Shimmy on acceleration can be caused by an inner CV axle joint. Sometimes it can feel like an 'up/down' shake instead of a side-to-side shake.
The right inner axle is the worst offender as it is about an inch or so away from the catalytic converter. After a few years, the CV grease can cook down into a varnish.
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Bearings may roar, but usually won't cause a shake.
Separation of lower control arm bushings can weaken dampening & rigidity and 'amplify' a shake.
The right inner axle is the worst offender as it is about an inch or so away from the catalytic converter. After a few years, the CV grease can cook down into a varnish.

Car Vibration When Accelerating: Suspect Your CV Joint
The cause of car vibrations can be difficult to pinpoint because of the vast number of potential causes. But if your car vibrates only when accelerating, you may have a CV joint issue. Address it and you won't be afraid to drive your car again.
Separation of lower control arm bushings can weaken dampening & rigidity and 'amplify' a shake.
