# 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.

## References

- 1.Ahn, J., Kuttler, K.L., Shillor, M.: Dynamic contact of two Gao beams. Electron. J. Diff. Eqn.
**2012**(194), 1–42 (2012)MATHGoogle Scholar - 2.Andrews, K.T., Shi, P., Shillor, M., Wright, S.: Thermoelastic contact with Barber’s heat exchange condition. Appl. Math. Optim.
**28**(1), 11–48 (1993)MathSciNetCrossRefMATHGoogle Scholar - 3.Andrews, K.T., Shillor, M.: Thermomechanical behaviour of a damageable beam in contact with two stops. Applicable Anal.
**85**(6–7), 845–865 (2006)MathSciNetCrossRefMATHGoogle Scholar - 4.Andrews, K.T., Kuttler, K. L., Shillor, M.: Dynamic Gao beam in contact with a reactive or rigid foundation. Chapter in Weimin Han, Stanislaw Migorski, and Mircea Sofonea (Eds), advances in variational and hemivariational inequalities with applications, advances in mechanics and mathematics (AMMA)
**33**, 225–248 (2015)Google Scholar - 5.Barber, J.R.: Stability of thermoelastic contact. Proc. IMech. Intl. Conference on Tribology, London, 981–986 (1987)Google Scholar
- 6.Bien, M.: Existence of global weak solutions for coupled thermoelasticity with Barber’s heat exchange condition. J. Applied Anal.
**9**(2), 163–185 (2003)MathSciNetCrossRefMATHGoogle Scholar - 7.Dumont, Y., Paoli, L.: Numerical simulation of a model of vibrations with joint clearance. Int. J. Computer Applications in Technol.
**33**(1), 45–53 (2008)CrossRefGoogle Scholar - 8.Han, W., Sofonea, M.: Quasistatic contact problems in viscoelasticity and viscoplasticity, studies in advanced mathematics 30. AMS-IP, Rhode Island (2002)CrossRefMATHGoogle Scholar
- 9.Khazaai, J.J., Qu, H., Shillor, M., Smith, L.: An electro-thermal MEMS Gripper with large tip opening and holding force: design and characterization. Sens. Transducers J.
**13**, Special Issue, 31–43 (2011)Google Scholar - 10.Kikuchi, N., Oden, J.T.: Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia (1988)Google Scholar
- 11.Klarbring, A., Mikelic, A., Shillor, M.: Frictional contact problems with normal compliance. Int. J. Engng. Sci.
**26**(8), 811–832 (1988)MathSciNetCrossRefMATHGoogle Scholar - 12.Kuttler, K.L., Shillor, M.: Dynamic contact with normal compliance wear and discontinuous friction coefficient. SIAM J. Math. Anal.
**34**(1), 1–27 (2002)MathSciNetCrossRefMATHGoogle Scholar - 13.Kuttler, K.L., Li, J., Shillor, M.: Existence for dynamic contact of a stochastic viscoelastic Gao Beam. Nonlinear Anal. RWA
**22**(4), 568–580 (2015)MathSciNetCrossRefMATHGoogle Scholar - 14.Kuttler, K.L., Li, J., Shillor, M.: Stochastic differential inclusions. submittedGoogle Scholar
- 15.Martins, J.A.C., Oden, J.T.: A numerical analysis of a class of problems in elastodynamics with friction. Comput. Meth. Appl. Mech. Engnr.
**40**, 327–360 (1983)MathSciNetCrossRefMATHGoogle Scholar - 16.Paoli, L.: An existence result for non-smooth vibro-impact problems
**211**(2), 247–281 (2005)Google Scholar - 17.Shillor, M., Sofonea, M., Telega, J.J.: Models and analysis of quasistatic contact, lecture notes in physics, 655. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
- 18.Steen, M., Barber, G.C., Shillor, M.: The heat exchange coefficient function in thermoelastic contact. Lubrication Eng.
**3**, 10–16 (2003)Google Scholar - 19.Strikwerda, J.: Finite difference schemes and partial differential equations. Wadsworth, New York (1989)MATHGoogle Scholar
- 20.Xu, X.: The
*N*-dimensional quasistatic problem of thermoelastic contact with Barber’s heat exchange condition. Adv. Math. Sci. Appl.**6**(2), 559–587 (1996)MathSciNetMATHGoogle Scholar - 21.Zeidler, E.: Nonlinear functional analysis and its applications II/B. Springer, New York (1990)CrossRefMATHGoogle Scholar