curve 1: Maximum translation = 1
Pitch Curve 2: Maximum translation = √5 ≈ 2.24
Pitch Curve 3: Maximum translation = 2.06
Pitch Curve 4: Maximum translation = 1.81
Therefore, Pitch Curve 2 (x(t) = cos t, y(t) = 2 sin t) is associated with the maximum translation of the follower during one full rotation of the cam.
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