The development of a theoretical principle for experimentally determining the rigidity parameters of nanometer-sized objects (nanoobjects), is discussed. The various rigidity parameters such as elastic moduli, bending rigidity, and flexural rigidity of nanoobjects can be determined using the phenomenon of dynamical quenching 'antiresonance' vibrations that provide capability to isolate the eigenfrequencies of a nanoobject under study from the entire spectrum of a nanoobjects-cantiliver system of an atomic force microscope. The redistribution of vibration eigenfrequencies of the system (composed of a cantilever and a nanoobjects) between eigenfrequencies of these objects helps to overcome the problem of frequency measurement of objects. As the characteristic sizes of nanosizerod crosssection decreases more in comparison of length of rod, the average value of the amplitude in a rod segment under study should be determined.