Abstract: Longitudinal vibration frequencies of the first to the sixth orders modes of stepped ultrasonic horns,with and without taper-transitional section,were calculated.Computations are performed with MATLAB programming based on the one-dimensional theory,and with ANSYS based on the finite element method(FEM),respectively.Systematic studies on the natural frequencies of horns are carried out,and some new rules obtained,with variations of the length ratio and section diameter ratio of the horn.Reduction of the length of thick rod or the length of transitional section can lower the first order natural frequencies of the horn,which is useful in adjusting the frequency response of a horn during machining processes.