If it is a manual transmission I would suspect you have clutch problems. Possibly gear lube or hydraulic clutch fluid has leaked onto the clutch surface. The only repair is to replace the clutch and repair the leak. When you let the clutch out this causes the clutch to chatter thus causing the shaking.
Your front, left or right axle needs to be replaced.
A worn out Universal joint in the rear driveshaft.
Check all front end components for wear. Then check the engine and transmission mounts as the engine could be shifting and causing the shake.
A vibration in the front end of a Volvo S60 while accelerating is due to imbalanced tires. Swap the tires until the right combination is achieved.
Shaking when accelerating and braking show a front end issue. Check the lug nuts to make sure your tires are bolted on tight. Check for play in front end components, like the wheel bearings and tie rod ends. While possibly causing the vibration, the front brakes usually only cause a shake when braking.
If your car shakes while accelerating check your motor mounts. Then check your transmission.
bad wheel bearings
Check for a busted engine or tranny mount.
hi it sounds like your universail joints or tail shaft
You rolled over a batch of chewing gum which then had gravel stuck to it.
A car accelerates from to at a rate of How far does it travel while accelerating
One possibility is you might have one bad tire or a u-joint.