OBJECTIVES:
Determine where the mass unbalance is located on a rotating shaft and then place additional masses on the rotor to balance it

BACKGROUND:
If you've ever wanted to know the theory behind balancing your tires but have been afraid to ask, look no further.  In this lab, you will balance a tire-like mass (but much smaller and metal--ok, so really nothing like a tire, but you can pretend) on a rotating shaft.  First, some theory.

A mass on a rotating shaft is imbalanced if the weight is not evenly distributed.  There are two types of imbalance--static and dynamic.  Static imbalance occurs when when there is a heavy spot on the tire; that is, as you go out radially from the tire's center, there is some point on a radius where the mass is greater than the others.  As the tire rotates, there is a centrifugal force that points away from the center of the tire.  If the tire has a heavy spot , then the sum of the forces does not equal zero.  This causes the tire to roll unevenly and induces a vertical vibration (as seen in the figure below).

In dynamic imbalance, the weight may be distributed evenly radially, but there is an imbalance in the width of the tire; so, when the sum of the moments about the center does not equal zero.  This causes the tire to wobble (as shown in the figure below).

Mechanics correct these imbalances by distributing lead weights on the tire rim in such a way as to cancel the effects of the heavy spot(s).  Imbalanced tires do not wear evenly and need to be replaced sooner than a balanced tire.

You can think of an imbalanced tire as a spring-mass system where the tire is the mass and the shaft is the spring.  In such systems, there is a known relationship between the driving force and the output characteristics.  One such relationship is the phase difference between input and output.  If, for example, you apply an input force that is much below the resonant frequency of the system, the input and output will be in phase.  That is, when you push the spring forward, the mass will also move forward.  If you drive the system above the resonant frequency, the input and output will be -180°out of phase; when you push the spring forward, the mass will be going backwards.  At resonance, the output lags the input by 90°.  Because I know you're dying to verify this for yourself, test it on any second order system (e.g., spring-mass, pendulum).

Back to tires...  So, the heavy spot causes the imbalance force that induces vibration of the tire (and shaft).  Let's call the heavy spot our input force and the maximum displacement of the tire/shaft the output.  This maximum displacement (for a given frequency) is called the high spot.  Assuming we have a transducer to measure shaft displacement, we should be able to plot displacement amplitude vs. shaft speed (frequency).  At resonance, the displacement should be at a maximum.  Combining this information and the phase difference between input and output, we can determine the location of the heavy spot.  At resonance, the high spot lags the heavy spot by 90 degrees.  In the figure below, the heavy spot and high spot plots are combined into one, more useful plot.

This polar plot (on the right) shows high spot magnitude at different phase angles (the dashed circle in the 4th quadrant).  The maximum high spot value (which occurs at resonance) is the point on the dashed circle which is furthest from the origin (~135°).  The heavy spot leads this point by 90°.  Although you might be tempted to add 90° to 135°(225°) to get the heavy spot, you must take into account the rotation direction.  In the figure above, the heavy spot is located at about 45° due to the counterclockwise rotation.  The vector pointing to the high spot is called the O vector. It graphically represents the response of the rotor to the original imbalance.

To balance the system, wouldn't it make sense to add a mass equal in magnitude but opposite in direction from the heavy spot direction (which is 90° ahead of the O vector)?  Fortunately, it is that simple!  Unfortunately, the amount of mass needed to balance the heavy spot is unknown.  To help us find the location and the amount of the correction weight, we must perform an intermediate step using a calibration weight.  After placing a calibration weight (~5-10% of rotor weight) at a given location on the tire (e.g., wherever makes you happy), generate a polar plot like the one above and find the new high spot.  The system's response will be different with the calibration weight and will yield a new "high spot" due to both the original imbalance, O, and the calibration weight, C.

The difference between O+C and O will yield the effect of C alone. NOTE - the diagram below doesn't show it this way, but the vector C should be 90° past the calibration weight location.

Translating C to the axis origin will make measurements easier.  Knowing the effect that the calibration weight has on the system allows us choose a proper correction weight.  The following figure contains the graphical and analytical (equations) methods for determining the location and magnitude of the correction weight. NOTE - the diagram below doesn't show it this way, but the correction weight should be 90° in front of vector -O, which would be 180° away from the original imbalance.

In words, this says that the angle between -O and C is equal to the angle between the correction weight and the calibration weight.  So if -O and C are 20° apart, then place the correction weight 20° from where you put the calibration weight (you take out the calibration weight before putting in the correction weight).  The amount of weight to add is equal to the ratio of the magnitueds of vectors -O and C times the amount of calibration weight you added.  The angles and magnitudes can be determined by plotting them and measuring with a ruler and compass.

Armed with this knowledge, you should now have no trouble securing that job at the local Big O or Discount Tire Company you always wanted!

MATERIAL AND METHODS:

1. Double click the icon for ADRE for Windows on the desktop.
2. Click FILE on the main menu, then OPEN, and then double click SINGLE.  This opens the proper configuration into ADRE.
3. Start the rotor kit by placing the speed controller in the "Slow Roll" position.
4. Set Ramp Rate to 3000 rpm / minute. Set the Max Speed to 3000 rpm (300 on the dial). Do not change the value of the max speed dial during the lab.
5. Click on "Store Enable" on the ADRE main menu.  After a pause and a message that says "settling," the data acquisition screen should show data values that are being acquired.
6. Ramp up the rotor kit to 3000 rpm.
7. Click the "Stop" to stop taking data. Click the "Close" button to close the data acquisition window.
8. Ramp down the rotor.  Once it's at its slow roll speed, stop the rotor.
9. Activate the polar plot by clicking on the polar plot button. Move the plot cursor noting the vertical response vector is at the top of the plot in blue. This is your high spot, which defines the O vector.
10. Plot this vector on polar paper.
11. Place a calibration weight wherever your little heart desires; although, to get the strongest measurements, you should place it about 90° away from the heavy spot.  Somewhere between 0.2 - 1.0 grams should work.
12. Find your O+C vector by repeating steps 3-10.
13. Connect the tip of the O vector to the tip of the O+C vector.  This is the C vector.
14. Redraw the C vector centered at the origin.
15. Calculate the position and angle of the correction weight.
16. Remove the calibration weight.
17. Repeat steps 3-10 with the correction weight in place.

ADDITIONAL RESOURCES:

•  Bently Nevada Rotor Kit setup
•  Discount Tire Co.'s explanation of tire balancing 
•  "Machinery Diagnostics Course," sections 5 & 7--white binder in Dynamics Lab office