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 post [Polynomial Cam Function (Part 3)], we explored the characteristics of the 5th-degree profile. Now, we put it to work.
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We will revisit the same die-press example from [Timing Diagram (Part 4)]. Originally, without overlap, the acceleration was a massive 4.154 m/s2. Using a Cycloid profile with overlap, we reduced it to 0.804 m/s2.
This time, we will use a Fifth-Degree (3-4-5) Polynomial combined with a Linear Cam Function. While the peak acceleration reduction might be similar to the Cycloid, the real advantage here is control—specifically, the ability to blend different motion segments seamlessly without stopping.
The Motion Strategy
Our constraints remain the same: Total displacement is 50 mm, and the die must dwell inside the hole for 30 mm.
Unit Conversion Note:
The equations we derived calculate velocity in mm/rad. To convert to real-world time domain:
• Velocity (mm/s) = Velocity (mm/rad) × Ï‰ (rad/s)
• Acceleration (mm/s2) = Acceleration (mm/rad2) × Ï‰2
The equations we derived calculate velocity in mm/rad. To convert to real-world time domain:
• Velocity (mm/s) = Velocity (mm/rad) × Ï‰ (rad/s)
• Acceleration (mm/s2) = Acceleration (mm/rad2) × Ï‰2
The Displacement Profile
Figure 1: The displacement profile divided into Linear (A-B) and Polynomial (B-C) sectors to optimize flow.
We have divided the motion into specific sectors to optimize the flow. Notice how we use a Linear segment (A-B) to handle the approach, connected to Polynomial segments for the heavy lifting.
| Sector | Function | Start Angle (°) | End Angle (°) | Start Height (mm) | End Height (mm) |
|---|---|---|---|---|---|
| A-B | Linear | 62 | 77 | 1.43 | 0.43 |
| B-C | Fifth-Degree | 77 | 206 | 0.43 | 50.00 |
| C-D | Dwell | 206 | 306 | 50.00 | 50.00 |
| D-A | Fifth-Degree | 306 | 62 | 50.00 | 1.43 |
Figure 2: The physical cam profile generated from the calculated displacement values.
Velocity Matching: The Secret to Smoothness
Figure 3: Velocity graph showing perfect continuity. The linear slope matches the polynomial start velocity exactly.
This is where the polynomial shines. To connect the Linear segment (A-B) to the Polynomial segment (B-C) without a shock, the velocities must match perfectly.
Calculating the Slope:
Slope (Velocity) = (0.43 - 1.43) / (77 - 62) = -0.0667 mm/deg
We input this value (-0.0667) as the Start Velocity (v0) for our polynomial calculation in sector B-C. This ensures a perfectly continuous transition with zero jerk spike.
Acceleration Analysis
Figure 4: Acceleration graph. We confirm there are no vertical spikes (Infinite Jerk) at the transition points.
Using motion simulation in Excel, we can visualize the result. We verify that there are no infinite spikes (Jerk) at the connecting points.
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Figure 5: Using Excel to calculate and verify the kinematic values before cutting metal.
Modern Application: Electronic Camming (E-Cam)
Today, physical cams are being replaced by Electronic Camming (E-Cam) profiles inside servo drives. However, the math remains identical.
Figure 6: Simulation of the die-press mechanism driven by the optimized polynomial profile.
By programming these 5th-degree polynomials into your servo controller, you can achieve smoother motion, higher speeds, and longer machine life—all without changing a single mechanical part.
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