Dynamic analysis of a flexible rotor supported by hydrodynamic bearings with uncertain parameters

Abstract

This paper is dedicated to the analysis of uncertainties affecting the dynamic behavior of a flexible rotor containing three rigid discs and supported by two cylindrical fluid film bearings. A stochastic method has been extensively used to model uncertain parameters, the so-called Monte Carlo simulation. However, in the present contribution, the inherent uncertainties of the bearings’ parameters (i.e., the oil viscosity as a function of the oil temperature and the radial clearance) are modeled by using a fuzzy logic based analysis. This alternative methodology seems to be more appropriate when the stochastic process that characterizes the uncertainties is unknown. The analysis procedure is confined to the time domain, being characterized by the envelopes of the rotor vibration responses. The hydrodynamic supporting forces are determined by considering a nonlinear model, which is based on the solution of the dimensionless Reynolds’ equation for cylindrical and short journal bearings. Finally, an experimental analysis is performed for comparison purposes. The most significant results are associated with the changes in the dynamic behavior of the rotating machine as a result from the supporting forces that are modified according to the uncertainties introduced in the bearings.

Appendices

Appendix 1

1.1 Differential evolution: a review

The DE algorithm is an optimization technique that belongs to the family of evolutionary computation, which differs from other evolutionary algorithms in the mutation and recombination schemes. As previously mentioned, DE executes its mutation operation by adding a weighted difference vector between two individuals to a third individual. Then, the mutated individuals will perform discrete crossover and greedy selection with the corresponding individuals from the last generation to produce the offspring. The key control parameters for DE are the population size (NP), the crossover constant (CR), and the so-called weight (F). The pseudo-code of DE algorithm is presented in the Fig. 11, in which P is the population of the current generation, and P’ is the population to be constructed for the next generation, C _[i] is the candidate solution with population index i, C _[i][j] is the jth entry in the solution vector of C _[i], and r is a random number between 0 and 1.

Fig. 11

The pseudo-code of DE algorithm

In [30], it was presented some simple rules for choosing the key parameters of DE for general applications. Normally, NP should be about 5–10 times the dimension of the problem (i.e., number of design variables). As for F, it lies in the range between 0.4 and 1.0. Initially, F = 0.5 can be tried, and then F and/or NP can be increased if the population converges prematurely. Price and Storn [31] proposed various mutation schemes for the generation of new candidate solutions by combining the vectors that are randomly chosen from the current population. In the applications of this paper, the rand/1 scheme was used.

Appendix 2

2.1 Sequential quadractic programming: a review

According to [22], the SQP algorithm is a direct method used for dealing with constrained minimization problems in which the search direction S is found by solving a sub problem with a quadratic objective function and linear constraints. For this aim, a quadratic approximation of the augmented objective function is created (i.e., from the association of the Lagrange multipliers λ with an exterior penalty technique) and a linear approximation for the constraints is written, as shows Eq. (5).

$$\begin{array}{*{20}c} {{\text{Minimize:}}\quad Q(S) = F(X) + \nabla F(X)^{T} S + \frac{1}{2}S^{T} BS} \\ \begin{aligned} {\text{Subject}}\,{\text{to:}}\quad \nabla g_{j} (X)^{T} S + \delta_{j} g_{j} (X) \le 0\quad j = 1,m \hfill \\ \quad \quad \nabla h_{k} (X)^{T} S + \bar{\delta }h_{k} (X) \le 0\quad k = 1,l \hfill \\ \end{aligned} \\ \end{array}$$

(5)

where X is the vector of design variables and B is initially an identity matrix that will be updated on subsequent iterations. The parameters δ _j and $\bar{\delta }$ are used to prevent inconsistences between the linearized constrains g _j and h _k (i.e., typically $0.9 \, \le \bar{\delta } \le 0.95$).

The δ _j parameter is defined as follows:

$$\begin{array}{*{20}l} {\delta_{j} = 1} \hfill & {if\;g_{j} (X) < 0} \hfill \\ {\delta_{j} = \bar{\delta }} \hfill & {if\;g_{j} (X) \ge 0} \hfill \\ \end{array}$$

(6)

The direction-finding problem described by Eq. (5) is actually a quadratic programming problem and special techniques should be used for its solution. The associated one-dimensional search is written from the determined search direction S and an exterior penalty function ϕ, as given by Eq. (7).

$$\phi = F(X) + \sum\limits_{j = 1}^{m} {u_{j} \{ \hbox{max} [0,\,g_{j} (X)]\} + } \sum\limits_{k = 1}^{l} {u_{m + k} \left| {h_{k} (X)} \right|}$$

(7)

where X = X ^q−1 + α _p S, u _j  = | λ _j | (j = 1, m + l) in the first iteration, u _j  = max [| λ _j |, 0.5 (u’ _j  + | λ _j |)] for the subsequent iterations, and u’ _j  = u _j from the previous iteration. In this case, α _p  = 1.