In today’s rapidly evolving and complex systems, advanced analytical tools are essential for ensuring safety, performance, and cost-effectiveness. Real-Time Principal Geodesic Analysis (RPGA) stands as a technique for analyzing nonlinear data on differentiable manifolds, addressing the limitations of traditional linear methods such as Principal Component Analysis (PCA). While PCA and Principal Geodesic Analysis (PGA) are widely used for condition monitoring in mechanical, aerospace, and civil engineering, they fall short when dealing with systems exhibiting uncertainties and dynamic fluctuations in curved, non-Euclidean spaces. To overcome this, RPGA adapts orthogonal transformation techniques to nonlinear settings, offering robust solutions where traditional methods fail. Over recent decades, advancements in real-time data-driven algorithms like RPCA, RCCA, and RSSA have provided reliable tools for analyzing complex multidimensional systems. However, RPGA emerges as the optimal solution for curved spaces, effectively extracting meaningful insights from stochastic, nonlinear data streams.
This paper highlights the transition from linear to nonlinear data analysis and showcases RPGA’s effectiveness through examples such as the stochastic geometric oscillator and the inverted spherical pendulum cart system navigating rough surfaces. By enabling reliable, real-time damage detection, RPGA delivers unparalleled precision in condition monitoring, making it an indispensable tool for systems operating in stochastic and dynamic environments.
A new framework has been developed to solve geometric stochastic differential equations (SDEs) arising from stochastic Hamiltonian dynamics. This framework takes into account the interaction of geometry, nonlinearity, and stochasticity while preserving the geometry of the manifold where the system evolves.
Accurate estimation of the response with lower-order errors is a crucial aspect of closing the fundamental gap in the advancement of non-Euclidean statistics. The use of this framework enables a better understanding of the behavior of complex systems under uncertain conditions, and has the potential to revolutionize the field of non-Euclidean statistics.
In addition, a symplectic numerical integration scheme for the non-linear stochastic oscillators evolving on a manifold is developed to preserve Hamiltonian’s drift arising from the stochastic nature of the system along with the manifold’s geometry. This will enable the simulation of the Hamiltonian systems over long time intervals without loosing its accuracy.
Numerous studies highlight the connection between music and brain activity, yet the therapeutic effects of Indian Classical Music (ICM) remain underexplored. This study investigates the brain’s response to two specific ragas—Yaman and Puria Dhanashree—which share common notes but differ in two counter notes, offering a unique case to examine the influence of subtle musical variations on brain dynamics. Using a 24-channel Electroencephalogram (EEG) cap and smartphone-based monitoring, brain responses were recorded from five volunteers. To capture the brain's dynamic evolution during live ICM stimuli, this work introduces automated approaches based on:
Energy and Mahalanobis distance measurements, accounting for input and output uncertainties.
A region-specific real-time algorithm utilizing eigen perturbation, providing insights into the time evolution of brain activity.
The findings reveal a significant reduction in beta band power, indicating a state of relaxation post-music exposure. Moreover, different brain regions were activated depending on the listener’s musical knowledge, underscoring the brain's nuanced processing of ICM. By offering evidence of ICM's impact on cognitive and emotional states, this study lays the foundation for integrating ICM into evidence-based music therapy. These methods have promising implications for mHealth applications, targeting cognitive, emotional, and psychological well-being.
A novel real-time modal identification algorithm has been developed, based on first-order eigen-perturbation and second-order separation techniques for linear vibrating systems with complex modes. The effectiveness of the algorithm is demonstrated through the use of both numerically simulated and benchmark case studies.
These case studies take into account various scenarios such as the formation of complex modes with closely spaced modes, non-proportional damping, dynamic changes in the damping matrix, and the addition of a secondary system over the primary structure. This work provides accurate, real-time identification of complex modes, even when system properties change dynamically.
This new algorithm represents a major advancement in the field of modal identification, providing valuable tools for understanding and managing the behavior of complex vibrating systems.
A stochastic bi-directional formulation for structural systems based on the concepts of correlated Wiener process is developed in this project. The framework enables the exploration of the effect of different levels of correlation in the input excitation on the structural response and its statistics. This is particularly important in the presence of torsional coupling, where the rotation of one component of the structure can have significant impacts on the behavior of other components. By taking into account these complex interactions, the framework provides a more comprehensive understanding of the structural behavior under uncertain conditions.
A new approach for estimating the mean square solution of nonlinear structural systems has been developed using a framework based on the Adomian method. This framework allows for easy calculation of second-order response statistics of stochastic nonlinear dynamical systems, eliminating the need for computationally intensive Monte Carlo simulations.
By enabling the estimation of the second-order response statistics, this framework provides important information about the behavior of nonlinear structural systems under uncertain conditions. This information can be used to improve the design and performance of these systems, and to make informed decisions about their operation and maintenance.
The unique feature of the FOEP-based algorithm is its quick convergence, allowing for near-instantaneous identification and assessment of the system's behavior, even in complex and challenging situations. This significantly improves the efficiency of the overall modal identification and damage detection process, providing timely and accurate results.