Belt Drives

Speed reducers are employed almost invariably to amplify torque rather than to reduce speed. The two most common speed reduction mechanisms in industry are belts (usually V-belts) and gears - though chains, hydrostatic transmissions or other drives may be used.

The efficiencies of belts are generally less than those of gears - that is why belts are not found in the main drive train of road vehicles where fuel economy is critical. However, shock absorption capacity, distance between shaft centres, accuracy required of shafts and mountings, tolerable vibration levels and so on may also need to be considered. Advantage of belt drives lies here

The transfer of power in a belt drive on the other hand relies critically on friction. The tensions   Fmin & Fmax in the two   strands ( the nominally straight parts of the belt not in contact with the pulleys ) cause a normal pressure over the belt- pulley contact, and it is the corresponding distributed friction whose moment about the pulley centre equilibrates the shaft torque   T - provided gross slip of the belt on the pulley surface does not occur due to friction breakaway.

The torque ratio equals the ideal ratio ( as may be seen from the free bodies ), but   creep results in the speed ratio being less than ideal. Creep - not to be confused with gross slip - is due to belt elements changing length as they travel between   Fmin & Fmax, and since the pulley is rigid then there must be relative motion between belt element and pulley.

Historically,   flat belts made from joined hides were first on the scene, however modern flat belts are of composite construction with cord reinforcement. They are particularly suitable for high speeds.

Classical banded ( ie. covered )   V-belts comprise cord tensile members located at the pitch line, embedded in a relatively soft matrix which is encased in a wear resistant cover..

Each component of a V-belt performs a particular function. The main load- carrying elements are the tensile members, often in the form of longitudinally stiff nylon/rayon cords located near the centroidal axis of the belt's cross-section,

These cords are embedded in a relatively soft elastomeric matrix whose main purpose is to channel the load from the contacts with the groove sides into the tensile members. The groove semi-angle lies usually in the range   17o ≤ β ≤ 19o. It should be noted that there is a gap ie. no contact at the bottom of the groove

The wedging action of a V-belt in a pulley groove results in a drive which is more compact than a flat belt drive, but short centre V-belt drives are not conducive to shock absorption.

Modern materials allow   cut belts to dispense with a separate cover. The belt illustrated also incorporates slots on the underside known as   cogging which alleviate deleterious bending stresses as the belt is forced to conform to pulley curvature. Cogging should not be confused with the teeth on timing belts

Synchronous or   timing belt drives are positive rather than friction drives as they rely on gear- like teeth on pulley and belt enabled by modern materials and manufacturing methods. They are mentioned here only for completeness - we shall not examine them further.

V-belts are available in a number of standard cross-sectional sizes, designated in order of increasing size A, B, etc, while wedge belts are designated variously as SPA, SPB, etc (or α, β etc in the US). Each size is suitable for a particular power range as suggested by the carpet diagrams. The regions of applicability for the various sizes in these diagrams overlap substantially.

As the belts are endless, only certain discrete standard pitch lengths are manufactured. The power demand very often necessitates a number of matched belts on multi-grooved pulleys, as illustrated above. Discrete dimensions apply also to off- the- shelf pulleys, which are available only with certain recommended pitch diameters and number of grooves.

A special pulley may be manufactured of course - but would cost more than a mass- produced commercial product. A pulley is referred to by its pitch diameter - other dimensions including its OD are available from suppliers' manuals which should be consulted also for local availability.

A drive comprises two pulleys, potentially with different values of   μ, β, and θ   - although the slack and tight strand tensions are common to both pulleys. The maximum tension ratio which the drive can support without slip on   either of the two pulleys is therefore :

3) ( Fmax - ρv2 ) / ( Fmin - ρv2 )  

≤ e ( fθ)min where   f ≡ μ cosecβ     and ∗( θ)min   =   min (   (fθ)1 , (fθ)2 )

If   f is the same for both pulleys, as is usual when both pulleys are grooved, then the smaller pulley will be limiting since   θ1 < θ2 , but this does not necessarily apply to   V-flat drives   (see below).

V-belts are available in a number of standard cross-sectional sizes, designated in order of increasing size A, B, etc, while wedge belts are designated variously as SPA, SPB, etc (or α, β etc in the US). Each size is suitable for a particular power range as suggested by the carpet diagrams. The regions of applicability for the various sizes in these diagrams overlap substantially.

As the belts are endless, only certain discrete standard pitch lengths are manufactured. The power demand very often necessitates a number of matched belts on multi-grooved pulleys, as illustrated above. Discrete dimensions apply also to off- the- shelf pulleys, which are available only with certain recommended pitch diameters and number of grooves. A special pulley may be manufactured of course - but would cost more than a mass- produced commercial product. A pulley is referred to by its pitch diameter - other dimensions including its OD are available from suppliers' manuals which should be consulted also for local availability.

Drive selection

Input to the V-belt drive selection process is the drive specification which defines :

the power capacity of the drive and the small pulley's shaft speed (rev/s)

Limits of pulley centre distance - once again the designer may have to define a tolerable band around a sole nominal value or, lacking even this, around 2D1 √( R+1) Limits of acceptable belt fatigue life: it is rare for these to be stated explicitly, and the designer may have to adopt limits in line with the duty (see the table here) and economic belt replacement frequency, or with the usual commercial expectation of 26 kh.

Drive selection involves choice of size (B, SPA etc) number and pitch length of the belt(s), and diameters of the two pulleys. As the design equations cannot be solved in closed form, and since many parameters are discrete, a trial- and- error selection process must be adopted. The steps illustrated below form a useful framework for this, and should be read from top to bottom.

The process starts with the choice of a suitable belt size - aided by a carpet diagram - with belt properties from Table 1 and with corresponding lists of readily available belt lengths and pulley diameters.

If a single V-belt is inadequate for power transmission then multiple belts and corresponding multi- grooved pulleys are necessary - this pulley is equipped with a tapered bush for axle clamping without the stress concentration associated with a key.

The rather extreme short-centre drive on the left illustrates a problem with multiple belts - how to ensure equitable load sharing between flexible belts whose as-manufactured dimensional tolerances are significantly looser than those of machined components for example. Two types of belt for avoiding mismatched lengths are shown :

The smaller pulley diameter is next selected, and is usually taken as small as possible to reduce potential pulley costs - consistent with acceptable belt life and effectivness. Belt speed and bounds on large pulley diameter follow from the desired speed ratio limits and equations ( 1) ie.   Rmin D1 ≤   D2 ≤   Rmax D1 .

If feasible, a large pulley   D2 is next chosen from the diameters available within this range, and belt pitch length limits   Lmin , Lmax corresponding to the stated centre distance limits   Cmin , Cmax worked out from ( 2a). If feasible, a belt length   Lmin ≤   L ≤   Lmax is next chosen from the lengths available, and the geometry finalised via ( 2b), ( 2c).

The sole design parameter which now remains to be chosen is the number of belts,   z. Rather than follow the scheme by working out limits to the number of belts consistent with stated life constraints, it is probably easier just to choose various belt numbers on a trial- and- error basis and evaluate belt life from ( 5a) until a suitable life eventuates.

Fatigue loading and Belt life

Every cross-section of the belt is subjected alternately to the strand tensions   Fmax and   Fmin and therefore undergoes fatigue. These tensions at least must be detemined before fatigue can be addressed.

In addition to the direct stress in the belt due to the peak tension,   Fmax , belt wrap around a pulley causes bending stresses. Invoking elastic bending theory,   σbending max = ymax E/R where R is the bending radius here. Bending stress is thus inversely proportional to pulley diameter   D - and although the belt will probably not behave elastically, this proportionality is a reasonable measure of damaging bending effects.

fat = endurance limit corresponding to the number of loadings No determined from the mean fatigue curve

max = maximum stress induced in the belt

u. = bending frequency of the belt = V/Lp

a = number of pulleys in the drive

Belt Life Determination

Modified procedure enables to determine the belt life by a simple relation using Minor’s cumulative damage theory in which at each amplitude of loading the damage is proportional to the cycle ratio which is the ratio of actual number of stressing to the total number of stressing for failure at that amplitude

So for each revolution two stress peaks are encountered and hence the total damage D=

1 2

b b

P

K KN

T T

Here T1 is the first stress peak and is given by

F1+(Fb)1 = F1+ (Kb)/d

And T2 = F2+(Fb)2 = F2+ (Kb)/D

Kb from 17-16 Table

The estimated life in hours =

63.6(10 )P PN L

V

Design Considerations

(P/z),load power per belt

(v)     velocity    

(D1 , D2 ), dimensions pulley diameters

(L) belt length and

(kθ) friction/arc factor        

material ie. belt section and corresponding properties (F, M, ρ, m) from Table 1        

degree of safety   or more appropriately in a fatigue situation, the drive life (T)

Specification Set

Belt Specification –Section , Inside length

Number of belts to be used j or Nb

Sheave diameters d and D

Center to Center distance C (actual)

Initial Tension Fi

Decision Set

Small sheave dia d

Large sheave dia D

Center distance

Belt cross section

Belt Length

Number of belts

Adequacy assessment

Factor of safety meets the minimum goal

Belt speed in the optimum range

Center distance variation within acceptable limit

Belt life meets the durability goal

Provision to sustain the initial tension

        load power per belt (P/z), velocity (v)

        dimensions

pulley diameters (D1 , D2 ), belt length (L) and friction/arc factor (kθ)

        material ie. belt section and corresponding properties (F, M, ρ, m) from Table 1

        degree of safety  

or more appropriately in a fatigue situation, the drive life (T)

The design power input to the selection process must incorporate a duty factor to allow for shock, high starting torques and other expected non-uniformities :-

( iii)       design power   =   actual nominal power duty factor (from the ∗table below eg.)

The   basic rating,   Po, is defined as the power capacity of a single belt of a particular size in a 1:1 drive, whose length is the reference length and whose life is the reference life. Thus, from ( 5b) : P  =kπ v [   F ( Lo / 2 v To )1/m - M/D   - ρv2 ]   ;     and using ( iv) :-

( v) Po     v [ K1 - K2/D - K3v2 - K4 log(v) ] ≅

The steps in designing with typical belt tables are as follows :-

Determine the design power

design power = actual power service ∗factor    

where the tabulated   service factor is a combination of the above duty factor, and a   life factor which caters for lives other than the reference life.

Select an appropriate belt size, guided by the carpet diagram of design power versus small pulley speed. This diagram is indicative, not prescriptive.

Ascertain the   power rating of a single belt of that size; this is a tabulated function of belt speed and small pulley diameter as noted above, the tabulation also making provision for the large pulley diameter by approximations similar to ( iv).

The   arc factor, again tabulated, caters for contact arcs other than 180o on the small pulley; it is just the ratio   kθ /kπ . A   length factor reflects the effect of a belt length other than the reference length; again the factor is tablulated. The number of necessary belts of that particular size follows from :       belt number   ≥   design power / ( belt power rating arc factor length ∗ ∗factor )

The general equation of belt service life can be written thu

x 3600ua H = a7N0

whence the belt service life is

N0 (--“h (14.20)

3600ua \maxJ

where N0 = base of fatigue tests assumed equal to 1O cycles

fat = endurance limit corresponding to the number of loadings No determined from the mean fatigue curve

max = maximum stress induced in the belt

u. = bending frequency of the belt = V/Lp

a = number of pulleys in the drive