Nonlinear Dynamics

, Volume 93, Issue 4, pp 2357–2378 | Cite as

Dynamic evolution of a primary resonance MEMS resonator under prebuckling pattern

  • Jianxin HanEmail author
  • Gang Jin
  • Qichang Zhang
  • Wei Wang
  • Baizhou Li
  • Houjun Qi
  • Jingjing Feng
Original Paper


This paper analytically investigates the dynamic evolution of the primary frequency response of a prebuckling microbeam-based resonator with \(\hbox {Z}_{2}\) symmetry. A doubly clamped straight microbeam actuated by two symmetric stationary electrodes is simplified as a time-varying capacitor model for qualitative analysis purpose. Nonlinearities induced by the midplane stretching of the microbeam and the electrostatic force are considered. During solution procedure, electrostatic force holds its original form without any Taylor series expansion, and only one assumption with a small ratio of AC to DC voltage is introduced. The average equation, frequency response, backbone curve and stability condition are determined, respectively, based on the method of multiple scales combined with homotopy concept. Results demonstrate for the first time that the frequency response includes two types of branches, namely low- energy branch and high-energy branch. As the increase ion AC excitation amplitude, both branches close to each other along the backbone curve until they intersect. Further analyses are then performed to investigate the details of the backbone curve and the frequency response equation. Analytical formulas to determine the hardening and softening switches of the frequency response and the intersection condition of the low- and high-energy branches are both deduced and examined in depth. Primary frequency response properties in pull-in and secondary pull-in case are classified and depicted through theoretical predictions via the method of multiple scales and then verified through numerical results via the finite difference method combined with Floquet theory. Finally, a specific case study based on equivalent lumped parameters via Galerkin method is presented. Excellent agreements between theoretical predictions and simulation results illustrate the effectiveness of the whole analyses.


MEMS \(\hbox {Z}_{2}\) symmetry Nonlinear vibration Multiple scales Homotopy 



This work was supported by the National Natural Science Foundation of China (Grant Nos. 11702192, 51505336, 11772218, 11602173, 11602169), Tianjin Research Program of Application Foundation and Advanced Technology (Grant Nos. 15JCQNJC05200, 16JCQNJC04700), Innovation Team Training Plan of Tianjin Universities and colleges (Grant No. TD13-5096), Tianjin Science and Technology Planning Project (Grant No. 15ZXZNGX00220), Scientific Research Program of Tianjin Education Committee (Grant No. JWK1602) and the Scientific Research Foundation of Tianjin University of Technology and Education (Grant Nos. KYQD16009, KYQD1701).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tianjin Key Laboratory of High Speed Cutting and Precision Machining, School of Mechanical EngineeringTianjin University of Technology and EducationTianjinChina
  2. 2.Tianjin Key Laboratory of Nonlinear Dynamics and Control, School of Mechanical EngineeringTianjin UniversityTianjinChina
  3. 3.Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, School of Mechanical EngineeringTianjin University of TechnologyTianjinChina

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