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.
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Aceleração máxima (came parabólico)
A aceleração constante máxima do seguidor de um came com movimento parabólico (aceleração constante) é a_max = (4·h·ω²) ÷ β², a partir da elevação total h, da velocidade angular do came ω e do ângulo de subida β. O movimento parabólico, ou de aceleração constante, é a lei de movimento que produz a menor aceleração máxima possível para uma dada elevação e tempo — por isso minimiza as forças de inércia de pico. Ele consiste em duas metades: na primeira, o seguidor acelera com aceleração constante (deslocamento parabólico crescente); na segunda, desacelera com a mesma aceleração constante (negativa), parando no topo. O nome 'parabólico' vem do diagrama de deslocamento, formado por duas parábolas. A grande vantagem é a baixa aceleração máxima; a desvantagem é que a aceleração salta bruscamente — de +a_max para −a_max no meio, e de zero para ±a_max nas extremidades — gerando jerk infinito nesses pontos, o que causa choques, ruído e vibração. Por isso, na prática, o parabólico puro é pouco usado em alta velocidade (apesar da baixa aceleração de pico), e prefere-se o cicloidal ou perfis modificados que suavizam essas transições. O parabólico é didático e útil quando a aceleração de pico é o fator limitante e as velocidades são moderadas. Informe a elevação, a velocidade angular e o ângulo de subida.
Related Tools
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.
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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