Nonlinear dynamic analysis of a tilting pad journal bearing subjected to pad fluttering

Abstract

This work brings an overview of the nonlinear dynamics of four-segment tilting pad journal bearings (TPJB) in load-between-pads configuration supported on flexible pivots. We focus primarily on analysing the sub-synchronous motion of statically unloaded pads, also called pad fluttering. In this particular case, the motion deviates between chaotic and periodic due to 1:4, 1:5, 1:6 and 1:7 internal resonances. The response is analysed with bifurcation diagrams and characterised with estimates of the largest Lyapunov exponents. Unlike previous works, we show that pad fluttering disappears at relative eccentricities lower than 0.34. This behaviour is enabled due to a fixed point trajectory that is not predicted by analysing static equilibria. The observation is crucial for high-speed rotating machinery supported on TPJBs, including pinions in multi-stage compressors and microturbines. The analysis is performed using a verified computational model which includes fluid-structure interaction and potential Hertzian contacts between individual parts of the TPJB. This model exploits local and auxiliary coordinate systems for a straightforward description of acting forces. Furthermore, we provide a detailed step-by-step discussion regarding the model formulation employing the finite difference method to solve hydrodynamic lubrication.

Appendices

Solution strategy

A flowchart describing the solution strategy and the order of calculation steps inside the used solver is depicted in Fig. 21. The depicted flowchart shows only the inner part of the loop in the k-th iteration step. An output function is different for each type of each analysis and has the following form with respect to the notation used in (1):

• 1. Static analysis—Static equilibrium points $\mathbf{q} = \left[ \mathbf{q}_r, \mathbf{q}_p\right] ^T$ are determined if $\mathbf{f}(\mathbf{q}) = \mathbf{0}$. The residuum function is defined as follows:

  $$\begin{aligned} \mathbf{f}(\mathbf{q}) = \begin{bmatrix} \mathbf{f}_{g,r} + \mathbf{f}_{c,r} \\ \mathbf{f}_{g,r} + \mathbf{f}_{c,p} \end{bmatrix}. \end{aligned}$$

  (34)

• 2. Analysis of dynamics—The system of the second order differential equations (1) is transformed to the system of the first order differential equations using the mass identity

  $$\begin{aligned} \begin{bmatrix} \mathbf{M}_r &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{M}_p \end{bmatrix} \begin{bmatrix} \dot{\mathbf{q}}_r \\ \dot{\mathbf{q}}_p \end{bmatrix} - \begin{bmatrix} \mathbf{M}_r &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{M}_p \end{bmatrix} \begin{bmatrix} \dot{\mathbf{q}}_r \\ \dot{\mathbf{q}}_p \end{bmatrix} = \begin{bmatrix} \mathbf{0} \\ \mathbf{0} \end{bmatrix}. \end{aligned}$$

  (35)

  The resulting system is solved for state-space vector $\mathbf{x}(t) = \left[ \dot{\mathbf{q}}_r(t),\dot{\mathbf{q}}_p(t), \mathbf{q}_r(t),\mathbf{q}_p(t)\right] ^T$. The derivative of the state-space vector, which is evaluated in each iteration step of solver ode15s, can be written in the matrix form

  $$\begin{aligned} \dot{\mathbf{x}}(t) = \begin{bmatrix} \begin{bmatrix} \mathbf{M}_r &{} \mathbf{0} \\ \mathbf{0} &{} \mathbf{M}_p \end{bmatrix}^{-1} \left( \begin{bmatrix} \mathbf{f}_{g,r} \\ \mathbf{f}_{g,p} \end{bmatrix} + \begin{bmatrix} \mathbf{f}_{un} \\ \mathbf{0} \end{bmatrix} + \begin{bmatrix} \mathbf{f}_{c,r} \\ \mathbf{f}_{c,p} \end{bmatrix} \right) \\ \dot{\mathbf{q}}_r \\ \dot{\mathbf{q}}_p \end{bmatrix}. \end{aligned}$$

  (36)

Fig. 21

Flowchart of solution strategy inside of used numerical solver

Determining the largest Lyapunov exponent from a time series

The largest Lyapunov exponent $\lambda _{\mathrm {max}}$ is estimated using an algorithm proposed by Wolf et al. [32]. The exact implementation of the procedure follows:

• 1. The response of the system is simulated sufficiently long. This condition is fulfilled if a simulated time series contains at least several hundred orbits, i.e. closed trajectories in the phase space.

• 2. The transient response is removed from the simulated time series. The amount of the removed data vary between 0.4 and 3.9 s depending on the exact character of the time series. Although removing the transient response is not necessary, it accelerates the temporal convergence of $\lambda _{\mathrm {max}}$ moderately.

• 3. The cropped time series is decimated using a lowpass Chebyshev Type I infinite impulse response filter of order 8. The decimation factor is a positive integer selected so that the decimated time series has 18 samples per orbit.

• 4. The decimated time series is analysed using Wolf’s algorithm [32] with the following parameters:

  • the dimension of the embedding phase space is 4,

  • the embedding delay is a third of an orbit,

  • the evolution time-step is a quarter of an orbit,

  • the minimum phase-space distance cut-off is defined by sequence $\lbrace 0,\,0.02,\,0.04,\,\dots ,\,0.5 \rbrace$, which is given in per cents from the maximum-to-minimum difference of the decimated time series,

  • the maximum phase-space distance cut-off is defined by sequence $\lbrace 8,\,8.25,\,8.5,\,\dots ,\,15 \rbrace$, which is given in per cents from the maximum-to-minimum difference of the decimated time series.

• 5. All combinations of the parameters yield 754 running estimates of $\lambda _{\mathrm {max}}$. The last 100 samples are taken from each running estimate to an array accommodating 75400 estimates of $\lambda _{\mathrm {max}}$ for each decimated time series. The array is further used to calculate the median and the mean difference from the median.