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A nonlinear model for dynamic performance analysis of gas foil bearing-rotor system considering frictional contacts

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Abstract

The dynamic behaviors of gas foil bearing-rotor system are greatly affected by the dam** resulting from friction inside foil structure. However, challenges exist in theoretical study when introducing the friction into the dynamic system. This paper presents an effective nonlinear model considering frictional contacts to study the dynamic performance of foil bearing-rotor system. The top foil and bump foil are modeled with the beam elements to consider the sagging effect and bump interactions. Instead of treating the structural dam** caused by friction as equivalent viscous dam**, the friction force is calculated by utilizing a regularization technique to capture its nonlinear behavior. The simulations are numerically implemented using a direct implicit integration scheme with consideration of unsteady hydrodynamic pressure, rotor motion, deflection of foil structure and their coupling relationships. Using this model, the dissipative characteristics and stability of foil bearing-rotor system are investigated. The results show that the friction leads to hysteretic loops and energy dissipation, and the stick motion becomes more prevalent with the increasing friction coefficient. The dynamic system exhibits rich and complicated nonlinear behaviors. With the change of system parameters, the rotor can experience periodic, quasi-periodic and multi-periodic motions. Only synchronous vibration occurs at low rotational speeds, but large-amplitude subsynchronous vibrations are dominant when rotational speed exceeds the onset speed of instability. Increasing the nominal clearance contributes to avoid the appearance of subsynchronous vibrations, resulting in a more stable system. The predictions can provide guidance for determining the operating and design parameters of foil bearing-rotor system in practice.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

A :

Cross-sectional area

C :

Nominal clearance

C :

Global dam** matrix

c t :

Smooth parameter

E :

Young’s modulus

F e :

Element film reaction force

F N :

Normal contact force

F p :

Hydrodynamic force

F p :

Hydrodynamic force vector

F T :

Tangential friction force

F ub :

Unbalance force

h :

Film thickness

H :

Dimensionless film thickness

h b :

Bump height

I :

Moment of inertia

I 2×2 :

2 × 2 Identity matrix

J :

Jacobian matrix

K :

Global stiffness matrix

k e :

Element stiffness matrix

L :

Bearing width

L e :

Element length

L e :

Transformation matrix

M :

Global mass matrix

m e :

Element mass matrix

m s :

Rotor mass

N 1-N 6 :

Shape functions

N b :

Number of bumps

p :

Pressure

P :

Dimensionless pressure

p a :

Ambient pressure

p θ :

Line pressure load

Q c :

Contact force vector

R :

Bearing radius

S :

Applied static load

s b :

Bump pitch

t :

Operating time

t b :

Bump foil thickness

t f :

Top foil thickness

u :

X-direction displacement

\(\dot{u}\) :

X-direction velocity

u :

Displacement vector

\({\dot{\mathbf{u}}}\) :

Velocity vector

\({\ddot{\mathbf{u}}}\) :

Acceleration vector

u c :

Nodal displacement vector of contact

u n :

Displacement vector of a node

u r :

Unbalance

\(\ddot{u}_{s}\) :

Rotor acceleration

v :

y-direction displacement

v T :

Deflection of top foil

z, Z :

Axial coordinate

α :

Penalty parameter

β :

Included angle

γ :

Proportional parameter

δ :

Newmark method parameter

ε x :

X-direction eccentricity ratio

ε y :

Y-direction eccentricity ratio

θ :

Circumferential angle

θ f :

Rotational degree of freedom

κ :

Newmark method parameter

μ :

Friction coefficient

μ a :

Viscosity

ν :

Excitation frequency ratio

ξ :

Reference coordinate

ρ :

Density

τ :

Dimensionless time

ω :

Rotational speed

ω e :

Excitation frequency

Λ:

Bearing number

B :

Bump foil

e :

Element

l :

The lth iteration

N :

Normal

P :

Point P

P1; P2:

Node P1; node P2

Q :

Node Q

T :

Top foil; transposition; tangential

x, X :

x or X direction

y,Y :

y or Y direction

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Funding

This work was supported by the National Key Research and Development Program of China (Grant No. 2022YFB3402703); the National Natural Science Foundation of China (Grant No. U2141210), (Grant No. 52005126); the Shenzhen Science and Technology Innovation Council (Grant No. JCYJ20220818103200001), and the Stable Support Program for Shenzhen Higher Education Institutions from Shenzhen Science and Technology Innovation Council (Grant No. GXWD20220811151458003).

Author information

Authors and Affiliations

  1. School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, 150001, China

    Xuewei Zhao & Yong Lu

  2. School of Mechanical Engineering and Automation, Harbin Institute of Technology, Shenzhen, Shenzhen, 518055, China

    Changlin Li & Jianjun Du

  3. Shenzhen Key Laboratory of Flexible Printed Electronics Technology, Shenzhen, 518055, China

    Changlin Li & Jianjun Du

Authors
  1. Xuewei Zhao
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  2. Changlin Li
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  3. Jianjun Du
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  4. Yong Lu
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Contributions

XZ: conceptualization, methodology, software, validation, formal analysis, investigation, data curation, visualization, writing—original draft. CL: data curation, visualization, funding acquisition, supervision, writing–review and editing. JD: conceptualization, funding acquisition, supervision, writing—review and editing. YL: visualization, writing—review and editing.

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Correspondence to Changlin Li or Jianjun Du.

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Appendices

Appendix A. The transformation matrix, mass matrix and stiffness matrix of beam element

$$ {\mathbf{L}}_{e} = \left[ {\begin{array}{*{20}c} {{\text{cos}}\beta } & {{\text{sin}}\beta } & 0 & 0 & 0 & 0 \\ { - {\text{sin}}\beta } & {{\text{cos}}\beta } & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\text{cos}}\beta } & {{\text{sin}}\beta } & 0 \\ 0 & 0 & 0 & { - {\text{sin}}\beta } & {{\text{cos}}\beta } & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
(A.1)
$$ {\mathbf{m}}_{e} = \frac{{\rho AL_{e} }}{2}\left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right],\,{\mathbf{k}}_{e} = \left[ {\begin{array}{*{20}c} {\frac{EA}{{L_{e} }}} & 0 & 0 & { - \frac{EA}{{L_{e} }}} & 0 & 0 \\ 0 & {\frac{12EI}{{L_{e}^{3} }}} & {\frac{6EI}{{L_{e}^{2} }}} & 0 & { - \frac{12EI}{{L_{e}^{3} }}} & {\frac{6EI}{{L_{e}^{2} }}} \\ 0 & {\frac{6EI}{{L_{e}^{2} }}} & {\frac{4EI}{{L_{e} }}} & 0 & { - \frac{6EI}{{L_{e}^{2} }}} & {\frac{2EI}{{L_{e} }}} \\ { - \frac{EA}{{L_{e} }}} & 0 & 0 & {\frac{EA}{{L_{e} }}} & 0 & 0 \\ 0 & { - \frac{12EI}{{L_{e}^{3} }}} & { - \frac{6EI}{{L_{e}^{2} }}} & 0 & {\frac{12EI}{{L_{e}^{3} }}} & { - \frac{6EI}{{L_{e}^{2} }}} \\ 0 & {\frac{6EI}{{L_{e}^{2} }}} & {\frac{2EI}{{L_{e} }}} & 0 & { - \frac{6EI}{{L_{e}^{2} }}} & {\frac{4EI}{{L_{e} }}} \\ \end{array} } \right] $$
(A.2)

where β is the included angle between the global coordinate system and the local coordinate system of the beam element, ρ is the density of the structure, A is cross-sectional area, E is the Young’s modulus and I is the moment of inertia.

Appendix B. The shape functions

$$ \begin{aligned} & N_{1} \left( \xi \right) = \frac{1}{2}\left( {1 - \xi } \right),\,N_{2} \left( \xi \right) = \frac{1}{2}\left( {1 + \xi } \right),\,N_{3} \left( \xi \right) = \frac{1}{4}\left( {1 - \xi } \right)^{2} \left( {2 + \xi } \right), \\ & N_{4} \left( \xi \right) = \frac{{L_{e} }}{8}\left( {1 - \xi } \right)^{2} \left( {1 + \xi } \right),\;N_{5} \left( \xi \right) = \frac{1}{4}\left( {1 + \xi } \right)^{2} \left( {2 - \xi } \right),\,N_{6} \left( \xi \right) = \frac{{L_{e} }}{8}\left( {1 + \xi } \right)^{2} \left( { - 1 + \xi } \right) \\ \end{aligned} $$
(B.1)

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Zhao, X., Li, C., Du, J. et al. A nonlinear model for dynamic performance analysis of gas foil bearing-rotor system considering frictional contacts. Nonlinear Dyn 112, 5975–5996 (2024). https://doi.org/10.1007/s11071-024-09309-0

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