Shaft Alignment Math

link to : math | math graphics  | outline 

Peterson Alignment kits utilize the rim-and-face method of shaft alignment. A general description of the procedure follows.

If you would like, start out with an outline of what's involved in doing an alignment.

Two dial indicators are used to obtain numbers from the "moveable" side of a pump-coupling-motor assembly. By using two dial indicators, one mounted perpendicular to the shaft (radial), and one parallel to the shaft (angular), we can correct for both a height difference (radial) between the two machines, and an angle difference between them (angularity).

The dials are held in place via a frame and tubing assembly originating on the "stationary" side of the coupling. For purposes of continuity and simplicity, the stationary side of the coupling will always be to the left. (Our Alignment Manager software allows you to choose which side of the coupling you would like to be the moveable side, but it will not be covered here).

It is then assumed that we need to calculate correction amounts (should there be any) to bring the moveable side (to the right of the coupling) into alignment with the left side. The pivot points (points at which the moveable side will be raised or lowered to bring it into alignment with the stationary (reference) side, are simply the motor bolts located (usually) at the four corners of the base (front and back "feet").

If we break down the alignment into (1) the vertical and horizontal planes and (2) angular and radial dial readings, we can describe each piece easier than trying to describe the whole process at once, and then put the pieces together to convey a sound alignment process.

Model #30RA in place

Vertical Plane
Let's first define the vertical plane. If you could take a thin, flat, clear surface, such as glass, and place this glass from the floor to the ceiling, running directly through the center of the shaft in the picture above, this would define the vertical plane with regard to our application. Movement in the vertical plane simply has to do with movement that is up and down.

Dial indicator readings for the vertical plane are those taken at 0° and 180°. (The 0° reading will always be zero).

Horizontal Plane
Similarly, the horizontal plane would be defined as a thin piece of glass placed through the center of the shaft and forming a sort of "table".

Dial indicator readings for the horizontal plane are those taken at 90° and 270°.

Radial and Angular Dial Readings
Starting at 0°, you rotate the shafts together with the alignment equipment mounted, and stop at 90°, 180° and 270°. As mentioned above, the readings at 0° and 180° are related, as are the readings at 90° and 270°. This will become apparent when we discuss the mathematical calculations these readings are used in.

First Stop - 90°
Our first stop is at 90°. The radial (perpendicular) dial indicator will register a positive, negative, or zero on its dial face. A positive number indicates that the plunger of the dial is getting pushed in, and thus the shaft is closer to us as we look at our equipment from the side. A negative number would indicate the shaft is farther away from us. A zero would indicate there is no change in the position from 0° (12 o'clock).

The angular (parallel) dial indicator will also have a positive, negative or zero on its dial face. A positive number would indicate that the target on which the dial plunger is resting has pushed in the plunger the amount shown on the dial face. Analysis then shows that the misaligned shaft is "cocked" in such a way that the back end of the shaft is closer to us than the front end nearest the coupling.

After the Readings are Taken
Once we have all of the dial indicator readings (radial and angular dial readings at 90°, 180° and 270°), lets look at how the math calculations will look based on the descriptions given above.

Our goal is to "sum" the misalignment due to the radial misalignment and the angular misalignment present. In the vertical plane, our equations will look like this:

VN=(R180-R0)/2 + D(F180-F0/H)
VF=(R180-R0)/2 + E(F180-F0/H)
(N stands for near, F stands for far)

The first portion, [ (R180-R0)/2 ] is the misalignment measured from the radial dial indictor (and is the same in both equations). The actual number taken off of the dial indicator is twice the misalignment, as explained by the following diagram:

The second portion of the equation is the measurement of the angular misalignment at front and back feet, given by D x (F180-F0/H) and  E x (F180-F0/H).

The last part of this segment given by F180-F0/H is the tangent of the angle measured by the dial indicator (tangent = opposite/hypotenuse). This number simply acts as a multiplier for any distance we want to measure the misalignment at. For this example we want to know the misalignment at distance "D" and "E", the distances to our front and back fee. By adding "D" and "E" to the equation, we now have the angular portion complete. Combining the two readings, our equations are complete.

Cleaning Up
Our final step is to apply our equations to the horizontal plane. The only difference is that our ending values will not be zero. We are shifting our plane by 90°, so our equations now look like this:

HN=(R270-R90)/2 + D(F270-F90/H)
HF=(R270-R90)/2 + E(F270-F90/H)
(N stands for near, F stands for far)

Instead of our ending values being taken from 180°, they are now taken from 270°. Instead of our ending values being taken from 0° where they were zero, they are being taken from 90°. The concept is still the same, only shifted 90°.

Math Explanations


Figure 1

FIGURE 1:  The stationary equipment will be to the left of the coupling. This will generally be a pump or any similar assembly which has outside connections attached to it, making it difficult or impossible to move.

Figure 2

FIGURE 2:  Radial offset is the amount of misalignment strictly from a standpoint of a height dimension.

Figure 3

FIGURE 3:  Angular offset is the amount of misalignment due to the perceived angle the shaft to be aligned is in relation to the stationary shaft. When the shaft has angularity, the misalignment will depend where the measurement is taken. Hence, shim requirements will be different at the front feet compared to the back feet.

Figure 4

FIGURE 4:  Front and back feet. The front feet are both motor bolts nearest the coupling assembly. Conversely, the back feet are both bolts farthest from the coupling assembly.

Figure 5

FIGURE 5:  Vertical and horizontal plane notations.

     Figure 6

FIGURE 6:  Rotation of the radial dial indicator and path for a positive dial movement

Figure 7

FIGURE 7:  We see how angularity is measured. A right triangle is formed, of which we know two of the dimensions. (1) The dial indicator reading is the base of the triangle (Adjacent) and is measured in thousandths. (2) The hypotenuse of the triangle is the "swing" of the dial indicator, defined as the diameter of the circle of revolution (see Figure 8, below), and is measured in inches. Simply put, these two numbers allow us to figure the misalignment per inch traveled away from the plunger point of the angular dial indicator.

We now have all of the math we need to figure misalignment in both the vertical and horizontal planes, as well as with respect to the angular and radial directions.

Figure 8

FIGURE 8:  This picture shows the outline of the diameter of circle of revolution, or the circle the angular indicator will scribe as it is rotated about the shaft centerline.

Alignment Outline

Your motivation to perform an "alignment" on your power transmission equipment may fall into one of three general categories:  (1) A New Installation;   (2) Re-alignment after coupling/bearing replacement; (3) Preventive maintenance.

Your goal is to accurately bring equipment on one side of the coupling (say the right side) into precise alignment with equipment on the opposite side (the left side, or reference).

To visualize what you need to do, imagine holding two pencil erasers (that are on the end of a new pencil) close together, but not touching each other. The reference pencil will be in your left hand, and the pencil you would like aligned with it is in your right hand. How can you accurately measure where the right hand pencil is in relation to the reference pencil on the left? That's where shaft alignment equipment comes into the picture.

Preliminary Inspection
If your equipment has been previously installed and is displaying obvious warning signs that something is wrong, such as excessive heat or vibration, or you've performed more than one replacement on this particular equipment, misalignment is definitely a candidate as the culprit. There could be more too it than that, however, and it's a good idea to check for what's known as "Soft Foot". Soft foot is the term commonly applied to that condition which exists when all four (4) of the machine feet are not supporting the weight of the machine. This condition is one of the major causes of frustration and lost productivity.

First Step
Careful measurements and technique play a critical role when preparing to do an alignment. Reference points and dial indicator readings are all part of package.

Once you are ready to perform the alignment procedure, as many as five (5) physical measurements must be taken from the equipment. This includes the (1) SWING DIAMETER, the distance to the (2) FRONT and (3) BACK feet of the moveable machine (to the right of the coupling), and the distance to the (4) FRONT and (5) BACK feet of the stationary machine (to the left of the coupling).

These measurements are generally done with a tape measure. Although you should be as accurate as possible when measuring from stated reference points, these initial physical measurements are not as critical as the dial indicator readings themselves.

Second Step
Once the initial physical measurements are taken and the alignment equipment is mounted and secured, you are then ready to "rotate" the alignment equipment on the shaft and coupling assembly. By doing this, you will obtain the three pairs of dial indicator numbers, which then describe the position of the moveable side of the coupling with respect to the stationary (or "reference") side.

Last Steps Involved
Once the physical dimensions and the dial indicator readings are collected, the last step is to plug these numbers into the formulae provided to determine the shims needed in the vertical direction, and any adjustment from side-to-side in the horizontal direction.

It is rare that your equipment will be in precise alignment after shimming for the first time. It may take at least two or three times to bring your equipment into alignment within an accepted tolerance range.

With the right maintenance equipment and a positive mental attitude, shaft alignment doesn't need to invoke the terror and agony once associated with the process.

link to Peterson alignment index page

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