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Chapter 17 Flexible Mechanical Elements Copyright © 2011 by The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Shigley’s Mechanical Engineering Design 9 th Edition in SI units Richard G. Budynas and J. Keith Nisbett Prepared by Kuei-Yuan Chan Associate Professor of Mechanical Engineering National Cheng Kung University

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17 Flexible Mechanical Elements Chapter Outline 17-1Belts 17-2Flat- and Round-Belt Drives 17-3V Belts 17-4Timing Belts 17-5Roller Chain 17-6Wire Rope 17-7Flexible Shafts

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3 Belts Most flexible elements do not have an infinite life. Characteristics of belts include They may be used for long center distances. Except for timing belts, there is some slip and creep, and so the angular-velocity ratio between the driving and driven shafts is neither constant nor exactly equal to the ratio of the pulley diameters. In some cases an idler or tension pulley can be used to avoid adjustments in center distance that are ordinarily necessitated by age or the installation of new belts. Flat-belt geometry.

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4 Belts (Cont.) Belt drives are either reversing or nonreversing. The shafts need not be at right angles as in a flat-belt drive with out- of-plane pulleys. In contrast with flat belts, V belts are used with similar sheaves and at shorter center distances. For timing belts, no initial tension is necessary, so that fixed-center drives may be used. The restriction on speeds has also been eliminated.

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5 Flat-Belt Drivers A flat-belt drive has an efficiency of about 98 percent, which is about the same as for a gear drive. On the other hand, the efficiency of a V-belt drive ranges from about 70 to 96 percent. When an open-belt drive is used, the contact angles are found to be The length of the belt is found by summing the two arc lengths with twice the distance between the beginning and end of contact. For crossed belt, the angle of wrap is the same for both pulleys and is The belt length for crossed belts is found to be

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6 Mechanics of Flat-Belt Drives Assuming that the friction force on the belt is proportional to the normal pressure along the arc of contact, a relationship between the tight side tension and slack side tension, follows F c is found as The tight side tension F 1 and the loose side tension F 2 on a pulley have the following additive components: A change in belt tension due to friction forces between the belt and pulley will cause the belt to elongate or contract and move relative to the surface of the pulley.

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7 Mechanics of Flat-Belt Drives (Cont.) Solving for the initial tension, we have The initial tension needs to be sufficient so that the difference between the F 1 and F 2 curve is 2T/D. Initial tension is the key to the functioning of the flat belt as intended. The weight of the belt itself can also provide the initial tension resulting in a dip. where d = dip, in L = center-to-center distance, ft w = weight per foot of the belt, lbf/ft F i = initial tension, lbf

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8 Analysis of Flat-Belt Drives The transmitted horsepower is given by Corrections on allowable tension give where (F 1 ) a = allowable largest tension, lbf b = belt width, in F a = manufacturer’s allowed tension, lbf/in C p = pulley correction factor (Table 17–4) C v = velocity correction factor The steps in analyzing a flat-belt drive can include Find exp(f φ) from belt-drive geometry and friction From belt geometry and speed find F c From T = H nom K s n d /n find necessary torque From torque T find the necessary (F 1 ) a − F 2 = 2T /D Find F 2 from (F 1 ) a − [(F 1 ) a − F 2 ] From Eq. (i) find the necessary initial tension Fi Check the friction development, f ′ < f. Use Eq. (17–7) solved for f ′: Find the factor of safety from n f s = H a /(H nom K s )

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9 Flat-Metal Belts Thin metal belts exhibit High strength-to-weight ratio Dimensional stability Accurate timing Usefulness to temperatures up to 700°F Good electrical and thermal conduction properties The selection of a metal flat belt can consist of the following steps:

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10 V Belts The cross-sectional dimensions of V belts have been standardized by manufacturers, with each section designated by a letter of the alphabet for sizes in inch dimensions. To specify a V belt, give the belt-section letter, followed by the inside circumference in inches. The pitch length is obtained by adding a quantity to the inside circumference. For best results, a V belt should be run quite fast: 20 m/s is a good speed. Trouble may be encountered if the belt runs much faster than 25 m/s or much slower than 5 m/s.

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11 Analysis of V Belts The analysis of a V-belt drive can consist of the following steps: Find V, Lp, C, φ, and exp(0.5123φ) Find H d, H a, and N b from H d /H a and round up Find F c, F, F 1, F 2, and F i, and n f s Find belt life in number of passes, or hours, if possible Pitch length : Allowable Power : where H a = allowable power, per belt, Table 17–12 K 1 = angle-of-wrap correction factor, Table 17–13 K 2 = belt length correction factor, Table 17–14 Design Power : where H nom is the nominal power, K s is the service factor given in Table 17–15, and n d is the design factor. Lifetime in hours :

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12 Timing Belts A timing belt does not stretch appreciably or slip and consequently transmits power at a constant angular-velocity ratio. Timing belts can operate over a very wide range of speeds, have efficiencies in the range of 97 to 99 percent, require no lubrication, and are quieter than chain drives. The five standard inch-series pitches available are listed in Table 17–18 with their letter designations. The design and selection process for timing belts is similar to that for V belts.

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13 Roller Chain Basic features of chain drives include a constant ratio, since no slippage or creep is involved; long life; and the ability to drive a number of shafts from a single source of power. The pitch diameter of the sprocket by D can be written The chain velocity V is defined as the number of feet coming off the sprocket per unit time. where N = number of sprocket teeth, p = chain pitch, in,n = sprocket speed, rev/min The maximum exit velocity of the chain is and the minimum exit velocity is

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14 Analysis of Roller Chains The chordal speed variation is For smooth operation at moderate and high speeds it is considered good practice to use a driving sprocket with at least 17 teeth and no more than 12 teeth. The maximum speed (rev/min) for a chain drive is limited by galling between the pin and the bushing. where F is the chain tension in pounds. Lubrication of roller chains is essential in order to obtain a long and trouble-free life.

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15 Wire Rope Wire rope is made with two types of winding, the regular lay and the lang-lay. A wire rope tension giving the same tensile stress as the sheave bending is called the equivalent bending load F b, given by A wire rope may fail because the static load exceeds the ultimate strength of the rope. For an average operation, use a factor of safety of 5. Factors of safety up to 8 or 9 are used if there is danger to human life and for very critical situations.

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16 Wire Rope (Cont.) The fatigue tensile strength in pounds for a specified life F f is where (p /S u ) = specified life, from Fig. 17–21 S u = ultimate tensile strength of the wires, psi D = sheave or winch drum diameter, in d = nominal wire rope size, in The equivalent bending load F b is where E r = Young’s modulus for the wire rope, Table 17–24 or 17–27, psi d w = diameter of the wires, in A m = metal cross-sectional area, Table 17–24 or 17–28, in2 D = sheave or winch drum diameter, in The static factor of safety n s is and the fatigue factor of safety n f is

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