The deck belt can cause vibration when the deck is running. Sometimes a flat spot gets burned into the belt should something stop a blade. bent blades can be a cause also. the deck belt is a part # 954-0371A. It has two traction belts. The belts could be worn, and the front drive pulley could be glazed plus the VS pulley in the back could be sluggish. Do an inspection first. Ideally watch over the idler pulley bearings- more so the rear one. If you replace traction belts, MTD wants you to replace the set. You can get by with doing a rear only, but a front only isn't good. There are a lot to do to replace the drive belts. You'll find a blind peg to hop over and tight clearances to get the belts through. Those #s are 954-04001a 954-04002. Lowes strangely enough has good $'s on the OEM belts if they have them. Good luck.

more likely one or more of the deck blades are out of balance. Re-balance/replace as needed.
Also check all blade quills for absence of loose brg play.

slipping going downhill is a separate issue - addressed by other reply from plum.