Parameter Identification of a Nonlinear Vertical Axis Rotating Machine: An Innovative Approach Based on Reduced Order Modeling and Data Assimilation

Academic Background

In modern engineering, the modeling of nonlinear dynamic systems is a crucial research area. However, such systems often involve parameters that are difficult to measure or estimate directly, and incorporating all relevant physical phenomena into mathematical models significantly increases computational costs. To address this issue, the Hybrid Twin model has emerged. The Hybrid Twin model combines the system’s physics-based mathematical model with empirical data collected from the actual system, enhancing the accuracy and reliability of parameter estimation and system behavior prediction through data assimilation techniques. Additionally, the use of Reduced Order Models (ROM) significantly reduces the computational burden of the entire process.

This study focuses on the parameter identification problem of Vertical Axis Rotating Machines (VARM), a common type of rotating machine whose vibration behavior is influenced by the flexibility of the shaft and support structures. Particularly, the nonlinear characteristics of the support structures make related parameters difficult to quantify. By combining Sparse Proper Generalized Decomposition (SPGD) and the Levenberg-Marquardt optimization technique, this study proposes a novel method that effectively identifies unknown parameters in complex nonlinear systems and offers significant computational advantages.

Source of the Paper

This paper was co-authored by Sima Rishmawi, Ludivine Moyne, Souheil Serroud, Sebastian Rodriguez, Francisco Chinesta, Oguzhan Tuysuz, and Frédérick P. Gosselin. The authors are affiliated with the Multi-Scale Mechanics Laboratory (LM2) at École Polytechnique de Montréal, the PIMM Laboratory at Arts et Métiers Institute of Technology in Paris, and the Virtual Manufacturing Research Laboratory at École Polytechnique de Montréal. The paper was accepted on February 18, 2025, and published in the journal Nonlinear Dynamics, with the DOI 10.1007/s11071-025-11032-3.

Research Process and Results

1. Research Process

The primary goal of this study is to identify unknown parameters of nonlinear bearing forces in VARM using the Hybrid Twin framework, combining SPGD and the Levenberg-Marquardt optimization technique. The research is divided into the following steps:

a) White-Box Model Construction

First, the researchers used Latin Hypercube Sampling to generate 1,000 random scenarios, each containing a combination of unknown parameters. These parameters include bearing stiffness, structural damping ratio, bearing damping ratio, and force coefficients. Next, the researchers employed the Harmonic-Modal Hybrid (HMH) method to solve the system responses for these scenarios. The HMH method solves nonlinear partial differential equations in the frequency domain using modal basis analysis, avoiding time integration schemes and enabling rapid computation.

To further reduce computational complexity, the researchers applied Singular Value Decomposition (SVD) to the response matrices obtained from the HMH solver, extracting the first few temporal modes to form a temporal reduced basis. Subsequently, the researchers constructed a parameterized reduced-order model based on SPGD, which can quickly predict system responses for any combination of parameters.

b) Black-Box Model Construction

In the actual system, the researchers collected displacement data from the VARM using sensors. These data were compared with simulated responses from the white-box model to provide a basis for subsequent parameter optimization.

c) Grey-Box Model Construction (Hybrid Twin)

At this stage, the researchers used the Levenberg-Marquardt optimization technique to estimate the optimal values of unknown parameters by minimizing the error between the simulated responses and the measured data. To ensure optimization accuracy, the researchers conducted 50 independent optimization trials and averaged the results using Monte Carlo simulations.

2. Research Results

a) Results of the Geometric Model

In the geometric model, the researchers successfully identified parameters such as bearing stiffness, structural damping ratio, bearing damping ratio, and force coefficients. The results showed that bearing stiffness and damping ratios remained consistent across different rotational speeds and eccentric masses, while force coefficients increased with rotational speed, consistent with theoretical predictions of centrifugal force. By comparing with measured data, the researchers verified the accuracy of the optimized parameters, demonstrating the effectiveness of the method in complex nonlinear systems.

b) Results of the Expansion Model

In the expansion model, the researchers further identified multiple parameters of the bearing stiffness function. By introducing a sparsity loss term, the researchers were able to assess the impact of each parameter on the system response and eliminate insignificant parameters. Ultimately, the researchers proposed a simplified bearing stiffness function model that accurately predicts the system’s dynamic response.

c) Construction of the Hybrid Twin

By combining the results of the geometric and expansion models, the researchers constructed a Hybrid Twin model. This model not only accurately predicts the behavior of the VARM under different operating conditions but also provides important insights for future system monitoring, maintenance, and failure prediction.

Conclusion and Significance

The innovation of this study lies in the combination of SPGD, HMH, and the Levenberg-Marquardt optimization technique, proposing an efficient method for parameter identification in nonlinear systems. This method is not only applicable to VARM but can also be extended to other complex systems, such as vertical-axis hydraulic turbines. By accurately estimating system parameters, the researchers provide critical insights for equipment maintenance planning and failure prediction, offering broad application value.

Research Highlights

1. Innovative Method: This study is the first to combine SPGD, HMH, and the Levenberg-Marquardt optimization technique, proposing an efficient method for parameter identification in nonlinear systems.
2. Computational Efficiency: Through reduced-order models and SPGD, the researchers significantly reduced computational complexity, enabling rapid parameter identification.
3. Broad Applicability: The method is not only applicable to VARM but can also be extended to other complex systems, such as vertical-axis hydraulic turbines, demonstrating broad application prospects.

Future Work

The researchers plan to apply this method to more complex systems, such as vertical-axis hydraulic turbines, and further explore its potential in transient response analysis and failure prediction. Additionally, the researchers will investigate other optimization techniques to improve the accuracy and efficiency of parameter identification.

Acknowledgments

This study was supported by funding from Hydro-Québec, Maya HTT, NSERC Alliance, Inovéé, IVADO, and Mitacs.

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Through this study, the researchers not only addressed the parameter identification problem in VARM but also provided new approaches and methods for modeling and optimizing complex nonlinear systems. In the future, the further development and application of this method will bring more innovation and breakthroughs to the engineering field.