A method and related device for machining a workpiece is provided, wherein material of the workpiece is removed in a machining region by a non-contact removal process. The workpiece is excited by means of an excitation oscillation having a natural resonant frequency of the workpiece, or a composite comprising the workpiece and a coupling element that is rigidly coupled to the workpiece is excited by means of an excitation oscillation having a joint natural resonant frequency of the workpiece and the coupling element. The natural resonant frequency is selected such that in the machining region an oscillation occurs that has a maximum oscillation amplitude in relation to the excitation amplitude of the excitation oscillation.