Analytical exploration of generic undamped mass–spring chains: A comprehensive study

Abstract

Analysis of structural vibrations presents challenges due to the coupling between different degrees of freedom, often preventing the derivation of analytical expressions for quantities such as poles, vibration amplitude, etc. While numerical methods are commonly employed to tackle such challenges, their purely numerical nature prevents achieving a comprehensive understanding of the system behaviour. This paper builds on a novel recursive transfer function formulation recently proposed by the authors to analytically characterise the dynamics of generic undamped mass-chain systems. This approach provides fundamental properties and analytical insights that cannot be obtained through current numerical approaches, representing a significant breakthrough in the understanding of vibrations and offering valuable insights for analysing complex vibrating systems. These results presented in this paper allow for a rapid assessment of the minimum phase nature of the vibration system through straightforward stability assessment, and present analytical expressions for sums and multiplications of natural frequencies squared by induction and Vieta’s formulas. Furthermore, it derives conditions for existence of repeated roots using direct analytical derivation of the characteristics equations, including purely imaginary roots and repeated roots at the origin, showing that repeated roots can only occur when some forms of spatial symmetries exist, and confirms bounds for the arithmetic mean of natural frequencies by using inequalities between geometric, arithmetic and quadratic means. Additionally, analytical conditions are established for systems with different degrees of freedom to share the same roots, providing a theoretical foundation for eigenvalue assignment in the time domain or pole placement in the frequency domain. This foundation enables the determination of desired natural frequencies from scratch using transparent analytical conditions, without relying on complex algorithms typically required by current inverse methodologies involving passive structural modifications or active controllers. Moreover, this paper also explores the relationship between independent and coupled vibrations, providing insights for comparing natural frequencies. Overall, this paper contributes to a deeper understanding of structural vibrations through analytical exploration and offers practical guidance for eigenvalue assignment and natural frequency analysis.