Tension in timing-belt drives

June 7, 2001
A urethane timing belt moves samples into position in this CAD simulation of an automated blood sampling machine.
 A urethane timing belt moves samples into position in this CAD simulation of an automated blood sampling machine.

Timing-belt drives transmit torque and motion from a driving to a driven pulley or force to a linear actuator. They may also convey a load placed on the belt surface.

A drive under load develops a difference in belt tension between the entering (tight) and leaving (slack) sides of the driver pulley. This effective tension, Te, is the force transmitted from the driver pulley to the belt and to the driven pulley or load:

where T1 and T2 = tight and slackside tensions. The effective tension or working force generated at the driver pulley overcomes the driven pulley's resistance to motion. The driving torque, M1, is related to the driven torque (load), M2, by:

where d1 = pitch diameter of the driver pulley, P2 = power required at the driven pulley, 1 and 2 = angular speeds of the driver and driven pulleys of pitch diameters d1 and d2, respectively, and = efficiency (typically about 0.94 to 0.96).

Shaft forces
A force equilibrium at the driver or driven pulley relates tight and slack-side tensions and the shaft reaction forces Fs1 or Fs2. In powertransmission drives, forces on both shafts are equal in magnitude:

where 1 = belt wrap angle around the driver pulley.

Belt pretension
Belt pretension (initial tension), Ti, is the tension set by an adjustable idler pulley. Pretension prevents belt slack-side sagging and ensures proper tooth meshing. In most cases, timing belts perform best when the magnitude of slackside tension is about 10 to 30% that of the effective tension.

Although generally not recommended, belt drives can work without an adjustment mechanism. This is possible because, after initial tensioning and straightening, belts tend not to elongate or creep. Overall belt length remains constant during operation regardless of loading conditions, provided belt sag and some other minor influences are neglected. However, reaction forces vary under load. And slack and tight side tensions not only depend on load and pretension, but on belt elasticity and structure stiffness as well.

A constant slack side tensioner is a better way to control belt tension in power-transmission drives and in some conveyors. Here, an adjustable, floating idler riding on the belt slack side compensates for a lengthening tight side. Slackside tension is maintained by an external tensioning force, Fe:

where e, = belt wrap angle about the idler pulley. This, and the expression for effective tension, combine to give tight-side tension, T1, and shaft reactions, Fs1 and Fs2.

Drives with constant, slack-side tension add an external load to the system and cannot be characterized by force analysis alone. Calculating tight and slack side tensions and shaft forces (two equations, three unknowns) for a given torque or effective tension, requires an additional relationship: belt elongation.

Total belt elongation equals that from pretension, neglecting belt sag and some factors that contribute little to elongation, such as belt bending resistance and radial shifting of the belt pitch line. Pulleys, shafts and mounting structures are assumed infinitely rigid for analysis purposes. Then, elongation is expressed by a geometric compatibility of deformation:

where DL11 and DL22 = tight and slack-side elongations due to T1 and T2, DLme = total elongation of the belt portion meshing with the driver (and driven) pulleys, and DL1i, DL2i, and DLmi = deformations from belt pretension, Ti.

In most cases, belt deformations at the pulleys during pretensioning and in operation are about equal (DLme ΔDLmi), so:

Tensile tests of properly loaded timing belts show stress is proportional to strain. Defining the stiffness of a unit long and wide belt as specific stiffness, csp, the belt stiffness coefficients on tight and slack sides, k1 and k2, are expressed by:

where L1 and L2 = unstretched lengths of the tight and slack sides, respectively, and b = belt width. Note that these expressions are similar to that for axial stiffness of a bar. Hooke's Law says that elongation equals tension divided by a stiffness coefficient provided that tension is constant over belt length:

Combining expressions for the stiffness coefficients with those for tight and slack side tensions gives:

where L = total belt length. These equations can be used to determine the shaft reactions, Fs1 and Fs2. In practice, a belt drive can be designed such that the desired slack-side tension equals about 10 to 30% of effective tension. This gives proper tooth meshing during operation. Then the expression for slack-side tension, T2, is used to calculate the correct pretension. As mentioned previously, the above relations apply only when these tensions are constant over belt length. In all other cases, elongations must be calculated according to the actual tension distribution.

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