Engine shimmy when it hits 40MPH

Do you feel it in the gas pedal or steering wheel?

Sounds like a bad tire. If vibration is in the steering wheel it should bea bad front tire. If it is your gas pedal it should be in rear. If the vibration gets worse when you apply the brake you have some warped rotors. Is there any noises that you hear when driving? Do you hear any noises going over bumps or potholes? You might have a bad motor mount. Here is the easy way to check that. Have someone watch the motor while you hold your brake pedal down. With brakes applied put it in drive and goose the gas. If the motor lifts up on one side that means you havea bad motor mount. While brake is still applied put trans in reverse. If the other side of engine raises up when you goose the gas you'll need both mounts replaced. If the mounts are bad that will cause a vibration issue, but that isn't something that would cause a shimmy type vibration. Look for spots on your rims that might have lost a wheel wieght.

When you're driving at 40 to 50 MPH and you feel the shimmy, take it out of gear and let the engine idle down. Do you still feel the same vibration with the engine slowed down? If not, it may be engine and or transmission mounts. If the vibration stays there, have the tires rotated an balanced. If the vibration continues, have the tires checked to make sure the tire cap is not letting go. I had a vibration I couldn't find in my car until I changed tires. Mine had less that 4,000 miles on them and looked new, but were 10 years old and the tire was starting to come apart. If you have hit road bumps or grand canyon size p;ot holes, you may have damaged the rims or tires. Have your front suspension checked by a trusted mechanic as you may have worn or damaged suspension parts.