Crossed-belt drive length

What does this quantity describe?

The common crossed-drive approximation is L ≈ 2C + π(D+d)/2 + (D+d)²/(4C). It represents two straight spans plus pulley contact arcs. D and d are pitch diameters measured along the neutral belt layer.

In an open drive, part of the pulley-size difference cancels geometrically, giving D−d in its correction. Crossed internal tangents pass according to the sum of both radii, so D+d controls the correction.

Formula and variables

L ≈ 2C + π(D+d)/2 + (D+d)²/(4C)

• L ≈ 2C + π(D+d)/2 + (D+d)²/(4C)
• C ≈ [K + √(K² − 2(D+d)²)]/4
• K = L − π(D+d)/2

Choose the inputs correctly

Use shaft center distance C and pitch diameters D and d rather than unchecked outside dimensions. The layout requires C > (D+d)/2. D is labelled as the larger pulley, although the equation itself is symmetric.

How to use the calculator

Select the target. Enter C, D and d for a preliminary length, or solve C for an available belt. Compare the approximation with an actual manufacturer pitch length, then accommodate the resulting center-distance adjustment and tensioning travel.

Worked example

For C = 600 mm, D = 250 mm and d = 100 mm, L ≈ 1,200 + 549.78 + 51.04 = 1,800.82 mm. The condition 600 mm > 175 mm is satisfied.

Understand the result and units

The result is geometric pitch length before installation stretch and product matching. An available nominal length near 1,801 mm may require a slightly different center distance. The two pulleys rotate in opposite directions in this simple crossed layout.

C, D, d and L are lengths and are converted through SI internally. Millimetres are customary for machine layouts. Input diameters and output length must use the same pitch-line convention.

When should a crossed layout be used?

An open belt gives equal pulley rotation directions and avoids added lateral twist at a crossing. A crossed flat belt reverses rotation and increases wrap, but experiences added twisting, possible belt contact at the crossing, and reverse bending.

Typical applications

Crossed flat belts can connect parallel shafts that must rotate oppositely. The calculation supports layout, replacement-belt preselection and a first estimate of adjustment travel.

Assumptions, limits and common mistakes

The approximation neglects belt thickness and elastic stretch. Dubbel also identifies twist stress and shorter life from reverse bending in crossed drives. Check current product-specific spacing and operating limits with the belt manufacturer.

Common mistake: Do not use the open-belt (D−d)² correction. Do not mix outside and pitch diameters. A geometric result is not automatically an orderable standard length, and crossed V-belts or timing belts must not be assumed acceptable without manufacturer approval.