BIZOMAT MAKEOS · Mechanics
Cam Follower Displacement, Velocity, Acceleration and Jerk (Double-Dwell Functions)
For a cam rise or fall between dwells using a simple harmonic, cycloidal, modified trapezoid, modified sine, 3-4-5 or 4-5-6-7 polynomial program, what are the follower displacement, velocity, acceleration and jerk at a cam angle, and what are their peak values at my cam speed?
Norton’s s v a j functions for double-dwell cams: simple harmonic s = (h/2)[1 − cos(πθ/β)], cycloidal s = h[θ/β − (1/2π) sin(2πθ/β)], the five-segment modified trapezoid and three-segment modified sine with their published coefficients, the 3-4-5 polynomial s = h[10(θ/β)³ − 15(θ/β)⁴ + 6(θ/β)⁵] and the 4-5-6-7 polynomial s = h[35(θ/β)⁴ − 84(θ/β)⁵ + 70(θ/β)⁶ − 20(θ/β)⁷], each with its exact derivatives, converted to time by ω, ω² and ω³, for a rise or a fall, with the peak values over the segment.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Cam program | choice | Displacement function used for the segment. |
| Segment | choice | Whether the follower rises or falls in this segment. |
| Total rise (lift) h | mm | Total follower lift over the segment. |
| Cam angle of the segment β | ° | Cam angle over which the segment takes place. |
| Cam angle into the segment θ | ° | Cam angle from the start of the segment at which to evaluate the motion. |
| Camshaft speed | rpm | Camshaft speed, taken as constant. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Cam program
- What it is
- Displacement program used for this rise or fall between dwells.
- Units and format
- Choose one of 6 options from the list: Simple harmonic; Cycloidal; Modified trapezoid; Modified sine; 3-4-5 polynomial; 4-5-6-7 polynomial.
- Example
- The form opens with “Cycloidal” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the peak velocity, acceleration and jerk.
- Get it right
- Modified sine suits heavy followers; cycloidal gives smooth jerk.
What this means: Segment
- What it is
- Whether the follower moves out (rise) or back (fall) in this segment.
- Units and format
- Choose one of 2 options from the list: Rise; Fall.
- Example
- The form opens with “Rise” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Reverses the signs of velocity, acceleration and jerk.
- Get it right
- A fall is the mirror of the rise.
What this means: Total rise (lift) h
- What it is
- Total lift of the follower over the segment.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All motion values rise in proportion.
- Get it right
- Take it from the cam timing diagram.
What this means: Cam angle of the segment β
- What it is
- Cam rotation angle over which the rise or fall happens.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.1° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A shorter angle raises acceleration with its square.
- Get it right
- Take it from the timing diagram, e.g. 90°.
What this means: Cam angle into the segment θ
- What it is
- Cam rotation from the start of the segment at which to read the motion.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 45° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets where the values are read.
- Get it right
- Enter half of β for the middle of the segment.
What this means: Camshaft speed
- What it is
- Rotational speed of the camshaft, taken as steady.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0 rpm to 1000000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 60 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Velocity rises with speed, acceleration with its square, jerk with its cube.
- Get it right
- Use the running speed of the machine.
Method and formulas
- x = θ/β. Simple harmonic: s = (h/2)(1 − cos πx); cycloidal: s = h(x − sin 2πx ÷ 2π) (eqs. 8.6, 8.12).
- Modified trapezoid: five pieces joined at β/8, 3β/8, 5β/8 and 7β/8 (eqs. 8.13a–e); modified sine: three pieces joined at β/8 and 7β/8 (eqs. 8.14a–c).
- 3-4-5: s = h(10x³ − 15x⁴ + 6x⁵) (eq. 8.20); 4-5-6-7: s = h(35x⁴ − 84x⁵ + 70x⁶ − 20x⁷) (eq. 8.21); v, a, j are the derivatives with respect to θ in radians.
- Time-based values: v × ω, a × ω², j × ω³ with ω = 2π rpm ÷ 60; a fall is h minus the rise with v, a, j reversed.
- Example: peak acceleration 2.5133 m/s² and peak jerk 63.165 m/s³.
Limitations
- These are double-dwell programs; the simple harmonic breaks the fundamental law of cam design at dwells because its acceleration is not zero at the ends.
- Peak values are found by evaluating the functions over the segment.
- The motion is of the follower; the cam profile also depends on the follower type and base circle.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Get the follower motion at any cam angle for a chosen cam program
- Compare peak velocity, acceleration and jerk of cam programs
- Size the follower spring from the peak negative acceleration
Who it is for
- Machine designers
- Mechanism designers
- Mechanical engineering students
What you get
- Displacement, velocity, acceleration and jerk at θ
- Peak velocity, acceleration and jerk over the segment
- The same per radian of cam rotation
Interpreting the result
- The cycloidal has smooth jerk; the modified trapezoid has the lowest peak acceleration of these; the modified sine the lowest peak velocity.
- Dynamic force is proportional to acceleration, so the peak negative acceleration sizes the return spring.
References
- Robert L. Norton, Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines, 2nd edition (McGraw-Hill, 1999), pp. 357–370, 380–382. Topic reference; no source tables, figures or examples are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Useful alongside · BIZOMAT original book
Move profiles and RMS torque
In Inertia and Flywheels by the Numbers
What it covers: How to turn a repeated move into acceleration, run, deceleration and dwell torques, check RMS torque against a motor's continuous rating and peak torque against its peak, and choose the profile shape.
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