# Modeling, Analysis and Simulations of a Dynamic Thermoviscoelastic Rod-Beam System

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## Abstract

This work models, analyses and simulates a coupled dynamic system consisting of a thermoviscoelastic rod and a linear viscoelastic beam. It is motivated by recent developments in MEMS systems, in particular the “V-shape” electro-thermal actuator that realizes large displacement and reliable contact in MEMS switches. The model consists of a system of three coupled partial differential equations for the beam’s and the rods’ displacements, and the rod’s temperature. Moreover, the rod may come in contact with a reactive foundation at one end, which is the main aspect of the actuating or switching property of the system. The thermal interaction at the contacting end of the rod is described by Barber’s heat exchange condition. The system is analyzed by setting it in an abstract form for which the existence of a weak solution is shown by using tools from the theory of variational inequalities and a fixed point theorem. A numerical algorithm for the system is constructed; its implementation yields computational depiction of the system’s behavior, with emphasis on the combined vibrations of the beam-rod system, dynamic contact force and thermal interaction.

### Keywords

Dynamic contact Thermoviscoelastic rod Normal compliance Barber’s heat exchange Simulations### Mathematics Subject Classification

35L53 74F05 74H15 74H45 74M15## Notes

### Acknowledgments

We would like to thank the referees for the very careful and thorough review of the paper, which improved the presentation and made it more readable.

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