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Example 6. An alloy of density 26.6 kN/m3 has an allowable stress of 70 MPa. For a flywheel speed of 200 rad/s, what is the largest possible average radius, overlooking the additional stress caused by the spokes From Eq. (7.31), V= g = 9.80(70)(1000) 26.6 (7.32)
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= 160.6 m/s and ra = V 160.6 = = 0.803 m 200
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(7.33)
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7.3.2 Rim-Type: With Bending If the effects of the spoke are taken into account, Timoshenko showed that the tensile stress in the thin rim (Fig. 7.6), including bending but neglecting the effect of the curvature on the bending stress and assuming spokes of constant cross-sectional area, can be calculated from beam theory using Castigliano s theorem [7.10], giving = where f1 = 1 2 sin2 sin 2 + 4 2 f2 = f1 1 2 (7.35) f4 Ara r 2 2 1 a 1 + f g 3 3Zr 4 f3 (7.34)
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Ar 2 f2 A a + f1 + I As
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(7.36)
and the section modulus of the rim Zr is positive for the outer face of the rim and negative for the inner face. The stress in the spokes is given by = r 2 2 4A 3r 2 a 3+ 2 6g f3As ra (7.37)
In addition, there will be bending stresses in the spokes as power is exchanged with the flywheel. This torque might be determined by the maximum shaft torque, by the minimum braking time from full speed, or, if the flywheel also serves as a pulley, by the difference in belt force between the power and slack sides. Depending on how thick the rim is, the spoke will behave as something between an end-loaded cantilever and a guided cantilever. For a thin rim, the maximum bending stress on the spokes will occur at the hub, assuming that they do not taper more than the usual 10 to 25 percent. This bending stress is = T(ra rh) Zs Ns ra (7.38)
If the flywheel serves as a pulley, the torque is not equally distributed to the spokes; the value for Ns should be halved in this case. For a flywheel with a large rim
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FLYWHEELS 7.16
MACHINE ELEMENTS THAT ABSORB AND STORE ENERGY
FIGURE 7.6 Rim-type flywheel. The angular position is measured from the center line bisecting the adjacent spoke locations. Points P1 through P4 correspond to the values given in Table 7.5.
section in comparison to the spokes, the spokes act as guided cantilevers and have a maximum bending stress that is half the value given by Eq. (7.38). Example 7. A cast-iron flywheel with a rectangular rim 5 in thick and 11 in wide has an average radius of 48 in. If the hub has a 5.2-in radius and each of the 8 spokes has an area of 11 in2, what are the maximum stresses at a speed of 400 rpm The maximum torque possible is 4000 lb ft at full speed. The area A = (5)(11) = 55 in2, = 2 (400)/60 = 41.9 rad/s, I = 11(5)3/12 = 114.6 in4, and Zr = 11(5)2/6 = 45.8 in 3. The angle is 22.5 degrees, or /8, and the maximum rim stress occurs at the inner surface of the rim with = /8. From Eqs. (7.34) through (7.36), using a density of 0.283 lb/in3, f1 = 1 2 sin2 ( /8) sin ( /4) /8 + = 1.274 4 2 (7.39) (7.40) (7.41)
f2 = 1.274 f3 =
1 = 7.600 10 4 /4
55(48)2 7.600 10 4 55 + 1.274 + = 7.11 114.6 11
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