Vibrating sound

Pop the hood and try to find the source of the noise before it quits. You might also consider having a mechanic listen to it. If you have anti-lock brakes, you might be hearing the pump replenish any pressurized brake fluid that may have leaked down. There might also be a pump for an automatic load-leveling system that's running.

On many vehicles the load-leveling pump won't run if a door is open so you might try leaving your door open to see if the noise is still heard. On some anti-lock brake systems the power brake assist comes from the pressurized brake fluid. Pumping the brake pedal a dozen times will deplete the pressure and the pump will turn on again and run for about ten seconds to build the pressure back up.

It is common for the system to do its thing each time you start the engine, but they do cause a real lot of headaches for people. The system will run also to raise the back end in response to how ever many dead bodies you have in there. When it starts to run repeatedly, one of the air springs or plastic hose going to it has a leak. Those can be fixed but it can take a while to find some leaks.

When the pump fails, and it will, you'll be in for a shock at the cost of repair, assuming the parts are still available. Most people elect to remove the system and install regular coil springs.

Usually the pump motor dies or one of the air springs pops a leak, then the rear of the car sits too low. No other damage normally occurs.