Steady-State and Bifurcation Analysis of Nonlinear Jumps in a Non-ideal Rotor System Using Magnetorheological Fluid Dampers

Abstract

Purpose

Non-ideal high-speed rotors often exhibit the Sommerfeld effect characterized by nonlinear jumps and eventually gets destabilized. This paper presents a bifurcation analysis to attenuate the jumps in a non-ideal internally damped DC motor-driven shaft-disk system via magnetorheological (MR) fluid damper.

Methods

A nonlinear hyperbolic tangent model of MR damper is proposed and linearized using equivalent linearization technique. To meet the demands of reliable engineering design, the system parameters of the proposed MR model are optimized using genetic algorithm (GA) with the consideration of parametric uncertainty. Then the system equations of the non-ideal rotor system are derived using Lagrangian formulation. Following, a characteristic equation of fifth-order polynomial in rotor speed is obtained through energy balance. The steady-state response is studied with the help of control current of the MR damper and subsequently verified through a bifurcation analysis with the help of root locus technique. A few results are also validated with earlier works.

Results

The nonlinear jumps are found to be attenuated as the control current of MR damper increases. The root locus technique confirms the jump phenomena though the existence of multiplicity of roots of the characteristic equation considering supply voltage as a gain. The proposed saddle-node bifurcation study confirms the cessation of the Sommerfeld effect when two unstable (saddle) points are found to be degenerated into a stable node at a specific bifurcation value of the MR control current.

Conclusion

The nonlinear jumps of non-ideal rotor can be attenuated by altering the control current of the MR damper. This study also suggests that MR-based semi-active strategy is more effective than the AMB-based active control as the former takes much lesser current to attenuate the jumps. MR-based attenuation is found to be safer and more reliable than its active counterpart, i.e., AMB as the overall natural frequency of MR-based non-ideal rotor is far more behind the instability threshold of the system.

Appendices

Appendix A

Equivalent Linearization Technique

The underlying idea of equivalent linearization is to replace the Eq. (3) with equivalent linear terms of z and $\dot{z}$ as:

$$F_{{_{{{\text{MR}}}} }}^{{{\text{Eq}}{.}}} = k_{{\text{e}}} z + c_{{\text{e}}} \dot{z},$$

(20)

where $c_{{\text{e}}}$ and $k_{{\text{e}}}$ are the equivalent damping and stiffness coefficients that best approximate the nonlinear force provided by the MR damper described in Eq. (1).

The error or the deficiency term may be expressed as:

$$e\left( {z,\dot{z}} \right) = \left( {F_{{_{{{\text{MR}}}} }}^{{{\text{Eq}}{.}}} - f_{{_{{{\text{MR}}}} }}^{z} } \right).$$

(21)

Neglecting higher-order terms, the hyperbolic tangent function may be approximated as $\tanh \left( {\delta \dot{z}} \right) \simeq \delta \dot{z} - \left( {\delta \dot{z}} \right)^{3} /3.$ Considering an arbitrary prescribed base motion $z = a_{0} \cos \left( {\overline{\omega }t} \right)$ and using the following criteria, the error $\left( {e\left( {z,\dot{z}} \right)} \right)$ given in Eq. (21) may be minimized:

$$\frac{\partial \left( E \right)}{{\partial k_{{\text{e}}} }} = 0,\frac{\partial \left( E \right)}{{\partial c_{{\text{e}}} }} = 0 \, \& \, \left[ {\frac{{\partial^{2} \left( E \right)}}{{\partial k_{{\text{e}}} \partial c_{{\text{e}}} }}} \right]^{2} - \left[ {\frac{{\partial^{2} \left( E \right)}}{{\partial k_{{\text{e}}}^{2} }}\frac{{\partial^{2} \left( E \right)}}{{\partial c_{{\text{e}}}^{2} }}} \right] < 0,$$

(22)

where $E = \int\limits_{0}^{{2\pi /\overline{\omega }}} {e\left( {z,\dot{z}} \right)^{2} {\text{d}}t} .$

Hence, the equivalent stiffness $\left( {k_{{\text{e}}} } \right)$ and damping $\left( {c_{{\text{e}}} } \right)$ coefficients can be evaluated as:

$$k_{{\text{e}}} = k_{0} + \frac{{\overline{\omega }\alpha }}{{\pi a_{0}^{2} }}\int\limits_{0}^{{2\pi /\overline{\omega }}} {\tanh \left( {\delta \dot{z}} \right)z{\text{d}}t} = k_{0} ,$$

(23)

$$c_{{\text{e}}} = c_{0} + \frac{\alpha }{{\pi \overline{\omega }a_{0}^{2} }}\int\limits_{0}^{{2\pi /\overline{\omega }}} {\tanh \left( {\delta \dot{z}} \right)} \, \dot{z}{\text{d}}t = c_{0} + \alpha \delta \left( {1 - \frac{{\delta^{2} a_{0}^{2} \overline{\omega }^{3} }}{3}} \right).$$

(24)

Appendix B

The coefficients of the transfer function given in Eq. (19) are as follows:

$$\begin{aligned} & N_{4} = m^{2} \mu_{{\text{m}}} ,N_{2} = - \mu_{{\text{m}}} \left( {2\overline{M}\overline{K} - \left( {\overline{R} - R_{{\text{i}}} } \right)_{{\text{d}}}^{2} } \right),N_{0} = \overline{K}^{2} \mu_{{\text{m}}} , \, D_{5} = \left( {m^{2} \left( {\mu_{{\text{m}}}^{2} + R_{{\text{m}}} R_{{\text{r}}} } \right) + m^{2} e^{2} R_{{\text{m}}} \overline{R}} \right), \\ & D_{3} = - \left( {\mu_{{\text{m}}}^{2} + R_{{\text{m}}} R_{{\text{r}}} } \right)\left( {2m\overline{K} - \left( {\overline{R} - R_{{\text{i}}} } \right)_{{\text{d}}}^{2} } \right){\text{ and }}D_{1} = \overline{K}^{2} \left( {\mu_{{\text{m}}}^{2} + R_{{\text{m}}} R_{{\text{r}}} } \right){. } \\ \end{aligned}$$