A novel multi-fidelity neural network for response prediction using rotor dynamics and model reduction

Abstract

Uncertainties in rotating machines are unavoidable, which affect their parameters and dynamic response. So, instead of employing deterministic models, data-driven meta-modeling techniques which incorporate unpredictability and randomness are necessary for the response variation analysis of rotating systems. The performance of the meta-model relies heavily on the quality and amount of the training dataset. In reality, however, only a tiny amount of high-fidelity data is obtainable from high-dimensional finite element simulation or experimental investigation, although low-cost low-fidelity data may be numerous. The objective of this paper is to develop a novel neural network model for multi-level response prediction by obtaining a high number of low-fidelity data quickly through model order reduction and a limited amount of high-fidelity data correctly from a full-order model. The accuracy of the meta-model is demonstrated by comparing against a classical deep neural network. Two different types of meta-model are established by using two model reduction techniques: Guyan reduction and modified system equivalent reduction expansion process. The performance of the model is demonstrated by employing frequency response variation characterization of a complex rotor as a case example. The results reveal that the multi-fidelity neural network performs better than the low-fidelity frequency response curves alone, which is observed to have a lot of inaccuracies. The deep neural network, on the other hand, is unable to reflect on the dynamic response of the full model. A regression of more than 90% shows that the meta-model has high effectiveness in properly predicting the frequency responses. The mean squared error values for the meta-model are found to be less than 0.1, which is typically regarded as acceptable. Frequency response curves of four test samples are selected at random for comparison. It is observed that the meta-model frequency response moves much closer to the full model than compared to that of the low-fidelity model reduction. The performance resilience of the model is tested by using five different training runs with random data splits. Minor changes in the values of logarithm mean absolute error and logarithm root mean squared error under different training runs show appropriate curve fitting and signify superior accuracy. It is concluded that the multi-fidelity neural network can reach a higher level of accuracy with a limited amount of high-fidelity data. The model effectively identifies both the linear and complex nonlinear correlation between the high-and low-fidelity data, resulting in enhanced efficacy in contrast to state-of-the-art methods.

Appendices

Appendix

Shaft element matrices

$$[M_{e}^{T} ]_{S} = \frac{{\uprho AL_{e} }}{{840(1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & {m_{2} } & {m_{3} } & 0 & 0 & {m_{4} } \\ 0 & {m_{1} } & { - m_{2} } & 0 & 0 & {m_{3} } & { - m_{4} } & 0 \\ 0 & { - m_{2} } & {m_{5} } & 0 & 0 & {m_{4} } & {m_{6} } & 0 \\ {m_{2} } & 0 & 0 & {m_{5} } & { - m_{4} } & 0 & 0 & {m_{6} } \\ {m_{3} } & 0 & 0 & { - m_{4} } & {m_{1} } & 0 & 0 & { - m_{2} } \\ 0 & {m_{3} } & {m_{4} } & 0 & 0 & {m_{1} } & {m_{2} } & 0 \\ 0 & { - m_{4} } & {m_{6} } & 0 & 0 & {m_{2} } & {m_{5} } & 0 \\ {m_{4} } & 0 & 0 & {m_{6} } & { - m_{2} } & 0 & 0 & {m_{5} } \\ \end{array} } \right]$$

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$$[M_{e}^{R} ]_{S} = \frac{{{\uprho {\rm I}}}}{{30L_{e} (1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} {m_{7} } & 0 & 0 & {m_{8} } & { - m_{7} } & 0 & 0 & {m_{8} } \\ 0 & {m_{7} } & { - m_{8} } & 0 & 0 & {m_{7} } & { - m_{8} } & 0 \\ 0 & { - m_{8} } & {m_{9} } & 0 & 0 & {m_{8} } & {m_{10} } & 0 \\ {m_{8} } & 0 & 0 & {m_{9} } & { - m_{8} } & 0 & 0 & {m_{10} } \\ { - m_{7} } & 0 & 0 & { - m_{8} } & {m_{7} } & 0 & 0 & { - m_{8} } \\ 0 & { - m_{7} } & {m_{8} } & 0 & 0 & {m_{7} } & {m_{8} } & 0 \\ 0 & { - m_{8} } & {m_{10} } & 0 & 0 & {m_{8} } & {m_{9} } & 0 \\ {m_{8} } & 0 & 0 & {m_{10} } & { - m_{8} } & 0 & 0 & {m_{9} } \\ \end{array} } \right]$$

(54)

$${[}G_{e} {]}_{S} = \frac{{{\uprho {\rm I}}}}{{15L_{e} (1 +\Phi )^{2} }}\left[ {\begin{array}{*{20}c} 0 & { - g_{1} } & {g_{2} } & 0 & 0 & {g_{1} } & {g_{2} } & 0 \\ {g_{1} } & 0 & 0 & {g_{2} } & { - g_{1} } & 0 & 0 & {g_{2} } \\ { - g_{2} } & 0 & 0 & { - g_{3} } & {g_{2} } & 0 & 0 & { - g_{4} } \\ 0 & { - g_{2} } & {g_{3} } & 0 & 0 & {g_{2} } & {g_{4} } & 0 \\ 0 & {g1} & { - g_{2} } & 0 & 0 & { - g_{1} } & { - g_{2} } & 0 \\ { - g_{1} } & 0 & 0 & { - g_{2} } & {g_{1} } & 0 & 0 & { - g_{2} } \\ { - g_{2} } & 0 & 0 & { - g_{4} } & {g_{2} } & 0 & 0 & { - g_{3} } \\ 0 & { - g_{2} } & {g_{4} } & 0 & 0 & {g_{2} } & {g_{3} } & 0 \\ \end{array} } \right]$$

(55)

$$[K_{e} ]_{S} = \frac{{{{\rm E}{\rm I}}}}{{L_{e}^{3} (1 +\Phi )}}\left[ {\begin{array}{*{20}c} {12} & 0 & 0 & {6L_{e} } & { - 12} & 0 & 0 & {6L_{e} } \\ 0 & {12} & { - 6L_{e} } & 0 & 0 & { - 12} & { - 6L_{e} } & 0 \\ 0 & { - 6L_{e} } & {(4 +\Phi )L_{e}^{2} } & 0 & 0 & {6L_{e} } & {(2 -\Phi )L_{e}^{2} } & 0 \\ {6L_{e} } & 0 & 0 & {(4 +\Phi )L_{e}^{2} } & { - 6L_{e} } & 0 & 0 & {(2 -\Phi )L_{e}^{2} } \\ { - 12} & 0 & 0 & { - 6L_{e} } & {12} & 0 & 0 & { - 6L_{e} } \\ 0 & { - 12} & {6L_{e} } & 0 & 0 & {12} & {6L_{e} } & 0 \\ 0 & { - 6L_{e} } & {(2 -\Phi )L_{e}^{2} } & 0 & 0 & {6L_{e} } & {(4 +\Phi )L_{e}^{2} } & 0 \\ {6L_{e} } & 0 & 0 & {(2 -\Phi )L_{e}^{2} } & { - 6L_{e} } & 0 & 0 & {(4 +\Phi )L_{e}^{2} } \\ \end{array} } \right]$$

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where,

$$\begin{array}{*{20}l} {m_{1} = 312 + 588\Phi + 280\Phi ^{2} } \hfill \\ {m_{2} = (44 + 77\Phi + 35\Phi ^{2} )L_{e} } \hfill \\ {m_{3} = 108 + 252\Phi + 140\Phi ^{2} } \hfill \\ {m_{4} = - (26 + 63\Phi + 35\Phi ^{2} )L_{e} \quad g_{1} = 36} \hfill \\ {m_{5} = (8 + 14\Phi + 7\Phi ^{2} )L_{e}^{2} \quad g_{2} = (3 - 15\Phi )L_{e} } \hfill \\ {m_{6} = - (6 + 14\Phi + 7\Phi ^{2} )L_{e}^{2} \quad } \hfill \\ {m_{7} = 36\quad g_{3} = (4 + 5\Phi + 10\Phi ^{2} )L_{e}^{2} } \hfill \\ {m_{8} = (3 - 15\Phi )L_{e} \quad g_{4} = ( - 1 - 5\Phi + 5\Phi ^{2} )L_{e}^{2} } \hfill \\ {m_{9} = (4 + 5\Phi + 10\Phi ^{2} )L_{e}^{2} } \hfill \\ {m_{10} = ( - 1 - 5\Phi + 5\Phi ^{2} )L_{e}^{2} } \hfill \\ \end{array}$$

Bearing matrices

$$[C]_{B} = \left[ {\begin{array}{*{20}c} {C_{xx} } & {C_{xy} } \\ {C_{yx} } & {C_{YY} } \\ \end{array} } \right]\quad [K]_{B} = \left[ {\begin{array}{*{20}c} {K_{xx} } & {K_{xy} } \\ {K_{yx} } & {K_{YY} } \\ \end{array} } \right]$$

(57)

Disk matrices

$$[M_{e} ]_{D} = \left[ {\begin{array}{*{20}c} {m_{d} } & 0 & 0 & 0 \\ 0 & {m_{d} } & 0 & 0 \\ 0 & 0 & {I_{d} } & 0 \\ 0 & 0 & 0 & {I_{d} } \\ \end{array} } \right]\quad [G_{e} ]_{D} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & { - I_{p} } \\ 0 & 0 & {I_{p} } & 0 \\ \end{array} } \right]$$

(58)