Vibration attenuation of rotor-bearing systems using smart electro-rheological elastomer supports

Abstract

Nowadays, capability of safe operation sufficiently away from the critical speeds is one of the most important design requirements of rotating machineries. The focus of this paper is the application of smart electro-rheological (ER) elastomers to rotor dynamics field to reduce the vibration level of a rotor system. A Jeffcott rotor, supported via two bearings at both ends augmented with ER elastomers, is considered. A finite element approach, based on the Rayleigh beam theory, is used to model the dynamics of the system, and the proposed model accounts for the rotary inertia, gyroscopic effects and shaft’s internal damping. The ER elastomer supports are simulated with four-parameter viscoelastic model. The simulation results reveal that the use of ER elastomer in the conventional bearing supports leads to downshifting of the critical speeds and a considerable reduction in its corresponding vibration amplitude. Also, the stability limit speed of the system is improved by employing the ER elastomer technology. To extend the stability region of the rotor system to higher operating rotational speeds, a simple on–off control strategy is employed. The proposed control scheme determines the required real-time voltage to be applied at ER elastomers and guarantees low vibration amplitude over a wide frequency range. The novel idea of using ER elastomers for vibration suppression of rotor systems can be fairly extended to other applications which suffer from unwanted high amplitude vibrations.

Appendices

Appendix A: shaft element matrices

$$ \left[ {\varvec{K}_{{\mathbf{B}}} } \right]_{\text{s}}^{\text{e}} = \frac{\text{EI}}{{l^{3} }}\left[ {\begin{array}{*{20}c} {12} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {12} & {} & {} & {} & {\text{sym}} & {} & {} \\ 0 & { - \,6l} & {4l^{2} } & {} & {} & {} & {} & {} \\ {6l} & 0 & 0 & {4l^{2} } & {} & {} & {} & {} \\ { - \,12} & 0 & 0 & { - \,6l} & {12} & {} & {} & {} \\ 0 & { - \,12} & {6l} & 0 & 0 & {12} & {} & {} \\ 0 & { - \,6l} & {2l^{2} } & 0 & 0 & {6l} & {4l^{2} } & {} \\ {6l} & 0 & 0 & {2l{}^{2}} & { - \,6l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right] $$

(A-1)

$$ \left[ {\varvec{K}_{{\mathbf{C}}} } \right]_{\text{s}}^{\text{e}} = \frac{\text{EI}}{{l^{3} }}\left[ {\begin{array}{*{20}c} 0 & {} & {} & {} & {} & {} & {} & {} \\ { - \,12} & 0 & {} & {} & {} & {\text{skew}} & {} & {} \\ {6l} & 0 & 0 & {} & {} & {\text{sym}} & {} & {} \\ 0 & {6l} & { - \,4l^{2} } & 0 & {} & {} & {} & {} \\ 0 & { - \,12} & {6l} & 0 & 0 & {} & {} & {} \\ {12} & 0 & 0 & {6l} & { - \,12} & 0 & {} & {} \\ {6l} & 0 & 0 & {2l^{2} } & { - \,6l} & 0 & 0 & {} \\ 0 & {6l} & { - \,2l^{2} } & 0 & 0 & { - \,6l} & { - \,4l^{2} } & 0 \\ \end{array} } \right] $$

(A-2)

$$ \left[ {\varvec{M}_{{\mathbf{T}}} } \right]_{\text{s}}^{\text{e}} = \frac{{m_{\text{e}} l}}{840}\left[ {\begin{array}{*{20}c} {312} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {312} & {} & {} & {} & {\text{sym}} & {} & {} \\ 0 & { - \,44l} & {8l^{2} } & {} & {} & {} & {} & {} \\ {44l} & 0 & 0 & {8l^{2} } & {} & {} & {} & {} \\ {108} & 0 & 0 & {26l} & {312} & {} & {} & {} \\ 0 & {108} & { - \,26l} & 0 & 0 & {312} & {} & {} \\ 0 & {26l} & { - \,6l^{2} } & 0 & 0 & {44l} & {8l^{2} } & {} \\ { - \,26l} & 0 & 0 & { - \,6l^{2} } & { - \,44l} & 0 & 0 & {8l^{2} } \\ \end{array}^{{}} } \right] $$

(A-3)

$$ \left[ {\varvec{M}_{{\mathbf{R}}} } \right]_{\text{s}}^{\text{e}} = \frac{{m_{\text{e}} r^{2} }}{120l}\left[ {\begin{array}{*{20}c} {36} & {} & {} & {} & {} & {} & {} & {} \\ 0 & {36} & {} & {} & {} & {\text{sym}} & {} & {} \\ 0 & { - \,3l} & {4l^{2} } & {} & {} & {} & {} & {} \\ {3l} & 0 & 0 & {4l^{2} } & {} & {} & {} & {} \\ { - \,36} & 0 & 0 & { - \,3l} & {36} & {} & {} & {} \\ 0 & { - \,36} & {3l} & 0 & 0 & {36} & {} & {} \\ 0 & { - \,3l} & { - \,l^{2} } & 0 & 0 & {3l} & {4l^{2} } & {} \\ {3l} & 0 & 0 & { - \,l^{2} } & { - \,3l} & 0 & 0 & {4l^{2} } \\ \end{array} } \right] $$

(A-4)

$$ \left[ \varvec{G} \right]_{\text{s}}^{\text{e}} = \frac{{m_{\text{e}} r^{2} }}{60l}\left[ {\begin{array}{*{20}c} 0 & {} & {} & {} & {} & {} & {} & {} \\ {36} & 0 & {} & {} & {} & {\text{skew}} & {} & {} \\ { - \,3l} & 0 & 0 & {} & {} & {\text{sym}} & {} & {} \\ 0 & { - \,3l} & {4l^{2} } & 0 & {} & {} & {} & {} \\ 0 & {36} & { - \,3l} & 0 & 0 & {} & {} & {} \\ { - \,36} & 0 & 0 & { - \,3l} & {36} & 0 & {} & {} \\ { - \,3l} & 0 & 0 & {l^{2} } & {3l} & 0 & 0 & {} \\ 0 & { - \,3l} & { - \,l^{2} } & 0 & 0 & {3l} & {4l^{2} } & 0 \\ \end{array} } \right] $$

(A-5)

In these equations,${\text{EI}}$ is the bending stiffness per unit curvature, $l$ is the length of the rotor element, $m_{\text{e}}$ is mass of the shaft element per unit length, and $r$ is the radius of the shaft element.

Appendix B: rigid disk element matrices

It has to be noted that since a disk is placed on a single node (having four DoFs), the size of the disk element matrices will be $4 \times 4$.

$$ \left[ {\varvec{M}_{{\mathbf{T}}} } \right]_{\text{d}}^{i} = \left[ {\begin{array}{*{20}c} {m_{\text{d}} } & 0 & 0 & 0 \\ 0 & {m_{\text{d}} } & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right] $$

(B-1)

$$ \left[ {\varvec{M}_{{\mathbf{R}}} } \right]_{\text{d}}^{i} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & {I_{\text{D}} } & 0 \\ 0 & 0 & 0 & {I_{\text{D}} } \\ \end{array} } \right] $$

(B-2)

$$ \left[ \varvec{G} \right]_{\text{d}}^{i} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {I_{\text{P}} } \\ 0 & 0 & { - \,I_{\text{P}} } & 0 \\ \end{array} } \right] $$

(B-3)

In these equations, $m_{\text{d}}$ is the mass of the disk, $I_{\text{D}}$ is its diametral mass moment of inertia, and $I_{\text{P}}$ is its polar mass moment of inertia.