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Practical Solutions to Machinery and Maintenance Vibration Problems

Chapter 7, Misalignment

Section 17, Basic Vector Understanding

For those with formal education as engineers or technicians, understanding vectorial addition should have been part of their early post high school training. However, many have not had much chance to use the addition of vectors in their work, and a short review will be helpful. Also, there are some who never did get an understanding of vectors in their training, and this explanation has to take them into account as well. The explanations do not have to be thoroughly understood, as a "gut level" understanding is enough. Those who do presently understand simple vector addition, can proceed directly to "Using Understanding of Vectorial Addition for Vibration Analysis."

In addition to magnitude, a vector expresses direction, usually measured as an angle at so many degrees from a horizontal line, other reference line, or so many degrees from another vector. There are several ways of adding vectors graphically, but the method that best expresses what is needed for vibration analysis, is the method of having all forces on a part expressed as originating from the point of intersection of a vertical and horizontal axis. This point is properly called the "origin." The above vector diagram shows two vectors originating from the origin. Vector A represents 6 oz•in of unbalance located at 70° from the 12:00 o'clock position. Vector B represents 4 oz•in of unbalance on the same rotor located at 280°. Obviously when the rotor is running the unbalance force that the bearings resist, it is the vectorial result (technically called the "resultant") of both the 4 oz•in and 6 oz•in. If they were both in the same line of action and exerted in the same direction, the resultant would be 10 oz•in. But as they exert their forces at different angular positions, the resultant force would be some other amount.

The simplest graphical way to express the resultant of these two unbalances would be as follows:

1. From the tip (end of arrowhead) of one vector, draw a light construction line parallel to the other vector. (In this example, from tip of vector A, parallel to direction of vector B.)

2. From tip of the other vector (vector B) draw a light construction line parallel to the first vector (vector A).

3. From the origin, draw a heavy line to the point of intersection of the two light construction lines. Add an "arrowhead" at its tip.

This is the resultant vector. Its length represents the resultant oz•in from the vectorial addition of vector A and vector B. Its angular position represents the direction of that resultant. In other words, vectorially, vector A plus vector B equals vector R (use the same scale as was used for vectors A and B.) The same resultant could be determined mathematically, but the foregoing gives a better visualization for those who have not had such training.


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