I recently purchased a 2011 Toyota Corolla LE with 73K miles. I love the responsiveness of the 1.8 liter engine and the gas mileage. After putting 1,500 miles on the car I began to notice that on the highway at speeds of 70 to 80 MPH the steering wheel shakes (not violently) and its shaky inside the car. Although the shaking is not violent enough to cause an accident it is concerning. I got the wheels balanced and rotated & an alignment. I still don’t feel an improvement on the highway. I was told at the alignment shop that I might want to replace the struts. The Corolla still has original struts and when I push down on the car to check the responsiveness of the struts they don’t seem very bouncy. KYB struts seem to be good quality struts for the price. What are your thoughts, have you seen this issue in Toyota Corollas.
The tires might be balanced and rotated, but that's not the end of it. The tires might be "out of round" (enough to cause vibration at high speed. They may have been poorly mounted on the rims, too.
Sometimes the problem is a sticky brake caliper (often a bit rusted, because the owner has not checked the brake fluid for water contamination. But that should produce some sort of smell, because of the brake pad contact.
The other thing it could be is that the engine is under stress, not enough power, particularly during acceleration (clogged engine air filter, for example). A scan tool should pick that up, one with live data (and short and long term fuel trim).
Ball joints are another possibility.
Having said all of that, many times a problem with the tires and/or mounting is the cause of high speed vibration.