Mean Friction Radius (Clutch)
Calculate the mean friction radius of a disc clutch or brake by uniform-wear theory, r_m = (D + d) ÷ 4, from the outer D and inner d diameters (m) of the friction annulus. The mean radius is the EFFECTIVE radius at which the resultant friction force is taken to act for torque calculation (T = μ·F·N·r_m). There are two classic assumptions for this radius: UNIFORM WEAR (assuming the disc has 'bedded in' and wears evenly, concentrating pressure at the inner radius; gives r_m = (D+d)/4, the simple mean of radii) and UNIFORM PRESSURE (new disc, constant pressure; gives r_m = (2/3)·(D³−d³)/(D²−d²), slightly larger). Uniform wear is most used in DESIGN, being conservative (slightly lower torque) and representing the run-in steady state. The mean radius shows an interesting design point: discs with a narrow friction annulus (D close to d, thin ring at large radius) have a high mean radius, transmitting more torque per unit force — so high-performance disc brakes use calipers acting near the disc edge (large radius). Enter the outer and inner diameters.
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Raio médio de atrito (embreagem)
O raio médio de atrito de uma embreagem ou freio de disco, pela teoria do desgaste uniforme, é r_m = (D + d) ÷ 4, a partir dos diâmetros externo D e interno d da coroa de atrito. O raio médio é o raio efetivo em que se considera atuar a força de atrito resultante para o cálculo do torque (T = μ·F·N·r_m). Há duas hipóteses clássicas: a de desgaste uniforme (assume que o disco já 'assentou' e desgasta por igual, levando a pressão a se concentrar no raio interno; resulta em r_m = (D+d)/4, a média simples dos raios) e a de pressão uniforme (disco novo, pressão constante; resulta em r_m = (2/3)·(D³−d³)/(D²−d²), ligeiramente maior). A hipótese de desgaste uniforme é a mais usada no projeto, por ser conservadora (dá torque um pouco menor) e por representar a condição de regime após o amaciamento. O raio médio revela um ponto interessante: discos com coroa de atrito estreita (D próximo de d, anel fino no raio grande) têm raio médio alto, transmitindo mais torque por unidade de força — por isso freios de disco de alto desempenho usam pinças atuando perto da borda do disco (raio grande). Informe os diâmetros externo e interno.
Related Tools
Clutch Max Pressure (Uniform Wear)
Calculate the maximum contact pressure in a disc clutch or brake by the uniform-wear assumption, p_max = 2·F ÷ (π·d·(D − d)), from the axial force F (N) and the outer D and inner d diameters (m); the result is in kPa. Under UNIFORM WEAR (the steady state, after disc run-in), pressure is NOT constant over the friction annulus: it is INVERSELY proportional to radius (highest at the inner radius, lowest at the outer), because wear — proportional to pressure × velocity — only becomes uniform if pressure falls with radius (since velocity grows with radius). So the MAXIMUM pressure occurs at the INNER radius (at diameter d), and that peak limits the design. The maximum pressure must be below the friction material's allowable (linings, pads, ceramic or sintered metallic materials — each with its limit, typically hundreds of kPa to a few MPa). Exceeding the allowable leads to accelerated wear, overheating and friction loss (fading). This calculation checks whether a clutch/brake, under the planned actuation force, operates within its material's pressure limit — an essential durability and safety check. Enter the axial force and the outer and inner diameters.
Disc Clutch Torque
Calculate the torque transmissible by a disc clutch (or brake), T = μ·F·N·r_m, from the friction coefficient μ, the axial clamping force F (N), the number of friction surfaces N and the mean friction radius r_m (m). A disc clutch transmits torque between two shafts by FRICTION between surfaces pressed together: an axial force F clamps the discs, and the friction at that interface, acting at the mean radius, generates the torque. The number of friction surfaces N multiplies the capacity — a single disc clutch has N=1 (one face) or N=2 (disc between two faces); MULTI-PLATE clutches (motorcycles, automatic transmissions) stack several discs with high N, transmitting large torque in compact space. The same principle applies to disc and clutch BRAKES: the braking (or transmitting) torque is μ·F·N·r_m. This is central in clutch and brake design: it sets the actuation force (pedal, spring, hydraulic actuator) needed to transmit/brake a given torque, and the area and number of discs. The design torque includes a service factor (1.2-3) over the nominal, to cover peaks and wear. Enter the friction coefficient, axial force, number of surfaces and mean radius.
Clutch Axial Force (Uniform Pressure)
Calculate the axial clamping force of a disc clutch or brake by the uniform-pressure assumption, F = p·(π/4)·(D² − d²), from the contact pressure p (Pa) and the outer D and inner d diameters (m) of the friction annulus. The axial force clamps the discs together (applied by springs in normally-engaged clutches, or by a hydraulic/pneumatic actuator). By the UNIFORM-PRESSURE assumption (valid for new discs, before wear), the force is simply the average contact pressure times the AREA of the friction annulus (the ring between outer and inner diameters). This force is the clutch/brake actuation parameter: it determines the transmissible torque (with friction and mean radius) and must be limited so the contact pressure does not exceed the friction material's allowable (which has a limit, above which it degrades, loses friction by overheating — fading — or wears fast). Design balances: enough axial force for the needed torque, but pressure within the material limit (setting the minimum area and disc count). Enter the contact pressure and the outer and inner diameters.
Brake Contact Pressure
Calculate the contact pressure between the shoe/pad and the drum/disc of a brake, p = F ÷ A, from the normal force F (N) and the friction material contact area A (m²); the result is in kPa. Contact pressure is the normal force distributed over the friction surface area, and one of the most important parameters in a brake's or clutch's DURABILITY and PERFORMANCE. It must be below the friction material's ALLOWABLE pressure (linings, organic, semi-metallic, ceramic or sintered metallic pads — each with its limit). Pressures ABOVE the allowable lead to accelerated wear, overheating and friction loss (fading), reducing material life and impairing braking. Very LOW pressures underuse the material (a bigger, costlier brake than needed). Contact pressure also relates to the p·v product (pressure × velocity), the key indicator of the friction contact's thermal intensity — friction materials have a p·v limit above which they overheat, and that limit often governs design. This simple check — comparing contact pressure with the material's allowable — is essential in brake and clutch design and in choosing the right friction material for the application. Enter the normal force and the contact area.
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.
V-Belt Tension Ratio
Calculate the maximum tension ratio of a V-belt at the slip limit, T₁/T₂ = e^(μ·θ ÷ sin β), from the friction coefficient μ, the wrap angle θ (radians) and the pulley groove half-angle β (degrees). This is the Euler-Eytelwein (capstan) equation with the V-belt correction. In a FLAT belt, the limit tension ratio is e^(μθ); but the V-belt has a clever advantage: it fits into a V-shaped GROOVE in the pulley, and when tensioned, is pulled INTO the groove, wedging against the two inclined walls. This WEDGE effect multiplies the normal force (and thus the friction) by a factor 1/sin β — since β is small (typically 17-19°, for a 34-38° groove), sin β is small and the EFFECTIVE friction (μ/sin β) is about 3 times the real friction! That is why V-belts transmit much more power than flat belts of the same size, with lower installation tension (sparing the bearings) and less slip — the reason for their huge popularity in industrial and automotive drives. The limit T₁/T₂ ratio sets the maximum effective tension (and thus power) the belt transmits before slipping. Enter the friction coefficient, the wrap angle and the groove half-angle.
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