The aim of this work is to develop theoretical grounds for experimental determination of the rigidity parameters of nanoobjects. An efficient way of determining elastic moduli used in macromechanics consists in measuring the eigenfrequencies of an object under testing. Details of applying this approach to nanoobjects are discussed. A method of experimental determination of the rigidity parameters is suggested that is based on dynamic damping of vibrations (antiresonance). This method provides the possibility of separating out the eigenfrequencies of the nanoobject from the frequency spectrum of a system composed of a nanoobject and the cantilever of an atomic force microscope.