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Emerging Trends in Symmetry Review Article

Representation-Theoretic Symmetry Reduction and Fuzzy-Grey Optimization of Modular Vibration Systems

  1. Tejas Bhushan N.B. Department of Chemistry, Regional Institute of Education (NCERT)
  2. Markala Karthik Department of Electrical and Electronics Engineering, SR University
  3. Mohammed Almakki School of Engineering, Architecture and Interior Design, Amity University Dubai
  4. Mohammed El Khider Department of General Undergraduate Curriculum Requirements, University of Dubai

Abstract

This paper presents a representation-theoretic framework for symmetry-aware vibration control in modular structural systems. Exploiting cyclic symmetry, the mass, damping, and stiffness operators are block-diagonalised into irreducible representations, reducing the full structural dynamics to a collection of lower-dimensional modal subsystems. This decomposition provides both computational efficiency and a rigorous mathematical description of symmetry-preserving dynamic behaviour. To account for imperfections arising in practical implementations, near-symmetry defects in stiffness and damping are quantified using projector-based measures defined on the corresponding invariant subspaces. An uncertainty band is introduced to model manufacturing tolerances, parameter variability, and control-induced perturbations, enabling the analysis of structural performance under bounded uncertainty. The resulting formulation captures deviations from ideal symmetry while retaining the underlying algebraic structure of the system. A multi-criteria optimisation framework is then developed to balance vibration attenuation, control effort, and symmetry preservation. These competing objectives are integrated through a fuzzy-grey relational model, producing a mathematically explicit objective function suitable for robust design and parameter tuning. The optimisation process identifies solutions that simultaneously enhance damping performance and limit symmetry degradation in the presence of uncertainty. A numerical study involving a six-module cyclic ring structure illustrates the effectiveness of the proposed approach. Results show that both symmetry-reduced retuning and the fuzzy-grey optimal design significantly improve vibration suppression compared with the baseline configuration. Moreover, the fuzzy-grey optimum achieves additional reductions in symmetry defect while maintaining favourable control characteristics. The proposed framework contributes an ETSY-aligned methodology in which symmetry, uncertainty quantification, and optimisation are unified through algebraic operators, representation theory, and high-density mathematical formulations, providing a systematic foundation for the design of robust modular structural systems.

Keywords

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