Equivalent linear systems of nonlinear systems

Tobias Friis*, Marius Tarpø, Evangelos I. Katsanos, Rune Brincker

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review


The dynamic behaviour of civil engineering structures is frequently nonlinear, but it is often engineering practice to assume linearity. Consequently, a link between the nonlinear and linear models is required to ensure appropriate correspondence between the true dynamic behaviour and the model, which can be achieved by enforcing an equivalence between the two. This paper focuses on estimating equivalent linear systems of simulated multi-degree-of-freedom systems with stiffness and damping nonlinearities subjected to broad-band or general excitation using three different approaches based on different equivalences and assumptions. The considered methods are: a method based on balancing the intensity of the excitation and the response in the frequency domain, a modal version of the so-called energy dissipation method, and an approach that estimates equivalent linear stiffness, viscous damping, and equivalent linear mode shapes directly through a least squares fit. Notably, the latter method enables the assessment of the common assumptions such as low damping, slowly varying amplitude and phase, invariant mode shapes, and proportional damping. Furthermore, using the aforementioned procedures, extensive numerical simulations enable a comparative assessment in terms of the related linear stiffness, damping, and mode shape estimation potential of nonlinear systems and the accompanying error of the mentioned assumptions. In general, it is concluded that the methods provide a reliable estimation and interpretation of equivalent linear systems.
Original languageEnglish
Article number115126
JournalJournal of Sound and Vibration
Number of pages17
Publication statusPublished - 2020


  • Equivalent linear stiffness and viscous damping
  • Equivalent linear mode shapes
  • Equivalent linearisation
  • Nonlinear dynamics
  • Friction and hysteretic damping

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