Timing Diagram (Part 3 - Cycloid Cam Profile Analysis)

In the previous post [Timing Diagram Part 2: Max Acceleration], we calculated the maximum forces acting on a die driven by a cycloid cam profile. We discovered a critical rule of physics: inertial forces are inversely proportional to the square of the time allowed for movement.

The Engineering Strategy:
If we can extend the indexing angle (time) by allowing Overlap Motion, we can drastically reduce wear. This is the heart of Predictive Maintenance—designing machines that inherently last longer.

1. The Cycloid Cam Profile

The Cycloidal motion curve is the industry standard for high-speed automation because it has zero acceleration at the start and end of the move. The displacement equation is:

  h = hm × [ (t / tm) - 1/(2π) × sin(2π × t / tm) ]

To solve for the Displacement Ratio (percentage of travel), we rearrange it:

  h / hm = (t / tm) - 0.159 × sin(6.28 × t / tm)

Figure 1: The "Soft Start." Notice that in the first 10% of time, the mechanism moves less than 1% of the distance.

2. Calculating Safe Overlap

Because the cycloid curve starts so slowly, we can safely start the die movement before the mill has completely stopped, provided we maintain clearance.

Step A: Analyze Mill Clearance

Indexing Angle: 150°
Stroke: 100 mm

At 10% of the indexing time (15°), the mill has only moved 0.65 mm. We will assume this small movement creates no collision risk.

Step B: Determine Available Time

The effective "busy" time of the mill is now reduced:
150° - (2 × 15°) = 120°

This leaves us with more remaining cycle time for the die:
360° - 120° = 240°

Subtracting the required 100° dwell at the bottom:
240° - 100° = 140° (Total Move Time)

Die Indexing Angle = 140° / 2 = 70 degrees

(This is already better than the 55° we calculated in Part 1, but we can maximize it further.)

3. Maximizing the Overlap

The die doesn't have to wait for the mill to stop. If we maintain a safety margin of 1 mm, the die can technically start moving down earlier.

When the die is at 20mm (clearance) - 1mm (margin) = 19 mm travel distance:

Displacement Ratio (h / h_m) = 19 / 50 = 38%

Standard Cycloid Motion Table
Time Ratio (t/t_m)	Angle (degrees)	Displacement (h/h_m)	Note
0.10	36°	0.65%	Soft Start
0.20	72°	5.0%
0.30	108°	14.5%
0.44	158°	38.0%	Overlap Point
0.50	180°	50.0%	Midpoint
0.60	216°	65.5%
1.00	360°	100%	End

Looking at the table above, 38% displacement corresponds to approximately 44% of the Indexing Time.

This means the die can already be 44% of the way through its cycle while the mill is still finishing its last 15 degrees.
The remaining "Safe Zone" angle is:
100% - 44% = 56%

Since we know this Safe Zone corresponds to our previously calculated 70 degrees, we can reverse calculate the full allowable angle:

New Indexing Angle = 70° / 0.56 = 125 degrees

Note: In practice, we would usually round down to 120 degrees to provide a 5-degree safety margin for assembly tolerances.

Timing diagram showing optimized overlap motion

4. The Result: 500% Acceleration Reduction

Let's recalculate the acceleration with our new optimized angle.

Speed (N): 2000 pcs/h
Cam Angle (B_m): 125° (Up from 55°!)
Lift (h_m): 0.05 m

Step 1: Calculate Time (t_m)
t_m = (10 × 125) / 2000 = 0.625 seconds

Step 2: Calculate Max Acceleration (a_max)
a_max = (2 × π × 0.05) / 0.625²

a_max = 0.804 m/s²

Comparison:
Without Overlap = 4.154 m/s²
With Overlap = 0.804 m/s²
Reduction Factor = 5.17 times!

By smart engineering, we reduced the force on the machine by over 500% without slowing down the production rate. This is the essence of high-efficiency industrial automation.

In the next post [Timing Diagram Part 4: Motion Simulation in Excel], we will visualize this difference using a custom simulation tool.

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