Timing Belt Theory

18
Gates Mectrol Timing Belt Theory 1 Timing Belt Theory

Transcript of Timing Belt Theory

Page 1: Timing Belt Theory

Gates Mectrol • Timing Belt Theory 1

Timing Belt Theory

Page 2: Timing Belt Theory

2

This paper presents a thoroughexplanation of geometric, loading anddeflection relationships of reinforcedurethane timing belts. It covers valuablebackground for the step by step selectionprocedure for the "Belt Sizing Guide"available on the Gates Mectrol website. Traditional understanding oftiming belt drives comes from powertransmission applications. However, theloading conditions on the belt differ

considerably between powertransmission applications and conveyingand linear positioning applications. Thispaper presents analysis of conveying andlinear positioning applications. Whereenlightening, reference will be made topower transmission and rotarypositioning drives. For simplicity, onlytwo pulley arrangements are consideredhere; however, the presented theory canbe extended to more complex systems.

Belt and Pulley Pitch

Belt pitch, p, is defined as the distancebetween the centerlines of two adjacentteeth and is measured at the belt pitch line(Fig. 1). The belt pitch line is identical tothe neutral bending axis of the belt andcoincides with the center line of the cords.

Pulley pitch is measured on the pitchcircle and is defined as the arc lengthbetween the centerlines of two adjacentpulley grooves (Figs. 1a and 1b). The pitchcircle coincides with the pitch line of thebelt while wrapped around the pulley. Intiming belt drives the pulley pitchdiameter, d, is larger than the pulley

Timing Belt Theory

Introduction

Geometric Relationships

Figure 1a. Belt and pulley mesh for inch seriesand metric T-series, HTD and STDseries geometry.

Figure 1b. Belt and pulley mesh for AT-series geometry.

Page 3: Timing Belt Theory

3

outside diameter, do. The pulley pitchdiameter is given by

dp z p=

⋅π

(1)

where p is the nominal pitch and zp thenumber of pulley teeth.The radial distance between pitch diameterand pulley outer diameter is called pitchdifferential, u, and has a standard value fora given belt section of inch pitch andmetric T series belts (see Table 1). Thepulley outside diameter can be expressedby

d d up z

uop= − =

⋅−2 2

π(2)

As inch pitch and metric T series belts aredesigned to ride on the top lands of pulleyteeth, the tolerance of the outside pulleydiameter may cause the pulley pitch todiffer from the nominal pitch (see Fig. 1a).

On the other hand, metric AT series beltsare designed to contact bottom lands (notthe top lands) of a pulley as shown in Fig.1b. Therefore, pulley pitch and pitchdiameter are affected by tolerance of thepulley root diameter, dr, which can beexpressed by

d d up z

ur rp

r= − =⋅

−2 2π

(3)

The radial distance between pitch diameter

Belt section p - beltpitch

H - beltheight

u - pitchdifferential

h - tooth height

XL in 0.200 0.090 0.010 0.050mm 5.1 2.3 0.3 1.3

L in 0.375 0.140 0.015 0.075mm 9.5 3.6 0.4 1.9

H in 0.500 0.160 0.027 0.090mm 12.7 4.1 0.7 2.3

XH in 0.875 0.440 0.055 0.250mm 22.2 11.2 1.4 6.4

T5 in 0.197 0.087 0.020 0.047mm 5.0 2.2 0.5 1.2

T10 in 0.394 0.177 0.039 0.098mm 10.0 4.5 1.0 2.5

T20 in 0.787 0.315 0.059 1.500mm 20.0 8.0 1.5 5.0

HTD 5 in 0.197 0.142 0.028 0.83mm 5.0 3.6 0.7 2.1

HTD 8 in 0.315 0.220 0.028 0.134mm 8.0 5.6 0.7 3.4

HTD 14 in 0.551 0.394 0.055 0.236mm 14.0 10.0 1.4 6.0

STD 5 in 0.197 0.134 0.028 0.075mm 5.0 3.4 0.7 1.9

STD 8 in 0.315 0.205 0.028 0.11mm 8.0 5.2 0.7 3.0

STD 14 in 0.551 0.402 0.055 0.209mm 14.0 10.2 1.4 5.3

Table 1

Page 4: Timing Belt Theory

4

and root diameter, ur, has a standard valuefor a particular AT series belt sections (seeTable 2).

Belt Length and Center Distance

Belt length, L, is measured along the pitchline and must equal a whole number ofbelt pitches (belt teeth), zb

L p zb= ⋅ (4)

Most linear actuators and conveyors aredesigned with two equal diameter pulleys.The relationship between belt length, L,center distance, C, and pitch diameter, d,is given by

L C d= ⋅ + ⋅2 π (5)For drives with two unequal pulleydiameters (Fig. 2) the followingrelationships can be written:Angle of wrap, θθθθ1, around the small pulley

θ12 122

= −⋅

⎛⎝⎜

⎞⎠⎟

arccosd d

C(6)

where d1 and d2 are the pitch diameters ofthe small and the large pulley,respectively.

Angle of wrap, θθθθ2, around the large pulley

θ π θ2 12= ⋅ − (7)

Span length, Ls

Figure 2. Belt drive with unequal pulley diameters.

Belt section p - beltpitch

H - beltheight

ur - pitchdifferential

h - toothheight

AT5 in 0.197 0.106 0.077 0.047mm 5.0 2.7 2.0 1.2

AT10 in 0.394 0.177 0.138 0.098mm 10.0 4.5 3.5 2.5

AT20 in 0.787 0.315 0.256 0.197mm 20.0 8.0 6.5 5.0

Table 2

Page 5: Timing Belt Theory

5

L Cs = ⋅ ⎛⎝⎜

⎞⎠⎟

sinθ1

2(8)

Belt length, L

( )

L Cd

d

= ⋅ ⋅ ⎛⎝⎜

⎞⎠⎟

+ ⋅

+ ⋅ − ⋅

22 2

22

11

1

12

sinθ θ

π θ

(9)

Since angle of wrap,θθθθ1, is a function of thecenter distance, C, Eq. (9) does not have aclosed form solution for C. It can be

solved using any of available numericalmethods.An approximation of the center distance asa function of the belt length is given by

( )C

Y Y d d≈

+ − ⋅ −22 1

22

4(10)

where( )

Y Ld d

= −⋅ +π 2 1

2

A timing belt transmits torque and motionfrom a driving to a driven pulley of apower transmission drive (Fig. 3), or aforce to a positioning platform of a linearactuator (Fig. 4). In conveyors it may alsocarry a load placed on its surface (Fig. 6).

Torque, Effective Tension, Tight andSlack Side Tension

During operation of belt drive under load adifference in belt tensions on the entering(tight) and leaving (slack) sides of the

driver pulley is developed. It is calledeffective tension, Te, and represents theforce transmitted from the driver pulley tothe belt

T T Te = −1 2 (11)

where T1 and T2 are the tight and slack sidetensions, respectively.

The driving torque, M (M1 in Fig. 3), isgiven by

Forces Acting in Timing Belt Drives

Figure 3. Power transmission and rotary positioning.

Page 6: Timing Belt Theory

6

M Td

e= ⋅2

(12)

where d (d1 in Fig. 3) is the pitch diameterof the driver pulley.The effective tension generated at thedriver pulley is the actual working forcethat overcomes the overall resistance to thebelt motion. It is necessary to identify andquantify the sum of the individual forcesacting on the belt that contribute to theeffective tension required at the driverpulley.

In power transmission drives (Fig. 3), theresistance to the motion occurs at thedriven pulley. The force transmitted fromthe belt to the driven pulley is equal to Te.

The following expressions for torquerequirement at the driver can be written

ηωηω

η

⋅=

⋅⋅⋅=

⋅=⋅=

1

2

22

12

2

1211 2

P

d

dP

d

dMdTM e

(13)

where M1 is the driving torque, M2 is thetorque requirement at the driven pulley, P2

is the power requirement at the drivenpulley, ωωωω1 and ωωωω2 are the angular speeds ofthe driver and driven pulley respectively,d1 and d2 are the pitch diameters of thedriver and driven pulley respectively, andηηηη is the efficiency of the belt drives(η = −094 0 96. . typically). The angular

speeds of the driver and driven pulley ωωωω1

and ωωωω2 are related in a following form:

ω ω2 11

2= ⋅ d

d(14)

The relationship between the angularspeeds and rotational speeds is given by

ωπ

1 21 2

30,,=

⋅n(15)

where n1 and n2 are rotational speeds ofthe driver and driven pulley in revolutionsper minute [rpm], and ωωωω1 and ωωωω2 areangular velocities of the driver and drivenpulley in radians per second.

In linear positioners (Fig. 4) the main loadacts at the positioning platform (slider). Itconsists of acceleration force Fa (linearacceleration of the slider), friction force ofthe linear bearing, Ff, external force (work

θ

1T

1T

s2F

aiF

sa

Fw

s

Fs1

eT

iT

fF

m

T i

d

F =m a

2L

Midler

T1

1L

2Tv, a

driver

d

Figure 4. Linear positioner - configuration I.

Page 7: Timing Belt Theory

7

load), Fw, component of weight of the slideFg parallel to the belt in inclined drives,inertial forces to accelerate belt, Fab, andthe idler pulley, Fai (rotation)

T F F F F F Fe a f w g ab ai= + + + + + (16)

The individual components of the effectivetension, Te, are given by

F m aa s= ⋅ (17)

where ms is the mass of the slider orplatform and a is the linear accelerationrate of the slider,

F m g Ff r s fi= ⋅ ⋅ ⋅ +μ βcos (18)

where μμμμr is the dynamic coefficient offriction of the linear bearing (usuallyavailable from the linear bearingmanufacturer), Ffi is a load independentresistance intrinsic to linear motion (sealdrag, preload resistance, viscous resistanceof the lubricant, etc.) and ββββ is the angle ofincline of the linear positioner,

F m gg s= ⋅ ⋅ sinβ (19)

Fw L b

gaab

b= ⋅ ⋅ ⋅ (20)

where L is the length of the belt, b is thewidth of the belt, wb is the specific weightof the belt and g is the gravity,

FJ

d

m d

daai

i i b= ⋅ ⋅ = ⋅ +⎛

⎝⎜⎜

⎠⎟⎟ ⋅

2

21

2

2

α(21)

where Ji is the inertia of the idler pulley, ααααis the angular acceleration of the idler, mi

is the mass of the idler, d is the diameter ofthe idler and db is the diameter of the idlerbore (if applicable).

An alternate linear positioner arrangementis shown in Fig. 5. This drive, the slidehouses the driver pulley and two idler rollsthat roll on the back of the belt. The slidermoves along the belt that has both endsclamped in stationary fixtures.

Similar to the linear positioning drive suchas configuration I” the effective tension iscomprised of linear acceleration force Fa,friction force of the linear bearing, Ff,external force (work load), Fw, componentof weight of the slide Fg parallel to the beltin inclined drives and inertial force toaccelerate the idler pulleys, Fai (rotation)

T F F F F Fe a f w g ai= + + + + ⋅2 (22)

The individual components of the effectivetension, Te, are given by

( )F m m m aa s p i= + + ⋅ ⋅2 (23)

2TTis2 mF

Fa

sm

v,a

i

2

2T

fF

s1Fd

Temp1

M

L

s2FdbiT1T

dimi

Fw

1L

T

Figure 5. Linear positioner - configuration II.

Page 8: Timing Belt Theory

8

where ms is the mass of the slider orplatform, mp is the mass of the driverpulley, mi is the mass of the idler rollersand a is the translational acceleration rateof the slider,

( )F m m m g

F

f r s p i

fi

= + + ⋅ ⋅

+

μ β2 cos(24)

where μμμμr is the dynamic coefficient offriction of the linear bearing, Ffi is a loadindependent resistance intrinsic to linearmotion and ββββ is the angle of incline of thelinear positioner,

( )F m m m gg s p i= + + ⋅ ⋅ ⋅2 sinβ (25)

where ββββ is the angle of incline of the linearpositioner,

ad

dm

d

JF

i

bi

iai ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=

⋅⋅⋅=

2

2

12

(26)

where Ji is the inertia of the idler pulleyreflected to the driver pulley, αααα is the

angular acceleration of the driver pulley,mi is the mass of the idler, d is thediameter of the driver, di is the diameter ofthe idler and db is the diameter of the idlerbore (if applicable).

In inclined conveyors in Fig. 6, theeffective tension has mainly two forces toovercome: friction and gravitationalforces. The component of the friction forcedue to the conveyed load, Ff, is given by

F N Wf kk

n

kk

nc c

= ⋅ = ⋅ ⋅= =

∑ ∑μ μ β( ) ( )cos1 1

(27)

where μμμμ is the friction coefficient betweenthe belt and the slider bed, N(k) is acomponent of weight, W(k), of a singleconveyed package perpendicular to thebelt, nc is the number of packages beingconveyed, index k designates the kth pieceof material along the belt and ββββ is theangle of incline. When conveying granularmaterials the friction force is given by

La

Lm

LwLwa

driver

idler

Fs2

s1F

TeM

v,a

Ff

faF

WN

F

(k)(k)

g(k)

1

1L*

*L2

1T

T2

iTT2

T2

Figure 6. Inclined conveyor with material accumulation.

Page 9: Timing Belt Theory

9

F w Lf m m= ⋅ ⋅μ βcos (28)

where wm is weight distribution over a unitof conveying length and Lm is theconveying length.

Some conveying applications includematerial accumulation (see Fig. 6). Herean additional friction component due tothe material sliding on the back surface ofthe belt is present and is given by

( )

( )

F N

W

fa kk

n

kk

n

a

a

= + ⋅

= + ⋅ ⋅

=

=

μ μ

μ μ β

11

11

( )

( )cos

(29)

where na is the number of packages beingaccumulated and μμμμ1 is the frictioncoefficient between belt and theaccumulated material. Similar to theexpression for conveying, Eq. (29) can berewritten as:

( )F w Lfa ma a= + ⋅ ⋅μ μ β1 cos (30)

where wma is weight distribution over aunit of accumulation length and La is theaccumulation length.

The gravitational load, Fg, is thecomponent of material weight parallel tothe belt

F Wg kk

n nc a

= ⋅=

+

∑sin ( )β1

(31)

Note that the Eq. (31) can be alsoexpressed as

( )F w L w Lg m m ma a= ⋅ + ⋅ ⋅sinβ (32)

In vacuum conveyors (Fig. 7) normally, themain resistance to the motion (thus themain component of the effective tension)consists of the friction force Ffv created bythe vacuum between the belt and sliderbed. Ffv is given by

F P Afv v= ⋅ ⋅μ (33)

where P is the magnitude of the vacuumpressure relative to the atmosphericpressure and Av is the total area of thevacuum openings in the slider bed. Auniformly distributed pressure accounts fora linear increase of the tight side tension asdepicted in Fig. 7.

vAP vacuum chamber

Ti

2T

T1

2L*

*L1

fvF v,a

M eTFs1s2F

idler driver

2T T2

Figure 7. Vacuum conveyor.

Page 10: Timing Belt Theory

10

Shaft forces

Force equilibrium at the driver or drivenpulley yields relationships between tightand slack side tensions and the shaftreaction forces Fs1 or Fs2. In powertransmission drives (see Fig. 3) the forceson both shafts are equal in magnitude andare given by

F T T T Ts1 2 12

22

1 2 12, cos= + − ⋅ ⋅ ⋅ θ (34)

where θθθθ1 is angle of belt wrap arounddriver pulley.Note that unlike power transmissiondrives, 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, Fs1, isgiven by

F T T T Ts1 12

22

1 2 12= + − ⋅ ⋅ ⋅ cosθ (35)

whenθ θ1 2 180≠ ≠ (unequal pulleydiameters) and by

F T Ts1 1 2= + (36)

when θ θ1 2 180= = (equal pulley

diameters), where θθθθ2 is angle of belt wraparound idler pulley.

The shaft force at the idler pulley, Fs2,when the load (conveyed material orslider) is moving toward the driver pulleyis given by

F T T T Ts2 22

22

2 2 22= + − ⋅ ⋅ ⋅" " cosθ (37)

whenθ θ1 2 180≠ ≠ or by

F T Ts2 2 2= + " (38)

when θ θ1 2 180= = . T2" is given by

T T Fai2 2" = + (39)

where Fai is given by Eq. (21).

However, when the load is moving awayfrom the driver pulley the shaft force at theidler pulley, Fs2, is given by

F T T T Ts2 12

12

1 1 12= + − ⋅ ⋅ ⋅' ' cosθ (40)

whenθ θ1 2 180≠ ≠ or by

F T Ts2 1 1= + ' (41)

when θ θ1 2 180= = . T1' is given by

T T Fai1 1' = − (42)

Eqs. (39) and (42) assume no friction inthe bearings supporting the idler pulley.Observe that during constant velocitymotion Eq. (38) can be expressed as

F Ts2 22= ⋅ (43)

The same applies to Eq. (41).

In linear positioning drives such as“configuration II” (shown in Fig. 5) theshaft force of the driver pulley, Fs1, isgiven by Eq. (35). The shaft forces on theidler rollers can be expressed by

F T T T T

F T T T T

s

s

2 12

12

1 1 2

1 22

22

2 2 2

2

2

' ' ' '

" " " "

cos

cos

= + − ⋅ ⋅ ⋅

= + − ⋅ ⋅ ⋅

θ

θ(44)

where Fs2' is the shaft force at the idler on

the side of the tight side tension,θ 2' is the

angle of belt wrap around the idler pulley

on the side of the tight side tension, Fs2" is

the shaft force at the idler on the side of

the slack side tension andθ 2" is the angle

of belt wrap around the idler pulley on theside of the slack side tension. Tension

Page 11: Timing Belt Theory

11

forces T1' and T2

" are given by Eqs. (39)and (42).

In the drive shown in Fig. 5 θ1 180= ° and

θ θ2 2 90' "= = ° , and the shaft force at thedriver pulley is given by Eq. (36) and theshaft forces at the idler rollers become

F T T

F T T

s

s

2 12

12

2 22

22

' '

" "

= +

= +(45)

Observe that in reversing drives (likelinear positioners in Fig. 4) the shaft forceat the idler pulley, Fs2, changes dependingon the direction of rotation of the driverpulley. For the same operating conditionsFs2 is larger when the slider moves awayfrom the driver pulley.

To determine tight and slack side tensionsas well as the shaft forces (2 equationswith 3 unknowns), given either the torqueM or the effective tension Te, an additionalequation is still required. This equationwill be obtained from analysis of belt pre-tension methods presented in the nextsection.

Belt Pre-tension

The pre-tension, Ti, (sometimes referred toas initial tension) is the belt tension in anidle drive (Fig. 8). When belt driveoperates under load tight side and slacksides develop. The pre-tension preventsthe slack side from sagging and ensuresproper tooth meshing. In most cases,timing belts perform best when themagnitude of the slack side tension, T2, is10% to 30% of the magnitude of theeffective tension, Te

( )T Te2 01 0 3∈ ⋅. ,..., . (46)

In order to determine the necessary pre-tension we need to examine a particulardrive configuration, loading conditionsand the pre-tensioning method.

To pre-tension a belt properly, anadjustable pulley or idler is required (Figs.3, 4, 6 and 7). In linear positioners whereopen-ended belts are used (Figs. 4 and 5)the pre-tension can also be attained bytensioning the ends of the belt. In Figs. 3to 7 the amount of initial tension isgraphically shown as the distance betweenthe belt and the dashed line.

Although generally not recommended, aconfiguration without a mechanism foradjusting the pre-tension may beimplemented. In this type of design, thecenter distance has to be determined in away that will ensure an adequate pre-tension after the belt is installed. Thismethod is possible because after the initialtensioning and straightening of the belt,there is practically no post-elongation(creep) of the belt. Consideration must begiven to belt elasticity, stiffness of thestructure and drive tolerances.

Drives with a fixed center distance areattained by locking the position of theadjustable shaft after pre-tensioning thebelt (Figs. 3, 4, 6 and 7). The overall beltlength remains constant during driveoperation regardless of the loadingconditions (belt sag and some other minorinfluences are neglected). The reactionforce on the locked shaft generally changesunder load. We will show later that theslack and tight side tensions depend notonly on the load and the pre-tension, butalso on the belt elasticity. Drives with afixed center distance are used in linearpositioning, conveying and powertransmission applications.

Page 12: Timing Belt Theory

12

Drives with a constant slack side tensionhave an adjustable idler tensioning theslack side which is not locked (floating)(Figs. 9 and 10). During operation, theconsistency of the slack side tension ismaintained by the external tensioningforce, Fe. The length increase of the tightside is compensated by a displacement ofthe idler. Drives with a constant slack sidetension may be considered for someconveying applications.

Resolving the Tension Forces

Drives with a constant slack side tensionhave an external load system, which can

be determined from force analysis alone.Force equilibrium at the idler gives

T TF

ie

e≈ =

⋅ ⎛⎝⎜

⎞⎠⎟

2

22

sinθ

(47)

where Fe is the external tensioning forceandθe 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, T1, aswell as the shaft reactions, Fs1 and Fs2.Drives with a fixed center distance (Figs.3, 4, 6 and 7) have an external loadsystem, which cannot be determined from

Figure 9. Power transmission drive with the constant slack side tension.

Figure 10. Vacuum conveyor with the constant slack side tension.

Page 13: Timing Belt Theory

13

force analysis alone. To calculate the belttension forces, T1 and T2, an additionalrelationship is required. This relationshipcan be derived from belt elongationanalysis. Pulleys, shafts and mountingstructures are assumed to have infiniterigidity. Neglecting the belt sag as well assome phenomena with little contribution(such as bending resistance of the belt andradial shifting of the pitch line explainedlater), the total elongation (deformation) ofthe belt operating under load is equal tothe total belt elongation resulting from thebelt pre-tension. This can be expressed bythe following equation of geometriccompatibility of deformation:

Δ Δ Δ

Δ Δ Δ

L L L

L L L

me

i i mi

11 22

1 2

+ + =

+ + (48)

where ΔΔΔΔL11 and ΔΔΔΔL22 are tight and slackside elongation due to T1 and T2,respectively, ΔΔΔΔLme is the total elongationof the belt portion meshing with the driver(and driven) pulley, ΔΔΔΔL1i, ΔΔΔΔL2i and ΔΔΔΔLmi

are the respective deformations caused bythe belt pre-tension, Ti.

For most practical cases the differencebetween the deformations of the belt incontact with both pulleys during pre-tension and during operation is negligible( Δ ΔL Lme mi≈ ). Eq. (48) can be simplified

Δ Δ Δ ΔL L L Li i11 22 1 2+ = + (49)

Tensile tests show that in the tension rangetiming belts are used, stress is proportionalto strain. Defining the stiffness of a unitlong and a unit wide belt as specificstiffness, csp, the stiffness coefficients ofthe belt on the tight and slack side, k1 andk2, are expressed by

k cb

L

k cb

L

sp

sp

11

22

= ⋅

= ⋅

(50)

where L1 and L2 are the unstretchedlengths of the tight and slack sides,respectively, and b is the belt width. Notethat the expressions in Eq. (50) have asimilar form to the formulation for the

axial stiffness of a bar k EA

l= ⋅ where E

is Young's modulus, A is cross sectionalarea and l is the length of the bar.

It is known that elongation equals tension

divided by stiffness coefficient, ΔLT

k= ,

provided the tension force is constant overthe belt length. Thus, Eq. (49) can beexpressed as

T

k

T

k

T

k

T

ki i1

1

2

2 1 2+ = + (51)

Combining expressions for the stiffnesscoefficients, Eq. (50) with Eq. (51) thetight and slack side tensions, T1 and T2,are given by

T T TL

L LT T

L

Li e i e12

1 2

2= ++

= + (52)

and

T T TL

L LT T

L

Li e i e21

1 2

1= −+

= − (53)

where L is the total belt length, L1 and L2

are the lengths of the tight and slack sidesrespectively. Using Eqs. (52) and (53) wecan find the shaft reaction forces, Fs1 andFs2.In practice, a belt drive can be designedsuch that the desired slack side tension, T2,is equal to 10% to 30% of the effective

Page 14: Timing Belt Theory

14

tension, Te (see Eq. (46)), which securesproper tooth meshing during belt driveoperation. Then Eq. (53) can be used tocalculate the pre-tension ensuring that theslack side tension is within 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 Lv overthe vacuum chamber length presented inFig. 7, caused by a linearly increasing belt

tension, equals the mean tension, T , where

TT T= +1 2

2, divided by the stiffness kv

where k cb

Lv spv

= of this belt portion, b is

the belt width, Lv is the length of thevacuum chamber T1 and T2 are thetensions at the beginning and end of thevacuum chamber stretch, respectively.Considering this, T1 and T2 can beexpressed by

T T TL

L

Li e

v

1

2 2max = +

+(54)

T T TL

L

Li e

v

2

1 2min = −

+(55)

Substituting L LLv

1 1 2* = + and

L LLv

2 2 2* = + Eqs. (54) and (55) can be

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 tension

calculated over the conveying andaccumulation length, L Lm a+ . The

distance, L ( 0 < < +L L Lm a ), from thebeginning of the conveying length to thelocation on the belt corresponding to the

mean tension, T , should be calculated. Themodified tight and slack lengths take onthe following form:

L L L L L

L L L

m a1 1

2 2

*

*

= + + −

= +(56)

Tooth Loading

Consider the belt in contact with the driverpulleys 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 kth tooth, thetension forces Tk and Tk+1 are balanced bythe force Ftk at the tooth flank (Fig. 11).The force equilibrium can be written as

T T Fk k tk+ + =+1 0 (57)

Figure 11. Tooth loading.

Page 15: Timing Belt Theory

15

In order to fulfill the equilibriumconditions, the belt tooth inclines andmoves radially outwards as shown in Fig.11. In addition to tooth deformation, toothshifting contributes to the relativedisplacement between belt and pulley,hence to the tooth stiffness.

Theoretically, the tooth stiffness increaseswith increasing belt tension over the tooth,

which has also been confirmedempirically. This results in the practicalrecommendation for linear actuators tooperate under high pre-tension in order toachieve higher stiffness, and hence, betterpositioning accuracy. However, to simplifythe calculations, a constant value for thetooth stiffness, kt, is used in the formulaspresented in the next section.

To determine the positioning error of alinear actuator, caused by an external forceat the slide, the stiffness of tight and slacksides as well as the stiffness of belt teethand cords along the arc of contact have tobe considered. Since tight and slack sidescan be considered as springs acting inparallel, their stiffness add linearly to forma resultant stiffness (spring) constant kr

k k k c bL L

L Lr s= + = ⋅ +⋅1 2

1 2

1 2(58)

In linear positioners (Figs. 12 and 13), thelength of tight and slack side and therefore

the resultant spring constant depend on theposition of the slide. The resultant stiffnessexhibits a minimum value at the positionwhere the difference between tight andslack side length is minimum.

To determine the resultant stiffness of thebelt teeth and cords in the teeth-in-mesharea, km, observe that the belt teeth aredeformed non-uniformly and act in aparallel like arrangement with reinforcingcord sections, but the belt sectionsbetween them are connected in series. Thesolution to the problem is involved and

Figure 12. Position error - linear positioner under static loading condition.

Positioning Error of Timing BeltDrives Due to Belt Elasticity

Page 16: Timing Belt Theory

16

beyond the scope of this paper but theresult is presented in Graph 1. Theordinate is made dimensionless bydividing km by the tooth stiffness kt. Graph1 shows that the gradient of km (indicatedby the slope of the curve) decreases withincreasing number of teeth in mesh.

Defining the ratiok

km

t as the virtual

number of teeth in mesh, zmv,corresponding to the actual number ofteeth in mesh zm the stiffness of the beltand the belt teeth around the arc of contactis

k z km mv t= ⋅ (59)

Observe that the virtual number of teeth inmesh, zmv, remains constant and equals 15for the actual number of belt teeth in meshzm≥ 15. The result of this is that themaximum number of teeth in mesh thatcarry load is 15.

In linear positioners the displacement ofthe slide due to the elasticity of the tightand slack sides has to be added to thedisplacement due to the elasticity of the

belt teeth and cords in the teeth-in-mesharea. Therefore, the total drive stiffness, k,is determined by the following formula:

1 1 1

k k kr m

= + (60)

In drives with a driven pulley (powertransmission drives) an additional term:

1

2kmmust be added to the right-hand side

of Eq. (60). This term introduces thestiffness of the belt and belt teeth aroundthe driven pulley.

The static positioning error, ΔΔΔΔx of a linearpositioner due to the elasticity of the beltcords and teeth is

ΔxF

kst= (61)

where Fst is the static (external) forceremaining at the slide. In Fig. 12, forexample, Fst is comprised of Ff and Fw,

and it is balanced by the static effectivetension Test at the driver pulley.The additional rotation angle, ΔΔΔΔϕϕϕϕ, of thedriving pulley necessary for exactpositioning of the slide is

Figure 13. Following error - linear positioner under dynamic loading condition.

Page 17: Timing Belt Theory

17

Δϕϕ

= =⋅

M

k

Td

k

est

2

(62)

Substituting M from Eq. (12) in Eq. (62)the relationship between linear stiffness, k,and rotational stiffness, kϕϕϕϕ, can beobtained

kd k

ϕ = ⋅2

4(63)

Observe that using pulleys with a largerpitch diameter increases the rotationalstiffness of the drive, but also increases thetorque on the pulley shaft and the inertia ofthe pulley.

Metric AT series belts have been designedfor high performance linear positionerapplications. Utilizing optimized, largertooth section and stronger steel reinforcingtension members these belts provide asignificant increase in tooth stiffness andoverall belt stiffness.

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18 20

Number of Teeth in Mesh, z m

Vir

tual

Num

ber

of T

eeth

in M

esh,

z mv

Graph 1. Correction for the number of teeth in mesh (virtual # of teeth in mesh vs.actual # of teeth in mesh)

Page 18: Timing Belt Theory

GATES MECTROL, INC.9 Northwestern Drive Salem, NH 03079, U.S.A. Tel. +1 (603) 890-1515Tel. +1 (800) 394-4844Fax +1 (603) 890-1616email: [email protected]

Gates Mectrol is a registered trademark of Gates MectrolIncorporated. All other trademarks used herein are the property of their respective owners.

© Copyright 2006 Gates Mectrol Incorporated. All rightsreserved. 10/06

GM_Belt Theory_06_US

www.gatesmectrol.com