Belt Transmission Ratio with Slip
Calculate a belt's real transmission ratio accounting for slip, i = (D ÷ d)·(1 − s/100), from the driving D and driven d pulley diameters (mm) and the slip percentage s (%). A belt's THEORETICAL transmission ratio is simply the pulley diameter ratio (D/d) — a large driving pulley turning a small driven one multiplies the rotation. But in practice, a belt drive is NOT exact like a gear drive (which has interlocking teeth): the belt transmits by FRICTION, and there is always a small SLIP between belt and pulleys. This slip has two components: ELASTIC slip (creep, inevitable, ~1-2%, from the belt stretching and contracting as tension changes between the two sides) and GROSS slip (occurring under overload, when the belt loses grip — undesirable and harmful). Slip makes the driven pulley's real rotation SLIGHTLY LOWER than theoretical, and the real transmission ratio a bit different from nominal. In applications needing exact synchronism (engine timing shafts, positioning), V-belt slip is unacceptable, and TIMING (toothed) belts or chains, which do not slip, are used. This calculation quantifies the slip effect on the transmission ratio. Enter the driving and driven pulley diameters and the slip percentage.
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Relação de transmissão com deslizamento (correia)
A relação de transmissão real de uma correia considerando o deslizamento é i = (D ÷ d)·(1 − s/100), a partir dos diâmetros da polia motora D e da movida d e do percentual de deslizamento s. A relação de transmissão teórica de uma correia é simplesmente a razão dos diâmetros das polias (D/d) — uma polia motora grande girando uma movida pequena multiplica a rotação. Mas, na prática, a transmissão por correia não é exata como a por engrenagens (que têm dentes que se encaixam): a correia transmite por atrito, e sempre há um pequeno deslizamento (escorregamento) entre a correia e as polias. Esse deslizamento tem duas componentes: o escorregamento elástico (creep, inevitável, ~1-2%, devido à correia esticar e contrair ao mudar de tração nos dois lados) e o escorregamento grosseiro (que ocorre sob sobrecarga, quando a correia perde aderência — indesejável e prejudicial). O deslizamento faz a rotação real da polia movida ser ligeiramente menor que a teórica, e a relação de transmissão real um pouco diferente da nominal. Em aplicações que exigem sincronismo exato (eixos de comando de motores, posicionamento), o deslizamento das correias em V é inaceitável, e usam-se correias sincronizadoras (dentadas) ou correntes, que não deslizam. Este cálculo quantifica o efeito do deslizamento na relação de transmissão. Informe os diâmetros das polias motora e movida e o percentual de deslizamento.
Related Tools
Belt Transmitted Power
Calculate the power transmitted by a belt, P = (T₁ − T₂)·v, from the tight-side tension T₁ (N), the slack-side tension T₂ (N) and the belt velocity v (m/s). In a belt drive, the driving pulley drags the belt by friction, creating a DIFFERENCE in tension between the two sides: the side that 'pulls' (tight side, T₁) is more tensioned than the side that 'follows' (slack side, T₂). This difference (T₁ − T₂), the effective tension or tangential force, is the net force that actually transmits motion; times the belt velocity, it gives the transmitted POWER. The larger the tension difference the belt can sustain without slipping (depending on friction, wrap angle and, in V-belts, the wedging effect of the pulley walls), the greater the transmissible power. Power also grows with belt velocity — so high-power drives use large pulleys and fast belts (up to a limit, since centrifugal tension reduces available friction at very high speeds). This is central in belt-drive design, present in almost every rotating machine: motors, fans, pumps, compressors, machine tools and vehicles. Enter the tight- and slack-side tensions and the belt velocity.
Belt Contact Arc
Calculate the contact-arc length of a belt on the smaller pulley, L_arc = (d ÷ 2)·θ, from the smaller pulley diameter d (mm) and the wrap angle θ (radians). The contact arc is the length of the belt portion actually in contact with the pulley (touching it), along the wrap angle — simply the pulley radius times the angle (in radians), the arc-length formula. This length matters for several reasons: it sets the CONTACT AREA between belt and pulley (with the width), governing contact pressure and friction distribution; it influences heating (friction × area) and wear of both belt and pulley; and it is relevant to elastic slip (creep), where the belt, changing tension from T₁ to T₂ along the arc, elastically stretches and contracts, sliding microscopically over the pulley — a small INEVITABLE slip (1-2%) occurring even without gross slipping, making the output speed always slightly below theoretical. A larger contact arc (bigger pulley or more wrap) distributes friction better and reduces the slip tendency. This calculation complements the geometric and friction analysis of a belt drive. Enter the smaller pulley diameter and the wrap angle.
Belt Wrap Angle
Calculate a belt's wrap (contact) angle on the smaller pulley, θ = π − 2·arcsin((D − d) ÷ (2·C)), from the larger D and smaller d pulley diameters (m) and the center distance C (m). The wrap angle is the angle of the arc over which the belt actually WRAPS the pulley, in contact with it — and it is a critical parameter, since it is along that arc that the friction (transmitting the force) acts. The LARGER the wrap angle, the greater the contact area and the greater the force the belt can transmit without slipping. In a drive between two DIFFERENT-DIAMETER pulleys, the belt wraps LESS around the smaller pulley (angle below 180°) and MORE around the larger — and slipping always starts on the pulley with LESS wrap (the smaller), which therefore limits capacity. The wrap angle decreases when the diameter difference grows or the center distance shrinks (close, very different pulleys 'wrap' little). So drives with large reduction (very different pulleys) or close centers have reduced capacity, and sometimes use an IDLER (tensioner) pulley to increase wrap. The wrap angle enters directly into the tension ratio (e^(μθ)) and the belt-count correction factors. Enter the pulley diameters and the center distance.
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.
Belt Centrifugal Tension
Calculate the centrifugal tension in a belt, T_c = m·v², from the mass per unit length m (kg/m) and the belt velocity v (m/s). When the belt wraps a pulley at high speed, its own mass, making the turn, generates a CENTRIFUGAL force tending to 'throw' the belt outward, LIFTING it off the pulley. This creates an additional tension throughout the belt (the centrifugal tension), the same at all points and not contributing to power transmission — it only 'steals' part of the belt's gripping capacity against the pulley. Centrifugal tension grows with the SQUARE of velocity, so it is negligible at low speeds but becomes important in fast belts. The effect is harmful: by lifting the belt off the pulley, centrifugal tension REDUCES the normal contact force and thus the friction available to transmit power — there is an OPTIMAL velocity above which increasing speed reduces transmissible power (the belt starts to 'float'). So belt speed has a practical limit (typically 25-30 m/s for conventional V-belts, more for special belts). Centrifugal tension must be added to the tensions to get the total tight- and slack-side tensions. Enter the mass per unit length and the velocity.
Belt Maximum Tension
Calculate a belt's maximum (tight-side) tension, T₁ = T_e·r ÷ (r − 1), from the effective tension T_e = T₁ − T₂ (the power-transmitting force, N) and the tension ratio r = T₁/T₂ (at the slip limit). Knowing the force the belt must transmit (the effective tension, from power and velocity) and the maximum tension ratio the belt sustains before slipping (from friction, wrap and, in V-belts, the wedge effect), the individual side tensions can be computed. The maximum tension T₁ (tight side) is the larger, and it SIZES the belt's strength (which must not break) and the load on the BEARINGS and pulley shafts (which feel the sum of both side tensions, bending the shaft). Knowing T₁ is essential to: check the belt resists (versus its tensile strength), size the bearings for the radial load imposed by the belt (which can be significant and shortens bearing life), and set the correct installation tension. The LOWER the tension ratio r (worse friction, less wrap), the HIGHER the T₁ needed for the same power — hence the advantage of V-belts (high r) in reducing loads. Enter the effective tension and the tension ratio.
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.