For engineers who already know the math—but still lose projects. For the last few years, I’ve been sharing technical guides here on Mechanical Design Handbook —how to size a motor, how to calculate fits, and (as you recently read) how to choose between timing belts and ball screws. But after 25 years in industrial automation, I realized something uncomfortable: Projects rarely fail because the math was wrong. They fail because: The client changed the scope three times in one week. A critical vendor lied about a shipping date (and no one verified it). The installation technician couldn’t fit a wrench into the gap we designed. University taught us the physics. It didn’t teach us the reality. That gap is why I wrote my new book, The Sheet Mechanic . This is not a textbook. It is a field manual for the messy, political, and chaotic space between the CAD model and the factory floor. It captures the systems I’ve used to survive industrial projec...
In the previous post [Timing Diagram (Part 1 - No Overlap Movement)], we determined that without overlap, our die must travel 50mm within a tight cam angle of just 55 degrees.
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Now, we must ask: What is the physical cost of this rapid movement?
To answer this, we calculate the Maximum Acceleration. In Machine Dynamics, acceleration is directly proportional to Force (F = m × a). High acceleration means high inertial forces, which lead to severe wear, vibration, and the need for expensive oversized servo motors.
Step 1: The Time Calculation
First, we need to convert our "Cam Angle" into actual "Time" in seconds.
Let:
N = Machine Speed (pieces per hour)
Bm = Indexing Angle (degrees)
Cycle time (sec) = 3600 / N
Indexing time tm = (Bm / 360) × Cycle time
Indexing time tm = (Bm / 360) × (3600 / N)
tm (sec) = (10 × Bm) / N
Step 2: Cycloid Cam Profile Equations
The die moves using a Cycloid Cam Profile, which is the industry standard for high-speed indexing because it has finite jerk.
Displacement Equation (h):
h = hm × [ t/tm - 1/(2Ï€) × sin(2Ï€ × t/tm) ]
Where:
hm = Maximum displacement (meters)
tm = Indexing time (seconds)
Ï€ = 3.14159...
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To find acceleration, we differentiate displacement twice.
Velocity (v = dh/dt):
v = (hm/tm) × [ 1 - cos(2Ï€ × t/tm) ]
Acceleration (a = dv/dt):
a = (2Ï€ × hm / tm2) × sin(2Ï€ × t/tm)
Step 3: The Maximum Acceleration Formula
The maximum acceleration (amplitude) occurs when the sine function equals 1. Therefore, the peak acceleration is:
amax = (2 × Ï€ × hm) / tm2
The Engineering Insight:
Notice that acceleration is inversely proportional to the square of the time (tm2). Since time is proportional to the cam angle (Bm), we arrive at a critical rule for Cam Design:
If you double the indexing angle, the maximum acceleration reduces by a factor of FOUR!
Notice that acceleration is inversely proportional to the square of the time (tm2). Since time is proportional to the cam angle (Bm), we arrive at a critical rule for Cam Design:
If you double the indexing angle, the maximum acceleration reduces by a factor of FOUR!
Step 4: Real-World Calculation (No Overlap)
Let's calculate the forces for our current "No Overlap" design from Part 1.
Parameters:
Machine Speed (N) = 2000 pcs/h
Cam Angle (Bm) = 55 deg (Very tight!)
Stroke (hm) = 50 mm = 0.05 m
1. Calculate Time (tm):
tm = (10 × 55) / 2000 = 0.275 seconds
2. Calculate Max Acceleration (amax):
amax = (2 × 3.1416 × 0.05) / 0.2752
amax = 4.154 m/s2
This is nearly 0.5 G of constant acceleration force, repeated 2,000 times an hour. This will likely cause premature wear on the cam followers and linkages.
In the Next Post (Part 3), we will apply the "Overlap" technique to increase the indexing angle and see how dramatically this acceleration number drops.
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