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Belt Theory
Friction Drives - V-Belts and V-Ribbed Belts
The figure to the left illustrates the basic idea of static friction. If the block weighs โwโ (perpendicular to the surface) and the force (parallel to the surface) โfโ can be applied to the side of the block before it begins moving, then the coefficient of static friction ฮผ๐ = ๐๐ค after the block begins to slip, the force โfโ can be reduced while still overcoming the kinetic friction ฮผ๐พ = (^) ๐ค๐ V-belts and flat belts employ the use of friction to transmit power while timing belts physically engage the pulley via teeth and grooves. A friction drive consists of two shafts connected by a belt that is drawn tight enough to grip each shaft and transmit torque. Generally, each shaft is fitted with a pulley which the belt is situated in. As the drive pulley turns, the belt transmits power to the driven pulley. The figure on the following page shows two views of a dual V-belt drive arrangement.
V-Ribbed Belts
V-ribbed, or Poly-V belts, are flat belts with a series of ribs running longitudinally along the driving face that fit into grooves in a sheave. These belts are relatively thin with a well-supported tensile member. They are thinner and weigh less than V-belts. Poly-V belts are a single unit utilizing an uninterrupted, full width tensile member which is completely supported. The drive load is evenly distributed across the width of the sheave, equalizing belt stress. A Poly-V belt also resists seating in the grooves, so speed ratios remain more consistent and output speed remains more uniform. These belts utilize the entire width of the pulley face, allowing more compact drives. Poly-V belts allow narrowing mounting clearances, need less center distance adjustment and require less take-up for tensioning. Narrow sheaves of smaller diameter can be used without sacrificing power capacity while reducing weight and increasing efficiency.
Advantages ๏ท Combines high power capacity of V-belts with the flexibility of flat belts ๏ท With a thin cross section and low weight, high speed ratios (up to 40:1) ๏ท Excel on small sheaves, at high speeds and with reverse bends ๏ท Generally run smoother than V-belts ๏ท Each belt is a single unit, no differential driving occurs (when load is carried unequally in a multi-belt drive) ๏ท No separate belts to turn over, slip or interfere with each other ๏ท Never any matching problems, the a V-ribbed belt is manufactured as a single unit ๏ท Track properly without the need for guides, flanges, crowns or deep grooves
V-Belts
The V-belt drive friction depends on several factors: ๏ท Total tension in the drive, including static and centrifugal tension ๏ท Coefficient of friction between the belt and pulley ๏ท Angle of contact, dependent on pulley diameters and the center distance ๏ท Centrifugal force lifting the belt off the pulley, produced by the rotation of the pulley ๏ท Angle of the โVโ in the pulley, which wedges the belt in place
Wedging Effect ๏ท V-Belts largest advantage over flat-belt drives is the utilization of wedging to increase the total friction between the belt and pulley without increasing the hub load or effective tension. ๏ท Tensioning a V-belt will cause the tensile members of the belt to exert resulting force R on the body of the belt. The body of the belt in turn exerts a total normal force N on the pulley groove. But because the bottom of the belt does not contact the bottom of the pulley, the sidewalls are where the force is exerted. ๏ท Balancing the forces gives: ๐
= ๐ sin ๐ฝ 2 o Therefore, ๐
โฒ^ = ๐ sin๐ฝ โ 2 For a 38ยฐ included sheave angle, N โ 2.92xR
๏ท For a flat belt drive, the friction between the belt and pulley for a small segment of belt is ๐น = ฮผ ร ๐
, while for a V- belt/sheave with a 38ยฐ angle, the friction would be ๐น = 2.92 ร ฮผ ร ๐
o The same hub load and stress in the tensile members gives V-belts nearly 3x the friction of flat belts. ๏ง For a 38ยฐ angle, assuming all other characteristics are the same.
Advantages of V-Belts ๏ท Reduced shock and vibration transmission ๏ท Quite and require no lubrication ๏ท Much cheaper, lighter, and cleaner than chains ๏ท Somewhat tolerant to misalignment and abuse ๏ท Easy installation and minimal maintenance
๏ท Simple to change shaft speed by changing pulley diameters ๐๐๐๐๐ ๐๐ ๐ท๐๐๐ฃ๐๐๐๐๐๐๐ ๐๐ ๐ท๐๐๐ฃ๐๐ = ๐ท๐๐๐๐๐ก๐๐ ๐๐ ๐ท๐๐๐ฃ๐๐๐ท๐๐๐๐๐ก๐๐ ๐๐ ๐ท๐๐๐ฃ๐๐
๏ท Flexible equipment design, many sizes and cross sections available ๏ท High efficiency, up to 98% ๏ท Extremely wide horsepower and speed ranges ๏ท Prevent severe power overload; can be used as a method of clutching ๏ท Belts and pulleys wear gradually, allowing for simple preventative maintenance
Transmission of Force - V-Belts and V-Ribbed Belts
If a belt is tensioned between two pulleys which are free to turn, the tension along the length of the belt is constant and equal to โTโ. However, when transmitting torque, the driven pulley will resist the motion and the driver pulley will have to pull on the โTight Sideโ in order to exert torque on the driven pulley. The tension in the โTight Sideโ can be called โT 1 โ. The total tension โTTโ must remain constant. Thus, if a โTight Sideโ is formed a โLoose Sideโ must also be formed. This โLoose Sideโ tension can be called โT 2 โ. Therefore, if โTTโ remains constant then ๐๐ = ๐ 1 + ๐ 2 at all times. The effectiveness of the drive depends on maintaining frictional contact between the belt and the pulleys.
๏ท Power is the rate of doing work. Power transmission by a V-belt is dependent on the effective tension โTeโ and the belt speed. The general power equation is ๐ = ๐ฃ(๐ 1 โ ๐ 2 ) = ๐ฃ๐๐ ๏ท When the limiting friction is developed around the arc of contact, the belt can transmit the maximum torque to the pulley or vice versa. ๏ท โT 1 โ & โT 2 โ are the tight and loose side tensions, respectively. โRโ is the resulting force perpendicular to the belt at a point. ฮผ is the coefficient of friction. โTโ is the tension at a given point. For simplification, โฮผRโ is not accounting for the wedging effect of a belt. ๏ท Analyzing the forces in the diagram tangentially for a small ฮฮ under low speed no-slip conditions leads to the following: o ฮผ๐
= (๐ + ฮT) cos 12 ฮฮ โ T cos 12 ฮฮ o Thus, taking the limit as ฮฮโ0 gives us ฮผ๐
= ฮT ๏ท Radially:
o ๐
= (๐ + ฮT) sin 12 ฮฮ + ๐ sin 12 ฮฮ o As ฮฮโ0, sin ฮฮ = ฮ and ฮT = 0 o Taking the limit, ๐
= ๐ฮฮ o Substituting with ฮผ๐
= ฮT, ฮ๐๐ = ฮผ๐ฉ o Integrating over the entire Angle of Contact (ฮธ), ln (๐ ๐^1 2 ) = ฮผ๐ฉ which gives ๐ ๐^12 = ๐ฮผ๐ฉ^ This formula governs the amount of power which can be transmitted before the belt slips.
๏ท This is the effective tension ratio between the tight and loose sides of the belt. For a belt drive between pulleys of different diameters, the angle of contact of the smaller pulley should be used.
Centrifugal Force & Tension
๏ท In many cases, the speeds at which V-belts are operated can cause significant centrifugal force. ๏ท Centrifugal force represents the effects of inertia which arise in connection with rotation and are experienced as an outward force away from the center of rotation. ๏ท Where ฯ is the weight of a unit length for a belt of a certain cross section and is equal to ฯ*A ; v is the belt speed.
๏ท The force on a belt element of length (ฮฮ*r) is given by ๐น๐ถ = ฯ โ A โ ๐ฃ^2
๐น๐ถ = [๐คโ๐โฮฮ๐ ] ร ๐ฃ
2 ๐ =^
๐คโ๐ฃ^2 โฮฮ ๐ ๏ท The forces must balance, so
2 ร ๐๐ถ ร sin 12 ฮฮ = ๐น๐ถ = ฯ โ A โ ๐ฃ^2 As ฮฮโ0, sin 12 ฮฮ = ๐ฉ 2
Therefore, ๐๐ถ = ๐คโ๐ฃ
2
๐ =^
ฯรAร๐ฃ^2 ๐ฉ ๏ท These additional forces must be accounted for when designing a drive. o The centrifugal force FC will reduce the amount of friction between the belt and the pulley, which could lead to excessive slip in extreme situations. o The centrifugal tension TC will increase the total tension in the belt. If a belt was installed with too much tension, this additional tension can lead to premature belt failure due to excessive stress in the tensile members. ๏ท Maximum power can be limited by centrifugal force.
o As previously mentioned, the power transmitted by a belt is ๐ = ๐ฃ(๐ 1 โ ๐ 2 ) = ๐ฃ๐๐ Additionally,
๐ 1 ๐ 2
= ๐ฮผ๐ฉ^ Solving ๐ ๐^1
2 = ๐ฮผ๐ฉ^ for T 2 , ๐ 2 = (^) ๐๐ฮผ๐ฉ^1 = ๐ 1 ๐โฮผ๐ฉ^ Substituting T 2 into the power equation gives us ๐ = ๐ฃ(๐ 1 โ ๐ 1 ๐โฮผ๐ฉ) = ๐ฃ๐ 1 (1 โ ๐โฮผ๐ฉ)
o The approximate power equation when considering centrifugal force is ๐ = ๐ฃ(๐ 1 โ ฯA๐ฃ^2 )(1 โ ๐โฮผ๐ฉ) Using
this equation and plotting the power vs speed for a nominal set of parameters results in the graph below. ๏ท At zero speed, the power is obviously zero. Initially, one might think that, assuming a constant torque, as the speed increases, the power able to be delivered will continue to increase. However, the centrifugal force of the belt will begin to decrease the effective normal force between the belt and the pulley, resulting in less traction. The belt will start to slip.
๏ท Taking the derivative at the critical speed, ๐๐๐๐ฃ = 0 =
(๐ 1 โ ฯA๐ฃ๐ถ^2 )(1 โ ๐โฮผ๐ฉ)
๏ท Therefore, ๐ฃ๐ถ = โ๐^1 โ 3ฯA the speed at which
maximum power can be delivered. ๏ท Critical speed must be calculated before using
๐๐๐๐ฅ = 23 (๐ฃ๐ถ ๐ 1 )(1 โ ๐โฮผ๐ฉ)๐ where n is the number of belts. ๏ท Note that these factors are usually only a concern at very high speeds. ๏ท While high speeds do not generally reduce the ability of a timing belt to transmit power, high tensions can result from large centrifugal forces.
0
25000
50000
75000
100000
0 20 40 60 80
Power (Watts)
Velocity (m/s)
2 Belts
1 Belt
๐๐ ๐๐ฃ
Critical Speed ๐ฃ๐ถ
Fatigue
๏ท As indicated above, the maximum stress in the belt is at the tight side entry point of the small pulley. The stress is a combination of the tight side tension and the bending stresses. o Elastic bending theory tells us that ๐๐๐๐๐๐๐๐ ๐๐๐ฅ = ๐ฆ๐๐๐ฅ^ โ ๐ธ โ๐
where ๐ฆ๐๐๐ฅ is the maximum distance to the neutral axis, E is the elastic modulus, and R is the radius of the curve. ๏ท The cyclical stress (seen in the above diagram) is what leads to the fatigue failure seen in belts. During each revolution of the belt, each infinitesimal belt segment goes through a double-peaked fatigue cycle (Peaks at B & F, minimum from C-E). Generally, cyclically loaded components follow an approximately log-log linear relationship between the life and load of the component. ๏ท Doubling the load a belt carries will not lead to half of the original life. The new life of the belt could be as little as 5%-10% of the original.
