Gates Mectrol Timing Belt Theory Timing Belt Theory This paper presents a thorough explanation - PDF document

Timing Belt Theory 1Timing Belt Theory This paper presents a thoroughexplanation of geometric, loading andurethane timing belts. It covers valuablebackground for the step by step selection Traditional understanding oftiming belt drives comes from powerloading conditions on the belt differconsiderably between powertransmission applications and conveyingand linear positioning applications. Thispaper presents analysis of conveying andlinear positioning applications. Whereenlightening, referenpositioning drives. For simplicity, onlytwo pulley arrangements are consideredhere; however, the presented theory canbe extended to more complex systems.Belt and Pulley Pitch(Fig. 1). The the neutral bending axis of the belt andpitch and is defined as the arc lengthpulley grooves (Figs. 1a and 1b). The pitchbelt while wrapped around the pulley. Intiming belt drives the pulley pitch, is larger than the pulleyTiming Belt Theory IntroductionGeometric Relationships Figure 1a.Belt and pulley meseries geometry.Figure 1b.Belt and pulley mesh for AT-series geometry. , is measured along the pitchLpzMost linear actuators and conveyors aredesigned with two equal diameter pulleys.The relationship between belt length, is given byLCd=+For drives with two unequal pulleydiameters (Fig. 2) the following arccos large pulley,respectively., around the large pulley=Š Figure 2.Belt drive with unequal pulley diameters.Belt sectionp pitch in0.1970.1060.0770.047 mm5.02.72.01.2 in0.3940.1770.1380.098 mm10.04.53.52.5 in0.7870.3150.2560.197 mm20.08.06.55.0 Table 2 6 in Fig. 3) is the pitch diameterof the driver pulley.The effective tension generated at thedriver pulley is the actual working forcebelt motion. It is necessary to identify andquantify the sum of the individual forcesacting on the belt pulley.In drives (Fig. 3), thedriven pulley. The force transmitted from pulley is equal to The following expressions for torque is the driving torque, torque requirement at the driven pulley, pulley, are the angular speeds ofpulley respectively,driver and driven pulley respectively, and is the efficiency of the belt drives094096.. typically). The angularspeeds of the driver and driven pulley are related in a following form: The relationship between the angularspeeds and rotational speeds is given by the driver and driven pulley in revolutionsper minute [rpm], and angular velocities of pulley in radians per second.In (Fig. 4) the main loadacts at the positioning platform (slider). Itthe linear bearing, , external force (work 1T1T saFw sF T iTfF mTi d F =m a idler 1 v, adriver Figure 4.Linear positioner - configuration I. pulley, Fmmmgfrspi=++2cos is the dynamic coefficient offriction of the linear bearing, is the angle of incline of theFmmmggspi=++2sinis the angle of incline of the linear reflected to the driver pulley, angular acceleration of the driver pulley,In inclined in Fig. 6, theeffective tension has mainly two forces toovercome: friction and gravitationaldue to the conveyed load,, is given byFNW==µµ()()(k)component of weight, (k), of a singleconveyed package perpendicular to the is the number of packages beingconveyed, index designates the kof material along the belt and angle of incline. When conveying granularmaterials the friction force is given by L a L m L w L driveridler (k)(k)g(k) 1L**L2 1TT2 iT T2 T2 Figure 6.Inclined conveyor with material accumulation. Shaft forcesForce equilibrium at the driver or drivenpulley yields relationships between tight. In powertransmission drives (see Fig. 3) the forceson both shafts are equal in magnitude andare given by FTTTT121121=+Šis angle of belt wrap arounddriver pulley.drives, both linear positioners (Fig 4) andconveyors (Figs. 6 and 7) have no drivenpulley - the second pulley is an idler.In conveyor and linear positioner drivesthe shaft force at the driver pulley,given by FTTTT121=+ŠFTT112is angle of belt wraparound idler pulley.The shaft force at the idler pulley,when the load (conveyed material orslider) is moving toward the driver pulleyis given by FTTTT222=+ŠFTT222 is given byTTF is given by Eq. (21).However, when the load is moving awayfrom the driver pulley the shaft force at theidler pulley,, is given by FTTTT111=+ŠFTT211 is given byTTFthe bearings supporting the idler pulley.Observe that during constant velocitymotion Eq. (38) can be expressed asIn linear positioning drives such asconfiguration IIŽ (shown in Fig. 5) theshaft force of the driver pulley, given by Eq. (35). The shaft forces on theidler rollers can be expressed by FTTTTFTTTT112222''''""""=+Š=+Šthe side of the tight side tension,angle of belt wrap around the idler pulleyon the side of the tight side tension, is the angleof belt wrap around the idler pulley on the have an adjustable idler tensioning the (floating)(Figs. 9 and 10). During operation, theconsistency of the slack side tension ismaintained by the external tensioning. The length increase of the tightside is compensated by a displacement oftension may be considered for someconveying applications.Resolving the Tension Forceshave an external load system, which canbe determined from force analysis alone.Force equilibrium at the idler gives is the external tensioning force is the wrap angle of the belt aroundthe idler (Figs. 9 and 10).Eq. (47) together with Eq. (11) can be usedto solve for the tight side tension, (Figs.3, 4, 6 and 7) have an external loadsystem, which cannot be determined from Figure 9. Figure 10.Vacuum conveyor with the constant slack side tension. proper tooth meshing during belt driveon ensuring that therecommended range.As mentioned before, Eqs. (51) through(53) apply when tight and slack sidetensions are constant over the length. In allother cases the elongation in Eq. (49) mustbe calculated according to the actualtension distribution. For example, theelongation of the conveying length the vacuum chamber length presented inFig. 7, caused by a linearly increasing belt , divided by the stiffness vsp is the length of thetensions at the beginning and end of thevacuum chamber stretch, respectively.Considering this, expressed by TTT TTTminSubstituting v* and expressed in the form of Eqs. (52) and(53), respectively.A similar analysis can be performed forthe conveyor drive in Fig. 6, with the beltelongation due to the mean tensioncalculated over the conveying andaccumulation length, +LLLbeginning of the conveying length to thelocation on the belt corresponding to themodified tight and slack lengths take onthe following form:LLLLLLLL=++Špulleys in belt drives presented in (Figs. 3to 10). Starting at the tight side, the belttension along the arc of contact decreaseswith every belt tooth. At the kk+1 (Fig. 11).&&&TTFkktk++=0(57) Figure 11.Tooth loading. beyond the scope of this paper but thedividing by the tooth stiffness 1 shows that the gradient of by the slope of the curve) decreases withincreasing number of teeth in mesh.Defining the ratio corresponding to the kzkmmvtmaximum number of teeth in mesh thatcarry load is 15.In linear positioners the displacement ofasticity of the tightdisplacement due to the elasticity of theis determined by the following formula: 111kkkIn drives with a driven pulley (power must be added to the right-hand sidethe driven pulley.positioner due to the elasticity of the belt is the static (external) forceremaining at the slide. In Fig. 12, forexample, at the driver pulley.The additional rotation angle, driving pulley necessary for exactpositioning of the slide is Figure 13.Following error - linear positioner under dynamic loading condition. GATES MECTROL, INC . Tel. +1 (603) 890-1515Tel. +1 (800) 394-4844 www.