Cam & Follower Calculator — Motion Profiles, Geometry & Contact Stress (Classical Kinematics)
The MechanixCalc cam & follower calculator generates the full 360° displacement, velocity and acceleration profile for a disk cam with any combination of five classical motion laws — simple harmonic (SHM), cycloidal, parabolic, and the 3-4-5 and 4-5-6-7 polynomial laws. Set the base-circle radius, follower lift, cam speed and the rise/dwell/return segment angles, and the tool instantly plots all three kinematic curves alongside a side-by-side motion-law comparison so you can see which law gives the smoothest acceleration at your operating speed.
Beyond kinematics, the tool checks cam geometry (maximum pressure angle and minimum radius of curvature with an automatic undercut warning), analyses the follower spring for liftoff and bounce risk, and computes the Hertz contact stress at the cam-follower interface for roller and flat-faced followers. It is aimed at machine-design engineers sizing indexing mechanisms, assembly-machine cams, or any cam-driven linkage who need a fast, defensible set of numbers without manually integrating motion equations.
What this calculator does
- Full 360° displacement, velocity and acceleration profiles for five motion laws: SHM, cycloidal, parabolic, 3-4-5 and 4-5-6-7 polynomial
- Side-by-side motion law comparison charts — instantly see how acceleration peaks differ between laws
- Pressure angle and minimum radius of curvature with automatic undercut detection for roller followers
- Knife, roller and flat follower types with eccentricity support
- Follower spring design — liftoff and bounce-risk analysis with minimum preload recommendation
- Hertz contact stress for roller and flat-face followers with safety factor against allowable
- Live polar cam-profile diagram and animated pressure-angle chart with 30° and 45° warning bands
Method & formulas
Motion-law kinematics
For each motion law the tool expresses follower displacement y as a function of the normalised cam angle τ = θ/β (0 ≤ τ ≤ 1, where β is the rise or return angle). Velocity and acceleration follow by differentiating with respect to cam rotation angle θ and scaling by the angular velocity ω. The cycloidal law is generally preferred for high-speed cams because it gives zero acceleration at both ends of the rise, avoiding impulsive loads at segment boundaries. The 3-4-5 polynomial also achieves zero velocity and acceleration at the endpoints; the 4-5-6-7 polynomial additionally zeroes the jerk.
y = h · [τ − sin(2πτ) / (2π)]where y = follower displacement; h = total lift; τ = θ/β = normalised cam angle; θ = cam rotation from start of rise; β = rise angle (rad)
a_max = 2π · h · ω² / β²where a_max = peak acceleration (mm/s²); h = lift (mm); ω = cam angular velocity (rad/s); β = rise angle (rad). Occurs at τ = 1/4.
Cam geometry — pressure angle and undercut
The pressure angle α is the angle between the follower's direction of motion and the normal to the pitch curve at the contact point. Large pressure angles cause excessive lateral (side) forces on the follower stem and increase friction and wear; the conventional design limit is α ≤ 30°. For a radial knife or roller follower with eccentricity e, the pressure angle at any cam angle follows from the follower velocity dh/dθ, the base-circle radius r_b and the current follower lift y. Undercutting occurs for roller followers when the cam radius of curvature ρ falls below the roller radius r_r, which causes the toolpath to self-intersect — the calculator flags this automatically.
tan(α) = (dy/dθ − e) / (√(R_0² − e²) + y)where α = pressure angle; dy/dθ = follower velocity per radian of cam rotation; e = follower eccentricity (mm); R_0 = prime-circle radius (mm) = r_b for knife/flat followers, r_b + r_r for roller followers; r_b = base-circle radius; r_r = roller radius; y = current follower displacement (mm)
Hertz contact stress and follower dynamics
For a roller follower in line contact with the cam, the Hertz contact half-width b and peak contact pressure p_max are computed from the equivalent radius R_eq (combining cam radius of curvature and roller radius), the normal contact force F_n, the face width L, and the reduced modulus E* for steel on steel. The allowable contact stress is 1200 MPa for hardened steel; the safety factor SF = 1200 / p_max is reported.
The follower-spring dynamics panel checks that the spring preload exceeds the inertia force of the follower mass at peak acceleration throughout the rotation. If the minimum contact force drops below zero the follower lifts off the cam; the tool reports the required preload and the maximum safe cam speed N_max that avoids bounce.
p_max = 2 · F_n / (π · b · L), b = √(4 · F_n · R_eq / (π · L · E*))where p_max = peak Hertz contact pressure (MPa); F_n = normal contact force (N); b = contact half-width (mm); L = face width (mm); R_eq = 1/(1/ρ_cam + 1/r_r) = equivalent contact radius (mm); E* = E / (2(1−ν²)) = reduced modulus (MPa, ≈ 113 GPa for steel)
Worked example
Find the peak follower acceleration for a disk cam with a cycloidal rise law, 20 mm lift, 120° rise angle and a cam speed of 300 rpm.
Given
- Motion lawCycloidal
- Follower lift h20 mm
- Rise angle β120° = 2π/3 rad ≈ 2.094 rad
- Cam speed N300 rpm
Result
- Peak follower acceleration a_max≈ 28,274 mm/s² (≈ 28.3 m/s²)
- Convert cam speed to angular velocity: ω = 2π × N / 60 = 2π × 300 / 60 = 10π ≈ 31.42 rad/s.
- Convert rise angle to radians: β = 120° × (π/180) = 2π/3 ≈ 2.094 rad.
- Apply the cycloidal peak-acceleration formula: a_max = 2π · h · ω² / β².
- Compute ω²: (10π)² = 100π² ≈ 986.96 (rad/s)².
- Compute β²: (2π/3)² = 4π²/9 ≈ 4.386 rad².
- Substitute: a_max = 2π × 20 × 986.96 / 4.386 ≈ 2π × 20 × 225.0 ≈ 2π × 4500 ≈ 28,274 mm/s².
Illustrative example — verify against your actual cam geometry, speed and follower mass. The calculator integrates the full 360° profile numerically and also reports velocity and contact force at every degree.
Frequently asked questions
Which standard does this cam & follower calculator use?
The motion-law kinematics (SHM, cycloidal, parabolic, 3-4-5 and 4-5-6-7 polynomial) are classical analytical results derived from first principles — there is no single ISO or DIN standard governing their formulation. The Hertz contact stress uses standard Hertz contact theory for line contact. The governing equations and their sources are shown in the generated PDF report alongside the numerical results.
Which motion law gives the best performance at high cam speeds?
Cycloidal and the 4-5-6-7 polynomial are the best choices for high-speed cams. Cycloidal gives zero acceleration at both the start and end of the rise, preventing impulsive loads at segment boundaries. The 4-5-6-7 polynomial additionally zeroes the jerk at both ends, suppressing resonance excitation. SHM and parabolic have finite acceleration discontinuities at segment boundaries and are better suited to low-to-moderate speeds.
What is the pressure angle limit for cam design?
The conventional design limit is 30° maximum pressure angle for translating followers. Above 30° the lateral (side) force on the follower stem rises sharply, increasing friction and guide wear. The calculator plots the full pressure-angle profile versus cam angle and highlights the 30° and 45° threshold bands; if the maximum exceeds 30°, it suggests increasing the base-circle radius or reducing the lift.
How does the undercut check work?
For a roller follower the cam profile is machined along the pitch curve offset inward by the roller radius. If the pitch-curve radius of curvature ρ falls below the roller radius r_r at any point, the inward offset self-intersects — a condition called undercutting — and the cam cannot be manufactured as specified. The calculator computes ρ at every degree and raises an undercut warning if ρ_min < r_r, with a prompt to increase the base-circle radius or reduce the roller size.
Is the cam & follower calculator free?
You can use it during a free 30-minute preview with no sign-up, and a free 14-day account trial unlocks every calculator with no credit card required. The branded PDF engineering report and saved calculations are part of a paid plan.
Related calculators
- Springs CalculatorDesign the follower return spring: rate, preload and fatigue life.
- Shaft Analysis (DIN 743)Analyse the camshaft for bending, torsion, fatigue and critical speed.
- Bearing Analysis (ISO 281)Carry the cam reaction force into bearing L10 life selection.
- Gear DesignSize the gear drive that transmits torque to the camshaft.
- Rotor DynamicsCheck for resonance if the camshaft is a high-speed rotor.
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