Follower Max Acceleration (SHM)
Calculate the maximum acceleration of a simple-harmonic-motion cam follower, a_max = (π²·h·ω²) ÷ (2·β²), from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Follower acceleration is perhaps the MOST important parameter in high-speed cam design, since it generates the INERTIA FORCES (F = m·a): the higher the acceleration, the greater the force the cam must apply to the follower (and the reaction back on the cam and bearings), the greater the tendency to vibration and follower 'jump', and the greater the contact stresses. In SHM, maximum acceleration occurs at the ENDS (start and finish of the rise), and — crucially — it has a DISCONTINUITY there (jumping from zero to maximum instantly), causing a shock and exciting vibrations. So for very high speeds, CYCLOIDAL motion is preferred (its acceleration is continuous, starting and ending at zero), despite cycloidal having a slightly higher peak acceleration. Acceleration grows with the SQUARE of the rotation ω — so doubling the rotation quadruples the inertia forces, and high-rpm engine cams are a design challenge. Enter the lift, the angular velocity and the rise angle.
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Maximum follower acceleration (SHM)
The maximum acceleration of the follower of a cam with simple harmonic motion is a_max = (π²·h·ω²) ÷ (2·β²), from the total lift h, the cam angular velocity ω and the rise angle β. Follower acceleration is arguably the most important parameter in high-speed cam design, because it is what generates the inertia forces (F = m·a): the higher the acceleration, the greater the force the cam has to apply to the follower (and which comes back as a reaction on the cam and its bearings), the stronger the tendency to vibrate and to follower jump, and the higher the contact stresses. Under SHM the peak acceleration occurs at the extremes of the rise (start and finish) and — crucially — it is discontinuous at those points, jumping instantaneously from zero to its maximum value, which delivers a shock and excites vibration. For very high speeds, therefore, cycloidal motion is preferred, since its acceleration is continuous, starting and ending at zero, even though the cycloidal peak acceleration is somewhat larger. Acceleration grows with the square of the shaft speed ω — doubling the speed quadruples the inertia forces, which is why cams for high-rpm engines are a design challenge. Enter the lift, the angular velocity and the rise angle.
Related Tools
Follower Max Velocity (SHM)
Calculate the maximum velocity of a simple-harmonic-motion cam follower, v_max = (π·h·ω) ÷ (2·β), from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). In simple harmonic motion, the follower velocity starts from zero (rest), rises to a MAXIMUM at mid-rise (when the follower passes mid-height) and returns to zero at the top. This peak matters for several reasons: it sets the speed the follower — and the coupled mass (valve, tool, part) — moves at, affecting inertia and dynamic forces; it influences cam-follower contact wear; and, with acceleration, it decides whether the follower can follow the cam without 'floating' (losing contact, jump, at high speeds). Maximum velocity grows linearly with the cam rotation ω and the lift h, and decreases with the rise angle β (more 'spread-out' rises are smoother). Comparing SHM with other motion laws (parabolic, cycloidal) by maximum velocity and acceleration is how the right law is chosen per application. Enter the lift, the cam angular velocity and the rise angle.
Max Velocity (Cycloidal Cam)
Calculate the maximum velocity of a cycloidal-motion cam follower, v_max = (2·h·ω) ÷ β, from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). In cycloidal motion, the follower velocity follows a smooth (1 − cosine) curve, starting from zero, reaching the MAXIMUM at mid-rise and returning to zero at the top — similar in shape to SHM, but with a slightly different profile ensuring acceleration continuity. The cycloidal maximum velocity (factor 2) is slightly HIGHER than SHM's (factor π/2 ≈ 1.57), reflecting that, to 'fit' the same lift in the same angle with smoother end accelerations, the mid velocity must be higher. Knowing the maximum velocity matters for the mechanism dynamics (the follower-mass kinetic energy, supplied then absorbed each cycle), for friction and wear at the cam-follower contact, and to check the system can follow the cam at high rpm. Comparing the maximum velocities and accelerations of the three classic laws (parabolic, SHM, cycloidal) is the basis of choosing the right cam profile per combination of load, speed and smoothness requirement. Enter the lift, the angular velocity and the rise angle.
Max Acceleration (Cycloidal Cam)
Calculate the maximum acceleration of a cycloidal-motion cam follower, a_max = (2π·h·ω²) ÷ β², from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Cycloidal motion is considered the BEST cam motion law for HIGH SPEEDS, and is standard in precision, high-rpm cams. Its decisive feature is that acceleration is a FULL SINE wave starting at zero, rising to a maximum, passing through zero, going to a minimum and returning to zero — i.e., acceleration is CONTINUOUS and starts and ends smoothly at ZERO at the ends, WITHOUT the discontinuities of SHM and parabolic. This means finite, continuous JERK, eliminating shocks and minimizing vibration excitation — the follower 'glides' smoothly without jolts. The price is a slightly HIGHER maximum acceleration than parabolic (2π ≈ 6.28 vs 4 in the factor) and SHM (π²/2 ≈ 4.93), but the dynamic SMOOTHNESS amply compensates at high speed. The name comes from the cycloid curve describing the displacement. Racing-engine valve cams, fast textile and packaging machines use cycloidal or derived (polynomial) profiles precisely to run at high rpm with low vibration. Enter the lift, the angular velocity and the rise angle.
Follower Displacement (Parabolic)
Calculate the displacement of a parabolic-motion cam follower, in the first half of the rise, s = 2·h·(θ/β)², from the total lift h (mm), the cam angle θ (rad, current position) and the rise angle β (rad). In parabolic (constant-acceleration) motion, the first HALF of the rise has the follower accelerating uniformly, and its displacement grows with the SQUARE of the angle — hence 'parabolic' (the s vs θ curve is a parabola). The formula s = 2h(θ/β)² holds for θ between 0 and β/2 (half the rise); in the second half, the follower decelerates and the curve is an inverted parabola completing the lift smoothly to h. This motion is the cam analog of a body in free fall (constant acceleration): just as distance traveled grows with the square of time, here displacement grows with the square of angle. The parabolic construction produces the lowest maximum acceleration among simple laws, but with infinite jerk at the junctions (start, middle and end), limiting its use at high speed. This calculation gives the follower position at any point of the first half, useful for tracing the cam profile and for kinematic analysis. Enter the lift, the current angle and the rise angle.
Max Acceleration (Parabolic Cam)
Calculate the (constant) maximum acceleration of a parabolic-motion (constant-acceleration) cam follower, a_max = (4·h·ω²) ÷ β², from the total lift h (mm), the cam angular velocity ω (rad/s) and the rise angle β (rad). Parabolic, or constant-acceleration, motion is the law producing the LOWEST possible maximum acceleration for a given lift and time — so it minimizes peak inertia forces. It consists of two halves: in the first, the follower accelerates with CONSTANT acceleration (rising parabolic displacement); in the second, it decelerates with the same constant (negative) acceleration, stopping at the top. The name 'parabolic' comes from the displacement diagram, formed by two parabolas. The great advantage is the low maximum acceleration; the drawback is that acceleration JUMPS abruptly — from +a_max to −a_max at the middle, and from zero to ±a_max at the ends — generating infinite JERK there, causing shocks, noise and vibration. So in practice pure parabolic is little used at high speed (despite low peak acceleration), and cycloidal or modified profiles that smooth these transitions are preferred. Parabolic is didactic and useful when peak acceleration is the limiting factor and speeds are moderate. Enter the lift, the angular velocity and the rise angle.
Follower Displacement (SHM)
Calculate the displacement of a simple-harmonic-motion (SHM) cam follower, s = (h/2)·(1 − cos(π·θ/β)), from the total lift h (mm), the cam angle θ (rad, current position) and the rise cam angle β (rad, ramp duration). A cam is a special-profiled mechanical element that, rotating, imposes a programmed motion on a FOLLOWER sliding or pivoting on it — the heart of engine valve trains, automatic machines, textile, printing and packaging equipment. Simple harmonic motion is a classic follower motion law: displacement follows a cosine, starting smoothly from rest, accelerating to mid-height and decelerating smoothly to rest at the top. It has continuous velocity and acceleration (no jumps), but acceleration is discontinuous at the ends (start and finish), causing a small shock — so SHM suits moderate speeds. The follower displacement diagram (s vs θ) is the starting point of cam-profile design: from it derive velocity, acceleration and jerk, which set the forces, vibrations and accuracy of the mechanism. Enter the lift, the current angle and the rise angle.
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