Model development for atomic force microscope stage mechanisms

Ralph C. Smith, Andrew G. Hatch, Tathagata De, Murti V. Salapaka, Ricardo C.H. Del Rosario, Julie K. Raye

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


In this paper, we develop nonlinear constitutive equations and resulting system models quantifying the nonlinear and hysteretic field-displacement relations inherent to lead zirconate titanate (PZT) devices employed in atomic force microscope stage mechanisms. We focus specifically on PZT rods utilizing d 33 motion and PZT shells driven in d 31 regimes, but the modeling framework is sufficiently general to accommodate a variety of drive geometries. In the first step of the model development, lattice-level energy relations are combined with stochastic homogenization techniques to construct nonlinear constitutive relations which accommodate the hysteresis inherent to ferroelectric compounds. Second, these constitutive relations are employed in classical rod and shell relations to construct system models appropriate for presently employed nanopositioner designs. The capability of the models for quantifying the frequency-dependent hysteresis inherent to the PZT stages is illustrated through comparison with experimental data.

Original languageEnglish (US)
Pages (from-to)1998-2026
Number of pages29
JournalSIAM Journal on Applied Mathematics
Issue number6
StatePublished - 2006


  • Atomic force microscope
  • Dynamics
  • Hysteresis model


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