Global static and dynamic tristability in electrostatically actuated initially curved and weakly coupled double microbeams

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In the present study, an expanded dynamic model of an electrostatically actuated double beam metastructure, composed of geometrically different beams, is formulated and solved. The two beams are connected via an elastic bridge in the form of a linear spring, forming a weak coupling. A reduced order (RO) model is derived using Galerkin's decomposition with buckling eigenmodes of a straight beam taken as the base functions. Through a case study of three different structures, it is found that weakly coupled and geometrically different sub-structures can manifest global tristability when introduced to a nonlinear, displacement-dependent, electrostatic actuation whilst exhibiting bistability under mechanical, displacement-independent, load. The stability of the structures was established via three different methods. One being the combined eigenvalues based calculation, known as the energy method. The second is a direct Lyapunov stability sweep. And the third method involves direct quasi-static finite elements (FE) based simulations, corroborating the combined stability properties of the structures. In addition, it was found that electrostatic tristability can be manifested in different configurations, with statically inaccessible stable branches. To corroborate the behaviour of the electrostatically actuated structures, quasi-static loading sequences were performed on the RO model, complemented by the dynamic trapping method, to account for all three stable equilibria. Furthermore, it was found that in contrast to rigidly coupled double beam structures, the current model can account for globally stable latching points as well. The study has therefore shown that the energy method can be used to characterise the stability of the proposed metastructure, while also revealing new abilities that can be manifested in a seemingly simple structure. Such findings can promote the usage of such structures in multi-valued micro-electromechanical systems (MEMS) devices, promoting further miniaturisation.

Original languageEnglish
Article number112342
JournalInternational Journal of Solids and Structures
StatePublished - 15 Aug 2023


  • Dynamic trapping
  • Elastic coupling
  • Energy method
  • Lyapunov stability
  • Stability mismatch

All Science Journal Classification (ASJC) codes

  • Modelling and Simulation
  • Materials Science(all)
  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics


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