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...
Figure 1: Mathematical coefficients determine the physical shape. Poor math leads to physical defects like the "dip" shown on the right.
In [Polynomial Cam Function (Derivation of Fifth-degree function) - Part 2], we derived the equations for the Fifth-Degree (3-4-5) Polynomial.
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Now, we apply this math to the real world of Mechanical Cam Design. The shape of the physical cam is determined by plotting these functions. Unlike a standard Cycloid curve, the polynomial allows us to manipulate the Start Velocity (v0) and End Velocity (v1) of the follower.
However, this flexibility requires careful design. If the coefficients are not balanced, the physical cam profile can develop "dips" or negative slopes, causing the mechanical linkage to behave unpredictably.
Case 1: Standard Dwell-to-Dwell (Zero Velocity)
Figure 2: The standard profile (v0=0, v1=0). Safe, smooth, and identical to a cycloid curve.
Here, we set v0 = 0 and v1 = 0.
Visually, this looks identical to a Cycloid curve. It is the safest profile for standard cam applications where the follower starts from a dwell (rest) and ends at a dwell.
Case 2: The "Dip" Danger (Negative Displacement)
Figure 3: WARNING: High start/end velocities cause a "Dip" (Negative Displacement), leading to vibration and crash.
This is a critical design trap.
Here, we set both velocities greater than zero (v0 = 1, v1 = 3). Even though the final lift (hm) is positive, the path takes a detour!
Notice how the curve dips below zero at the start?
In a physical cam, this means the profile would be cut inwards. The follower would actually move backward first to gain momentum. If your mechanism isn't designed for this (e.g., a one-way clutch or limited clearance), this will cause a mechanical crash or severe vibration.
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Case 3: Smooth Forward Motion (Optimized)
Figure 4: Optimized coefficients (v1 reduced) eliminate the dip, creating smooth forward motion for flying transfers.
Here, we adjust the parameters (v0 = 1, v1 = 0.5).
By lowering the target end velocity, we eliminate the "dip." The cam follower now moves continuously forward. This is ideal for machines where the cam must synchronize with a linear conveyor belt for a "flying transfer" operation.
Case 4: Intentional Reciprocating Motion
Figure 5: Reciprocating motion. The follower overshoots and returns, useful for specific oscillating mechanisms.
Sometimes, you want the mechanism to oscillate.
Here, we set (v0 = 1, v1 = -1). The follower pushes forward past the target height, stops, and returns. Because this is defined by a 5th-degree polynomial, the acceleration remains finite throughout the reversal, meaning no infinite jerk and less wear on the cam surface.
More advanced applications of the Fifth-Degree Polynomial will be discussed in the next post.
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