V-Belts and V-Ribbed Belts: Transmission of Power with Friction and Positive Engagement, Study notes of Design

Download V-Belts and V-Ribbed Belts: Transmission of Power with Friction and Positive Engagement and more Study notes Design in PDF only on Docsity! 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 “w” (perpendicular to the surface) and the force (parallel to the surface) “f” can be applied to the side of the block before it begins moving, then the coefficient of static friction µ𝑆 = 𝑓 𝑤 after the block begins to slip, the force “f” can be reduced while still overcoming the kinetic friction µ𝐾 = 𝑓 𝑤 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  Combines high power capacity of V-belts with the flexibility of flat belts  With a thin cross section and low weight, high speed ratios (up to 40:1)  Excel on small sheaves, at high speeds and with reverse bends  Generally run smoother than V-belts  Each belt is a single unit, no differential driving occurs (when load is carried unequally in a multi-belt drive)  No separate belts to turn over, slip or interfere with each other  Never any matching problems, the a V-ribbed belt is manufactured as a single unit  Track properly without the need for guides, flanges, crowns or deep grooves V-Belts The V-belt drive friction depends on several factors:  Total tension in the drive, including static and centrifugal tension  Coefficient of friction between the belt and pulley  Angle of contact, dependent on pulley diameters and the center distance  Centrifugal force lifting the belt off the pulley, produced by the rotation of the pulley  Angle of the “V” in the pulley, which wedges the belt in place Wedging Effect  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.  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.  Balancing the forces gives: 𝑅 = 𝑁 sin 𝛽 2 o Therefore, 𝑅′ = 𝑁 sin 𝛽 2⁄ For a 38° included sheave angle, N ≈ 2.92xR  For a flat belt drive, the friction between the belt and pulley for a small segment of belt is 𝐹 = µ × 𝑅, while for a V- belt/sheave with a 38° angle, the friction would be 𝐹 = 2.92 × µ × 𝑅 o The same hub load and stress in the tensile members gives V-belts nearly 3x the friction of flat belts.  For a 38° angle, assuming all other characteristics are the same. Advantages of V-Belts  Reduced shock and vibration transmission  Quite and require no lubrication  Much cheaper, lighter, and cleaner than chains  Somewhat tolerant to misalignment and abuse  Easy installation and minimal maintenance  Simple to change shaft speed by changing pulley diameters 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐷𝑟𝑖𝑣𝑒𝑟 𝑆𝑝𝑒𝑒𝑑 𝑜𝑓 𝐷𝑟𝑖𝑣𝑒𝑛 = 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝐷𝑟𝑖𝑣𝑒𝑛 𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝐷𝑟𝑖𝑣𝑒𝑟  Flexible equipment design, many sizes and cross sections available  High efficiency, up to 98%  Extremely wide horsepower and speed ranges  Prevent severe power overload; can be used as a method of clutching  Belts and pulleys wear gradually, allowing for simple preventative maintenance Centrifugal Force & Tension  In many cases, the speeds at which V-belts are operated can cause significant centrifugal force.  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.  Where ω is the weight of a unit length for a belt of a certain cross section and is equal to ρ*A ; v is the belt speed.  The force on a belt element of length (ΔΘ*r) is given by 𝐹𝐶 = ρ ∗ A ∗ 𝑣2 𝐹𝐶 = [ 𝑤∗𝑟∗ΔΘ 𝑔 ] × 𝑣2 𝑟 = 𝑤∗𝑣2∗ΔΘ 𝑔  The forces must balance, so 2 × 𝑇𝐶 × sin 1 2 ΔΘ = 𝐹𝐶 = ρ ∗ A ∗ 𝑣2 As ΔΘ→0, sin 1 2 ΔΘ = 𝛩 2 Therefore, 𝑇𝐶 = 𝑤∗𝑣2 𝑔 = ρ×A×𝑣2 𝛩  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.  Maximum power can be limited by centrifugal force. o As previously mentioned, the power transmitted by a belt is 𝑃 = 𝑣(𝑇1 − 𝑇2) = 𝑣𝑇𝑒 Additionally, 𝑇1 𝑇2 = 𝑒µ𝛩 Solving 𝑇1 𝑇2 = 𝑒µ𝛩 for T2, 𝑇2 = 𝑇1 𝑒µ𝛩 = 𝑇1𝑒−µ𝛩 Substituting T2 into the power equation gives us 𝑃 = 𝑣(𝑇1 − 𝑇1𝑒−µ𝛩) = 𝑣𝑇1(1 − 𝑒−µ𝛩) o The approximate power equation when considering centrifugal force is 𝑃 = 𝑣(𝑇1 − ρA𝑣2)(1 − 𝑒−µ𝛩) Using this equation and plotting the power vs speed for a nominal set of parameters results in the graph below.  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.  Taking the derivative at the critical speed, 𝑑𝑃 𝑑𝑣 = 0 = (𝑇1 − ρA𝑣𝐶 2)(1 − 𝑒−µ𝛩)  Therefore, 𝑣𝐶 = √ 𝑇1 3ρA⁄ the speed at which maximum power can be delivered.  Critical speed must be calculated before using 𝑃𝑚𝑎𝑥 = 2 3 (𝑣𝐶𝑇1)(1 − 𝑒−µ𝛩)𝑛 where n is the number of belts.  Note that these factors are usually only a concern at very high speeds.  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 P o w e r (W at ts ) Velocity (m/s) 2 Belts 1 Belt 𝑑𝑃 𝑑𝑣 Critical Speed 𝑣𝐶 Creep and Shear  Creep when referring to belt drives is not the same as when used in regards to other engineering applications. The change in length of the belt over time is called stretch. Creep is not the same as slip. Creep is defined as a loss of driven speed as the result of alternate lengthening and shortening of each belt segment as it cycles through the tight and loose side tensions.  Whenever a belt passes around a pulley with different entering and exiting tensions, i.e. any driving or driven pulley, belt creep occurs. The belt will be slightly longer on the tight side than on the loose side. As the belt travels throughout the arc of contact, the length of a belt segment will slowly change from the tight side length to the loose side length as tension decreases. This motion relative to the pulley is creep and leads to some sidewall wear.  A critical assumption of creep theory is that the belt is elastic enough to allow for large enough differences between tight and loose side nominal segment lengths that any shear stress within the belt can be considered negligible. (Developed for round belts with significant stretch and flat belts thin enough for minimal shear stress.)  Shear theory assumes that the stretch experienced by a belt as it alternates between tight and loose side tensions is negligible compared to the shear stress within the belt. (Developed as a result of high strength tensile members limiting cordline stretch.)  The diagram to the left shows the stress within the tensile members as a belt travels around the pulleys shown. “FC” is the tension induced by centrifugal force. “Fi” is the initial static tension of the belt. “ΔF” is tension difference induced by the transmission of torque. Note that the maximum stress in the tensile members occurs at the entry point (tight side) of the small pulley.  The solid curve shows the stress as described by shear theory while the dashed curve shows the stress described by creep theory. Point “A” is the tight side; point “D” is the loose side. All other points are the tangential entry or exit points of the belt on the pulley. The vertical jumps in stress on the shear stress curve indicate the additional stress imposed on the members of the belt when conforming to the curve of the pulley assuming relatively inextensible tensile members. Those tension forces are balanced within the belt by compression stress within the body of the belt. Note that the stress imposed by the smaller pulley is larger due to the smaller radius the belt must conform to. Both cases result in fluctuating stresses within the belt, leading to fatigue and eventual failure. Fatigue  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 𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔 𝑚𝑎𝑥 = 𝑦𝑚𝑎𝑥 ∗ 𝐸 𝑅⁄ where 𝑦𝑚𝑎𝑥 is the maximum distance to the neutral axis, E is the elastic modulus, and R is the radius of the curve.  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.  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.