This happens when I start to accelerate from a complete stop or under 40 mph when I accelerate somewhat hard
My car has 156000 miles.
My car has an automatic transmission.
It sounds like you may possibly have a broken motor mount. As you may know, motor mounts are steel mounted brackets that secure the motor and transmission to the chassis of the vehicle. They have rubber internals in them to help isolate engine vibration from being transmitted to the vehicle’s frame. Over time, these rubber internals will develop cracks and eventually break or deteriorate causing the mount to no longer function as it was originally designed to. When these fail, you may notice the vehicle vibrating more than usual which can result in additional damage to other components if not addressed. I would suggest having a professional from YourMechanic come to your location to diagnose and repair your vehicle.