Rear end shakes between 50&60 MPH

[jjbax073]

2011 base model with new tires that have been rotated and balanced by 2 nationwide service centers. Nobody seems to have a clue as to what is causing the vibration. I would love to get it fixed if I could find someone who has an idea of where to look.

TIA

[Phil T]

Are the "Nationwide tyre centres" different places as I had a tyre fitted and they put the weights in the opposite place they should have been. Felt like driving on bricks over 40mph.

[tfc]

Get the suspension checked maybe a rubber Bush or mounting has worn check the shock absorbers

[Ravenrico]

It sounds like unbalanced rear wheel(s) to me.  I’d look at the simplest - and cheapest - possibilities first.  
Can you swap the rear wheels with the fronts?  If the problem moves to the front of the car then that points to a wheel balancing issue. 

[YarisHybrid2016]

Get the wheels (re-)balanced - ALL of them!

[TonyHSD]

Finding a source of vibrations in any car is probably the most difficult task to any mechanic. The reason for that is because in most if not all cases of vibration problems there is no fault code to begin with, secondly vibrations can be caused by anything that rotates in the car but also from any rubber bush too.  
When looking for vibrations first you will need to pinpoint from where exactly they come from, often can be mistaken front with rear axle.
Rear wheels or suspension caused vibrations will be felt on the floor and seat base, and pedals. They will come on certain speeds only.
Front axle vibrations are more pronounced at the steering wheel first, then seat and the rest. 
Vibrations caused by engine misfire are common in Toyota cars with egr system, they will be felt at 40 or 60 mph mostly and when engine is not loaded. Vibrations caused by engine or transmission mounts will be noticed also when the car is stationary and you rise rpm or shift gears or try to move the car with parking brake engaged. 
Another common fault that causes a vibrations and mechanics fail to find are stuck slider pins and seized brake callipers. These can cause violent vibrations indeed. Extra heat to the brake components and reddish residue on the wheels from a burnt out pad material. 
Ball joints, suspension arm bushing, and tie rod ends are also known to cause vibrations too. 
Tyres: here it’s another science involved. Poor quality tyres even brand new can cause vibrations and not possible to be balanced. 
Wheel alignment, out of spec wheel alignment can cause vibrations, however this never comes along and it’s accomplished by unstable behaviour on higher speeds , pulling to side, uneven tyre wear. 
I can right even more but that’s it for now and most important things to be checked. Finding vibrations are often best as diy project as it requires a lots of time for research and check, something if you pay a garage may cost more than the car itself. 
Good luck 

[Dippy]

Will any owner nearby allow you to swap (a pair of) wheels for 30 mins?