Body vibration in a BMW with a diesel engine - how to fix it?

I have a BMW E90 with a 2000 ccm engine I am a new owner. I have recently noticed that there has been a strong body vibration especially at idle. As I raise the engine speed even at idle I also feel very strong vibrations at higher revs. In addition, there has been a squeaking sound when extinguishi

Body vibration in a BMW with a diesel engine - how to fix it?
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BMW driver question

I have a BMW 3 E90 with a 2000 ccm engine I am its new owner. I have recently noticed that there has been a strong body vibration especially at idle. As I raise the engine speed even at idle, I also feel a very strong vibration at higher revs. In addition, when extinguishing the engine, there has been a squeaking sound from the front of the engine. What could be the cause?

The problem affects the BMW 3 E90, but it also happens in other models.

wibracje diesel BMW
BMW 3 E90 | By OSX (Own work) [Public domain], via Wikimedia Commons

Response from BM Cars service

The symptoms described may indicate an emerging problem with the pulley (vibration damper). The pulley, from which the engine's accessory drives are distributed by means of multi-ribbed belts, thanks to its design also acts as a crankshaft vibration damper. In the course of intensive use, after a mileage of about 200,000 kilometers, the damping layer from the wheel axle is vulcanized. Vibrations of the wheel itself then occur, which are transmitted through the engine and engine cushions to the body.

In addition, when extinguishing and starting the engine, there may be belt slip noises and increased vibration and even knocking. The defect is particularly dangerous to the continued operation of the car, as if left unrepaired it can lead to the pulley breaking off while the engine is running and cause numerous damages in the engine compartment. During the visual inspection, it is also necessary to check the condition of the engine and transmission cushions.

wibracje diesel BMW
BMW pulley
wibracje diesel BMW
Drive shaft and crankshaft in BMW

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BMW 320d E90 | By Thomas doerfer (Own work)[CC BY 3.0], via Wikimedia Commons