2 17 Flexible Mechanical Elements Chapter Outline17-1Belts17-2Flat- and Round-Belt Drives17-3V Belts17-4Timing Belts17-5Roller Chain17-6Wire Rope17-7Flexible Shafts1-1Design1-2Mechanical Engineering Design
3 Belts Most flexible elements do not have an infinite life. Characteristics of belts includeThey 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.
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.
5 Flat-Belt DriversA 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 beThe 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 isThe belt length for crossed belts is found to be
6 Mechanics of Flat-Belt Drives 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.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, followsFc is found asThe tight side tension F1 and the loose side tension F2 on a pulley have the following additive components:
7 Mechanics of Flat-Belt Drives (Cont.) Solving for the initial tension, we haveThe initial tension needs to be sufficient so that the difference between the F1 and F2 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, inL = center-to-center distance, ftw = weight per foot of the belt, lbf/ftFi = initial tension, lbf
8 Analysis of Flat-Belt Drives The transmitted horsepower is given byFind exp(f φ) from belt-drive geometry and frictionFrom belt geometry and speed find FcFrom T = HnomKsnd /n find necessary torqueCorrections on allowable tension giveFrom torque T find the necessary (F1)a − F2 = 2T /Dwhere (F1)a = allowable largest tension, lbfFind F2 from (F1)a − [(F1)a − F2]From Eq. (i) find the necessary initial tension Fib = belt width, inFa = manufacturer’s allowed tension, lbf/inCheck the friction development, f ′ < f . Use Eq. (17–7) solved for f ′:Cp = pulley correction factor (Table 17–4)Cv = velocity correction factorFind the factor of safety fromThe steps in analyzing a flat-belt drive can includenf s = Ha /(HnomKs)
9 Flat-Metal Belts Thin metal belts exhibit High strength-to-weight ratioDimensional stabilityAccurate timingUsefulness to temperatures up to 700°FGood electrical and thermal conduction propertiesThe selection of a metal flat belt can consist of the following steps:
10 V BeltsThe 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 .
11 Analysis of V BeltsThe 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 upFind Fc, F, F1, F2, and Fi , and nf sFind belt life in number of passes, or hours, if possiblePitch 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 17–14Design 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 :
12 Timing BeltsA 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.
13 Roller ChainBasic 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 writtenThe 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/minThe maximum exit velocity of the chain isand the minimum exit velocity is
14 Analysis of Roller Chains The chordal speed variation isFor 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.
15 Wire RopeWire 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 byA 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.
16 Wire Rope (Cont.)The fatigue tensile strength in pounds for a specified life Ff iswhere (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, inThe equivalent bending load Fb iswhere 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, inThe static factor of safety ns isand the fatigue factor of safety nf is