Direct Parametrisation of Invariant Manifolds for Non-Autonomous Forced Systems Including Superharmonic Resonances

Core Concepts

Efficiently deriving invariant manifolds for non-autonomous systems with superharmonic resonances.

Abstract

The content discusses a method to derive reduced-order models for nonlinear systems using the direct parametrisation of invariant manifolds. It introduces an algorithm that extends the dimension of parametrising coordinates to include forcing terms, enabling the derivation of efficient reduced-order models for systems exhibiting superharmonic resonance. The approach is validated on academic test cases involving beams and arches, demonstrating its effectiveness in handling 3:1 and 2:1 superharmonic resonances. The method overcomes limitations of previous approaches by relaxing assumptions on non-autonomous terms and allowing arbitrary order expansions. It also addresses time-dependent invariant manifolds due to external forcing, providing a comprehensive solution applicable to finite element problems with higher forcing levels.

Stats

"It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances." "The resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency." "The results shown highlight the need to use a high-order expansion for the ε-order forcing term." "For some extreme conditions, this might not be sufficient." "The main limitation of this approach is to provide in the reduced dynamics only terms that are proportional to the forcing amplitude at power 1." "In conjunction with these developments, efficient open-source softwares implementing the developments have been released in order to share the method."

Quotes

"It is numerically demonstrated that the method generates efficient ROMs for problems involving 3:1 and 2:1 superharmonic resonances." "The resonance relationships appearing through the homological equations involve multiple occurrences of the forcing frequency."

Key Insights Distilled From

Direct parametrisation of invariant manifolds for non-autonomous forced systems including superharmonic resonances

by Ales... at arxiv.org 03-19-2024

https://arxiv.org/pdf/2306.09860.pdf

Deeper Inquiries

How can this method be extended beyond mechanical vibrations

The method described in the context above, which involves direct parametrisation of invariant manifolds for non-autonomous systems, can be extended beyond mechanical vibrations to a wide range of fields. One potential application is in structural engineering, where it can be used to analyze and model the behavior of complex structures under varying external forces. This could be particularly useful in designing resilient and efficient structures that can withstand dynamic loading conditions. In the field of aerospace engineering, this method could be applied to study the dynamics of aircraft and spacecraft subjected to non-uniform aerodynamic forces during flight. By accurately capturing the nonlinear responses of these vehicles, engineers can optimize their designs for improved performance and safety. Furthermore, this approach could also find applications in biological systems modeling. For instance, it could be used to analyze the response of biological tissues or organs to external stimuli or perturbations. Understanding how these systems behave under different conditions is crucial for advancements in medical research and healthcare technology. Overall, by extending this method beyond mechanical vibrations, researchers can gain valuable insights into a diverse range of dynamic systems across various disciplines.

What are potential drawbacks or limitations of using arbitrary order expansions

While arbitrary order expansions offer flexibility and accuracy in solving complex problems involving nonlinear dynamics like those seen in mechanical vibrations with geometric nonlinearity, there are some potential drawbacks or limitations associated with using them: Computational Complexity: As the order of expansion increases, so does the computational complexity involved in solving the equations. Higher-order expansions require more computational resources and time for processing. Convergence Issues: In some cases, higher-order expansions may lead to convergence issues due to numerical instabilities or inaccuracies introduced during calculations. Ensuring convergence becomes challenging as the order increases. Increased Memory Usage: Storing coefficients for higher-order terms requires more memory space which can become a limitation when dealing with large-scale problems or limited computing resources. Overfitting: There is a risk of overfitting data when using high-order expansions without proper validation techniques. This may result in models that perform well on training data but fail to generalize effectively on unseen data.

How does this research impact other fields outside mathematics

This research has significant implications beyond mathematics and theoretical physics into practical applications across various fields: 1- Engineering Applications: Structural Engineering: Improved understanding of nonlinear behaviors helps design safer buildings. Aerospace Engineering: Enhances aircraft design by predicting responses under varying loads. 2- Biomedical Applications: Biomechanics: Enables better modeling of tissue responses under different biomechanical forces. Medical Device Design: Optimizes medical devices based on accurate simulations. 3- Environmental Science: Climate Modeling: Provides insights into complex climate system interactions. 4- Robotics: Control Systems: Enhances robot control algorithms for better performance. 5- Material Science: - Material Properties Prediction : Helps predict material properties under different stress scenarios By advancing our understanding through mathematical methods like direct parametrisation techniques applied here , we pave way towards innovation across multiple domains leading us towards advanced technologies benefiting society as whole .