ABSTRACT Group theory has profoundly advanced physics and chemistry in systems with symmetries. Yet its use in structural engineering applications has not yet been fully explored beyond the aesthetics of symmetric designs. This work addresses two significant gaps that have limited the broader adoption of group‐theoretic methods in structural vibration analysis and clarifies their implications for structural design when multiple eigenvalues arise. First, a problem‐independent approach is presented with detailed derivations for constructing group representations for symmetric structures directly from the ‐invariance for structural vibration analysis. This method applies effectively to both dihedral groups and the higher‐order Platonic groups, including tetrahedral (), octahedral (), and icosahedral () symmetries. The method used in this work scales well with structural complexity and enables both explicit and canonical block diagonalizations. Second, this work provides a comprehensive guide to applying finite‐point‐group representations in structural vibration analysis and proves that the dimensions of irreducible representations determine eigenfrequency multiplicities. Although no optimization is performed in this work, this theoretical result has direct implications for structural optimization, resolving longstanding misconceptions about the coalescence of eigenvalues by showing that symmetry is the origin of repeated eigenfrequencies. The theoretical developments are validated on truss structures with dihedral and higher‐order symmetries, accurately predicting their eigenfrequency distributions.
Sun et al. (Fri,) studied this question.
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