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Electric motor motor noise diagnosis: this article describes the causes, diagnosis, cure & prevention of noises traced to the electric motor component of an oil burner, air conditioner, or other electric-motor driven equipment. The sketch of an oil burner shown above is provided courtesy of Carson Dunlop Associates.
For HVAC compressor motors that hum
see HUMMING sounds from A/C or Heat Pump system
2. Motor shaft end play
Excessive electric motor noise can be caused by excessive motor shaft end play - the distance the shaft can move in and out of the motor.
Beckett (1989)  notes that a new electric AC motor will have no more than .035" of movement in and out. Push the shaft all the way "in", and measure its outwards movement. (Automotive feeler gauges can sometimes assist).
If electric motor shaft end play is more than .060" and if noise is a problem, the motor could be suspect.
3. Motor is improperly mounted or aligned
Motor vibration noise can be caused by misalignment of the motor due to its mounting position or due to bad shaft couplings to driven devices such as fans or fuel units.
We find this problem also on hot water heating circulator motors.
Realign the motor or replace a bad coupling. - Beckett (1989) 
4. Electrical phase problems
If the motor is a 3-phase unit running only on one phase there may be an open circuit, blown fuse/tripped breaker, or unbalanced voltages. Any of these conditions can cause motor vibration.
Check for open circuit, blown fuse, tripped breaker, uneven voltages on the three phases. - Beckett (1989) 
Bad motor bearings cause vibration at the shaft support at either end of the motor.
Check for bearing damage (loose, side play, noise, visible damage); Usually motor bearing damage is caused by improper mounting or motor misalignment.
We've seen a whole series of hot water heating system circulator motors that were destroyed and replaced, one after another until finally a more experienced tech noted that the motor mount was askew. Beckett (1989) 
7. Loose coupling between electric motor and driven appliance
A bad coupling between the motor drive shaft and the driven fan, fuel unit, or other component can cause vibration and noise as well as motor damage.
Replace the bad coupling; If there is a loose sheave, tighten the set screws or replace that component.
Adapted & expanded from Beckett (1989)
Other Noises at HVAC Equipment
Watch out: some noises at HVAC equipment could be a sign of dangerous operation. For example, rumbling, stumbling noisy oil burners, especially accompanied by smoke, or oil smells, could be a sign of pending PUFFBACKS, OIL BURNER.
Squealing motors may be traced to failed bearings, overheating, and on occasion the risk of an electrical fire.
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 Beckett Oil Burners, "Burner Motor Service Facts", Beckett Corporation, Technical Information, 15 June 1989 [copy on file as PDF].
 "Noise and Moisture, Using scare tactics” may be necessary when installing a new heating system", George Lanthier, Fuel Oil News, M2MEDIA360, 1030 W. Higgins Road Suite 230, Park Ridge, IL 60068 847-720-5600
02/1/2006. According to Fuel Oil News, George Lanthier owns Firedragon Enterprises and is the author of nine books on oil heating and heating systems. He is a teaching consultant and expert witness on oil heating systems. He can be contacted at 132 Lowell Street, Arlington, MA 02474-2756. His phone number is (781) 646-2584 and he can be faxed at (781) 641-7099. He can also be contacted through his Web site at www.FiredragonEnt.com.
Carson, Dunlop & Associates Ltd., 120 Carlton Street Suite 407, Toronto ON M5A 4K2. Tel: (416) 964-9415 1-800-268-7070 Email: firstname.lastname@example.org. The firm provides professional home inspection services & home inspection education & publications. Alan Carson is a past president of ASHI, the American Society of Home Inspectors. Thanks to Alan Carson and Bob Dunlop, for permission for InspectAPedia to use text excerpts from The Home Reference Book & illustrations from The Illustrated Home. Carson Dunlop Associates' provides extensive home inspection education and report writing material.
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The Home Reference Book - the Encyclopedia of Homes, Carson Dunlop & Associates, Toronto, Ontario, 25th Ed., 2012, is a bound volume of more than 450 illustrated pages that assist home inspectors and home owners in the inspection and detection of problems on buildings. The text is intended as a reference guide to help building owners operate and maintain their home effectively. Field inspection worksheets are included at the back of the volume.
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