Parametric Instability of an Electromechanically Coupled Rotor-Bearing System Subjected to Periodic Axial Loads: A Preliminary Theoretical Analysis

Abstract

The parametric instability of an electromechanically coupled single-span rotor-bearing system subjected to periodic axial loads is studied. Here, the rotor system is equipped with two piezoelectric dampers, which has been developed in our previous work. The so-called electromechanically coupled characteristic is namely derived from that damper. By using assumed mode method and Lagrange equation, the equations of motion are derived. The multiple scales method is utilized to obtain the analytical instability boundaries. Numerical simulations based on the discrete state transition matrix method (DSTM) are conducted to verify the analytical results. With the comparison between analytical results and simulated results, we find that the additional combination instability regions are created due to the usage of piezoelectric dampers.

Appendices

Appendix A

The detailed expressions of ${\tilde{\mathbf{M}}}$, ${\tilde{\mathbf{G}}}$, ${\tilde{\mathbf{K}}}$, $\stackrel{\mathrm{\sim }}{\text{D}}$, $\stackrel{\mathrm{\sim }}{\text{S}}$, ${{\tilde{\mathbf{F}}}}_{\text{v}}$, ${{\tilde{\mathbf{F}}}}_{\text{w}}$ are given as follows:

where

$$ \begin{gathered} \tilde{m}_{ij} = \int_{0}^{L} {\rho A\;\phi_{i} \left( x \right)\phi_{j} \left( x \right){\text{d}}x} + \int_{0}^{L} {\rho I_{D} \phi^{\prime}_{i} \left( x \right)\phi^{\prime}_{j} \left( x \right){\text{d}}x} + M\phi_{i} \left( {x_{0} } \right)\phi_{j} \left( {x_{0} } \right) + J_{d} \phi^{\prime}_{i} \left( {x_{0} } \right)\phi^{\prime}_{j} \left( {x_{0} } \right) \hfill \\ \tilde{m}_{55} = \tilde{m}_{66} = 2\cot^{2} \beta \theta_{p}^{2} L \hfill \\ \tilde{g}_{ij} = \int_{0}^{L} {\rho \Omega I_{p} \phi^{\prime}_{i} \left( x \right)\phi^{\prime}_{j} \left( x \right){\text{d}}x} + J_{p} \Omega \phi^{\prime}_{i} \left( {x_{0} } \right)\phi^{\prime}_{j} \left( {x_{0} } \right)\;\;\left( {i,j = 1,\;2} \right) \hfill \\ \end{gathered} $$

where

$$ \begin{gathered} \tilde{k}_{ij} = \int_{0}^{L} {EI_{D} \phi^{\prime\prime}_{i} \left( x \right)\phi^{\prime\prime}_{j} \left( x \right){\text{d}}x} + k_{eq} \left[ {\phi_{i} \left( 0 \right)\phi_{j} \left( 0 \right) + \phi_{i} \left( L \right)\phi_{j} \left( L \right)} \right] - \chi P_{cr} \tilde{s}_{ij} \hfill \\ \tilde{k}_{55} = \tilde{k}_{66} = k_{eq2} + \frac{{2\cot^{2} \beta \theta_{p}^{2} }}{C},\;\;\tilde{k}_{13} = \tilde{k}_{31} = - k_{eq} \phi_{1} \left( 0 \right),\;\;\tilde{k}_{14} = \tilde{k}_{41} = - k_{eq} \phi_{1} \left( L \right),\;\; \hfill \\ \tilde{k}_{23} = \tilde{k}_{32} = - k_{eq} \phi_{2} \left( 0 \right),\;\;\tilde{k}_{24} = \tilde{k}_{42} = - k_{eq} \phi_{2} \left( L \right) \hfill \\ \tilde{d}_{ij} = \eta \left[ {\phi_{i} \left( 0 \right)\phi_{j} \left( 0 \right) + \phi_{i} \left( L \right)\phi_{j} \left( L \right)} \right] \hfill \\ \tilde{d}_{55} = \tilde{d}_{66} = 2\cot^{2} \beta \theta_{p}^{2} R,\;\;\tilde{d}_{13} = \tilde{d}_{31} = - \eta \phi_{1} \left( 0 \right),\;\;\tilde{d}_{14} = \tilde{d}_{41} = - \eta \phi_{1} \left( L \right),\;\; \hfill \\ \tilde{d}_{23} = \tilde{d}_{32} = - \eta \phi_{2} \left( 0 \right),\;\;\tilde{d}_{24} = \tilde{d}_{42} = - \eta \phi_{2} \left( L \right) \hfill \\ \end{gathered} $$

and

$$ \left\{ \begin{gathered} {\tilde{\mathbf{F}}}_{v} = M\Omega^{2} e\left[ {\phi_{1} \left( {x_{0} } \right)\cos \left( {\Omega t + \phi } \right),\;\phi_{2} \left( {x_{0} } \right)\cos \left( {\Omega t + \phi } \right),\;0,\;0,\;0,\;0} \right]^{{\text{T}}} \hfill \\ {\tilde{\mathbf{F}}}_{w} = M\Omega^{2} e\left[ {\phi_{1} \left( {x_{0} } \right)\sin \left( {\Omega t + \phi } \right),\;\phi_{2} \left( {x_{0} } \right)\sin \left( {\Omega t + \phi } \right),\;0,\;0,\;0,\;0} \right]^{{\text{T}}} \hfill \\ \end{gathered} \right. $$

Appendix B

For simplicity, only the mode vector r_i is derived here and the derivation process of s_k is similar. In Eq. (21a), the matrix equation can be detailedly expressed as

$$ \left( {\begin{array}{*{20}c} { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{11} + \alpha_{1} } & {\omega_{Fi} \overline{g}_{12} } & \cdots & {\omega_{Fi} \overline{g}_{1N} } \\ {\omega_{Fi} \overline{g}_{21} } & { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{11} + \alpha_{1} } & \cdots & {\omega_{Fi} \overline{g}_{2N} } \\ \vdots & \vdots & \ddots & \vdots \\ {\omega_{Fi} \overline{g}_{N1} } & {\omega_{Fi} \overline{g}_{N2} } & \cdots & { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{NN} + \alpha_{N} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r_{1i} } \\ {r_{2i} } \\ \vdots \\ {r_{Ni} } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} 0 \\ 0 \\ \vdots \\ 0 \\ \end{array} } \right) $$

Assume r_1i = 1, then the other elements of r_i can be calculated by the following N − 1 linear algebraic equations

$$ \begin{gathered} \left( {\begin{array}{*{20}c} { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{22} + \alpha_{2} } & {\omega_{Fi} \overline{g}_{22} } & \cdots & {\omega_{Fi} \overline{g}_{2N} } \\ {\omega_{Fi} \overline{g}_{32} } & { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{33} + \alpha_{3} } & \cdots & {\omega_{Fi} \overline{g}_{3N} } \\ \vdots & \vdots & \ddots & \vdots \\ {\omega_{Fi} \overline{g}_{N2} } & {\omega_{Fi} \overline{g}_{N3} } & \cdots & { - \omega_{Fi}^{2} + \omega_{Fi} \overline{g}_{NN} + \alpha_{N} } \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {r_{2i} } \\ {r_{3i} } \\ \vdots \\ {r_{Ni} } \\ \end{array} } \right) \hfill \\ = \left( {\begin{array}{*{20}c} { - \omega_{Fi} \overline{g}_{21} } \\ { - \omega_{Fi} \overline{g}_{31} } \\ \vdots \\ { - \omega_{Fi} \overline{g}_{N1} } \\ \end{array} } \right) \hfill \\ \end{gathered} $$