Experimental verification: a multi-objective optimization method for inversion technology of hydrodynamic journal bearings

Abstract

Aiming at the key design variables of journal bearings, a novel optimization scheme is proposed to minimize oil leakage and power loss. For the first time, the inversion technology is introduced into the multi-objective optimization genetic algorithm under thermohydrodynamics. Using the hybrid optimization method (sequential quadratic programming and multi-objective optimization genetic algorithm) and the pareto optimal frontier method, the journal bearing model under the oil supply condition of oil groove (Model A) and oil hole (Model B) is optimized. More importantly, the oil leakage (Q_L) formula is exhaustively deduced, and good prediction results are obtained by simulating the data in literature. The optimization test results show that compared with the maximum errors (13% and 25%) of the power loss and leakage flow prediction results in literature, the maximum errors of this prediction model are 8% and 14%, respectively. In addition, compared with hybrid optimization method, the pareto optimal frontier has better advantages under inversion technology. Both methods can give good prediction results. The accuracy of this model is proved by comparing experimental data in the literature.

Appendices

Appendix

Inversion process (IP)

It can be seen from Fig. 2 that inversion technology is embed in the process of computing power loss (P_L) and side leakage (Q_L). The basic principle is similar to the constrained linear least squares inversion method. In particular, prior information r is introduced in addition to the starting (or initial) model m⁰. At the same time, the data equation is scaled with a diagonal weighting matrix W = σ⁻¹I, which makes the solution process stable.

Considering p parameters, r is the prior data, the constraint equation in the form of Dm = r can be expressed as:

$$Dm=\left[\begin{array}{cccc}1& & & \\ & 1& & \\ & & \ddots & \\ & & & 1\end{array}\right]\left[\begin{array}{c}{m}_{1}\\ {m}_{2}\\ \vdots \\ {m}_{p}\end{array}\right]=\left[\begin{array}{c}{r}_{1}\\ {r}_{2}\\ \vdots \\ {r}_{p}\end{array}\right]$$

(28)

In order to minimize the difference between adjacent parameters to the minimum smoothness, the Twoney–Tikhonoy smoothness measure is taken, which is described by:

$$Dm=\left[\begin{array}{cccc}1& -1& & \\ & 1& -1& \\ & & \ddots & -1\\ & & & 1\end{array}\right]\left[\begin{array}{c}{m}_{1}\\ {m}_{2}\\ \vdots \\ {m}_{p}\end{array}\right]=\left[\begin{array}{c}{r}_{1}\\ {r}_{2}\\ \vdots \\ {r}_{p}\end{array}\right]$$

(29)

Our aim is to bias m towards r, so given a limited set of inaccurate observations, find the true solution (considering data and model errors) among all equivalent solutions and make it fit the observations, and satisfy the reliable estimation of the model parameters. Mathematically speaking, the above problem is equivalent to a very small deviation of the prediction error and the final solution from the specified constraints, which is described as:

$$L = \left( {Wd - Wf\left( m \right)} \right)^{T} \left( {Wd - Wf\left( m \right)} \right) + \left( {\beta \left[ {Dm - r} \right]} \right)^{T} \left( {\beta \left[ {Dm - r} \right]} \right)$$

(30)

If $f\left(m\right)$ is continuous and differentiable. Then it can be expanded with respect to the initial model m⁰ using Taylor's theorem to obtain a linear approximation of Eq. (30), which is given by:

$$L = \left( {Wy - WAx} \right)^{T} \left( {Wy - WAx} \right) + \left\{ {\left[ {D\left( {m^{0} + x} \right) - r} \right]^{T} \beta^{T} \beta \left[ {D\left( {m^{0} + x} \right) - r} \right]} \right\}$$

(31)

Make B = β^Tβ, expand Eq. (31), reset the partial differential to 0, and finally get the bias solution, which is illustrated by:

$$x = \left[ {\left( {WA} \right)^{T} WA + B} \right]^{ - 1} \left[ {\left( {WA} \right)^{T} Wy + B\left\{ {r - m^{0} } \right\}} \right]$$

(32)

The corresponding iterative formula can be denoted as:

$$m^{k + 1} = m^{k} + \left[ {\left( {WA} \right)^{T} WA + B} \right]^{ - 1} \left[ {\left( {WA} \right)^{T} Wy + B\left\{ {r - m^{k} } \right\}} \right]$$

(33)

If the prior information is doubtful (or not credible), then the constraint needs to be reset, that is, $r={\left[\mathrm{0,0},\cdots ,0\right]}^{T}$, and all elements of d are set to equal constants (0 < β < 1), so that all parameters have equal weights. In this case, β can be conveniently replaced by a single-valued indeterminate multiplier α, resulting in a parameter-corrected solution, which is given by:

$$x_{s} = \left[ {\left( {WA} \right)^{T} + \beta^{2} I} \right]^{ - 1} \left[ {\left( {WA} \right)^{T} Wy - \beta^{2} m^{0} } \right]$$

(34)

The corresponding iterative formula can be denoted as:

$$m^{k + 1} = m^{k} + \left[ {\left( {WA} \right)^{T} + \beta^{2} I} \right]^{ - 1} \left[ {\left( {WA} \right)^{T} Wy - \beta^{2} m^{k} } \right]$$

(35)

Considering D = 1, therefore, β²I is used to control the solution step size, while β²m^k helps to reduce its position towards the zero vector r.