<p class=\"MsoNormal\" style=\"margin-bottom: .0001pt; text-align: justify; text-indent: 1.0cm;\"><span style=\"font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\">The dynamics of vertical unbalanced gyroscope rotor with hard nonlinear resilient characteristics is considered. For the purpose of investigation of rotor vibrations, its dynamic and mathematical model is constructed. The expressions of rotor energy, potential energy of the resilient mounting, external couples as well as dissipation function of </span><span style=\"font-size: 10pt; line-height: 115%; font-family: \'Times New Roman\', serif; background-position: initial initial; background-repeat: initial initial;\" lang=\"EN\">Rayleigh</span><span style=\"font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\">are made. The motion equations are recorded as equations of</span><span class=\"apple-converted-space\"><span style=\"font-family: \'Times New Roman\', serif; color: #444444; background-position: initial initial; background-repeat: initial initial;\" lang=\"EN-US\">&nbsp;</span></span><span style=\"font-size: 10pt; line-height: 115%; font-family: \'Times New Roman\', serif; background-position: initial initial; background-repeat: initial initial;\" lang=\"EN-US\">Lagrange of the 2<sup>nd</sup> kind</span><span style=\"font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\">. The expressions of amplitude and phase of the main harmonic component are expressed using expansion of solutions to the equations of expressed vibrations as well as harmonic balance to the series</span><span style=\"font-size: 9.0pt; mso-bidi-font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\"> of </span><span style=\"font-size: 10pt; line-height: 115%; font-family: \'Times New Roman\', serif; background-position: initial initial; background-repeat: initial initial;\" lang=\"EN-US\">Fourier</span><span style=\"font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\">. The equations as variations, and then the equations of Hill type, are recorded to study hardiness on the basis of motion equations. According to the theory of <span style=\"background-position: initial initial; background-repeat: initial initial;\">Floquet,</span> these equations are solved, and the stability criterion is found using the harmonic balance method. We concentrate on study of skewing influence, mass imbalance, disk thickness, size of nonlinear characteristics of the resilient mounting as well as external buffering to amplitude and phase and frequency characteristics as well as area boundaries of variability of the main resonance vibration. On the basis of analysis of research results the particularities of dependencies of the amplitude and phase of the main resonance vibration and instability region boundaries are accentuated and described from the angle between the imbalance lines, disk thickness, external resistance and shaft speed without the hard nonlinear resilient characteristics of the rotor. They can be important for development of optimal control of resonance vibrations, determination of optimal construction parameters, working speed range and rotor balancing methods in the pre-design works. The main working expressions are presented in a compact and non-dimensional form.</span></p>
 
<p class=\"MsoNormal\" style=\"text-align: justify; text-indent: 1.0cm;\"><span style=\"font-size: 10.0pt; line-height: 115%; font-family: \'Times New Roman\',\'serif\'; mso-ansi-language: EN-US;\" lang=\"EN-US\">Keywords: gyroscope rotor, nonlinear characteristics, resonance vibration, instability region, mass imbalance, disk skewing.&nbsp;</span></p>