Alternating-current motors are often designed to run at a constant speed, typically either 1750 or 3500 rpm. One such motor for a power tool weighs 20 lb and is to be mounted at the end of a steel cantilever beam. Static-force calculations and space considerations suggest that a beam 6 in. long, 4 in. wide, and 3/8 in. thick would be suitable. Use E = 3 × 107 psi for the modulus of elasticity of steel. Use ( = 15.2 slug/ft3 for the density of steel. The rotating part of the motor weighs 1 lb and has an eccentricity of 0.01 ft. (This distance can be determined by standard balancing methods or applying the analysis in this example in reverse, using a vibration test.)
The damping ratio for such beams is difficult to determine but is usually very small, say, ( ≤ 0.1. Estimate the amplitude of vibration of the beam at steady state. Consider both speeds.