Determining Stiffness of an Absorber and Natural Frequencies of a Rotating Machine

Determining Stiffness of the Absorber:

Using the formula provided, we can calculate the stiffness of the absorber to eliminate vibrations at the operating speed of 400 rpm:

k = (400 * 2π/60)² * 125 lb

k = (400 * 2π/60)² * 125

k = (400 * 2 * 3.14159 / 60)² * 125

k = (41.8879)² * 125

k = 1758.10 * 125

k = 219,762.50 lb/rad²

Determining Natural Frequencies after Absorber Installation:

To calculate the natural frequencies after the absorber is installed on the machine, we need to subtract the absorber's mass from the total machine's mass. The machine weighs 500 lb, so the total mass after installation would be 500 lb - 125 lb = 375 lb.

Substitute this new total mass into the formula to find the natural frequencies:

ω_n = (√k / m)^1/2

ω_n = (√219,762.50 / 375)^1/2

ω_n = (√586.36)

ω_n = 24.21 rad/s

To express this frequency in rpm, we can convert it back by multiplying it by 60/2π:

24.21 * 60 / 2π = 230.63 rpm

Therefore, the natural frequency after the absorber is installed on the machine is 230.63 rpm.