1999 Dodge Caravan Shaking when excelerating between 35-45

My Caravan shakes when excelerating between 35-45 mph. The wheels & tires have been replaced & balanced (more than once) & no motor mounts are broken. I don't feel the shaking in the steering wheel or when decelerating through the same speeds. The vehicle was in an accident before I purchased it on the front passenger side fender. I do know that the front sway bars are broken, but was told that wouldn't cause the vibration. The vehicle handles ok, but the shaking is really bothersome, because I'm not sure if it is dangerous to drive in that condition. Please help give me some sort of idea what else could be causing the problem. Thank you!