Belt-Drive Problems
Belt-drive problems, which include shaft misalignment, pulley misalignment, belt wear, belt resonance, belts too tight, belts too loose, pulley eccentricity and bent shafts, can be relatively straight forward to detect but can be far more difficult to specifically diagnose and correct. That is mainly due to the wide variety of problems that can occur in the installation and assembling of the belt drive, the difficulty of doing field testing on belts and the possibility of other influences (i.e. the base) having some effect. 
It is important to realize that some of the belt-drive vibration problems listed above do NOT cause vibration at belt related frequencies. Problems due to the shafts or pulleys (misalignment, eccentricity, etc.) cause vibration at 1x rpm of the component with the problem (i.e. eccentric pulley on the fan causes vibration at 1x rpm of the fan). Worn belts, on the other hand, will cause vibration at harmonics of belt running speed.
The good news, especially in the case of component (belt and pulley) wear, is that belts and pulleys are typically relatively easy to inspect and inexpensive to replace. The bad news is that outside of that, they're often difficult to correct. One positive development in recent years has been the availability of laser alignment units for belt drives for a moderate price. Unfortunately, in more cases than not the old string & straight edge is still the alignment method used for belt drives. The first step to identifying a belt problem is to determine the belt speed.
Determining the Belt Speed:
Obtaining belt speed can be a bit difficult but there are a few tricks. Some methods are listed here:
Calculate it.
  • It can be calculated mathematically if you know some of the variables: belt length, pitch diameters, center distances, etc.; but usually that is not the case. The formulas are listed below. 
Measure it.
  • Detecting it with a strobe light is very difficult since it is usually a slow flash rate and the mark used may be unreliable (lettering on the belt, etc.). 
  • A photoeye will be very accurate but will require proper setup and a mark applied to the belts. 
  • A "lasertach" would be the best option for an accurate belt rpm since it does not require a traditional "mark" - a good one will operate on pattern recognition.
Estimate it. With a bit a practice and understanding of a simple technique, an analyst can actually extract the probable belt rpm from the spectrum. One important requirement for this technique to be successful - there must be at least some vibration at belt-related frequencies. The following steps should be used:
  • First, identify any driver and driven related peaks (1x rpm and harmonics). Label them or make a mental note of which peaks they are.
  • Second, imagine cutting the belt in half and wrapping it around one of the pulleys. How many times will it wrap around - twice ? three times ? This will give you a very rough estimate in your mind of the belt speed (if it wraps 3 times, the speed would be 1/3 of that pulley's speed).
  • Finally, do the following:
    • Display your velocity spectrum on a logarithmic scale.
    • Move your spectrum cursor to your estimated belt rpm and turn on the harmonics. 
    • Move your cursor left and right in the smallest increments possible (some software allows movement of 1/10th of a line of resolution - this helps with identifying harmonics) and try to get the harmonics to line up on top of any significant but previously unidentified amplitude peaks. 
    • If there are significant belt related peaks on the spectrum, you should be able to get them lined up at some point. 
    • If you cannot find any pattern of previously unidentified peaks of significant amplitude, that means one of two things:
      • Either you do not have the spectrum resolution necessary, or;
      • There is no significant belt vibration (in which case, why do we need to know the belt rpm ?).
Formulas for Calculating Belt Frequencies:
If you know:
  • The Belt Length
    • - and -
  • Either of the Pulley RPMs and Its Diameter:
    • Driver Pulley Diameter and Speed
            or
      Driven Pulley Diameter and Speed
You can calculate belt RPM with the following:

3.14 x PS1 x PD1/BL = Belt RPM

-  or  -

3.14 x PS2 x PD2/BL = Belt RPM

Variable Definitions:
PS = Pulley rpm (PS1 = Driver Pulley Speed, PS2 = Driven Pulley Speed)
PD = Pulley diameter (PD1 = Driver Pulley Dia., PD2 = Driven Pulley Dia)
SD = Distance between shaft centers
BL = Belt Length

- OR -
If you only know the pulley sizes and diameters, you can roughly calculate belt length and plug it into the formula above by using the following:

Belt Length = 1.57 x (PD1 + PD2) + 2(SD)

In other words, 2x the center to center distance plus 1/2 the circumference of each pulley will provide the belt length.