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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Deslocamento do seguidor (MHS)
O deslocamento do seguidor de um came com movimento harmônico simples (MHS) é s = (h/2)·(1 − cos(π·θ/β)), a partir da elevação total h, do ângulo de came θ (posição atual) e do ângulo de came para a subida β (duração da rampa). O came é um elemento mecânico com perfil especial que, ao girar, impõe um movimento programado a um seguidor que desliza ou pivota apoiado nele — é o coração de comandos de válvulas de motores, máquinas automáticas, têxteis, gráficas e de embalagem. O movimento harmônico simples é uma das leis de movimento clássicas: o deslocamento segue uma cossenoide, partindo suave do repouso, acelerando até a meia-altura e desacelerando suavemente até o repouso no topo. Ele tem velocidade e aceleração contínuas (sem saltos), mas a aceleração tem descontinuidade nas extremidades, gerando um pequeno choque — por isso o MHS é adequado para velocidades moderadas. O diagrama de deslocamento do seguidor (s × θ) é o ponto de partida do projeto do perfil do came: dele se derivam a velocidade, a aceleração e o jerk, que determinam as forças, as vibrações e a precisão do mecanismo. Informe a elevação, o ângulo atual e o ângulo de subida.
Related Tools
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 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.
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.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.