HVAC - fan makes a noise when it kicks on

Was at Mom's house today and noticed that her furnace fan makes a kind of vibrating/humming noise the first few seconds as it starts up. It's a Gas furnace (Trane) and a 10 year old house.
Recently I had to replace the blower motor in my 22 year old Trane and cleaned everything. When I start my fan, there's an initial buzzing from the transformer and the fan comes on right away. The transformer buzzing gets quieter after 2 seconds.
I know that blower motors may need oil from time to time, but then if it's in need of lube, why does it make a noise only on start? After it gets spinning I only hear air blowing.
The noise from Mom's sounds "wobbly", very much like a fan that isn't aligned or attached right but then evens out from centrifugal force as the rpm's increase.
Could the blower have come a bit loose from the frame or the frame from the housing? I won't be back there for a few days but wanted to get some ideas on what to look for if possible. Or could dust just be gumming up the works and making the motor work harder. I'm not sure if it's ever been cleaned.
Also, the labored startup of the fan seems to last a little longer each time, like whatever is going wrong is getting worse.
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wall-e wrote:

Sometimes the motor capacitor will fail and cause the motor to labor on startup.
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Thanks CJT, I didn't think about the capacitor.
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The blower and fan set up is as simple as one can get. Just disassemble the motor and fan assemblies and do a thorough cleanup and the furnace will be good for another ten years or so. The maintenance stuff to look for will be the fan's axle and axle bearings. I had overtightened my fanbelt before and this caused the shaft to wear into the bearings and ovalled it. To correct the fan's startup and stop "clunk noise" I tightened the fanbelt further. This of course caused further wear. By the time I took the fan assembly apart the shaft had worn down more than half its diameter, and the bearring holes oversized. Should you replace the axle and bearings do not add any lubrication. The bronze bearings are meant to run dry. Lube, especially grease is commonly dried by the heat. This caused the bearings to sieze.
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PaPaPeng wrote:

That's simply incorrect. The bronze (sleeve bearings) are oilable. The metal sleeve is microscopically perforated so that the oil in the wick can seep through.
hvacrmedic
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wrote:

If the sleeves aren't sintered bronze, then are they lop-sided? ;->
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