Practical Solutions to Machinery and Maintenance Vibration Problems
Chapter 2, Mechanical Resonance
Section 6, Resonance at Frequencies other than the Natural Frequency or First Critical
Resonance at the natural frequency of a part is usually referred to as the first. resonance frequency or 1st critical speed. Resonance in the same part can again occur at higher frequencies, with the next highest frequency called the second resonance -- the next the third, and so on. For the same vibration energy input, the vibration displacement amplitude at one of the higher criticals (such as second and third) is usually smaller than at the 1st resonance frequency. However, the frequencies are higher, and therefore the resulting vibrations in velocity units show that they are still harmful to the machine.
A common misconception is that the higher resonances always occur at full integer multiples of the natural frequency, such as 2 x, 3 x, or 4 x or 3 x, 5 x, or 7 x the natural frequency. This can be true in special situations, but the next resonance frequency is dependent on many conditions and factors. The mathematics involved is quite complicated, often best left to the machinery manufacturer. Yet, to get at least a rough idea of the numerical value of some of the multiples of the natural frequency, consider the following examples. They are over-simplified compared to the typical machinery complications, but they do show that for most situations, the second resonance frequency is most often not twice the frequency of the 1st; the third is not three times the 1st, and so on. These are based on the simple beam theory and are accurate enough only for beams having a length to depth ratio of 10 to 1 or more. (Source: "A Vibration Manual for Engineers," by R.I. McGoldrick, Navy Department, David Taylor, Model Basin, Washington, D.C., Dec. 1957.)
For a uniform beam with clamped ends, the second resonance is 52/32 x the 1st. The third is 72/32 x the 1st. The fourth is 92/32 x the 1st, and so on. For a shaft, rather than a beam, plain bearings act closest to "clamped ends." A uniform beam with simple support at the ends produces higher resonant frequencies above the natural frequency of 22, 32, 42, etc., times the natural frequency. For a shaft, narrow rolling element bearings act closest to "simple support."
A uniform cantilevered beam doesn't produce its second resonance frequency until 6.27 x its 1st; its third resonance 17.6 x its 1st, and so on. To show how complicated it can get, consider a continuous shaft in uniformly spaced bearings. Its natural frequency is given approximately by the formula for a simply supported beam, whereas the second critical speed has the critical frequency of a beam with clamped ends.
For a person working on practical vibration problems on already built machinery or structures, the above simply indicates to not be surprised when upper resonance frequencies are not simple, uniformly-spaced multiples of the 1st resonance. Yet, there are situations wherein the second resonance is 2 x the 1st; the third is 3 x the 1st, and so on (e.g. long slender parts such as tie rods, very long, small diameter papermachine rolls, etc.).
This textbook contains only part of the information in our Practical Vibration Analysis seminar.