Considering　the　interaction　between　visco-elastic　soils　and　a　variable　section　impedance　pile　with　arbitrary　bottom　boundaries,the　dynamic　response　at　the　pile　head　to　a　arbitrary　dynamic　torque　is　theoretically　investigated.　By　Laplace　transforms　and　the　transmit　property　of　impedance　function,　the　analytical　expression　of　the　impedance　function　and　solutions　of　the　displacement　and　velocity　response　function　in　frequency　domain　and　in　time　domain　at　the　pile　head　are　derived.　The　characteristic　of　interaction　between　soil　layers　is　also　theoretically　investigated　here.