9.1.1 Case study: Moment-of-inertia calculations

A steel coupling that resembles a short thick-walled tube has a thickness of 6 in and inner and outer diameters of 12 and 30 in, respectively. The material density is 0.283 lb/in ³. The coupling has 8 bolt holes that are 3 in in diameter on a bolt circle diameter of 25 in. A gear ring of width equal to the coupling length and of an equivalent solid radial thickness of 1 in is shrunk onto the coupling outer diameter (OD). The equivalent density of the gear is 0.2 lb/in ³. What is the polar moment of inertia of the coupling without coupling bolts installed, expressed in both U.S. Customary and SI units? How precisely can such inertia calculations be made?

Solution to Case Study 9.1.1. The calculation is performed in three steps using the formulas given in Table 6.1. The polar moment of inertia of an annulus is first calculated to represent a solid coupling (no bolt holes). To this is added the polar moment of inertia of another annulus to represent the shrunk-on gear. The formula for a thin ring in Table 6.1 could be used in this step, but the result would be approximate and has no computational advantage over the annulus formula. In this case, using the thin-ring formula would overestimate the ring s polar moment of inertia by 4.8 percent. Finally, the polar moment of each filled coupling hole is...