2001 Accura MDX. Shimmy in the steering that starts about 60 MPH. It has a 30 second cycle. It increases in intensity for about 30 seconds, then rapidly decreases in intensity. Then the cycle begins again.
Have you had the alignment and balance of all four wheels checked?
How old are the tires?
Do You Live In The “Salt Belt?” Has This Car Had A Recall Performed On The Front Damper Springs?
Cars in salt belt states may qualify. This could be a starting point for locating the shimmy. Call a dealer if you are unsure about the recall. Have your VIN (vehicle Identification Number) handy. Maybe those fresh springs and an inspection could help.
Possibly fresh tires and a balance would be next, as McParadise advises.