Forced vibrations of damped single degree freedom systems?

They don't, a single m-spring-damper system (1 DOF) will only ever vibrate at a single frequency. The damped free vibration (your first example) is cause by the initial displacement force k*x whereas in a forced vibration system the vibration is created by a time variable force f(t). Simply put, a 1 DOF system only vibrates in 1 mode