Adamped harmonic oscillator driven by a harmonic external force maintains a steady oscillatory motion. is energy lost to friction in the motion? if so. what keeps the oscillator moving?

Answered

Explanation:

When you are saying "damped harmonic oscillator" it basically implies the motion of the oscillator will be damped, so the oscillation should decay with time.

For example, consider a harmonic oscillator inside water. so here the motion of the oscillator will be damped due to the friction with the water.

So, yes energy will be lost to friction in the motion.

Ok, now for your case, you are giving an external harmonic force, that will keep the oscillator moving. Also, it maintains a steady oscillatory motion, implies that whatever damping is causing because of the friction, is exactly nullified by the external harmonic force.  Hence you get the steady oscillation.