Cam Pressure Angle
Calculate the pressure angle of a radial translating-follower cam, α = arctan((ds/dθ) ÷ (R_b + s)), from the displacement derivative with respect to angle ds/dθ (mm/rad, the profile 'slope'), the base circle radius R_b (mm) and the follower displacement s (mm). The pressure angle is the angle between the direction of the FORCE the cam applies to the follower (normal to the profile, at the contact point) and the direction of the follower MOTION. It is a critical design parameter: the LARGER the pressure angle, the greater the LATERAL force component (perpendicular to follower motion), which does no useful work but pushes the follower against its guides, causing friction, wear and possibly JAMMING the follower if excessive. The rule of thumb limits the pressure angle to about 30° (less for translating followers with long guides). The pressure angle depends on the profile (ds/dθ, steeper = larger angle), the base radius (larger cams have smaller angles and smoother operation) and the displacement. So when the pressure angle comes out excessive, the solution is to INCREASE the base circle radius (bigger cam) — at the cost of more space, mass and peripheral speed. Controlling the pressure angle is essential for smooth, durable operation. Enter the displacement derivative, the base radius and the displacement.
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Cam pressure angle
The pressure angle of a cam with a radial translating follower is α = arctan((ds/dθ) ÷ (R_b + s)), obtained from the derivative of the displacement with respect to the cam angle ds/dθ (the slope of the profile), the base circle radius R_b and the follower displacement s. It is the angle between the direction of the force the cam applies to the follower (normal to the profile at the contact point) and the direction of the follower motion. It is a critical design parameter: the larger the pressure angle, the larger the side component of the force (perpendicular to the motion), which does no useful work but presses the follower against its guides, generating friction and wear, and can even jam the follower (jamming) when it grows excessive. The rule of thumb limits the pressure angle to about 30° (less for translating followers with long guides). The pressure angle depends on the profile (ds/dθ, the steeper the larger the angle), on the base radius (bigger cams have smaller angles and run more smoothly) and on the displacement. That is why, when the pressure angle comes out excessive, the fix is to increase the base circle radius (a bigger cam), at the cost of more space, mass and surface speed. Keeping the pressure angle under control is essential for smooth, durable operation. Enter the displacement derivative, the base radius and the displacement.
Related Tools
Gear Base Diameter
Calculate the base circle diameter of an involute gear, d_b = d·cos(φ), from the pitch diameter d (mm) and the pressure angle φ (degrees). The base circle is the circle from which the INVOLUTE tooth profile is generated — the standard profile of modern gears. The involute is the curve traced by the tip of a string unwinding from a cylinder: that cylinder is exactly the base circle. The entire active tooth profile (the part that actually transmits force) is ABOVE the base circle; below it there is no involute profile. The base diameter is fundamental in gear geometry because it defines the involute profile and, with it, key properties: the LINE OF ACTION (the line tangent to both base circles of the mesh, along which tooth contact travels, always in the same direction — why involute gears transmit uniform motion), the base pitch and the contact ratio. The relation d_b = d·cos(φ) shows that the pressure angle is the angle between the line of action and the tangent to the pitch circles. It is an essential parameter in designing and manufacturing (generating) involute gears. Enter the pitch diameter and the pressure angle.
Cam Pitch Radius
Calculate the pitch radius of a roller-follower cam, R_p = R_b + R_r, from the base circle radius R_b (mm) and the follower roller radius R_r (mm). In ROLLER-follower cams (a bearing rolling on the cam profile, reducing friction versus flat-face or knife-edge followers), two important curves are distinguished: the real PROFILE of the cam (the physical surface the roller touches) and the PITCH curve, the locus of the roller CENTER as it follows the cam. The pitch curve is designed first (from the displacement diagram), and the real profile is obtained by 'offsetting' the roller radius from the pitch curve. The pitch radius, at the base position, is the sum of the base circle radius and the roller radius. This distinction is fundamental for a practical reason: the roller radius cannot exceed the smallest RADIUS OF CURVATURE of the pitch curve in CONCAVE regions, or the roller does not 'fit' and the cam gets an incorrect profile (undercutting), distorting the motion. So the choice of roller radius and base radius is coupled to the cam geometry. The pitch radius also enters the pressure-angle and peripheral-speed calculations. Enter the base circle radius and the roller radius.
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.
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.
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.