KEVOS® MAKEOS · Mechanics
Cam Follower Force Analysis (Return Spring Rate, Normal Force with Guide Friction, Cam Torque, Surface Error)
For a disc cam driving a translating roller follower held by a return spring, what spring rate keeps the follower on the cam, what normal force does the cam carry at a chosen angle once friction in the follower guide is included, what torque does the cam shaft need there, and how much would a small bump on the cam surface add to the acceleration?
Force analysis of a radial translating roller follower by the handbook method: the follower acceleration a = h y″ ω² ÷ β² for the motion law, the equivalent mass m = mf + ms ÷ 3 + me, the inertia force R = m a at the point of greatest deceleration (θ = 0.75 β cycloidal, β simple harmonic, β ÷ 2 parabolic), the spring force Fs = k R − mf g − Fe − Ff with your dynamic factor k, the spring rate Ks = (Fs − preload) ÷ ya, then at your chosen angle the pressure angle, the load P = m a + (y Ks + preload) + mf g + Fe, the normal force Fn = P ÷ [cos φ − μ sin φ (2 l1 + l2 − μ d) ÷ l2] with friction in the guide bushing, the cam torque To = (Rmin + y) Fn sin φ, the radius of curvature, and the extra acceleration Δa = 2 e (6 N ÷ ε)² from a surface error of height e over ε degrees.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Rise motion | choice | Motion of the follower during the rise. |
| Rise h | mm | Total rise of the follower. |
| Cam angle for the rise β | ° | Cam rotation during which the rise takes place. |
| Cam speed N | rpm | Rotational speed of the cam. |
| Least radius to the pitch curve Rmin | mm | Least radius from the cam centre to the pitch curve. |
| Cam angle to analyse θ (from the start of the rise) | ° | Cam angle, from the start of the rise, at which to work out the normal force and torque. |
| Masses and loads | ||
| Follower mass mf | kg | Mass of the follower. |
| Return spring mass ms | kg | Mass of the return spring; one third of it moves with the follower. |
| External mass moved me | kg | Mass of the external parts the follower pushes. |
| External force resisting the follower Fe | N | External force resisting the follower motion. |
| Other friction force Ff | N | Friction force from guide bushings and other sources, other than the pressure-angle friction worked out here. |
| Spring preload at Rmin | N | Preload of the return spring when the follower is at its lowest position. |
| Dynamic factor on the inertia force | Factor on the inertia force for elasticity and backlash; the handbook says at least 2 for parabolic and 1.05 for cycloidal motion. | |
| Follower guide | ||
| Friction coefficient in the guide bushing μ | Coefficient of friction between the follower stem and its guide bushing. | |
| Distance from roller to bushing l1 | mm | Distance from the roller centre to the near end of the guide bushing. |
| Length of the bushing l2 | mm | Length of the guide bushing. |
| Follower stem diameter d | mm | Diameter of the follower stem. |
| Surface error | ||
| Height of a bump or dent e | mm | Height of a bump (or depth of a dent) left on the cam surface. |
| Width of the error ε | ° | Width of the error, as cam rotation. |
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: Rise motion
- What it is
- Motion law the follower follows during the rise.
- Units and format
- Choose one of 3 options from the list: Cycloidal; Simple harmonic; Parabolic (constant acceleration).
- Example
- The form opens with “Cycloidal” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Changes the accelerations, pressure angles and curvature.
- Get it right
- Cycloidal motion gives the lowest dynamic forces at high speed.
What this means: Rise h
- What it is
- Total rise of the follower from its lowest to its highest position.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 25.4 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Larger rises raise the pressure angle and the accelerations.
- Get it right
- Take it from the motion the machine needs.
What this means: Cam angle for the rise β
- What it is
- Cam rotation during which the follower makes its rise.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Accelerations fall with the square of the rise angle.
- Get it right
- A longer rise angle lowers the accelerations and the pressure angle.
What this means: Cam speed N
- What it is
- Rotational speed of the cam shaft.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0.000001 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 900 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Inertia forces rise with the square of the speed.
- Get it right
- Use the highest speed the machine runs at.
What this means: Least radius to the pitch curve Rmin
- What it is
- Least radius from the cam centre to the pitch curve.
- 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 39.624 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the pressure angle and the torque arm.
- Get it right
- Take it from the cam size tool or the cam drawing.
What this means: Cam angle to analyse θ (from the start of the r…
- What it is
- Cam angle from the start of the rise at which to work out the forces.
- 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 the rise, pressure angle and acceleration used.
- Get it right
- Use the angle of the largest pressure angle, or check several angles.
Masses and loads
What this means: Follower mass mf
- What it is
- Mass of the follower, roller and stem that move with the cam.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0.000001 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.9072 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the inertia force and the spring needed.
- Get it right
- Weigh the parts or take them from the model.
What this means: Return spring mass ms
- What it is
- Mass of the return spring; a third of it moves with the follower.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Adds a third of its mass to the moving mass.
- Get it right
- Weigh the spring; enter 0 if it is small.
What this means: External mass moved me
- What it is
- Mass of the external parts the follower pushes, such as a slide or piston.
- Units and format
- Enter a number in kg: kilograms. The form accepts 0 kg to 100000 kg. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 kg as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the inertia force.
- Get it right
- Enter 0 if the follower moves nothing else.
What this means: External force resisting the follower Fe
- What it is
- External force resisting the follower as it rises.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 44.48 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the normal force and lowers the spring force needed.
- Get it right
- For example a process load or a counterbalance; enter 0 if none.
What this means: Other friction force Ff
- What it is
- Friction from guide bushings and other sources, besides the pressure-angle friction.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lowers the spring force needed to hold the follower.
- Get it right
- Enter 0 when it is small or unknown.
What this means: Spring preload at Rmin
- What it is
- Force of the return spring with the follower at its lowest position.
- Units and format
- Enter a number in N: newtons. The form accepts 0 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 160.1 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Lowers the spring rate needed and raises the cam load.
- Get it right
- Allow some preload for the set the spring takes in service.
What this means: Dynamic factor on the inertia force
- What it is
- Factor on the calculated inertia force for elasticity and backlash.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.05 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the spring force needed.
- Get it right
- The handbook says at least 2 for parabolic and 1.05 for cycloidal motion.
Follower guide
What this means: Friction coefficient in the guide bushing μ
- What it is
- Coefficient of friction between the follower stem and its bushing.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 1. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.05 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the normal force on the cam.
- Get it right
- About 0.05 for a lubricated steel stem; it rises sharply if the stem bends.
What this means: Distance from roller to bushing l1
- What it is
- Distance from the roller centre to the near end of the guide bushing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 38.1 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer overhang raises the friction force.
- Get it right
- Keep it short so the stem does not bend.
What this means: Length of the bushing l2
- What it is
- Length of the guide bushing that supports the follower stem.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 101.6 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A longer bushing lowers the normal force.
- Get it right
- A longer bushing lowers the friction from the pressure angle.
What this means: Follower stem diameter d
- What it is
- Diameter of the follower stem in the bushing.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 19.05 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Changes the friction term slightly.
- Get it right
- Measure the stem or take it from the drawing.
Surface error
What this means: Height of a bump or dent e
- What it is
- Height of a bump, or depth of a dent, on the cam surface.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.0254 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The extra acceleration rises in proportion.
- Get it right
- From chatter marks or poor blending; a few micrometres on good cams.
What this means: Width of the error ε
- What it is
- Width of the surface error, measured as cam rotation.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 90°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The extra acceleration rises with the inverse square of the width.
- Get it right
- Short errors are the most harmful.
Method and formulas
- ω = π N ÷ 30 rad/s; a = h y″(θ ÷ β) ω² ÷ β² (β in radians); m = mf + ms ÷ 3 + me.
- Greatest deceleration at θa = 0.75 β (cycloidal), β (simple harmonic) or β ÷ 2 (parabolic); R = m |a|; Fs = kd R − mf g − Fe − Ff; Ks = (Fs − preload) ÷ ya.
- At θ: tan φ = (dr ÷ dθ) ÷ (Rmin + y); P = m a + (y Ks + preload) + mf g + Fe.
- Fn = P ÷ [cos φ − μ sin φ (2 l1 + l2 − μ d) ÷ l2]; To = (Rmin + y) Fn sin φ.
- ρ = (r² + r′²)^1.5 ÷ (r² + 2 r′² − r r″); Δa = 2 e (6 N ÷ ε)².
- Example: a spring rate of 9.9521 N/mm keeps the follower on the cam; at θ = 45° the cam carries 539.29 N and needs 13.303 N·m.
Limitations
- The analysis treats the follower as rigid; the dynamic factor allows for elasticity and backlash.
- The spring is checked only at the point of greatest deceleration, as the handbook does; check other points if the spring is weak.
- A follower stem that bends can jam in its bushing and raise the friction coefficient sharply.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size the return spring that keeps a follower on a high-speed cam
- Find the normal force on the cam and the drive torque at a point of the rise
- Judge how accurately a high-speed cam must be made
Who it is for
- Machine designers
- Automation and packaging engineers
- Toolmakers
What you get
- Greatest deceleration, where it occurs, and the inertia force
- Spring force and spring rate to keep the follower on the cam
- Pressure angle, acceleration and load at the chosen angle
- Normal force on the cam with guide friction
- Cam torque at the chosen angle
- Radius of curvature there and the extra acceleration from a surface error
Interpreting the result
- A small surface error at high speed can multiply the acceleration many times; high-speed cams must be made accurately.
- The cam torque resists during the rise and assists during the return.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2301–2305. Topic reference; no source tables or recommended settings are reproduced.
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