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For a constant-thickness disk without a central hole, the radial and tangential stresses are then given by r = 0 1 t = 0 1 and the maximum stress, at r = 0, is r,max = t,max = 0 For a disk with a central hole, the stresses are given by r = 0 1 t = 0 1 The maximum radial stress, at r = r 2 r2 r2 i i 2 + 2 ro ro r 2 1 + 3 r 2 r 2 r 2 i i 2 + 2 + 3 + ro ro r 2 (7.62) (7.61) r2 2 ro 1 + 3 r 2 2 3 + ro (7.59)
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r,max = 0 1 and the maximum tangential stress, at r = ri , is t,max = 0 2 + The inertia is J=
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(7.64)
2 2 r 2 i 2 3 + ro
(7.65)
z 4 (r o r 4) i 2g
(7.66)
Example 8. A steel disk of uniform thickness and outer radius 0.6 m rotates at 30 rad/s. Find the maximum stress in the disk if it has an integral shaft, neglecting stress rises due to the geometry change at the shaft. What is the maximum disk stress if the disk is bored for a shaft of 0.025-m radius The stress is independent of thickness. From Eqs. (7.58) and (7.61) with = 76.5 kN/m3 and = 0.3, r,max = t,max = 0 = 76.5(0.6)2(30)2(3 + 0.3) = 1.043 MPa 8(9.80)(1000) (7.67) 0.6(0.025) = 0.122 m.
For the bored disk, the maximum radial stress occurs at r = From Eq. (7.64), r,max = 1.043 1 0.025 0.6
= 0.958 MPa
(7.68)
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FLYWHEELS 7.20
MACHINE ELEMENTS THAT ABSORB AND STORE ENERGY
The maximum tangential stress occurs at the hub. From Eq. (7.65), t,max = 1.043 2 + = 2.09 MPa 2 2(0.3) (0.025)2 (0.6)2 3 + 0.3 (7.69)
7.4 FLYWHEELS FOR ENERGY STORAGE
The flywheel can be used as an energy reservoir, with energy being supplied at a slow constant rate or when it is available and being withdrawn when desired. A flywheel might, for example, be used to give good acceleration to an automobile that is underpowered by present standards. Regenerative breaking, power storage for peak-demand periods, and mechanical replacements for battery banks are all potential uses for the flywheel. The high charging and discharging rates of a flywheel system give it an advantage over other portable sources of power, such as batteries. Although the concepts developed in the previous sections are still true for energy-storage flywheels, the purpose is now to store as much kinetic energy, 0.5J 2, as possible. In most applications, the flywheel speed does not vary over 50 percent, so that only about 75 percent of this total energy is actually recoverable. The design of the ordinary flywheel is usually dictated by the allowable diameter, governed by the machine size, and the maximum speed, governed by the practicalities of a speedincreasing drive and higher bearing speeds. These constraints can result in a low peripheral speed, causing the economics to favor a rim-type flywheel design. The economics change with the energy-storage flywheel, since (1) larger values of total stored energy are usually involved, requiring heavier flywheels or more energy per unit weight of flywheel, (2) the weight of a heavy flywheel and the correspondingly heavy bearings and other components may be unacceptable, especially in mobile applications, and (3) the design constraints imposed in a machine where the flywheel limits the speed variation can be relaxed when the flywheel is the main component, encouraging optimization. Depending on the application, the energy per dollar, energy per weight, or energy per swept volume is usually maximized [7.1].
7.4.1 Isotropic and Anisotropic Designs The stress equations for the thin disk given in Sec. 7.3.3 can be solved with r = t to give the shape for a fully stressed thin isotropic disk with no central bore: z = z0 exp 2 r2 2gSy (7.70)
where Sy = allowable strength, for example, the yield strength of the material. Define the energy stored per unit weight as R = Fs Sy (7.71)
where Fs is a dimensionless factor that depends only on the shape of the flywheel. Using Eqs. (7.46) through (7.49), it turns out that the efficiency or geometric shape
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