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: Deriving the "secret sauce" of smooth motion control requires calculus.
In [Polynomial Cam Function (Introduction) - Part 1], we discussed the fundamental law of cam design: continuity of acceleration.
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In this post, we are going to derive the exact equations for the Fifth-Degree Polynomial Cam Function. This is the mathematical foundation used in Motion Control Algorithms for high-end servo drives to ensure smooth, jerk-free movement.
The General Equations
We start with the general polynomial equation. To make the math handleable, we normalize the input angle as a ratio (x = θ / β), where x goes from 0 to 1.
s = C0 + C1x + C2x2 + C3x3 + C4x4 + C5x5
Where:
s = Displacement (mm)
x = Ratio of cam angle (θ / β)
β = Total angle in sector (rad)
To find Velocity (v) and Acceleration (a), we differentiate with respect to the angle.
(Note: The chain rule applies. v is in mm/rad. To convert to mm/s, multiply by angular velocity ω).
Velocity Equation (v = ds/dθ):
v = (1/β) · [ C1 + 2C2x + 3C3x2 + 4C4x3 + 5C5x4 ]
Acceleration Equation (a = dv/dθ):
a = (1/β2) · [ 2C2 + 6C3x + 12C4x2 + 20C5x3 ]
Applying Boundary Conditions
To solve for the 6 constants (C0 to C5), we need 6 known conditions. This is where Kinematics Analysis comes into play. We define what happens at the start (x=0) and end (x=1) of the motion.
Figure 2: Visualizing the 6 boundary conditions. Note the ability to define non-zero start/end velocities (v0, v1).
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Start Conditions (x=0)
We start at zero displacement with zero acceleration. Crucially, we allow a non-zero velocity (v0).
- s = 0
- v = v0
- a = 0
End Conditions (x=1)
We end at the total lift height (hm) with a target end velocity (v1) and zero acceleration.
- s = hm
- v = v1
- a = 0
Solving for Constants
Step 1: Apply Start Conditions (x=0)
By substituting x=0 into the general equations, terms with x vanish immediately. This quickly solves the first three constants:
- From Displacement: C0 = 0
- From Acceleration: C2 = 0
- From Velocity: v0 = C1 / β → C1 = βv0
Step 2: Apply End Conditions (x=1)
Now we have a system of 3 linear equations with 3 unknowns (C3, C4, C5). Solving this simultaneous system (via substitution or matrix methods handled by engineering software) yields the final coefficients:
The Solved Coefficients
C1 = βv0
C3 = 10h - β(6v0 + 4v1)
C4 = -15h + β(8v0 + 7v1)
C5 = 6h - β(3v0 + 3v1)
The Final Polynomial Equation
Therefore, the 5th-degree polynomial cam function that satisfies all boundary conditions is:
Figure 3: The final mathematical model for smooth cam motion control.
s = C1x + C3x3 + C4x4 + C5x5
With this equation derived, we can now plug in any start/end velocity we want. This is critical for designing "Flying Shears" or "Rotary Knives" where the tool must match the speed of the product line perfectly.
In the next post, we will analyze the characteristics of this curve and graph the results.
Continue to Part 3:
Polynomial Cam Function (Analysis & Graphs) - Part 3
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