Kinematics & Dynamics

Cam Follower Calculator — Motion, Contact Force & Pressure Angle

Analyze a radial roller cam follower from lift, motion law, cam speed, follower mass, spring data and geometry to calculate acceleration, contact force, closure margin, pressure angle, side force, contact-loss speed and undercut risk.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

Enter the known values and review the calculated result

Deterministic calculation
01
Parameters

Input parameters

Use consistent values and select the intended engineering units.

Cam motion program

Follower & spring

Radial roller-follower geometry

Design screening

02
Output

Results

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Calculation tools Save, export and manage your results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Cam follower motion and contact equations

The calculator models an in-line radial roller follower over one complete cam revolution. Rise and return can use different motion laws.

Angular speed

ω = 2πn / 60

where ω is cam angular speed [rad/s] and n is cam speed [rpm].

Normalized follower motion

For a rise of duration β:

u = θ / β
y = h · q(u)

For the return:

y = h · [1 − q(u)]

The implemented normalized displacement functions are:

  • Simple harmonic: q(u) = [1 − cos(πu)] / 2
  • Cycloidal: q(u) = u − sin(2πu) / (2π)
  • Constant acceleration/deceleration: q(u) = 2u2 for u < 0.5, and q(u) = 1 − 2(1 − u)2 for u ≥ 0.5
  • 3-4-5 polynomial: q(u) = 10u3 − 15u4 + 6u5
  • 4-5-6-7 polynomial: q(u) = 35u4 − 84u5 + 70u6 − 20u7

Follower velocity, acceleration and jerk

v = ω · dy/dθ
a = ω2 · d2y/dθ2
j = ω3 · d3y/dθ3

Pressure angle

Rp = Rb + Rr
φ = atan(|dy/dθ| / (Rp + y))

Rb is the cam base-circle radius and Rr is the roller radius. If an allowable pressure angle is entered, the calculated maximum pressure angle is checked against that user-defined limit.

Spring force and follower closure force

Fspring = F0 + ky
Fclosure = Fspring + ma

The minimum value of Fclosure over the complete cycle is reported as the follower closure margin.

Cam–roller normal force and side force

While the calculated closure force is positive:

Fn = Fclosure / cos(φ)
Fside = Fn · sin(φ)

Minimum preload required

F0,min = max[0, max(−ky − ma)]

This is the minimum spring preload predicted by the implemented model to keep the closure force from becoming negative at the current cam speed.

Approximate contact-loss speed

At portions of the motion where d2y/dθ2 < 0:

ωcrit = √[(F0 + ky) / (m · (−d2y/dθ2))]

The minimum value over the cycle is converted to:

ncrit = ωcrit · 60 / (2π)

This is a contact-loss estimate for the implemented follower model, not a structural resonance critical speed.

Pitch-curve curvature and roller undercut screening

With r = Rp + y:

ρp = [(r2 + (dy/dθ)2)3/2] / [r2 + 2(dy/dθ)2 − r(d2y/dθ2)]

The calculator finds the minimum positive pitch-curve radius and evaluates the margin ρp,min − Rr. A non-positive margin is flagged as roller-profile undercut risk.

02
Application

When to use this calculator

When to use this calculator

Use this calculator for preliminary kinematic and dynamic screening of an in-line radial cam with a translating roller follower and spring closure.

It is useful when you need to:

  • compare cycloidal, simple harmonic, constant-acceleration, 3-4-5 polynomial and 4-5-6-7 polynomial cam motion laws;
  • calculate maximum follower velocity, acceleration and finite jerk for a defined rise–dwell–return program;
  • estimate the normal cam–roller contact force from spring force, follower inertia and pressure angle;
  • check whether the follower remains positively loaded against the cam at the selected speed;
  • estimate the minimum spring preload required to maintain follower closure;
  • estimate the first cam speed at which the implemented model predicts contact loss;
  • evaluate maximum pressure angle and follower side force;
  • screen roller geometry for pitch-curve curvature and undercut risk.

This calculator is not a Hertzian contact-stress, cam torque, lubrication, wear, fatigue or detailed valve-train dynamics solver. It also does not generate manufacturing coordinates for the final cam profile.

03
Decision support

How to interpret the result

How to interpret the results

Maximum acceleration

Maximum acceleration is the largest absolute follower acceleration generated by the selected lift, rise and return angles, motion laws and cam speed. Because acceleration is proportional to ω2, increasing cam speed can rapidly increase follower inertia loads.

Maximum cam–roller contact force

This is the largest positive normal force calculated from the spring-plus-inertia closure force and the instantaneous pressure angle. It is a load result, not a contact-stress result.

Minimum follower closure force

The closure force is F0 + ky + ma. A positive minimum means the evaluated spring–mass model remains positively loaded against the cam throughout the sampled cycle. A value at or below zero produces a contact-loss flag.

Minimum required preload

F0,min estimates the preload needed to maintain non-negative follower closure at the current speed. Comparing it with the entered preload shows how much preload margin is available within this simplified model.

Approximate contact-loss speed

ncrit is the first calculated cam speed at which negative follower acceleration can overcome the available spring force. The ratio n/ncrit indicates proximity to that predicted contact-loss condition. It should not be interpreted as a resonance ratio.

Pressure angle and side force

Maximum pressure angle is calculated from the slope of the follower displacement curve and the roller-center pitch radius. A larger pressure angle increases the lateral component of the calculated normal force. If a pressure-angle limit is entered, exceeding it produces a warning.

Pitch-curve curvature and undercut margin

The calculator reports the minimum positive pitch-curve radius and subtracts the roller radius to obtain the undercut margin. A positive margin passes this geometric screening criterion. A zero or negative margin produces an undercut-risk flag.

Jerk and motion-law discontinuities

For motion laws without the implemented acceleration-discontinuity flag, the calculator reports the maximum finite jerk found during numerical evaluation. Simple harmonic and constant acceleration/deceleration motion are instead flagged for acceleration discontinuity / impulsive-jerk risk.

Overall status

  • Safe: no modeled contact loss or undercut is detected, the optional pressure-angle limit is not exceeded, and no acceleration-discontinuity flag is active.
  • Warning: the optional pressure-angle limit is exceeded or the selected motion law triggers the acceleration-discontinuity flag.
  • Invalid: the model predicts contact loss or roller undercut, or the entered data fail input validation.

A favorable result confirms only the criteria evaluated by this calculator. It does not by itself verify contact stress, fatigue life, lubrication, vibration, spring durability, guide strength or complete cam-system safety.

04
Worked case

Calculation example

Example — cycloidal roller cam follower

Consider a radial roller follower with a 20 mm lift. Both rise and return use cycloidal motion. The cam operates at 1200 rpm and the spring-loaded follower is checked for acceleration, contact force, pressure angle, contact retention and undercut risk.

Inputs

  • Cam lift, h = 20 mm
  • Rise angle, βrise = 120°
  • Top dwell angle = 60°
  • Return angle, βreturn = 120°
  • Rise motion law = Cycloidal
  • Return motion law = Cycloidal
  • Cam speed, n = 1200 rpm
  • Effective follower mass, m = 0.50 kg
  • Spring stiffness, k = 10 N/mm
  • Spring preload, F0 = 100 N
  • Cam base-circle radius, Rb = 40 mm
  • Roller radius, Rr = 10 mm
  • Pressure-angle limit = 30°

Basic conversions

ω = 2π · 1200 / 60 = 125.66 rad/s

Rp = Rb + Rr = 40 + 10 = 50 mm

The programmed rise, top dwell and return occupy 300°, leaving a 60° bottom dwell.

Calculated results

  • Maximum follower speed: 2.400 m/s
  • Maximum follower acceleration: 452.39 m/s2
  • Maximum finite follower jerk: 170,547 m/s3
  • Maximum cam–roller contact force: 355.07 N
  • Minimum follower closure force: 53.02 N
  • Maximum pressure angle: 17.85°
  • Maximum follower side force: 92.21 N
  • Minimum required preload at 1200 rpm: 46.98 N
  • Approximate first contact-loss speed: 1334.6 rpm
  • Contact-speed utilization, n/ncrit: 0.899
  • Minimum positive pitch-curve radius: 47.77 mm
  • Roller undercut margin: 37.77 mm

Interpretation

The minimum closure force remains positive, so the implemented model does not predict follower separation at 1200 rpm. The calculated speed is about 89.9% of the estimated first contact-loss speed, however, so increasing cam speed would reduce the available closure margin.

The maximum pressure angle of 17.85° remains below the entered 30° limit, and the positive curvature margin indicates no roller undercut condition in this screening calculation.

The result is a preliminary rigid-body and spring-force assessment. Detailed cam design can still require contact-stress, material, lubrication, fatigue, vibration and manufacturing verification.

05
Model boundaries

Assumptions and limitations

Assumptions and limitations

  • The model represents an in-line translating radial roller follower with zero follower offset.
  • The pitch radius is modeled as Rp = Rb + Rr.
  • Cam rotational speed is constant over the analyzed revolution.
  • The follower is represented by a single effective translating mass.
  • The follower spring is linear, with constant stiffness k and preload F0 at zero lift.
  • Follower closure force includes spring force and translational inertia only. Gravity, process loads, gas loads, friction, damping and other external forces are not included.
  • The normal contact force is calculated from the positive closure force and pressure angle; the model does not include elastic contact deformation.
  • The optional pressure-angle limit is supplied by the user. The calculator does not automatically select a design limit for a particular machine, speed, guide arrangement or service condition.
  • The approximate critical speed is a follower contact-loss estimate, not a natural-frequency or resonance calculation.
  • Pitch-curve curvature provides a geometric roller-undercut screen. The calculator does not generate or validate complete manufacturing cam-profile coordinates.
  • Hertzian contact stress, surface fatigue, pitting, lubrication, wear, roller-bearing stress, camshaft deflection and drive torque are outside the implemented model.
  • The cam cycle is evaluated numerically using 7200 angular intervals, so reported extrema are numerical approximations rather than symbolic continuous-domain maxima.
  • Simple harmonic and constant acceleration/deceleration motion are explicitly flagged by the implementation for acceleration discontinuity / impulsive-jerk risk.
  • Rise angle must be greater than 0° and less than 360°, return angle must be greater than 0° and less than 360°, and rise + top dwell + return must not exceed 360°.
  • Final cam design should additionally verify all relevant material, contact, fatigue, lubrication, vibration, spring and manufacturing requirements for the actual mechanism.
06
Questions

Frequently asked questions

What does this cam follower calculator calculate?
It calculates follower velocity, acceleration and jerk together with cam–roller normal force, minimum closure force, pressure angle, side force, required spring preload, an approximate contact-loss speed and a roller-undercut screening result for an in-line radial roller follower.
Which cam motion laws are supported?
Rise and return can independently use cycloidal, 3-4-5 polynomial, 4-5-6-7 polynomial, simple harmonic, or constant acceleration/deceleration motion. The selected law changes the displacement derivatives and therefore the follower velocity, acceleration, jerk and dynamic contact force.
How is cam follower contact loss detected?
The calculator evaluates Fclosure = F0 + ky + ma throughout the complete cam cycle. If the minimum calculated closure force is zero or negative, the current operating point is flagged for contact loss.
What is the minimum spring preload result?
Minimum preload is the calculated zero-lift spring preload required to prevent the modeled closure force from becoming negative at the selected cam speed. It is based on the entered spring stiffness, follower mass and prescribed motion and does not include external loads that are absent from the model.
What does the cam pressure angle result mean?
Pressure angle describes the direction of the calculated cam–roller normal force relative to the follower motion axis. In this model, a larger pressure angle also increases the lateral follower force for the same axial closure force. The calculator compares the maximum value with the optional limit entered by the user; it does not impose a universal pressure-angle limit automatically.
How does the calculator check for roller follower undercut?
It calculates the positive radius of curvature of the roller-center pitch curve around the cam cycle. If the minimum positive pitch-curve radius is less than or equal to the roller radius, the geometry is flagged for undercut risk. The reported curvature margin is ρp,min − Rr.
Why can simple harmonic or constant-acceleration motion produce a jerk warning?
The implemented model identifies these motion laws as having acceleration discontinuities at one or more transitions. In that case the calculator reports the acceleration-discontinuity / impulsive-jerk risk and does not present a finite peak-jerk value as a complete description of the transition.
Is the calculated critical speed a resonance critical speed?
No. The reported ncrit is the approximate first speed at which this spring–mass follower model predicts loss of positive closure force. It is not a natural-frequency or structural-resonance calculation.
Does this calculator calculate Hertzian contact stress?
No. It calculates the normal cam–roller force but does not calculate Hertzian contact pressure, subsurface stress, pitting life, lubrication-film behavior or roller-bearing stress. Those checks require additional geometry and material data.
Does the calculator generate the final cam profile for manufacturing?
No. Base-circle radius, roller radius and pitch-curve curvature are used for pressure-angle and undercut screening, but the calculator does not output cam-profile coordinates, CAD geometry or machining data.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.