Identifying Equations of Motion in Vibrating Structures from Measurements Using a Data-Driven, Energy-Based Approach

Centrala begrepp

This paper presents a novel data-driven method called Energy-Based Dual-Phase Dynamics Identification (EDDI) for identifying equations of motion in nonlinear vibrating structures directly from measurements, leveraging energy relationships to model damping and stiffness forces.

Sammanfattning

Bibliographic Information

Lopez, C., Singh, A., Naranjo, A., Moore, K. J. (Year). A Data-Driven, Energy-based Approach for Identifying Equations of Motion in Vibrating Structures Directly from Measurements. [Journal Name, Volume(Issue), Page Range].

Research Objective

This research paper introduces a new data-driven method, called the Energy-Based Dual-Phase Dynamics Identification (EDDI) method, to identify the nonlinear dynamics of single-degree-of-freedom (SDOF) oscillators directly from measurements. The primary objective is to accurately capture and model the nonlinear damping and stiffness characteristics of vibrating structures using only free-response measurements and the mass of the oscillator, without requiring prior knowledge of the system's dynamics.

Methodology

The EDDI method consists of two phases: damping-model identification and stiffness-model identification. In the first phase, the method leverages the equivalence of kinetic and mechanical energies at zero displacement to compute the energy dissipated by the system. This information is then used to identify a suitable mathematical model for the nonlinear damping forces. The second phase utilizes the computed dissipated energy to determine the mechanical energy, which is subsequently employed to obtain a reformulated Lagrangian. By analyzing the Lagrangian, the conservative forces acting on the oscillator are determined, leading to the identification of a mathematical model for the nonlinear stiffness of the system. The method is demonstrated using both simulated data from a Duffing oscillator and a damped pendulum, as well as experimental data from two different physical SDOF oscillator setups.

Key Findings

The EDDI method successfully identified the nonlinear damping and stiffness parameters for all simulated and experimental cases considered. The identified models accurately reproduced the dynamic behavior of the systems, as evidenced by the close agreement between simulated and measured responses in both the time and frequency domains. The method proved effective in capturing both smooth nonlinearities, such as those arising from geometric effects, and more complex softening-hardening behavior observed in systems with specific material properties.

Main Conclusions

The EDDI method presents a powerful and versatile approach for identifying the equations of motion for nonlinear SDOF systems directly from measurements. Its data-driven nature and reliance on fundamental energy principles make it applicable to a wide range of engineering applications where accurate system identification is crucial. The method's ability to effectively capture and model complex nonlinear behavior makes it a valuable tool for understanding and predicting the dynamics of vibrating structures.

Significance

This research significantly contributes to the field of nonlinear system identification by providing a novel and robust method for characterizing the dynamics of vibrating structures. The EDDI method's data-driven approach and ability to handle strong nonlinearities address a critical challenge in structural dynamics, enabling more accurate modeling and analysis of complex systems.

Limitations and Future Research

While the EDDI method demonstrates promising results for SDOF systems, its current formulation is not directly applicable to multi-degree-of-freedom (MDOF) systems. Future research should focus on extending the EDDI framework to handle MDOF systems, potentially by employing signal decomposition techniques to analyze individual modes. Additionally, further investigation is needed to assess the method's robustness and accuracy when dealing with non-smooth nonlinearities, such as those encountered in systems with impacts or discontinuities.

Statistik

The mass of the simulated Duffing oscillator is 0.05 kg.
The linear damping coefficient of the simulated Duffing oscillator is 0.5 Ns/m.
The nonlinear damping coefficient of the simulated Duffing oscillator is 4000 Ns/m³.
The linear stiffness coefficient of the simulated Duffing oscillator is 300 N/m.
The nonlinear stiffness coefficient of the simulated Duffing oscillator is 3 × 100 N/m³.
The mass of the damped pendulum is 2 kg.
The length of the damped pendulum is 0.8 m.
The damping coefficient of the damped pendulum is 0.1 kg/s².
The mass of the experimental Duffing oscillator with smooth nonlinearity is 0.815 kg.
The mass of the experimental Duffing oscillator with softening-hardening nonlinearity is 0.088 kg.

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A Data-Driven, Energy-based Approach for Identifying Equations of Motion in Vibrating Structures Directly from Measurements

by Cris... på arxiv.org 10-24-2024

https://arxiv.org/pdf/2410.17845.pdf

Djupare frågor

How could the EDDI method be adapted to identify time-varying system parameters in non-stationary vibrating structures?

The EDDI method, as described, focuses on systems with time-invariant parameters. To adapt it for non-stationary vibrating structures with time-varying parameters, several modifications could be explored:

Short-Time Windowing: Instead of analyzing the entire response signal at once, divide it into short, overlapping time windows. Within each window, assume the system parameters remain approximately constant. Apply the EDDI method to each window to obtain time-localized estimates of the damping and stiffness parameters. This approach would result in parameter trajectories over time, revealing their variation.

Adaptive Parameter Estimation: Incorporate techniques from adaptive control and online system identification. Instead of solving for constant parameters, formulate the problem with time-varying coefficients (e.g., b(t), k(t)). Employ recursive algorithms like Recursive Least Squares (RLS) or Kalman Filtering to update the parameter estimates as new data becomes available. This approach requires modifications to the core EDDI equations to accommodate time-varying terms.

Basis Function Expansion: Represent the time-varying parameters using a set of basis functions (e.g., polynomials, Fourier series, wavelets). Project the parameter variations onto this basis, effectively transforming the problem from identifying time-varying coefficients to identifying constant coefficients for the basis functions. The EDDI method can then be applied to estimate these constant coefficients, and the time-varying parameters can be reconstructed from the basis function expansion.

Hybrid Methods: Combine the EDDI method with other techniques specifically designed for non-stationary systems, such as Empirical Mode Decomposition (EMD) or Hilbert-Huang Transform (HHT). These methods can decompose the response signal into intrinsic mode functions (IMFs) that represent different time scales of the system's behavior. The EDDI method could then be applied to each IMF separately to capture the time-varying dynamics associated with different frequency bands.

Challenges in adapting EDDI for non-stationary systems include:

Increased computational complexity due to analyzing multiple time windows or implementing recursive algorithms.
Selection of appropriate window sizes or adaptation rates to balance accuracy and tracking ability.
Potential for overfitting if the parameter variations are not smooth or if the chosen model complexity is too high.

Could the reliance on energy relationships in the EDDI method limit its applicability to systems with significant energy exchange with the environment, and how might this limitation be addressed?

Yes, the EDDI method's reliance on energy relationships can be a limiting factor when dealing with systems experiencing significant energy exchange with the environment. This is because the method assumes that energy dissipation is primarily due to internal damping mechanisms. If external forces or energy sources are present, the estimated energy balance will be inaccurate, leading to errors in identifying the system parameters.
Here are some potential ways to address this limitation:

Input Energy Estimation: If the external forces acting on the system are measurable, their contribution to the energy balance can be estimated and incorporated into the EDDI equations. This would involve calculating the work done by the external forces and adjusting the energy dissipation calculations accordingly.

System Isolation: Experimentally, efforts can be made to isolate the system from external disturbances as much as possible. This could involve using vibration isolation platforms, minimizing air resistance, or conducting experiments in controlled environments.

Energy Flow Modeling: For cases where external energy exchange is unavoidable or even desired, consider extending the EDDI framework to explicitly model the energy flow between the system and its surroundings. This would involve incorporating additional terms in the energy balance equations to account for energy input and output.

Adaptive Filtering: Implement adaptive filtering techniques to separate the system's response from external disturbances in the measured data. This could involve using algorithms like Adaptive Noise Cancellation (ANC) or Independent Component Analysis (ICA) to identify and remove the unwanted components from the signal before applying the EDDI method.

Hybrid Modeling: Combine the EDDI method with other system identification techniques that are less sensitive to external disturbances. For example, use a black-box model to capture the overall input-output behavior of the system, and then use the EDDI method to refine the model by focusing on the free-response dynamics.

The choice of approach would depend on the specific application and the nature of the energy exchange with the environment.

How can the insights gained from identifying nonlinear equations of motion using the EDDI method be leveraged to develop more effective vibration control strategies in practical engineering applications?

Identifying accurate nonlinear equations of motion using the EDDI method can significantly benefit the development of more effective vibration control strategies in various engineering applications:

Model-Based Control Design: Accurate nonlinear models are crucial for designing high-performance controllers. With the identified equations of motion, engineers can employ advanced control techniques like feedback linearization, sliding mode control, or model predictive control to suppress unwanted vibrations more effectively than using linear approximations.

Nonlinearity Compensation: By understanding the specific nonlinear characteristics of the system, control strategies can be tailored to compensate for them. This could involve designing nonlinear damping or stiffness elements to counteract the inherent nonlinearities, leading to improved vibration attenuation.

Performance Optimization: The identified models allow for simulating and evaluating different control strategies under various operating conditions. This enables engineers to optimize controller parameters, actuator placement, and other design choices to achieve desired performance specifications, such as minimizing settling time, reducing peak amplitudes, or suppressing specific frequency bands.

Fault Detection and Diagnosis: Deviations between the predicted behavior of the identified model and the actual system response can indicate potential faults or damage. By monitoring these discrepancies, it's possible to develop model-based fault detection and diagnosis systems for vibrating structures, enabling early detection of problems and preventing catastrophic failures.

Digital Twin Development: The identified nonlinear models can be used to create high-fidelity digital twins of vibrating structures. These digital representations can be used for virtual testing, what-if analyses, and control strategy development in a safe and cost-effective manner.

Here are some specific examples of how EDDI-identified models can improve vibration control:

Precision Machine Tools: Suppress vibrations caused by nonlinear cutting forces to improve machining accuracy and surface finish.
Aerospace Structures: Design active control systems for aircraft wings or helicopter rotors to mitigate vibrations induced by aerodynamic nonlinearities.
Civil Engineering Structures: Develop semi-active or active damping systems for buildings and bridges to enhance their resilience to earthquakes and wind loads.
Micro- and Nano-Systems: Control vibrations in MEMS and NEMS devices, where nonlinearities are often significant due to scaling effects.
By leveraging the insights from accurate nonlinear models, engineers can design smarter, more adaptive, and more effective vibration control strategies, leading to safer, more reliable, and higher-performance engineering systems.