Dynamic characteristics of flexible rotor with squeeze film damper excited by two frequencies

Abstract

This paper is focused on the dynamic characteristics of a flexible rotor with squeeze film damper excited by two frequencies. The multiple harmonic balance method and Runge–Kutta method are combined to analyze the periodic solution and quasi-periodic solution of the system. The nonlinear characteristics discussed are fastened on the resonance region corresponding, respectively, to the rigid body translation mode and first bending mode. In the former region, the motion of disk center shows the ‘hard spring’ characteristic, and the combination frequencies are dominated by the difference between double low excitation frequency and high excitation frequency and the difference between double high excitation frequency and low excitation frequency. In the latter region, however, the combination frequencies are dominated by the difference between high excitation frequency and double low excitation frequency and the difference between triple low excitation frequency and double high excitation frequency. Moreover, the motion of disk center shows a kind of ‘cross’, ‘soft spring’ or ‘hard spring’ characteristics with the variation of the ratio of the two excitation frequencies. Besides, the independent quasi-periodic solution coexists with the periodic solution in these cases. The system is sensitive to the ratio of excitation frequencies, and it could have two independent quasi-periodic solutions in some conditions. The results in this paper provide a reveal of nonlinear characteristics in this type of double excitation nonlinear rotor system.

Appendices

Appendix 1: List of symbols

$a_{j,k ,l} , b_{j,k,l}$ :

Harmonic coefficients of dimensionless displacements, dimensionless

A :

Periodic matrix, dimensionless

B :

Dimensionless bearing parameter of squeeze film damper, dimensionless

c :

Clearance of squeeze film damper, m

$c_{j,k,l} ,d_{j,k,l}$ :

Harmonic coefficients of dimensionless nonlinear forces, dimensionless

$c_{1}$ :

Damping coefficient corresponding to transverse deformation of the shaft, N s/m

$c_{2}$, $c_{3}$ :

Damping coefficients corresponding to coupling deformation of the shaft, N s

$c_{4 }$ :

Damping coefficient corresponding to angular deformation of the shaft, N m s

$c_{5}$, $c_{6}$ :

Damping coefficients of the supports, N s/m

D :

Monodromy matrix, dimensionless

$F_{\mathrm{cx}}$, $F_{\mathrm{cy}}$ :

Oil film forces in the Cartesian coordinate system, N

$\bar{{F}}_{\mathrm{cx}} ,\bar{{F}}_{\mathrm{cy}}$ :

Dimensionless oil film forces in the Cartesian coordinate system, dimensionless

$F_{\mathrm{r}}$, $F_{{\uptau } }$ :

Oil film forces in the polar coordinate system, N

$\bar{{F}}_{\mathrm{r}} ,\bar{{F}}_{{{\uptau } }}$ :

Dimensionless oil film forces in the polar coordinate system, dimensionless

G :

Vector of harmonic balance terms, dimensionless

J :

Jacobi matrix, dimensionless

$J_{\mathrm{d}}$ :

Equivalent equatorial moment of inertia of the disk, kg $\hbox {m}^{2}$

$J_{\mathrm{p}}$ :

Equivalent polar moment of inertia of the disk, kg $\hbox {m}^{2}$

k, l :

Harmonic parameters, dimensionless

$k_{\mathrm{rr}}$ :

Stiffness coefficient corresponding to transverse deformation of the shaft, N/m

$k_{\mathrm{r}{\upvarphi }}$, $k_{{\upvarphi } \mathrm{r}}$ :

Stiffness coefficients corresponding to coupling deformation of the shaft, N

$k_{{\upvarphi } {\upvarphi } }$ :

Stiffness coefficient corresponding to angular deformation of the shaft, N m

$k_{\mathrm{a}}$, $k_{\mathrm{b}}$ :

Stiffness coefficient of the supports, N/m

l :

Length of the shaft, m

$l_{1}$ :

Distance from the disk to the left support, m

$l_{2}$ :

Distance from the disk to the right support, m

$l_{3}$ :

Distance from the disk to the intershaft bearing, m

L :

Length of squeeze film damper, m

m, n :

Parameters of discrete points in time domain, dimensionless

$m_{\mathrm{o}}$ :

Equivalent mass of the rotor, kg

$m_{\mathrm{a}}$ :

Equivalent mass of the left journal, kg

$m_{\mathrm{b}}$ :

Equivalent mass of the right journal, kg

M :

Maximal harmonic parameter, dimensionless

N :

Number of discrete points in time domain, dimensionless

P :

Vector of harmonic coefficients of dimensionless displacements, dimensionless

$q_{\mathrm{j}}$ :

Dimensionless displacement (j=1$\sim$8), dimensionless

Q :

Vector of harmonic coefficients of dimensionless nonlinear forces, dimensionless

r :

Radial displacement of journal, m

R :

Radius of the journal, m

s :

Dimensionless arc length, dimensionless

t :

Time, s

U :

Vector of state variables, dimensionless

$U_{1}$, $U_{2}$, $U_{3}$ :

Dimensionless unbalance value, dimensionless

x, y :

Displacements of center of disk, m

$x_{\mathrm{a}}$, $y_{\mathrm{a}}$ :

Displacements of center of left journal, m

$x_{\mathrm{b}}$, $y_{\mathrm{b}}$ :

Displacements of center of right journal, m

$\alpha _{0}$, $\alpha _{1}$, $\alpha _{2}$ :

Dimensionless mass, dimensionless

$\gamma _{1}$, $\gamma _{2}$ :

Ratio of distance $l_{1}$, $l_{2}$ to length l, dimensionless

$\delta _{1}$, $\delta _{2}$ :

Unbalance value, kg m

$\varepsilon$ :

Dimensionless radial displacement of journal, dimensionless

$\zeta _{\mathrm{j}}$ :

Dimensionless damping coefficients (j=1$\sim$6), dimensionless

$\eta$ :

Ratio of moment of inertia $J_{\mathrm{p}}$ to moment of inertia $J_{\mathrm{d}}$, dimensionless

$\theta$ :

Journal position angle measured from line of journal centers, dimensionless

$\theta _{1}$, $\theta _{2}$ :

Angles from line of journal centers to start and end of positive pressure region, dimensionless

$\theta _{\mathrm{x}}$, $\theta _{\mathrm{y}}$ :

Angles of disk rotate along x axis and y axis, dimensionless

$\kappa _{\mathrm{j}}$ :

Dimensionless stiffness coefficients (j=1$\sim$6), dimensionless

$\lambda$ :

Bifurcation parameter, dimensionless

$\mu$ :

Viscosity coefficient of oil, N s/m$^{2}$

$\xi$ :

Ratio of high excitation frequency $\varOmega _{2}$ to low excitation frequency $\varOmega _{1}$, dimensionless

$\tau$, $\tau _{1}$, $\tau _{2}$ :

Dimensionless time, dimensionless

$\psi$ :

angular displacement of journal, dimensionless

$\omega _{1}$, $\omega _{2}$ :

Dimensionless low excitation frequency and dimensionless high excitation frequency, dimensionless

$\varOmega _{0}$ :

Reference rotational speed

$\varOmega _{1}$, $\varOmega _{2}$ :

Low-pressure rotor rotational speed and high-pressure rotor rotational speed, rad/s

Appendix 2

$$\begin{aligned}&g_{3,k,l} =\left[ {\kappa _1 -\left( {k+l\xi } \right) ^{2}} \right] a_{2kl} +\zeta _1 \left( {k+l\xi } \right) b_{2kl} \nonumber \\&\quad +\,\kappa _2 a_{3kl} +\zeta _2 \left( {k+l\xi } \right) b_{3kl} -\left( {\kappa _2 +\gamma _2 \kappa _1 } \right) a_{6kl} \\&\quad -\left( {\zeta _2 +\gamma _2 \zeta _1 } \right) \left( {k+l\xi } \right) b_{6kl} +\left( {\kappa _2 -\gamma _1 \kappa _1 } \right) a_{8kl} \nonumber \\&\quad +\left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) b_{8kl} =0 \\&g_{4,k,l} =-\zeta _1 \left( {k+l\xi } \right) a_{2kl} +\left[ {\kappa _1 -\left( {k+l\xi } \right) ^{2}} \right] b_{2kl} \nonumber \\&\quad -\zeta _2 \left( {k+l\xi } \right) a_{3kl} +\kappa _2 b_{3kl} +\left( {\zeta _2 +\gamma _2 \zeta _1 } \right) \left( {k+l\xi } \right) a_{6kl} \\&\quad -\left( {\kappa _2 +\gamma _2 \kappa _1 } \right) b_{6kl} -\left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) a_{8kl} \nonumber \\&\quad +\left( {\kappa _2 -\gamma _1 \kappa _1 } \right) b_{8kl} -U_1 \phi _1 -\xi ^{2}U_2 \phi _2 =0 \\&g_{5,k,l} =\kappa _3 a_{2kl} +\zeta _3 \left( {k+l\xi } \right) b_{2kl} +\left[ {\kappa _4 -\alpha _0 \left( {k+l\xi } \right) ^{2}} \right] a_{3kl} \nonumber \\&\quad +\zeta _4 \left( {k+l\xi } \right) b_{3kl} +\alpha _0 \eta \left( {k+l\xi } \right) b_{4kl} \\&\quad -\left( {\kappa _4 +\gamma _2 \kappa _3 } \right) a_{6kl} \nonumber \\&\quad -\left( {\zeta _4 +\gamma _2 \zeta _3 } \right) \left( {k+l\xi } \right) b_{6kl} +\left( {\kappa _4 -\gamma _1 \kappa _3 } \right) a_{8kl} \nonumber \\&\quad +\left( {\zeta _4 -\gamma _1 \zeta _3 } \right) \left( {k+l\xi } \right) b_{8kl} =0 \\&g_{6,k,l} =-\zeta _3 \left( {k+l\xi } \right) a_{2kl} +\kappa _3 b_{2kl} -\zeta _4 \left( {k+l\xi } \right) a_{3kl} \nonumber \\&\quad +\left[ {\kappa _4 -\alpha _0 \left( {k+l\xi } \right) ^{2}} \right] b_{3kl} -\alpha _0 \eta \left( {k+l\xi } \right) a_{4kl} \\&\quad +\left( {\zeta _4 +\gamma _2 \zeta _3 } \right) \left( {k+l\xi } \right) a_{6kl} -\left( {\kappa _4 +\gamma _2 \kappa _3 } \right) b_{6kl} \nonumber \\&\quad -\left( {\zeta _4 -\gamma _1 \zeta _3 } \right) \left( {k+l\xi } \right) a_{8kl} +\left( {\kappa _4 -\gamma _1 \kappa _3 } \right) b_{8kl} \nonumber \\&\quad -\xi ^{2}U_3 \phi _2 =0 \\&g_{7,k,l} =-\kappa _3 a_{1kl} -\zeta _3 \left( {k+l\xi } \right) b_{1kl} -\alpha _0 \eta \left( {k+l\xi } \right) b_{3kl}\nonumber \\&\quad +\left[ {\kappa _4 -\left( {k+l\xi } \right) ^{2}} \right] a_{4kl} +\zeta _4 \left( {k+l\xi } \right) b_{4kl} \nonumber \\&\quad +\left( {\kappa _4 +\gamma _2 \kappa _3 } \right) a_{5kl} +\left( {\zeta _4 +\gamma _2 \zeta _3 } \right) \left( {k+l\xi } \right) b_{5kl} \nonumber \\&\quad -\left( {\kappa _4 -\gamma _1 \kappa _3 } \right) a_{7kl} -\left( {\zeta _4 -\gamma _1 \zeta _3 } \right) \left( {k+l\xi } \right) b_{7kl} \nonumber \\&\quad +\xi ^{2}U_3 \phi _2 =0 \\&g_{8,k,l} =\zeta _3 \left( {k+l\xi } \right) a_{1kl} - \kappa _3 b_{1kl} +\alpha _0 \eta \left( {k+l\xi } \right) a_{3kl} \nonumber \\&\quad -\zeta _4 \left( {k+l\xi } \right) a_{4kl} +\left[ {\kappa _4 -\left( {k+l\xi } \right) ^{2}} \right] b_{4kl} \\&\quad -\left( {\zeta _4 +\gamma _2 \zeta _3 } \right) \left( {k+l\xi } \right) a_{5kl} +\left( {\kappa _4 +\gamma _2 \kappa _3 } \right) b_{5kl} \nonumber \\&\quad +\left( {\zeta _4 -\gamma _1 \zeta _3 } \right) \left( {k+l\xi } \right) a_{7kl} -\left( {\kappa _4 -\gamma _1 \kappa _3 } \right) b_{7kl} =0 \nonumber \\&g_{9,k,l} =-\gamma _2 \kappa _1 a_{1kl} -\gamma _2 \zeta _1 \left( {k+l\xi } \right) b_{1kl} +\gamma _2 \kappa _2 a_{4kl} +\gamma _2\zeta _2 \nonumber \\&\quad \left( {k+l\xi } \right) b_{4kl} +\left[ {\left( {\kappa _5 +\gamma _2 \kappa _2 +\gamma _2^2 \kappa _1 } \right) -\alpha _1 \left( {k+l\xi } \right) ^{2}} \right] a_{5kl} \\&\quad +\left[ {\zeta _5 +\gamma _2 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) } \right] \left( {k+l\xi } \right) b_{5kl} -\gamma _2 \left( {\kappa _2 -\gamma _1 \kappa _1 } \right) \nonumber \\&\quad a_{7kl} -\gamma _2 \left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) b_{7kl} +Bc_{5kl} =0 \\&g_{10,k,l} =\gamma _2 \zeta _1 \left( {k+l\xi } \right) a_{1kl} -\gamma _2 \kappa _1 b_{1kl} -\gamma _2 \zeta _2 \left( {k+l\xi } \right) a_{4kl} \nonumber \\&\quad +\,\gamma _2 \kappa _2 b_{4kl} -\left[ {\gamma _2 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) +\zeta _5 } \right] \left( {k+l\xi } \right) a_{5kl} \\&\quad +\,\left[ {\left( {\kappa _5 +\gamma _2 \kappa _2 +\gamma _2^2 \kappa _1 } \right) -\alpha _1 \left( {k+l\xi } \right) ^{2}} \right] b_{5kl} \nonumber \\&\quad -\,\gamma _2 \left( {\kappa _2 -\gamma _1 \kappa _1 } \right) b_{7kl} +\gamma _2 \left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) a_{7kl} \nonumber \\&\quad +\,Bd_{5kl} =0 \end{aligned}$$

$$\begin{aligned}&g_{11,k,l} =-\gamma _2 \kappa _1 a_{2kl} -\gamma _2 \zeta _1 \left( {k+l\xi } \right) b_{2kl} \nonumber \\&\quad -\gamma _2 \kappa _2 a_{3kl} -\gamma _2 \zeta _2 \left( {k+l\xi } \right) b_{3kl} \nonumber \\&\quad +\left[ {\left( {\kappa _5 +\gamma _2 \kappa _2 +\gamma _2^2 \kappa _1 } \right) -\alpha _1 \left( {k+l\xi } \right) ^{2}} \right] a_{6kl} \\&\quad +\left[ {\zeta _5 +\gamma _2 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) } \right] \left( {k+l\xi } \right) b_{6kl}\nonumber \\&\quad -\gamma _2 \left( {\kappa _2 -\gamma _1 \kappa _1 } \right) a_{8kl} -\gamma _2 \left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) b_{8kl} \nonumber \\&\quad +Bc_{6kl} =0 \\&g_{12,k,l} =\gamma _2 \zeta _1 \left( {k+l\xi } \right) a_{2kl} -\gamma _2 \kappa _1 b_{2kl}\nonumber \\&\quad \quad +\,\gamma _2 \zeta _2 \left( {k+l\xi } \right) a_{3kl} \nonumber \\&\quad -\gamma _2 \kappa _2 b_{3kl} -\left[ {\gamma _2 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) +\zeta _5 } \right] \left( {k+l\xi } \right) a_{6kl} \\&\quad +\left[ {\left( {\kappa _5 +\gamma _2 \kappa _2 +\gamma _2^2 \kappa _1 } \right) -\alpha _1 \left( {k+l\xi } \right) ^{2}} \right] b_{6kl} \nonumber \\&\quad +\gamma _2 \left( {\zeta _2 -\gamma _1 \zeta _1 } \right) \left( {k+l\xi } \right) a_{8kl} -\gamma _2 \left( {\kappa _2 -\gamma _1 \kappa _1 } \right) b_{8kl} \nonumber \\&\quad +Bd_{6kl} =0 \\&g_{13,k,l} =-\gamma _1 \kappa _1 a_{1kl} -\gamma _1 \zeta _1 \left( {k+l\xi } \right) b_{1kl} \\&\quad +\gamma _1 \kappa _2 a_{4kl} +\gamma _1 \zeta _2 \left( {k+l\xi } \right) b_{4kl} +\gamma _1 \left( {\kappa _2 +\gamma _2 \kappa _1 } \right) a_{5kl} \\&\quad +\gamma _1 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) \left( {k+l\xi } \right) b_{5kl} \\&\quad +\left[ {\left( {\kappa _6 -\gamma _1 \kappa _2 +\gamma _1^2 \kappa _1 } \right) -\alpha _2 \left( {k+l\xi } \right) ^{2}} \right] a_{7kl} \\&\quad +\left[ {\gamma _1 \left( {-\zeta _2 +\gamma _1 \zeta _1 } \right) +\zeta _6 } \right] \left( {k+l\xi } \right) b_{7kl} =0 \\&g_{14,k,l} =\gamma _1 \zeta _1 \left( {k+l\xi } \right) a_{1kl} -\gamma _1 \kappa _1 b_{1kl}\\&\quad \quad -\gamma _1 \zeta _2 \left( {k+l\xi } \right) a_{4kl} \\&\quad +\gamma _1 \kappa _2 b_{4kl} -\gamma _1 \left( {\zeta _2 +\gamma _2 \zeta _1 } \right) \left( {k+l\xi } \right) a_{5kl} \\&\quad +\gamma _1 \left( {\kappa _2 +\gamma _2 \kappa _1 } \right) b_{5kl} \\&\quad +\left[ {\left( {\kappa _6 -\gamma _1 \kappa _2 +\gamma _1^2 \kappa _1 } \right) -\alpha _2 \left( {k+l\xi } \right) ^{2}} \right] b_{7kl}\\&\quad -\left[ {\gamma _1 \left( {-\zeta _2 +\gamma _1 \zeta _1 } \right) +\zeta _6 } \right] \left( {k+l\xi } \right) a_{7kl} =0 \\ \end{aligned}$$