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Bifurcation analysis of a rigid-rotor squeeze film damper system with unsymmetrical stiffness supports

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Abstract

This paper is focused on the relationship between rigid body translation and rigid body precession in a squeeze film damper–rigid-rotor system with unsymmetrical stiffness supports. Two cases are considered: the precession motion non-resonance and internal resonance when translation motion occur primary resonant. In the first case, the amplitudes of translation and precession can be connected with an integration parameter about rotor parameters such as geometric size, stiffness, and mass. Some combination of system parameters will make the amplitudes of precession motion reach the same magnitude of the amplitudes of translation motion, so this integration parameter become an index to reflect the precessional motion degree, it is can be used to judge the feasibility of neglecting processional motion and simplifying asymmetry rotor as reasonable symmetry model. In the second case, the translation motion and precessional motion are strongly coupled, the vibration energy transfer between two kind of motion and the system occur internal resonant, which is possible appear in the rotor with cantilever disk. Each of case may appear nonlinear phenomenon, which is closely related with system parameters. The bifurcation analysis by using singularity methods is carried out to delimit the range of applicative operation parameters to avoid harmful phenomenon in unsymmetrical rotor system. The results of this paper provide a theoretical foundation for the convenient model simplification judgment and parameters optimization of the squeeze film damper-unsymmetrical rotor systems.

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Abbreviations

\(a_{1,2}\) :

Non-dimensional amplitudes in complex coordinates

\(\bar{{a}}_{1,2} \) :

Non-dimensional amplitudes in principal coordinates

B :

Non-dimensional bearing parameter

\({B}'\) :

Non-dimensional bearing parameter

c :

Clearance of squeeze film damper (m)

\({c}'\) :

Damping of the supports (N s/m)

e :

Radial displacement of journal (m)

\(F_{cx}\) :

Oil film force in x-direction (N)

\(F_{cy}\) :

Oil film force in y-direction (N)

\(F_{r}\) :

Oil film force in radial direction (N)

\(F_{t}\) :

Oil film force in tangential direction (N)

\(\bar{{F}}_r \) :

Non-dimensional oil film force in radial direction

\(\bar{{F}}_t \) :

Non-dimensional oil film force in tangential direction

\(J_{d}\) :

Equivalent equatorial moment of inertia (\(\hbox {kg m}^{2}\))

\(J_{p}\) :

Equivalent polar moment of inertia (\(\hbox {kg m}^{2}\))

K :

Ratio of stiffness coefficient

\(k_{1}\) :

Equivalent stiffness coefficient of the left support (N/m)

\(k_{2}\) :

Equivalent stiffness coefficient of the right support (N/m)

L :

Length of squeeze film damper (m)

\(l_{1}\) :

Distance from the disk to the left support (m)

\(l_{2}\) :

Distance from the disk to the right support (m)

m :

Equivalent mass of the rotor (kg)

\(p_{1,2}\) :

Elements of modal matrix

\(Q_{1,2}\) :

Non-dimensional displacements in complex coordinates

\(\bar{{Q}}_{1,2} \) :

Non-dimensional displacements in principal coordinates

\(q_{11,22}\) :

Elements of inverse matrix of modal matrix

R :

Radius of the journal (m)

r :

Non-dimensional radial displacement of journal

t :

Time (s)

U :

Non-dimensional integrated unbalance value

\(\alpha _{1,2} \) :

Non-dimensional stiffness parameters

\(\gamma \) :

Length parameter

\(\delta \) :

Unbalance value (kg m)

\(\zeta \) :

Non-dimensional damping parameter

\(\eta \) :

Non-dimensional moment of inertia parameter

\(\theta _{1,2} \) :

Phase angles of vibration responses

\(\kappa \) :

Non-dimensional stiffness parameters

\(\mu \) :

Kinetic viscosity coefficient of oil (\(\hbox {N s/m}^{2}\))

\(\xi _{1,2} \) :

Non-dimensional damping parameters

\(\tau \) :

Non-dimensional time

\(\psi \) :

Angular displacement of journal (rad)

\(\omega \) :

Rotor speed (rad/s)

\(\omega _{1,2}\) :

Principal frequencies

\(\omega _c \) :

Natural frequency of rigid rotor on retainer springs (rad/s)

\(\varOmega \) :

Speed parameter

\(\bar{{\varOmega }}\) :

Ratio of rotor speed

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Acknowledgements

The authors would like to acknowledge the financial supports from the National Basic Research Program (973 Program) of China (Grant No. 2015CB057400). Funding was provided by China Postdoctoral Science Foundation (Grant No. 2016M590277), National Natural Science Foundation of China (Grant No. 11602070).

Author information

Authors and Affiliations

  1. School of Astronautics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Huizheng Chen, Lei Hou & Yushu Chen

  2. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Lei Hou

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  1. Huizheng Chen
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Corresponding author

Correspondence to Lei Hou.

Appendix

Appendix

$$\begin{aligned} X_1= & {} \left[ {-2p_1 \zeta -\zeta \left( {1-\gamma } \right) +\frac{p_1 \left( {q_{22} -1} \right) }{q_{11} }\left( {\xi _{1} -\xi _{2} } \right) +\frac{(q_{22} -1)}{q_{11} }\left( {\xi _{1} +\gamma \xi _{2} } \right) } \right] \frac{\omega _1 a_1 }{p_1 +1}\\&+\left( {\frac{q_{22} -1}{q_{11} }{B}'-B} \right) \bar{{F}}_t\\ X_2= & {} \frac{2\left( {1-\bar{{\varOmega }}} \right) }{q_{11} }\frac{\omega _1 ^{2}}{p_1 }\frac{a_1 }{p_1 +1}+\frac{\left( {q_{22} -1} \right) }{q_{11} }\eta \bar{{\varOmega }}\frac{a_1 }{p_1 +1}\omega _1 ^{2}\\&+\left[ {B-\frac{(q_{22} -1)}{q_{11} }{B}'} \right] \bar{{F}}_r\\ X_3= & {} \left[ {-2p_1 \zeta -\zeta \left( {1-\gamma } \right) +\frac{p_1 \left( {q_{22} -1} \right) }{q_{11} }\left( {\xi _{1} -\xi _{2} } \right) +\frac{(q_{22} -1)}{q_{11} }\left( {\xi _{1} +\gamma \xi _{2} } \right) } \right] \frac{\omega _1 }{p_1 +1}\\&+\left( {\frac{q_{22} -1}{q_{11} }{B}'-B} \right) \frac{\partial \bar{{F}}_t }{\partial a_1 }\\ X_4= & {} \frac{2\left( {1-\bar{{\varOmega }}} \right) }{p_1 q_{11} }\frac{\omega _1 ^{2}}{p_1 +1}+\frac{\left( {q_{22} -1} \right) }{q_{11} }\eta \bar{{\varOmega }}\frac{\omega _1 ^{2}}{p_1 +1}+\left[ {B-\frac{(q_{22} -1)}{q_{11} }{B}'} \right] \frac{\partial \bar{{F}}_r }{\partial a_1 }\\ S_1= & {} -2 \omega _2 a_2 \left( {\sigma _1 +\sigma _2 } \right) +\eta \sigma _1 a_2 +2 \sigma _2 \omega _2 a_2 + \frac{2{B}' \left( {a_1 +a_2 } \right) a_2 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}, S_2 =\left( {\xi _1 -\xi _2 } \right) \omega _1 a_1 \\&- \frac{\pi {B}'a_1 }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}\\ S_3= & {} \left( {\alpha _1 -\alpha _2 } \right) a_1 + \frac{2{B}' \left( {a_1 +a_2 } \right) a_1 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}},\\ S_4= & {} \left( {\xi _1 +\gamma \xi _2 } \right) \omega _2 a_2 -\frac{\pi {B}'a_2 }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}},\\ S_5= & {} \left( {\xi _1 -\xi _2 } \right) \omega _1 a_1 + \frac{\pi {B}'a_1 }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}, S_6 = \frac{2{B}' a_2 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}+ \frac{8{B}' \left( {a_1 +a_2 } \right) ^{2}a_2 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}},\\ S_7= & {} \left( {\xi _1 -\xi _2 } \right) \omega _1 - \frac{\pi {B}'}{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}- \frac{3\pi {B}' \left( {a_1 + a_2 } \right) a_1 }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}},\\ S_8= & {} \alpha _1 -\alpha _2 + \frac{2{B}'\left( {2a_1 +a_2 } \right) }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}+ \frac{8{B}' \left( {a_1 +a_2 } \right) ^{2}a_1 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}},\\ S_9= & {} \frac{3\pi {B}'a_2 \left( { a_1 + a_2 } \right) }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}},\\ S_{10}= & {} \left( {\xi _1 -\xi _2 } \right) \omega _1 + \frac{\pi {B}'}{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}+ \frac{3\pi {B}' \left( {a_1 + a_2 } \right) a_1 }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}},\\ S_{11}= & {} -2\omega 2 \left( {\sigma _1 +\sigma _2 } \right) +\eta \sigma _1 +2 \sigma _2 \omega _2 + \frac{2{B}' \left( {a_1 +2a_2 } \right) }{\left( {1-\left( {a1+a2} \right) ^{2}} \right) ^{2}}+ \frac{8{B}' \left( {a_1 +a_2 } \right) ^{2}a_2 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}},\\ S_{12}= & {} -\frac{3\pi {B}'a_1 \left( {a_1 + a_2 } \right) }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}},\\ S_{13}= & {} \frac{2{B}' a_1 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}+ \frac{8{B}' \left( {a_1 +a_2 } \right) ^{2}a_1 }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}},\\ S_{14}= & {} \left( {\xi _1 +\gamma \xi _2 } \right) \omega _2 - \frac{\pi {B}' }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}-\frac{3\pi {B}'a_2 \left( { a_1 + a_2 } \right) }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}}, \end{aligned}$$
$$\begin{aligned} Q_1= & {} 2 \zeta \omega _1 a_1 +\zeta \left( {1-\gamma } \right) \omega _2 a_2 +\frac{B\pi \left( {a_1 +a_2 } \right) }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}},\\ Q_2= & {} 2 \zeta \omega _1 +\frac{3\pi B \left( {a_1 +a_2 } \right) ^{2}}{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}}+\frac{B\pi }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}},\\ Q_3= & {} 2 \omega _1 a_1 \sigma _1 -\kappa a_2 - \frac{2B\left( {a_1 +a_2 } \right) ^{2}}{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}},\\ Q_4= & {} 2 \omega _1 \sigma _1 - \frac{4B\left( {a_1 +a_2 } \right) }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}- \frac{8B\left( {a_1 +a_2 } \right) ^{3}}{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}},\\ Q_5= & {} \left( {1-\gamma } \right) \zeta \omega _2 + \frac{B\pi }{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3/2}}+ \frac{3B\pi \left( {a_1 +a_2 } \right) ^{2}}{2\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{5/2}},\\ Q_6= & {} -\kappa - \frac{4B\left( {a_1 +a_2 } \right) }{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}- \frac{8B\left( {a_1 +a_2 } \right) ^{3}}{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{3}}.\\ G_{1x}= & {} 2 Q_1 Q_2 +2 Q_3 Q_4 ,\quad G_{1y} =2 Q_1 Q_5 +2 Q_3 Q_6 ,\\ G_{2x}= & {} 2\left( {S_1 S_2 +S_3 S_4 } \right) \left( {S_2 S_6 +S_1 S_7 +S_4 S_8 +S_3 S_9 } \right) +2\left( {-S_4 S_5 +S_1 S_3 } \right) \\&\times \left( {-S_5 S_9 +S_4 S_{10} +S_3 S_6 +S_1 S_8 } \right) \\&+\,2\left( {S_2 S_5 -S_3 ^{2}} \right) \left( {S_5 S_7 +S_2 S_{10} -2S_3 S_8 } \right) \\ G_{2y}= & {} 2\left( {S_1 S_2 +S_3 S_4 } \right) \left( {S_2 S_{11} +S_1 S_{12} +S_4 S_{13} +S_3 S_{14} } \right) +2\left( {-S_4 S_5 +S_1 S_3 } \right) \\&\times \left( {-S_5 S_{14} -S_4 S_{12} +S_3 S_{11} +S_1 S_{13} } \right) \\&+\,2\left( {S_2 S_5 -S_3 ^{2}} \right) \left( {S_5 S_{12} -S_2 S_{12} -2S_3 S_{13} } \right) \\ G_{1\varOmega }= & {} -4 U^{2}\left( {1+\sigma _1 } \right) ^{3}+4 \left( {2 \omega _1 a_1 \sigma _1 -\kappa a_2 -2 \frac{B\left( {a_1 +a_2 } \right) ^{2}}{\left( {1-\left( {a_1 +a_2 } \right) ^{2}} \right) ^{2}}} \right) \omega _1 a_1 ,\\ G_{2\varOmega }= & {} 2\left( {-2 \omega _2 a_2 +\eta a_2 } \right) \left[ { \left( {S_1 S_2 } \right. \left. {+S_3 S_4 } \right) S_2 +S_3 \left( {-S_4 S_5 } \right. \left. {+S_1 S_3 } \right) } \right] ,\\ f_1= & {} G_{1xx} {a}'_1 {a}'_1 +2G_{1xy} {a}'_1 {a}'_2 +G_{1yy} {a}'_2 {a}'_2 , \quad f_2 =G_{2xx} {a}'_1 {a}'_1 +2G_{2xy} {a}'_1 {a}'_2 +G_{2yy} {a}'_2 {a}'_2 ,\\ G_{1xx}= & {} \frac{\partial G_{1x} }{\partial a_1 }, \quad G_{1xy} =\frac{\partial G_{1x} }{\partial a_2 }, \quad G_{1yy} =\frac{\partial G_{1y} }{\partial a_2 }, \quad G_{2xx} =\frac{\partial G_{2x} }{\partial a_1 }, \quad G_{2xy} =\frac{\partial G_{2x} }{\partial a_2 }\\ G_{2yy}= & {} \frac{\partial G_{2y} }{\partial a_2 },\hbox { and} {a}'_1, {a}'_2\hbox { satisfies }G_{1x} {a}'_1 +G_{1y} {a}'_2 =0 \end{aligned}$$

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Chen, H., Hou, L. & Chen, Y. Bifurcation analysis of a rigid-rotor squeeze film damper system with unsymmetrical stiffness supports. Arch Appl Mech 87, 1347–1364 (2017). https://doi.org/10.1007/s00419-017-1254-9

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