Page 51 - Rotating Machinery Pratical Solutions to Unbalance and Misalignment

Resonance and Beat Frequencies

supports was measured to be 12 feet. What is the velocity of the
wave? What frequencies should be avoided so that the pipe will
not resonate at its natural frequency or one of the first three har-
monics?

Step 1. Since the pipe is vibrating at its first harmonic, n = 1, and
substituting into Equation (3.3), λ = 24 feet. From Equation (3.4),
the velocity is found to be v = (24)(2) = 48 feet per second.

Step 2. By rearranging Equation (3.4) f = v/λ. The second har-
monic λ = L, and then the third harmonic λ = 2L/3. Therefore,
2 3
the second harmonic frequency is 96 Hz. and the third harmonic
frequency is 144 Hz. Thus the three frequencies to be avoided are
48 ±10%, 96 ±10%, and 144 ±10% Hz.
If a frequency near these existed, consideration should be
given to relocate the distance between pipe supports. Obviously,
the distances 6 feet and 4 feet should be avoided.

Example 3.4
A steel shaft is simply supported with 4 feet between sup-
ports. The shaft is struck at its end and allowed to vibrate freely.
What should be the recorded frequency?

Step 1. Since steel weighs 480 pounds per cubic foot, the mass of
a cubic foot is (480/32.2 × 12))/1728 or .00071888 pounds mass per
cubic inch. The modulus of elasticity for steel is approximately 30
6
× 10 . Using Equation (3-5) the velocity is found to be 17,023.5
inches per second.

Step 2. Since the shaft is vibrating at its first critical or natural
frequency, λ = 2L. Since L equals 4 feet, λ = 8 feet or 96 inches.
Rearranging Equation (3-4), f = v/λ or the frequency anticipated
is approximately 177.3 Hz.
For a machine, or machine component, vibrating at resonance
or one of its harmonics, there are only three basic methods that
can be employed to correct the vibration.