Two-to-one internal resonance in microscanners

Mohammed F. Daqaq, Eihab M. Abdel-Rahman, Ali H. Nayfeh

Research output: Contribution to journalArticlepeer-review


To realize large scanning angles, torsional microscanners are normally excited at their natural frequencies. Usually, a bias DC voltage is also applied to scan around a desired nonzero tilt angle. As a result, a deep understanding of the mirror's response to a DC-shifted primary resonance excitation is imperative. Along these lines, we use the method of multiple scales to obtain a second-order nonlinear approximate analytical solution of the mirror steady-state response. We show that the response of the mirror exhibits a softening-type behavior that increases as the magnitude of the DC component increases. For a given mirror, we can also identify a DC voltage range wherein the mirror exhibits a two-to-one internal resonance between the first two modes; that is, ω 2≈2ω 1. To analyze the mirror behavior within that range, we first treat the case where the excitation frequency is near the first-mode frequency; that is, Ω≈ω 1. Then we treat the case where the excitation frequency is near the second-mode frequency; that is, Ω≈ω 2. We analyze the stability of the response and compare the analytical results to numerical solutions obtained via long-time integration of the equations of motion. We show that, due to the internal resonance, the mirror exhibits complex dynamic behavior characterized by aperiodic responses to primary resonance excitations. This behavior results in undesirable oscillations that are detrimental to the mirror performance, namely bringing the target point in and out of focus and resulting in distorted images.

Original languageEnglish (US)
Pages (from-to)231-251
Number of pages21
JournalNonlinear Dynamics
Issue number1-2
StatePublished - Jul 2009


  • Method of multiple scales
  • Microscanner
  • Nonlinear interactions

ASJC Scopus subject areas

  • Mechanical Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering
  • Control and Systems Engineering


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