The Karhunen-Loève Transform was established to find structures in random process data. Nonlinear dynamical systems often appear to have uncorrelated output in case of chaotic behavior. This analogy leads to the idea of analyzing nonlinear dynamical systems with methods developed for random processes. The Karhunen-Loève Transform provides a basis for different approaches to the investigation of these systems. This paper gives an introduction to the mathematical concept and an overview of popular Karhunen-Loève Transform applications. It focuses on approaches to state monitoring of nonlinear dynamical systems based on experimental data.

The term "discrete element method" (DEM) in engineering science comprises various approaches to model physical systems by agglomerates of free particles. While shapes, sizes and properties of particles may vary, in most DEM models, particles are not confined by constraints, but subject to applied forces derived from potential fields and/or contact laws. This general approach allows for widespread use of DEM models for physical phenomena including gas dynamics, granular flow, fracture and impact analysis. However, its characteristic feature, combining particle restraints and forces into applied forces, does not only provide for flexible adaption of DEM to different physics, but also creates the most limiting restriction: Evaluation of the applied forces for each particle is computational expensive restraining the time sequence and sample size for numerical analyses. As an ansatz to circumvent this obstacle for a class of DEM models, we propose a model order reduction method based on coherency in the dynamics of particles. While initial flexibility of DEM is conserved, computational effort can be reduced significantly.

Inspection and maintenance of railway networks is a complex and expensive task. Special measurement vehicles are used to record the geometrical properties of railway lines within required time intervals. Due to the extent of measurement data the quality of railway track is evaluated considering only a few parameters. Although safety and comfort of wheel-rail-systems depend on the dynamical behavior, current inspection vehicles are not equipped to measure dynamic properties.In this paper, we will discuss a novel approach to evaluate the quality of railway tracks based on wheel-rail dynamics: Wheelset dynamics of subway trains are analyzed by Karhunen-Loève Transformation to extract the principal dynamics from the collected measurement data.

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This paper focuses on the identification of dominant coherent structures in the nonlinear dynamic behavior of railway wheelsets by Karhunen‐Loève‐Transform (KLT). The ultimate goal is the characterization of the rolling quality of railway wheelsets. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)