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Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

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Abstract

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincaré maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping (\(p=0.5, 2, 3\)) suspension as for the linear damping (\(p=1\)) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model (\(p=0.5\)) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping (\(p=2\)) shows similar results to linear damping with some special damping coefficients. The cubic damping (\(p=3\)) shows more stable response than the linear damping in general.

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Abbreviations

\(b=\frac{\rho }{\delta }\) :

Dimensionless static unbalance of rotor

\(c_0\) :

Damping coefficient of the supported structure (N  (s/m)\(^{p}\))

\(c_2\) :

Damping coefficient of the rotor disk (N s/m)

\(e\) :

Dimensional eccentricity of journal (m)

\(F_r ,F_t\) :

Oil film force in radial and tangential directions (N)

\(F_x ,F_y\) :

Oil film force in X and Y directions (N)

\(g\) :

Acceleration of gravity (\(\hbox {m/s}^{2}\))

\(h\) :

Film thickness (m)

\(k_0\) :

Stiffness coefficient of the supported structure (N/m)

\(k_2\) :

Stiffness coefficient of shaft (N/m)

\(L\) :

Axial length of bearing (m)

\(m_0\) :

Mass of bearing (kg)

\(m_2\) :

Mass of rotor (kg)

\(O_0 ,O_1 ,O_2\) :

Geometric centers of bearing, journal and rotor

\(O_m\) :

Gravity center of rotor

\(p\) :

Damping exponent

\(P_m=\frac{m_0 }{m_2 }\) :

Dimensionless mass ratio

\(P_k=\frac{k_0 }{k_2 }\) :

Dimensionless stiffness ratio

\(P_r=\frac{m_2 g}{\delta k_2 }\) :

Dimensionless gravity parameter

\(R\) :

Radius of bearing (m)

\(s=\sqrt{{\omega ^{2}}/{\omega _n^{2}}}\) :

Dimensionless rotational speed ratio

\(X_i ,Y_i ,Z_i (i=0,1,2)\) :

Coordinates of \(O_0 ,O_1 ,O_2 \) (m)

\(x_i ,y_i ,z_i=\frac{X_i }{\delta },\frac{Y_i }{\delta },\frac{Z_i }{\delta }(i=0,1,2)\) :

Dimensionless coordinates

\(\alpha =\frac{\delta ^{3}\sqrt{k_2 m_2 }}{\mu RL^{3}}\) :

Dimensionless parameter

\(\beta \) :

Unit velocity (1 m/s)

\(\gamma =\frac{\delta \omega _n }{\beta }\) :

Dimensionless velocity coefficient

\(\rho \) :

Dimensional static unbalance of rotor (m)

\(\delta \) :

Radial clearance of bearing (m)

\(\varepsilon =\frac{e}{\delta }\) :

Dimensionless eccentricity of journal

\(\zeta _0=\frac{c_0 \beta ^{p-1}}{2\sqrt{k_0 m_0 }}\) :

Dimensionless damping coefficient

\(\zeta _2=\frac{c_2 }{2\sqrt{k_2 m_2 }}\) :

Dimensionless damping coefficient

\(\mu \) :

Dynamic viscosity of lubricant (Pa s)

\(\phi =\omega t\) :

Rotational angle (rad)

\(\varphi \) :

Attitude angle from the gravity direction (rad)

\(\omega \) :

Angular speed of rotor (rad/s)

\(\omega _n=\sqrt{{k_2 }/{m_2 }}\) :

Natural frequency

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Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 50975199). Financial supports from China Scholarship Council are gratefully acknowledged, and this work was completed, while the author Shuai Yan was a visiting scholar in Duke University.

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Authors and Affiliations

  1. Key Laboratory of Advanced Ceramics and Machining Technology of Ministry of Education, Tianjin University, Tianjin, 300072, China

    Shuai Yan & Bin Lin

  2. Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, 27708, USA

    Earl H. Dowell

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  1. Shuai Yan
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  2. Earl H. Dowell
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  3. Bin Lin
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Correspondence to Bin Lin.

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Yan, S., Dowell, E.H. & Lin, B. Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings. Nonlinear Dyn 78, 1435–1450 (2014). https://doi.org/10.1007/s11071-014-1526-7

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  • DOI: https://doi.org/10.1007/s11071-014-1526-7

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