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Chapter 17Flexible Mechanical Elements

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Shigley’s Mechanical Engineering Design9th Edition in SI units

Richard G. Budynas and J. Keith Nisbett

17 Flexible Mechanical Elements

Chapter Outline

17-1 Belts

17-2 Flat- and Round-Belt Drives

17-3 V Belts

17-4 Timing Belts

17-5 Roller Chain

17-6 Wire Rope

17-7 Flexible Shafts

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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.

Belts

• Flat-belt geometry.

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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.

Belts (Cont.)

Non-reversing open belt

Reversing crossed belt

Reversing open belt

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Flat-Belt Drivers

• 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

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

• Fc is found as

• The tight side tension F1 and the loose side tension F2 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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Mechanics of Flat-Belt Drives

• Fc is found as

• The tight side tension F1 and the loose side tension F2 on a pulley have the following additive components:

• Velocity:

• Weight:

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 F1 and F2curve 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, inL = center-to-center distance, ftw = weight per foot of the belt, lbf/ftFi = initial tension, lbf

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Analysis of Flat-Belt Drives

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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Flat-Metal Belts

Flat-Metal Belts

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Flat-Metal Belts

Flat-Metal Belts

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Flat-Metal Belts

Flat-Metal Belts

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Flat-Metal Belts

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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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 Hd , Ha , and Nb from Hd /Ha and round up� Find Fc, F, F1, F2, and Fi , and nf s� Find belt life in number of passes, or hours, if possible

• Pitch length :

• Allowable Power : where Ha = allowable power, per belt, Table 17–12K1 = angle-of-wrap correction factor, Table 17–13K2 = belt length correction factor, Table 17–14• Design Power :where Hnom is the nominal power, Ks is the service factor given in Table 17–15, and nd is

the design factor.• Lifetime in hours :

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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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

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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Wire Rope

Wire Rope

(Strands)

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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 Fb, 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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Wire Rope (Cont.)

• The fatigue tensile strength in pounds for a specified life Ff is

where (p /Su) = specified life, from Fig. 17–21Su = ultimate tensile strength of the wires, psiD = sheave or winch drum diameter, ind = nominal wire rope size, in

• The equivalent bending load Fb is

where Er = Young’s modulus for the wire rope, Table 17–24 or 17–27, psidw = diameter of the wires, inAm = metal cross-sectional area, Table 17–24 or 17–28, in2D = sheave or winch drum diameter, in

• The static factor of safety ns is

and the fatigue factor of safety nf is

[1] Budynas, Nusbett, “Shigley’s Mechanical Engineering Design”, 9th Edition, Chapter 17.

[2] Hamrock, Schmid, Jacobson, “Fundamentalsof Machine Elements”, 2nd Edition, Chapter 19.

References