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Q: Car shakes when I slow down.

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On the interstate I had to put the gas to the floor to speed up and get past someone then after that the vehicle shook and vibrated when I slowed down.

My car has 164000 miles.
My car has an automatic transmission.

Hello. The imbalance of rotating parts on a car, such as your wheel and tire assemblies, is a function of speed but also "positioning". When you were traveling at a steady state rate, say 65 MPH, the suspension was clearly capable of withstanding whatever residual imbalance exists in the rotating assemblies and thus the residual imbalance is essentially invisible to you (not felt).

Nothing that rotates on a car is ever perfectly balanced. Rather, the components are balanced just to the point where the residual imbalance creates ONLY vibration that is below the threshold of detection. The imbalance is there; it’s so small though your car suspension doesn’t transmit it through (the steering) wheel or vehicle frame. What is happening in your case when you suddenly accelerate and then completely unload the drive train (by taking your foot off the gas and coating) is you are changing the position of the rotating elements (wheels and tires) such that the imbalance that does exist is "magnified" and thence transmitted through the vehicle. The practical causes of this are worn tires and worn suspension parts that are not holding the wheel/tire assemblies in the same fixed position during coasting as during acceleration. That is the likely cause.

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