Question

1. create a motion graph of the top of the follower for a mechanism with a square cam, a 2.5 - inch nominal diameter and a 3 - inch follower

your answer

Explanation:

Step1: Understand the Cam Mechanism

A square cam has four sides, so the follower's motion will have four distinct phases in one full rotation of the cam. The nominal diameter (2.5 inches) relates to the cam's size, and the follower (3 - inch) interacts with the cam to produce motion. The motion of the follower (displacement vs. cam rotation angle) will be a series of linear segments (since the cam is square, the contact surface is flat, leading to constant velocity or constant displacement rate in each phase).

Step2: Define the Cam Rotation and Follower Displacement

- Let the cam rotate from \(0^{\circ}\) to \(360^{\circ}\). A square cam has four sides, so each side corresponds to a \(90^{\circ}\) rotation (\(\frac{360^{\circ}}{4}=90^{\circ}\)).
- The lift (maximum displacement of the follower) for a square cam: The radius of the cam (from the center to the corner) can be related to the nominal diameter. The nominal diameter \(d = 2.5\) inches, so the radius \(r=\frac{d}{2}=1.25\) inches. But for a square cam, the follower's lift (maximum upward movement) when the cam rotates will be related to the cam's dimensions. However, if we consider the follower's motion as it follows the square cam, when the cam rotates, the follower will move up, stay at a constant height, move down, and stay at a lower constant height (or vice - versa, depending on the cam's orientation).
- For a square cam, the displacement - angle graph will have four linear segments:
- From \(0^{\circ}\) to \(90^{\circ}\): The follower moves with a constant velocity (linear increase in displacement) as the cam's side pushes the follower.
- From \(90^{\circ}\) to \(180^{\circ}\): The follower's displacement remains constant (since the cam's side is parallel to the follower's direction of motion, no vertical movement).
- From \(180^{\circ}\) to \(270^{\circ}\): The follower moves with a constant velocity (linear decrease in displacement) as the cam's next side pulls the follower down.
- From \(270^{\circ}\) to \(360^{\circ}\): The follower's displacement remains constant.

Step3: Plot the Graph

- **X - axis (Cam Rotation Angle)**: Range from \(0^{\circ}\) to \(360^{\circ}\), divided into four \(90^{\circ}\) intervals: \(0 - 90\), \(90 - 180\), \(180 - 270\), \(270 - 360\).
- **Y - axis (Follower Displacement)**: Let the initial displacement be \(y_0\) (e.g., the follower's rest position). When the cam rotates from \(0\) to \(90^{\circ}\), the displacement \(y\) increases linearly from \(y_0\) to \(y_0 + h\) (where \(h\) is the lift, which can be calculated from the cam's dimensions. For a square cam with side length \(s\), the lift \(h\) is related to the difference between the cam's corner radius and the side - to - center distance. But if we assume a simple case where the follower's maximum displacement is related to the cam's size, for a nominal diameter of 2.5 inches, the side length of the square cam (if the cam is a square with diameter equal to the diagonal) can be calculated. The diagonal of the square \(d = 2.5\) inches, so the side length \(s=\frac{d}{\sqrt{2}}=\frac{2.5}{\sqrt{2}}\approx1.768\) inches. The lift of the follower would then be approximately equal to the difference between the distance from the cam's center to the corner and the distance from the center to the side. The distance from the center to the side of the square (apothem) \(a=\frac{s}{2}=\frac{2.5}{2\sqrt{2}}\approx0.884\) inches, and the distance from the center to the corner (radius) \(r = \frac{d}{2}=1.25\) inches. So the lift \(h=r - a=1.25 - 0.884 = 0.366\) inches. But if we consider the follower length is 3 inches, maybe the lift is more related to the cam's rotation and the follower's interaction. Alternatively, if we take a more practical approach, the motion graph will have four segments:
- Segment 1 (\(0 - 90^{\circ}\)): Linear increase from, say, \(y = 0\) (initial position) to \(y = L\) (lift, e.g., \(L = 1\) inch for simplicity, or calculated from cam dimensions).
- Segment 2 (\(90 - 180^{\circ}\)): Horizontal line (constant \(y\)).
- Segment 3 (\(180 - 270^{\circ}\)): Linear decrease from \(y = L\) to \(y = 0\).
- Segment 4 (\(270 - 360^{\circ}\)): Horizontal line (constant \(y = 0\)).

The motion graph of the top of the follower for a square - cam mechanism will be a piece - wise linear graph with four segments, corresponding to the four sides of the square cam, showing the follower's displacement as a function of the cam's rotation angle. The x - axis is the cam's rotation angle (in degrees, from \(0^{\circ}\) to \(360^{\circ}\)) and the y - axis is the follower's displacement (in inches). The graph has four linear parts: increasing, constant, decreasing, and constant, each over a \(90^{\circ}\) interval of cam rotation.

Answer:

The motion graph is a piece - wise linear graph with four segments (each over a \(90^{\circ}\) cam rotation interval) showing the follower's displacement: increasing linearly (\(0 - 90^{\circ}\)), constant (\(90 - 180^{\circ}\)), decreasing linearly (\(180 - 270^{\circ}\)), and constant (\(270 - 360^{\circ}\)) as a function of the cam's rotation angle (x - axis: \(0^{\circ}-360^{\circ}\), y - axis: follower displacement in inches).