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RESONANCE-Theory and Practice
by Charles Jackson

### ABSTRACT

This paper reviews the analysis of a simple spring-mass-damped system. Case histories from industrial machines are used to illustrate the analysis of data from resonant machines and piping. Methods for evaluation of the amplification factor (damping) are illustrated with case histories.

### PREVIEW

“INTRODUCTION
Every machine or structure has a Natural Frequency, a physical property of mass, stiffness, and damping. Every machine or structure is not necessarily in “resonance”. Webster did good:

a) The quality of state of being resonant
b) ( I): a vibration of large amplitude in a mechanical or electrical system caused by a relatively small periodic stimulus of the same or nearly the same period as the natural vibration period of the system (2): the state of adjustment that produces resonance in a mechanical or electrical system.

Resonance can be good or bad. For machines or structures it is generally bad. The stimulus can be very small- “self excited” in some “instability” cases.

A vector diagram can be used to represent forced vibration with viscous damping. Three (3) diagrams will be shown later representing the condition of a frequency well below resonant frequency, at resonance, and a frequency well above resonance. These diagrams will attempt to correlate the Amplification Factor and phase changes as they relate to a Bode’ diagram or a balancing chart using Bode’ and Polar Data.

There are certain factors to understand about the Force Vector as it explains many things about vibration (acceleration, velocity, & displacement) and a rotor’s response during acceleration or de¬celeration . The Vector Diagram above represents a rotor operating below its 1st resonance (critical)(natural frequency).

1. The displacement (vibration) lags the exciting force (e.g. mass unbalance) by the angle Φ, which can vary between 0 & 180°
2. The spring force is always opposite in direction to the displacement.
3. The damping force lags the displacement by 90 degrees and hence is opposite in direction to the velocity.
4. The inertia force is in phase with the displacement and opposite in direction to the acceleration.

HINT: The “lag” or “lead” really depends on a person’s frame of reference. I like to think of velocity leading displacement by 90 degrees and the acceleration leading the velocity by 90 degrees. With instrumentation, one should determine whether the instrument vendor has built his equipment to “compensate” (do) a 90 degree phase shift ,e.g., displacement-to- velocity “shift”.

AMPLIFICATION (MAGNIFICATION) FACTOR vs. FREQUENCY vs PHASE:
Below is a forced response plot of a damped system, it represents the response X/Xo plotted against the frequency ratio, ω/ωn, [@ 1.0 = resonance]. Further, the phase change is shown for different percentages of damping ,C, respective to critical damping, Cc.”