Effect of a parametric damping on nonlinear dynamics of a symmetric heavy gyroscope

Chaotic dynamics, coexistence of attractors and nonlinear resonances are some of the most complex phenomena that affect the operation and performance of complex systems in general, and rotating machines in particular. This work deals with the analysis of the route to chaos and the coexistence of the attractors of a parametrically damped gyroscope. After having modeled the dynamics of the gyroscope under the influence of the control force, using numerical simulations, the types of resonances are sought, and then the amplitudes and the frequencies of the oscillations are determined. Subsequently, the study and the control of chaotic dynamics and the coexistence of gyroscope attractors are carried out using bifurcation diagrams, Lyapunov exponents and phase portraits using red fourth-order Runge–Kutta algorithm. It is shown that a high value of the parameter \(\alpha\) $\alpha$ considerably reduces the amplitude and the resonant frequency of the gyroscope and therefore can eliminate the resonance of the gyroscope when \(\Omega =\omega\) $\mathrm{\Omega }=\omega$ , \(\Omega =2\omega\) $\mathrm{\Omega }=2\omega$ and \(\Omega =3\omega\) $\mathrm{\Omega }=3\omega$ , while a large value of the parameter \(\beta\) $\beta$ or of the amplitude f of periodic excitation accentuates the resonance of the gyroscope in these cases. Similarly, an increase in the amplitude h of the parametric damping increases the resonance amplitude. Finally, the chaotic behavior and the coexistence of the attractors of the gyroscope studied depend on the values of each of the parameters \(\alpha\) $\alpha$ , \(\beta , f\) $\beta ,f$ , h and the frequencies of the parametric excitation and the damping parametric force.