Corolla engine shaking

Hi! I have a 2003 Toyota Corolla that occasionally (sometimes once a week, sometimes 4 times a day) will start making a low grumbling in the engine anywhere between 40 - 70 mph. When I push on the break it becomes significantly louder - it’s kinda like my engine is shaking, but it doesn’t make my entire car shake. After it starts, the only way to make it go away is to slow down to 20 mph - sometimes this works and other times I’ll have to stop completely. I’d like to bring it in to get it fixed, but since it doesn’t happen all the time, I’m not sure the mechanic will be able to recreate the problem. Any thoughts??

I’d start with the motor mounts.

Pushing on the brake making it worse could indicate a vacuum hose problem. Try to get a situation that you can replicate consistently, it will help in the diagnosis.

I think your problem is in the suspension or wheel bearings.