Hopf bifurcation analysis of asymmetrical rotating shafts

Abstract

The Hopf and double Hopf bifurcations analysis of asymmetrical rotating shafts with stretching nonlinearity are investigated. The shaft is simply supported and is composed of viscoelastic material. The rotary inertia and gyroscopic effect are considered, but, shear deformation is neglected. To consider the viscoelastic behavior of the shaft, the Kelvin–Voigt model is used. Hopf bifurcations occur due to instability caused by internal damping. To analyze the dynamics of the system in the vicinity of Hopf bifurcations, the center manifold theory is utilized. The standard normal forms of Hopf bifurcations for symmetrical and asymmetrical shafts are obtained. It is shown that the symmetrical shafts have double zero eigenvalues in the absence of external damping, but asymmetrical shafts do not have. The asymmetrical shaft in the absence of external damping has a saddle point, therefore the system is unstable. Also, for symmetrical and asymmetrical shafts, in the presence of external damping at the critical speeds, supercritical Hopf bifurcations occur. The amplitude of periodic solution due to supercritical Hopf bifurcations for symmetrical and asymmetrical shafts for the higher modes would be different, due to shaft asymmetry. Consequently, the effect of shaft asymmetry in the higher modes is considerable. Also, the amplitude of periodic solutions for symmetrical shafts with rotary inertia effect is higher than those of without one. In addition, the dynamic behavior of the system in the vicinity of double Hopf bifurcation is investigated. It is seen that in this case depending on the damping and rotational speed, the sink, source, or saddle equilibrium points occur in the system.

Appendix

$$\begin{aligned}&\Theta _1 =\frac{\left( c+(1+\Delta )\mu \omega _k^2 \right) }{(I_q +\omega _k \Delta _I )} ,\quad \Theta _2 =\frac{\omega _k^2 (1+\Delta )}{(I_q +\omega _k \Delta _I )} ,\\&\Theta _3 =\frac{(I_q -\omega _k I_p -\omega _k \Delta _I )}{(I_q +\omega _k \Delta _I )} \\&\Theta _4 =\frac{(2I_q -\omega _k I_p )}{(I_q +\omega _k \Delta _I )} ,\quad \Theta _5 =\frac{c}{(I_q +\omega _k \Delta _I )} ,\\&\Theta _6 =\frac{\omega _k \alpha }{(I_q +\omega _k \Delta _I )} , \\&\Theta _7 =\frac{\left( c+(1-\Delta )\mu \omega _k^2 \right) }{(I_q -\omega _k \Delta _I )}\Theta _8 =\frac{\omega _k^2 (1-\Delta )}{(I_q -\omega _k \Delta _I )} ,\\&\Theta _9 =\frac{(I_q -\omega _k I_p +\omega _k \Delta _I )}{(I_q -\omega _k \Delta _I )} \\&\Theta _{10} =\frac{(2I_q -\omega _k I_p )}{(I_q -\omega _k \Delta _I )} ,\quad \Theta _{11} =\frac{c}{(I_q -\omega _k \Delta _I )} ,\\&\Theta _{12} =\frac{\omega _k \alpha }{(I_q -\omega _k \Delta _I )} \\&C_1 =-\omega (\sigma )d_{30} -2c_{30} \delta (\sigma )-\omega (\sigma )c_{21} +a_1 \\&C_2 =-d_{21} \omega (\sigma )-2c_{21} \delta (\sigma )-2c_{12} \omega (\sigma )\\&\quad +3c_{30} \omega (\sigma )+b_1 \\&C_3 =-d_{12} \omega (\sigma )-3c_{03} \omega (\sigma )+2c_{21} \omega (\sigma )\\&\quad \quad -2c_{12} \delta (\sigma )+a_1 \\&C_4 =-d_{03} \omega (\sigma )+c_{12} \omega (\sigma )-2c_{03} \delta (\sigma )+b_1 \\&C_5 =c_{30} \omega (\sigma )-2d_{30} \delta (\sigma )-\omega (\sigma )d_{21} -b_2 \\&C_6 =c_{21} \omega (\sigma )-2d_{21} \delta (\sigma )-2d_{12} \omega (\sigma )\\&+3d_{30} \omega (\sigma )+a_2 \\&C_7 =c_{12} \omega (\sigma )-3d_{03} \omega (\sigma )\\&\quad +2d_{21} \omega (\sigma )-2d_{12} \delta (\sigma )-b_2 \\&C_8 =c_{03} \omega (\sigma )+d_{12} \omega (\sigma )-2d_{03} \delta (\sigma )+a_2 \\&\Lambda _1 =\frac{-\Theta _{6,n} \omega _n \sigma _1 }{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\Lambda }}_1 =\frac{-\Theta _{6,n+1} \omega _{n+1} \sigma _3 }{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\xi _1 =\frac{-\Theta _{6,n} \omega _{n+1} \sigma _1 }{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\xi }}_1 =\frac{-\Theta _{6,n+1} \omega _n \sigma _3 }{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\Gamma _1 =\frac{\Theta _{12,n} \omega _n (\sigma _2 -\omega _{\mathrm{cr},n} )}{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\Gamma }}_1 =\frac{\Theta _{12,n+1} \omega _{n+1} (\sigma _4 -\omega _{\mathrm{cr},n+1} )}{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\eta _1 \!=\!\frac{\Theta _{12,n} \omega _{n+1} (\sigma _2 -\omega _{\mathrm{cr},n} )}{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\eta }}_1 =\frac{\Theta _{12,n+1} \omega _n (\sigma _4 -\omega _{\mathrm{cr},n+1} )}{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\Lambda _2 =\frac{-\Theta _{12,n} \omega _n \sigma _1 }{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\Lambda }}_2 =\frac{-\Theta _{12,n+1} \omega _{n+1} \sigma _3 }{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\xi _2 =\frac{-\Theta _{12,n} \omega _{n+1} \sigma _1 }{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\xi }}_2 =\frac{-\Theta _{12,n+1} \omega _n \sigma _3 }{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\Gamma _2 =\frac{\Theta _{6,n} \omega _n (\sigma _2 -\omega _{\mathrm{cr},n} )}{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\Gamma }}_2 =\frac{\Theta _{6,n+1} \omega _{n+1} (\sigma _4 -\omega _{\mathrm{cr},n+1} )}{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\&\eta _2 =\frac{\Theta _{6,n} \omega _{n+1} (\sigma _2 -\omega _{\mathrm{cr},n} )}{(\sigma _2 -\omega _{\mathrm{cr},n} )^{2}+\sigma _1^2 },\quad \hat{{\eta }}_2 =\frac{\Theta _{6,n+1} \omega _n (\sigma _4 -\omega _{\mathrm{cr},n+1} )}{(\sigma _4 -\omega _{\mathrm{cr},n+1} )^{2}+\sigma _3^2 } \\ \end{aligned}$$