Vibration Analysis of Flexible Shafts with Active Magnetic Bearings

Abstract

In this paper, primary resonance of a flexible rotor levitated by active magnetic bearings (AMBs) is investigated. The shaft has a linear flexural motion, but the AMB has nonlinear characteristic. Although the equations of motion are linear, the boundary conditions are nonlinear. The renormalization group method is directly applied to partial differential equations of motion with nonlinear boundary conditions. The effect of bearing parameters on the frequency response of system is investigated, and it is shown that near the primary resonances only the forward modes are excited. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon and softening nonlinearity.

Appendix

• 1. Dimensionless quantities are defined as

  $$ \begin{aligned} s^{*} & = \frac{x}{l},\,\,v^{*} = \frac{v}{l},\,\,w^{*} = \frac{w}{l},\,\,t^{*} = \sqrt {\frac{{D_{66} }}{{ml^{4} }}} t,\,\,C^{*} = \frac{{l^{2} }}{{\sqrt {mD_{66} } }}C,\,\,I_{2}^{*} = \frac{{I_{D} }}{{ml^{2} }},\,\,\Omega ^{*} = \sqrt {\frac{{ml^{4} }}{{D_{66} }}}\Omega ,\,\, \\ \alpha_{1}^{*} & = \frac{{l^{3} }}{{D_{66} }}\alpha_{1} ,\,\,\alpha_{2}^{*} = \frac{{l^{5} }}{{D_{66} }}\alpha_{2} ,\,\alpha_{3}^{*} = \frac{{l^{5} }}{{D_{66} }}\alpha_{3} ,\,\,\mu_{1}^{*} = \,\frac{l}{{\sqrt {mD_{66} } }}\mu_{1} ,\,\,\mu_{2}^{*} = \,\frac{{l^{3} }}{{\sqrt {mD_{66} } }}\mu_{2} ,\,\, \\ \mu_{3}^{*} & = \,\frac{{l^{3} }}{{\sqrt {mD_{66} } }}\mu_{3} ,\,\,\mu_{4}^{*} = \frac{l}{m}\mu_{4} ,\,\,\mu_{5}^{*} = \frac{l}{m}\mu_{5} ,\,\,\mu_{6}^{*} = \,\frac{{l^{3} }}{{\sqrt {mD_{66} } }}\mu_{6} . \\ \end{aligned} $$

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• 2. Functions defined in Eqs. (34)–(35) are:

  $$ \begin{aligned} H_{1} & = I\beta_{\text{f}} \mu_{1} A_{1} \phi (0) + \frac{1}{4}\left[ {\left( {3\bar{A}_{1} A_{1}^{2} + 6\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\alpha_{2} + \left( {\bar{A}_{1} A_{1}^{2} + 2\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\alpha_{3} } \right. \\ & \quad + \,I\left( {\bar{A}_{1} A_{1}^{2} + 2\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\beta_{\text{f}} \mu_{2} + I\left( {2\beta_{\text{f}}^{{}} \bar{A}_{2} A_{2} A_{1}^{{}} - 4\beta_{\text{b}}^{{}} \bar{A}_{2} A_{1}^{{}} A_{2} + 3\beta_{\text{f}}^{{}} A_{1}^{2} \bar{A}_{1}^{{}} } \right)\mu_{3} \\ & \quad + \,\left( {2\beta_{\text{b}}^{2} \bar{A}_{2} A_{1}^{{}} A_{2} + 3\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{2} - 4\beta_{\text{f}}^{{}} \beta_{\text{b}}^{{}} A_{2} \bar{A}_{2} A_{1}^{{}} } \right)\mu_{4} + \left( {\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{2} + 2\beta_{\text{b}}^{2} A_{2} \bar{A}_{2} A_{1}^{{}} } \right)\mu_{5} \\ & \quad \left. { + \,I\left( {2A_{1} A_{2} \bar{A}_{2} \beta_{\text{b}} - A_{1}^{2} \bar{A}_{1} \beta_{\text{f}} } \right)\mu_{6} } \right]\phi^{3} (0), \\ H_{2} & = - I\beta_{\text{b}} \mu_{1} A_{2} \phi (0) + \frac{1}{4}\left[ {\left( {3\bar{A}_{2} A_{2}^{2} + 6\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\alpha_{2} + \left( {\bar{A}_{2} A_{2}^{2} + 2\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\alpha_{3} } \right. \\ & \quad - \,I\left( {\bar{A}_{2} A_{2}^{2} + 2\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\beta_{\text{b}} \mu_{2} - I\left( {2\beta_{\text{b}}^{{}} \bar{A}_{1} A_{2} A_{1}^{{}} - 4\beta_{f}^{{}} \bar{A}_{1} A_{1}^{{}} A_{2} + 3\beta_{\text{b}}^{{}} A_{2}^{2} \bar{A}_{2}^{{}} } \right)\mu_{3} \\ & \quad + \,\left( {2\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{{}} A_{2} + 3\beta_{\text{b}}^{2} \bar{A}_{2} A_{2}^{2} - 4\beta_{\text{b}}^{{}} \beta_{\text{f}}^{{}} A_{2} \bar{A}_{1} A_{1}^{{}} } \right)\mu_{4} + \left( {\beta_{\text{b}}^{2} \bar{A}_{2} A_{2}^{2} + 2\beta_{\text{f}}^{2} A_{2} \bar{A}_{1} A_{1}^{{}} } \right)\mu_{5} \\ & \quad \left. { - \,I\left( {2A_{1} A_{2} \bar{A}_{1} \beta_{\text{f}} - A_{2}^{2} \bar{A}_{2} \beta_{\text{b}} } \right)\mu_{6} } \right]\phi^{3} (0), \\ H_{3} & = - I\beta_{\text{f}} \mu_{1} A_{1} \phi (1) - \frac{1}{4}\left[ {\left( {3\bar{A}_{1} A_{1}^{2} + 6\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\alpha_{2} + \left( {\bar{A}_{1} A_{1}^{2} + 2\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\alpha_{3} } \right. \\ & \quad + \,I\left( {\bar{A}_{1} A_{1}^{2} + 2\bar{A}_{2} A_{1}^{{}} A_{2} } \right)\beta_{\text{f}} \mu_{2} + I\left( {2\beta_{\text{f}}^{{}} \bar{A}_{2} A_{2} A_{1}^{{}} - 4\beta_{\text{b}}^{{}} \bar{A}_{2} A_{1}^{{}} A_{2} + 3\beta_{\text{f}}^{{}} A_{1}^{2} \bar{A}_{1}^{{}} } \right)\mu_{3} \\ & \quad + \,\left( {2\beta_{\text{b}}^{2} \bar{A}_{2} A_{1}^{{}} A_{2} + 3\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{2} - 4\beta_{\text{f}}^{{}} \beta_{\text{b}}^{{}} A_{2} \bar{A}_{2} A_{1}^{{}} } \right)\mu_{4} + \left( {\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{2} + 2\beta_{\text{b}}^{2} A_{2} \bar{A}_{2} A_{1}^{{}} } \right)\mu_{5} \\ & \quad \left. { + \,I\left( {2A_{1} A_{2} \bar{A}_{2} \beta_{\text{b}} - A_{1}^{2} \bar{A}_{1} \beta_{\text{f}} } \right)\mu_{6} } \right]\phi^{3} (1), \\ H_{4} & = I\beta_{\text{b}} \mu_{1} A_{2} \phi (1) - \frac{1}{4}\left[ {\left( {3\bar{A}_{2} A_{2}^{2} + 6\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\alpha_{2} + \left( {\bar{A}_{2} A_{2}^{2} + 2\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\alpha_{3} } \right. \\ & \quad - \,I\left( {\bar{A}_{2} A_{2}^{2} + 2\bar{A}_{1} A_{1}^{{}} A_{2} } \right)\beta_{\text{b}} \mu_{2} - I\left( {2\beta_{\text{b}}^{{}} \bar{A}_{1} A_{2} A_{1}^{{}} - 4\beta_{\text{f}}^{{}} \bar{A}_{1} A_{1}^{{}} A_{2} + 3\beta_{\text{b}}^{{}} A_{2}^{2} \bar{A}_{2}^{{}} } \right)\mu_{3} \\ & \quad + \,\left( {2\beta_{\text{f}}^{2} \bar{A}_{1} A_{1}^{{}} A_{2} + 3\beta_{\text{b}}^{2} \bar{A}_{2} A_{2}^{2} - 4\beta_{\text{b}}^{{}} \beta_{\text{f}}^{{}} A_{2} \bar{A}_{1} A_{1}^{{}} } \right)\mu_{4} + \left( {\beta_{\text{b}}^{2} \bar{A}_{2} A_{2}^{2} + 2\beta_{\text{f}}^{2} A_{2} \bar{A}_{1} A_{1}^{{}} } \right)\mu_{5} \\ & \quad \left. { - \,I\left( {2A_{1} A_{2} \bar{A}_{1} \beta_{\text{f}} - A_{2}^{2} \bar{A}_{2} \beta_{\text{b}} } \right)\mu_{6} } \right]\phi^{3} (1), \\ \end{aligned} $$

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