Belt Center Distance Calculator
Compute center distance between two pulleys for a given belt length using Liter's approximation.
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Belt length and center distance: L = 2·C + π·(D₁+D₂)/2 + (D₁−D₂)²/(4C)
For an open belt drive running over two pulleys, the belt length is L = 2·C + π·(D₁+D₂)/2 + (D₁−D₂)²/(4C), where C is the center distance between shafts and D₁, D₂ are the pulley pitch diameters. Going the other way, finding C from a known L, means solving a quadratic. Take D₁ = 200 mm, D₂ = 100 mm and L = 1200 mm, and the center distance comes out at C ≈ 366 mm. There are several belt types to know. V-belts come in sections A, B, C and D, and the wider the section, the more torque it carries. Poly-V belts use multiple small ribs and turn up in alternators and car accessories. Timing belts are toothed and do not slip, which is why engine camshafts and CNC machines use them. And there are plain flat belts. Tension matters a lot here. Run a belt too loose and it slips and overheats; run it too tight and you overload the bearings and wear the belt out early.
Applications
Stationary electric motors driving pumps, fans and compressors. Industrial machinery like lathes, mills and conveyor lines. Automotive timing belts and accessory belts for the alternator, water pump and A/C. Motorcycles and CR-class bicycles, where a Gates Carbon Drive toothed belt takes the place of the usual chain. Agricultural equipment, and household appliances such as washing machines.
FAQ
What's the difference between L and C? L is the total circumference of the belt. C is the straight-line distance between the shaft centers. Since the belt wraps around both pulleys, L always ends up larger than 2·C.
Can I shorten C to fit a smaller belt? Only up to a point. The belt needs a minimum wrap angle on the smaller pulley, usually at least 120°, to transmit torque without slipping. Rather than crowd the pulleys, many drives add an idler tensioner instead.
Does the formula apply to timing belts? The geometry holds, but timing belts only come in discrete tooth counts. So after you find L, round it to the nearest standard belt length and recompute C.
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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.
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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.