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

  • Mohammed AL Rkabi
  • Hamid MoeenfardEmail author
  • Jalil Rezaeepazhand
Technical Paper
  • 36 Downloads

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.

Keywords

Rotor dynamics Electro-rheological elastomer Smart supports Finite element method Vibration suppression On–off control strategy 

List of symbols

\( \left[ {\varvec{K}_{{\mathbf{B}}} } \right]_{\text{s}}^{\text{e}} \)

Bending stiffness matrix for shaft element

\( \left[ {\varvec{K}_{{\mathbf{C}}} } \right]_{\text{s}}^{\text{e}} \)

Circulatory matrix for shaft element

\( \left[ {\varvec{M}_{{\mathbf{T}}} } \right]_{\text{s}}^{\text{e}} \), \( \left[ {\varvec{M}_{{\mathbf{T}}} } \right]_{\text{d}}^{i} \)

Translational mass matrix for shaft element and disk

\( \left[ \varvec{C} \right]_{\text{b}}^{i} \)

Bearing damping matrix

\( \left\{ F \right\}_{ \sup }^{i} \)

Load vector for support

\( \left[ \varvec{G} \right]_{\text{s}}^{\text{e}} \), \( \left[ \varvec{G} \right]_{\text{d}}^{i} \)

Gyroscopic matrices for shaft element and disk

\( \left[ \varvec{K} \right]_{\text{b}}^{i} \)

Bearing stiffness matrix

\( \left[ \varvec{K} \right]_{ \sup }^{i} \)

Support stiffness matrix

\( \left\{ q \right\}_{\text{b}}^{i} \), \( \left\{ q \right\}_{\text{d}}^{i} \)

Generalized displacement vector for the bearing node and ith disk node, respectively

\( \left\{ q \right\}_{ \sup }^{i} \)

Generalized displacement vector for the support

\( \left\{ F \right\}_{\text{s}}^{\text{e}} \), \( \left\{ F \right\}_{\text{d}}^{i} \), \( \left\{ F \right\}_{\text{b}}^{i} \)

Element load vector for shaft elements, ith disk and bearing

\( \left\{ {F_{\text{c}} } \right\}, \left\{ {F_{\text{s}} } \right\} \)

Unbalance forces associated with \(\cos {{\Omega }} t\) and \( \sin {{\Omega }} t \), respectively

\( \left\{ {q_{\text{c}} } \right\},\left\{ {q_{\text{s}} } \right\} \)

Unbalance response associated with \( \cos {{\Omega }} t \) and \( \sin {{\Omega }} t \), respectively

\( \left\{ q \right\}_{\text{s}}^{\text{e}} \)

Nodal displacement vector for a shaft finite element

\( y_{\text{d}} \), \( z_{\text{d}} \)

Mass center eccentricities of the disk in Y and Z directions measured at t = 0

\( D_{\text{m}} \)

Mean diameter of ER elastomer ring

\( G^{\prime} \)

Storage modulus

\( G^{\prime\prime} \)

Loss modulus

\( G^{*} \)

Complex shear modulus

\( \left\{ F \right\} \)

System load vector

\( I_{\text{P}} \), \( I_{\text{D}} \)

Polar and diametral mass moments of inertia of the disk

\( b \), \( h \)

Width and height of the ER elastomer ring, respectively

\( c^{*} \)

Damping of ER elastomer ring

\( c_{yy} \), \( c_{yz} \), \( c_{zy} \), \( c_{zz} \)

Elements bearing damping matrix

\( k^{*} \)

Stiffness of ER elastomer ring

\( k_{yy} \), \( k_{yz} \), \( k_{zy} \), \( k_{zz} \)

Elements of bearing stiffness matrix

\( m_{\text{A}} \)

Assembly mass of ball bearing and inner steel ring

\( m_{\text{d}} \)

Mass of the disk

\( m_{\text{e}} \)

Mass of the shaft element per unit length

\( \left\{ q \right\} \)

Generalized displacement vector for the entire system

\( v, w \)

Translational displacement of a point on rotor along Y and Z axes, respectively

\( \left[ \varvec{D} \right] \)

System damping matrix

\( \left[ \varvec{K} \right] \)

System stiffness matrix

\( \left[ \varvec{M} \right] \)

System mass matrix

\( {{\Omega }} \)

Spin speed of rotor about its axis

\( \eta_{\upupsilon} \)

Viscous internal damping coefficient

\( \theta ,\psi \)

Slope of elastic line of rotor about Y and Z axes, respectively

\( {\text{EI}} \)

Bending stiffness per unit curvature

\( F\left( {{\Omega }} \right) \)

Excitation force in frequency domain

\( {\text{E}} \)

Electric field strength

\( \omega \)

Natural frequency

\( f\left( {\text{t}} \right) \)

Excitation force in time domain

\( j \)

\( \sqrt { - 1} \)

\( k \)

Ball bearing stiffness

\( l \)

Length of rotor element

\( n \)

Number of elastomer rings in parallel

\( t \)

Time

\( \eta \)

Loss factor

\( \beta \)

Ratio of width-to-height of the ER elastomer

\( r \)

Radius of the shaft element

\( k_{\text{L}} \)

Form factor influencing the stiffness of the ring

\( k_{\text{eq}} \)

Equivalent dynamic stiffness

Notes

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Mohammed AL Rkabi
    • 1
    • 2
  • Hamid Moeenfard
    • 2
    Email author
  • Jalil Rezaeepazhand
    • 3
  1. 1.Department of Mechanical EngineeringFerdowsi University of MashhadMashhadIran
  2. 2.Center of Excellence in Soft Computing and Intelligent Information ProcessingFerdowsi University of MashhadMashhadIran
  3. 3.Smart and Composite Structures Lab., Department of Mechanical EngineeringFerdowsi University of MashhadMashhadIran

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