A burning smell from a dryer can have many causes, including lint buildup, a failing thermostat, a faulty motor, or a belt:
Lint Dryer lint is flammable and can get very hot, even to the point of catching fire. You can check for lint buildup in the lint trap, inside the dryer, underneath, around, and behind the dryer, and in the vent hose. Use a vacuum cleaner with a hose attachment to remove lint from inside or around the dryer. After removing lint, run an empty cycle and see if the smell persists.
Drive belt If the smell is similar to burning rubber, the drive belt may have slipped off the drum and onto the heating element, where it can dry out or melt.
Thermostat a failing thermostat can cause the dryer to overheat, which can lead to a burning smell.
Motor a worn-out dryer motor can cause friction between grinding parts, which can lead to a smoky smell. A failing motor can also be very noisy. A qualified technician should replace a dryer motor.
Heating element a n open heating element can short against metal inside the dryer cabinet, causing a burning smell. You might also smell something like metallic odor or burning wiring insulation. If you smell something burning, turn off the dryer and have a technician examine it.
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